From 326cb531606ca93236cc4406afc496525054616c Mon Sep 17 00:00:00 2001
From: krahets <krahets@163.com>
Date: Fri, 6 Dec 2024 07:19:06 +0800
Subject: [PATCH] deploy

---
 en/sitemap.xml.gz                             | Bin 1008 -> 1008 bytes
 sitemap.xml.gz                                | Bin 1011 -> 1011 bytes
 .../contribution.assets/edit_markdown.png     | Bin 444155 -> 89985 bytes
 .../vscode_extension_installation.png         | Bin 475802 -> 146194 bytes
 .../vscode_installation.png                   | Bin 488611 -> 124802 bytes
 .../chapter_appendix/terminology/index.html   |   5 +++++
 .../suggestions.assets/download_code.png      | Bin 395104 -> 109029 bytes
 .../pythontutor_example.png                   | Bin 395258 -> 75912 bytes
 zh-hant/search/search_index.json              |   2 +-
 zh-hant/sitemap.xml.gz                        | Bin 1010 -> 1010 bytes
 10 files changed, 6 insertions(+), 1 deletion(-)

diff --git a/en/sitemap.xml.gz b/en/sitemap.xml.gz
index 02c3c525cc9f82201768b58c425454f82555fdc4..c2b4b2d2ea4faeeec1d2d06228be4701baf2f46f 100644
GIT binary patch
delta 15
Wcmeys{(+rMzMF$XLun)1D`o&F9|WBM

delta 15
Wcmeys{(+rMzMF$%JKILKSIht^?FA43

diff --git a/sitemap.xml.gz b/sitemap.xml.gz
index 24d0361351935c58405f61660ac3e18c21ef6452..23e7d18a4d56daf94968805b7404ca3451ea85c7 100644
GIT binary patch
delta 15
Wcmey&{+XRkzMF$XP-!FETV?<!?*xwk

delta 15
Wcmey&{+XRkzMF$%G22GAx6A-4&jkel

diff --git a/zh-hant/chapter_appendix/contribution.assets/edit_markdown.png b/zh-hant/chapter_appendix/contribution.assets/edit_markdown.png
index d91ab3a2ab64347489f2b8b192b873dcaa98f785..377d87967e6d5c3db04d5e0c29a956051c9dbed1 100644
GIT binary patch
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diff --git a/zh-hant/chapter_appendix/terminology/index.html b/zh-hant/chapter_appendix/terminology/index.html
index a08077162..70b24a755 100644
--- a/zh-hant/chapter_appendix/terminology/index.html
+++ b/zh-hant/chapter_appendix/terminology/index.html
@@ -3748,6 +3748,11 @@
 <td>雜湊表</td>
 </tr>
 <tr>
+<td>hash set</td>
+<td>哈希集合</td>
+<td>雜湊集合</td>
+</tr>
+<tr>
 <td>bucket</td>
 <td>桶</td>
 <td>桶</td>
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diff --git a/zh-hant/search/search_index.json b/zh-hant/search/search_index.json
index 896a21f13..60b3c6c07 100644
--- a/zh-hant/search/search_index.json
+++ b/zh-hant/search/search_index.json
@@ -1 +1 @@
-{"config":{"lang":["en"],"separator":"[\\s\\u200b\\u3000\\-\u3001\u3002\uff0c\uff0e\uff1f\uff01\uff1b]+","pipeline":["stemmer"]},"docs":[{"location":"chapter_appendix/","title":"\u7b2c 16 \u7ae0 \u00a0 \u9644\u9304","text":""},{"location":"chapter_appendix/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>16.1 \u00a0 \u7a0b\u5f0f\u8a2d\u8a08\u74b0\u5883\u5b89\u88dd</li> <li>16.2 \u00a0 \u4e00\u8d77\u53c3\u8207\u5275\u4f5c</li> <li>16.3 \u00a0 \u8853\u8a9e\u8868</li> </ul>"},{"location":"chapter_appendix/contribution/","title":"16.2 \u00a0 \u4e00\u8d77\u53c3\u8207\u5275\u4f5c","text":"<p>\u7531\u65bc\u7b46\u8005\u80fd\u529b\u6709\u9650\uff0c\u66f8\u4e2d\u96e3\u514d\u5b58\u5728\u4e00\u4e9b\u907a\u6f0f\u548c\u932f\u8aa4\uff0c\u8acb\u60a8\u8ad2\u89e3\u3002\u5982\u679c\u60a8\u767c\u73fe\u4e86\u7b46\u8aa4\u3001\u9023\u7d50\u5931\u6548\u3001\u5167\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u7fa9\u3001\u89e3\u91cb\u4e0d\u6e05\u6670\u6216\u884c\u6587\u7d50\u69cb\u4e0d\u5408\u7406\u7b49\u554f\u984c\uff0c\u8acb\u5354\u52a9\u6211\u5011\u9032\u884c\u4fee\u6b63\uff0c\u4ee5\u7d66\u8b80\u8005\u63d0\u4f9b\u66f4\u512a\u8cea\u7684\u5b78\u7fd2\u8cc7\u6e90\u3002</p> <p>\u6240\u6709\u64b0\u7a3f\u4eba\u7684 GitHub ID \u5c07\u5728\u672c\u66f8\u5009\u5eab\u3001\u7db2\u9801\u7248\u548c PDF \u7248\u7684\u4e3b\u9801\u4e0a\u9032\u884c\u5c55\u793a\uff0c\u4ee5\u611f\u8b1d\u4ed6\u5011\u5c0d\u958b\u6e90\u793e\u7fa4\u7684\u7121\u79c1\u5949\u737b\u3002</p> <p>\u958b\u6e90\u7684\u9b45\u529b</p> <p>\u7d19\u8cea\u5716\u66f8\u7684\u5169\u6b21\u5370\u5237\u7684\u9593\u9694\u6642\u9593\u5f80\u5f80\u8f03\u4e45\uff0c\u5167\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002</p> <p>\u800c\u5728\u672c\u958b\u6e90\u66f8\u4e2d\uff0c\u5167\u5bb9\u66f4\u8fed\u7684\u6642\u9593\u88ab\u7e2e\u77ed\u81f3\u6578\u65e5\u751a\u81f3\u5e7e\u500b\u5c0f\u6642\u3002</p>"},{"location":"chapter_appendix/contribution/#1","title":"1. \u00a0 \u5167\u5bb9\u5fae\u8abf","text":"<p>\u5982\u5716 16-3 \u6240\u793a\uff0c\u6bcf\u500b\u9801\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u201c\u7de8\u8f2f\u5716\u793a\u201d\u3002\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9a5f\u4fee\u6539\u6587\u5b57\u6216\u7a0b\u5f0f\u78bc\u3002</p> <ol> <li>\u9ede\u9078\u201c\u7de8\u8f2f\u5716\u793a\u201d\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u5009\u5eab\u201d\u7684\u63d0\u793a\uff0c\u8acb\u540c\u610f\u8a72\u64cd\u4f5c\u3002</li> <li>\u4fee\u6539 Markdown \u6e90\u6a94\u6848\u5167\u5bb9\uff0c\u6aa2\u67e5\u5167\u5bb9\u7684\u6b63\u78ba\u6027\uff0c\u4e26\u5118\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7d71\u4e00\u3002</li> <li>\u5728\u9801\u9762\u5e95\u90e8\u586b\u5beb\u4fee\u6539\u8aaa\u660e\uff0c\u7136\u5f8c\u9ede\u9078\u201cPropose file change\u201d\u6309\u9215\u3002\u9801\u9762\u8df3\u8f49\u5f8c\uff0c\u9ede\u9078\u201cCreate pull request\u201d\u6309\u9215\u5373\u53ef\u767c\u8d77\u62c9\u53d6\u8acb\u6c42\u3002</li> </ol> <p></p> <p> \u5716 16-3 \u00a0 \u9801\u9762\u7de8\u8f2f\u6309\u9375 </p> <p>\u5716\u7247\u7121\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u900f\u904e\u65b0\u5efa Issue \u6216\u8a55\u8ad6\u7559\u8a00\u4f86\u63cf\u8ff0\u554f\u984c\uff0c\u6211\u5011\u6703\u76e1\u5feb\u91cd\u65b0\u7e6a\u88fd\u4e26\u66ff\u63db\u5716\u7247\u3002</p>"},{"location":"chapter_appendix/contribution/#2","title":"2. \u00a0 \u5167\u5bb9\u5275\u4f5c","text":"<p>\u5982\u679c\u60a8\u6709\u8208\u8da3\u53c3\u8207\u6b64\u958b\u6e90\u5c08\u6848\uff0c\u5305\u62ec\u5c07\u7a0b\u5f0f\u78bc\u7ffb\u8b6f\u6210\u5176\u4ed6\u7a0b\u5f0f\u8a9e\u8a00\u3001\u64f4\u5c55\u6587\u7ae0\u5167\u5bb9\u7b49\uff0c\u90a3\u9ebc\u9700\u8981\u5be6\u65bd\u4ee5\u4e0b Pull Request \u5de5\u4f5c\u6d41\u7a0b\u3002</p> <ol> <li>\u767b\u5165 GitHub \uff0c\u5c07\u672c\u66f8\u7684\u7a0b\u5f0f\u78bc\u5009\u5eab Fork \u5230\u500b\u4eba\u5e33\u865f\u4e0b\u3002</li> <li>\u9032\u5165\u60a8\u7684 Fork \u5009\u5eab\u7db2\u9801\uff0c\u4f7f\u7528 <code>git clone</code> \u547d\u4ee4\u5c07\u5009\u5eab\u514b\u9686\u81f3\u672c\u5730\u3002</li> <li>\u5728\u672c\u5730\u9032\u884c\u5167\u5bb9\u5275\u4f5c\uff0c\u4e26\u9032\u884c\u5b8c\u6574\u6e2c\u8a66\uff0c\u9a57\u8b49\u7a0b\u5f0f\u78bc\u7684\u6b63\u78ba\u6027\u3002</li> <li>\u5c07\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u5f8c Push \u81f3\u9060\u7aef\u5009\u5eab\u3002</li> <li>\u91cd\u65b0\u6574\u7406\u5009\u5eab\u7db2\u9801\uff0c\u9ede\u9078\u201cCreate pull request\u201d\u6309\u9215\u5373\u53ef\u767c\u8d77\u62c9\u53d6\u8acb\u6c42\u3002</li> </ol>"},{"location":"chapter_appendix/contribution/#3-docker","title":"3. \u00a0 Docker \u90e8\u7f72","text":"<p>\u5728 <code>hello-algo</code> \u6839\u76ee\u9304\u4e0b\uff0c\u57f7\u884c\u4ee5\u4e0b Docker \u6307\u4ee4\u78bc\uff0c\u5373\u53ef\u5728 <code>http://localhost:8000</code> \u8a2a\u554f\u672c\u5c08\u6848\uff1a</p> <pre><code>docker-compose up -d\n</code></pre> <p>\u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u522a\u9664\u90e8\u7f72\uff1a</p> <pre><code>docker-compose down\n</code></pre>"},{"location":"chapter_appendix/installation/","title":"16.1 \u00a0 \u7a0b\u5f0f\u8a2d\u8a08\u74b0\u5883\u5b89\u88dd","text":""},{"location":"chapter_appendix/installation/#1611-ide","title":"16.1.1 \u00a0 \u5b89\u88dd IDE","text":"<p>\u63a8\u85a6\u4f7f\u7528\u958b\u6e90\u3001\u8f15\u91cf\u7684 VS Code \u4f5c\u70ba\u672c\u5730\u6574\u5408\u958b\u767c\u74b0\u5883\uff08IDE\uff09\u3002\u8a2a\u554f VS Code \u5b98\u7db2\uff0c\u6839\u64da\u4f5c\u696d\u7cfb\u7d71\u9078\u64c7\u76f8\u61c9\u7248\u672c\u7684 VS Code \u9032\u884c\u4e0b\u8f09\u548c\u5b89\u88dd\u3002</p> <p></p> <p> \u5716 16-1 \u00a0 \u5f9e\u5b98\u7db2\u4e0b\u8f09 VS Code </p> <p>VS Code \u64c1\u6709\u5f37\u5927\u7684\u64f4\u5c55\u5305\u751f\u614b\u7cfb\u7d71\uff0c\u652f\u6301\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u7684\u57f7\u884c\u548c\u9664\u932f\u3002\u4ee5 Python \u70ba\u4f8b\uff0c\u5b89\u88dd\u201cPython Extension Pack\u201d\u64f4\u5c55\u5305\u4e4b\u5f8c\uff0c\u5373\u53ef\u9032\u884c Python \u7a0b\u5f0f\u78bc\u9664\u932f\u3002\u5b89\u88dd\u6b65\u9a5f\u5982\u5716 16-2 \u6240\u793a\u3002</p> <p></p> <p> \u5716 16-2 \u00a0 \u5b89\u88dd VS Code \u64f4\u5c55\u5305 </p>"},{"location":"chapter_appendix/installation/#1612","title":"16.1.2 \u00a0 \u5b89\u88dd\u8a9e\u8a00\u74b0\u5883","text":""},{"location":"chapter_appendix/installation/#1-python","title":"1. \u00a0 Python \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Miniconda3 \uff0c\u9700\u8981 Python 3.10 \u6216\u66f4\u65b0\u7248\u672c\u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>python</code> \uff0c\u5b89\u88dd Python Extension Pack \u3002</li> <li>\uff08\u53ef\u9078\uff09\u5728\u547d\u4ee4\u5217\u8f38\u5165 <code>pip install black</code> \uff0c\u5b89\u88dd\u7a0b\u5f0f\u78bc\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#2-cc","title":"2. \u00a0 C/C++ \u74b0\u5883","text":"<ol> <li>Windows \u7cfb\u7d71\u9700\u8981\u5b89\u88dd MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff1bMacOS \u81ea\u5e36 Clang \uff0c\u7121\u9808\u5b89\u88dd\u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>c++</code> \uff0c\u5b89\u88dd C/C++ Extension Pack \u3002</li> <li>\uff08\u53ef\u9078\uff09\u958b\u555f Settings \u9801\u9762\uff0c\u641c\u5c0b <code>Clang_format_fallback Style</code> \u7a0b\u5f0f\u78bc\u683c\u5f0f\u5316\u9078\u9805\uff0c\u8a2d\u5b9a\u70ba <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code> \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#3-java","title":"3. \u00a0 Java \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd OpenJDK\uff08\u7248\u672c\u9700\u6eff\u8db3 &gt; JDK 9\uff09\u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>java</code> \uff0c\u5b89\u88dd Extension Pack for Java \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#4-c","title":"4. \u00a0 C# \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd .Net 8.0 \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>C# Dev Kit</code> \uff0c\u5b89\u88dd C# Dev Kit \uff08\u914d\u7f6e\u6559\u7a0b\uff09\u3002</li> <li>\u4e5f\u53ef\u4f7f\u7528 Visual Studio\uff08\u5b89\u88dd\u6559\u7a0b\uff09\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#5-go","title":"5. \u00a0 Go \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd go \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>go</code> \uff0c\u5b89\u88dd Go \u3002</li> <li>\u6309\u5feb\u6377\u9375 <code>Ctrl + Shift + P</code> \u64a5\u51fa\u547d\u4ee4\u6b04\uff0c\u8f38\u5165 go \uff0c\u9078\u64c7 <code>Go: Install/Update Tools</code> \uff0c\u5168\u90e8\u52fe\u9078\u4e26\u5b89\u88dd\u5373\u53ef\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#6-swift","title":"6. \u00a0 Swift \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Swift \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>swift</code> \uff0c\u5b89\u88dd Swift for Visual Studio Code \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#7-javascript","title":"7. \u00a0 JavaScript \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Node.js \u3002</li> <li>\uff08\u53ef\u9078\uff09\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>Prettier</code> \uff0c\u5b89\u88dd\u7a0b\u5f0f\u78bc\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#8-typescript","title":"8. \u00a0 TypeScript \u74b0\u5883","text":"<ol> <li>\u540c JavaScript \u74b0\u5883\u5b89\u88dd\u6b65\u9a5f\u3002</li> <li>\u5b89\u88dd TypeScript Execute (tsx) \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>typescript</code> \uff0c\u5b89\u88dd Pretty TypeScript Errors \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#9-dart","title":"9. \u00a0 Dart \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Dart \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>dart</code> \uff0c\u5b89\u88dd Dart \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#10-rust","title":"10. \u00a0 Rust \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Rust \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>rust</code> \uff0c\u5b89\u88dd rust-analyzer \u3002</li> </ol>"},{"location":"chapter_appendix/terminology/","title":"16.3 \u00a0 \u8853\u8a9e\u8868","text":"<p>\u8868 16-1 \u5217\u51fa\u4e86\u66f8\u4e2d\u51fa\u73fe\u7684\u91cd\u8981\u8853\u8a9e\uff0c\u503c\u5f97\u6ce8\u610f\u4ee5\u4e0b\u5e7e\u9ede\u3002</p> <ul> <li>\u5efa\u8b70\u8a18\u4f4f\u540d\u8a5e\u7684\u82f1\u6587\u53eb\u6cd5\uff0c\u4ee5\u4fbf\u95b1\u8b80\u82f1\u6587\u6587\u737b\u3002</li> <li>\u90e8\u5206\u540d\u8a5e\u5728\u7c21\u9ad4\u4e2d\u6587\u548c\u7e41\u9ad4\u4e2d\u6587\u4e0b\u7684\u53eb\u6cd5\u4e0d\u540c\u3002</li> </ul> <p> \u8868 16-1 \u00a0 \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u91cd\u8981\u540d\u8a5e </p> English \u7c21\u9ad4\u4e2d\u6587 \u7e41\u9ad4\u4e2d\u6587 algorithm \u7b97\u6cd5 \u6f14\u7b97\u6cd5 data structure \u6570\u636e\u7ed3\u6784 \u8cc7\u6599\u7d50\u69cb code \u4ee3\u7801 \u7a0b\u5f0f\u78bc file \u6587\u4ef6 \u6a94\u6848 function \u51fd\u6570 \u51fd\u5f0f method \u65b9\u6cd5 \u65b9\u6cd5 variable \u53d8\u91cf \u8b8a\u6578 asymptotic complexity analysis \u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 \u6f38\u8fd1\u8907\u96dc\u5ea6\u5206\u6790 time complexity \u65f6\u95f4\u590d\u6742\u5ea6 \u6642\u9593\u8907\u96dc\u5ea6 space complexity \u7a7a\u95f4\u590d\u6742\u5ea6 \u7a7a\u9593\u8907\u96dc\u5ea6 loop \u5faa\u73af \u8ff4\u5708 iteration \u8fed\u4ee3 \u8fed\u4ee3 recursion \u9012\u5f52 \u905e\u8ff4 tail recursion \u5c3e\u9012\u5f52 \u5c3e\u905e\u8ff4 recursion tree \u9012\u5f52\u6811 \u905e\u8ff4\u6a39 big-\\(O\\) notation \u5927 \\(O\\) \u8bb0\u53f7 \u5927 \\(O\\) \u8a18\u865f asymptotic upper bound \u6e10\u8fd1\u4e0a\u754c \u6f38\u8fd1\u4e0a\u754c sign-magnitude \u539f\u7801 \u539f\u78bc 1\u2019s complement \u53cd\u7801 \u4e00\u88dc\u6578 2\u2019s complement \u8865\u7801 \u4e8c\u88dc\u6578 array \u6570\u7ec4 \u9663\u5217 index \u7d22\u5f15 \u7d22\u5f15 linked list \u94fe\u8868 \u93c8\u7d50\u4e32\u5217 linked list node, list node \u94fe\u8868\u8282\u70b9 \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede head node \u5934\u8282\u70b9 \u982d\u7bc0\u9ede tail node \u5c3e\u8282\u70b9 \u5c3e\u7bc0\u9ede list \u5217\u8868 \u4e32\u5217 dynamic array \u52a8\u6001\u6570\u7ec4 \u52d5\u614b\u9663\u5217 hard disk \u786c\u76d8 \u786c\u789f random-access memory (RAM) \u5185\u5b58 \u8a18\u61b6\u9ad4 cache memory \u7f13\u5b58 \u5feb\u53d6 cache miss \u7f13\u5b58\u672a\u547d\u4e2d \u5feb\u53d6\u672a\u547d\u4e2d cache hit rate \u7f13\u5b58\u547d\u4e2d\u7387 \u5feb\u53d6\u547d\u4e2d\u7387 stack \u6808 \u5806\u758a top of the stack \u6808\u9876 \u5806\u758a\u9802 bottom of the stack \u6808\u5e95 \u5806\u758a\u5e95 queue \u961f\u5217 \u4f47\u5217 double-ended queue \u53cc\u5411\u961f\u5217 \u96d9\u5411\u4f47\u5217 front of the queue \u961f\u9996 \u4f47\u5217\u9996 rear of the queue \u961f\u5c3e \u4f47\u5217\u5c3e hash table \u54c8\u5e0c\u8868 \u96dc\u6e4a\u8868 bucket \u6876 \u6876 hash function \u54c8\u5e0c\u51fd\u6570 \u96dc\u6e4a\u51fd\u5f0f hash collision \u54c8\u5e0c\u51b2\u7a81 \u96dc\u6e4a\u885d\u7a81 load factor \u8d1f\u8f7d\u56e0\u5b50 \u8ca0\u8f09\u56e0\u5b50 separate chaining \u94fe\u5f0f\u5730\u5740 \u93c8\u7d50\u4f4d\u5740 open addressing \u5f00\u653e\u5bfb\u5740 \u958b\u653e\u5b9a\u5740 linear probing \u7ebf\u6027\u63a2\u6d4b \u7dda\u6027\u63a2\u67e5 lazy deletion \u61d2\u5220\u9664 \u61f6\u522a\u9664 binary tree \u4e8c\u53c9\u6811 \u4e8c\u5143\u6a39 tree node \u6811\u8282\u70b9 \u6a39\u7bc0\u9ede left-child node \u5de6\u5b50\u8282\u70b9 \u5de6\u5b50\u7bc0\u9ede right-child node \u53f3\u5b50\u8282\u70b9 \u53f3\u5b50\u7bc0\u9ede parent node \u7236\u8282\u70b9 \u7236\u7bc0\u9ede left subtree \u5de6\u5b50\u6811 \u5de6\u5b50\u6a39 right subtree \u53f3\u5b50\u6811 \u53f3\u5b50\u6a39 root node \u6839\u8282\u70b9 \u6839\u7bc0\u9ede leaf node \u53f6\u8282\u70b9 \u8449\u7bc0\u9ede edge \u8fb9 \u908a level \u5c42 \u5c64 degree \u5ea6 \u5ea6 height \u9ad8\u5ea6 \u9ad8\u5ea6 depth \u6df1\u5ea6 \u6df1\u5ea6 perfect binary tree \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u5b8c\u7f8e\u4e8c\u5143\u6a39 complete binary tree \u5b8c\u5168\u4e8c\u53c9\u6811 \u5b8c\u5168\u4e8c\u5143\u6a39 full binary tree \u5b8c\u6ee1\u4e8c\u53c9\u6811 \u5b8c\u6eff\u4e8c\u5143\u6a39 balanced binary tree \u5e73\u8861\u4e8c\u53c9\u6811 \u5e73\u8861\u4e8c\u5143\u6a39 binary search tree \u4e8c\u53c9\u641c\u7d22\u6811 \u4e8c\u5143\u641c\u5c0b\u6a39 AVL tree AVL \u6811 AVL \u6a39 red-black tree \u7ea2\u9ed1\u6811 \u7d05\u9ed1\u6a39 level-order traversal \u5c42\u5e8f\u904d\u5386 \u5c64\u5e8f\u8d70\u8a2a breadth-first traversal \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a depth-first traversal \u6df1\u5ea6\u4f18\u5148\u904d\u5386 \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a binary search tree \u4e8c\u53c9\u641c\u7d22\u6811 \u4e8c\u5143\u641c\u5c0b\u6a39 balanced binary search tree \u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 \u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39 balance factor \u5e73\u8861\u56e0\u5b50 \u5e73\u8861\u56e0\u5b50 heap \u5806 \u5806\u7a4d max heap \u5927\u9876\u5806 \u5927\u9802\u5806\u7a4d min heap \u5c0f\u9876\u5806 \u5c0f\u9802\u5806\u7a4d priority queue \u4f18\u5148\u961f\u5217 \u512a\u5148\u4f47\u5217 heapify \u5806\u5316 \u5806\u7a4d\u5316 top-\\(k\\) problem Top-\\(k\\) \u95ee\u9898 Top-\\(k\\) \u554f\u984c graph \u56fe \u5716 vertex \u9876\u70b9 \u9802\u9ede undirected graph \u65e0\u5411\u56fe \u7121\u5411\u5716 directed graph \u6709\u5411\u56fe \u6709\u5411\u5716 connected graph \u8fde\u901a\u56fe \u9023\u901a\u5716 disconnected graph \u975e\u8fde\u901a\u56fe \u975e\u9023\u901a\u5716 weighted graph \u6709\u6743\u56fe \u6709\u6b0a\u5716 adjacency \u90bb\u63a5 \u9130\u63a5 path \u8def\u5f84 \u8def\u5f91 in-degree \u5165\u5ea6 \u5165\u5ea6 out-degree \u51fa\u5ea6 \u51fa\u5ea6 adjacency matrix \u90bb\u63a5\u77e9\u9635 \u9130\u63a5\u77e9\u9663 adjacency list \u90bb\u63a5\u8868 \u9130\u63a5\u8868 breadth-first search \u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 \u5ee3\u5ea6\u512a\u5148\u641c\u5c0b depth-first search \u6df1\u5ea6\u4f18\u5148\u641c\u7d22 \u6df1\u5ea6\u512a\u5148\u641c\u5c0b binary search \u4e8c\u5206\u67e5\u627e \u4e8c\u5206\u641c\u5c0b searching algorithm \u641c\u7d22\u7b97\u6cd5 \u641c\u5c0b\u6f14\u7b97\u6cd5 sorting algorithm \u6392\u5e8f\u7b97\u6cd5 \u6392\u5e8f\u6f14\u7b97\u6cd5 selection sort \u9009\u62e9\u6392\u5e8f \u9078\u64c7\u6392\u5e8f bubble sort \u5192\u6ce1\u6392\u5e8f \u6ce1\u6cab\u6392\u5e8f insertion sort \u63d2\u5165\u6392\u5e8f \u63d2\u5165\u6392\u5e8f quick sort \u5feb\u901f\u6392\u5e8f \u5feb\u901f\u6392\u5e8f merge sort \u5f52\u5e76\u6392\u5e8f \u5408\u4f75\u6392\u5e8f heap sort \u5806\u6392\u5e8f \u5806\u7a4d\u6392\u5e8f bucket sort \u6876\u6392\u5e8f \u6876\u6392\u5e8f counting sort \u8ba1\u6570\u6392\u5e8f \u8a08\u6578\u6392\u5e8f radix sort \u57fa\u6570\u6392\u5e8f \u57fa\u6578\u6392\u5e8f divide and conquer \u5206\u6cbb \u5206\u6cbb hanota problem \u6c49\u8bfa\u5854\u95ee\u9898 \u6cb3\u5167\u5854\u554f\u984c backtracking algorithm \u56de\u6eaf\u7b97\u6cd5 \u56de\u6eaf\u6f14\u7b97\u6cd5 constraint \u7ea6\u675f \u7d04\u675f solution \u89e3 \u89e3 state \u72b6\u6001 \u72c0\u614b pruning \u526a\u679d \u526a\u679d permutations problem \u5168\u6392\u5217\u95ee\u9898 \u5168\u6392\u5217\u554f\u984c subset-sum problem \u5b50\u96c6\u548c\u95ee\u9898 \u5b50\u96c6\u5408\u554f\u984c \\(n\\)-queens problem \\(n\\) \u7687\u540e\u95ee\u9898 \\(n\\) \u7687\u540e\u554f\u984c dynamic programming \u52a8\u6001\u89c4\u5212 \u52d5\u614b\u898f\u5283 initial state \u521d\u59cb\u72b6\u6001 \u521d\u59cb\u72c0\u614b state-transition equation \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \u72c0\u614b\u8f49\u79fb\u65b9\u7a0b knapsack problem \u80cc\u5305\u95ee\u9898 \u80cc\u5305\u554f\u984c edit distance problem \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 \u7de8\u8f2f\u8ddd\u96e2\u554f\u984c greedy algorithm \u8d2a\u5fc3\u7b97\u6cd5 \u8caa\u5a6a\u6f14\u7b97\u6cd5"},{"location":"chapter_array_and_linkedlist/","title":"\u7b2c 4 \u7ae0 \u00a0 \u9663\u5217\u8207\u93c8\u7d50\u4e32\u5217","text":"<p>Abstract</p> <p>\u8cc7\u6599\u7d50\u69cb\u7684\u4e16\u754c\u5982\u540c\u4e00\u5835\u539a\u5be6\u7684\u78da\u7246\u3002</p> <p>\u9663\u5217\u7684\u78da\u584a\u6574\u9f4a\u6392\u5217\uff0c\u9010\u500b\u7dca\u8cbc\u3002\u93c8\u7d50\u4e32\u5217\u7684\u78da\u584a\u5206\u6563\u5404\u8655\uff0c\u9023\u7dda\u7684\u85e4\u8513\u81ea\u7531\u5730\u7a7f\u68ad\u65bc\u78da\u7e2b\u4e4b\u9593\u3002</p>"},{"location":"chapter_array_and_linkedlist/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>4.1 \u00a0 \u9663\u5217</li> <li>4.2 \u00a0 \u93c8\u7d50\u4e32\u5217</li> <li>4.3 \u00a0 \u4e32\u5217</li> <li>4.4 \u00a0 \u8a18\u61b6\u9ad4\u8207\u5feb\u53d6 *</li> <li>4.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/","title":"4.1 \u00a0 \u9663\u5217","text":"<p>\u9663\u5217\uff08array\uff09\u662f\u4e00\u7a2e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u5176\u5c07\u76f8\u540c\u578b\u5225\u7684\u5143\u7d20\u5132\u5b58\u5728\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e2d\u3002\u6211\u5011\u5c07\u5143\u7d20\u5728\u9663\u5217\u4e2d\u7684\u4f4d\u7f6e\u7a31\u70ba\u8a72\u5143\u7d20\u7684\u7d22\u5f15\uff08index\uff09\u3002\u5716 4-1 \u5c55\u793a\u4e86\u9663\u5217\u7684\u4e3b\u8981\u6982\u5ff5\u548c\u5132\u5b58\u65b9\u5f0f\u3002</p> <p></p> <p> \u5716 4-1 \u00a0 \u9663\u5217\u5b9a\u7fa9\u8207\u5132\u5b58\u65b9\u5f0f </p>"},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1 \u00a0 \u9663\u5217\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/array/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u9663\u5217","text":"<p>\u6211\u5011\u53ef\u4ee5\u6839\u64da\u9700\u6c42\u9078\u7528\u9663\u5217\u7684\u5169\u7a2e\u521d\u59cb\u5316\u65b9\u5f0f\uff1a\u7121\u521d\u59cb\u503c\u3001\u7d66\u5b9a\u521d\u59cb\u503c\u3002\u5728\u672a\u6307\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u6cc1\u4e0b\uff0c\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u6703\u5c07\u9663\u5217\u5143\u7d20\u521d\u59cb\u5316\u70ba \\(0\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code># \u521d\u59cb\u5316\u9663\u5217\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]  \n</code></pre> array.cpp<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\n// \u5132\u5b58\u5728\u5806\u758a\u4e0a\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// \u5132\u5b58\u5728\u5806\u7a4d\u4e0a\uff08\u9700\u8981\u624b\u52d5\u91cb\u653e\u7a7a\u9593\uff09\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u9577\u5ea6\u6642\uff08[5]int\uff09\u70ba\u9663\u5217\uff0c\u4e0d\u6307\u5b9a\u9577\u5ea6\u6642\uff08[]int\uff09\u70ba\u5207\u7247\n// \u7531\u65bc Go \u7684\u9663\u5217\u88ab\u8a2d\u8a08\u70ba\u5728\u7de8\u8b6f\u671f\u78ba\u5b9a\u9577\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u6578\u4f86\u6307\u5b9a\u9577\u5ea6\n// \u70ba\u4e86\u65b9\u4fbf\u5be6\u73fe\u64f4\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c07\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u9663\u5217\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nlet arr: [i32; 5] = [0; 5]; // [0, 0, 0, 0, 0]\nlet slice: &amp;[i32] = &amp;[0; 5];\n// \u5728 Rust \u4e2d\uff0c\u6307\u5b9a\u9577\u5ea6\u6642\uff08[i32; 5]\uff09\u70ba\u9663\u5217\uff0c\u4e0d\u6307\u5b9a\u9577\u5ea6\u6642\uff08&amp;[i32]\uff09\u70ba\u5207\u7247\n// \u7531\u65bc Rust \u7684\u9663\u5217\u88ab\u8a2d\u8a08\u70ba\u5728\u7de8\u8b6f\u671f\u78ba\u5b9a\u9577\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u6578\u4f86\u6307\u5b9a\u9577\u5ea6\n// Vector \u662f Rust \u4e00\u822c\u60c5\u6cc1\u4e0b\u7528\u4f5c\u52d5\u614b\u9663\u5217\u7684\u578b\u5225\n// \u70ba\u4e86\u65b9\u4fbf\u5be6\u73fe\u64f4\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c07 vector \u770b\u4f5c\u9663\u5217\uff08array\uff09\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nvar arr = IntArray(5) // { 0, 0, 0, 0, 0 }\nvar nums = intArrayOf(1, 3, 2, 5, 4)\n</code></pre> array.rb<pre><code># \u521d\u59cb\u5316\u9663\u5217\narr = Array.new(5, 0)\nnums = [1, 3, 2, 5, 4]\n</code></pre> array.zig<pre><code>// \u521d\u59cb\u5316\u9663\u5217\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#2","title":"2. \u00a0 \u8a2a\u554f\u5143\u7d20","text":"<p>\u9663\u5217\u5143\u7d20\u88ab\u5132\u5b58\u5728\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e2d\uff0c\u9019\u610f\u5473\u8457\u8a08\u7b97\u9663\u5217\u5143\u7d20\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7d66\u5b9a\u9663\u5217\u8a18\u61b6\u9ad4\u4f4d\u5740\uff08\u9996\u5143\u7d20\u8a18\u61b6\u9ad4\u4f4d\u5740\uff09\u548c\u67d0\u500b\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u5716 4-2 \u6240\u793a\u7684\u516c\u5f0f\u8a08\u7b97\u5f97\u5230\u8a72\u5143\u7d20\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\uff0c\u5f9e\u800c\u76f4\u63a5\u8a2a\u554f\u8a72\u5143\u7d20\u3002</p> <p></p> <p> \u5716 4-2 \u00a0 \u9663\u5217\u5143\u7d20\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u8a08\u7b97 </p> <p>\u89c0\u5bdf\u5716 4-2 \uff0c\u6211\u5011\u767c\u73fe\u9663\u5217\u9996\u500b\u5143\u7d20\u7684\u7d22\u5f15\u70ba \\(0\\) \uff0c\u9019\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89ba\uff0c\u56e0\u70ba\u5f9e \\(1\\) \u958b\u59cb\u8a08\u6578\u6703\u66f4\u81ea\u7136\u3002\u4f46\u5f9e\u4f4d\u5740\u8a08\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8cea\u4e0a\u662f\u8a18\u61b6\u9ad4\u4f4d\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u500b\u5143\u7d20\u7684\u4f4d\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u5b83\u7684\u7d22\u5f15\u70ba \\(0\\) \u662f\u5408\u7406\u7684\u3002</p> <p>\u5728\u9663\u5217\u4e2d\u8a2a\u554f\u5143\u7d20\u975e\u5e38\u9ad8\u6548\uff0c\u6211\u5011\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u5167\u96a8\u6a5f\u8a2a\u554f\u9663\u5217\u4e2d\u7684\u4efb\u610f\u4e00\u500b\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"\u96a8\u6a5f\u8a2a\u554f\u5143\u7d20\"\"\"\n    # \u5728\u5340\u9593 [0, len(nums)-1] \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    random_index = random.randint(0, len(nums) - 1)\n    # \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u5340\u9593 [0, size) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    int randomIndex = rand() % size;\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint randomAccess(int[] nums) {\n    // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // \u5728\u5340\u9593 [0, nums.Length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    int randomIndex = random.Next(nums.Length);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    randomIndex := rand.Intn(len(nums))\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // \u5728\u5340\u9593 [0, nums.count) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    let randomIndex = nums.indices.randomElement()!\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfunction randomAccess(nums) {\n    // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n    // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint randomAccess(List&lt;int&gt; nums) {\n  // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n  int randomIndex = Random().nextInt(nums.length);\n  // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // \u5728\u5340\u9593 [0, nums.len()) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u5340\u9593 [0, size) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    int randomIndex = rand() % size;\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfun randomAccess(nums: IntArray): Int {\n    // \u5728\u5340\u9593 [0, nums.size) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 ###\ndef random_access(nums)\n  # \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n  random_index = Random.rand(0...nums.length)\n\n  # \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n  nums[random_index]\nend\n</code></pre> array.zig<pre><code>// \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n    // \u5728\u5340\u9593 [0, nums.len) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6574\u6578\n    var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    var randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#3","title":"3. \u00a0 \u63d2\u5165\u5143\u7d20","text":"<p>\u9663\u5217\u5143\u7d20\u5728\u8a18\u61b6\u9ad4\u4e2d\u662f\u201c\u7dca\u6328\u8457\u7684\u201d\uff0c\u5b83\u5011\u4e4b\u9593\u6c92\u6709\u7a7a\u9593\u518d\u5b58\u653e\u4efb\u4f55\u8cc7\u6599\u3002\u5982\u5716 4-3 \u6240\u793a\uff0c\u5982\u679c\u60f3\u5728\u9663\u5217\u4e2d\u9593\u63d2\u5165\u4e00\u500b\u5143\u7d20\uff0c\u5247\u9700\u8981\u5c07\u8a72\u5143\u7d20\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u4e4b\u5f8c\u518d\u628a\u5143\u7d20\u8ce6\u503c\u7d66\u8a72\u7d22\u5f15\u3002</p> <p></p> <p> \u5716 4-3 \u00a0 \u9663\u5217\u63d2\u5165\u5143\u7d20\u793a\u4f8b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u65bc\u9663\u5217\u7684\u9577\u5ea6\u662f\u56fa\u5b9a\u7684\uff0c\u56e0\u6b64\u63d2\u5165\u4e00\u500b\u5143\u7d20\u5fc5\u5b9a\u6703\u5c0e\u81f4\u9663\u5217\u5c3e\u90e8\u5143\u7d20\u201c\u4e1f\u5931\u201d\u3002\u6211\u5011\u5c07\u9019\u500b\u554f\u984c\u7684\u89e3\u6c7a\u65b9\u6848\u7559\u5728\u201c\u4e32\u5217\u201d\u7ae0\u7bc0\u4e2d\u8a0e\u8ad6\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"\u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num\"\"\"\n    # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).reversed() {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // \u5c07 _num \u8ce6\u7d66 index \u8655\u5143\u7d20\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &amp;mut [i32], num: i32, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num ###\ndef insert(nums, num, index)\n  # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n  nums[index] = num\nend\n</code></pre> array.zig<pre><code>// \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    var i = nums.len - 1;\n    while (i &gt; index) : (i -= 1) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#4","title":"4. \u00a0 \u522a\u9664\u5143\u7d20","text":"<p>\u540c\u7406\uff0c\u5982\u5716 4-4 \u6240\u793a\uff0c\u82e5\u60f3\u522a\u9664\u7d22\u5f15 \\(i\\) \u8655\u7684\u5143\u7d20\uff0c\u5247\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\u3002</p> <p></p> <p> \u5716 4-4 \u00a0 \u9663\u5217\u522a\u9664\u5143\u7d20\u793a\u4f8b </p> <p>\u8acb\u6ce8\u610f\uff0c\u522a\u9664\u5143\u7d20\u5b8c\u6210\u5f8c\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u8b8a\u5f97\u201c\u7121\u610f\u7fa9\u201d\u4e86\uff0c\u6240\u4ee5\u6211\u5011\u7121\u9808\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"\u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20\"\"\"\n    # \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfn remove(nums: &amp;mut [i32], index: usize) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u4f54\u7528\u4e86 remove \u95dc\u9375\u8a5e\nvoid removeItem(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfun remove(nums: IntArray, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 ###\ndef remove(nums, index)\n  # \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n  for i in index...(nums.length - 1)\n    nums[i] = nums[i + 1]\n  end\nend\n</code></pre> array.zig<pre><code>// \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    var i = index;\n    while (i &lt; nums.len - 1) : (i += 1) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u7e3d\u7684\u4f86\u770b\uff0c\u9663\u5217\u7684\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u9ede\u3002</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u9ad8\uff1a\u9663\u5217\u7684\u63d2\u5165\u548c\u522a\u9664\u7684\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u70ba\u9663\u5217\u9577\u5ea6\u3002</li> <li>\u4e1f\u5931\u5143\u7d20\uff1a\u7531\u65bc\u9663\u5217\u7684\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u5f8c\uff0c\u8d85\u51fa\u9663\u5217\u9577\u5ea6\u7bc4\u570d\u7684\u5143\u7d20\u6703\u4e1f\u5931\u3002</li> <li>\u8a18\u61b6\u9ad4\u6d6a\u8cbb\uff1a\u6211\u5011\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u500b\u6bd4\u8f03\u9577\u7684\u9663\u5217\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u9019\u6a23\u5728\u63d2\u5165\u8cc7\u6599\u6642\uff0c\u4e1f\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u201c\u7121\u610f\u7fa9\u201d\u7684\uff0c\u4f46\u9019\u6a23\u505a\u6703\u9020\u6210\u90e8\u5206\u8a18\u61b6\u9ad4\u7a7a\u9593\u6d6a\u8cbb\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#5","title":"5. \u00a0 \u8d70\u8a2a\u9663\u5217","text":"<p>\u5728\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\uff0c\u6211\u5011\u65e2\u53ef\u4ee5\u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u8d70\u8a2a\u7372\u53d6\u9663\u5217\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"\u8d70\u8a2a\u9663\u5217\"\"\"\n    count = 0\n    # \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for i in range(len(nums)):\n        count += nums[i]\n    # \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for num in nums:\n        count += num\n    # \u540c\u6642\u8d70\u8a2a\u8cc7\u6599\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfunc traverse(nums []int) {\n    count := 0\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for _, num := range nums {\n        count += num\n    }\n    // \u540c\u6642\u8d70\u8a2a\u8cc7\u6599\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for num in nums {\n        count += num\n    }\n    // \u540c\u6642\u8d70\u8a2a\u8cc7\u6599\u7d22\u5f15\u548c\u5143\u7d20\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfunction traverse(nums) {\n    let count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* \u8d70\u8a2a\u9663\u5217\u5143\u7d20 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n  for (int _num in nums) {\n    count += _num;\n  }\n  // \u900f\u904e forEach \u65b9\u6cd5\u8d70\u8a2a\u9663\u5217\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (j in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### \u8d70\u8a2a\u9663\u5217 ###\ndef traverse(nums)\n  count = 0\n\n  # \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n  for num in nums\n    count += num\n  end\nend\n</code></pre> array.zig<pre><code>// \u8d70\u8a2a\u9663\u5217\nfn traverse(nums: []i32) void {\n    var count: i32 = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    var i: i32 = 0;\n    while (i &lt; nums.len) : (i += 1) {\n        count += nums[i];\n    }\n    count = 0;\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (nums) |num| {\n        count += num;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#6","title":"6. \u00a0 \u67e5\u8a62\u5143\u7d20","text":"<p>\u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20\u9700\u8981\u8d70\u8a2a\u9663\u5217\uff0c\u6bcf\u8f2a\u5224\u65b7\u5143\u7d20\u503c\u662f\u5426\u5339\u914d\uff0c\u82e5\u5339\u914d\u5247\u8f38\u51fa\u5c0d\u61c9\u7d22\u5f15\u3002</p> <p>\u56e0\u70ba\u9663\u5217\u662f\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u6240\u4ee5\u4e0a\u8ff0\u67e5\u8a62\u64cd\u4f5c\u88ab\u7a31\u70ba\u201c\u7dda\u6027\u67e5\u8a62\u201d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target)\n            return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> array.zig<pre><code>// \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n    for (nums, 0..) |num, i| {\n        if (num == target) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#7","title":"7. \u00a0 \u64f4\u5bb9\u9663\u5217","text":"<p>\u5728\u8907\u96dc\u7684\u7cfb\u7d71\u74b0\u5883\u4e2d\uff0c\u7a0b\u5f0f\u96e3\u4ee5\u4fdd\u8b49\u9663\u5217\u4e4b\u5f8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u662f\u53ef\u7528\u7684\uff0c\u5f9e\u800c\u7121\u6cd5\u5b89\u5168\u5730\u64f4\u5c55\u9663\u5217\u5bb9\u91cf\u3002\u56e0\u6b64\u5728\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\uff0c\u9663\u5217\u7684\u9577\u5ea6\u662f\u4e0d\u53ef\u8b8a\u7684\u3002</p> <p>\u5982\u679c\u6211\u5011\u5e0c\u671b\u64f4\u5bb9\u9663\u5217\uff0c\u5247\u9700\u91cd\u65b0\u5efa\u7acb\u4e00\u500b\u66f4\u5927\u7684\u9663\u5217\uff0c\u7136\u5f8c\u628a\u539f\u9663\u5217\u5143\u7d20\u4f9d\u6b21\u8907\u88fd\u5230\u65b0\u9663\u5217\u3002\u9019\u662f\u4e00\u500b \\(O(n)\\) \u7684\u64cd\u4f5c\uff0c\u5728\u9663\u5217\u5f88\u5927\u7684\u60c5\u6cc1\u4e0b\u975e\u5e38\u8017\u6642\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"\u64f4\u5c55\u9663\u5217\u9577\u5ea6\"\"\"\n    # \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    res = [0] * (len(nums) + enlarge)\n    # \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res\n</code></pre> array.cpp<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    int *res = new int[size + enlarge];\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    delete[] nums;\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.java<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    int[] res = new int[nums.length + enlarge];\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.cs<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    int[] res = new int[nums.Length + enlarge];\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.go<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    res := make([]int, len(nums)+enlarge)\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for i, num := range nums {\n        res[i] = num\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res\n}\n</code></pre> array.swift<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res\n}\n</code></pre> array.js<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\n// \u8acb\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52d5\u614b\u9663\u5217\uff0c\u53ef\u4ee5\u76f4\u63a5\u64f4\u5c55\n// \u70ba\u4e86\u65b9\u4fbf\u5b78\u7fd2\uff0c\u672c\u51fd\u5f0f\u5c07 Array \u770b\u4f5c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u7684\u9663\u5217\nfunction extend(nums, enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.ts<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\n// \u8acb\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52d5\u614b\u9663\u5217\uff0c\u53ef\u4ee5\u76f4\u63a5\u64f4\u5c55\n// \u70ba\u4e86\u65b9\u4fbf\u5b78\u7fd2\uff0c\u672c\u51fd\u5f0f\u5c07 Array \u770b\u4f5c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u7684\u9663\u5217\nfunction extend(nums: number[], enlarge: number): number[] {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.dart<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n  return res;\n}\n</code></pre> array.rs<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nfn extend(nums: &amp;[i32], enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\n    for i in 0..nums.len() {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    res\n}\n</code></pre> array.c<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u521d\u59cb\u5316\u64f4\u5c55\u5f8c\u7684\u7a7a\u9593\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.kt<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    val res = IntArray(nums.size + enlarge)\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res\n}\n</code></pre> array.rb<pre><code>### \u64f4\u5c55\u9663\u5217\u9577\u5ea6 ###\n# \u8acb\u6ce8\u610f\uff0cRuby \u7684 Array \u662f\u52d5\u614b\u9663\u5217\uff0c\u53ef\u4ee5\u76f4\u63a5\u64f4\u5c55\n# \u70ba\u4e86\u65b9\u4fbf\u5b78\u7fd2\uff0c\u672c\u51fd\u5f0f\u5c07 Array \u770b\u4f5c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u7684\u9663\u5217\ndef extend(nums, enlarge)\n  # \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n  res = Array.new(nums.length + enlarge, 0)\n\n  # \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n  res\nend\n</code></pre> array.zig<pre><code>// \u64f4\u5c55\u9663\u5217\u9577\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n    @memset(res, 0);\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    std.mem.copy(i32, res, nums);\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2 \u00a0 \u9663\u5217\u7684\u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u9663\u5217\u5132\u5b58\u5728\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u5167\uff0c\u4e14\u5143\u7d20\u578b\u5225\u76f8\u540c\u3002\u9019\u7a2e\u505a\u6cd5\u5305\u542b\u8c50\u5bcc\u7684\u5148\u9a57\u8cc7\u8a0a\uff0c\u7cfb\u7d71\u53ef\u4ee5\u5229\u7528\u9019\u4e9b\u8cc7\u8a0a\u4f86\u6700\u4f73\u5316\u8cc7\u6599\u7d50\u69cb\u7684\u64cd\u4f5c\u6548\u7387\u3002</p> <ul> <li>\u7a7a\u9593\u6548\u7387\u9ad8\uff1a\u9663\u5217\u70ba\u8cc7\u6599\u5206\u914d\u4e86\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u584a\uff0c\u7121\u9808\u984d\u5916\u7684\u7d50\u69cb\u958b\u92b7\u3002</li> <li>\u652f\u6301\u96a8\u6a5f\u8a2a\u554f\uff1a\u9663\u5217\u5141\u8a31\u5728 \\(O(1)\\) \u6642\u9593\u5167\u8a2a\u554f\u4efb\u4f55\u5143\u7d20\u3002</li> <li>\u5feb\u53d6\u5340\u57df\u6027\uff1a\u7576\u8a2a\u554f\u9663\u5217\u5143\u7d20\u6642\uff0c\u8a08\u7b97\u6a5f\u4e0d\u50c5\u6703\u8f09\u5165\u5b83\uff0c\u9084\u6703\u5feb\u53d6\u5176\u5468\u570d\u7684\u5176\u4ed6\u8cc7\u6599\uff0c\u5f9e\u800c\u85c9\u52a9\u9ad8\u901f\u5feb\u53d6\u4f86\u63d0\u5347\u5f8c\u7e8c\u64cd\u4f5c\u7684\u57f7\u884c\u901f\u5ea6\u3002</li> </ul> <p>\u9023\u7e8c\u7a7a\u9593\u5132\u5b58\u662f\u4e00\u628a\u96d9\u5203\u528d\uff0c\u5176\u5b58\u5728\u4ee5\u4e0b\u4fb7\u9650\u6027\u3002</p> <ul> <li>\u63d2\u5165\u8207\u522a\u9664\u6548\u7387\u4f4e\uff1a\u7576\u9663\u5217\u4e2d\u5143\u7d20\u8f03\u591a\u6642\uff0c\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u9700\u8981\u79fb\u52d5\u5927\u91cf\u7684\u5143\u7d20\u3002</li> <li>\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff1a\u9663\u5217\u5728\u521d\u59cb\u5316\u5f8c\u9577\u5ea6\u5c31\u56fa\u5b9a\u4e86\uff0c\u64f4\u5bb9\u9663\u5217\u9700\u8981\u5c07\u6240\u6709\u8cc7\u6599\u8907\u88fd\u5230\u65b0\u9663\u5217\uff0c\u958b\u92b7\u5f88\u5927\u3002</li> <li>\u7a7a\u9593\u6d6a\u8cbb\uff1a\u5982\u679c\u9663\u5217\u5206\u914d\u7684\u5927\u5c0f\u8d85\u904e\u5be6\u969b\u6240\u9700\uff0c\u90a3\u9ebc\u591a\u9918\u7684\u7a7a\u9593\u5c31\u88ab\u6d6a\u8cbb\u4e86\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3 \u00a0 \u9663\u5217\u5178\u578b\u61c9\u7528","text":"<p>\u9663\u5217\u662f\u4e00\u7a2e\u57fa\u790e\u4e14\u5e38\u898b\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u65e2\u983b\u7e41\u61c9\u7528\u5728\u5404\u985e\u6f14\u7b97\u6cd5\u4e4b\u4e2d\uff0c\u4e5f\u53ef\u7528\u65bc\u5be6\u73fe\u5404\u7a2e\u8907\u96dc\u8cc7\u6599\u7d50\u69cb\u3002</p> <ul> <li>\u96a8\u6a5f\u8a2a\u554f\uff1a\u5982\u679c\u6211\u5011\u60f3\u96a8\u6a5f\u62bd\u53d6\u4e00\u4e9b\u6a23\u672c\uff0c\u90a3\u9ebc\u53ef\u4ee5\u7528\u9663\u5217\u5132\u5b58\uff0c\u4e26\u751f\u6210\u4e00\u500b\u96a8\u6a5f\u5e8f\u5217\uff0c\u6839\u64da\u7d22\u5f15\u5be6\u73fe\u96a8\u6a5f\u62bd\u6a23\u3002</li> <li>\u6392\u5e8f\u548c\u641c\u5c0b\uff1a\u9663\u5217\u662f\u6392\u5e8f\u548c\u641c\u5c0b\u6f14\u7b97\u6cd5\u6700\u5e38\u7528\u7684\u8cc7\u6599\u7d50\u69cb\u3002\u5feb\u901f\u6392\u5e8f\u3001\u5408\u4f75\u6392\u5e8f\u3001\u4e8c\u5206\u641c\u5c0b\u7b49\u90fd\u4e3b\u8981\u5728\u9663\u5217\u4e0a\u9032\u884c\u3002</li> <li>\u67e5\u8a62\u8868\uff1a\u7576\u9700\u8981\u5feb\u901f\u67e5\u8a62\u4e00\u500b\u5143\u7d20\u6216\u5176\u5c0d\u61c9\u95dc\u4fc2\u6642\uff0c\u53ef\u4ee5\u4f7f\u7528\u9663\u5217\u4f5c\u70ba\u67e5\u8a62\u8868\u3002\u5047\u5982\u6211\u5011\u60f3\u5be6\u73fe\u5b57\u5143\u5230 ASCII \u78bc\u7684\u5c0d\u6620\uff0c\u5247\u53ef\u4ee5\u5c07\u5b57\u5143\u7684 ASCII \u78bc\u503c\u4f5c\u70ba\u7d22\u5f15\uff0c\u5c0d\u61c9\u7684\u5143\u7d20\u5b58\u653e\u5728\u9663\u5217\u4e2d\u7684\u5c0d\u61c9\u4f4d\u7f6e\u3002</li> <li>\u6a5f\u5668\u5b78\u7fd2\uff1a\u795e\u7d93\u7db2\u8def\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9663\u3001\u5f35\u91cf\u4e4b\u9593\u7684\u7dda\u6027\u4ee3\u6578\u904b\u7b97\uff0c\u9019\u4e9b\u8cc7\u6599\u90fd\u662f\u4ee5\u9663\u5217\u7684\u5f62\u5f0f\u69cb\u5efa\u7684\u3002\u9663\u5217\u662f\u795e\u7d93\u7db2\u8def\u7a0b\u5f0f\u8a2d\u8a08\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u8cc7\u6599\u7d50\u69cb\u5be6\u73fe\uff1a\u9663\u5217\u53ef\u4ee5\u7528\u65bc\u5be6\u73fe\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u5806\u7a4d\u3001\u5716\u7b49\u8cc7\u6599\u7d50\u69cb\u3002\u4f8b\u5982\uff0c\u5716\u7684\u9130\u63a5\u77e9\u9663\u8868\u793a\u5be6\u969b\u4e0a\u662f\u4e00\u500b\u4e8c\u7dad\u9663\u5217\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2 \u00a0 \u93c8\u7d50\u4e32\u5217","text":"<p>\u8a18\u61b6\u9ad4\u7a7a\u9593\u662f\u6240\u6709\u7a0b\u5f0f\u7684\u516c\u5171\u8cc7\u6e90\uff0c\u5728\u4e00\u500b\u8907\u96dc\u7684\u7cfb\u7d71\u57f7\u884c\u74b0\u5883\u4e0b\uff0c\u7a7a\u9592\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u53ef\u80fd\u6563\u843d\u5728\u8a18\u61b6\u9ad4\u5404\u8655\u3002\u6211\u5011\u77e5\u9053\uff0c\u5132\u5b58\u9663\u5217\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u5fc5\u9808\u662f\u9023\u7e8c\u7684\uff0c\u800c\u7576\u9663\u5217\u975e\u5e38\u5927\u6642\uff0c\u8a18\u61b6\u9ad4\u53ef\u80fd\u7121\u6cd5\u63d0\u4f9b\u5982\u6b64\u5927\u7684\u9023\u7e8c\u7a7a\u9593\u3002\u6b64\u6642\u93c8\u7d50\u4e32\u5217\u7684\u9748\u6d3b\u6027\u512a\u52e2\u5c31\u9ad4\u73fe\u51fa\u4f86\u4e86\u3002</p> <p>\u93c8\u7d50\u4e32\u5217\uff08linked list\uff09\u662f\u4e00\u7a2e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u5176\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\u90fd\u662f\u4e00\u500b\u7bc0\u9ede\u7269\u4ef6\uff0c\u5404\u500b\u7bc0\u9ede\u900f\u904e\u201c\u5f15\u7528\u201d\u76f8\u9023\u7dda\u3002\u5f15\u7528\u8a18\u9304\u4e86\u4e0b\u4e00\u500b\u7bc0\u9ede\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\uff0c\u900f\u904e\u5b83\u53ef\u4ee5\u5f9e\u7576\u524d\u7bc0\u9ede\u8a2a\u554f\u5230\u4e0b\u4e00\u500b\u7bc0\u9ede\u3002</p> <p>\u93c8\u7d50\u4e32\u5217\u7684\u8a2d\u8a08\u4f7f\u5f97\u5404\u500b\u7bc0\u9ede\u53ef\u4ee5\u5206\u6563\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u5404\u8655\uff0c\u5b83\u5011\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u7121\u9808\u9023\u7e8c\u3002</p> <p></p> <p> \u5716 4-5 \u00a0 \u93c8\u7d50\u4e32\u5217\u5b9a\u7fa9\u8207\u5132\u5b58\u65b9\u5f0f </p> <p>\u89c0\u5bdf\u5716 4-5 \uff0c\u93c8\u7d50\u4e32\u5217\u7684\u7d44\u6210\u55ae\u4f4d\u662f\u7bc0\u9ede\uff08node\uff09\u7269\u4ef6\u3002\u6bcf\u500b\u7bc0\u9ede\u90fd\u5305\u542b\u5169\u9805\u8cc7\u6599\uff1a\u7bc0\u9ede\u7684\u201c\u503c\u201d\u548c\u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u201c\u5f15\u7528\u201d\u3002</p> <ul> <li>\u93c8\u7d50\u4e32\u5217\u7684\u9996\u500b\u7bc0\u9ede\u88ab\u7a31\u70ba\u201c\u982d\u7bc0\u9ede\u201d\uff0c\u6700\u5f8c\u4e00\u500b\u7bc0\u9ede\u88ab\u7a31\u70ba\u201c\u5c3e\u7bc0\u9ede\u201d\u3002</li> <li>\u5c3e\u7bc0\u9ede\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5b83\u5728 Java\u3001C++ \u548c Python \u4e2d\u5206\u5225\u88ab\u8a18\u70ba <code>null</code>\u3001<code>nullptr</code> \u548c <code>None</code> \u3002</li> <li>\u5728 C\u3001C++\u3001Go \u548c Rust \u7b49\u652f\u6301\u6307\u6a19\u7684\u8a9e\u8a00\u4e2d\uff0c\u4e0a\u8ff0\u201c\u5f15\u7528\u201d\u61c9\u88ab\u66ff\u63db\u70ba\u201c\u6307\u6a19\u201d\u3002</li> </ul> <p>\u5982\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\uff0c\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede <code>ListNode</code> \u9664\u4e86\u5305\u542b\u503c\uff0c\u9084\u9700\u984d\u5916\u5132\u5b58\u4e00\u500b\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002\u56e0\u6b64\u5728\u76f8\u540c\u8cc7\u6599\u91cf\u4e0b\uff0c\u93c8\u7d50\u4e32\u5217\u6bd4\u9663\u5217\u4f54\u7528\u66f4\u591a\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class ListNode:\n    \"\"\"\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # \u7bc0\u9ede\u503c\n        self.next: ListNode | None = None # \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct ListNode {\n    int val;         // \u7bc0\u9ede\u503c\n    ListNode *next;  // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n    ListNode(int x) : val(x), next(nullptr) {}  // \u5efa\u69cb\u5b50\n};\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    int val;        // \u7bc0\u9ede\u503c\n    ListNode next;  // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u5efa\u69cb\u5b50\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode(int x) {  //\u5efa\u69cb\u5b50\n    int val = x;         // \u7bc0\u9ede\u503c\n    ListNode? next;      // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntype ListNode struct {\n    Val  int       // \u7bc0\u9ede\u503c\n    Next *ListNode // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n}\n\n// NewListNode \u5efa\u69cb\u5b50\uff0c\u5efa\u7acb\u4e00\u500b\u65b0\u7684\u93c8\u7d50\u4e32\u5217\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var next: ListNode? // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n\n    init(x: Int) { // \u5efa\u69cb\u5b50\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // \u7bc0\u9ede\u503c\n        this.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // \u7bc0\u9ede\u503c\n        this.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n  int val; // \u7bc0\u9ede\u503c\n  ListNode? next; // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n  ListNode(this.val, [this.next]); // \u5efa\u69cb\u5b50\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u7bc0\u9ede\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct ListNode {\n    int val;               // \u7bc0\u9ede\u503c\n    struct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n} ListNode;\n\n/* \u5efa\u69cb\u5b50 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\n// \u5efa\u69cb\u5b50\nclass ListNode(x: Int) {\n    val _val: Int = x          // \u7bc0\u9ede\u503c\n    val next: ListNode? = null // \u6307\u5411\u4e0b\u4e00\u500b\u7bc0\u9ede\u7684\u5f15\u7528\n}\n</code></pre> <pre><code># \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\nclass ListNode\n  attr_accessor :val  # \u7bc0\u9ede\u503c\n  attr_accessor :next # \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n\n  def initialize(val=0, next_node=nil)\n    @val = val\n    @next = next_node\n  end\nend\n</code></pre> <pre><code>// \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u7bc0\u9ede\u503c\n        next: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n\n        // \u5efa\u69cb\u5b50\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1 \u00a0 \u93c8\u7d50\u4e32\u5217\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217","text":"<p>\u5efa\u7acb\u93c8\u7d50\u4e32\u5217\u5206\u70ba\u5169\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\u7269\u4ef6\uff0c\u7b2c\u4e8c\u6b65\u662f\u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\u95dc\u4fc2\u3002\u521d\u59cb\u5316\u5b8c\u6210\u5f8c\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5f9e\u93c8\u7d50\u4e32\u5217\u7684\u982d\u7bc0\u9ede\u51fa\u767c\uff0c\u900f\u904e\u5f15\u7528\u6307\u5411 <code>next</code> \u4f9d\u6b21\u8a2a\u554f\u6240\u6709\u7bc0\u9ede\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code># \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\\\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nval n0 = ListNode(1)\nval n1 = ListNode(3)\nval n2 = ListNode(2)\nval n3 = ListNode(5)\nval n4 = ListNode(4)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rb<pre><code># \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nn0 = ListNode.new(1)\nn1 = ListNode.new(3)\nn2 = ListNode.new(2)\nn3 = ListNode.new(5)\nn4 = ListNode.new(4)\n# \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.zig<pre><code>// \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = &amp;n1;\nn1.next = &amp;n2;\nn2.next = &amp;n3;\nn3.next = &amp;n4;\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u9663\u5217\u6574\u9ad4\u662f\u4e00\u500b\u8b8a\u6578\uff0c\u6bd4\u5982\u9663\u5217 <code>nums</code> \u5305\u542b\u5143\u7d20 <code>nums[0]</code> \u548c <code>nums[1]</code> \u7b49\uff0c\u800c\u93c8\u7d50\u4e32\u5217\u662f\u7531\u591a\u500b\u7368\u7acb\u7684\u7bc0\u9ede\u7269\u4ef6\u7d44\u6210\u7684\u3002\u6211\u5011\u901a\u5e38\u5c07\u982d\u7bc0\u9ede\u7576\u4f5c\u93c8\u7d50\u4e32\u5217\u7684\u4ee3\u7a31\uff0c\u6bd4\u5982\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u4e2d\u7684\u93c8\u7d50\u4e32\u5217\u53ef\u8a18\u4f5c\u93c8\u7d50\u4e32\u5217 <code>n0</code> \u3002</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#2","title":"2. \u00a0 \u63d2\u5165\u7bc0\u9ede","text":"<p>\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u63d2\u5165\u7bc0\u9ede\u975e\u5e38\u5bb9\u6613\u3002\u5982\u5716 4-6 \u6240\u793a\uff0c\u5047\u8a2d\u6211\u5011\u60f3\u5728\u76f8\u9130\u7684\u5169\u500b\u7bc0\u9ede <code>n0</code> \u548c <code>n1</code> \u4e4b\u9593\u63d2\u5165\u4e00\u500b\u65b0\u7bc0\u9ede <code>P</code> \uff0c\u5247\u53ea\u9700\u6539\u8b8a\u5169\u500b\u7bc0\u9ede\u5f15\u7528\uff08\u6307\u6a19\uff09\u5373\u53ef\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</p> <p>\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5728\u9663\u5217\u4e2d\u63d2\u5165\u5143\u7d20\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u5728\u5927\u8cc7\u6599\u91cf\u4e0b\u7684\u6548\u7387\u8f03\u4f4e\u3002</p> <p></p> <p> \u5716 4-6 \u00a0 \u93c8\u7d50\u4e32\u5217\u63d2\u5165\u7bc0\u9ede\u793a\u4f8b </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"\u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede _p ###\n# Ruby \u7684 `p` \u662f\u4e00\u500b\u5167\u5efa\u51fd\u5f0f\uff0c `P` \u662f\u4e00\u500b\u5e38\u6578\uff0c\u6240\u4ee5\u53ef\u4ee5\u4f7f\u7528 `_p` \u4ee3\u66ff\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n    var n1 = n0.?.next;\n    P.?.next = n1;\n    n0.?.next = P;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#3","title":"3. \u00a0 \u522a\u9664\u7bc0\u9ede","text":"<p>\u5982\u5716 4-7 \u6240\u793a\uff0c\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u522a\u9664\u7bc0\u9ede\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u8b8a\u4e00\u500b\u7bc0\u9ede\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\u5373\u53ef\u3002</p> <p>\u8acb\u6ce8\u610f\uff0c\u5118\u7ba1\u5728\u522a\u9664\u64cd\u4f5c\u5b8c\u6210\u5f8c\u7bc0\u9ede <code>P</code> \u4ecd\u7136\u6307\u5411 <code>n1</code> \uff0c\u4f46\u5be6\u969b\u4e0a\u8d70\u8a2a\u6b64\u93c8\u7d50\u4e32\u5217\u5df2\u7d93\u7121\u6cd5\u8a2a\u554f\u5230 <code>P</code> \uff0c\u9019\u610f\u5473\u8457 <code>P</code> \u5df2\u7d93\u4e0d\u518d\u5c6c\u65bc\u8a72\u93c8\u7d50\u4e32\u5217\u4e86\u3002</p> <p></p> <p> \u5716 4-7 \u00a0 \u93c8\u7d50\u4e32\u5217\u522a\u9664\u7bc0\u9ede </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"\u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    if n0.borrow().next.is_none() {\n        return;\n    };\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\n// \u6ce8\u610f\uff1astdio.h \u4f54\u7528\u4e86 remove \u95dc\u9375\u8a5e\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfun remove(n0: ListNode?) {\n    if (n0?.next == null)\n        return\n    // n0 -&gt; P -&gt; n1\n    val p = n0.next\n    val n1 = p?.next\n    n0.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> linked_list.zig<pre><code>// \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede\nfn remove(n0: ?*inc.ListNode(i32)) void {\n    if (n0.?.next == null) return;\n    // n0 -&gt; P -&gt; n1\n    var P = n0.?.next;\n    var n1 = P.?.next;\n    n0.?.next = n1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#4","title":"4. \u00a0 \u8a2a\u554f\u7bc0\u9ede","text":"<p>\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u8a2a\u554f\u7bc0\u9ede\u7684\u6548\u7387\u8f03\u4f4e\u3002\u5982\u4e0a\u4e00\u7bc0\u6240\u8ff0\uff0c\u6211\u5011\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u4e0b\u8a2a\u554f\u9663\u5217\u4e2d\u7684\u4efb\u610f\u5143\u7d20\u3002\u93c8\u7d50\u4e32\u5217\u5247\u4e0d\u7136\uff0c\u7a0b\u5f0f\u9700\u8981\u5f9e\u982d\u7bc0\u9ede\u51fa\u767c\uff0c\u9010\u500b\u5411\u5f8c\u8d70\u8a2a\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6a19\u7bc0\u9ede\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u7684\u7b2c \\(i\\) \u500b\u7bc0\u9ede\u9700\u8981\u8ff4\u5708 \\(i - 1\\) \u8f2a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n    if index &lt;= 0 {\n        return head;\n    };\n    if let Some(node) = &amp;head.borrow().next {\n        return access(node.clone(), index - 1);\n    }\n\n    return head;\n}\n</code></pre> linked_list.c<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        if (h == null)\n            return null\n        h = h.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> linked_list.zig<pre><code>// \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n    var head = node;\n    var i: i32 = 0;\n    while (i &lt; index) : (i += 1) {\n        head = head.?.next;\n        if (head == null) return null;\n    }\n    return head;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#5","title":"5. \u00a0 \u67e5\u8a62\u7bc0\u9ede","text":"<p>\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\uff0c\u67e5\u8a62\u5176\u4e2d\u503c\u70ba <code>target</code> \u7684\u7bc0\u9ede\uff0c\u8f38\u51fa\u8a72\u7bc0\u9ede\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u7d22\u5f15\u3002\u6b64\u904e\u7a0b\u4e5f\u5c6c\u65bc\u7dda\u6027\u67e5\u8a62\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T, index: i32) -&gt; i32 {\n    if head.borrow().val == target {\n        return index;\n    };\n    if let Some(node) = &amp;head.borrow_mut().next {\n        return find(node.clone(), target, index + 1);\n    }\n    return -1;\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h._val == target)\n            return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n    var head = node;\n    var index: i32 = 0;\n    while (head != null) {\n        if (head.?.val == target) return index;\n        head = head.?.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","title":"4.2.2 \u00a0 \u9663\u5217 vs. \u93c8\u7d50\u4e32\u5217","text":"<p>\u8868 4-1 \u7e3d\u7d50\u4e86\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u7684\u5404\u9805\u7279\u9ede\u4e26\u5c0d\u6bd4\u4e86\u64cd\u4f5c\u6548\u7387\u3002\u7531\u65bc\u5b83\u5011\u63a1\u7528\u5169\u7a2e\u76f8\u53cd\u7684\u5132\u5b58\u7b56\u7565\uff0c\u56e0\u6b64\u5404\u7a2e\u6027\u8cea\u548c\u64cd\u4f5c\u6548\u7387\u4e5f\u5448\u73fe\u5c0d\u7acb\u7684\u7279\u9ede\u3002</p> <p> \u8868 4-1 \u00a0 \u9663\u5217\u8207\u93c8\u7d50\u4e32\u5217\u7684\u6548\u7387\u5c0d\u6bd4 </p> \u9663\u5217 \u93c8\u7d50\u4e32\u5217 \u5132\u5b58\u65b9\u5f0f \u9023\u7e8c\u8a18\u61b6\u9ad4\u7a7a\u9593 \u5206\u6563\u8a18\u61b6\u9ad4\u7a7a\u9593 \u5bb9\u91cf\u64f4\u5c55 \u9577\u5ea6\u4e0d\u53ef\u8b8a \u53ef\u9748\u6d3b\u64f4\u5c55 \u8a18\u61b6\u9ad4\u6548\u7387 \u5143\u7d20\u4f54\u7528\u8a18\u61b6\u9ad4\u5c11\u3001\u4f46\u53ef\u80fd\u6d6a\u8cbb\u7a7a\u9593 \u5143\u7d20\u4f54\u7528\u8a18\u61b6\u9ad4\u591a \u8a2a\u554f\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \u65b0\u589e\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \u522a\u9664\u5143\u7d20 \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3 \u00a0 \u5e38\u898b\u93c8\u7d50\u4e32\u5217\u578b\u5225","text":"<p>\u5982\u5716 4-8 \u6240\u793a\uff0c\u5e38\u898b\u7684\u93c8\u7d50\u4e32\u5217\u578b\u5225\u5305\u62ec\u4e09\u7a2e\u3002</p> <ul> <li>\u55ae\u5411\u93c8\u7d50\u4e32\u5217\uff1a\u5373\u524d\u9762\u4ecb\u7d39\u7684\u666e\u901a\u93c8\u7d50\u4e32\u5217\u3002\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\u5169\u9805\u8cc7\u6599\u3002\u6211\u5011\u5c07\u9996\u500b\u7bc0\u9ede\u7a31\u70ba\u982d\u7bc0\u9ede\uff0c\u5c07\u6700\u5f8c\u4e00\u500b\u7bc0\u9ede\u7a31\u70ba\u5c3e\u7bc0\u9ede\uff0c\u5c3e\u7bc0\u9ede\u6307\u5411\u7a7a <code>None</code> \u3002</li> <li>\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\uff1a\u5982\u679c\u6211\u5011\u4ee4\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u7684\u5c3e\u7bc0\u9ede\u6307\u5411\u982d\u7bc0\u9ede\uff08\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5247\u5f97\u5230\u4e00\u500b\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u3002\u5728\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u4e2d\uff0c\u4efb\u610f\u7bc0\u9ede\u90fd\u53ef\u4ee5\u8996\u4f5c\u982d\u7bc0\u9ede\u3002</li> <li>\u96d9\u5411\u93c8\u7d50\u4e32\u5217\uff1a\u8207\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u76f8\u6bd4\uff0c\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u8a18\u9304\u4e86\u5169\u500b\u65b9\u5411\u7684\u5f15\u7528\u3002\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede\u5b9a\u7fa9\u540c\u6642\u5305\u542b\u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\uff08\u4e0b\u4e00\u500b\u7bc0\u9ede\uff09\u548c\u524d\u9a45\u7bc0\u9ede\uff08\u4e0a\u4e00\u500b\u7bc0\u9ede\uff09\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002\u76f8\u8f03\u65bc\u55ae\u5411\u93c8\u7d50\u4e32\u5217\uff0c\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u66f4\u5177\u9748\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u5169\u500b\u65b9\u5411\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\uff0c\u4f46\u76f8\u61c9\u5730\u4e5f\u9700\u8981\u4f54\u7528\u66f4\u591a\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class ListNode:\n    \"\"\"\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u7bc0\u9ede\u503c\n        self.next: ListNode | None = None  # \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n        self.prev: ListNode | None = None  # \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct ListNode {\n    int val;         // \u7bc0\u9ede\u503c\n    ListNode *next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n    ListNode *prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u5efa\u69cb\u5b50\n};\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    int val;        // \u7bc0\u9ede\u503c\n    ListNode next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u5efa\u69cb\u5b50\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode(int x) {  // \u5efa\u69cb\u5b50\n    int val = x;    // \u7bc0\u9ede\u503c\n    ListNode next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntype DoublyListNode struct {\n    Val  int             // \u7bc0\u9ede\u503c\n    Next *DoublyListNode // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n    Prev *DoublyListNode // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n}\n\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var next: ListNode? // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    var prev: ListNode? // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n\n    init(x: Int) { // \u5efa\u69cb\u5b50\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // \u7bc0\u9ede\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // \u7bc0\u9ede\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    int val;        // \u7bc0\u9ede\u503c\n    ListNode? next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode? prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode(this.val, [this.next, this.prev]);  // \u5efa\u69cb\u5b50\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u578b\u5225 */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u7bc0\u9ede\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n}\n\n/* \u5efa\u69cb\u5b50 */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct ListNode {\n    int val;               // \u7bc0\u9ede\u503c\n    struct ListNode *next; // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n    struct ListNode *prev; // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n} ListNode;\n\n/* \u5efa\u69cb\u5b50 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\n// \u5efa\u69cb\u5b50\nclass ListNode(x: Int) {\n    val _val: Int = x           // \u7bc0\u9ede\u503c\n    val next: ListNode? = null  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    val prev: ListNode? = null  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n}\n</code></pre> <pre><code># \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\nclass ListNode\n  attr_accessor :val    # \u7bc0\u9ede\u503c\n  attr_accessor :next   # \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n  attr_accessor :prev   # \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n\n  def initialize(val=0, next_node=nil, prev_node=nil)\n    @val = val\n    @next = next_node\n    @prev = prev_node\n  end\nend\n</code></pre> <pre><code>// \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u7bc0\u9ede\u503c\n        next: ?*Self = null, // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n        prev: ?*Self = null, // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n\n        // \u5efa\u69cb\u5b50\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n</code></pre> <p></p> <p> \u5716 4-8 \u00a0 \u5e38\u898b\u93c8\u7d50\u4e32\u5217\u7a2e\u985e </p>"},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4 \u00a0 \u93c8\u7d50\u4e32\u5217\u5178\u578b\u61c9\u7528","text":"<p>\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u901a\u5e38\u7528\u65bc\u5be6\u73fe\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u548c\u5716\u7b49\u8cc7\u6599\u7d50\u69cb\u3002</p> <ul> <li>\u5806\u758a\u8207\u4f47\u5217\uff1a\u7576\u63d2\u5165\u548c\u522a\u9664\u64cd\u4f5c\u90fd\u5728\u93c8\u7d50\u4e32\u5217\u7684\u4e00\u7aef\u9032\u884c\u6642\uff0c\u5b83\u8868\u73fe\u7684\u7279\u6027\u70ba\u5148\u9032\u5f8c\u51fa\uff0c\u5c0d\u61c9\u5806\u758a\uff1b\u7576\u63d2\u5165\u64cd\u4f5c\u5728\u93c8\u7d50\u4e32\u5217\u7684\u4e00\u7aef\u9032\u884c\uff0c\u522a\u9664\u64cd\u4f5c\u5728\u93c8\u7d50\u4e32\u5217\u7684\u53e6\u4e00\u7aef\u9032\u884c\uff0c\u5b83\u8868\u73fe\u7684\u7279\u6027\u70ba\u5148\u9032\u5148\u51fa\uff0c\u5c0d\u61c9\u4f47\u5217\u3002</li> <li>\u96dc\u6e4a\u8868\uff1a\u93c8\u5f0f\u4f4d\u5740\u662f\u89e3\u6c7a\u96dc\u6e4a\u885d\u7a81\u7684\u4e3b\u6d41\u65b9\u6848\u4e4b\u4e00\uff0c\u5728\u8a72\u65b9\u6848\u4e2d\uff0c\u6240\u6709\u885d\u7a81\u7684\u5143\u7d20\u90fd\u6703\u88ab\u653e\u5230\u4e00\u500b\u93c8\u7d50\u4e32\u5217\u4e2d\u3002</li> <li>\u5716\uff1a\u9130\u63a5\u8868\u662f\u8868\u793a\u5716\u7684\u4e00\u7a2e\u5e38\u7528\u65b9\u5f0f\uff0c\u5176\u4e2d\u5716\u7684\u6bcf\u500b\u9802\u9ede\u90fd\u8207\u4e00\u500b\u93c8\u7d50\u4e32\u5217\u76f8\u95dc\u806f\uff0c\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\u90fd\u4ee3\u8868\u8207\u8a72\u9802\u9ede\u76f8\u9023\u7684\u5176\u4ed6\u9802\u9ede\u3002</li> </ul> <p>\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5e38\u7528\u65bc\u9700\u8981\u5feb\u901f\u67e5\u8a62\u524d\u4e00\u500b\u548c\u5f8c\u4e00\u500b\u5143\u7d20\u7684\u5834\u666f\u3002</p> <ul> <li>\u9ad8\u968e\u8cc7\u6599\u7d50\u69cb\uff1a\u6bd4\u5982\u5728\u7d05\u9ed1\u6a39\u3001B \u6a39\u4e2d\uff0c\u6211\u5011\u9700\u8981\u8a2a\u554f\u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\uff0c\u9019\u53ef\u4ee5\u900f\u904e\u5728\u7bc0\u9ede\u4e2d\u5132\u5b58\u4e00\u500b\u6307\u5411\u7236\u7bc0\u9ede\u7684\u5f15\u7528\u4f86\u5be6\u73fe\uff0c\u985e\u4f3c\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u3002</li> <li>\u700f\u89bd\u5668\u6b77\u53f2\uff1a\u5728\u7db2\u9801\u700f\u89bd\u5668\u4e2d\uff0c\u7576\u7528\u6236\u9ede\u9078\u524d\u9032\u6216\u5f8c\u9000\u6309\u9215\u6642\uff0c\u700f\u89bd\u5668\u9700\u8981\u77e5\u9053\u4f7f\u7528\u8005\u8a2a\u554f\u904e\u7684\u524d\u4e00\u500b\u548c\u5f8c\u4e00\u500b\u7db2\u9801\u3002\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7684\u7279\u6027\u4f7f\u5f97\u9019\u7a2e\u64cd\u4f5c\u8b8a\u5f97\u7c21\u55ae\u3002</li> <li>LRU \u6f14\u7b97\u6cd5\uff1a\u5728\u5feb\u53d6\u6dd8\u6c70\uff08LRU\uff09\u6f14\u7b97\u6cd5\u4e2d\uff0c\u6211\u5011\u9700\u8981\u5feb\u901f\u627e\u5230\u6700\u8fd1\u6700\u5c11\u4f7f\u7528\u7684\u8cc7\u6599\uff0c\u4ee5\u53ca\u652f\u6301\u5feb\u901f\u65b0\u589e\u548c\u522a\u9664\u7bc0\u9ede\u3002\u9019\u6642\u5019\u4f7f\u7528\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5c31\u975e\u5e38\u5408\u9069\u3002</li> </ul> <p>\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u5e38\u7528\u65bc\u9700\u8981\u9031\u671f\u6027\u64cd\u4f5c\u7684\u5834\u666f\uff0c\u6bd4\u5982\u4f5c\u696d\u7cfb\u7d71\u7684\u8cc7\u6e90\u6392\u7a0b\u3002</p> <ul> <li>\u6642\u9593\u7247\u8f2a\u8f49\u6392\u7a0b\u6f14\u7b97\u6cd5\uff1a\u5728\u4f5c\u696d\u7cfb\u7d71\u4e2d\uff0c\u6642\u9593\u7247\u8f2a\u8f49\u6392\u7a0b\u6f14\u7b97\u6cd5\u662f\u4e00\u7a2e\u5e38\u898b\u7684 CPU \u6392\u7a0b\u6f14\u7b97\u6cd5\uff0c\u5b83\u9700\u8981\u5c0d\u4e00\u7d44\u7a0b\u5e8f\u9032\u884c\u8ff4\u5708\u3002\u6bcf\u500b\u7a0b\u5e8f\u88ab\u8ce6\u4e88\u4e00\u500b\u6642\u9593\u7247\uff0c\u7576\u6642\u9593\u7247\u7528\u5b8c\u6642\uff0cCPU \u5c07\u5207\u63db\u5230\u4e0b\u4e00\u500b\u7a0b\u5e8f\u3002\u9019\u7a2e\u8ff4\u5708\u64cd\u4f5c\u53ef\u4ee5\u900f\u904e\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u4f86\u5be6\u73fe\u3002</li> <li>\u8cc7\u6599\u7de9\u885d\u5340\uff1a\u5728\u67d0\u4e9b\u8cc7\u6599\u7de9\u885d\u5340\u7684\u5be6\u73fe\u4e2d\uff0c\u4e5f\u53ef\u80fd\u6703\u4f7f\u7528\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u3002\u6bd4\u5982\u5728\u97f3\u8a0a\u3001\u5f71\u7247\u64ad\u653e\u5668\u4e2d\uff0c\u8cc7\u6599\u6d41\u53ef\u80fd\u6703\u88ab\u5206\u6210\u591a\u500b\u7de9\u885d\u584a\u4e26\u653e\u5165\u4e00\u500b\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\uff0c\u4ee5\u4fbf\u5be6\u73fe\u7121\u7e2b\u64ad\u653e\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/list/","title":"4.3 \u00a0 \u4e32\u5217","text":"<p>\u4e32\u5217\uff08list\uff09\u662f\u4e00\u500b\u62bd\u8c61\u7684\u8cc7\u6599\u7d50\u69cb\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u5143\u7d20\u7684\u6709\u5e8f\u96c6\u5408\uff0c\u652f\u6301\u5143\u7d20\u8a2a\u554f\u3001\u4fee\u6539\u3001\u65b0\u589e\u3001\u522a\u9664\u548c\u8d70\u8a2a\u7b49\u64cd\u4f5c\uff0c\u7121\u9808\u4f7f\u7528\u8005\u8003\u616e\u5bb9\u91cf\u9650\u5236\u7684\u554f\u984c\u3002\u4e32\u5217\u53ef\u4ee5\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u6216\u9663\u5217\u5be6\u73fe\u3002</p> <ul> <li>\u93c8\u7d50\u4e32\u5217\u5929\u7136\u53ef\u4ee5\u770b\u4f5c\u4e00\u500b\u4e32\u5217\uff0c\u5176\u652f\u6301\u5143\u7d20\u589e\u522a\u67e5\u6539\u64cd\u4f5c\uff0c\u4e26\u4e14\u53ef\u4ee5\u9748\u6d3b\u52d5\u614b\u64f4\u5bb9\u3002</li> <li>\u9663\u5217\u4e5f\u652f\u6301\u5143\u7d20\u589e\u522a\u67e5\u6539\uff0c\u4f46\u7531\u65bc\u5176\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u53ea\u80fd\u770b\u4f5c\u4e00\u500b\u5177\u6709\u9577\u5ea6\u9650\u5236\u7684\u4e32\u5217\u3002</li> </ul> <p>\u7576\u4f7f\u7528\u9663\u5217\u5be6\u73fe\u4e32\u5217\u6642\uff0c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u7684\u6027\u8cea\u6703\u5c0e\u81f4\u4e32\u5217\u7684\u5be6\u7528\u6027\u964d\u4f4e\u3002\u9019\u662f\u56e0\u70ba\u6211\u5011\u901a\u5e38\u7121\u6cd5\u4e8b\u5148\u78ba\u5b9a\u9700\u8981\u5132\u5b58\u591a\u5c11\u8cc7\u6599\uff0c\u5f9e\u800c\u96e3\u4ee5\u9078\u64c7\u5408\u9069\u7684\u4e32\u5217\u9577\u5ea6\u3002\u82e5\u9577\u5ea6\u904e\u5c0f\uff0c\u5247\u5f88\u53ef\u80fd\u7121\u6cd5\u6eff\u8db3\u4f7f\u7528\u9700\u6c42\uff1b\u82e5\u9577\u5ea6\u904e\u5927\uff0c\u5247\u6703\u9020\u6210\u8a18\u61b6\u9ad4\u7a7a\u9593\u6d6a\u8cbb\u3002</p> <p>\u70ba\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u52d5\u614b\u9663\u5217\uff08dynamic array\uff09\u4f86\u5be6\u73fe\u4e32\u5217\u3002\u5b83\u7e7c\u627f\u4e86\u9663\u5217\u7684\u5404\u9805\u512a\u9ede\uff0c\u4e26\u4e14\u53ef\u4ee5\u5728\u7a0b\u5f0f\u57f7\u884c\u904e\u7a0b\u4e2d\u9032\u884c\u52d5\u614b\u64f4\u5bb9\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u7684\u6a19\u6e96\u5eab\u63d0\u4f9b\u7684\u4e32\u5217\u662f\u57fa\u65bc\u52d5\u614b\u9663\u5217\u5be6\u73fe\u7684\uff0c\u4f8b\u5982 Python \u4e2d\u7684 <code>list</code> \u3001Java \u4e2d\u7684 <code>ArrayList</code> \u3001C++ \u4e2d\u7684 <code>vector</code> \u548c C# \u4e2d\u7684 <code>List</code> \u7b49\u3002\u5728\u63a5\u4e0b\u4f86\u7684\u8a0e\u8ad6\u4e2d\uff0c\u6211\u5011\u5c07\u628a\u201c\u4e32\u5217\u201d\u548c\u201c\u52d5\u614b\u9663\u5217\u201d\u8996\u70ba\u7b49\u540c\u7684\u6982\u5ff5\u3002</p>"},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1 \u00a0 \u4e32\u5217\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e32\u5217","text":"<p>\u6211\u5011\u901a\u5e38\u4f7f\u7528\u201c\u7121\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u9019\u5169\u7a2e\u521d\u59cb\u5316\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u521d\u59cb\u5316\u4e32\u5217\n# \u7121\u521d\u59cb\u503c\nnums1: list[int] = []\n# \u6709\u521d\u59cb\u503c\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 nums\n// \u7121\u521d\u59cb\u503c\nvector&lt;int&gt; nums1;\n// \u6709\u521d\u59cb\u503c\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u9663\u5217\u7684\u5143\u7d20\u578b\u5225\u9700\u70ba int[] \u7684\u5305\u88dd\u985e\u5225 Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nnums1 := []int{}\n// \u6709\u521d\u59cb\u503c\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nlet nums1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nconst nums1 = [];\n// \u6709\u521d\u59cb\u503c\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nconst nums1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// \u6709\u521d\u59cb\u503c\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nvar nums1 = listOf&lt;Int&gt;()\n// \u6709\u521d\u59cb\u503c\nvar numbers = arrayOf(1, 3, 2, 5, 4)\nvar nums = numbers.toMutableList()\n</code></pre> list.rb<pre><code># \u521d\u59cb\u5316\u4e32\u5217\n# \u7121\u521d\u59cb\u503c\nnums1 = []\n# \u6709\u521d\u59cb\u503c\nnums = [1, 3, 2, 5, 4]\n</code></pre> list.zig<pre><code>// \u521d\u59cb\u5316\u4e32\u5217\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&amp;[_]i32{ 1, 3, 2, 5, 4 });\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#2","title":"2. \u00a0 \u8a2a\u554f\u5143\u7d20","text":"<p>\u4e32\u5217\u672c\u8cea\u4e0a\u662f\u9663\u5217\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u5167\u8a2a\u554f\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u8a2a\u554f\u5143\u7d20\nnum: int = nums[1]  # \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0    # \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.cpp<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nint num = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.java<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nint num = nums.get(1);  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums.set(1, 0);  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.cs<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nint num = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list_test.go<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nnum := nums[1]  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0     // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.swift<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nlet num = nums[1] // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.js<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nconst num = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.ts<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nconst num: number = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.dart<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nint num = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.rs<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nlet num: i32 = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;             // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nval num = nums[1]       // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0             // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.rb<pre><code># \u8a2a\u554f\u5143\u7d20\nnum = nums[1] # \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0 # \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.zig<pre><code>// \u8a2a\u554f\u5143\u7d20\nvar num = nums.items[1]; // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n// \u66f4\u65b0\u5143\u7d20\nnums.items[1] = 0; // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#3","title":"3. \u00a0 \u63d2\u5165\u8207\u522a\u9664\u5143\u7d20","text":"<p>\u76f8\u8f03\u65bc\u9663\u5217\uff0c\u4e32\u5217\u53ef\u4ee5\u81ea\u7531\u5730\u65b0\u589e\u8207\u522a\u9664\u5143\u7d20\u3002\u5728\u4e32\u5217\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u522a\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u8207\u9663\u5217\u76f8\u540c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u6e05\u7a7a\u4e32\u5217\nnums.clear()\n\n# \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n# \u522a\u9664\u5143\u7d20\nnums.pop(3)        # \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.cpp<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.insert(nums.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.erase(nums.begin() + 3);      // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.java<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.remove(3);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.cs<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.Clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.Insert(3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.RemoveAt(3);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list_test.go<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums = nil\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums = append(nums[:3], nums[4:]...) // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.swift<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.removeAll()\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.remove(at: 3) // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.js<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6); // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.splice(3, 1);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.ts<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6); // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.splice(3, 1);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.dart<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.removeAt(3); // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.rs<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.remove(3);    // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.remove(3);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.rb<pre><code># \u6e05\u7a7a\u4e32\u5217\nnums.clear\n\n# \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\nnums &lt;&lt; 1\nnums &lt;&lt; 3\nnums &lt;&lt; 2\nnums &lt;&lt; 5\nnums &lt;&lt; 4\n\n# \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6) # \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n# \u522a\u9664\u5143\u7d20\nnums.delete_at(3) # \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.zig<pre><code>// \u6e05\u7a7a\u4e32\u5217\nnums.clearRetainingCapacity();\n\n// \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\ntry nums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n// \u522a\u9664\u5143\u7d20\n_ = nums.orderedRemove(3); // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#4","title":"4. \u00a0 \u8d70\u8a2a\u4e32\u5217","text":"<p>\u8207\u9663\u5217\u4e00\u6a23\uff0c\u4e32\u5217\u53ef\u4ee5\u6839\u64da\u7d22\u5f15\u8d70\u8a2a\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u8d70\u8a2a\u5404\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nvar count = 0\nfor (i in nums.indices) {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\nfor (num in nums) {\n    count += num\n}\n</code></pre> list.rb<pre><code># \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217\ncount = 0\nfor i in 0...nums.length\n    count += nums[i]\nend\n\n# \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20\ncount = 0\nfor num in nums\n    count += num\nend\n</code></pre> list.zig<pre><code>// \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; nums.items.len) : (i += 1) {\n    count += nums[i];\n}\n\n// \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20\ncount = 0;\nfor (nums.items) |num| {\n    count += num;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#5","title":"5. \u00a0 \u62fc\u63a5\u4e32\u5217","text":"<p>\u7d66\u5b9a\u4e00\u500b\u65b0\u4e32\u5217 <code>nums1</code> \uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5176\u62fc\u63a5\u5230\u539f\u4e32\u5217\u7684\u5c3e\u90e8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u62fc\u63a5\u5169\u500b\u4e32\u5217\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.cpp<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.cs<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list_test.go<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.swift<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.js<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.ts<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.dart<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.rs<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nval nums1 = intArrayOf(6, 8, 7, 10, 9).toMutableList()\nnums.addAll(nums1)  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.rb<pre><code># \u62fc\u63a5\u5169\u500b\u4e32\u5217\nnums1 = [6, 8, 7, 10, 9]\nnums += nums1\n</code></pre> list.zig<pre><code>// \u62fc\u63a5\u5169\u500b\u4e32\u5217\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&amp;[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#6","title":"6. \u00a0 \u6392\u5e8f\u4e32\u5217","text":"<p>\u5b8c\u6210\u4e32\u5217\u6392\u5e8f\u5f8c\uff0c\u6211\u5011\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u9663\u5217\u985e\u5225\u6f14\u7b97\u6cd5\u984c\u4e2d\u7d93\u5e38\u8003\u67e5\u7684\u201c\u4e8c\u5206\u641c\u5c0b\u201d\u548c\u201c\u96d9\u6307\u6a19\u201d\u6f14\u7b97\u6cd5\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u6392\u5e8f\u4e32\u5217\nnums.sort()  # \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cpp<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nsort(nums.begin(), nums.end());  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.java<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nCollections.sort(nums);  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cs<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.Sort(); // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list_test.go<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nsort.Ints(nums)  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.swift<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort() // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.js<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.ts<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.dart<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort(); // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rs<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort(); // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort() // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rb<pre><code># \u6392\u5e8f\u4e32\u5217\nnums = nums.sort { |a, b| a &lt;=&gt; b } # \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.zig<pre><code>// \u6392\u5e8f\u4e32\u5217\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2 \u00a0 \u4e32\u5217\u5be6\u73fe","text":"<p>\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u5167\u5efa\u4e86\u4e32\u5217\uff0c\u4f8b\u5982 Java\u3001C++\u3001Python \u7b49\u3002\u5b83\u5011\u7684\u5be6\u73fe\u6bd4\u8f03\u8907\u96dc\uff0c\u5404\u500b\u53c3\u6578\u7684\u8a2d\u5b9a\u4e5f\u975e\u5e38\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u64f4\u5bb9\u500d\u6578\u7b49\u3002\u611f\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u67e5\u95b1\u539f\u59cb\u78bc\u9032\u884c\u5b78\u7fd2\u3002</p> <p>\u70ba\u4e86\u52a0\u6df1\u5c0d\u4e32\u5217\u5de5\u4f5c\u539f\u7406\u7684\u7406\u89e3\uff0c\u6211\u5011\u5617\u8a66\u5be6\u73fe\u4e00\u500b\u7c21\u6613\u7248\u4e32\u5217\uff0c\u5305\u62ec\u4ee5\u4e0b\u4e09\u500b\u91cd\u9ede\u8a2d\u8a08\u3002</p> <ul> <li>\u521d\u59cb\u5bb9\u91cf\uff1a\u9078\u53d6\u4e00\u500b\u5408\u7406\u7684\u9663\u5217\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u5011\u9078\u64c7 10 \u4f5c\u70ba\u521d\u59cb\u5bb9\u91cf\u3002</li> <li>\u6578\u91cf\u8a18\u9304\uff1a\u5ba3\u544a\u4e00\u500b\u8b8a\u6578 <code>size</code> \uff0c\u7528\u65bc\u8a18\u9304\u4e32\u5217\u7576\u524d\u5143\u7d20\u6578\u91cf\uff0c\u4e26\u96a8\u8457\u5143\u7d20\u63d2\u5165\u548c\u522a\u9664\u5be6\u6642\u66f4\u65b0\u3002\u6839\u64da\u6b64\u8b8a\u6578\uff0c\u6211\u5011\u53ef\u4ee5\u5b9a\u4f4d\u4e32\u5217\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65b7\u662f\u5426\u9700\u8981\u64f4\u5bb9\u3002</li> <li>\u64f4\u5bb9\u6a5f\u5236\uff1a\u82e5\u63d2\u5165\u5143\u7d20\u6642\u4e32\u5217\u5bb9\u91cf\u5df2\u6eff\uff0c\u5247\u9700\u8981\u9032\u884c\u64f4\u5bb9\u3002\u5148\u6839\u64da\u64f4\u5bb9\u500d\u6578\u5efa\u7acb\u4e00\u500b\u66f4\u5927\u7684\u9663\u5217\uff0c\u518d\u5c07\u7576\u524d\u9663\u5217\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52d5\u81f3\u65b0\u9663\u5217\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u5011\u898f\u5b9a\u6bcf\u6b21\u5c07\u9663\u5217\u64f4\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_list.py<pre><code>class MyList:\n    \"\"\"\u4e32\u5217\u985e\u5225\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._capacity: int = 10  # \u4e32\u5217\u5bb9\u91cf\n        self._arr: list[int] = [0] * self._capacity  # \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n        self._size: int = 0  # \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n        self._extend_ratio: int = 2  # \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4e32\u5217\u5bb9\u91cf\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"\u8a2a\u554f\u5143\u7d20\"\"\"\n        # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"\u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\"\"\"\n        # \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"\u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        # \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"\u522a\u9664\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        num = self._arr[index]\n        # \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self._size -= 1\n        # \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num\n\n    def extend_capacity(self):\n        \"\"\"\u4e32\u5217\u64f4\u5bb9\"\"\"\n        # \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 _extend_ratio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6709\u6548\u9577\u5ea6\u7684\u4e32\u5217\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n  private:\n    int *arr;             // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    int arrCapacity = 10; // \u4e32\u5217\u5bb9\u91cf\n    int arrSize = 0;      // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    int extendRatio = 2;   // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u6790\u69cb\u65b9\u6cd5 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    int size() {\n        return arrSize;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    void add(int num) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size() == capacity())\n            extendCapacity();\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba Vector \u7528\u65bc\u5217\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private int[] arr; // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private int capacity = 10; // \u4e32\u5217\u5bb9\u91cf\n    private int size = 0; // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private int extendRatio = 2; // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09 */\n    public int size() {\n        return size;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    public int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    public void add(int num) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size == capacity())\n            extendCapacity();\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    public void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        capacity = arr.length;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    public int[] toArray() {\n        int size = size();\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private int[] arr;           // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private int arrCapacity = 10;    // \u4e32\u5217\u5bb9\u91cf\n    private int arrSize = 0;         // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private readonly int extendRatio = 2;  // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    public int Get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    public void Add(int num) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    public void ExtendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba arrCapacity * extendRatio \u7684\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        arrCapacity = arr.Length;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    public int[] ToArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* \u4e32\u5217\u985e\u5225 */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // \u4e32\u5217\u5bb9\u91cf\n        arr:         make([]int, 10), // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n        arrSize:     0,               // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n        extendRatio: 2,               // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n    }\n}\n\n/* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* \u8a2a\u554f\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n    // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    l.arrSize++\n}\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    l.arrSize++\n}\n\n/* \u522a\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    num := l.arr[index]\n    // \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    l.arrSize--\n    // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n    return num\n}\n\n/* \u4e32\u5217\u64f4\u5bb9 */\nfunc (l *myList) extendCapacity() {\n    // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n    l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u9577\u5ea6\u7684\u4e32\u5217 */\nfunc (l *myList) toArray() []int {\n    // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private var arr: [Int] // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private var _capacity: Int // \u4e32\u5217\u5bb9\u91cf\n    private var _size: Int // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private let extendRatio: Int // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    func get(index: Int) -&gt; Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5247\u4e1f\u64f2\u932f\u8aa4\uff0c\u4e0b\u540c\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    func add(num: Int) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        _size += 1\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        _size += 1\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        let num = arr[index]\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        _size -= 1\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    func extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        _capacity = arr.count\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    #arr = new Array(); // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    #capacity = 10; // \u4e32\u5217\u5bb9\u91cf\n    #size = 0; // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    #extendRatio = 2; // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    size() {\n        return this.#size;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    get(index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.#arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.#arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    add(num) {\n        // \u5982\u679c\u9577\u5ea6\u7b49\u65bc\u5bb9\u91cf\uff0c\u5247\u9700\u8981\u64f4\u5bb9\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c07\u65b0\u5143\u7d20\u65b0\u589e\u5230\u4e32\u5217\u5c3e\u90e8\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.#arr[index];\n        // \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        this.#size--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        this.#capacity = this.#arr.length;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    toArray() {\n        let size = this.size();\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private arr: Array&lt;number&gt;; // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private _capacity: number = 10; // \u4e32\u5217\u5bb9\u91cf\n    private _size: number = 0; // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private extendRatio: number = 2; // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    public get(index: number): number {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    public add(num: number): void {\n        // \u5982\u679c\u9577\u5ea6\u7b49\u65bc\u5bb9\u91cf\uff0c\u5247\u9700\u8981\u64f4\u5bb9\n        if (this._size === this._capacity) this.extendCapacity();\n        // \u5c07\u65b0\u5143\u7d20\u65b0\u589e\u5230\u4e32\u5217\u5c3e\u90e8\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.arr[index];\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        this._size--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    public extendCapacity(): void {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba size \u7684\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        this._capacity = this.arr.length;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    public toArray(): number[] {\n        let size = this.size();\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n  late List&lt;int&gt; _arr; // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n  int _capacity = 10; // \u4e32\u5217\u5bb9\u91cf\n  int _size = 0; // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n  int _extendRatio = 2; // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n  /* \u5efa\u69cb\u5b50 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n  int size() =&gt; _size;\n\n  /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n  int capacity() =&gt; _capacity;\n\n  /* \u8a2a\u554f\u5143\u7d20 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    return _arr[index];\n  }\n\n  /* \u66f4\u65b0\u5143\u7d20 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    _arr[index] = _num;\n  }\n\n  /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n  void add(int _num) {\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    _size++;\n  }\n\n  /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if (_size == _capacity) extendCapacity();\n    // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    _size++;\n  }\n\n  /* \u522a\u9664\u5143\u7d20 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    int _num = _arr[index];\n    // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    _size--;\n    // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n    return _num;\n  }\n\n  /* \u4e32\u5217\u64f4\u5bb9 */\n  void extendCapacity() {\n    // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 _extendRatio \u500d\u7684\u65b0\u9663\u5217\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // \u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n    List.copyRange(_newNums, 0, _arr);\n    // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n    _arr = _newNums;\n    // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n    _capacity = _arr.length;\n  }\n\n  /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* \u4e32\u5217\u985e\u5225 */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    capacity: usize,     // \u4e32\u5217\u5bb9\u91cf\n    size: usize,         // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    extend_ratio: usize, // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* \u5efa\u69cb\u5b50 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = Vec::new();\n        vec.resize(capacity, 0);\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        return self.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    pub fn add(&amp;mut self, num: i32) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self.size += 1;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self.size += 1;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        let num = self.arr[index];\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for j in (index..self.size - 1) {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self.size -= 1;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    pub fn extend_capacity(&amp;mut self) {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extend_ratio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        self.capacity = new_capacity;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    pub fn to_array(&amp;mut self) -&gt; Vec&lt;i32&gt; {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* \u4e32\u5217\u985e\u5225 */\ntypedef struct {\n    int *arr;        // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    int capacity;    // \u4e32\u5217\u5bb9\u91cf\n    int size;        // \u4e32\u5217\u5927\u5c0f\n    int extendRatio; // \u4e32\u5217\u6bcf\u6b21\u64f4\u5bb9\u7684\u500d\u6578\n} MyList;\n\n/* \u5efa\u69cb\u5b50 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* \u7372\u53d6\u4e32\u5217\u9577\u5ea6 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u64f4\u5bb9\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u64f4\u5bb9\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* \u522a\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u4f54\u7528\u4e86 remove \u95dc\u9375\u8a5e\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* \u4e32\u5217\u64f4\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n    // \u5148\u5206\u914d\u7a7a\u9593\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // \u8907\u88fd\u820a\u8cc7\u6599\u5230\u65b0\u8cc7\u6599\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // \u91cb\u653e\u820a\u8cc7\u6599\n    free(temp);\n\n    // \u66f4\u65b0\u65b0\u8cc7\u6599\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* \u5c07\u4e32\u5217\u8f49\u63db\u70ba Array \u7528\u65bc\u5217\u5370 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private var capacity: Int = 10 // \u4e32\u5217\u5bb9\u91cf\n    private var size: Int = 0 // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private var extendRatio: Int = 2 // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09 */\n    fun size(): Int {\n        return size\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    fun get(index: Int): Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    fun add(num: Int) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size++\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size == capacity())\n            extendCapacity()\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size++\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        val num = arr[index]\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size--\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    fun extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        arr = arr.copyOf(capacity() * extendRatio)\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        capacity = arr.size\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    fun toArray(): IntArray {\n        val size = size()\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### \u4e32\u5217\u985e\u5225 ###\nclass MyList\n  attr_reader :size       # \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n  attr_reader :capacity   # \u7372\u53d6\u4e32\u5217\u5bb9\u91cf\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new(capacity)\n  end\n\n  ### \u8a2a\u554f\u5143\u7d20 ###\n  def get(index)\n    # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### \u8a2a\u554f\u5143\u7d20 ###\n  def set(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 ###\n  def add(num)\n    # \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    @size += 1\n  end\n\n  ### \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 ###\n  def insert(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n\n    # \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    extend_capacity if size == capacity\n\n    # \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j]\n    end\n    @arr[index] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    @size += 1\n  end\n\n  ### \u522a\u9664\u5143\u7d20 ###\n  def remove(index)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    @size -= 1\n\n    # \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n    num\n  end\n\n  ### \u4e32\u5217\u64f4\u5bb9 ###\n  def extend_capacity\n    # \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extend_ratio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n    @capacity = arr.length\n  end\n\n  ### \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 ###\n  def to_array\n    sz = size\n    # \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    arr = Array.new(sz)\n    for i in 0...sz\n      arr[i] = get(i)\n    end\n    arr\n  end\nend\n</code></pre> my_list.zig<pre><code>// \u4e32\u5217\u985e\u5225\nfn MyList(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        arr: []T = undefined,                        // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n        arrCapacity: usize = 10,                     // \u4e32\u5217\u5bb9\u91cf\n        numSize: usize = 0,                           // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n        extendRatio: usize = 2,                       // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined, // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u4e32\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n            @memset(self.arr, @as(T, 0));\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n        pub fn size(self: *Self) usize {\n            return self.numSize;\n        }\n\n        // \u7372\u53d6\u4e32\u5217\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.arrCapacity;\n        }\n\n        // \u8a2a\u554f\u5143\u7d20\n        pub fn get(self: *Self, index: usize) T {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            return self.arr[index];\n        }  \n\n        // \u66f4\u65b0\u5143\u7d20\n        pub fn set(self: *Self, index: usize, num: T) void {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            self.arr[index] = num;\n        }  \n\n        // \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\n        pub fn add(self: *Self, num: T) !void {\n            // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            self.arr[self.size()] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n            self.numSize += 1;\n        }  \n\n        // \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\n        pub fn insert(self: *Self, index: usize, num: T) !void {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n            var j = self.size() - 1;\n            while (j &gt;= index) : (j -= 1) {\n                self.arr[j + 1] = self.arr[j];\n            }\n            self.arr[index] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n            self.numSize += 1;\n        }\n\n        // \u522a\u9664\u5143\u7d20\n        pub fn remove(self: *Self, index: usize) T {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            var num = self.arr[index];\n            // \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n            var j = index;\n            while (j &lt; self.size() - 1) : (j += 1) {\n                self.arr[j] = self.arr[j + 1];\n            }\n            // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n            self.numSize -= 1;\n            // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n            return num;\n        }\n\n        // \u4e32\u5217\u64f4\u5bb9\n        pub fn extendCapacity(self: *Self) !void {\n            // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba size * extendRatio \u7684\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n            var newCapacity = self.capacity() * self.extendRatio;\n            var extend = try self.mem_allocator.alloc(T, newCapacity);\n            @memset(extend, @as(T, 0));\n            // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n            std.mem.copy(T, extend, self.arr);\n            self.arr = extend;\n            // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n            self.arrCapacity = newCapacity;\n        }\n\n        // \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217\n        pub fn toArray(self: *Self) ![]T {\n            // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n            var arr = try self.mem_allocator.alloc(T, self.size());\n           @memset(arr, @as(T, 0));\n            for (arr, 0..) |*num, i| {\n                num.* = self.get(i);\n            }\n            return arr;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 \u8a18\u61b6\u9ad4\u8207\u5feb\u53d6 *","text":"<p>\u5728\u672c\u7ae0\u7684\u524d\u5169\u7bc0\u4e2d\uff0c\u6211\u5011\u63a2\u8a0e\u4e86\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u9019\u5169\u7a2e\u57fa\u790e\u4e14\u91cd\u8981\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u5b83\u5011\u5206\u5225\u4ee3\u8868\u4e86\u201c\u9023\u7e8c\u5132\u5b58\u201d\u548c\u201c\u5206\u6563\u5132\u5b58\u201d\u5169\u7a2e\u7269\u7406\u7d50\u69cb\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u7269\u7406\u7d50\u69cb\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u6c7a\u5b9a\u4e86\u7a0b\u5f0f\u5c0d\u8a18\u61b6\u9ad4\u548c\u5feb\u53d6\u7684\u4f7f\u7528\u6548\u7387\uff0c\u9032\u800c\u5f71\u97ff\u6f14\u7b97\u6cd5\u7a0b\u5f0f\u7684\u6574\u9ad4\u6548\u80fd\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","title":"4.4.1 \u00a0 \u8a08\u7b97\u6a5f\u5132\u5b58\u88dd\u7f6e","text":"<p>\u8a08\u7b97\u6a5f\u4e2d\u5305\u62ec\u4e09\u7a2e\u985e\u578b\u7684\u5132\u5b58\u88dd\u7f6e\uff1a\u786c\u789f\uff08hard disk\uff09\u3001\u8a18\u61b6\u9ad4\uff08random-access memory, RAM\uff09\u3001\u5feb\u53d6\uff08cache memory\uff09\u3002\u8868 4-2 \u5c55\u793a\u4e86\u5b83\u5011\u5728\u8a08\u7b97\u6a5f\u7cfb\u7d71\u4e2d\u7684\u4e0d\u540c\u89d2\u8272\u548c\u6548\u80fd\u7279\u9ede\u3002</p> <p> \u8868 4-2 \u00a0 \u8a08\u7b97\u6a5f\u7684\u5132\u5b58\u88dd\u7f6e </p> \u786c\u789f \u8a18\u61b6\u9ad4 \u5feb\u53d6 \u7528\u9014 \u9577\u671f\u5132\u5b58\u8cc7\u6599\uff0c\u5305\u62ec\u4f5c\u696d\u7cfb\u7d71\u3001\u7a0b\u5f0f\u3001\u6a94\u6848\u7b49 \u81e8\u6642\u5132\u5b58\u7576\u524d\u57f7\u884c\u7684\u7a0b\u5f0f\u548c\u6b63\u5728\u8655\u7406\u7684\u8cc7\u6599 \u5132\u5b58\u7d93\u5e38\u8a2a\u554f\u7684\u8cc7\u6599\u548c\u6307\u4ee4\uff0c\u6e1b\u5c11 CPU \u8a2a\u554f\u8a18\u61b6\u9ad4\u7684\u6b21\u6578 \u6613\u5931\u6027 \u65b7\u96fb\u5f8c\u8cc7\u6599\u4e0d\u6703\u4e1f\u5931 \u65b7\u96fb\u5f8c\u8cc7\u6599\u6703\u4e1f\u5931 \u65b7\u96fb\u5f8c\u8cc7\u6599\u6703\u4e1f\u5931 \u5bb9\u91cf \u8f03\u5927\uff0cTB \u7d1a\u5225 \u8f03\u5c0f\uff0cGB \u7d1a\u5225 \u975e\u5e38\u5c0f\uff0cMB \u7d1a\u5225 \u901f\u5ea6 \u8f03\u6162\uff0c\u5e7e\u767e\u5230\u5e7e\u5343 MB/s \u8f03\u5feb\uff0c\u5e7e\u5341 GB/s \u975e\u5e38\u5feb\uff0c\u5e7e\u5341\u5230\u5e7e\u767e GB/s \u50f9\u683c \u8f03\u4fbf\u5b9c\uff0c\u5e7e\u6bdb\u5230\u5e7e\u5143 / GB \u8f03\u8cb4\uff0c\u5e7e\u5341\u5230\u5e7e\u767e\u5143 / GB \u975e\u5e38\u8cb4\uff0c\u96a8 CPU \u6253\u5305\u8a08\u50f9 <p>\u6211\u5011\u53ef\u4ee5\u5c07\u8a08\u7b97\u6a5f\u5132\u5b58\u7cfb\u7d71\u60f3\u8c61\u70ba\u5716 4-9 \u6240\u793a\u7684\u91d1\u5b57\u5854\u7d50\u69cb\u3002\u8d8a\u9760\u8fd1\u91d1\u5b57\u5854\u9802\u7aef\u7684\u5132\u5b58\u88dd\u7f6e\u7684\u901f\u5ea6\u8d8a\u5feb\u3001\u5bb9\u91cf\u8d8a\u5c0f\u3001\u6210\u672c\u8d8a\u9ad8\u3002\u9019\u7a2e\u591a\u5c64\u7d1a\u7684\u8a2d\u8a08\u4e26\u975e\u5076\u7136\uff0c\u800c\u662f\u8a08\u7b97\u6a5f\u79d1\u5b78\u5bb6\u548c\u5de5\u7a0b\u5e2b\u5011\u7d93\u904e\u6df1\u601d\u719f\u616e\u7684\u7d50\u679c\u3002</p> <ul> <li>\u786c\u789f\u96e3\u4ee5\u88ab\u8a18\u61b6\u9ad4\u53d6\u4ee3\u3002\u9996\u5148\uff0c\u8a18\u61b6\u9ad4\u4e2d\u7684\u8cc7\u6599\u5728\u65b7\u96fb\u5f8c\u6703\u4e1f\u5931\uff0c\u56e0\u6b64\u5b83\u4e0d\u9069\u5408\u9577\u671f\u5132\u5b58\u8cc7\u6599\uff1b\u5176\u6b21\uff0c\u8a18\u61b6\u9ad4\u7684\u6210\u672c\u662f\u786c\u789f\u7684\u5e7e\u5341\u500d\uff0c\u9019\u4f7f\u5f97\u5b83\u96e3\u4ee5\u5728\u6d88\u8cbb\u8005\u5e02\u5834\u666e\u53ca\u3002</li> <li>\u5feb\u53d6\u7684\u5927\u5bb9\u91cf\u548c\u9ad8\u901f\u5ea6\u96e3\u4ee5\u517c\u5f97\u3002\u96a8\u8457 L1\u3001L2\u3001L3 \u5feb\u53d6\u7684\u5bb9\u91cf\u9010\u6b65\u589e\u5927\uff0c\u5176\u7269\u7406\u5c3a\u5bf8\u6703\u8b8a\u5927\uff0c\u8207 CPU \u6838\u5fc3\u4e4b\u9593\u7684\u7269\u7406\u8ddd\u96e2\u6703\u8b8a\u9060\uff0c\u5f9e\u800c\u5c0e\u81f4\u8cc7\u6599\u50b3\u8f38\u6642\u9593\u589e\u52a0\uff0c\u5143\u7d20\u8a2a\u554f\u5ef6\u9072\u8b8a\u9ad8\u3002\u5728\u7576\u524d\u6280\u8853\u4e0b\uff0c\u591a\u5c64\u7d1a\u7684\u5feb\u53d6\u7d50\u69cb\u662f\u5bb9\u91cf\u3001\u901f\u5ea6\u548c\u6210\u672c\u4e4b\u9593\u7684\u6700\u4f73\u5e73\u8861\u9ede\u3002</li> </ul> <p></p> <p> \u5716 4-9 \u00a0 \u8a08\u7b97\u6a5f\u5132\u5b58\u7cfb\u7d71 </p> <p>Tip</p> <p>\u8a08\u7b97\u6a5f\u7684\u5132\u5b58\u5c64\u6b21\u7d50\u69cb\u9ad4\u73fe\u4e86\u901f\u5ea6\u3001\u5bb9\u91cf\u548c\u6210\u672c\u4e09\u8005\u4e4b\u9593\u7684\u7cbe\u5999\u5e73\u8861\u3002\u5be6\u969b\u4e0a\uff0c\u9019\u7a2e\u6b0a\u8861\u666e\u904d\u5b58\u5728\u65bc\u6240\u6709\u5de5\u696d\u9818\u57df\uff0c\u5b83\u8981\u6c42\u6211\u5011\u5728\u4e0d\u540c\u7684\u512a\u52e2\u548c\u9650\u5236\u4e4b\u9593\u627e\u5230\u6700\u4f73\u5e73\u8861\u9ede\u3002</p> <p>\u7e3d\u7684\u4f86\u8aaa\uff0c\u786c\u789f\u7528\u65bc\u9577\u671f\u5132\u5b58\u5927\u91cf\u8cc7\u6599\uff0c\u8a18\u61b6\u9ad4\u7528\u65bc\u81e8\u6642\u5132\u5b58\u7a0b\u5f0f\u57f7\u884c\u4e2d\u6b63\u5728\u8655\u7406\u7684\u8cc7\u6599\uff0c\u800c\u5feb\u53d6\u5247\u7528\u65bc\u5132\u5b58\u7d93\u5e38\u8a2a\u554f\u7684\u8cc7\u6599\u548c\u6307\u4ee4\uff0c\u4ee5\u63d0\u9ad8\u7a0b\u5f0f\u57f7\u884c\u6548\u7387\u3002\u4e09\u8005\u5171\u540c\u5354\u4f5c\uff0c\u78ba\u4fdd\u8a08\u7b97\u6a5f\u7cfb\u7d71\u9ad8\u6548\u57f7\u884c\u3002</p> <p>\u5982\u5716 4-10 \u6240\u793a\uff0c\u5728\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u8cc7\u6599\u6703\u5f9e\u786c\u789f\u4e2d\u88ab\u8b80\u53d6\u5230\u8a18\u61b6\u9ad4\u4e2d\uff0c\u4f9b CPU \u8a08\u7b97\u4f7f\u7528\u3002\u5feb\u53d6\u53ef\u4ee5\u770b\u4f5c CPU \u7684\u4e00\u90e8\u5206\uff0c\u5b83\u900f\u904e\u667a\u6167\u5730\u5f9e\u8a18\u61b6\u9ad4\u8f09\u5165\u8cc7\u6599\uff0c\u7d66 CPU \u63d0\u4f9b\u9ad8\u901f\u7684\u8cc7\u6599\u8b80\u53d6\uff0c\u5f9e\u800c\u986f\u8457\u63d0\u5347\u7a0b\u5f0f\u7684\u57f7\u884c\u6548\u7387\uff0c\u6e1b\u5c11\u5c0d\u8f03\u6162\u7684\u8a18\u61b6\u9ad4\u7684\u4f9d\u8cf4\u3002</p> <p></p> <p> \u5716 4-10 \u00a0 \u786c\u789f\u3001\u8a18\u61b6\u9ad4\u548c\u5feb\u53d6\u4e4b\u9593\u7684\u8cc7\u6599\u6d41\u901a </p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","title":"4.4.2 \u00a0 \u8cc7\u6599\u7d50\u69cb\u7684\u8a18\u61b6\u9ad4\u6548\u7387","text":"<p>\u5728\u8a18\u61b6\u9ad4\u7a7a\u9593\u5229\u7528\u65b9\u9762\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u5404\u81ea\u5177\u6709\u512a\u52e2\u548c\u4fb7\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u8a18\u61b6\u9ad4\u662f\u6709\u9650\u7684\uff0c\u4e14\u540c\u4e00\u584a\u8a18\u61b6\u9ad4\u4e0d\u80fd\u88ab\u591a\u500b\u7a0b\u5f0f\u5171\u4eab\uff0c\u56e0\u6b64\u6211\u5011\u5e0c\u671b\u8cc7\u6599\u7d50\u69cb\u80fd\u5920\u5118\u53ef\u80fd\u9ad8\u6548\u5730\u5229\u7528\u7a7a\u9593\u3002\u9663\u5217\u7684\u5143\u7d20\u7dca\u5bc6\u6392\u5217\uff0c\u4e0d\u9700\u8981\u984d\u5916\u7684\u7a7a\u9593\u4f86\u5132\u5b58\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\uff0c\u56e0\u6b64\u7a7a\u9593\u6548\u7387\u66f4\u9ad8\u3002\u7136\u800c\uff0c\u9663\u5217\u9700\u8981\u4e00\u6b21\u6027\u5206\u914d\u8db3\u5920\u7684\u9023\u7e8c\u8a18\u61b6\u9ad4\u7a7a\u9593\uff0c\u9019\u53ef\u80fd\u5c0e\u81f4\u8a18\u61b6\u9ad4\u6d6a\u8cbb\uff0c\u9663\u5217\u64f4\u5bb9\u4e5f\u9700\u8981\u984d\u5916\u7684\u6642\u9593\u548c\u7a7a\u9593\u6210\u672c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u93c8\u7d50\u4e32\u5217\u4ee5\u201c\u7bc0\u9ede\u201d\u70ba\u55ae\u4f4d\u9032\u884c\u52d5\u614b\u8a18\u61b6\u9ad4\u5206\u914d\u548c\u56de\u6536\uff0c\u63d0\u4f9b\u4e86\u66f4\u5927\u7684\u9748\u6d3b\u6027\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5728\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u96a8\u8457\u53cd\u8986\u7533\u8acb\u8207\u91cb\u653e\u8a18\u61b6\u9ad4\uff0c\u7a7a\u9592\u8a18\u61b6\u9ad4\u7684\u788e\u7247\u5316\u7a0b\u5ea6\u6703\u8d8a\u4f86\u8d8a\u9ad8\uff0c\u5f9e\u800c\u5c0e\u81f4\u8a18\u61b6\u9ad4\u7684\u5229\u7528\u6548\u7387\u964d\u4f4e\u3002\u9663\u5217\u7531\u65bc\u5176\u9023\u7e8c\u7684\u5132\u5b58\u65b9\u5f0f\uff0c\u76f8\u5c0d\u4e0d\u5bb9\u6613\u5c0e\u81f4\u8a18\u61b6\u9ad4\u788e\u7247\u5316\u3002\u76f8\u53cd\uff0c\u93c8\u7d50\u4e32\u5217\u7684\u5143\u7d20\u662f\u5206\u6563\u5132\u5b58\u7684\uff0c\u5728\u983b\u7e41\u7684\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u4e2d\uff0c\u66f4\u5bb9\u6613\u5c0e\u81f4\u8a18\u61b6\u9ad4\u788e\u7247\u5316\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","title":"4.4.3 \u00a0 \u8cc7\u6599\u7d50\u69cb\u7684\u5feb\u53d6\u6548\u7387","text":"<p>\u5feb\u53d6\u96d6\u7136\u5728\u7a7a\u9593\u5bb9\u91cf\u4e0a\u9060\u5c0f\u65bc\u8a18\u61b6\u9ad4\uff0c\u4f46\u5b83\u6bd4\u8a18\u61b6\u9ad4\u5feb\u5f97\u591a\uff0c\u5728\u7a0b\u5f0f\u57f7\u884c\u901f\u5ea6\u4e0a\u8d77\u8457\u81f3\u95dc\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7531\u65bc\u5feb\u53d6\u7684\u5bb9\u91cf\u6709\u9650\uff0c\u53ea\u80fd\u5132\u5b58\u4e00\u5c0f\u90e8\u5206\u983b\u7e41\u8a2a\u554f\u7684\u8cc7\u6599\uff0c\u56e0\u6b64\u7576 CPU \u5617\u8a66\u8a2a\u554f\u7684\u8cc7\u6599\u4e0d\u5728\u5feb\u53d6\u4e2d\u6642\uff0c\u5c31\u6703\u767c\u751f\u5feb\u53d6\u672a\u547d\u4e2d\uff08cache miss\uff09\uff0c\u6b64\u6642 CPU \u4e0d\u5f97\u4e0d\u5f9e\u901f\u5ea6\u8f03\u6162\u7684\u8a18\u61b6\u9ad4\u4e2d\u8f09\u5165\u6240\u9700\u8cc7\u6599\u3002</p> <p>\u986f\u7136\uff0c\u201c\u5feb\u53d6\u672a\u547d\u4e2d\u201d\u8d8a\u5c11\uff0cCPU \u8b80\u5beb\u8cc7\u6599\u7684\u6548\u7387\u5c31\u8d8a\u9ad8\uff0c\u7a0b\u5f0f\u6548\u80fd\u4e5f\u5c31\u8d8a\u597d\u3002\u6211\u5011\u5c07 CPU \u5f9e\u5feb\u53d6\u4e2d\u6210\u529f\u7372\u53d6\u8cc7\u6599\u7684\u6bd4\u4f8b\u7a31\u70ba\u5feb\u53d6\u547d\u4e2d\u7387\uff08cache hit rate\uff09\uff0c\u9019\u500b\u6307\u6a19\u901a\u5e38\u7528\u4f86\u8861\u91cf\u5feb\u53d6\u6548\u7387\u3002</p> <p>\u70ba\u4e86\u5118\u53ef\u80fd\u9054\u5230\u66f4\u9ad8\u7684\u6548\u7387\uff0c\u5feb\u53d6\u6703\u63a1\u53d6\u4ee5\u4e0b\u8cc7\u6599\u8f09\u5165\u6a5f\u5236\u3002</p> <ul> <li>\u5feb\u53d6\u884c\uff1a\u5feb\u53d6\u4e0d\u662f\u55ae\u500b\u4f4d\u5143\u7d44\u5730\u5132\u5b58\u8207\u8f09\u5165\u8cc7\u6599\uff0c\u800c\u662f\u4ee5\u5feb\u53d6\u884c\u70ba\u55ae\u4f4d\u3002\u76f8\u6bd4\u65bc\u55ae\u500b\u4f4d\u5143\u7d44\u7684\u50b3\u8f38\uff0c\u5feb\u53d6\u884c\u7684\u50b3\u8f38\u5f62\u5f0f\u66f4\u52a0\u9ad8\u6548\u3002</li> <li>\u9810\u53d6\u6a5f\u5236\uff1a\u8655\u7406\u5668\u6703\u5617\u8a66\u9810\u6e2c\u8cc7\u6599\u8a2a\u554f\u6a21\u5f0f\uff08\u4f8b\u5982\u9806\u5e8f\u8a2a\u554f\u3001\u56fa\u5b9a\u6b65\u9577\u8df3\u8e8d\u8a2a\u554f\u7b49\uff09\uff0c\u4e26\u6839\u64da\u7279\u5b9a\u6a21\u5f0f\u5c07\u8cc7\u6599\u8f09\u5165\u81f3\u5feb\u53d6\u4e4b\u4e2d\uff0c\u5f9e\u800c\u63d0\u5347\u547d\u4e2d\u7387\u3002</li> <li>\u7a7a\u9593\u5340\u57df\u6027\uff1a\u5982\u679c\u4e00\u500b\u6578\u64da\u88ab\u8a2a\u554f\uff0c\u90a3\u9ebc\u5b83\u9644\u8fd1\u7684\u8cc7\u6599\u53ef\u80fd\u8fd1\u671f\u4e5f\u6703\u88ab\u8a2a\u554f\u3002\u56e0\u6b64\uff0c\u5feb\u53d6\u5728\u8f09\u5165\u67d0\u4e00\u8cc7\u6599\u6642\uff0c\u4e5f\u6703\u8f09\u5165\u5176\u9644\u8fd1\u7684\u8cc7\u6599\uff0c\u4ee5\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> <li>\u6642\u9593\u5340\u57df\u6027\uff1a\u5982\u679c\u4e00\u500b\u6578\u64da\u88ab\u8a2a\u554f\uff0c\u90a3\u9ebc\u5b83\u5728\u4e0d\u4e45\u7684\u5c07\u4f86\u5f88\u53ef\u80fd\u518d\u6b21\u88ab\u8a2a\u554f\u3002\u5feb\u53d6\u5229\u7528\u9019\u4e00\u539f\u7406\uff0c\u900f\u904e\u4fdd\u7559\u6700\u8fd1\u8a2a\u554f\u904e\u7684\u8cc7\u6599\u4f86\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u5c0d\u5feb\u53d6\u7684\u5229\u7528\u6548\u7387\u662f\u4e0d\u540c\u7684\uff0c\u4e3b\u8981\u9ad4\u73fe\u5728\u4ee5\u4e0b\u5e7e\u500b\u65b9\u9762\u3002</p> <ul> <li>\u4f54\u7528\u7a7a\u9593\uff1a\u93c8\u7d50\u4e32\u5217\u5143\u7d20\u6bd4\u9663\u5217\u5143\u7d20\u4f54\u7528\u7a7a\u9593\u66f4\u591a\uff0c\u5c0e\u81f4\u5feb\u53d6\u4e2d\u5bb9\u7d0d\u7684\u6709\u6548\u8cc7\u6599\u91cf\u66f4\u5c11\u3002</li> <li>\u5feb\u53d6\u884c\uff1a\u93c8\u7d50\u4e32\u5217\u8cc7\u6599\u5206\u6563\u5728\u8a18\u61b6\u9ad4\u5404\u8655\uff0c\u800c\u5feb\u53d6\u662f\u201c\u6309\u884c\u8f09\u5165\u201d\u7684\uff0c\u56e0\u6b64\u8f09\u5165\u5230\u7121\u6548\u8cc7\u6599\u7684\u6bd4\u4f8b\u66f4\u9ad8\u3002</li> <li>\u9810\u53d6\u6a5f\u5236\uff1a\u9663\u5217\u6bd4\u93c8\u7d50\u4e32\u5217\u7684\u8cc7\u6599\u8a2a\u554f\u6a21\u5f0f\u66f4\u5177\u201c\u53ef\u9810\u6e2c\u6027\u201d\uff0c\u5373\u7cfb\u7d71\u66f4\u5bb9\u6613\u731c\u51fa\u5373\u5c07\u88ab\u8f09\u5165\u7684\u8cc7\u6599\u3002</li> <li>\u7a7a\u9593\u5340\u57df\u6027\uff1a\u9663\u5217\u88ab\u5132\u5b58\u5728\u96c6\u4e2d\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e2d\uff0c\u56e0\u6b64\u88ab\u8f09\u5165\u8cc7\u6599\u9644\u8fd1\u7684\u8cc7\u6599\u66f4\u6709\u53ef\u80fd\u5373\u5c07\u88ab\u8a2a\u554f\u3002</li> </ul> <p>\u7e3d\u9ad4\u800c\u8a00\uff0c\u9663\u5217\u5177\u6709\u66f4\u9ad8\u7684\u5feb\u53d6\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u5728\u64cd\u4f5c\u6548\u7387\u4e0a\u901a\u5e38\u512a\u65bc\u93c8\u7d50\u4e32\u5217\u3002\u9019\u4f7f\u5f97\u5728\u89e3\u6c7a\u6f14\u7b97\u6cd5\u554f\u984c\u6642\uff0c\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u8cc7\u6599\u7d50\u69cb\u5f80\u5f80\u66f4\u53d7\u6b61\u8fce\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9ad8\u5feb\u53d6\u6548\u7387\u4e26\u4e0d\u610f\u5473\u8457\u9663\u5217\u5728\u6240\u6709\u60c5\u6cc1\u4e0b\u90fd\u512a\u65bc\u93c8\u7d50\u4e32\u5217\u3002\u5be6\u969b\u61c9\u7528\u4e2d\u9078\u64c7\u54ea\u7a2e\u8cc7\u6599\u7d50\u69cb\uff0c\u61c9\u6839\u64da\u5177\u9ad4\u9700\u6c42\u4f86\u6c7a\u5b9a\u3002\u4f8b\u5982\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u90fd\u53ef\u4ee5\u5be6\u73fe\u201c\u5806\u758a\u201d\u8cc7\u6599\u7d50\u69cb\uff08\u4e0b\u4e00\u7ae0\u6703\u8a73\u7d30\u4ecb\u7d39\uff09\uff0c\u4f46\u5b83\u5011\u9069\u7528\u65bc\u4e0d\u540c\u5834\u666f\u3002</p> <ul> <li>\u5728\u505a\u6f14\u7b97\u6cd5\u984c\u6642\uff0c\u6211\u5011\u6703\u50be\u5411\u65bc\u9078\u64c7\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\uff0c\u56e0\u70ba\u5b83\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u64cd\u4f5c\u6548\u7387\u548c\u96a8\u6a5f\u8a2a\u554f\u7684\u80fd\u529b\uff0c\u4ee3\u50f9\u50c5\u662f\u9700\u8981\u9810\u5148\u70ba\u9663\u5217\u5206\u914d\u4e00\u5b9a\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> <li>\u5982\u679c\u8cc7\u6599\u91cf\u975e\u5e38\u5927\u3001\u52d5\u614b\u6027\u5f88\u9ad8\u3001\u5806\u758a\u7684\u9810\u671f\u5927\u5c0f\u96e3\u4ee5\u4f30\u8a08\uff0c\u90a3\u9ebc\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a\u66f4\u52a0\u5408\u9069\u3002\u93c8\u7d50\u4e32\u5217\u80fd\u5920\u5c07\u5927\u91cf\u8cc7\u6599\u5206\u6563\u5132\u5b58\u65bc\u8a18\u61b6\u9ad4\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u4e26\u4e14\u907f\u514d\u4e86\u9663\u5217\u64f4\u5bb9\u7522\u751f\u7684\u984d\u5916\u958b\u92b7\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/","title":"4.5 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_array_and_linkedlist/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u662f\u5169\u7a2e\u57fa\u672c\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u5206\u5225\u4ee3\u8868\u8cc7\u6599\u5728\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u4e2d\u7684\u5169\u7a2e\u5132\u5b58\u65b9\u5f0f\uff1a\u9023\u7e8c\u7a7a\u9593\u5132\u5b58\u548c\u5206\u6563\u7a7a\u9593\u5132\u5b58\u3002\u5169\u8005\u7684\u7279\u9ede\u5448\u73fe\u51fa\u4e92\u88dc\u7684\u7279\u6027\u3002</li> <li>\u9663\u5217\u652f\u6301\u96a8\u6a5f\u8a2a\u554f\u3001\u4f54\u7528\u8a18\u61b6\u9ad4\u8f03\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u522a\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u5f8c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u3002</li> <li>\u93c8\u7d50\u4e32\u5217\u900f\u904e\u66f4\u6539\u5f15\u7528\uff08\u6307\u6a19\uff09\u5be6\u73fe\u9ad8\u6548\u7684\u7bc0\u9ede\u63d2\u5165\u8207\u522a\u9664\uff0c\u4e14\u53ef\u4ee5\u9748\u6d3b\u8abf\u6574\u9577\u5ea6\uff1b\u4f46\u7bc0\u9ede\u8a2a\u554f\u6548\u7387\u4f4e\u3001\u4f54\u7528\u8a18\u61b6\u9ad4\u8f03\u591a\u3002\u5e38\u898b\u7684\u93c8\u7d50\u4e32\u5217\u578b\u5225\u5305\u62ec\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u3001\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u3001\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u3002</li> <li>\u4e32\u5217\u662f\u4e00\u7a2e\u652f\u6301\u589e\u522a\u67e5\u6539\u7684\u5143\u7d20\u6709\u5e8f\u96c6\u5408\uff0c\u901a\u5e38\u57fa\u65bc\u52d5\u614b\u9663\u5217\u5be6\u73fe\u3002\u5b83\u4fdd\u7559\u4e86\u9663\u5217\u7684\u512a\u52e2\uff0c\u540c\u6642\u53ef\u4ee5\u9748\u6d3b\u8abf\u6574\u9577\u5ea6\u3002</li> <li>\u4e32\u5217\u7684\u51fa\u73fe\u5927\u5e45\u63d0\u9ad8\u4e86\u9663\u5217\u7684\u5be6\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5c0e\u81f4\u90e8\u5206\u8a18\u61b6\u9ad4\u7a7a\u9593\u6d6a\u8cbb\u3002</li> <li>\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u8cc7\u6599\u4e3b\u8981\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u4e2d\u3002\u9663\u5217\u53ef\u63d0\u4f9b\u66f4\u9ad8\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u6548\u7387\uff0c\u800c\u93c8\u7d50\u4e32\u5217\u5247\u5728\u8a18\u61b6\u9ad4\u4f7f\u7528\u4e0a\u66f4\u52a0\u9748\u6d3b\u3002</li> <li>\u5feb\u53d6\u900f\u904e\u5feb\u53d6\u884c\u3001\u9810\u53d6\u6a5f\u5236\u4ee5\u53ca\u7a7a\u9593\u5340\u57df\u6027\u548c\u6642\u9593\u5340\u57df\u6027\u7b49\u8cc7\u6599\u8f09\u5165\u6a5f\u5236\uff0c\u70ba CPU \u63d0\u4f9b\u5feb\u901f\u8cc7\u6599\u8a2a\u554f\uff0c\u986f\u8457\u63d0\u5347\u7a0b\u5f0f\u7684\u57f7\u884c\u6548\u7387\u3002</li> <li>\u7531\u65bc\u9663\u5217\u5177\u6709\u66f4\u9ad8\u7684\u5feb\u53d6\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u901a\u5e38\u6bd4\u93c8\u7d50\u4e32\u5217\u66f4\u9ad8\u6548\u3002\u5728\u9078\u64c7\u8cc7\u6599\u7d50\u69cb\u6642\uff0c\u61c9\u6839\u64da\u5177\u9ad4\u9700\u6c42\u548c\u5834\u666f\u505a\u51fa\u6070\u7576\u9078\u64c7\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u9663\u5217\u5132\u5b58\u5728\u5806\u758a\u4e0a\u548c\u5132\u5b58\u5728\u5806\u7a4d\u4e0a\uff0c\u5c0d\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u662f\u5426\u6709\u5f71\u97ff\uff1f</p> <p>\u5132\u5b58\u5728\u5806\u758a\u4e0a\u548c\u5806\u7a4d\u4e0a\u7684\u9663\u5217\u90fd\u88ab\u5132\u5b58\u5728\u9023\u7e8c\u8a18\u61b6\u9ad4\u7a7a\u9593\u5167\uff0c\u8cc7\u6599\u64cd\u4f5c\u6548\u7387\u57fa\u672c\u4e00\u81f4\u3002\u7136\u800c\uff0c\u5806\u758a\u548c\u5806\u7a4d\u5177\u6709\u5404\u81ea\u7684\u7279\u9ede\uff0c\u5f9e\u800c\u5c0e\u81f4\u4ee5\u4e0b\u4e0d\u540c\u9ede\u3002</p> <ol> <li>\u5206\u914d\u548c\u91cb\u653e\u6548\u7387\uff1a\u5806\u758a\u662f\u4e00\u584a\u8f03\u5c0f\u7684\u8a18\u61b6\u9ad4\uff0c\u5206\u914d\u7531\u7de8\u8b6f\u5668\u81ea\u52d5\u5b8c\u6210\uff1b\u800c\u5806\u7a4d\u8a18\u61b6\u9ad4\u76f8\u5c0d\u66f4\u5927\uff0c\u53ef\u4ee5\u5728\u7a0b\u5f0f\u78bc\u4e2d\u52d5\u614b\u5206\u914d\uff0c\u66f4\u5bb9\u6613\u788e\u7247\u5316\u3002\u56e0\u6b64\uff0c\u5806\u7a4d\u4e0a\u7684\u5206\u914d\u548c\u91cb\u653e\u64cd\u4f5c\u901a\u5e38\u6bd4\u5806\u758a\u4e0a\u7684\u6162\u3002</li> <li>\u5927\u5c0f\u9650\u5236\uff1a\u5806\u758a\u8a18\u61b6\u9ad4\u76f8\u5c0d\u8f03\u5c0f\uff0c\u5806\u7a4d\u7684\u5927\u5c0f\u4e00\u822c\u53d7\u9650\u65bc\u53ef\u7528\u8a18\u61b6\u9ad4\u3002\u56e0\u6b64\u5806\u7a4d\u66f4\u52a0\u9069\u5408\u5132\u5b58\u5927\u578b\u9663\u5217\u3002</li> <li>\u9748\u6d3b\u6027\uff1a\u5806\u758a\u4e0a\u7684\u9663\u5217\u7684\u5927\u5c0f\u9700\u8981\u5728\u7de8\u8b6f\u6642\u78ba\u5b9a\uff0c\u800c\u5806\u7a4d\u4e0a\u7684\u9663\u5217\u7684\u5927\u5c0f\u53ef\u4ee5\u5728\u57f7\u884c\u6642\u52d5\u614b\u78ba\u5b9a\u3002</li> </ol> <p>Q\uff1a\u70ba\u4ec0\u9ebc\u9663\u5217\u8981\u6c42\u76f8\u540c\u578b\u5225\u7684\u5143\u7d20\uff0c\u800c\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u537b\u6c92\u6709\u5f37\u8abf\u76f8\u540c\u578b\u5225\u5462\uff1f</p> <p>\u93c8\u7d50\u4e32\u5217\u7531\u7bc0\u9ede\u7d44\u6210\uff0c\u7bc0\u9ede\u4e4b\u9593\u900f\u904e\u5f15\u7528\uff08\u6307\u6a19\uff09\u9023\u7dda\uff0c\u5404\u500b\u7bc0\u9ede\u53ef\u4ee5\u5132\u5b58\u4e0d\u540c\u578b\u5225\u7684\u8cc7\u6599\uff0c\u4f8b\u5982 <code>int</code>\u3001<code>double</code>\u3001<code>string</code>\u3001<code>object</code> \u7b49\u3002</p> <p>\u76f8\u5c0d\u5730\uff0c\u9663\u5217\u5143\u7d20\u5247\u5fc5\u9808\u662f\u76f8\u540c\u578b\u5225\u7684\uff0c\u9019\u6a23\u624d\u80fd\u900f\u904e\u8a08\u7b97\u504f\u79fb\u91cf\u4f86\u7372\u53d6\u5c0d\u61c9\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u9663\u5217\u540c\u6642\u5305\u542b <code>int</code> \u548c <code>long</code> \u5169\u7a2e\u578b\u5225\uff0c\u55ae\u500b\u5143\u7d20\u5206\u5225\u4f54\u7528 4 \u4f4d\u5143\u7d44\u548c 8 \u4f4d\u5143\u7d44 \uff0c\u6b64\u6642\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8a08\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u70ba\u9663\u5217\u4e2d\u5305\u542b\u4e86\u5169\u7a2e\u201c\u5143\u7d20\u9577\u5ea6\u201d\u3002</p> <pre><code># \u5143\u7d20\u8a18\u61b6\u9ad4\u4f4d\u5740 = \u9663\u5217\u8a18\u61b6\u9ad4\u4f4d\u5740\uff08\u9996\u5143\u7d20\u8a18\u61b6\u9ad4\u4f4d\u5740\uff09 + \u5143\u7d20\u9577\u5ea6 * \u5143\u7d20\u7d22\u5f15\n</code></pre> <p>Q\uff1a\u522a\u9664\u7bc0\u9ede <code>P</code> \u5f8c\uff0c\u662f\u5426\u9700\u8981\u628a <code>P.next</code> \u8a2d\u70ba <code>None</code> \u5462\uff1f</p> <p>\u4e0d\u4fee\u6539 <code>P.next</code> \u4e5f\u53ef\u4ee5\u3002\u5f9e\u8a72\u93c8\u7d50\u4e32\u5217\u7684\u89d2\u5ea6\u770b\uff0c\u5f9e\u982d\u7bc0\u9ede\u8d70\u8a2a\u5230\u5c3e\u7bc0\u9ede\u5df2\u7d93\u4e0d\u6703\u9047\u5230 <code>P</code> \u4e86\u3002\u9019\u610f\u5473\u8457\u7bc0\u9ede <code>P</code> \u5df2\u7d93\u5f9e\u93c8\u7d50\u4e32\u5217\u4e2d\u522a\u9664\u4e86\uff0c\u6b64\u6642\u7bc0\u9ede <code>P</code> \u6307\u5411\u54ea\u88e1\u90fd\u4e0d\u6703\u5c0d\u8a72\u93c8\u7d50\u4e32\u5217\u7522\u751f\u5f71\u97ff\u3002</p> <p>\u5f9e\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\uff08\u505a\u984c\uff09\u7684\u89d2\u5ea6\u770b\uff0c\u4e0d\u65b7\u958b\u6c92\u6709\u95dc\u4fc2\uff0c\u53ea\u8981\u4fdd\u8b49\u7a0b\u5f0f\u7684\u908f\u8f2f\u662f\u6b63\u78ba\u7684\u5c31\u884c\u3002\u5f9e\u6a19\u6e96\u5eab\u7684\u89d2\u5ea6\u770b\uff0c\u65b7\u958b\u66f4\u52a0\u5b89\u5168\u3001\u908f\u8f2f\u66f4\u52a0\u6e05\u6670\u3002\u5982\u679c\u4e0d\u65b7\u958b\uff0c\u5047\u8a2d\u88ab\u522a\u9664\u7bc0\u9ede\u672a\u88ab\u6b63\u5e38\u56de\u6536\uff0c\u90a3\u9ebc\u5b83\u6703\u5f71\u97ff\u5f8c\u7e7c\u7bc0\u9ede\u7684\u8a18\u61b6\u9ad4\u56de\u6536\u3002</p> <p>Q\uff1a\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u63d2\u5165\u548c\u522a\u9664\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u522a\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u7684\u6642\u9593\u67e5\u8a62\u5143\u7d20\uff0c\u90a3\u70ba\u4ec0\u9ebc\u6642\u9593\u8907\u96dc\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f</p> <p>\u5982\u679c\u662f\u5148\u67e5\u8a62\u5143\u7d20\u3001\u518d\u522a\u9664\u5143\u7d20\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u78ba\u5be6\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u93c8\u7d50\u4e32\u5217\u7684 \\(O(1)\\) \u589e\u522a\u7684\u512a\u52e2\u53ef\u4ee5\u5728\u5176\u4ed6\u61c9\u7528\u4e0a\u5f97\u5230\u9ad4\u73fe\u3002\u4f8b\u5982\uff0c\u96d9\u5411\u4f47\u5217\u9069\u5408\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\uff0c\u6211\u5011\u7dad\u8b77\u4e00\u500b\u6307\u6a19\u8b8a\u6578\u59cb\u7d42\u6307\u5411\u982d\u7bc0\u9ede\u3001\u5c3e\u7bc0\u9ede\uff0c\u6bcf\u6b21\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002</p> <p>Q\uff1a\u5716\u201c\u93c8\u7d50\u4e32\u5217\u5b9a\u7fa9\u8207\u5132\u5b58\u65b9\u5f0f\u201d\u4e2d\uff0c\u6dfa\u85cd\u8272\u7684\u5132\u5b58\u7bc0\u9ede\u6307\u6a19\u662f\u4f54\u7528\u4e00\u584a\u8a18\u61b6\u9ad4\u4f4d\u5740\u55ce\uff1f\u9084\u662f\u548c\u7bc0\u9ede\u503c\u5404\u4f54\u4e00\u534a\u5462\uff1f</p> <p>\u8a72\u793a\u610f\u5716\u53ea\u662f\u5b9a\u6027\u8868\u793a\uff0c\u5b9a\u91cf\u8868\u793a\u9700\u8981\u6839\u64da\u5177\u9ad4\u60c5\u6cc1\u9032\u884c\u5206\u6790\u3002</p> <ul> <li>\u4e0d\u540c\u578b\u5225\u7684\u7bc0\u9ede\u503c\u4f54\u7528\u7684\u7a7a\u9593\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 <code>int</code>\u3001<code>long</code>\u3001<code>double</code> \u548c\u4f8b\u9805\u7269\u4ef6\u7b49\u3002</li> <li>\u6307\u6a19\u8b8a\u6578\u4f54\u7528\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u5927\u5c0f\u6839\u64da\u6240\u4f7f\u7528\u7684\u4f5c\u696d\u7cfb\u7d71\u53ca\u7de8\u8b6f\u74b0\u5883\u800c\u5b9a\uff0c\u5927\u591a\u70ba 8 \u4f4d\u5143\u7d44\u6216 4 \u4f4d\u5143\u7d44\u3002</li> </ul> <p>Q\uff1a\u5728\u4e32\u5217\u672b\u5c3e\u65b0\u589e\u5143\u7d20\u662f\u5426\u6642\u6642\u523b\u523b\u90fd\u70ba \\(O(1)\\) \uff1f</p> <p>\u5982\u679c\u65b0\u589e\u5143\u7d20\u6642\u8d85\u51fa\u4e32\u5217\u9577\u5ea6\uff0c\u5247\u9700\u8981\u5148\u64f4\u5bb9\u4e32\u5217\u518d\u65b0\u589e\u3002\u7cfb\u7d71\u6703\u7533\u8acb\u4e00\u584a\u65b0\u7684\u8a18\u61b6\u9ad4\uff0c\u4e26\u5c07\u539f\u4e32\u5217\u7684\u6240\u6709\u5143\u7d20\u642c\u904b\u904e\u53bb\uff0c\u9019\u6642\u5019\u6642\u9593\u8907\u96dc\u5ea6\u5c31\u6703\u662f \\(O(n)\\) \u3002</p> <p>Q\uff1a\u201c\u4e32\u5217\u7684\u51fa\u73fe\u6975\u5927\u5730\u63d0\u9ad8\u4e86\u9663\u5217\u7684\u5be6\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5c0e\u81f4\u90e8\u5206\u8a18\u61b6\u9ad4\u7a7a\u9593\u6d6a\u8cbb\u201d\uff0c\u9019\u88e1\u7684\u7a7a\u9593\u6d6a\u8cbb\u662f\u6307\u984d\u5916\u589e\u52a0\u7684\u8b8a\u6578\u5982\u5bb9\u91cf\u3001\u9577\u5ea6\u3001\u64f4\u5bb9\u500d\u6578\u6240\u4f54\u7684\u8a18\u61b6\u9ad4\u55ce\uff1f</p> <p>\u9019\u88e1\u7684\u7a7a\u9593\u6d6a\u8cbb\u4e3b\u8981\u6709\u5169\u65b9\u9762\u542b\u7fa9\uff1a\u4e00\u65b9\u9762\uff0c\u4e32\u5217\u90fd\u6703\u8a2d\u5b9a\u4e00\u500b\u521d\u59cb\u9577\u5ea6\uff0c\u6211\u5011\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u9019\u9ebc\u591a\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u70ba\u4e86\u9632\u6b62\u983b\u7e41\u64f4\u5bb9\uff0c\u64f4\u5bb9\u4e00\u822c\u6703\u4e58\u4ee5\u4e00\u500b\u4fc2\u6578\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u9019\u6a23\u4e00\u4f86\uff0c\u4e5f\u6703\u51fa\u73fe\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u5011\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6eff\u5b83\u5011\u3002</p> <p>Q\uff1a\u5728 Python \u4e2d\u521d\u59cb\u5316 <code>n = [1, 2, 3]</code> \u5f8c\uff0c\u9019 3 \u500b\u5143\u7d20\u7684\u4f4d\u5740\u662f\u76f8\u9023\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316 <code>m = [2, 1, 3]</code> \u6703\u767c\u73fe\u5b83\u5011\u6bcf\u500b\u5143\u7d20\u7684 id \u4e26\u4e0d\u662f\u9023\u7e8c\u7684\uff0c\u800c\u662f\u5206\u5225\u8ddf <code>n</code> \u4e2d\u7684\u76f8\u540c\u3002\u9019\u4e9b\u5143\u7d20\u7684\u4f4d\u5740\u4e0d\u9023\u7e8c\uff0c\u90a3\u9ebc <code>m</code> \u9084\u662f\u9663\u5217\u55ce\uff1f</p> <p>\u5047\u5982\u628a\u4e32\u5217\u5143\u7d20\u63db\u6210\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede <code>n = [n1, n2, n3, n4, n5]</code> \uff0c\u901a\u5e38\u60c5\u6cc1\u4e0b\u9019 5 \u500b\u7bc0\u9ede\u7269\u4ef6\u4e5f\u5206\u6563\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u5404\u8655\u3002\u7136\u800c\uff0c\u7d66\u5b9a\u4e00\u500b\u4e32\u5217\u7d22\u5f15\uff0c\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u5167\u7372\u53d6\u7bc0\u9ede\u8a18\u61b6\u9ad4\u4f4d\u5740\uff0c\u5f9e\u800c\u8a2a\u554f\u5230\u5c0d\u61c9\u7684\u7bc0\u9ede\u3002\u9019\u662f\u56e0\u70ba\u9663\u5217\u4e2d\u5132\u5b58\u7684\u662f\u7bc0\u9ede\u7684\u5f15\u7528\uff0c\u800c\u975e\u7bc0\u9ede\u672c\u8eab\u3002</p> <p>\u8207\u8a31\u591a\u8a9e\u8a00\u4e0d\u540c\uff0cPython \u4e2d\u7684\u6578\u5b57\u4e5f\u88ab\u5305\u88dd\u70ba\u7269\u4ef6\uff0c\u4e32\u5217\u4e2d\u5132\u5b58\u7684\u4e0d\u662f\u6578\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5c0d\u6578\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u5011\u6703\u767c\u73fe\u5169\u500b\u9663\u5217\u4e2d\u7684\u76f8\u540c\u6578\u5b57\u64c1\u6709\u540c\u4e00\u500b id \uff0c\u4e26\u4e14\u9019\u4e9b\u6578\u5b57\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u7121\u9808\u9023\u7e8c\u3002</p> <p>Q\uff1aC++ STL \u88e1\u9762\u7684 <code>std::list</code> \u5df2\u7d93\u5be6\u73fe\u4e86\u96d9\u5411\u93c8\u7d50\u4e32\u5217\uff0c\u4f46\u597d\u50cf\u4e00\u4e9b\u6f14\u7b97\u6cd5\u66f8\u4e0a\u4e0d\u600e\u9ebc\u76f4\u63a5\u4f7f\u7528\u5b83\uff0c\u662f\u4e0d\u662f\u56e0\u70ba\u6709\u4ec0\u9ebc\u4fb7\u9650\u6027\u5462\uff1f</p> <p>\u4e00\u65b9\u9762\uff0c\u6211\u5011\u5f80\u5f80\u66f4\u9752\u775e\u4f7f\u7528\u9663\u5217\u5be6\u73fe\u6f14\u7b97\u6cd5\uff0c\u800c\u53ea\u5728\u5fc5\u8981\u6642\u624d\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\uff0c\u4e3b\u8981\u6709\u5169\u500b\u539f\u56e0\u3002</p> <ul> <li>\u7a7a\u9593\u958b\u92b7\uff1a\u7531\u65bc\u6bcf\u500b\u5143\u7d20\u9700\u8981\u5169\u500b\u984d\u5916\u7684\u6307\u6a19\uff08\u4e00\u500b\u7528\u65bc\u524d\u4e00\u500b\u5143\u7d20\uff0c\u4e00\u500b\u7528\u65bc\u5f8c\u4e00\u500b\u5143\u7d20\uff09\uff0c\u6240\u4ee5 <code>std::list</code> \u901a\u5e38\u6bd4 <code>std::vector</code> \u66f4\u4f54\u7528\u7a7a\u9593\u3002</li> <li>\u5feb\u53d6\u4e0d\u53cb\u597d\uff1a\u7531\u65bc\u8cc7\u6599\u4e0d\u662f\u9023\u7e8c\u5b58\u653e\u7684\uff0c\u56e0\u6b64 <code>std::list</code> \u5c0d\u5feb\u53d6\u7684\u5229\u7528\u7387\u8f03\u4f4e\u3002\u4e00\u822c\u60c5\u6cc1\u4e0b\uff0c<code>std::vector</code> \u7684\u6548\u80fd\u6703\u66f4\u597d\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5fc5\u8981\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\u7684\u60c5\u6cc1\u4e3b\u8981\u662f\u4e8c\u5143\u6a39\u548c\u5716\u3002\u5806\u758a\u548c\u4f47\u5217\u5f80\u5f80\u6703\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u63d0\u4f9b\u7684 <code>stack</code> \u548c <code>queue</code> \uff0c\u800c\u975e\u93c8\u7d50\u4e32\u5217\u3002</p> <p>Q\uff1a\u64cd\u4f5c <code>res = [[0]] * n</code> \u751f\u6210\u4e86\u4e00\u500b\u4e8c\u7dad\u4e32\u5217\uff0c\u5176\u4e2d\u6bcf\u4e00\u500b <code>[0]</code> \u90fd\u662f\u7368\u7acb\u7684\u55ce\uff1f</p> <p>\u4e0d\u662f\u7368\u7acb\u7684\u3002\u6b64\u4e8c\u7dad\u4e32\u5217\u4e2d\uff0c\u6240\u6709\u7684 <code>[0]</code> \u5be6\u969b\u4e0a\u662f\u540c\u4e00\u500b\u7269\u4ef6\u7684\u5f15\u7528\u3002\u5982\u679c\u6211\u5011\u4fee\u6539\u5176\u4e2d\u4e00\u500b\u5143\u7d20\uff0c\u6703\u767c\u73fe\u6240\u6709\u7684\u5c0d\u61c9\u5143\u7d20\u90fd\u6703\u96a8\u4e4b\u6539\u8b8a\u3002</p> <p>\u5982\u679c\u5e0c\u671b\u4e8c\u7dad\u4e32\u5217\u4e2d\u7684\u6bcf\u500b <code>[0]</code> \u90fd\u662f\u7368\u7acb\u7684\uff0c\u53ef\u4ee5\u4f7f\u7528 <code>res = [[0] for _ in range(n)]</code> \u4f86\u5be6\u73fe\u3002\u9019\u7a2e\u65b9\u5f0f\u7684\u539f\u7406\u662f\u521d\u59cb\u5316\u4e86 \\(n\\) \u500b\u7368\u7acb\u7684 <code>[0]</code> \u4e32\u5217\u7269\u4ef6\u3002</p> <p>Q\uff1a\u64cd\u4f5c <code>res = [0] * n</code> \u751f\u6210\u4e86\u4e00\u500b\u4e32\u5217\uff0c\u5176\u4e2d\u6bcf\u4e00\u500b\u6574\u6578 0 \u90fd\u662f\u7368\u7acb\u7684\u55ce\uff1f</p> <p>\u5728\u8a72\u4e32\u5217\u4e2d\uff0c\u6240\u6709\u6574\u6578 0 \u90fd\u662f\u540c\u4e00\u500b\u7269\u4ef6\u7684\u5f15\u7528\u3002\u9019\u662f\u56e0\u70ba Python \u5c0d\u5c0f\u6574\u6578\uff08\u901a\u5e38\u662f -5 \u5230 256\uff09\u63a1\u7528\u4e86\u5feb\u53d6\u6c60\u6a5f\u5236\uff0c\u4ee5\u4fbf\u6700\u5927\u5316\u7269\u4ef6\u8907\u7528\uff0c\u5f9e\u800c\u63d0\u5347\u6548\u80fd\u3002</p> <p>\u96d6\u7136\u5b83\u5011\u6307\u5411\u540c\u4e00\u500b\u7269\u4ef6\uff0c\u4f46\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u7368\u7acb\u4fee\u6539\u4e32\u5217\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\uff0c\u9019\u662f\u56e0\u70ba Python \u7684\u6574\u6578\u662f\u201c\u4e0d\u53ef\u8b8a\u7269\u4ef6\u201d\u3002\u7576\u6211\u5011\u4fee\u6539\u67d0\u500b\u5143\u7d20\u6642\uff0c\u5be6\u969b\u4e0a\u662f\u5207\u63db\u70ba\u53e6\u4e00\u500b\u7269\u4ef6\u7684\u5f15\u7528\uff0c\u800c\u4e0d\u662f\u6539\u8b8a\u539f\u6709\u7269\u4ef6\u672c\u8eab\u3002</p> <p>\u7136\u800c\uff0c\u7576\u4e32\u5217\u5143\u7d20\u662f\u201c\u53ef\u8b8a\u7269\u4ef6\u201d\u6642\uff08\u4f8b\u5982\u4e32\u5217\u3001\u5b57\u5178\u6216\u985e\u5225\u4f8b\u9805\u7b49\uff09\uff0c\u4fee\u6539\u67d0\u500b\u5143\u7d20\u6703\u76f4\u63a5\u6539\u8b8a\u8a72\u7269\u4ef6\u672c\u8eab\uff0c\u6240\u6709\u5f15\u7528\u8a72\u7269\u4ef6\u7684\u5143\u7d20\u90fd\u6703\u7522\u751f\u76f8\u540c\u8b8a\u5316\u3002</p>"},{"location":"chapter_backtracking/","title":"\u7b2c 13 \u7ae0 \u00a0 \u56de\u6eaf","text":"<p>Abstract</p> <p>\u6211\u5011\u5982\u540c\u8ff7\u5bae\u4e2d\u7684\u63a2\u7d22\u8005\uff0c\u5728\u524d\u9032\u7684\u9053\u8def\u4e0a\u53ef\u80fd\u6703\u9047\u5230\u56f0\u96e3\u3002</p> <p>\u56de\u6eaf\u7684\u529b\u91cf\u8b93\u6211\u5011\u80fd\u5920\u91cd\u65b0\u958b\u59cb\uff0c\u4e0d\u65b7\u5617\u8a66\uff0c\u6700\u7d42\u627e\u5230\u901a\u5f80\u5149\u660e\u7684\u51fa\u53e3\u3002</p>"},{"location":"chapter_backtracking/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>13.1 \u00a0 \u56de\u6eaf\u6f14\u7b97\u6cd5</li> <li>13.2 \u00a0 \u5168\u6392\u5217\u554f\u984c</li> <li>13.3 \u00a0 \u5b50\u96c6\u548c\u554f\u984c</li> <li>13.4 \u00a0 N \u7687\u540e\u554f\u984c</li> <li>13.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/","title":"13.1 \u00a0 \u56de\u6eaf\u6f14\u7b97\u6cd5","text":"<p>\u56de\u6eaf\u6f14\u7b97\u6cd5\uff08backtracking algorithm\uff09\u662f\u4e00\u7a2e\u900f\u904e\u7aae\u8209\u4f86\u89e3\u6c7a\u554f\u984c\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5f9e\u4e00\u500b\u521d\u59cb\u72c0\u614b\u51fa\u767c\uff0c\u66b4\u529b\u641c\u5c0b\u6240\u6709\u53ef\u80fd\u7684\u89e3\u6c7a\u65b9\u6848\uff0c\u7576\u9047\u5230\u6b63\u78ba\u7684\u89e3\u5247\u5c07\u5176\u8a18\u9304\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5617\u8a66\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9078\u64c7\u90fd\u7121\u6cd5\u627e\u5230\u89e3\u70ba\u6b62\u3002</p> <p>\u56de\u6eaf\u6f14\u7b97\u6cd5\u901a\u5e38\u63a1\u7528\u201c\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u201d\u4f86\u8d70\u8a2a\u89e3\u7a7a\u9593\u3002\u5728\u201c\u4e8c\u5143\u6a39\u201d\u7ae0\u7bc0\u4e2d\uff0c\u6211\u5011\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u5f8c\u5e8f\u8d70\u8a2a\u90fd\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u3002\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5229\u7528\u524d\u5e8f\u8d70\u8a2a\u69cb\u9020\u4e00\u500b\u56de\u6eaf\u554f\u984c\uff0c\u9010\u6b65\u77ad\u89e3\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p>\u4f8b\u984c\u4e00</p> <p>\u7d66\u5b9a\u4e00\u68f5\u4e8c\u5143\u6a39\uff0c\u641c\u5c0b\u4e26\u8a18\u9304\u6240\u6709\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\uff0c\u8acb\u8fd4\u56de\u7bc0\u9ede\u4e32\u5217\u3002</p> <p>\u5c0d\u65bc\u6b64\u984c\uff0c\u6211\u5011\u524d\u5e8f\u8d70\u8a2a\u9019\u68f5\u6a39\uff0c\u4e26\u5224\u65b7\u7576\u524d\u7bc0\u9ede\u7684\u503c\u662f\u5426\u70ba \\(7\\) \uff0c\u82e5\u662f\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u7684\u503c\u52a0\u5165\u7d50\u679c\u4e32\u5217 <code>res</code> \u4e4b\u4e2d\u3002\u76f8\u95dc\u904e\u7a0b\u5be6\u73fe\u5982\u5716 13-1 \u548c\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # \u8a18\u9304\u89e3\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // \u8a18\u9304\u89e3\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // \u8a18\u9304\u89e3\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // \u8a18\u9304\u89e3\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // \u8a18\u9304\u89e3\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.as_ref());\n        pre_order(res, node.borrow().right.as_ref());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.kt<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    if (root._val == 7) {\n        // \u8a18\u9304\u89e3\n        res!!.add(root)\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n}\n</code></pre> preorder_traversal_i_compact.rb<pre><code>### \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 ###\ndef pre_order(root)\n  return unless root\n\n  # \u8a18\u9304\u89e3\n  $res &lt;&lt; root if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\nend\n</code></pre> preorder_traversal_i_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 13-1 \u00a0 \u5728\u524d\u5e8f\u8d70\u8a2a\u4e2d\u641c\u7d22\u7bc0\u9ede </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1311","title":"13.1.1 \u00a0 \u5617\u8a66\u8207\u56de\u9000","text":"<p>\u4e4b\u6240\u4ee5\u7a31\u4e4b\u70ba\u56de\u6eaf\u6f14\u7b97\u6cd5\uff0c\u662f\u56e0\u70ba\u8a72\u6f14\u7b97\u6cd5\u5728\u641c\u5c0b\u89e3\u7a7a\u9593\u6642\u6703\u63a1\u7528\u201c\u5617\u8a66\u201d\u8207\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u7576\u6f14\u7b97\u6cd5\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\u9047\u5230\u67d0\u500b\u72c0\u614b\u7121\u6cd5\u7e7c\u7e8c\u524d\u9032\u6216\u7121\u6cd5\u5f97\u5230\u6eff\u8db3\u689d\u4ef6\u7684\u89e3\u6642\uff0c\u5b83\u6703\u64a4\u92b7\u4e0a\u4e00\u6b65\u7684\u9078\u64c7\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72c0\u614b\uff0c\u4e26\u5617\u8a66\u5176\u4ed6\u53ef\u80fd\u7684\u9078\u64c7\u3002</p> <p>\u5c0d\u65bc\u4f8b\u984c\u4e00\uff0c\u8a2a\u554f\u6bcf\u500b\u7bc0\u9ede\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5617\u8a66\u201d\uff0c\u800c\u8d8a\u904e\u8449\u7bc0\u9ede\u6216\u8fd4\u56de\u7236\u7bc0\u9ede\u7684 <code>return</code> \u5247\u8868\u793a\u201c\u56de\u9000\u201d\u3002</p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u56de\u9000\u4e26\u4e0d\u50c5\u50c5\u5305\u62ec\u51fd\u5f0f\u8fd4\u56de\u3002\u70ba\u89e3\u91cb\u9019\u4e00\u9ede\uff0c\u6211\u5011\u5c0d\u4f8b\u984c\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002</p> <p>\u4f8b\u984c\u4e8c</p> <p>\u5728\u4e8c\u5143\u6a39\u4e2d\u641c\u7d22\u6240\u6709\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\uff0c\u8acb\u8fd4\u56de\u6839\u7bc0\u9ede\u5230\u9019\u4e9b\u7bc0\u9ede\u7684\u8def\u5f91\u3002</p> <p>\u5728\u4f8b\u984c\u4e00\u7a0b\u5f0f\u78bc\u7684\u57fa\u790e\u4e0a\uff0c\u6211\u5011\u9700\u8981\u85c9\u52a9\u4e00\u500b\u4e32\u5217 <code>path</code> \u8a18\u9304\u8a2a\u554f\u904e\u7684\u7bc0\u9ede\u8def\u5f91\u3002\u7576\u8a2a\u554f\u5230\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\u6642\uff0c\u5247\u8907\u88fd <code>path</code> \u4e26\u65b0\u589e\u9032\u7d50\u679c\u4e32\u5217 <code>res</code> \u3002\u8d70\u8a2a\u5b8c\u6210\u5f8c\uff0c<code>res</code> \u4e2d\u5132\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c\"\"\"\n    if root is None:\n        return\n    # \u5617\u8a66\n    path.append(root)\n    if root.val == 7:\n        # \u8a18\u9304\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // \u5617\u8a66\n    path.add(root);\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // \u5617\u8a66\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // \u5617\u8a66\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8a18\u9304\u89e3\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u5617\u8a66\n    path.append(root)\n    if root.val == 7 {\n        // \u8a18\u9304\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push(root);\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push(root);\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // \u5617\u8a66\n  path.add(root);\n  if (root.val == 7) {\n    // \u8a18\u9304\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5617\u8a66\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8a18\u9304\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // \u56de\u9000\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // \u5617\u8a66\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.kt<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    // \u5617\u8a66\n    path!!.add(root)\n    if (root._val == 7) {\n        // \u8a18\u9304\u89e3\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // \u56de\u9000\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_ii_compact.rb<pre><code>### \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c ###\ndef pre_order(root)\n  return unless root\n\n  # \u5617\u8a66\n  $path &lt;&lt; root\n\n  # \u8a18\u9304\u89e3\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # \u56de\u9000\n  $path.pop\nend\n</code></pre> preorder_traversal_ii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5728\u6bcf\u6b21\u201c\u5617\u8a66\u201d\u4e2d\uff0c\u6211\u5011\u900f\u904e\u5c07\u7576\u524d\u7bc0\u9ede\u65b0\u589e\u9032 <code>path</code> \u4f86\u8a18\u9304\u8def\u5f91\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u5011\u9700\u8981\u5c07\u8a72\u7bc0\u9ede\u5f9e <code>path</code> \u4e2d\u5f48\u51fa\uff0c\u4ee5\u6062\u5fa9\u672c\u6b21\u5617\u8a66\u4e4b\u524d\u7684\u72c0\u614b\u3002</p> <p>\u89c0\u5bdf\u5716 13-2 \u6240\u793a\u7684\u904e\u7a0b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5617\u8a66\u548c\u56de\u9000\u7406\u89e3\u70ba\u201c\u524d\u9032\u201d\u8207\u201c\u64a4\u92b7\u201d\uff0c\u5169\u500b\u64cd\u4f5c\u4e92\u70ba\u9006\u5411\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 13-2 \u00a0 \u5617\u8a66\u8207\u56de\u9000 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1312","title":"13.1.2 \u00a0 \u526a\u679d","text":"<p>\u8907\u96dc\u7684\u56de\u6eaf\u554f\u984c\u901a\u5e38\u5305\u542b\u4e00\u500b\u6216\u591a\u500b\u7d04\u675f\u689d\u4ef6\uff0c\u7d04\u675f\u689d\u4ef6\u901a\u5e38\u53ef\u7528\u65bc\u201c\u526a\u679d\u201d\u3002</p> <p>\u4f8b\u984c\u4e09</p> <p>\u5728\u4e8c\u5143\u6a39\u4e2d\u641c\u7d22\u6240\u6709\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\uff0c\u8acb\u8fd4\u56de\u6839\u7bc0\u9ede\u5230\u9019\u4e9b\u7bc0\u9ede\u7684\u8def\u5f91\uff0c\u4e26\u8981\u6c42\u8def\u5f91\u4e2d\u4e0d\u5305\u542b\u503c\u70ba \\(3\\) \u7684\u7bc0\u9ede\u3002</p> <p>\u70ba\u4e86\u6eff\u8db3\u4ee5\u4e0a\u7d04\u675f\u689d\u4ef6\uff0c\u6211\u5011\u9700\u8981\u65b0\u589e\u526a\u679d\u64cd\u4f5c\uff1a\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\uff0c\u82e5\u9047\u5230\u503c\u70ba \\(3\\) \u7684\u7bc0\u9ede\uff0c\u5247\u63d0\u524d\u8fd4\u56de\uff0c\u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09\"\"\"\n    # \u526a\u679d\n    if root is None or root.val == 3:\n        return\n    # \u5617\u8a66\n    path.append(root)\n    if root.val == 7:\n        # \u8a18\u9304\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid preOrder(TreeNode root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.add(root);\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid PreOrder(TreeNode? root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // \u526a\u679d\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // \u5617\u8a66\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8a18\u9304\u89e3\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfunc preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // \u5617\u8a66\n    path.append(root)\n    if root.val == 7 {\n        // \u8a18\u9304\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfunction preOrder(root, path, res) {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push(root);\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push(root);\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // \u5617\u8a66\n  path.add(root);\n  if (root.val == 7) {\n    // \u8a18\u9304\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    // \u526a\u679d\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5617\u8a66\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8a18\u9304\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // \u56de\u9000\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.kt<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfun preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    if (root == null || root._val == 3) {\n        return\n    }\n    // \u5617\u8a66\n    path!!.add(root)\n    if (root._val == 7) {\n        // \u8a18\u9304\u89e3\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // \u56de\u9000\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_iii_compact.rb<pre><code>### \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 ###\ndef pre_order(root)\n  # \u526a\u679d\n  return if !root || root.val == 3\n\n  # \u5617\u8a66\n  $path.append(root)\n\n  # \u8a18\u9304\u89e3\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # \u56de\u9000\n  $path.pop\nend\n</code></pre> preorder_traversal_iii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u201c\u526a\u679d\u201d\u662f\u4e00\u500b\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8a5e\u3002\u5982\u5716 13-3 \u6240\u793a\uff0c\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\uff0c\u6211\u5011\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\u7684\u641c\u5c0b\u5206\u652f\uff0c\u907f\u514d\u8a31\u591a\u7121\u610f\u7fa9\u7684\u5617\u8a66\uff0c\u5f9e\u800c\u63d0\u9ad8\u4e86\u641c\u5c0b\u6548\u7387\u3002</p> <p></p> <p> \u5716 13-3 \u00a0 \u6839\u64da\u7d04\u675f\u689d\u4ef6\u526a\u679d </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1313","title":"13.1.3 \u00a0 \u6846\u67b6\u7a0b\u5f0f\u78bc","text":"<p>\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5617\u8a66\u5c07\u56de\u6eaf\u7684\u201c\u5617\u8a66\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u9ad4\u6846\u67b6\u63d0\u7149\u51fa\u4f86\uff0c\u63d0\u5347\u7a0b\u5f0f\u78bc\u7684\u901a\u7528\u6027\u3002</p> <p>\u5728\u4ee5\u4e0b\u6846\u67b6\u7a0b\u5f0f\u78bc\u4e2d\uff0c<code>state</code> \u8868\u793a\u554f\u984c\u7684\u7576\u524d\u72c0\u614b\uff0c<code>choices</code> \u8868\u793a\u7576\u524d\u72c0\u614b\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9078\u64c7\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6\"\"\"\n    # \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if is_solution(state):\n        # \u8a18\u9304\u89e3\n        record_solution(state, res)\n        # \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices:\n        # \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (IsSolution(state)) {\n        // \u8a18\u9304\u89e3\n        RecordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    foreach (Choice choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if isSolution(state) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if isSolution(state: state) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state: state, res: &amp;res)\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n  if (isSolution(state)) {\n    // \u8a18\u9304\u89e3\n    recordSolution(state, res);\n    // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (Choice choice in choices) {\n    // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if is_solution(state) {\n        // \u8a18\u9304\u89e3\n        record_solution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res, numRes);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; numChoices; i++) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, &amp;choices[i])) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfun backtrack(state: State?, choices: List&lt;Choice?&gt;, res: List&lt;State?&gt;?) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 ###\ndef backtrack(state, choices, res)\n    # \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if is_solution?(state)\n        # \u8a18\u9304\u89e3\n        record_solution(state, res)\n        return\n    end\n\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices\n        # \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid?(state, choice)\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice)\n        end\n    end\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u57fa\u65bc\u6846\u67b6\u7a0b\u5f0f\u78bc\u4f86\u89e3\u6c7a\u4f8b\u984c\u4e09\u3002\u72c0\u614b <code>state</code> \u70ba\u7bc0\u9ede\u8d70\u8a2a\u8def\u5f91\uff0c\u9078\u64c7 <code>choices</code> \u70ba\u7576\u524d\u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u548c\u53f3\u5b50\u7bc0\u9ede\uff0c\u7d50\u679c <code>res</code> \u662f\u8def\u5f91\u4e32\u5217\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"\u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"\u8a18\u9304\u89e3\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"\u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u66f4\u65b0\u72c0\u614b\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u6062\u5fa9\u72c0\u614b\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09\"\"\"\n    # \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if is_solution(state):\n        # \u8a18\u9304\u89e3\n        record_solution(state, res)\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices:\n        # \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice)\n            # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, [choice.left, choice.right], res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (TreeNode *choice : choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (TreeNode choice : choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (IsSolution(state)) {\n        // \u8a18\u9304\u89e3\n        RecordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    foreach (TreeNode choice in choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            MakeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* \u8a18\u9304\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, append([]*TreeNode{}, *state...))\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if isSolution(state) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for _, choice := range *choices {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* \u8a18\u9304\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state: &amp;state, choice: choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfunction backtrack(state, choices, res) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n  if (isSolution(state)) {\n    // \u8a18\u9304\u89e3\n    recordSolution(state, res);\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (TreeNode? choice in choices) {\n    // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      makeChoice(state, choice);\n      // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, [choice!.left, choice.right], res);\n      // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.last().unwrap().borrow().val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nfn record_solution(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    res.push(state.clone());\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return choice.is_some() &amp;&amp; choice.unwrap().borrow().val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfn backtrack(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    choices: &amp;Vec&lt;Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if is_solution(state) {\n        // \u8a18\u9304\u89e3\n        record_solution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for &amp;choice in choices.iter() {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice.unwrap().clone());\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(\n                state,\n                &amp;vec![\n                    choice.unwrap().borrow().left.as_ref(),\n                    choice.unwrap().borrow().right.as_ref(),\n                ],\n                res,\n            );\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice.unwrap().clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid backtrack(TreeNode *choices[2]) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution()) {\n        // \u8a18\u9304\u89e3\n        recordSolution();\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.kt<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfun isSolution(state: MutableList&lt;TreeNode?&gt;): Boolean {\n    return state.isNotEmpty() &amp;&amp; state[state.size - 1]?._val == 7\n}\n\n/* \u8a18\u9304\u89e3 */\nfun recordSolution(state: MutableList&lt;TreeNode?&gt;?, res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;) {\n    res.add(state!!.toMutableList())\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfun isValid(state: MutableList&lt;TreeNode?&gt;?, choice: TreeNode?): Boolean {\n    return choice != null &amp;&amp; choice._val != 3\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfun makeChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.add(choice)\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfun undoChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfun backtrack(\n    state: MutableList&lt;TreeNode?&gt;,\n    choices: MutableList&lt;TreeNode?&gt;,\n    res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;\n) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, mutableListOf(choice!!.left, choice.right), res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.rb<pre><code>### \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 ###\ndef is_solution?(state)\n  !state.empty? &amp;&amp; state.last.val == 7\nend\n\n### \u8a18\u9304\u89e3 ###\ndef record_solution(state, res)\n  res &lt;&lt; state.dup\nend\n\n### \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 ###\ndef is_valid?(state, choice)\n  choice &amp;&amp; choice.val != 3\nend\n\n### \u66f4\u65b0\u72c0\u614b ###\ndef make_choice(state, choice)\n  state &lt;&lt; choice\nend\n\n### \u6062\u5fa9\u72c0\u614b ###\ndef undo_choice(state, choice)\n  state.pop\nend\n\n### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 ###\ndef backtrack(state, choices, res)\n  # \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n  record_solution(state, res) if is_solution?(state)\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for choice in choices\n    # \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n    if is_valid?(state, choice)\n      # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      make_choice(state, choice)\n      # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, [choice.left, choice.right], res)\n      # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      undo_choice(state, choice)\n    end\n  end\nend\n</code></pre> preorder_traversal_iii_template.zig<pre><code>[class]{}-[func]{isSolution}\n\n[class]{}-[func]{recordSolution}\n\n[class]{}-[func]{isValid}\n\n[class]{}-[func]{makeChoice}\n\n[class]{}-[func]{undoChoice}\n\n[class]{}-[func]{backtrack}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6839\u64da\u984c\u610f\uff0c\u6211\u5011\u5728\u627e\u5230\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\u5f8c\u61c9\u8a72\u7e7c\u7e8c\u641c\u5c0b\uff0c\u56e0\u6b64\u9700\u8981\u5c07\u8a18\u9304\u89e3\u4e4b\u5f8c\u7684 <code>return</code> \u8a9e\u53e5\u522a\u9664\u3002\u5716 13-4 \u5c0d\u6bd4\u4e86\u4fdd\u7559\u6216\u522a\u9664 <code>return</code> \u8a9e\u53e5\u7684\u641c\u5c0b\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 13-4 \u00a0 \u4fdd\u7559\u8207\u522a\u9664 return \u7684\u641c\u5c0b\u904e\u7a0b\u5c0d\u6bd4 </p> <p>\u76f8\u6bd4\u57fa\u65bc\u524d\u5e8f\u8d70\u8a2a\u7684\u7a0b\u5f0f\u78bc\u5be6\u73fe\uff0c\u57fa\u65bc\u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6\u7684\u7a0b\u5f0f\u78bc\u5be6\u73fe\u96d6\u7136\u986f\u5f97\u56c9\u5506\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u56de\u6eaf\u554f\u984c\u53ef\u4ee5\u5728\u8a72\u6846\u67b6\u4e0b\u89e3\u6c7a\u3002\u6211\u5011\u53ea\u9700\u6839\u64da\u5177\u9ad4\u554f\u984c\u4f86\u5b9a\u7fa9 <code>state</code> \u548c <code>choices</code> \uff0c\u4e26\u5be6\u73fe\u6846\u67b6\u4e2d\u7684\u5404\u500b\u65b9\u6cd5\u5373\u53ef\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1314","title":"13.1.4 \u00a0 \u5e38\u7528\u8853\u8a9e","text":"<p>\u70ba\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u6211\u5011\u7e3d\u7d50\u4e00\u4e0b\u56de\u6eaf\u6f14\u7b97\u6cd5\u4e2d\u5e38\u7528\u8853\u8a9e\u7684\u542b\u7fa9\uff0c\u4e26\u5c0d\u7167\u4f8b\u984c\u4e09\u7d66\u51fa\u5c0d\u61c9\u793a\u4f8b\uff0c\u5982\u8868 13-1 \u6240\u793a\u3002</p> <p> \u8868 13-1 \u00a0 \u5e38\u898b\u7684\u56de\u6eaf\u6f14\u7b97\u6cd5\u8853\u8a9e </p> \u540d\u8a5e \u5b9a\u7fa9 \u4f8b\u984c\u4e09 \u89e3\uff08solution\uff09 \u89e3\u662f\u6eff\u8db3\u554f\u984c\u7279\u5b9a\u689d\u4ef6\u7684\u7b54\u6848\uff0c\u53ef\u80fd\u6709\u4e00\u500b\u6216\u591a\u500b \u6839\u7bc0\u9ede\u5230\u7bc0\u9ede \\(7\\) \u7684\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\u7684\u6240\u6709\u8def\u5f91 \u7d04\u675f\u689d\u4ef6\uff08constraint\uff09 \u7d04\u675f\u689d\u4ef6\u662f\u554f\u984c\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u689d\u4ef6\uff0c\u901a\u5e38\u7528\u65bc\u526a\u679d \u8def\u5f91\u4e2d\u4e0d\u5305\u542b\u7bc0\u9ede \\(3\\) \u72c0\u614b\uff08state\uff09 \u72c0\u614b\u8868\u793a\u554f\u984c\u5728\u67d0\u4e00\u6642\u523b\u7684\u60c5\u6cc1\uff0c\u5305\u62ec\u5df2\u7d93\u505a\u51fa\u7684\u9078\u64c7 \u7576\u524d\u5df2\u8a2a\u554f\u7684\u7bc0\u9ede\u8def\u5f91\uff0c\u5373 <code>path</code> \u7bc0\u9ede\u4e32\u5217 \u5617\u8a66\uff08attempt\uff09 \u5617\u8a66\u662f\u6839\u64da\u53ef\u7528\u9078\u64c7\u4f86\u63a2\u7d22\u89e3\u7a7a\u9593\u7684\u904e\u7a0b\uff0c\u5305\u62ec\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\uff0c\u6aa2\u67e5\u662f\u5426\u70ba\u89e3 \u905e\u8ff4\u8a2a\u554f\u5de6\uff08\u53f3\uff09\u5b50\u7bc0\u9ede\uff0c\u5c07\u7bc0\u9ede\u65b0\u589e\u9032 <code>path</code> \uff0c\u5224\u65b7\u7bc0\u9ede\u7684\u503c\u662f\u5426\u70ba \\(7\\) \u56de\u9000\uff08backtracking\uff09 \u56de\u9000\u6307\u9047\u5230\u4e0d\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\u7684\u72c0\u614b\u6642\uff0c\u64a4\u92b7\u524d\u9762\u505a\u51fa\u7684\u9078\u64c7\uff0c\u56de\u5230\u4e0a\u4e00\u500b\u72c0\u614b \u7576\u8d8a\u904e\u8449\u7bc0\u9ede\u3001\u7d50\u675f\u7bc0\u9ede\u8a2a\u554f\u3001\u9047\u5230\u503c\u70ba \\(3\\) \u7684\u7bc0\u9ede\u6642\u7d42\u6b62\u641c\u5c0b\uff0c\u51fd\u5f0f\u8fd4\u56de \u526a\u679d\uff08pruning\uff09 \u526a\u679d\u662f\u6839\u64da\u554f\u984c\u7279\u6027\u548c\u7d04\u675f\u689d\u4ef6\u907f\u514d\u7121\u610f\u7fa9\u7684\u641c\u5c0b\u8def\u5f91\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u5c0b\u6548\u7387 \u7576\u9047\u5230\u503c\u70ba \\(3\\) \u7684\u7bc0\u9ede\u6642\uff0c\u5247\u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b <p>Tip</p> <p>\u554f\u984c\u3001\u89e3\u3001\u72c0\u614b\u7b49\u6982\u5ff5\u662f\u901a\u7528\u7684\uff0c\u5728\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52d5\u614b\u898f\u5283\u3001\u8caa\u5a6a\u7b49\u6f14\u7b97\u6cd5\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1315","title":"13.1.5 \u00a0 \u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u56de\u6eaf\u6f14\u7b97\u6cd5\u672c\u8cea\u4e0a\u662f\u4e00\u7a2e\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u6f14\u7b97\u6cd5\uff0c\u5b83\u5617\u8a66\u6240\u6709\u53ef\u80fd\u7684\u89e3\u6c7a\u65b9\u6848\u76f4\u5230\u627e\u5230\u6eff\u8db3\u689d\u4ef6\u7684\u89e3\u3002\u9019\u7a2e\u65b9\u6cd5\u7684\u512a\u9ede\u5728\u65bc\u80fd\u5920\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u6c7a\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002</p> <p>\u7136\u800c\uff0c\u5728\u8655\u7406\u5927\u898f\u6a21\u6216\u8005\u8907\u96dc\u554f\u984c\u6642\uff0c\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u57f7\u884c\u6548\u7387\u53ef\u80fd\u96e3\u4ee5\u63a5\u53d7\u3002</p> <ul> <li>\u6642\u9593\uff1a\u56de\u6eaf\u6f14\u7b97\u6cd5\u901a\u5e38\u9700\u8981\u8d70\u8a2a\u72c0\u614b\u7a7a\u9593\u7684\u6240\u6709\u53ef\u80fd\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u9054\u5230\u6307\u6578\u968e\u6216\u968e\u4e58\u968e\u3002</li> <li>\u7a7a\u9593\uff1a\u5728\u905e\u8ff4\u547c\u53eb\u4e2d\u9700\u8981\u5132\u5b58\u7576\u524d\u7684\u72c0\u614b\uff08\u4f8b\u5982\u8def\u5f91\u3001\u7528\u65bc\u526a\u679d\u7684\u8f14\u52a9\u8b8a\u6578\u7b49\uff09\uff0c\u7576\u6df1\u5ea6\u5f88\u5927\u6642\uff0c\u7a7a\u9593\u9700\u6c42\u53ef\u80fd\u6703\u8b8a\u5f97\u5f88\u5927\u3002</li> </ul> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u6f14\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u5c0b\u554f\u984c\u548c\u7d04\u675f\u6eff\u8db3\u554f\u984c\u7684\u6700\u4f73\u89e3\u6c7a\u65b9\u6848\u3002\u5c0d\u65bc\u9019\u4e9b\u554f\u984c\uff0c\u7531\u65bc\u7121\u6cd5\u9810\u6e2c\u54ea\u4e9b\u9078\u64c7\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u5011\u5fc5\u9808\u5c0d\u6240\u6709\u53ef\u80fd\u7684\u9078\u64c7\u9032\u884c\u8d70\u8a2a\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u95dc\u9375\u662f\u5982\u4f55\u6700\u4f73\u5316\u6548\u7387\uff0c\u5e38\u898b\u7684\u6548\u7387\u6700\u4f73\u5316\u65b9\u6cd5\u6709\u5169\u7a2e\u3002</p> <ul> <li>\u526a\u679d\uff1a\u907f\u514d\u641c\u5c0b\u90a3\u4e9b\u80af\u5b9a\u4e0d\u6703\u7522\u751f\u89e3\u7684\u8def\u5f91\uff0c\u5f9e\u800c\u7bc0\u7701\u6642\u9593\u548c\u7a7a\u9593\u3002</li> <li>\u555f\u767c\u5f0f\u641c\u5c0b\uff1a\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8a08\u503c\uff0c\u5f9e\u800c\u512a\u5148\u641c\u5c0b\u6700\u6709\u53ef\u80fd\u7522\u751f\u6709\u6548\u89e3\u7684\u8def\u5f91\u3002</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/#1316","title":"13.1.6 \u00a0 \u56de\u6eaf\u5178\u578b\u4f8b\u984c","text":"<p>\u56de\u6eaf\u6f14\u7b97\u6cd5\u53ef\u7528\u65bc\u89e3\u6c7a\u8a31\u591a\u641c\u5c0b\u554f\u984c\u3001\u7d04\u675f\u6eff\u8db3\u554f\u984c\u548c\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\u3002</p> <p>\u641c\u5c0b\u554f\u984c\uff1a\u9019\u985e\u554f\u984c\u7684\u76ee\u6a19\u662f\u627e\u5230\u6eff\u8db3\u7279\u5b9a\u689d\u4ef6\u7684\u89e3\u6c7a\u65b9\u6848\u3002</p> <ul> <li>\u5168\u6392\u5217\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u500b\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7d44\u5408\u3002</li> <li>\u5b50\u96c6\u548c\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u500b\u96c6\u5408\u548c\u4e00\u500b\u76ee\u6a19\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u70ba\u76ee\u6a19\u548c\u7684\u5b50\u96c6\u3002</li> <li>\u6cb3\u5167\u5854\u554f\u984c\uff1a\u7d66\u5b9a\u4e09\u6839\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5713\u76e4\uff0c\u8981\u6c42\u5c07\u6240\u6709\u5713\u76e4\u5f9e\u4e00\u6839\u67f1\u5b50\u79fb\u52d5\u5230\u53e6\u4e00\u6839\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52d5\u4e00\u500b\u5713\u76e4\uff0c\u4e14\u4e0d\u80fd\u5c07\u5927\u5713\u76e4\u653e\u5728\u5c0f\u5713\u76e4\u4e0a\u3002</li> </ul> <p>\u7d04\u675f\u6eff\u8db3\u554f\u984c\uff1a\u9019\u985e\u554f\u984c\u7684\u76ee\u6a19\u662f\u627e\u5230\u6eff\u8db3\u6240\u6709\u7d04\u675f\u689d\u4ef6\u7684\u89e3\u3002</p> <ul> <li>\\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76e4\u4e0a\u653e\u7f6e \\(n\\) \u500b\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u5011\u4e92\u4e0d\u653b\u64ca\u3002</li> <li>\u6578\u7368\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7db2\u683c\u4e2d\u586b\u5165\u6578\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u500b \\(3 \\times 3\\) \u5b50\u7db2\u683c\u4e2d\u7684\u6578\u5b57\u4e0d\u91cd\u8907\u3002</li> <li>\u5716\u8457\u8272\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u500b\u7121\u5411\u5716\uff0c\u7528\u6700\u5c11\u7684\u984f\u8272\u7d66\u5716\u7684\u6bcf\u500b\u9802\u9ede\u8457\u8272\uff0c\u4f7f\u5f97\u76f8\u9130\u9802\u9ede\u984f\u8272\u4e0d\u540c\u3002</li> </ul> <p>\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\uff1a\u9019\u985e\u554f\u984c\u7684\u76ee\u6a19\u662f\u5728\u4e00\u500b\u7d44\u5408\u7a7a\u9593\u4e2d\u627e\u5230\u6eff\u8db3\u67d0\u4e9b\u689d\u4ef6\u7684\u6700\u512a\u89e3\u3002</p> <ul> <li>0-1 \u80cc\u5305\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u7d44\u7269\u54c1\u548c\u4e00\u500b\u80cc\u5305\uff0c\u6bcf\u500b\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u50f9\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5167\uff0c\u9078\u64c7\u7269\u54c1\u4f7f\u5f97\u7e3d\u50f9\u503c\u6700\u5927\u3002</li> <li>\u65c5\u884c\u5546\u554f\u984c\uff1a\u5728\u4e00\u500b\u5716\u4e2d\uff0c\u5f9e\u4e00\u500b\u9ede\u51fa\u767c\uff0c\u8a2a\u554f\u6240\u6709\u5176\u4ed6\u9ede\u6070\u597d\u4e00\u6b21\u5f8c\u8fd4\u56de\u8d77\u9ede\uff0c\u6c42\u6700\u77ed\u8def\u5f91\u3002</li> <li>\u6700\u5927\u5718\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u500b\u7121\u5411\u5716\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u5716\uff0c\u5373\u5b50\u5716\u4e2d\u7684\u4efb\u610f\u5169\u500b\u9802\u9ede\u4e4b\u9593\u90fd\u6709\u908a\u76f8\u9023\u3002</li> </ul> <p>\u8acb\u6ce8\u610f\uff0c\u5c0d\u65bc\u8a31\u591a\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\uff0c\u56de\u6eaf\u4e0d\u662f\u6700\u512a\u89e3\u6c7a\u65b9\u6848\u3002</p> <ul> <li>0-1 \u80cc\u5305\u554f\u984c\u901a\u5e38\u4f7f\u7528\u52d5\u614b\u898f\u5283\u89e3\u6c7a\uff0c\u4ee5\u9054\u5230\u66f4\u9ad8\u7684\u6642\u9593\u6548\u7387\u3002</li> <li>\u65c5\u884c\u5546\u662f\u4e00\u500b\u8457\u540d\u7684 NP-Hard \u554f\u984c\uff0c\u5e38\u7528\u89e3\u6cd5\u6709\u907a\u50b3\u6f14\u7b97\u6cd5\u548c\u87fb\u7fa4\u6f14\u7b97\u6cd5\u7b49\u3002</li> <li>\u6700\u5927\u5718\u554f\u984c\u662f\u5716\u8ad6\u4e2d\u7684\u4e00\u500b\u7d93\u5178\u554f\u984c\uff0c\u53ef\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7b49\u555f\u767c\u5f0f\u6f14\u7b97\u6cd5\u4f86\u89e3\u6c7a\u3002</li> </ul>"},{"location":"chapter_backtracking/n_queens_problem/","title":"13.4 \u00a0 n \u7687\u540e\u554f\u984c","text":"<p>Question</p> <p>\u6839\u64da\u570b\u969b\u8c61\u68cb\u7684\u898f\u5247\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u64ca\u8207\u540c\u8655\u4e00\u884c\u3001\u4e00\u5217\u6216\u4e00\u689d\u659c\u7dda\u4e0a\u7684\u68cb\u5b50\u3002\u7d66\u5b9a \\(n\\) \u500b\u7687\u540e\u548c\u4e00\u500b \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5c0b\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u9593\u7121\u6cd5\u76f8\u4e92\u653b\u64ca\u7684\u64fa\u653e\u65b9\u6848\u3002</p> <p>\u5982\u5716 13-15 \u6240\u793a\uff0c\u7576 \\(n = 4\\) \u6642\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u5169\u500b\u89e3\u3002\u5f9e\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76e4\u5171\u6709 \\(n^2\\) \u500b\u683c\u5b50\uff0c\u7d66\u51fa\u4e86\u6240\u6709\u7684\u9078\u64c7 <code>choices</code> \u3002\u5728\u9010\u500b\u653e\u7f6e\u7687\u540e\u7684\u904e\u7a0b\u4e2d\uff0c\u68cb\u76e4\u72c0\u614b\u5728\u4e0d\u65b7\u5730\u8b8a\u5316\uff0c\u6bcf\u500b\u6642\u523b\u7684\u68cb\u76e4\u5c31\u662f\u72c0\u614b <code>state</code> \u3002</p> <p></p> <p> \u5716 13-15 \u00a0 4 \u7687\u540e\u554f\u984c\u7684\u89e3 </p> <p>\u5716 13-16 \u5c55\u793a\u4e86\u672c\u984c\u7684\u4e09\u500b\u7d04\u675f\u689d\u4ef6\uff1a\u591a\u500b\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u3001\u540c\u4e00\u689d\u5c0d\u89d2\u7dda\u4e0a\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5c0d\u89d2\u7dda\u5206\u70ba\u4e3b\u5c0d\u89d2\u7dda <code>\\</code> \u548c\u6b21\u5c0d\u89d2\u7dda <code>/</code> \u5169\u7a2e\u3002</p> <p></p> <p> \u5716 13-16 \u00a0 n \u7687\u540e\u554f\u984c\u7684\u7d04\u675f\u689d\u4ef6 </p>"},{"location":"chapter_backtracking/n_queens_problem/#1","title":"1. \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565","text":"<p>\u7687\u540e\u7684\u6578\u91cf\u548c\u68cb\u76e4\u7684\u884c\u6578\u90fd\u70ba \\(n\\) \uff0c\u56e0\u6b64\u6211\u5011\u5bb9\u6613\u5f97\u5230\u4e00\u500b\u63a8\u8ad6\uff1a\u68cb\u76e4\u6bcf\u884c\u90fd\u5141\u8a31\u4e14\u53ea\u5141\u8a31\u653e\u7f6e\u4e00\u500b\u7687\u540e\u3002</p> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u6211\u5011\u53ef\u4ee5\u63a1\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u5f9e\u7b2c\u4e00\u884c\u958b\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u500b\u7687\u540e\uff0c\u76f4\u81f3\u6700\u5f8c\u4e00\u884c\u7d50\u675f\u3002</p> <p>\u5716 13-17 \u6240\u793a\u70ba 4 \u7687\u540e\u554f\u984c\u7684\u9010\u884c\u653e\u7f6e\u904e\u7a0b\u3002\u53d7\u756b\u5e45\u9650\u5236\uff0c\u5716 13-17 \u50c5\u5c55\u958b\u4e86\u7b2c\u4e00\u884c\u7684\u5176\u4e2d\u4e00\u500b\u641c\u5c0b\u5206\u652f\uff0c\u4e26\u4e14\u5c07\u4e0d\u6eff\u8db3\u5217\u7d04\u675f\u548c\u5c0d\u89d2\u7dda\u7d04\u675f\u7684\u65b9\u6848\u90fd\u9032\u884c\u4e86\u526a\u679d\u3002</p> <p></p> <p> \u5716 13-17 \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565 </p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u9010\u884c\u653e\u7f6e\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73fe\u591a\u500b\u7687\u540e\u7684\u6240\u6709\u641c\u5c0b\u5206\u652f\u3002</p>"},{"location":"chapter_backtracking/n_queens_problem/#2","title":"2. \u00a0 \u5217\u8207\u5c0d\u89d2\u7dda\u526a\u679d","text":"<p>\u70ba\u4e86\u6eff\u8db3\u5217\u7d04\u675f\uff0c\u6211\u5011\u53ef\u4ee5\u5229\u7528\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u5e03\u6797\u578b\u9663\u5217 <code>cols</code> \u8a18\u9304\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u6c7a\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u5011\u900f\u904e <code>cols</code> \u5c07\u5df2\u6709\u7687\u540e\u7684\u5217\u9032\u884c\u526a\u679d\uff0c\u4e26\u5728\u56de\u6eaf\u4e2d\u52d5\u614b\u66f4\u65b0 <code>cols</code> \u7684\u72c0\u614b\u3002</p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u77e9\u9663\u7684\u8d77\u9ede\u4f4d\u65bc\u5de6\u4e0a\u89d2\uff0c\u5176\u4e2d\u884c\u7d22\u5f15\u5f9e\u4e0a\u5230\u4e0b\u589e\u52a0\uff0c\u5217\u7d22\u5f15\u5f9e\u5de6\u5230\u53f3\u589e\u52a0\u3002</p> <p>\u90a3\u9ebc\uff0c\u5982\u4f55\u8655\u7406\u5c0d\u89d2\u7dda\u7d04\u675f\u5462\uff1f\u8a2d\u68cb\u76e4\u4e2d\u67d0\u500b\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u70ba \\((row, col)\\) \uff0c\u9078\u5b9a\u77e9\u9663\u4e2d\u7684\u67d0\u689d\u4e3b\u5c0d\u89d2\u7dda\uff0c\u6211\u5011\u767c\u73fe\u8a72\u5c0d\u89d2\u7dda\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u6e1b\u5217\u7d22\u5f15\u90fd\u76f8\u7b49\uff0c\u5373\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u6240\u6709\u683c\u5b50\u7684 \\(row - col\\) \u70ba\u6046\u5b9a\u503c\u3002</p> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u5982\u679c\u5169\u500b\u683c\u5b50\u6eff\u8db3 \\(row_1 - col_1 = row_2 - col_2\\) \uff0c\u5247\u5b83\u5011\u4e00\u5b9a\u8655\u5728\u540c\u4e00\u689d\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u3002\u5229\u7528\u8a72\u898f\u5f8b\uff0c\u6211\u5011\u53ef\u4ee5\u85c9\u52a9\u5716 13-18 \u6240\u793a\u7684\u9663\u5217 <code>diags1</code> \u8a18\u9304\u6bcf\u689d\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002</p> <p>\u540c\u7406\uff0c\u6b21\u5c0d\u89d2\u7dda\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 \\(row + col\\) \u662f\u6046\u5b9a\u503c\u3002\u6211\u5011\u540c\u6a23\u4e5f\u53ef\u4ee5\u85c9\u52a9\u9663\u5217 <code>diags2</code> \u4f86\u8655\u7406\u6b21\u5c0d\u89d2\u7dda\u7d04\u675f\u3002</p> <p></p> <p> \u5716 13-18 \u00a0 \u8655\u7406\u5217\u7d04\u675f\u548c\u5c0d\u89d2\u7dda\u7d04\u675f </p>"},{"location":"chapter_backtracking/n_queens_problem/#3","title":"3. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u8acb\u6ce8\u610f\uff0c\\(n\\) \u7dad\u65b9\u9663\u4e2d \\(row - col\\) \u7684\u7bc4\u570d\u662f \\([-n + 1, n - 1]\\) \uff0c\\(row + col\\) \u7684\u7bc4\u570d\u662f \\([0, 2n - 2]\\) \uff0c\u6240\u4ee5\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\u7684\u6578\u91cf\u90fd\u70ba \\(2n - 1\\) \uff0c\u5373\u9663\u5217 <code>diags1</code> \u548c <code>diags2</code> \u7684\u9577\u5ea6\u90fd\u70ba \\(2n - 1\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e\"\"\"\n    # \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # \u8d70\u8a2a\u6240\u6709\u5217\n    for col in range(n):\n        # \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"\u6c42\u89e3 n \u7687\u540e\"\"\"\n    # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    diags1 = [False] * (2 * n - 1)  # \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    diags2 = [False] * (2 * n - 1)  # \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags1 = new boolean[2 * n - 1]; // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags2 = new boolean[2 * n - 1]; // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    bool[] diags1 = new bool[2 * n - 1]; // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool[] diags2 = new bool[2 * n - 1]; // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for col := 0; col &lt; n; col++ {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunc nQueens(n int) [][][]string {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if row == n {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for col in 0 ..&lt; n {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunction nQueens(n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u5217\n  for (int col = 0; col &lt; n; col++) {\n    // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // \u653e\u7f6e\u4e0b\u4e00\u884c\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfn backtrack(\n    row: usize,\n    n: usize,\n    state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool],\n    diags1: &amp;mut [bool],\n    diags2: &amp;mut [bool],\n) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if row == n {\n        res.push(state.clone());\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for col in 0..n {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = vec![vec![\"#\".to_string(); n]; n];\n    let mut cols = vec![false; n]; // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    let mut diags1 = vec![false; 2 * n - 1]; // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut diags2 = vec![false; 2 * n - 1]; // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(\n        0,\n        n,\n        &amp;mut state,\n        &amp;mut res,\n        &amp;mut cols,\n        &amp;mut diags1,\n        &amp;mut diags2,\n    );\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfun backtrack(\n    row: Int,\n    n: Int,\n    state: MutableList&lt;MutableList&lt;String&gt;&gt;,\n    res: MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;,\n    cols: BooleanArray,\n    diags1: BooleanArray,\n    diags2: BooleanArray\n) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        val copyState = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n        for (sRow in state) {\n            copyState.add(sRow.toMutableList())\n        }\n        res.add(copyState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (col in 0..&lt;n) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        val diag1 = row - col + n - 1\n        val diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\"\n            diags2[diag2] = true\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            diags2[diag2] = false\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfun nQueens(n: Int): MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    val state = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val row = mutableListOf&lt;String&gt;()\n        for (j in 0..&lt;n) {\n            row.add(\"#\")\n        }\n        state.add(row)\n    }\n    val cols = BooleanArray(n) // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    val diags1 = BooleanArray(2 * n - 1) // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    val diags2 = BooleanArray(2 * n - 1) // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    val res = mutableListOf&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;()\n\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n}\n</code></pre> n_queens.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e ###\ndef backtrack(row, n, state, res, cols, diags1, diags2)\n  # \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n  if row == n\n    res &lt;&lt; state.map { |row| row.dup }\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u5217\n  for col in 0...n\n    # \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n    diag1 = row - col + n - 1\n    diag2 = row + col\n    # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n    if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]\n      # \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n      state[row][col] = \"Q\"\n      cols[col] = diags1[diag1] = diags2[diag2] = true\n      # \u653e\u7f6e\u4e0b\u4e00\u884c\n      backtrack(row + 1, n, state, res, cols, diags1, diags2)\n      # \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n      state[row][col] = \"#\"\n      cols[col] = diags1[diag1] = diags2[diag2] = false\n    end\n  end\nend\n\n### \u6c42\u89e3 n \u7687\u540e ###\ndef n_queens(n)\n  # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n  state = Array.new(n) { Array.new(n, \"#\") }\n  cols = Array.new(n, false) # \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n  diags1 = Array.new(2 * n - 1, false) # \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n  diags2 = Array.new(2 * n - 1, false) # \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n  res = []\n  backtrack(0, n, state, res, cols, diags1, diags2)\n\n  res\nend\n</code></pre> n_queens.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{nQueens}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u616e\u5217\u7d04\u675f\uff0c\u5247\u5f9e\u7b2c\u4e00\u884c\u5230\u6700\u5f8c\u4e00\u884c\u5206\u5225\u6709 \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u500b\u9078\u64c7\uff0c\u4f7f\u7528 \\(O(n!)\\) \u6642\u9593\u3002\u7576\u8a18\u9304\u89e3\u6642\uff0c\u9700\u8981\u8907\u88fd\u77e9\u9663 <code>state</code> \u4e26\u65b0\u589e\u9032 <code>res</code> \uff0c\u8907\u88fd\u64cd\u4f5c\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002\u56e0\u6b64\uff0c\u7e3d\u9ad4\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n! \\cdot n^2)\\) \u3002\u5be6\u969b\u4e0a\uff0c\u6839\u64da\u5c0d\u89d2\u7dda\u7d04\u675f\u7684\u526a\u679d\u4e5f\u80fd\u5920\u5927\u5e45\u7e2e\u5c0f\u641c\u5c0b\u7a7a\u9593\uff0c\u56e0\u800c\u641c\u5c0b\u6548\u7387\u5f80\u5f80\u512a\u65bc\u4ee5\u4e0a\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <p>\u9663\u5217 <code>state</code> \u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u9593\uff0c\u9663\u5217 <code>cols</code>\u3001<code>diags1</code> \u548c <code>diags2</code> \u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u9593\u3002\u6700\u5927\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002\u56e0\u6b64\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/","title":"13.2 \u00a0 \u5168\u6392\u5217\u554f\u984c","text":"<p>\u5168\u6392\u5217\u554f\u984c\u662f\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u4e00\u500b\u5178\u578b\u61c9\u7528\u3002\u5b83\u7684\u5b9a\u7fa9\u662f\u5728\u7d66\u5b9a\u4e00\u500b\u96c6\u5408\uff08\u5982\u4e00\u500b\u9663\u5217\u6216\u5b57\u4e32\uff09\u7684\u60c5\u6cc1\u4e0b\uff0c\u627e\u51fa\u5176\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u8868 13-2 \u5217\u8209\u4e86\u5e7e\u500b\u793a\u4f8b\u8cc7\u6599\uff0c\u5305\u62ec\u8f38\u5165\u9663\u5217\u548c\u5c0d\u61c9\u7684\u6240\u6709\u6392\u5217\u3002</p> <p> \u8868 13-2 \u00a0 \u5168\u6392\u5217\u793a\u4f8b </p> \u8f38\u5165\u9663\u5217 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1321","title":"13.2.1 \u00a0 \u7121\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u8f38\u5165\u4e00\u500b\u6574\u6578\u9663\u5217\uff0c\u5176\u4e2d\u4e0d\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u5f9e\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u904e\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9078\u64c7\u7684\u7d50\u679c\u3002\u5047\u8a2d\u8f38\u5165\u9663\u5217\u70ba \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u5011\u5148\u9078\u64c7 \\(1\\) \uff0c\u518d\u9078\u64c7 \\(3\\) \uff0c\u6700\u5f8c\u9078\u64c7 \\(2\\) \uff0c\u5247\u7372\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u92b7\u4e00\u500b\u9078\u64c7\uff0c\u4e4b\u5f8c\u7e7c\u7e8c\u5617\u8a66\u5176\u4ed6\u9078\u64c7\u3002</p> <p>\u5f9e\u56de\u6eaf\u7a0b\u5f0f\u78bc\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9078\u96c6\u5408 <code>choices</code> \u662f\u8f38\u5165\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72c0\u614b <code>state</code> \u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9078\u64c7\u7684\u5143\u7d20\u3002\u8acb\u6ce8\u610f\uff0c\u6bcf\u500b\u5143\u7d20\u53ea\u5141\u8a31\u88ab\u9078\u64c7\u4e00\u6b21\uff0c\u56e0\u6b64 <code>state</code> \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u61c9\u8a72\u662f\u552f\u4e00\u7684\u3002</p> <p>\u5982\u5716 13-5 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u641c\u5c0b\u904e\u7a0b\u5c55\u958b\u6210\u4e00\u68f5\u905e\u8ff4\u6a39\uff0c\u6a39\u4e2d\u7684\u6bcf\u500b\u7bc0\u9ede\u4ee3\u8868\u7576\u524d\u72c0\u614b <code>state</code> \u3002\u5f9e\u6839\u7bc0\u9ede\u958b\u59cb\uff0c\u7d93\u904e\u4e09\u8f2a\u9078\u64c7\u5f8c\u5230\u9054\u8449\u7bc0\u9ede\uff0c\u6bcf\u500b\u8449\u7bc0\u9ede\u90fd\u5c0d\u61c9\u4e00\u500b\u6392\u5217\u3002</p> <p></p> <p> \u5716 13-5 \u00a0 \u5168\u6392\u5217\u7684\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_backtracking/permutations_problem/#1","title":"1. \u00a0 \u91cd\u8907\u9078\u64c7\u526a\u679d","text":"<p>\u70ba\u4e86\u5be6\u73fe\u6bcf\u500b\u5143\u7d20\u53ea\u88ab\u9078\u64c7\u4e00\u6b21\uff0c\u6211\u5011\u8003\u616e\u5f15\u5165\u4e00\u500b\u5e03\u6797\u578b\u9663\u5217 <code>selected</code> \uff0c\u5176\u4e2d <code>selected[i]</code> \u8868\u793a <code>choices[i]</code> \u662f\u5426\u5df2\u88ab\u9078\u64c7\uff0c\u4e26\u57fa\u65bc\u5b83\u5be6\u73fe\u4ee5\u4e0b\u526a\u679d\u64cd\u4f5c\u3002</p> <ul> <li>\u5728\u505a\u51fa\u9078\u64c7 <code>choice[i]</code> \u5f8c\uff0c\u6211\u5011\u5c31\u5c07 <code>selected[i]</code> \u8ce6\u503c\u70ba \\(\\text{True}\\) \uff0c\u4ee3\u8868\u5b83\u5df2\u88ab\u9078\u64c7\u3002</li> <li>\u8d70\u8a2a\u9078\u64c7\u4e32\u5217 <code>choices</code> \u6642\uff0c\u8df3\u904e\u6240\u6709\u5df2\u88ab\u9078\u64c7\u7684\u7bc0\u9ede\uff0c\u5373\u526a\u679d\u3002</li> </ul> <p>\u5982\u5716 13-6 \u6240\u793a\uff0c\u5047\u8a2d\u6211\u5011\u7b2c\u4e00\u8f2a\u9078\u64c7 1 \uff0c\u7b2c\u4e8c\u8f2a\u9078\u64c7 3 \uff0c\u7b2c\u4e09\u8f2a\u9078\u64c7 2 \uff0c\u5247\u9700\u8981\u5728\u7b2c\u4e8c\u8f2a\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f2a\u526a\u6389\u5143\u7d20 1 \u548c\u5143\u7d20 3 \u7684\u5206\u652f\u3002</p> <p></p> <p> \u5716 13-6 \u00a0 \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b </p> <p>\u89c0\u5bdf\u5716 13-6 \u767c\u73fe\uff0c\u8a72\u526a\u679d\u64cd\u4f5c\u5c07\u641c\u5c0b\u7a7a\u9593\u5927\u5c0f\u5f9e \\(O(n^n)\\) \u6e1b\u5c0f\u81f3 \\(O(n!)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u60f3\u6e05\u695a\u4ee5\u4e0a\u8cc7\u8a0a\u4e4b\u5f8c\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u7a0b\u5f0f\u78bc\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u70ba\u4e86\u7e2e\u77ed\u6574\u9ad4\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u4e0d\u55ae\u7368\u5be6\u73fe\u6846\u67b6\u7a0b\u5f0f\u78bc\u4e2d\u7684\u5404\u500b\u51fd\u5f0f\uff0c\u800c\u662f\u5c07\u5b83\u5011\u5c55\u958b\u5728 <code>backtrack()</code> \u51fd\u5f0f\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n    # \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if not selected[i]:\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = True\n            state.append(choice)\n            # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.push_back(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.add(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.Add(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if !(*selected)[i] {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrackI(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if !selected[i] {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true\n            state.append(choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n    if (!selected[i]) {\n      // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      selected[i] = true;\n      state.add(choice);\n      // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if !selected[i] {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true\n            state.add(choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfun permutationsI(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_i.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I ###\ndef backtrack(state, choices, selected, res)\n  # \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  choices.each_with_index do |choice, i|\n    # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n    unless selected[i]\n      # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      selected[i] = true\n      state &lt;&lt; choice\n      # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, choices, selected, res)\n      # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### \u5168\u6392\u5217 I ###\ndef permutations_i(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> permutations_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsI}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_backtracking/permutations_problem/#1322","title":"13.2.2 \u00a0 \u8003\u616e\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u8f38\u5165\u4e00\u500b\u6574\u6578\u9663\u5217\uff0c\u9663\u5217\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u8907\u7684\u6392\u5217\u3002</p> <p>\u5047\u8a2d\u8f38\u5165\u9663\u5217\u70ba \\([1, 1, 2]\\) \u3002\u70ba\u4e86\u65b9\u4fbf\u5340\u5206\u5169\u500b\u91cd\u8907\u5143\u7d20 \\(1\\) \uff0c\u6211\u5011\u5c07\u7b2c\u4e8c\u500b \\(1\\) \u8a18\u70ba \\(\\hat{1}\\) \u3002</p> <p>\u5982\u5716 13-7 \u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u662f\u91cd\u8907\u7684\u3002</p> <p></p> <p> \u5716 13-7 \u00a0 \u91cd\u8907\u6392\u5217 </p> <p>\u90a3\u9ebc\u5982\u4f55\u53bb\u9664\u91cd\u8907\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u8003\u616e\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u96c6\u5408\uff0c\u76f4\u63a5\u5c0d\u6392\u5217\u7d50\u679c\u9032\u884c\u53bb\u91cd\u3002\u7136\u800c\u9019\u6a23\u505a\u4e0d\u5920\u512a\u96c5\uff0c\u56e0\u70ba\u751f\u6210\u91cd\u8907\u6392\u5217\u7684\u641c\u5c0b\u5206\u652f\u6c92\u6709\u5fc5\u8981\uff0c\u61c9\u7576\u63d0\u524d\u8b58\u5225\u4e26\u526a\u679d\uff0c\u9019\u6a23\u53ef\u4ee5\u9032\u4e00\u6b65\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u89c0\u5bdf\u5716 13-8 \uff0c\u5728\u7b2c\u4e00\u8f2a\u4e2d\uff0c\u9078\u64c7 \\(1\\) \u6216\u9078\u64c7 \\(\\hat{1}\\) \u662f\u7b49\u50f9\u7684\uff0c\u5728\u9019\u5169\u500b\u9078\u64c7\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u8907\u7684\u3002\u56e0\u6b64\u61c9\u8a72\u628a \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f2a\u9078\u64c7 \\(2\\) \u4e4b\u5f8c\uff0c\u7b2c\u4e8c\u8f2a\u9078\u64c7\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u6703\u7522\u751f\u91cd\u8907\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u61c9\u5c07\u7b2c\u4e8c\u8f2a\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u6211\u5011\u7684\u76ee\u6a19\u662f\u5728\u67d0\u4e00\u8f2a\u9078\u64c7\u4e2d\uff0c\u4fdd\u8b49\u591a\u500b\u76f8\u7b49\u7684\u5143\u7d20\u50c5\u88ab\u9078\u64c7\u4e00\u6b21\u3002</p> <p></p> <p> \u5716 13-8 \u00a0 \u91cd\u8907\u6392\u5217\u526a\u679d </p>"},{"location":"chapter_backtracking/permutations_problem/#2_1","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5728\u4e0a\u4e00\u984c\u7684\u7a0b\u5f0f\u78bc\u7684\u57fa\u790e\u4e0a\uff0c\u6211\u5011\u8003\u616e\u5728\u6bcf\u4e00\u8f2a\u9078\u64c7\u4e2d\u958b\u555f\u4e00\u500b\u96dc\u6e4a\u96c6\u5408 <code>duplicated</code> \uff0c\u7528\u65bc\u8a18\u9304\u8a72\u8f2a\u4e2d\u5df2\u7d93\u5617\u8a66\u904e\u7684\u5143\u7d20\uff0c\u4e26\u5c07\u91cd\u8907\u5143\u7d20\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n    # \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if not selected[i] and choice not in duplicated:\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice)  # \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = True\n            state.append(choice)\n            # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.emplace(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push_back(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.add(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.Add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.Add(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrackII(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if !selected[i], !duplicated.contains(choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.insert(choice) // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.append(choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      duplicated.add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n      selected[i] = true;\n      state.add(choice);\n      // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.insert(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated[choice] = true; // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    val duplicated = HashSet&lt;Int&gt;()\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice) // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.add(choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfun permutationsII(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_ii.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II ###\ndef backtrack(state, choices, selected, res)\n  # \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  duplicated = Set.new\n  choices.each_with_index do |choice, i|\n    # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n    if !selected[i] &amp;&amp; !duplicated.include?(choice)\n      # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      duplicated.add(choice)\n      selected[i] = true\n      state &lt;&lt; choice\n      # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, choices, selected, res)\n      # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### \u5168\u6392\u5217 II ###\ndef permutations_ii(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> permutations_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsII}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5047\u8a2d\u5143\u7d20\u5169\u5169\u4e4b\u9593\u4e92\u4e0d\u76f8\u540c\uff0c\u5247 \\(n\\) \u500b\u5143\u7d20\u5171\u6709 \\(n!\\)  \u7a2e\u6392\u5217\uff08\u968e\u4e58\uff09\uff1b\u5728\u8a18\u9304\u7d50\u679c\u6642\uff0c\u9700\u8981\u8907\u88fd\u9577\u5ea6\u70ba \\(n\\) \u7684\u4e32\u5217\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n!n)\\) \u3002</p> <p>\u6700\u5927\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002<code>selected</code> \u4f7f\u7528 \\(O(n)\\) \u7a7a\u9593\u3002\u540c\u4e00\u6642\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u500b <code>duplicated</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u9593\u3002\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#3","title":"3. \u00a0 \u5169\u7a2e\u526a\u679d\u5c0d\u6bd4","text":"<p>\u8acb\u6ce8\u610f\uff0c\u96d6\u7136 <code>selected</code> \u548c <code>duplicated</code> \u90fd\u7528\u65bc\u526a\u679d\uff0c\u4f46\u5169\u8005\u7684\u76ee\u6a19\u4e0d\u540c\u3002</p> <ul> <li>\u91cd\u8907\u9078\u64c7\u526a\u679d\uff1a\u6574\u500b\u641c\u5c0b\u904e\u7a0b\u4e2d\u53ea\u6709\u4e00\u500b <code>selected</code> \u3002\u5b83\u8a18\u9304\u7684\u662f\u7576\u524d\u72c0\u614b\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u5176\u4f5c\u7528\u662f\u907f\u514d\u67d0\u500b\u5143\u7d20\u5728 <code>state</code> \u4e2d\u91cd\u8907\u51fa\u73fe\u3002</li> <li>\u76f8\u7b49\u5143\u7d20\u526a\u679d\uff1a\u6bcf\u8f2a\u9078\u64c7\uff08\u6bcf\u500b\u547c\u53eb\u7684 <code>backtrack</code> \u51fd\u5f0f\uff09\u90fd\u5305\u542b\u4e00\u500b <code>duplicated</code> \u3002\u5b83\u8a18\u9304\u7684\u662f\u5728\u672c\u8f2a\u8d70\u8a2a\uff08<code>for</code> \u8ff4\u5708\uff09\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9078\u64c7\u904e\uff0c\u5176\u4f5c\u7528\u662f\u4fdd\u8b49\u76f8\u7b49\u5143\u7d20\u53ea\u88ab\u9078\u64c7\u4e00\u6b21\u3002</li> </ul> <p>\u5716 13-9 \u5c55\u793a\u4e86\u5169\u500b\u526a\u679d\u689d\u4ef6\u7684\u751f\u6548\u7bc4\u570d\u3002\u6ce8\u610f\uff0c\u6a39\u4e2d\u7684\u6bcf\u500b\u7bc0\u9ede\u4ee3\u8868\u4e00\u500b\u9078\u64c7\uff0c\u5f9e\u6839\u7bc0\u9ede\u5230\u8449\u7bc0\u9ede\u7684\u8def\u5f91\u4e0a\u7684\u5404\u500b\u7bc0\u9ede\u69cb\u6210\u4e00\u500b\u6392\u5217\u3002</p> <p></p> <p> \u5716 13-9 \u00a0 \u5169\u7a2e\u526a\u679d\u689d\u4ef6\u7684\u4f5c\u7528\u7bc4\u570d </p>"},{"location":"chapter_backtracking/subset_sum_problem/","title":"13.3 \u00a0 \u5b50\u96c6\u548c\u554f\u984c","text":""},{"location":"chapter_backtracking/subset_sum_problem/#1331","title":"13.3.1 \u00a0 \u7121\u91cd\u8907\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u6b63\u6574\u6578\u9663\u5217 <code>nums</code> \u548c\u4e00\u500b\u76ee\u6a19\u6b63\u6574\u6578 <code>target</code> \uff0c\u8acb\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7d44\u5408\uff0c\u4f7f\u5f97\u7d44\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u65bc <code>target</code> \u3002\u7d66\u5b9a\u9663\u5217\u7121\u91cd\u8907\u5143\u7d20\uff0c\u6bcf\u500b\u5143\u7d20\u53ef\u4ee5\u88ab\u9078\u53d6\u591a\u6b21\u3002\u8acb\u4ee5\u4e32\u5217\u5f62\u5f0f\u8fd4\u56de\u9019\u4e9b\u7d44\u5408\uff0c\u4e32\u5217\u4e2d\u4e0d\u61c9\u5305\u542b\u91cd\u8907\u7d44\u5408\u3002</p> <p>\u4f8b\u5982\uff0c\u8f38\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6a19\u6574\u6578 \\(9\\) \uff0c\u89e3\u70ba \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u5169\u9ede\u3002</p> <ul> <li>\u8f38\u5165\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u7121\u9650\u6b21\u91cd\u8907\u9078\u53d6\u3002</li> <li>\u5b50\u96c6\u4e0d\u5340\u5206\u5143\u7d20\u9806\u5e8f\uff0c\u6bd4\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u500b\u5b50\u96c6\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#1","title":"1. \u00a0 \u53c3\u8003\u5168\u6392\u5217\u89e3\u6cd5","text":"<p>\u985e\u4f3c\u65bc\u5168\u6392\u5217\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u904e\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9078\u64c7\u7684\u7d50\u679c\uff0c\u4e26\u5728\u9078\u64c7\u904e\u7a0b\u4e2d\u5be6\u6642\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u7576\u5143\u7d20\u548c\u7b49\u65bc <code>target</code> \u6642\uff0c\u5c31\u5c07\u5b50\u96c6\u8a18\u9304\u81f3\u7d50\u679c\u4e32\u5217\u3002</p> <p>\u800c\u8207\u5168\u6392\u5217\u554f\u984c\u4e0d\u540c\u7684\u662f\uff0c\u672c\u984c\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u7121\u9650\u6b21\u9078\u53d6\uff0c\u56e0\u6b64\u7121\u9808\u85c9\u52a9 <code>selected</code> \u5e03\u6797\u4e32\u5217\u4f86\u8a18\u9304\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9078\u64c7\u3002\u6211\u5011\u53ef\u4ee5\u5c0d\u5168\u6392\u5217\u7a0b\u5f0f\u78bc\u9032\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u984c\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if total == target:\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i in range(len(choices)):\n        # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if total + choices[i] &gt; target:\n            continue\n        # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res)\n        # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09\"\"\"\n    state = []  # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    total = 0  # \u5b50\u96c6\u548c\n    res = []  # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push_back(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    int total = 0;           // \u5b50\u96c6\u548c\n    vector&lt;vector&lt;int&gt;&gt; res; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.Add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        Backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i := 0; i &lt; len(*choices); i++ {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        *state = append(*state, (*choices)[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    total := 0              // \u5b50\u96c6\u548c\n    res := make([][]int, 0) // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if total == target {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i in choices.indices {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    let total = 0 // \u5b50\u96c6\u548c\n    var res: [[Int]] = [] // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, total, choices, res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (int i = 0; i &lt; choices.length; i++) {\n    // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n    state.add(choices[i]);\n    // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target, total + choices[i], choices, res);\n    // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  int total = 0; // \u5143\u7d20\u548c\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    total: i32,\n    choices: &amp;[i32],\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if total == target {\n        res.push(state.clone());\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i in 0..choices.len() {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    let total = 0; // \u5b50\u96c6\u548c\n    let mut res = Vec::new(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(&amp;mut state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state[stateSize++] = choices[i];\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // \u521d\u59cb\u5316\u89e3\u7684\u6578\u91cf\u70ba0\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    total: Int,\n    choices: IntArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (i in choices.indices) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfun subsetSumINaive(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    val total = 0 // \u5b50\u96c6\u548c\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n}\n</code></pre> subset_sum_i_naive.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I ###\ndef backtrack(state, target, total, choices, res)\n  # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if total == target\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for i in 0...choices.length\n    # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n    next if total + choices[i] &gt; target\n    # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n    state &lt;&lt; choices[i]\n    # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target, total + choices[i], choices, res)\n    # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.pop\n  end\nend\n\n### \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09###\ndef subset_sum_i_naive(nums, target)\n  state = [] # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  total = 0 # \u5b50\u96c6\u548c\n  res = [] # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, total, nums, res)\n  res\nend\n</code></pre> subset_sum_i_naive.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumINaive}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5411\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u8f38\u5165\u9663\u5217 \\([3, 4, 5]\\) \u548c\u76ee\u6a19\u5143\u7d20 \\(9\\) \uff0c\u8f38\u51fa\u7d50\u679c\u70ba \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u96d6\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u70ba \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u8907\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002</p> <p>\u9019\u662f\u56e0\u70ba\u641c\u5c0b\u904e\u7a0b\u662f\u5340\u5206\u9078\u64c7\u9806\u5e8f\u7684\uff0c\u7136\u800c\u5b50\u96c6\u4e0d\u5340\u5206\u9078\u64c7\u9806\u5e8f\u3002\u5982\u5716 13-10 \u6240\u793a\uff0c\u5148\u9078 \\(4\\) \u5f8c\u9078 \\(5\\) \u8207\u5148\u9078 \\(5\\) \u5f8c\u9078 \\(4\\) \u662f\u4e0d\u540c\u7684\u5206\u652f\uff0c\u4f46\u5c0d\u61c9\u540c\u4e00\u500b\u5b50\u96c6\u3002</p> <p></p> <p> \u5716 13-10 \u00a0 \u5b50\u96c6\u641c\u5c0b\u8207\u8d8a\u754c\u526a\u679d </p> <p>\u70ba\u4e86\u53bb\u9664\u91cd\u8907\u5b50\u96c6\uff0c\u4e00\u7a2e\u76f4\u63a5\u7684\u601d\u8def\u662f\u5c0d\u7d50\u679c\u4e32\u5217\u9032\u884c\u53bb\u91cd\u3002\u4f46\u9019\u500b\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u6709\u5169\u65b9\u9762\u539f\u56e0\u3002</p> <ul> <li>\u7576\u9663\u5217\u5143\u7d20\u8f03\u591a\uff0c\u5c24\u5176\u662f\u7576 <code>target</code> \u8f03\u5927\u6642\uff0c\u641c\u5c0b\u904e\u7a0b\u6703\u7522\u751f\u5927\u91cf\u7684\u91cd\u8907\u5b50\u96c6\u3002</li> <li>\u6bd4\u8f03\u5b50\u96c6\uff08\u9663\u5217\uff09\u7684\u7570\u540c\u975e\u5e38\u8017\u6642\uff0c\u9700\u8981\u5148\u6392\u5e8f\u9663\u5217\uff0c\u518d\u6bd4\u8f03\u9663\u5217\u4e2d\u6bcf\u500b\u5143\u7d20\u7684\u7570\u540c\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#2","title":"2. \u00a0 \u91cd\u8907\u5b50\u96c6\u526a\u679d","text":"<p>\u6211\u5011\u8003\u616e\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\u900f\u904e\u526a\u679d\u9032\u884c\u53bb\u91cd\u3002\u89c0\u5bdf\u5716 13-11 \uff0c\u91cd\u8907\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u9806\u5e8f\u9078\u64c7\u9663\u5217\u5143\u7d20\u6642\u7522\u751f\u7684\uff0c\u4f8b\u5982\u4ee5\u4e0b\u60c5\u6cc1\u3002</p> <ol> <li>\u7576\u7b2c\u4e00\u8f2a\u548c\u7b2c\u4e8c\u8f2a\u5206\u5225\u9078\u64c7 \\(3\\) \u548c \\(4\\) \u6642\uff0c\u6703\u751f\u6210\u5305\u542b\u9019\u5169\u500b\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8a18\u70ba \\([3, 4, \\dots]\\) \u3002</li> <li>\u4e4b\u5f8c\uff0c\u7576\u7b2c\u4e00\u8f2a\u9078\u64c7 \\(4\\) \u6642\uff0c\u5247\u7b2c\u4e8c\u8f2a\u61c9\u8a72\u8df3\u904e \\(3\\) \uff0c\u56e0\u70ba\u8a72\u9078\u64c7\u7522\u751f\u7684\u5b50\u96c6 \\([4, 3, \\dots]\\) \u548c\u7b2c <code>1.</code> \u6b65\u4e2d\u751f\u6210\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u8907\u3002</li> </ol> <p>\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c64\u7684\u9078\u64c7\u90fd\u662f\u5f9e\u5de6\u5230\u53f3\u88ab\u9010\u500b\u5617\u8a66\u7684\uff0c\u56e0\u6b64\u8d8a\u9760\u53f3\u7684\u5206\u652f\u88ab\u526a\u6389\u7684\u8d8a\u591a\u3002</p> <ol> <li>\u524d\u5169\u8f2a\u9078\u64c7 \\(3\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([3, 5, \\dots]\\) \u3002</li> <li>\u524d\u5169\u8f2a\u9078\u64c7 \\(4\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([4, 5, \\dots]\\) \u3002</li> <li>\u82e5\u7b2c\u4e00\u8f2a\u9078\u64c7 \\(5\\) \uff0c\u5247\u7b2c\u4e8c\u8f2a\u61c9\u8a72\u8df3\u904e \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u70ba\u5b50\u96c6 \\([5, 3, \\dots]\\) \u548c \\([5, 4, \\dots]\\) \u8207\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\u4e2d\u63cf\u8ff0\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u8907\u3002</li> </ol> <p></p> <p> \u5716 13-11 \u00a0 \u4e0d\u540c\u9078\u64c7\u9806\u5e8f\u5c0e\u81f4\u7684\u91cd\u8907\u5b50\u96c6 </p> <p>\u7e3d\u7d50\u4f86\u770b\uff0c\u7d66\u5b9a\u8f38\u5165\u9663\u5217 \\([x_1, x_2, \\dots, x_n]\\) \uff0c\u8a2d\u641c\u5c0b\u904e\u7a0b\u4e2d\u7684\u9078\u64c7\u5e8f\u5217\u70ba \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) \uff0c\u5247\u8a72\u9078\u64c7\u5e8f\u5217\u9700\u8981\u6eff\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4e0d\u6eff\u8db3\u8a72\u689d\u4ef6\u7684\u9078\u64c7\u5e8f\u5217\u90fd\u6703\u9020\u6210\u91cd\u8907\uff0c\u61c9\u7576\u526a\u679d\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#3","title":"3. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u70ba\u5be6\u73fe\u8a72\u526a\u679d\uff0c\u6211\u5011\u521d\u59cb\u5316\u8b8a\u6578 <code>start</code> \uff0c\u7528\u65bc\u6307\u793a\u8d70\u8a2a\u8d77\u59cb\u9ede\u3002\u7576\u505a\u51fa\u9078\u64c7 \\(x_{i}\\) \u5f8c\uff0c\u8a2d\u5b9a\u4e0b\u4e00\u8f2a\u5f9e\u7d22\u5f15 \\(i\\) \u958b\u59cb\u8d70\u8a2a\u3002\u9019\u6a23\u505a\u5c31\u53ef\u4ee5\u8b93\u9078\u64c7\u5e8f\u5217\u6eff\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u5f9e\u800c\u4fdd\u8b49\u5b50\u96c6\u552f\u4e00\u3002</p> <p>\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u5011\u9084\u5c0d\u7a0b\u5f0f\u78bc\u9032\u884c\u4e86\u4ee5\u4e0b\u5169\u9805\u6700\u4f73\u5316\u3002</p> <ul> <li>\u5728\u958b\u555f\u641c\u5c0b\u524d\uff0c\u5148\u5c07\u9663\u5217 <code>nums</code> \u6392\u5e8f\u3002\u5728\u8d70\u8a2a\u6240\u6709\u9078\u64c7\u6642\uff0c\u7576\u5b50\u96c6\u548c\u8d85\u904e <code>target</code> \u6642\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\uff0c\u56e0\u70ba\u5f8c\u908a\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e <code>target</code> \u3002</li> <li>\u7701\u53bb\u5143\u7d20\u548c\u8b8a\u6578 <code>total</code> \uff0c\u900f\u904e\u5728 <code>target</code> \u4e0a\u57f7\u884c\u6e1b\u6cd5\u4f86\u7d71\u8a08\u5143\u7d20\u548c\uff0c\u7576 <code>target</code> \u7b49\u65bc \\(0\\) \u6642\u8a18\u9304\u89e3\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    # \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        # \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0:\n            break\n        # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res)\n        # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\n    state = []  # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5c0d nums \u9032\u884c\u6392\u5e8f\n    start = 0  # \u8d70\u8a2a\u8d77\u59cb\u9ede\n    res = []  # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0;                  // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        Backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    start := 0              // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    res := make([][]int, 0) // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for i in choices.indices.dropFirst(start) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    let start = 0 // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    var res: [[Int]] = [] // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums, target) {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    const start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    const start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n    // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target - choices[i], choices, i, res);\n    // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n  int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    let start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    let mut res = Vec::new(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0;                           // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (i in start..&lt;choices.size) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfun subsetSumI(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort() // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    val start = 0 // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_i.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I ###\ndef backtrack(state, target, choices, start, res)\n  # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  # \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n  for i in start...choices.length\n    # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n    # \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n    break if target - choices[i] &lt; 0\n    # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n    state &lt;&lt; choices[i]\n    # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target - choices[i], choices, i, res)\n    # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.pop\n  end\nend\n\n### \u6c42\u89e3\u5b50\u96c6\u548c I ###\ndef subset_sum_i(nums, target)\n  state = [] # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  nums.sort! # \u5c0d nums \u9032\u884c\u6392\u5e8f\n  start = 0 # \u8d70\u8a2a\u8d77\u59cb\u9ede\n  res = [] # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> subset_sum_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumI}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 13-12 \u6240\u793a\u70ba\u5c07\u9663\u5217 \\([3, 4, 5]\\) \u548c\u76ee\u6a19\u5143\u7d20 \\(9\\) \u8f38\u5165\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u5f8c\u7684\u6574\u9ad4\u56de\u6eaf\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 13-12 \u00a0 \u5b50\u96c6\u548c I \u56de\u6eaf\u904e\u7a0b </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1332","title":"13.3.2 \u00a0 \u8003\u616e\u91cd\u8907\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u6b63\u6574\u6578\u9663\u5217 <code>nums</code> \u548c\u4e00\u500b\u76ee\u6a19\u6b63\u6574\u6578 <code>target</code> \uff0c\u8acb\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7d44\u5408\uff0c\u4f7f\u5f97\u7d44\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u65bc <code>target</code> \u3002\u7d66\u5b9a\u9663\u5217\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u6bcf\u500b\u5143\u7d20\u53ea\u53ef\u88ab\u9078\u64c7\u4e00\u6b21\u3002\u8acb\u4ee5\u4e32\u5217\u5f62\u5f0f\u8fd4\u56de\u9019\u4e9b\u7d44\u5408\uff0c\u4e32\u5217\u4e2d\u4e0d\u61c9\u5305\u542b\u91cd\u8907\u7d44\u5408\u3002</p> <p>\u76f8\u6bd4\u65bc\u4e0a\u984c\uff0c\u672c\u984c\u7684\u8f38\u5165\u9663\u5217\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u9019\u5f15\u5165\u4e86\u65b0\u7684\u554f\u984c\u3002\u4f8b\u5982\uff0c\u7d66\u5b9a\u9663\u5217 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6a19\u5143\u7d20 \\(9\\) \uff0c\u5247\u73fe\u6709\u7a0b\u5f0f\u78bc\u7684\u8f38\u51fa\u7d50\u679c\u70ba \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u51fa\u73fe\u4e86\u91cd\u8907\u5b50\u96c6\u3002</p> <p>\u9020\u6210\u9019\u7a2e\u91cd\u8907\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f2a\u4e2d\u88ab\u591a\u6b21\u9078\u64c7\u3002\u5728\u5716 13-13 \u4e2d\uff0c\u7b2c\u4e00\u8f2a\u5171\u6709\u4e09\u500b\u9078\u64c7\uff0c\u5176\u4e2d\u5169\u500b\u90fd\u70ba \\(4\\) \uff0c\u6703\u7522\u751f\u5169\u500b\u91cd\u8907\u7684\u641c\u5c0b\u5206\u652f\uff0c\u5f9e\u800c\u8f38\u51fa\u91cd\u8907\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f2a\u7684\u5169\u500b \\(4\\) \u4e5f\u6703\u7522\u751f\u91cd\u8907\u5b50\u96c6\u3002</p> <p></p> <p> \u5716 13-13 \u00a0 \u76f8\u7b49\u5143\u7d20\u5c0e\u81f4\u7684\u91cd\u8907\u5b50\u96c6 </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u70ba\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u6211\u5011\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f2a\u4e2d\u53ea\u80fd\u88ab\u9078\u64c7\u4e00\u6b21\u3002\u5be6\u73fe\u65b9\u5f0f\u6bd4\u8f03\u5de7\u5999\uff1a\u7531\u65bc\u9663\u5217\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u9130\u7684\u3002\u9019\u610f\u5473\u8457\u5728\u67d0\u8f2a\u9078\u64c7\u4e2d\uff0c\u82e5\u7576\u524d\u5143\u7d20\u8207\u5176\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u5247\u8aaa\u660e\u5b83\u5df2\u7d93\u88ab\u9078\u64c7\u904e\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u904e\u7576\u524d\u5143\u7d20\u3002</p> <p>\u8207\u6b64\u540c\u6642\uff0c\u672c\u984c\u898f\u5b9a\u6bcf\u500b\u9663\u5217\u5143\u7d20\u53ea\u80fd\u88ab\u9078\u64c7\u4e00\u6b21\u3002\u5e78\u904b\u7684\u662f\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u5229\u7528\u8b8a\u6578 <code>start</code> \u4f86\u6eff\u8db3\u8a72\u7d04\u675f\uff1a\u7576\u505a\u51fa\u9078\u64c7 \\(x_{i}\\) \u5f8c\uff0c\u8a2d\u5b9a\u4e0b\u4e00\u8f2a\u5f9e\u7d22\u5f15 \\(i + 1\\) \u958b\u59cb\u5411\u5f8c\u8d70\u8a2a\u3002\u9019\u6a23\u65e2\u80fd\u53bb\u9664\u91cd\u8907\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u8907\u9078\u64c7\u5143\u7d20\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#2_1","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    # \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    # \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        # \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0:\n            break\n        # \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\n    state = []  # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5c0d nums \u9032\u884c\u6392\u5e8f\n    start = 0  # \u8d70\u8a2a\u8d77\u59cb\u9ede\n    res = []  # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0;                  // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    start := 0              // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    res := make([][]int, 0) // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for i in choices.indices.dropFirst(start) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    let start = 0 // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    var res: [[Int]] = [] // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums, target) {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    const start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    const start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n  // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n    // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n  int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    let start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    let mut res = Vec::new(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u8df3\u904e\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // \u958b\u59cb\u56de\u6eaf\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (i in start..&lt;choices.size) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfun subsetSumII(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort() // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    val start = 0 // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_ii.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II ###\ndef backtrack(state, target, choices, start, res)\n  # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  # \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n  # \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n  for i in start...choices.length\n    # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n    # \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n    break if target - choices[i] &lt; 0\n    # \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n    next if i &gt; start &amp;&amp; choices[i] == choices[i - 1]\n    # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n    state &lt;&lt; choices[i]\n    # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target - choices[i], choices, i + 1, res)\n    # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.pop\n  end\nend\n\n### \u6c42\u89e3\u5b50\u96c6\u548c II ###\ndef subset_sum_ii(nums, target)\n  state = [] # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  nums.sort! # \u5c0d nums \u9032\u884c\u6392\u5e8f\n  start = 0 # \u8d70\u8a2a\u8d77\u59cb\u9ede\n  res = [] # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> subset_sum_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumII}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 13-14 \u5c55\u793a\u4e86\u9663\u5217 \\([4, 4, 5]\\) \u548c\u76ee\u6a19\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u904e\u7a0b\uff0c\u5171\u5305\u542b\u56db\u7a2e\u526a\u679d\u64cd\u4f5c\u3002\u8acb\u4f60\u5c07\u5716\u793a\u8207\u7a0b\u5f0f\u78bc\u8a3b\u91cb\u76f8\u7d50\u5408\uff0c\u7406\u89e3\u6574\u500b\u641c\u5c0b\u904e\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u7a2e\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002</p> <p></p> <p> \u5716 13-14 \u00a0 \u5b50\u96c6\u548c II \u56de\u6eaf\u904e\u7a0b </p>"},{"location":"chapter_backtracking/summary/","title":"13.5 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_backtracking/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u672c\u8cea\u662f\u7aae\u8209\u6cd5\uff0c\u900f\u904e\u5c0d\u89e3\u7a7a\u9593\u9032\u884c\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u4f86\u5c0b\u627e\u7b26\u5408\u689d\u4ef6\u7684\u89e3\u3002\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\uff0c\u9047\u5230\u6eff\u8db3\u689d\u4ef6\u7684\u89e3\u5247\u8a18\u9304\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u8d70\u8a2a\u5b8c\u6210\u5f8c\u7d50\u675f\u3002</li> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u641c\u5c0b\u904e\u7a0b\u5305\u62ec\u5617\u8a66\u8207\u56de\u9000\u5169\u500b\u90e8\u5206\u3002\u5b83\u900f\u904e\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u4f86\u5617\u8a66\u5404\u7a2e\u9078\u64c7\uff0c\u7576\u9047\u5230\u4e0d\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\u7684\u60c5\u6cc1\u6642\uff0c\u5247\u64a4\u92b7\u4e0a\u4e00\u6b65\u7684\u9078\u64c7\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72c0\u614b\uff0c\u4e26\u7e7c\u7e8c\u5617\u8a66\u5176\u4ed6\u9078\u64c7\u3002\u5617\u8a66\u8207\u56de\u9000\u662f\u5169\u500b\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002</li> <li>\u56de\u6eaf\u554f\u984c\u901a\u5e38\u5305\u542b\u591a\u500b\u7d04\u675f\u689d\u4ef6\uff0c\u5b83\u5011\u53ef\u7528\u65bc\u5be6\u73fe\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7d50\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u5c0b\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u5c0b\u6548\u7387\u3002</li> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u65bc\u89e3\u6c7a\u641c\u5c0b\u554f\u984c\u548c\u7d04\u675f\u6eff\u8db3\u554f\u984c\u3002\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\u96d6\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u6f14\u7b97\u6cd5\u89e3\u6c7a\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u6548\u7387\u66f4\u9ad8\u6216\u6548\u679c\u66f4\u597d\u7684\u89e3\u6cd5\u3002</li> <li>\u5168\u6392\u5217\u554f\u984c\u65e8\u5728\u641c\u5c0b\u7d66\u5b9a\u96c6\u5408\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u5011\u85c9\u52a9\u4e00\u500b\u9663\u5217\u4f86\u8a18\u9304\u6bcf\u500b\u5143\u7d20\u662f\u5426\u88ab\u9078\u64c7\uff0c\u526a\u6389\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\u7684\u641c\u5c0b\u5206\u652f\uff0c\u78ba\u4fdd\u6bcf\u500b\u5143\u7d20\u53ea\u88ab\u9078\u64c7\u4e00\u6b21\u3002</li> <li>\u5728\u5168\u6392\u5217\u554f\u984c\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff0c\u5247\u6700\u7d42\u7d50\u679c\u6703\u51fa\u73fe\u91cd\u8907\u6392\u5217\u3002\u6211\u5011\u9700\u8981\u7d04\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f2a\u4e2d\u53ea\u80fd\u88ab\u9078\u64c7\u4e00\u6b21\uff0c\u9019\u901a\u5e38\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u96c6\u5408\u4f86\u5be6\u73fe\u3002</li> <li>\u5b50\u96c6\u548c\u554f\u984c\u7684\u76ee\u6a19\u662f\u5728\u7d66\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u70ba\u76ee\u6a19\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u5340\u5206\u5143\u7d20\u9806\u5e8f\uff0c\u800c\u641c\u5c0b\u904e\u7a0b\u6703\u8f38\u51fa\u6240\u6709\u9806\u5e8f\u7684\u7d50\u679c\uff0c\u7522\u751f\u91cd\u8907\u5b50\u96c6\u3002\u6211\u5011\u5728\u56de\u6eaf\u524d\u5c07\u8cc7\u6599\u9032\u884c\u6392\u5e8f\uff0c\u4e26\u8a2d\u5b9a\u4e00\u500b\u8b8a\u6578\u4f86\u6307\u793a\u6bcf\u4e00\u8f2a\u7684\u8d70\u8a2a\u8d77\u59cb\u9ede\uff0c\u5f9e\u800c\u5c07\u751f\u6210\u91cd\u8907\u5b50\u96c6\u7684\u641c\u5c0b\u5206\u652f\u9032\u884c\u526a\u679d\u3002</li> <li>\u5c0d\u65bc\u5b50\u96c6\u548c\u554f\u984c\uff0c\u9663\u5217\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u6703\u7522\u751f\u91cd\u8907\u96c6\u5408\u3002\u6211\u5011\u5229\u7528\u9663\u5217\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u689d\u4ef6\uff0c\u900f\u904e\u5224\u65b7\u76f8\u9130\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5be6\u73fe\u526a\u679d\uff0c\u5f9e\u800c\u78ba\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f2a\u4e2d\u53ea\u80fd\u88ab\u9078\u4e2d\u4e00\u6b21\u3002</li> <li>\\(n\\) \u7687\u540e\u554f\u984c\u65e8\u5728\u5c0b\u627e\u5c07 \\(n\\) \u500b\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76e4\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u5169\u5169\u4e4b\u9593\u7121\u6cd5\u653b\u64ca\u5c0d\u65b9\u3002\u8a72\u554f\u984c\u7684\u7d04\u675f\u689d\u4ef6\u6709\u884c\u7d04\u675f\u3001\u5217\u7d04\u675f\u3001\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\u7d04\u675f\u3002\u70ba\u6eff\u8db3\u884c\u7d04\u675f\uff0c\u6211\u5011\u63a1\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8b49\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u500b\u7687\u540e\u3002</li> <li>\u5217\u7d04\u675f\u548c\u5c0d\u89d2\u7dda\u7d04\u675f\u7684\u8655\u7406\u65b9\u5f0f\u985e\u4f3c\u3002\u5c0d\u65bc\u5217\u7d04\u675f\uff0c\u6211\u5011\u5229\u7528\u4e00\u500b\u9663\u5217\u4f86\u8a18\u9304\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u5f9e\u800c\u6307\u793a\u9078\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5c0d\u65bc\u5c0d\u89d2\u7dda\u7d04\u675f\uff0c\u6211\u5011\u85c9\u52a9\u5169\u500b\u9663\u5217\u4f86\u5206\u5225\u8a18\u9304\u8a72\u4e3b\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96e3\u9ede\u5728\u65bc\u627e\u8655\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5c0d\u89d2\u7dda\u4e0a\u683c\u5b50\u6eff\u8db3\u7684\u884c\u5217\u7d22\u5f15\u898f\u5f8b\u3002</li> </ul>"},{"location":"chapter_backtracking/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u600e\u9ebc\u7406\u89e3\u56de\u6eaf\u548c\u905e\u8ff4\u7684\u95dc\u4fc2\uff1f</p> <p>\u7e3d\u7684\u4f86\u770b\uff0c\u56de\u6eaf\u662f\u4e00\u7a2e\u201c\u6f14\u7b97\u6cd5\u7b56\u7565\u201d\uff0c\u800c\u905e\u8ff4\u66f4\u50cf\u662f\u4e00\u500b\u201c\u5de5\u5177\u201d\u3002</p> <ul> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u5be6\u73fe\u3002\u7136\u800c\uff0c\u56de\u6eaf\u662f\u905e\u8ff4\u7684\u61c9\u7528\u5834\u666f\u4e4b\u4e00\uff0c\u662f\u905e\u8ff4\u5728\u641c\u5c0b\u554f\u984c\u4e2d\u7684\u61c9\u7528\u3002</li> <li>\u905e\u8ff4\u7684\u7d50\u69cb\u9ad4\u73fe\u4e86\u201c\u5b50\u554f\u984c\u5206\u89e3\u201d\u7684\u89e3\u984c\u7bc4\u5f0f\uff0c\u5e38\u7528\u65bc\u89e3\u6c7a\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52d5\u614b\u898f\u5283\uff08\u8a18\u61b6\u5316\u905e\u8ff4\uff09\u7b49\u554f\u984c\u3002</li> </ul>"},{"location":"chapter_computational_complexity/","title":"\u7b2c 2 \u7ae0 \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<p>Abstract</p> <p>\u8907\u96dc\u5ea6\u5206\u6790\u7336\u5982\u6d69\u701a\u7684\u6f14\u7b97\u6cd5\u5b87\u5b99\u4e2d\u7684\u6642\u7a7a\u56ae\u5c0e\u3002</p> <p>\u5b83\u5e36\u9818\u6211\u5011\u5728\u6642\u9593\u8207\u7a7a\u9593\u9019\u5169\u500b\u7dad\u5ea6\u4e0a\u6df1\u5165\u63a2\u7d22\uff0c\u5c0b\u627e\u66f4\u512a\u96c5\u7684\u89e3\u6c7a\u65b9\u6848\u3002</p>"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>2.1 \u00a0 \u6f14\u7b97\u6cd5\u6548\u7387\u8a55\u4f30</li> <li>2.2 \u00a0 \u8fed\u4ee3\u8207\u905e\u8ff4</li> <li>2.3 \u00a0 \u6642\u9593\u8907\u96dc\u5ea6</li> <li>2.4 \u00a0 \u7a7a\u9593\u8907\u96dc\u5ea6</li> <li>2.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/","title":"2.2 \u00a0 \u8fed\u4ee3\u8207\u905e\u8ff4","text":"<p>\u5728\u6f14\u7b97\u6cd5\u4e2d\uff0c\u91cd\u8907\u57f7\u884c\u67d0\u500b\u4efb\u52d9\u662f\u5f88\u5e38\u898b\u7684\uff0c\u5b83\u8207\u8907\u96dc\u5ea6\u5206\u6790\u606f\u606f\u76f8\u95dc\u3002\u56e0\u6b64\uff0c\u5728\u4ecb\u7d39\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u4e4b\u524d\uff0c\u6211\u5011\u5148\u4f86\u4e86\u89e3\u5982\u4f55\u5728\u7a0b\u5f0f\u4e2d\u5be6\u73fe\u91cd\u8907\u57f7\u884c\u4efb\u52d9\uff0c\u5373\u5169\u7a2e\u57fa\u672c\u7684\u7a0b\u5f0f\u63a7\u5236\u7d50\u69cb\uff1a\u8fed\u4ee3\u3001\u905e\u8ff4\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","title":"2.2.1 \u00a0 \u8fed\u4ee3","text":"<p>\u8fed\u4ee3\uff08iteration\uff09\u662f\u4e00\u7a2e\u91cd\u8907\u57f7\u884c\u67d0\u500b\u4efb\u52d9\u7684\u63a7\u5236\u7d50\u69cb\u3002\u5728\u8fed\u4ee3\u4e2d\uff0c\u7a0b\u5f0f\u6703\u5728\u6eff\u8db3\u4e00\u5b9a\u7684\u689d\u4ef6\u4e0b\u91cd\u8907\u57f7\u884c\u67d0\u6bb5\u7a0b\u5f0f\u78bc\uff0c\u76f4\u5230\u9019\u500b\u689d\u4ef6\u4e0d\u518d\u6eff\u8db3\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","title":"1. \u00a0 for \u8ff4\u5708","text":"<p><code>for</code> \u8ff4\u5708\u662f\u6700\u5e38\u898b\u7684\u8fed\u4ee3\u5f62\u5f0f\u4e4b\u4e00\uff0c\u9069\u5408\u5728\u9810\u5148\u77e5\u9053\u8fed\u4ee3\u6b21\u6578\u6642\u4f7f\u7528\u3002</p> <p>\u4ee5\u4e0b\u51fd\u5f0f\u57fa\u65bc <code>for</code> \u8ff4\u5708\u5be6\u73fe\u4e86\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff0c\u6c42\u548c\u7d50\u679c\u4f7f\u7528\u8b8a\u6578 <code>res</code> \u8a18\u9304\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0cPython \u4e2d <code>range(a, b)</code> \u5c0d\u61c9\u7684\u5340\u9593\u662f\u201c\u5de6\u9589\u53f3\u958b\u201d\u7684\uff0c\u5c0d\u61c9\u7684\u8d70\u8a2a\u7bc4\u570d\u70ba \\(a, a + 1, \\dots, b-1\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for \u8ff4\u5708\"\"\"\n    res = 0\n    # \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for \u8ff4\u5708 */\nint forLoop(int n) {\n    int res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for \u8ff4\u5708 */\nint forLoop(int n) {\n    int res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for \u8ff4\u5708 */\nint ForLoop(int n) {\n    int res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for \u8ff4\u5708 */\nfunc forLoop(n int) int {\n    res := 0\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for \u8ff4\u5708 */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for \u8ff4\u5708 */\nfunction forLoop(n) {\n    let res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for \u8ff4\u5708 */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for \u8ff4\u5708 */\nint forLoop(int n) {\n  int res = 0;\n  // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for \u8ff4\u5708 */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for \u8ff4\u5708 */\nint forLoop(int n) {\n    int res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for \u8ff4\u5708 */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### for \u8ff4\u5708 ###\ndef for_loop(n)\n  res = 0\n\n  # \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n  for i in 1..n\n    res += i\n  end\n\n  res\nend\n</code></pre> iteration.zig<pre><code>// for \u8ff4\u5708\nfn forLoop(n: usize) i32 {\n    var res: i32 = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        res = res + @as(i32, @intCast(i));\n    }\n    return res;\n} \n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-1 \u662f\u8a72\u6c42\u548c\u51fd\u5f0f\u7684\u6d41\u7a0b\u6846\u5716\u3002</p> <p></p> <p> \u5716 2-1 \u00a0 \u6c42\u548c\u51fd\u5f0f\u7684\u6d41\u7a0b\u6846\u5716 </p> <p>\u6b64\u6c42\u548c\u51fd\u5f0f\u7684\u64cd\u4f5c\u6578\u91cf\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8aaa\u6210\u201c\u7dda\u6027\u95dc\u4fc2\u201d\u3002\u5be6\u969b\u4e0a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u63cf\u8ff0\u7684\u5c31\u662f\u9019\u500b\u201c\u7dda\u6027\u95dc\u4fc2\u201d\u3002\u76f8\u95dc\u5167\u5bb9\u5c07\u6703\u5728\u4e0b\u4e00\u7bc0\u4e2d\u8a73\u7d30\u4ecb\u7d39\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","title":"2. \u00a0 while \u8ff4\u5708","text":"<p>\u8207 <code>for</code> \u8ff4\u5708\u985e\u4f3c\uff0c<code>while</code> \u8ff4\u5708\u4e5f\u662f\u4e00\u7a2e\u5be6\u73fe\u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5728 <code>while</code> \u8ff4\u5708\u4e2d\uff0c\u7a0b\u5f0f\u6bcf\u8f2a\u90fd\u6703\u5148\u6aa2\u67e5\u689d\u4ef6\uff0c\u5982\u679c\u689d\u4ef6\u70ba\u771f\uff0c\u5247\u7e7c\u7e8c\u57f7\u884c\uff0c\u5426\u5247\u5c31\u7d50\u675f\u8ff4\u5708\u3002</p> <p>\u4e0b\u9762\u6211\u5011\u7528 <code>while</code> \u8ff4\u5708\u4f86\u5be6\u73fe\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while \u8ff4\u5708\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    # \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n:\n        res += i\n        i += 1  # \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u8ff4\u5708 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u8ff4\u5708 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u8ff4\u5708 */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u8ff4\u5708 */\nfunc whileLoop(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    i := 1\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u8ff4\u5708 */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i\n        i += 1 // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u8ff4\u5708 */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u8ff4\u5708 */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u8ff4\u5708 */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n  // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n  while (i &lt;= n) {\n    res += i;\n    i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u8ff4\u5708 */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i;\n        i += 1; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u8ff4\u5708 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u8ff4\u5708 */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i\n        i++ // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while \u8ff4\u5708 ###\ndef while_loop(n)\n  res = 0\n  i = 1 # \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n\n  # \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n  while i &lt;= n\n    res += i\n    i += 1 # \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n  end\n\n  res\nend\n</code></pre> iteration.zig<pre><code>// while \u8ff4\u5708\nfn whileLoop(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += @intCast(i);\n        i += 1;\n    }\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p><code>while</code> \u8ff4\u5708\u6bd4 <code>for</code> \u8ff4\u5708\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\u3002\u5728 <code>while</code> \u8ff4\u5708\u4e2d\uff0c\u6211\u5011\u53ef\u4ee5\u81ea\u7531\u5730\u8a2d\u8a08\u689d\u4ef6\u8b8a\u6578\u7684\u521d\u59cb\u5316\u548c\u66f4\u65b0\u6b65\u9a5f\u3002</p> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u689d\u4ef6\u8b8a\u6578 \\(i\\) \u6bcf\u8f2a\u9032\u884c\u5169\u6b21\u66f4\u65b0\uff0c\u9019\u7a2e\u60c5\u6cc1\u5c31\u4e0d\u592a\u65b9\u4fbf\u7528 <code>for</code> \u8ff4\u5708\u5be6\u73fe\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    # \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n:\n        res += i\n        # \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    i := 1\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n  // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n  while (i &lt;= n) {\n    res += i;\n    // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09###\ndef while_loop_ii(n)\n  res = 0\n  i = 1 # \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n\n  # \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n  while i &lt;= n\n    res += i\n    # \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    i += 1\n    i *= 2\n  end\n\n  res\nend\n</code></pre> iteration.zig<pre><code>//  while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += @intCast(i);\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u7e3d\u7684\u4f86\u8aaa\uff0c<code>for</code> \u8ff4\u5708\u7684\u7a0b\u5f0f\u78bc\u66f4\u52a0\u7dca\u6e4a\uff0c<code>while</code> \u8ff4\u5708\u66f4\u52a0\u9748\u6d3b\uff0c\u5169\u8005\u90fd\u53ef\u4ee5\u5be6\u73fe\u8fed\u4ee3\u7d50\u69cb\u3002\u9078\u64c7\u4f7f\u7528\u54ea\u4e00\u500b\u61c9\u8a72\u6839\u64da\u7279\u5b9a\u554f\u984c\u7684\u9700\u6c42\u4f86\u6c7a\u5b9a\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","title":"3. \u00a0 \u5de2\u72c0\u8ff4\u5708","text":"<p>\u6211\u5011\u53ef\u4ee5\u5728\u4e00\u500b\u8ff4\u5708\u7d50\u69cb\u5167\u5de2\u72c0\u53e6\u4e00\u500b\u8ff4\u5708\u7d50\u69cb\uff0c\u4e0b\u9762\u4ee5 <code>for</code> \u8ff4\u5708\u70ba\u4f8b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"\u96d9\u5c64 for \u8ff4\u5708\"\"\"\n    res = \"\"\n    # \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfunction nestedForLoop(n) {\n    let res = '';\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nchar *nestedForLoop(int n) {\n    // n * n \u70ba\u5c0d\u61c9\u9ede\u6578\u91cf\uff0c\"(i, j), \" \u5c0d\u61c9\u5b57\u4e32\u9577\u6700\u5927\u70ba 6+10*2\uff0c\u52a0\u4e0a\u6700\u5f8c\u4e00\u500b\u7a7a\u5b57\u5143 \\0 \u7684\u984d\u5916\u7a7a\u9593\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>### \u96d9\u5c64 for \u8ff4\u5708 ###\ndef nested_for_loop(n)\n  res = \"\"\n\n  # \u8ff4\u5708 i = 1, 2, ..., n-1, n\n  for i in 1..n\n    # \u8ff4\u5708 j = 1, 2, ..., n-1, n\n    for j in 1..n\n      res += \"(#{i}, #{j}), \"\n    end\n  end\n\n  res\nend\n</code></pre> iteration.zig<pre><code>// \u96d9\u5c64 for \u8ff4\u5708\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n    var res = std.ArrayList(u8).init(allocator);\n    defer res.deinit();\n    var buffer: [20]u8 = undefined;\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (1..n+1) |j| {\n            var _str = try std.fmt.bufPrint(&amp;buffer, \"({d}, {d}), \", .{i, j});\n            try res.appendSlice(_str);\n        }\n    }\n    return res.toOwnedSlice();\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-2 \u662f\u8a72\u5de2\u72c0\u8ff4\u5708\u7684\u6d41\u7a0b\u6846\u5716\u3002</p> <p></p> <p> \u5716 2-2 \u00a0 \u5de2\u72c0\u8ff4\u5708\u7684\u6d41\u7a0b\u6846\u5716 </p> <p>\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u51fd\u5f0f\u7684\u64cd\u4f5c\u6578\u91cf\u8207 \\(n^2\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8aaa\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u548c\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u6210\u201c\u5e73\u65b9\u95dc\u4fc2\u201d\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u7e7c\u7e8c\u65b0\u589e\u5de2\u72c0\u8ff4\u5708\uff0c\u6bcf\u4e00\u6b21\u5de2\u72c0\u90fd\u662f\u4e00\u6b21\u201c\u5347\u7dad\u201d\uff0c\u5c07\u6703\u4f7f\u6642\u9593\u8907\u96dc\u5ea6\u63d0\u9ad8\u81f3\u201c\u7acb\u65b9\u95dc\u4fc2\u201d\u201c\u56db\u6b21\u65b9\u95dc\u4fc2\u201d\uff0c\u4ee5\u6b64\u985e\u63a8\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","title":"2.2.2 \u00a0 \u905e\u8ff4","text":"<p>\u905e\u8ff4\uff08recursion\uff09\u662f\u4e00\u7a2e\u6f14\u7b97\u6cd5\u7b56\u7565\uff0c\u900f\u904e\u51fd\u5f0f\u547c\u53eb\u81ea\u8eab\u4f86\u89e3\u6c7a\u554f\u984c\u3002\u5b83\u4e3b\u8981\u5305\u542b\u5169\u500b\u968e\u6bb5\u3002</p> <ol> <li>\u905e\uff1a\u7a0b\u5f0f\u4e0d\u65b7\u6df1\u5165\u5730\u547c\u53eb\u81ea\u8eab\uff0c\u901a\u5e38\u50b3\u5165\u66f4\u5c0f\u6216\u66f4\u7c21\u5316\u7684\u53c3\u6578\uff0c\u76f4\u5230\u9054\u5230\u201c\u7d42\u6b62\u689d\u4ef6\u201d\u3002</li> <li>\u8ff4\uff1a\u89f8\u767c\u201c\u7d42\u6b62\u689d\u4ef6\u201d\u5f8c\uff0c\u7a0b\u5f0f\u5f9e\u6700\u6df1\u5c64\u7684\u905e\u8ff4\u51fd\u5f0f\u958b\u59cb\u9010\u5c64\u8fd4\u56de\uff0c\u532f\u805a\u6bcf\u4e00\u5c64\u7684\u7d50\u679c\u3002</li> </ol> <p>\u800c\u5f9e\u5be6\u73fe\u7684\u89d2\u5ea6\u770b\uff0c\u905e\u8ff4\u7a0b\u5f0f\u78bc\u4e3b\u8981\u5305\u542b\u4e09\u500b\u8981\u7d20\u3002</p> <ol> <li>\u7d42\u6b62\u689d\u4ef6\uff1a\u7528\u65bc\u6c7a\u5b9a\u4ec0\u9ebc\u6642\u5019\u7531\u201c\u905e\u201d\u8f49\u201c\u8ff4\u201d\u3002</li> <li>\u905e\u8ff4\u547c\u53eb\uff1a\u5c0d\u61c9\u201c\u905e\u201d\uff0c\u51fd\u5f0f\u547c\u53eb\u81ea\u8eab\uff0c\u901a\u5e38\u8f38\u5165\u66f4\u5c0f\u6216\u66f4\u7c21\u5316\u7684\u53c3\u6578\u3002</li> <li>\u8fd4\u56de\u7d50\u679c\uff1a\u5c0d\u61c9\u201c\u8ff4\u201d\uff0c\u5c07\u7576\u524d\u905e\u8ff4\u5c64\u7d1a\u7684\u7d50\u679c\u8fd4\u56de\u81f3\u4e0a\u4e00\u5c64\u3002</li> </ol> <p>\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u53ea\u9700\u547c\u53eb\u51fd\u5f0f <code>recur(n)</code>  \uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210 \\(1 + 2 + \\dots + n\\) \u7684\u8a08\u7b97\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"\u905e\u8ff4\"\"\"\n    # \u7d42\u6b62\u689d\u4ef6\n    if n == 1:\n        return 1\n    # \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    res = recur(n - 1)\n    # \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* \u905e\u8ff4 */\nint recur(int n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    int res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* \u905e\u8ff4 */\nint recur(int n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    int res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* \u905e\u8ff4 */\nint Recur(int n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    int res = Recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* \u905e\u8ff4 */\nfunc recur(n int) int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    res := recur(n - 1)\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* \u905e\u8ff4 */\nfunc recur(n: Int) -&gt; Int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    let res = recur(n: n - 1)\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* \u905e\u8ff4 */\nfunction recur(n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n === 1) return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    const res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* \u905e\u8ff4 */\nfunction recur(n: number): number {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n === 1) return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    const res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* \u905e\u8ff4 */\nint recur(int n) {\n  // \u7d42\u6b62\u689d\u4ef6\n  if (n == 1) return 1;\n  // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n  int res = recur(n - 1);\n  // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* \u905e\u8ff4 */\nfn recur(n: i32) -&gt; i32 {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 1 {\n        return 1;\n    }\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    let res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* \u905e\u8ff4 */\nint recur(int n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    int res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* \u905e\u8ff4 */\nfun recur(n: Int): Int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1\n    // \u905e: \u905e\u8ff4\u547c\u53eb\n    val res = recur(n - 1)\n    // \u8ff4: \u8fd4\u56de\u7d50\u679c\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>### \u905e\u8ff4 ###\ndef recur(n)\n  # \u7d42\u6b62\u689d\u4ef6\n  return 1 if n == 1\n  # \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n  res = recur(n - 1)\n  # \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n  n + res\nend\n</code></pre> recursion.zig<pre><code>// \u905e\u8ff4\u51fd\u5f0f\nfn recur(n: i32) i32 {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1) {\n        return 1;\n    }\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    var res: i32 = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-3 \u5c55\u793a\u4e86\u8a72\u51fd\u5f0f\u7684\u905e\u8ff4\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 2-3 \u00a0 \u6c42\u548c\u51fd\u5f0f\u7684\u905e\u8ff4\u904e\u7a0b </p> <p>\u96d6\u7136\u5f9e\u8a08\u7b97\u89d2\u5ea6\u770b\uff0c\u8fed\u4ee3\u8207\u905e\u8ff4\u53ef\u4ee5\u5f97\u5230\u76f8\u540c\u7684\u7d50\u679c\uff0c\u4f46\u5b83\u5011\u4ee3\u8868\u4e86\u5169\u7a2e\u5b8c\u5168\u4e0d\u540c\u7684\u601d\u8003\u548c\u89e3\u6c7a\u554f\u984c\u7684\u7bc4\u5f0f\u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u5730\u89e3\u6c7a\u554f\u984c\u3002\u5f9e\u6700\u57fa\u790e\u7684\u6b65\u9a5f\u958b\u59cb\uff0c\u7136\u5f8c\u4e0d\u65b7\u91cd\u8907\u6216\u7d2f\u52a0\u9019\u4e9b\u6b65\u9a5f\uff0c\u76f4\u5230\u4efb\u52d9\u5b8c\u6210\u3002</li> <li>\u905e\u8ff4\uff1a\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u5730\u89e3\u6c7a\u554f\u984c\u3002\u5c07\u539f\u554f\u984c\u5206\u89e3\u70ba\u66f4\u5c0f\u7684\u5b50\u554f\u984c\uff0c\u9019\u4e9b\u5b50\u554f\u984c\u548c\u539f\u554f\u984c\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\u3002\u63a5\u4e0b\u4f86\u5c07\u5b50\u554f\u984c\u7e7c\u7e8c\u5206\u89e3\u70ba\u66f4\u5c0f\u7684\u5b50\u554f\u984c\uff0c\u76f4\u5230\u57fa\u672c\u60c5\u6cc1\u6642\u505c\u6b62\uff08\u57fa\u672c\u60c5\u6cc1\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff09\u3002</li> </ul> <p>\u4ee5\u4e0a\u8ff0\u6c42\u548c\u51fd\u5f0f\u70ba\u4f8b\uff0c\u8a2d\u554f\u984c \\(f(n) = 1 + 2 + \\dots + n\\) \u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u5728\u8ff4\u5708\u4e2d\u6a21\u64ec\u6c42\u548c\u904e\u7a0b\uff0c\u5f9e \\(1\\) \u8d70\u8a2a\u5230 \\(n\\) \uff0c\u6bcf\u8f2a\u57f7\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5373\u53ef\u6c42\u5f97 \\(f(n)\\) \u3002</li> <li>\u905e\u8ff4\uff1a\u5c07\u554f\u984c\u5206\u89e3\u70ba\u5b50\u554f\u984c \\(f(n) = n + f(n-1)\\) \uff0c\u4e0d\u65b7\uff08\u905e\u8ff4\u5730\uff09\u5206\u89e3\u4e0b\u53bb\uff0c\u76f4\u81f3\u57fa\u672c\u60c5\u6cc1 \\(f(1) = 1\\) \u6642\u7d42\u6b62\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1","title":"1. \u00a0 \u547c\u53eb\u5806\u758a","text":"<p>\u905e\u8ff4\u51fd\u5f0f\u6bcf\u6b21\u547c\u53eb\u81ea\u8eab\u6642\uff0c\u7cfb\u7d71\u90fd\u6703\u70ba\u65b0\u958b\u555f\u7684\u51fd\u5f0f\u5206\u914d\u8a18\u61b6\u9ad4\uff0c\u4ee5\u5132\u5b58\u5340\u57df\u6027\u8b8a\u6578\u3001\u547c\u53eb\u4f4d\u5740\u548c\u5176\u4ed6\u8cc7\u8a0a\u7b49\u3002\u9019\u5c07\u5c0e\u81f4\u5169\u65b9\u9762\u7684\u7d50\u679c\u3002</p> <ul> <li>\u51fd\u5f0f\u7684\u4e0a\u4e0b\u6587\u8cc7\u6599\u90fd\u5132\u5b58\u5728\u7a31\u70ba\u201c\u5806\u758a\u5e40\u7a7a\u9593\u201d\u7684\u8a18\u61b6\u9ad4\u5340\u57df\u4e2d\uff0c\u76f4\u81f3\u51fd\u5f0f\u8fd4\u56de\u5f8c\u624d\u6703\u88ab\u91cb\u653e\u3002\u56e0\u6b64\uff0c\u905e\u8ff4\u901a\u5e38\u6bd4\u8fed\u4ee3\u66f4\u52a0\u8017\u8cbb\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> <li>\u905e\u8ff4\u547c\u53eb\u51fd\u5f0f\u6703\u7522\u751f\u984d\u5916\u7684\u958b\u92b7\u3002\u56e0\u6b64\u905e\u8ff4\u901a\u5e38\u6bd4\u8ff4\u5708\u7684\u6642\u9593\u6548\u7387\u66f4\u4f4e\u3002</li> </ul> <p>\u5982\u5716 2-4 \u6240\u793a\uff0c\u5728\u89f8\u767c\u7d42\u6b62\u689d\u4ef6\u524d\uff0c\u540c\u6642\u5b58\u5728 \\(n\\) \u500b\u672a\u8fd4\u56de\u7684\u905e\u8ff4\u51fd\u5f0f\uff0c\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \u3002</p> <p></p> <p> \u5716 2-4 \u00a0 \u905e\u8ff4\u547c\u53eb\u6df1\u5ea6 </p> <p>\u5728\u5be6\u969b\u4e2d\uff0c\u7a0b\u5f0f\u8a9e\u8a00\u5141\u8a31\u7684\u905e\u8ff4\u6df1\u5ea6\u901a\u5e38\u662f\u6709\u9650\u7684\uff0c\u904e\u6df1\u7684\u905e\u8ff4\u53ef\u80fd\u5c0e\u81f4\u5806\u758a\u6ea2\u4f4d\u932f\u8aa4\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","title":"2. \u00a0 \u5c3e\u905e\u8ff4","text":"<p>\u6709\u8da3\u7684\u662f\uff0c\u5982\u679c\u51fd\u5f0f\u5728\u8fd4\u56de\u524d\u7684\u6700\u5f8c\u4e00\u6b65\u624d\u9032\u884c\u905e\u8ff4\u547c\u53eb\uff0c\u5247\u8a72\u51fd\u5f0f\u53ef\u4ee5\u88ab\u7de8\u8b6f\u5668\u6216\u76f4\u8b6f\u5668\u6700\u4f73\u5316\uff0c\u4f7f\u5176\u5728\u7a7a\u9593\u6548\u7387\u4e0a\u8207\u8fed\u4ee3\u76f8\u7576\u3002\u9019\u7a2e\u60c5\u6cc1\u88ab\u7a31\u70ba\u5c3e\u905e\u8ff4\uff08tail recursion\uff09\u3002</p> <ul> <li>\u666e\u901a\u905e\u8ff4\uff1a\u7576\u51fd\u5f0f\u8fd4\u56de\u5230\u4e0a\u4e00\u5c64\u7d1a\u7684\u51fd\u5f0f\u5f8c\uff0c\u9700\u8981\u7e7c\u7e8c\u57f7\u884c\u7a0b\u5f0f\u78bc\uff0c\u56e0\u6b64\u7cfb\u7d71\u9700\u8981\u5132\u5b58\u4e0a\u4e00\u5c64\u547c\u53eb\u7684\u4e0a\u4e0b\u6587\u3002</li> <li>\u5c3e\u905e\u8ff4\uff1a\u905e\u8ff4\u547c\u53eb\u662f\u51fd\u5f0f\u8fd4\u56de\u524d\u7684\u6700\u5f8c\u4e00\u500b\u64cd\u4f5c\uff0c\u9019\u610f\u5473\u8457\u51fd\u5f0f\u8fd4\u56de\u5230\u4e0a\u4e00\u5c64\u7d1a\u5f8c\uff0c\u7121\u9808\u7e7c\u7e8c\u57f7\u884c\u5176\u4ed6\u64cd\u4f5c\uff0c\u56e0\u6b64\u7cfb\u7d71\u7121\u9808\u5132\u5b58\u4e0a\u4e00\u5c64\u51fd\u5f0f\u7684\u4e0a\u4e0b\u6587\u3002</li> </ul> <p>\u4ee5\u8a08\u7b97 \\(1 + 2 + \\dots + n\\) \u70ba\u4f8b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u7d50\u679c\u8b8a\u6578 <code>res</code> \u8a2d\u70ba\u51fd\u5f0f\u53c3\u6578\uff0c\u5f9e\u800c\u5be6\u73fe\u5c3e\u905e\u8ff4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"\u5c3e\u905e\u8ff4\"\"\"\n    # \u7d42\u6b62\u689d\u4ef6\n    if n == 0:\n        return res\n    # \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* \u5c3e\u905e\u8ff4 */\nint tailRecur(int n, int res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* \u5c3e\u905e\u8ff4 */\nint tailRecur(int n, int res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* \u5c3e\u905e\u8ff4 */\nint TailRecur(int n, int res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* \u5c3e\u905e\u8ff4 */\nfunc tailRecur(n int, res int) int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* \u5c3e\u905e\u8ff4 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* \u5c3e\u905e\u8ff4 */\nfunction tailRecur(n, res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* \u5c3e\u905e\u8ff4 */\nfunction tailRecur(n: number, res: number): number {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* \u5c3e\u905e\u8ff4 */\nint tailRecur(int n, int res) {\n  // \u7d42\u6b62\u689d\u4ef6\n  if (n == 0) return res;\n  // \u5c3e\u905e\u8ff4\u547c\u53eb\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* \u5c3e\u905e\u8ff4 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 0 {\n        return res;\n    }\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* \u5c3e\u905e\u8ff4 */\nint tailRecur(int n, int res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* \u5c3e\u905e\u8ff4 */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // \u65b0\u589e tailrec \u95dc\u9375\u8a5e\uff0c\u4ee5\u958b\u555f\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>### \u5c3e\u905e\u8ff4 ###\ndef tail_recur(n, res)\n  # \u7d42\u6b62\u689d\u4ef6\n  return res if n == 0\n  # \u5c3e\u905e\u8ff4\u547c\u53eb\n  tail_recur(n - 1, res + n)\nend\n</code></pre> recursion.zig<pre><code>// \u5c3e\u905e\u8ff4\u51fd\u5f0f\nfn tailRecur(n: i32, res: i32) i32 {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0) {\n        return res;\n    }\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5c3e\u905e\u8ff4\u7684\u57f7\u884c\u904e\u7a0b\u5982\u5716 2-5 \u6240\u793a\u3002\u5c0d\u6bd4\u666e\u901a\u905e\u8ff4\u548c\u5c3e\u905e\u8ff4\uff0c\u5169\u8005\u7684\u6c42\u548c\u64cd\u4f5c\u7684\u57f7\u884c\u9ede\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li>\u666e\u901a\u905e\u8ff4\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u8ff4\u201d\u7684\u904e\u7a0b\u4e2d\u57f7\u884c\u7684\uff0c\u6bcf\u5c64\u8fd4\u56de\u5f8c\u90fd\u8981\u518d\u57f7\u884c\u4e00\u6b21\u6c42\u548c\u64cd\u4f5c\u3002</li> <li>\u5c3e\u905e\u8ff4\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u905e\u201d\u7684\u904e\u7a0b\u4e2d\u57f7\u884c\u7684\uff0c\u201c\u8ff4\u201d\u7684\u904e\u7a0b\u53ea\u9700\u5c64\u5c64\u8fd4\u56de\u3002</li> </ul> <p></p> <p> \u5716 2-5 \u00a0 \u5c3e\u905e\u8ff4\u904e\u7a0b </p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u8a31\u591a\u7de8\u8b6f\u5668\u6216\u76f4\u8b6f\u5668\u4e26\u4e0d\u652f\u6301\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\u3002\u4f8b\u5982\uff0cPython \u9810\u8a2d\u4e0d\u652f\u6301\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff0c\u56e0\u6b64\u5373\u4f7f\u51fd\u5f0f\u662f\u5c3e\u905e\u8ff4\u5f62\u5f0f\uff0c\u4ecd\u7136\u53ef\u80fd\u6703\u9047\u5230\u5806\u758a\u6ea2\u4f4d\u554f\u984c\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","title":"3. \u00a0 \u905e\u8ff4\u6a39","text":"<p>\u7576\u8655\u7406\u8207\u201c\u5206\u6cbb\u201d\u76f8\u95dc\u7684\u6f14\u7b97\u6cd5\u554f\u984c\u6642\uff0c\u905e\u8ff4\u5f80\u5f80\u6bd4\u8fed\u4ee3\u7684\u601d\u8def\u66f4\u52a0\u76f4\u89c0\u3001\u7a0b\u5f0f\u78bc\u66f4\u52a0\u6613\u8b80\u3002\u4ee5\u201c\u8cbb\u6ce2\u90a3\u5951\u6578\u5217\u201d\u70ba\u4f8b\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u8cbb\u6ce2\u90a3\u5951\u6578\u5217 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) \uff0c\u6c42\u8a72\u6578\u5217\u7684\u7b2c \\(n\\) \u500b\u6578\u5b57\u3002</p> <p>\u8a2d\u8cbb\u6ce2\u90a3\u5951\u6578\u5217\u7684\u7b2c \\(n\\) \u500b\u6578\u5b57\u70ba \\(f(n)\\) \uff0c\u6613\u5f97\u5169\u500b\u7d50\u8ad6\u3002</p> <ul> <li>\u6578\u5217\u7684\u524d\u5169\u500b\u6578\u5b57\u70ba \\(f(1) = 0\\) \u548c \\(f(2) = 1\\) \u3002</li> <li>\u6578\u5217\u4e2d\u7684\u6bcf\u500b\u6578\u5b57\u662f\u524d\u5169\u500b\u6578\u5b57\u7684\u548c\uff0c\u5373 \\(f(n) = f(n - 1) + f(n - 2)\\) \u3002</li> </ul> <p>\u6309\u7167\u905e\u63a8\u95dc\u4fc2\u9032\u884c\u905e\u8ff4\u547c\u53eb\uff0c\u5c07\u524d\u5169\u500b\u6578\u5b57\u4f5c\u70ba\u7d42\u6b62\u689d\u4ef6\uff0c\u4fbf\u53ef\u5beb\u51fa\u905e\u8ff4\u7a0b\u5f0f\u78bc\u3002\u547c\u53eb <code>fib(n)</code> \u5373\u53ef\u5f97\u5230\u8cbb\u6ce2\u90a3\u5951\u6578\u5217\u7684\u7b2c \\(n\\) \u500b\u6578\u5b57\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"\u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4\"\"\"\n    # \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # \u8fd4\u56de\u7d50\u679c f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint fib(int n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint fib(int n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint Fib(int n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfunc fib(n int) int {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfunc fib(n: Int) -&gt; Int {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfunction fib(n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfunction fib(n: number): number {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint fib(int n) {\n  // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // \u8fd4\u56de\u7d50\u679c f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfn fib(n: i32) -&gt; i32 {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint fib(int n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfun fib(n: Int): Int {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res\n}\n</code></pre> recursion.rb<pre><code>### \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 ###\ndef fib(n)\n  # \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n  return n - 1 if n == 1 || n == 2\n  # \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n  res = fib(n - 1) + fib(n - 2)\n  # \u8fd4\u56de\u7d50\u679c f(n)\n  res\nend\n</code></pre> recursion.zig<pre><code>// \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\nfn fib(n: i32) i32 {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 or n == 2) {\n        return n - 1;\n    }\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    var res: i32 = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u5728\u51fd\u5f0f\u5167\u905e\u8ff4\u547c\u53eb\u4e86\u5169\u500b\u51fd\u5f0f\uff0c\u9019\u610f\u5473\u8457\u5f9e\u4e00\u500b\u547c\u53eb\u7522\u751f\u4e86\u5169\u500b\u547c\u53eb\u5206\u652f\u3002\u5982\u5716 2-6 \u6240\u793a\uff0c\u9019\u6a23\u4e0d\u65b7\u905e\u8ff4\u547c\u53eb\u4e0b\u53bb\uff0c\u6700\u7d42\u5c07\u7522\u751f\u4e00\u68f5\u5c64\u6578\u70ba \\(n\\) \u7684\u905e\u8ff4\u6a39\uff08recursion tree\uff09\u3002</p> <p></p> <p> \u5716 2-6 \u00a0 \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\u7684\u905e\u8ff4\u6a39 </p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u905e\u8ff4\u9ad4\u73fe\u4e86\u201c\u5c07\u554f\u984c\u5206\u89e3\u70ba\u66f4\u5c0f\u5b50\u554f\u984c\u201d\u7684\u601d\u7dad\u7bc4\u5f0f\uff0c\u9019\u7a2e\u5206\u6cbb\u7b56\u7565\u81f3\u95dc\u91cd\u8981\u3002</p> <ul> <li>\u5f9e\u6f14\u7b97\u6cd5\u89d2\u5ea6\u770b\uff0c\u641c\u5c0b\u3001\u6392\u5e8f\u3001\u56de\u6eaf\u3001\u5206\u6cbb\u3001\u52d5\u614b\u898f\u5283\u7b49\u8a31\u591a\u91cd\u8981\u6f14\u7b97\u6cd5\u7b56\u7565\u76f4\u63a5\u6216\u9593\u63a5\u5730\u61c9\u7528\u4e86\u9019\u7a2e\u601d\u7dad\u65b9\u5f0f\u3002</li> <li>\u5f9e\u8cc7\u6599\u7d50\u69cb\u89d2\u5ea6\u770b\uff0c\u905e\u8ff4\u5929\u7136\u9069\u5408\u8655\u7406\u93c8\u7d50\u4e32\u5217\u3001\u6a39\u548c\u5716\u7684\u76f8\u95dc\u554f\u984c\uff0c\u56e0\u70ba\u5b83\u5011\u975e\u5e38\u9069\u5408\u7528\u5206\u6cbb\u601d\u60f3\u9032\u884c\u5206\u6790\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#223","title":"2.2.3 \u00a0 \u5169\u8005\u5c0d\u6bd4","text":"<p>\u7e3d\u7d50\u4ee5\u4e0a\u5167\u5bb9\uff0c\u5982\u8868 2-1 \u6240\u793a\uff0c\u8fed\u4ee3\u548c\u905e\u8ff4\u5728\u5be6\u73fe\u3001\u6548\u80fd\u548c\u9069\u7528\u6027\u4e0a\u6709\u6240\u4e0d\u540c\u3002</p> <p> \u8868 2-1 \u00a0 \u8fed\u4ee3\u8207\u905e\u8ff4\u7279\u9ede\u5c0d\u6bd4 </p> \u8fed\u4ee3 \u905e\u8ff4 \u5be6\u73fe\u65b9\u5f0f \u8ff4\u5708\u7d50\u69cb \u51fd\u5f0f\u547c\u53eb\u81ea\u8eab \u6642\u9593\u6548\u7387 \u6548\u7387\u901a\u5e38\u8f03\u9ad8\uff0c\u7121\u51fd\u5f0f\u547c\u53eb\u958b\u92b7 \u6bcf\u6b21\u51fd\u5f0f\u547c\u53eb\u90fd\u6703\u7522\u751f\u958b\u92b7 \u8a18\u61b6\u9ad4\u4f7f\u7528 \u901a\u5e38\u4f7f\u7528\u56fa\u5b9a\u5927\u5c0f\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593 \u7d2f\u7a4d\u51fd\u5f0f\u547c\u53eb\u53ef\u80fd\u4f7f\u7528\u5927\u91cf\u7684\u5806\u758a\u5e40\u7a7a\u9593 \u9069\u7528\u554f\u984c \u9069\u7528\u65bc\u7c21\u55ae\u8ff4\u5708\u4efb\u52d9\uff0c\u7a0b\u5f0f\u78bc\u76f4\u89c0\u3001\u53ef\u8b80\u6027\u597d \u9069\u7528\u65bc\u5b50\u554f\u984c\u5206\u89e3\uff0c\u5982\u6a39\u3001\u5716\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u7b49\uff0c\u7a0b\u5f0f\u78bc\u7d50\u69cb\u7c21\u6f54\u3001\u6e05\u6670 <p>Tip</p> <p>\u5982\u679c\u611f\u89ba\u4ee5\u4e0b\u5167\u5bb9\u7406\u89e3\u56f0\u96e3\uff0c\u53ef\u4ee5\u5728\u8b80\u5b8c\u201c\u5806\u758a\u201d\u7ae0\u7bc0\u5f8c\u518d\u4f86\u8907\u7fd2\u3002</p> <p>\u90a3\u9ebc\uff0c\u8fed\u4ee3\u548c\u905e\u8ff4\u5177\u6709\u4ec0\u9ebc\u5167\u5728\u95dc\u806f\u5462\uff1f\u4ee5\u4e0a\u8ff0\u905e\u8ff4\u51fd\u5f0f\u70ba\u4f8b\uff0c\u6c42\u548c\u64cd\u4f5c\u5728\u905e\u8ff4\u7684\u201c\u8ff4\u201d\u968e\u6bb5\u9032\u884c\u3002\u9019\u610f\u5473\u8457\u6700\u521d\u88ab\u547c\u53eb\u7684\u51fd\u5f0f\u5be6\u969b\u4e0a\u662f\u6700\u5f8c\u5b8c\u6210\u5176\u6c42\u548c\u64cd\u4f5c\u7684\uff0c\u9019\u7a2e\u5de5\u4f5c\u6a5f\u5236\u8207\u5806\u758a\u7684\u201c\u5148\u5165\u5f8c\u51fa\u201d\u539f\u5247\u7570\u66f2\u540c\u5de5\u3002</p> <p>\u4e8b\u5be6\u4e0a\uff0c\u201c\u547c\u53eb\u5806\u758a\u201d\u548c\u201c\u5806\u758a\u5e40\u7a7a\u9593\u201d\u9019\u985e\u905e\u8ff4\u8853\u8a9e\u5df2\u7d93\u6697\u793a\u4e86\u905e\u8ff4\u8207\u5806\u758a\u4e4b\u9593\u7684\u5bc6\u5207\u95dc\u4fc2\u3002</p> <ol> <li>\u905e\uff1a\u7576\u51fd\u5f0f\u88ab\u547c\u53eb\u6642\uff0c\u7cfb\u7d71\u6703\u5728\u201c\u547c\u53eb\u5806\u758a\u201d\u4e0a\u70ba\u8a72\u51fd\u5f0f\u5206\u914d\u65b0\u7684\u5806\u758a\u5e40\uff0c\u7528\u65bc\u5132\u5b58\u51fd\u5f0f\u7684\u5340\u57df\u6027\u8b8a\u6578\u3001\u53c3\u6578\u3001\u8fd4\u56de\u4f4d\u5740\u7b49\u8cc7\u6599\u3002</li> <li>\u8ff4\uff1a\u7576\u51fd\u5f0f\u5b8c\u6210\u57f7\u884c\u4e26\u8fd4\u56de\u6642\uff0c\u5c0d\u61c9\u7684\u5806\u758a\u5e40\u6703\u88ab\u5f9e\u201c\u547c\u53eb\u5806\u758a\u201d\u4e0a\u79fb\u9664\uff0c\u6062\u5fa9\u4e4b\u524d\u51fd\u5f0f\u7684\u57f7\u884c\u74b0\u5883\u3002</li> </ol> <p>\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u547c\u53eb\u5806\u758a\u7684\u884c\u70ba\uff0c\u5f9e\u800c\u5c07\u905e\u8ff4\u8f49\u5316\u70ba\u8fed\u4ee3\u5f62\u5f0f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4\"\"\"\n    # \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    stack = []\n    res = 0\n    # \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for i in range(n, 0, -1):\n        # \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.append(i)\n    # \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while stack:\n        # \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (int i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (!stack.empty()) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (int i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (!stack.isEmpty()) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint ForLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (int i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.Push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (stack.Count &gt; 0) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfunc forLoopRecur(n int) int {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    stack := list.New()\n    res := 0\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for i := n; i &gt; 0; i-- {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.PushBack(i)\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    for stack.Len() != 0 {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    var stack: [Int] = []\n    var res = 0\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for i in (1 ... n).reversed() {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.append(i)\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while !stack.isEmpty {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfunction forLoopRecur(n) {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    const stack = [];\n    let res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (let i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (stack.length) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfunction forLoopRecur(n: number): number {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a \n    const stack: number[] = [];\n    let res: number = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (let i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (stack.length) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint forLoopRecur(int n) {\n  // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n  for (int i = n; i &gt; 0; i--) {\n    // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n    stack.add(i);\n  }\n  // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n  while (!stack.isEmpty) {\n    // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for i in (1..=n).rev() {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while !stack.is_empty() {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint forLoopRecur(int n) {\n    int stack[1000]; // \u85c9\u52a9\u4e00\u500b\u5927\u9663\u5217\u4f86\u6a21\u64ec\u5806\u758a\n    int top = -1;    // \u5806\u758a\u9802\u7d22\u5f15\n    int res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (int i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack[1 + top++] = i;\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (top &gt;= 0) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfun forLoopRecur(n: Int): Int {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // \u905e: \u905e\u8ff4\u547c\u53eb\n    for (i in n downTo 0) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i)\n    }\n    // \u8ff4: \u8fd4\u56de\u7d50\u679c\n    while (stack.isNotEmpty()) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>### \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 ###\ndef for_loop_recur(n)\n  # \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n  stack = []\n  res = 0\n\n  # \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n  for i in n.downto(0)\n    # \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n    stack &lt;&lt; i\n  end\n  # \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n  while !stack.empty?\n    res += stack.pop\n  end\n\n  # res = 1+2+3+...+n\n  res\nend\n</code></pre> recursion.zig<pre><code>// \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4\nfn forLoopRecur(comptime n: i32) i32 {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    var stack: [n]i32 = undefined;\n    var res: i32 = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    var i: usize = n;\n    while (i &gt; 0) {\n        stack[i - 1] = @intCast(i);\n        i -= 1;\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    var index: usize = n;\n    while (index &gt; 0) {\n        index -= 1;\n        res += stack[index];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\uff0c\u7576\u905e\u8ff4\u8f49\u5316\u70ba\u8fed\u4ee3\u5f8c\uff0c\u7a0b\u5f0f\u78bc\u8b8a\u5f97\u66f4\u52a0\u8907\u96dc\u4e86\u3002\u5118\u7ba1\u8fed\u4ee3\u548c\u905e\u8ff4\u5728\u5f88\u591a\u60c5\u6cc1\u4e0b\u53ef\u4ee5\u4e92\u76f8\u8f49\u5316\uff0c\u4f46\u4e0d\u4e00\u5b9a\u503c\u5f97\u9019\u6a23\u505a\uff0c\u6709\u4ee5\u4e0b\u5169\u9ede\u539f\u56e0\u3002</p> <ul> <li>\u8f49\u5316\u5f8c\u7684\u7a0b\u5f0f\u78bc\u53ef\u80fd\u66f4\u52a0\u96e3\u4ee5\u7406\u89e3\uff0c\u53ef\u8b80\u6027\u66f4\u5dee\u3002</li> <li>\u5c0d\u65bc\u67d0\u4e9b\u8907\u96dc\u554f\u984c\uff0c\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\u7684\u884c\u70ba\u53ef\u80fd\u975e\u5e38\u56f0\u96e3\u3002</li> </ul> <p>\u7e3d\u4e4b\uff0c\u9078\u64c7\u8fed\u4ee3\u9084\u662f\u905e\u8ff4\u53d6\u6c7a\u65bc\u7279\u5b9a\u554f\u984c\u7684\u6027\u8cea\u3002\u5728\u7a0b\u5f0f\u8a2d\u8a08\u5be6\u8e10\u4e2d\uff0c\u6b0a\u8861\u5169\u8005\u7684\u512a\u52a3\u4e26\u6839\u64da\u60c5\u5883\u9078\u64c7\u5408\u9069\u7684\u65b9\u6cd5\u81f3\u95dc\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 \u6f14\u7b97\u6cd5\u6548\u7387\u8a55\u4f30","text":"<p>\u5728\u6f14\u7b97\u6cd5\u8a2d\u8a08\u4e2d\uff0c\u6211\u5011\u5148\u5f8c\u8ffd\u6c42\u4ee5\u4e0b\u5169\u500b\u5c64\u9762\u7684\u76ee\u6a19\u3002</p> <ol> <li>\u627e\u5230\u554f\u984c\u89e3\u6cd5\uff1a\u6f14\u7b97\u6cd5\u9700\u8981\u5728\u898f\u5b9a\u7684\u8f38\u5165\u7bc4\u570d\u5167\u53ef\u9760\u5730\u6c42\u5f97\u554f\u984c\u7684\u6b63\u78ba\u89e3\u3002</li> <li>\u5c0b\u6c42\u6700\u512a\u89e3\u6cd5\uff1a\u540c\u4e00\u500b\u554f\u984c\u53ef\u80fd\u5b58\u5728\u591a\u7a2e\u89e3\u6cd5\uff0c\u6211\u5011\u5e0c\u671b\u627e\u5230\u5118\u53ef\u80fd\u9ad8\u6548\u7684\u6f14\u7b97\u6cd5\u3002</li> </ol> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u5728\u80fd\u5920\u89e3\u6c7a\u554f\u984c\u7684\u524d\u63d0\u4e0b\uff0c\u6f14\u7b97\u6cd5\u6548\u7387\u5df2\u6210\u70ba\u8861\u91cf\u6f14\u7b97\u6cd5\u512a\u52a3\u7684\u4e3b\u8981\u8a55\u50f9\u6307\u6a19\uff0c\u5b83\u5305\u62ec\u4ee5\u4e0b\u5169\u500b\u7dad\u5ea6\u3002</p> <ul> <li>\u6642\u9593\u6548\u7387\uff1a\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u7684\u9577\u77ed\u3002</li> <li>\u7a7a\u9593\u6548\u7387\uff1a\u6f14\u7b97\u6cd5\u4f54\u7528\u8a18\u61b6\u9ad4\u7a7a\u9593\u7684\u5927\u5c0f\u3002</li> </ul> <p>\u7c21\u800c\u8a00\u4e4b\uff0c\u6211\u5011\u7684\u76ee\u6a19\u662f\u8a2d\u8a08\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u3002\u800c\u6709\u6548\u5730\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\u81f3\u95dc\u91cd\u8981\uff0c\u56e0\u70ba\u53ea\u6709\u9019\u6a23\uff0c\u6211\u5011\u624d\u80fd\u5c07\u5404\u7a2e\u6f14\u7b97\u6cd5\u9032\u884c\u5c0d\u6bd4\uff0c\u9032\u800c\u6307\u5c0e\u6f14\u7b97\u6cd5\u8a2d\u8a08\u8207\u6700\u4f73\u5316\u904e\u7a0b\u3002</p> <p>\u6548\u7387\u8a55\u4f30\u65b9\u6cd5\u4e3b\u8981\u5206\u70ba\u5169\u7a2e\uff1a\u5be6\u969b\u6e2c\u8a66\u3001\u7406\u8ad6\u4f30\u7b97\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1 \u00a0 \u5be6\u969b\u6e2c\u8a66","text":"<p>\u5047\u8a2d\u6211\u5011\u73fe\u5728\u6709\u6f14\u7b97\u6cd5 <code>A</code> \u548c\u6f14\u7b97\u6cd5 <code>B</code> \uff0c\u5b83\u5011\u90fd\u80fd\u89e3\u6c7a\u540c\u4e00\u554f\u984c\uff0c\u73fe\u5728\u9700\u8981\u5c0d\u6bd4\u9019\u5169\u500b\u6f14\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u627e\u4e00\u81fa\u8a08\u7b97\u6a5f\uff0c\u57f7\u884c\u9019\u5169\u500b\u6f14\u7b97\u6cd5\uff0c\u4e26\u76e3\u63a7\u8a18\u9304\u5b83\u5011\u7684\u57f7\u884c\u6642\u9593\u548c\u8a18\u61b6\u9ad4\u4f54\u7528\u60c5\u6cc1\u3002\u9019\u7a2e\u8a55\u4f30\u65b9\u5f0f\u80fd\u5920\u53cd\u6620\u771f\u5be6\u60c5\u6cc1\uff0c\u4f46\u4e5f\u5b58\u5728\u8f03\u5927\u7684\u4fb7\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u96e3\u4ee5\u6392\u9664\u6e2c\u8a66\u74b0\u5883\u7684\u5e72\u64fe\u56e0\u7d20\u3002\u786c\u9ad4\u914d\u7f6e\u6703\u5f71\u97ff\u6f14\u7b97\u6cd5\u7684\u6548\u80fd\u8868\u73fe\u3002\u6bd4\u5982\u4e00\u500b\u6f14\u7b97\u6cd5\u7684\u4e26\u884c\u5ea6\u8f03\u9ad8\uff0c\u90a3\u9ebc\u5b83\u5c31\u66f4\u9069\u5408\u5728\u591a\u6838 CPU \u4e0a\u57f7\u884c\uff0c\u4e00\u500b\u6f14\u7b97\u6cd5\u7684\u8a18\u61b6\u9ad4\u64cd\u4f5c\u5bc6\u96c6\uff0c\u90a3\u9ebc\u5b83\u5728\u9ad8\u6548\u80fd\u8a18\u61b6\u9ad4\u4e0a\u7684\u8868\u73fe\u5c31\u6703\u66f4\u597d\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u6f14\u7b97\u6cd5\u5728\u4e0d\u540c\u7684\u6a5f\u5668\u4e0a\u7684\u6e2c\u8a66\u7d50\u679c\u53ef\u80fd\u662f\u4e0d\u4e00\u81f4\u7684\u3002\u9019\u610f\u5473\u8457\u6211\u5011\u9700\u8981\u5728\u5404\u7a2e\u6a5f\u5668\u4e0a\u9032\u884c\u6e2c\u8a66\uff0c\u7d71\u8a08\u5e73\u5747\u6548\u7387\uff0c\u800c\u9019\u662f\u4e0d\u73fe\u5be6\u7684\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5c55\u958b\u5b8c\u6574\u6e2c\u8a66\u975e\u5e38\u8017\u8cbb\u8cc7\u6e90\u3002\u96a8\u8457\u8f38\u5165\u8cc7\u6599\u91cf\u7684\u8b8a\u5316\uff0c\u6f14\u7b97\u6cd5\u6703\u8868\u73fe\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5728\u8f38\u5165\u8cc7\u6599\u91cf\u8f03\u5c0f\u6642\uff0c\u6f14\u7b97\u6cd5 <code>A</code> \u7684\u57f7\u884c\u6642\u9593\u6bd4\u6f14\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u800c\u5728\u8f38\u5165\u8cc7\u6599\u91cf\u8f03\u5927\u6642\uff0c\u6e2c\u8a66\u7d50\u679c\u53ef\u80fd\u6070\u6070\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u70ba\u4e86\u5f97\u5230\u6709\u8aaa\u670d\u529b\u7684\u7d50\u8ad6\uff0c\u6211\u5011\u9700\u8981\u6e2c\u8a66\u5404\u7a2e\u898f\u6a21\u7684\u8f38\u5165\u8cc7\u6599\uff0c\u800c\u9019\u9700\u8981\u8017\u8cbb\u5927\u91cf\u7684\u8a08\u7b97\u8cc7\u6e90\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2 \u00a0 \u7406\u8ad6\u4f30\u7b97","text":"<p>\u7531\u65bc\u5be6\u969b\u6e2c\u8a66\u5177\u6709\u8f03\u5927\u7684\u4fb7\u9650\u6027\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u8003\u616e\u50c5\u900f\u904e\u4e00\u4e9b\u8a08\u7b97\u4f86\u8a55\u4f30\u6f14\u7b97\u6cd5\u7684\u6548\u7387\u3002\u9019\u7a2e\u4f30\u7b97\u65b9\u6cd5\u88ab\u7a31\u70ba\u6f38\u8fd1\u8907\u96dc\u5ea6\u5206\u6790\uff08asymptotic complexity analysis\uff09\uff0c\u7c21\u7a31\u8907\u96dc\u5ea6\u5206\u6790\u3002</p> <p>\u8907\u96dc\u5ea6\u5206\u6790\u80fd\u5920\u9ad4\u73fe\u6f14\u7b97\u6cd5\u57f7\u884c\u6240\u9700\u7684\u6642\u9593\u548c\u7a7a\u9593\u8cc7\u6e90\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u4e4b\u9593\u7684\u95dc\u4fc2\u3002\u5b83\u63cf\u8ff0\u4e86\u96a8\u8457\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u7684\u589e\u52a0\uff0c\u6f14\u7b97\u6cd5\u57f7\u884c\u6240\u9700\u6642\u9593\u548c\u7a7a\u9593\u7684\u589e\u9577\u8da8\u52e2\u3002\u9019\u500b\u5b9a\u7fa9\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5176\u5206\u70ba\u4e09\u500b\u91cd\u9ede\u4f86\u7406\u89e3\u3002</p> <ul> <li>\u201c\u6642\u9593\u548c\u7a7a\u9593\u8cc7\u6e90\u201d\u5206\u5225\u5c0d\u61c9\u6642\u9593\u8907\u96dc\u5ea6\uff08time complexity\uff09\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\uff08space complexity\uff09\u3002</li> <li>\u201c\u96a8\u8457\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u7684\u589e\u52a0\u201d\u610f\u5473\u8457\u8907\u96dc\u5ea6\u53cd\u6620\u4e86\u6f14\u7b97\u6cd5\u57f7\u884c\u6548\u7387\u8207\u8f38\u5165\u8cc7\u6599\u9ad4\u91cf\u4e4b\u9593\u7684\u95dc\u4fc2\u3002</li> <li>\u201c\u6642\u9593\u548c\u7a7a\u9593\u7684\u589e\u9577\u8da8\u52e2\u201d\u8868\u793a\u8907\u96dc\u5ea6\u5206\u6790\u95dc\u6ce8\u7684\u4e0d\u662f\u57f7\u884c\u6642\u9593\u6216\u4f54\u7528\u7a7a\u9593\u7684\u5177\u9ad4\u503c\uff0c\u800c\u662f\u6642\u9593\u6216\u7a7a\u9593\u589e\u9577\u7684\u201c\u5feb\u6162\u201d\u3002</li> </ul> <p>\u8907\u96dc\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5be6\u969b\u6e2c\u8a66\u65b9\u6cd5\u7684\u5f0a\u7aef\uff0c\u9ad4\u73fe\u5728\u4ee5\u4e0b\u5e7e\u500b\u65b9\u9762\u3002</p> <ul> <li>\u5b83\u7121\u9700\u5be6\u969b\u57f7\u884c\u7a0b\u5f0f\u78bc\uff0c\u66f4\u52a0\u7da0\u8272\u7bc0\u80fd\u3002</li> <li>\u5b83\u7368\u7acb\u65bc\u6e2c\u8a66\u74b0\u5883\uff0c\u5206\u6790\u7d50\u679c\u9069\u7528\u65bc\u6240\u6709\u57f7\u884c\u5e73\u81fa\u3002</li> <li>\u5b83\u53ef\u4ee5\u9ad4\u73fe\u4e0d\u540c\u8cc7\u6599\u91cf\u4e0b\u7684\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u8cc7\u6599\u91cf\u4e0b\u7684\u6f14\u7b97\u6cd5\u6548\u80fd\u3002</li> </ul> <p>Tip</p> <p>\u5982\u679c\u4f60\u4ecd\u5c0d\u8907\u96dc\u5ea6\u7684\u6982\u5ff5\u611f\u5230\u56f0\u60d1\uff0c\u7121\u9808\u64d4\u5fc3\uff0c\u6211\u5011\u6703\u5728\u5f8c\u7e8c\u7ae0\u7bc0\u4e2d\u8a73\u7d30\u4ecb\u7d39\u3002</p> <p>\u8907\u96dc\u5ea6\u5206\u6790\u70ba\u6211\u5011\u63d0\u4f9b\u4e86\u4e00\u628a\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6a19\u5c3a\u201d\uff0c\u4f7f\u6211\u5011\u53ef\u4ee5\u8861\u91cf\u57f7\u884c\u67d0\u500b\u6f14\u7b97\u6cd5\u6240\u9700\u7684\u6642\u9593\u548c\u7a7a\u9593\u8cc7\u6e90\uff0c\u5c0d\u6bd4\u4e0d\u540c\u6f14\u7b97\u6cd5\u4e4b\u9593\u7684\u6548\u7387\u3002</p> <p>\u8907\u96dc\u5ea6\u662f\u500b\u6578\u5b78\u6982\u5ff5\uff0c\u5c0d\u65bc\u521d\u5b78\u8005\u53ef\u80fd\u6bd4\u8f03\u62bd\u8c61\uff0c\u5b78\u7fd2\u96e3\u5ea6\u76f8\u5c0d\u8f03\u9ad8\u3002\u5f9e\u9019\u500b\u89d2\u5ea6\u770b\uff0c\u8907\u96dc\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9069\u5408\u4f5c\u70ba\u6700\u5148\u4ecb\u7d39\u7684\u5167\u5bb9\u3002\u7136\u800c\uff0c\u7576\u6211\u5011\u8a0e\u8ad6\u67d0\u500b\u8cc7\u6599\u7d50\u69cb\u6216\u6f14\u7b97\u6cd5\u7684\u7279\u9ede\u6642\uff0c\u96e3\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u57f7\u884c\u901f\u5ea6\u548c\u7a7a\u9593\u4f7f\u7528\u60c5\u6cc1\u3002</p> <p>\u7d9c\u4e0a\u6240\u8ff0\uff0c\u5efa\u8b70\u4f60\u5728\u6df1\u5165\u5b78\u7fd2\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u4e4b\u524d\uff0c\u5148\u5c0d\u8907\u96dc\u5ea6\u5206\u6790\u5efa\u7acb\u521d\u6b65\u7684\u77ad\u89e3\uff0c\u4ee5\u4fbf\u80fd\u5920\u5b8c\u6210\u7c21\u55ae\u6f14\u7b97\u6cd5\u7684\u8907\u96dc\u5ea6\u5206\u6790\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 \u7a7a\u9593\u8907\u96dc\u5ea6","text":"<p>\u7a7a\u9593\u8907\u96dc\u5ea6\uff08space complexity\uff09\u7528\u65bc\u8861\u91cf\u6f14\u7b97\u6cd5\u4f54\u7528\u8a18\u61b6\u9ad4\u7a7a\u9593\u96a8\u8457\u8cc7\u6599\u91cf\u8b8a\u5927\u6642\u7684\u589e\u9577\u8da8\u52e2\u3002\u9019\u500b\u6982\u5ff5\u8207\u6642\u9593\u8907\u96dc\u5ea6\u975e\u5e38\u985e\u4f3c\uff0c\u53ea\u9700\u5c07\u201c\u57f7\u884c\u6642\u9593\u201d\u66ff\u63db\u70ba\u201c\u4f54\u7528\u8a18\u61b6\u9ad4\u7a7a\u9593\u201d\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#241","title":"2.4.1 \u00a0 \u6f14\u7b97\u6cd5\u76f8\u95dc\u7a7a\u9593","text":"<p>\u6f14\u7b97\u6cd5\u5728\u57f7\u884c\u904e\u7a0b\u4e2d\u4f7f\u7528\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u5e7e\u7a2e\u3002</p> <ul> <li>\u8f38\u5165\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u6f14\u7b97\u6cd5\u7684\u8f38\u5165\u8cc7\u6599\u3002</li> <li>\u66ab\u5b58\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u6f14\u7b97\u6cd5\u5728\u57f7\u884c\u904e\u7a0b\u4e2d\u7684\u8b8a\u6578\u3001\u7269\u4ef6\u3001\u51fd\u5f0f\u4e0a\u4e0b\u6587\u7b49\u8cc7\u6599\u3002</li> <li>\u8f38\u51fa\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u6f14\u7b97\u6cd5\u7684\u8f38\u51fa\u8cc7\u6599\u3002</li> </ul> <p>\u4e00\u822c\u60c5\u6cc1\u4e0b\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u7684\u7d71\u8a08\u7bc4\u570d\u662f\u201c\u66ab\u5b58\u7a7a\u9593\u201d\u52a0\u4e0a\u201c\u8f38\u51fa\u7a7a\u9593\u201d\u3002</p> <p>\u66ab\u5b58\u7a7a\u9593\u53ef\u4ee5\u9032\u4e00\u6b65\u5283\u5206\u70ba\u4e09\u500b\u90e8\u5206\u3002</p> <ul> <li>\u66ab\u5b58\u8cc7\u6599\uff1a\u7528\u65bc\u5132\u5b58\u6f14\u7b97\u6cd5\u57f7\u884c\u904e\u7a0b\u4e2d\u7684\u5404\u7a2e\u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u7b49\u3002</li> <li>\u5806\u758a\u5e40\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u547c\u53eb\u51fd\u5f0f\u7684\u4e0a\u4e0b\u6587\u8cc7\u6599\u3002\u7cfb\u7d71\u5728\u6bcf\u6b21\u547c\u53eb\u51fd\u5f0f\u6642\u90fd\u6703\u5728\u5806\u758a\u9802\u90e8\u5efa\u7acb\u4e00\u500b\u5806\u758a\u5e40\uff0c\u51fd\u5f0f\u8fd4\u56de\u5f8c\uff0c\u5806\u758a\u5e40\u7a7a\u9593\u6703\u88ab\u91cb\u653e\u3002</li> <li>\u6307\u4ee4\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u7de8\u8b6f\u5f8c\u7684\u7a0b\u5f0f\u6307\u4ee4\uff0c\u5728\u5be6\u969b\u7d71\u8a08\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8a08\u3002</li> </ul> <p>\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5f0f\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u6642\uff0c\u6211\u5011\u901a\u5e38\u7d71\u8a08\u66ab\u5b58\u8cc7\u6599\u3001\u5806\u758a\u5e40\u7a7a\u9593\u548c\u8f38\u51fa\u8cc7\u6599\u4e09\u90e8\u5206\uff0c\u5982\u5716 2-15 \u6240\u793a\u3002</p> <p></p> <p> \u5716 2-15 \u00a0 \u6f14\u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u95dc\u7a7a\u9593 </p> <p>\u76f8\u95dc\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class Node:\n    \"\"\"\u985e\u5225\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # \u7bc0\u9ede\u503c\n        self.next: Node | None = None  # \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n\ndef function() -&gt; int:\n    \"\"\"\u51fd\u5f0f\"\"\"\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n\ndef algorithm(n) -&gt; int:  # \u8f38\u5165\u8cc7\u6599\n    A = 0                 # \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff0c\u4e00\u822c\u7528\u5927\u5beb\u5b57\u6bcd\u8868\u793a\uff09\n    b = 0                 # \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    node = Node(0)        # \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    c = function()        # \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return A + b + c      # \u8f38\u51fa\u8cc7\u6599\n</code></pre> <pre><code>/* \u7d50\u69cb\u9ad4 */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* \u51fd\u5f0f */\nint func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f38\u5165\u8cc7\u6599\n    const int a = 0;          // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    int b = 0;                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    Node* node = new Node(0); // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    int c = func();           // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* \u51fd\u5f0f */\nint function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f38\u5165\u8cc7\u6599\n    final int a = 0;          // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    int b = 0;                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    Node node = new Node(0);  // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    int c = function();       // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* \u51fd\u5f0f */\nint Function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint Algorithm(int n) {        // \u8f38\u5165\u8cc7\u6599\n    const int a = 0;          // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    int b = 0;                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    Node node = new(0);       // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    int c = Function();       // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u7d50\u69cb\u9ad4 */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* \u5efa\u7acb node \u7d50\u69cb\u9ad4  */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* \u51fd\u5f0f */\nfunc function() int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n int) int { // \u8f38\u5165\u8cc7\u6599\n    const a = 0             // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    b := 0                  // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    newNode(0)              // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    c := function()         // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c        // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* \u51fd\u5f0f */\nfunc function() -&gt; Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // \u8f38\u5165\u8cc7\u6599\n    let a = 0             // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    var b = 0             // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    let node = Node(x: 0) // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    let c = function()    // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c      // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // \u7bc0\u9ede\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u5f0f */\nfunction constFunc() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n) {       // \u8f38\u5165\u8cc7\u6599\n    const a = 0;              // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    let b = 0;                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    const node = new Node(0); // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    const c = constFunc();    // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // \u7bc0\u9ede\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u5f0f */\nfunction constFunc(): number {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n: number): number { // \u8f38\u5165\u8cc7\u6599\n    const a = 0;                        // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    let b = 0;                          // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    const node = new Node(0);           // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    const c = constFunc();              // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;                   // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* \u51fd\u5f0f */\nint function() {\n  // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n  return 0;\n}\n\nint algorithm(int n) {  // \u8f38\u5165\u8cc7\u6599\n  const int a = 0;      // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n  int b = 0;            // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n  Node node = Node(0);  // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n  int c = function();   // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n  return a + b + c;     // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u7d50\u69cb\u9ad4 */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* \u5efa\u7acb Node \u7d50\u69cb\u9ad4 */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* \u51fd\u5f0f */\nfn function() -&gt; i32 {      \n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // \u8f38\u5165\u8cc7\u6599\n    const a: i32 = 0;               // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    let mut b = 0;                  // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    let node = Node::new(0);        // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    let c = function();             // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;               // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u51fd\u5f0f */\nint func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) { // \u8f38\u5165\u8cc7\u6599\n    const int a = 0;   // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    int b = 0;         // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    int c = func();    // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;  // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node(var _val: Int) {\n    var next: Node? = null\n}\n\n/* \u51fd\u5f0f */\nfun function(): Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfun algorithm(n: Int): Int { // \u8f38\u5165\u8cc7\u6599\n    val a = 0                // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    var b = 0                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    val node = Node(0)       // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    val c = function()       // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>### \u985e\u5225 ###\nclass Node\n    attr_accessor :val      # \u7bc0\u9ede\u503c\n    attr_accessor :next     # \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n\n    def initialize(x)\n        @val = x\n    end\nend\n\n### \u51fd\u5f0f ###\ndef function\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    0\nend\n\n### \u6f14\u7b97\u6cd5 ###\ndef algorithm(n)        # \u8f38\u5165\u8cc7\u6599\n    a = 0               # \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    b = 0               # \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    node = Node.new(0)  # \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    c = function        # \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    a + b + c           # \u8f38\u51fa\u8cc7\u6599\nend\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#242","title":"2.4.2 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u7a7a\u9593\u8907\u96dc\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u8207\u6642\u9593\u8907\u96dc\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u9700\u5c07\u7d71\u8a08\u7269\u4ef6\u5f9e\u201c\u64cd\u4f5c\u6578\u91cf\u201d\u8f49\u70ba\u201c\u4f7f\u7528\u7a7a\u9593\u5927\u5c0f\u201d\u3002</p> <p>\u800c\u8207\u6642\u9593\u8907\u96dc\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u5011\u901a\u5e38\u53ea\u95dc\u6ce8\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u3002\u9019\u662f\u56e0\u70ba\u8a18\u61b6\u9ad4\u7a7a\u9593\u662f\u4e00\u9805\u786c\u6027\u8981\u6c42\uff0c\u6211\u5011\u5fc5\u9808\u78ba\u4fdd\u5728\u6240\u6709\u8f38\u5165\u8cc7\u6599\u4e0b\u90fd\u6709\u8db3\u5920\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u9810\u7559\u3002</p> <p>\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u5169\u5c64\u542b\u7fa9\u3002</p> <ol> <li>\u4ee5\u6700\u5dee\u8f38\u5165\u8cc7\u6599\u70ba\u6e96\uff1a\u7576 \\(n &lt; 10\\) \u6642\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \uff1b\u4f46\u7576 \\(n &gt; 10\\) \u6642\uff0c\u521d\u59cb\u5316\u7684\u9663\u5217 <code>nums</code> \u4f54\u7528 \\(O(n)\\) \u7a7a\u9593\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</li> <li>\u4ee5\u6f14\u7b97\u6cd5\u57f7\u884c\u4e2d\u7684\u5cf0\u503c\u8a18\u61b6\u9ad4\u70ba\u6e96\uff1a\u4f8b\u5982\uff0c\u7a0b\u5f0f\u5728\u57f7\u884c\u6700\u5f8c\u4e00\u884c\u4e4b\u524d\uff0c\u4f54\u7528 \\(O(1)\\) \u7a7a\u9593\uff1b\u7576\u521d\u59cb\u5316\u9663\u5217 <code>nums</code> \u6642\uff0c\u7a0b\u5f0f\u4f54\u7528 \\(O(n)\\) \u7a7a\u9593\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    val a = 0                    // O(1)\n    val b = IntArray(10000)      // O(1)\n    if (n &gt; 10) {\n        val nums = IntArray(n)   // O(n)\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 0                           # O(1)\n    b = Array.new(10000)            # O(1)\n    nums = Array.new(n) if n &gt; 10   # O(n)\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u5728\u905e\u8ff4\u51fd\u5f0f\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7d71\u8a08\u5806\u758a\u5e40\u7a7a\u9593\u3002\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def function() -&gt; int:\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef loop(n: int):\n    \"\"\"\u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1)\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"\u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  return recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfn recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>fun function(): Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfun loop(n: Int) {\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfun recur(n: Int) {\n    if (n == 1) return\n    return recur(n - 1)\n}\n</code></pre> <pre><code>def function\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    0\nend\n\n### \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) ###\ndef loop(n)\n    (0...n).each { function }\nend\n\n### \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) ###\ndef recur(n)\n    return if n == 1\n    recur(n - 1)\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u51fd\u5f0f <code>loop()</code> \u548c <code>recur()</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n)\\) \uff0c\u4f46\u7a7a\u9593\u8907\u96dc\u5ea6\u4e0d\u540c\u3002</p> <ul> <li>\u51fd\u5f0f <code>loop()</code> \u5728\u8ff4\u5708\u4e2d\u547c\u53eb\u4e86 \\(n\\) \u6b21 <code>function()</code> \uff0c\u6bcf\u8f2a\u4e2d\u7684 <code>function()</code> \u90fd\u8fd4\u56de\u4e26\u91cb\u653e\u4e86\u5806\u758a\u5e40\u7a7a\u9593\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u4ecd\u70ba \\(O(1)\\) \u3002</li> <li>\u905e\u8ff4\u51fd\u5f0f <code>recur()</code> \u5728\u57f7\u884c\u904e\u7a0b\u4e2d\u6703\u540c\u6642\u5b58\u5728 \\(n\\) \u500b\u672a\u8fd4\u56de\u7684 <code>recur()</code> \uff0c\u5f9e\u800c\u4f54\u7528 \\(O(n)\\) \u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002</li> </ul>"},{"location":"chapter_computational_complexity/space_complexity/#243","title":"2.4.3 \u00a0 \u5e38\u898b\u578b\u5225","text":"<p>\u8a2d\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff0c\u5716 2-16 \u5c55\u793a\u4e86\u5e38\u898b\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u578b\u5225\uff08\u5f9e\u4f4e\u5230\u9ad8\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline \\text{\u5e38\u6578\u968e} &lt; \\text{\u5c0d\u6578\u968e} &lt; \\text{\u7dda\u6027\u968e} &lt; \\text{\u5e73\u65b9\u968e} &lt; \\text{\u6307\u6578\u968e} \\end{aligned} \\] <p></p> <p> \u5716 2-16 \u00a0 \u5e38\u898b\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u578b\u5225 </p>"},{"location":"chapter_computational_complexity/space_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6578\u968e \\(O(1)\\)","text":"<p>\u5e38\u6578\u968e\u5e38\u898b\u65bc\u6578\u91cf\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7121\u95dc\u7684\u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u8ff4\u5708\u4e2d\u521d\u59cb\u5316\u8b8a\u6578\u6216\u547c\u53eb\u51fd\u5f0f\u800c\u4f54\u7528\u7684\u8a18\u61b6\u9ad4\uff0c\u5728\u9032\u5165\u4e0b\u4e00\u8ff4\u5708\u5f8c\u5c31\u6703\u88ab\u91cb\u653e\uff0c\u56e0\u6b64\u4e0d\u6703\u7d2f\u7a4d\u4f54\u7528\u7a7a\u9593\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u4ecd\u70ba \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"\u51fd\u5f0f\"\"\"\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef constant(n: int):\n    \"\"\"\u5e38\u6578\u968e\"\"\"\n    # \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for _ in range(n):\n        c = 0\n    # \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* \u51fd\u5f0f */\nint func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid constant(int n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u51fd\u5f0f */\nint function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid constant(int n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u51fd\u5f0f */\nint Function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid Constant(int n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u51fd\u5f0f */\nfunc function() int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\n/* \u5e38\u6578\u968e */\nfunc spaceConstant(n int) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* \u51fd\u5f0f */\n@discardableResult\nfunc function() -&gt; Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6578\u968e */\nfunc constant(n: Int) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* \u51fd\u5f0f */\nfunction constFunc() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nfunction constant(n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u51fd\u5f0f */\nfunction constFunc(): number {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nfunction constant(n: number): void {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u51fd\u5f0f */\nint function() {\n  // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid constant(int n) {\n  // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u51fd\u5f0f */\nfn function() -&gt; i32 {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\n#[allow(unused)]\nfn constant(n: i32) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for i in 0..n {\n        let c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u51fd\u5f0f */\nint func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid constant(int n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u51fd\u5f0f */\nfun function(): Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6578\u968e */\nfun constant(n: Int) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### \u51fd\u5f0f ###\ndef function\n  # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n  0\nend\n\n### \u5e38\u6578\u968e ###\ndef constant(n)\n  # \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n  a = 0\n  nums = [0] * 10000\n  node = ListNode.new\n\n  # \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n  (0...n).each { c = 0 }\n  # \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n  (0...n).each { function }\nend\n</code></pre> space_complexity.zig<pre><code>// \u51fd\u5f0f\nfn function() i32 {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n// \u5e38\u6578\u968e\nfn constant(n: i32) void {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const a: i32 = 0;\n    var b: i32 = 0;\n    var nums = [_]i32{0}**10000;\n    var node = inc.ListNode(i32){.val = 0};\n    var i: i32 = 0;\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    while (i &lt; n) : (i += 1) {\n        var c: i32 = 0;\n        _ = c;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        _ = function();\n    }\n    _ = a;\n    _ = b;\n    _ = nums;\n    _ = node;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_computational_complexity/space_complexity/#2-on","title":"2. \u00a0 \u7dda\u6027\u968e \\(O(n)\\)","text":"<p>\u7dda\u6027\u968e\u5e38\u898b\u65bc\u5143\u7d20\u6578\u91cf\u8207 \\(n\\) \u6210\u6b63\u6bd4\u7684\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\u7b49\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"\u7dda\u6027\u968e\"\"\"\n    # \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    nums = [0] * n\n    # \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* \u7dda\u6027\u968e */\nvoid linear(int n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    vector&lt;int&gt; nums(n);\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u7dda\u6027\u968e */\nvoid linear(int n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    int[] nums = new int[n];\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7dda\u6027\u968e */\nvoid Linear(int n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    int[] nums = new int[n];\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u7dda\u6027\u968e */\nfunc spaceLinear(n int) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    _ = make([]int, n)\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7dda\u6027\u968e */\nfunc linear(n: Int) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    let nums = Array(repeating: 0, count: n)\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* \u7dda\u6027\u968e */\nfunction linear(n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    const nums = new Array(n);\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7dda\u6027\u968e */\nfunction linear(n: number): void {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    const nums = new Array(n);\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7dda\u6027\u968e */\nvoid linear(int n) {\n  // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7dda\u6027\u968e */\n#[allow(unused)]\nfn linear(n: i32) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    let mut nums = vec![0; n as usize];\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u96dc\u6e4a\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u65bc uthash.h \u5be6\u73fe\n} HashTable;\n\n/* \u7dda\u6027\u968e */\nvoid linear(int n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // \u8a18\u61b6\u9ad4\u91cb\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // \u8a18\u61b6\u9ad4\u91cb\u653e\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7dda\u6027\u968e */\nfun linear(n: Int) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    val nums = Array(n) { 0 }\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### \u7dda\u6027\u968e ###\ndef linear(n)\n  # \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n  nums = Array.new(n, 0)\n\n  # \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n  hmap = {}\n  for i in 0...n\n    hmap[i] = i.to_s\n  end\nend\n</code></pre> space_complexity.zig<pre><code>// \u7dda\u6027\u968e\nfn linear(comptime n: i32) !void {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    var nums = [_]i32{0}**n;\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        try nodes.append(i);\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n    defer map.deinit();\n    var j: i32 = 0;\n    while (j &lt; n) : (j += 1) {\n        const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n        defer std.heap.page_allocator.free(string);\n        try map.put(i, string);\n    }\n    _ = nums;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 2-17 \u6240\u793a\uff0c\u6b64\u51fd\u5f0f\u7684\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u5373\u540c\u6642\u5b58\u5728 \\(n\\) \u500b\u672a\u8fd4\u56de\u7684 <code>linear_recur()</code> \u51fd\u5f0f\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u5806\u758a\u5e40\u7a7a\u9593\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"\u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    print(\"\u905e\u8ff4 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"\u905e\u8ff4 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid linearRecur(int n) {\n    System.out.println(\"\u905e\u8ff4 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"\u905e\u8ff4 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"\u905e\u8ff4 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc linearRecur(n: Int) {\n    print(\"\u905e\u8ff4 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction linearRecur(n) {\n    console.log(`\u905e\u8ff4 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction linearRecur(n: number): void {\n    console.log(`\u905e\u8ff4 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid linearRecur(int n) {\n  print('\u905e\u8ff4 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn linear_recur(n: i32) {\n    println!(\"\u905e\u8ff4 n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid linearRecur(int n) {\n    printf(\"\u905e\u8ff4 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfun linearRecur(n: Int) {\n    println(\"\u905e\u8ff4 n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef linear_recur(n)\n  puts \"\u905e\u8ff4 n = #{n}\"\n  return if n == 1\n  linear_recur(n - 1)\nend\n</code></pre> space_complexity.zig<pre><code>// \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn linearRecur(comptime n: i32) void {\n    std.debug.print(\"\u905e\u8ff4 n = {}\\n\", .{n});\n    if (n == 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-17 \u00a0 \u905e\u8ff4\u51fd\u5f0f\u7522\u751f\u7684\u7dda\u6027\u968e\u7a7a\u9593\u8907\u96dc\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u968e \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u968e\u5e38\u898b\u65bc\u77e9\u9663\u548c\u5716\uff0c\u5143\u7d20\u6578\u91cf\u8207 \\(n\\) \u6210\u5e73\u65b9\u95dc\u4fc2\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"\u5e73\u65b9\u968e\"\"\"\n    # \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u968e */\nvoid quadratic(int n) {\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u968e */\nvoid quadratic(int n) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    int[][] numMatrix = new int[n][n];\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u968e */\nvoid Quadratic(int n) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    int[,] numMatrix = new int[n, n];\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u968e */\nfunc spaceQuadratic(n int) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u968e */\nfunc quadratic(n: Int) {\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u968e */\nfunction quadratic(n) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u968e */\nfunction quadratic(n: number): void {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u968e */\nvoid quadratic(int n) {\n  // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u968e */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u968e */\nvoid quadratic(int n) {\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // \u8a18\u61b6\u9ad4\u91cb\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u968e */\nfun quadratic(n: Int) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    val numMatrix = arrayOfNulls&lt;Array&lt;Int&gt;?&gt;(n)\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    val numList = mutableListOf&lt;MutableList&lt;Int&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### \u5e73\u65b9\u968e ###\ndef quadratic(n)\n  # \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n  Array.new(n) { Array.new(n, 0) }\nend\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u968e\nfn quadratic(n: i32) !void {\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n        defer tmp.deinit();\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            try tmp.append(0);\n        }\n        try nodes.append(tmp);\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 2-18 \u6240\u793a\uff0c\u8a72\u51fd\u5f0f\u7684\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u5728\u6bcf\u500b\u905e\u8ff4\u51fd\u5f0f\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u500b\u9663\u5217\uff0c\u9577\u5ea6\u5206\u5225\u70ba \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u5e73\u5747\u9577\u5ea6\u70ba \\(n / 2\\) \uff0c\u56e0\u6b64\u7e3d\u9ad4\u4f54\u7528 \\(O(n^2)\\) \u7a7a\u9593\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    if n &lt;= 0:\n        return 0\n    # \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"\u905e\u8ff4 n = \" &lt;&lt; n &lt;&lt; \" \u4e2d\u7684 nums \u9577\u5ea6 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"\u905e\u8ff4 n = \" + n + \" \u4e2d\u7684 nums \u9577\u5ea6 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"\u905e\u8ff4 n = \" + n + \" \u4e2d\u7684 nums \u9577\u5ea6 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"\u905e\u8ff4 n = %d \u4e2d\u7684 nums \u9577\u5ea6 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"\u905e\u8ff4 n = \\(n) \u4e2d\u7684 nums \u9577\u5ea6 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u905e\u8ff4 n = ${n} \u4e2d\u7684 nums \u9577\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u905e\u8ff4 n = ${n} \u4e2d\u7684 nums \u9577\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('\u905e\u8ff4 n = $n \u4e2d\u7684 nums \u9577\u5ea6 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"\u905e\u8ff4 n = {} \u4e2d\u7684 nums \u9577\u5ea6 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"\u905e\u8ff4 n = %d \u4e2d\u7684 nums \u9577\u5ea6 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    val nums = Array(n) { 0 }\n    println(\"\u905e\u8ff4 n = $n \u4e2d\u7684 nums \u9577\u5ea6 = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef quadratic_recur(n)\n  return 0 unless n &gt; 0\n\n  # \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n  nums = Array.new(n, 0)\n  quadratic_recur(n - 1)\nend\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n    if (n &lt;= 0) return 0;\n    var nums = [_]i32{0}**n;\n    std.debug.print(\"\u905e\u8ff4 n = {} \u4e2d\u7684 nums \u9577\u5ea6 = {}\\n\", .{n, nums.len});\n    return quadraticRecur(n - 1);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-18 \u00a0 \u905e\u8ff4\u51fd\u5f0f\u7522\u751f\u7684\u5e73\u65b9\u968e\u7a7a\u9593\u8907\u96dc\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6578\u968e \\(O(2^n)\\)","text":"<p>\u6307\u6578\u968e\u5e38\u898b\u65bc\u4e8c\u5143\u6a39\u3002\u89c0\u5bdf\u5716 2-19 \uff0c\u5c64\u6578\u70ba \\(n\\) \u7684\u201c\u6eff\u4e8c\u5143\u6a39\u201d\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\(2^n - 1\\) \uff0c\u4f54\u7528 \\(O(2^n)\\) \u7a7a\u9593\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"\u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>### \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09###\ndef build_tree(n)\n  return if n == 0\n\n  TreeNode.new.tap do |root|\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n  end\nend\n</code></pre> space_complexity.zig<pre><code>// \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n    if (n == 0) return null;\n    const root = try mem_allocator.create(inc.TreeNode(i32));\n    root.init(0);\n    root.left = try buildTree(mem_allocator, n - 1);\n    root.right = try buildTree(mem_allocator, n - 1);\n    return root;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-19 \u00a0 \u6eff\u4e8c\u5143\u6a39\u7522\u751f\u7684\u6307\u6578\u968e\u7a7a\u9593\u8907\u96dc\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","title":"5. \u00a0 \u5c0d\u6578\u968e \\(O(\\log n)\\)","text":"<p>\u5c0d\u6578\u968e\u5e38\u898b\u65bc\u5206\u6cbb\u6f14\u7b97\u6cd5\u3002\u4f8b\u5982\u5408\u4f75\u6392\u5e8f\uff0c\u8f38\u5165\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217\uff0c\u6bcf\u8f2a\u905e\u8ff4\u5c07\u9663\u5217\u5f9e\u4e2d\u9ede\u8655\u5283\u5206\u70ba\u5169\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u70ba \\(\\log n\\) \u7684\u905e\u8ff4\u6a39\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002</p> <p>\u518d\u4f8b\u5982\u5c07\u6578\u5b57\u8f49\u5316\u70ba\u5b57\u4e32\uff0c\u8f38\u5165\u4e00\u500b\u6b63\u6574\u6578 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6578\u70ba \\(\\lfloor \\log_{10} n \\rfloor + 1\\) \uff0c\u5373\u5c0d\u61c9\u5b57\u4e32\u9577\u5ea6\u70ba \\(\\lfloor \\log_{10} n \\rfloor + 1\\) \uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log_{10} n + 1) = O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#244","title":"2.4.4 \u00a0 \u6b0a\u8861\u6642\u9593\u8207\u7a7a\u9593","text":"<p>\u7406\u60f3\u60c5\u6cc1\u4e0b\uff0c\u6211\u5011\u5e0c\u671b\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u90fd\u80fd\u9054\u5230\u6700\u512a\u3002\u7136\u800c\u5728\u5be6\u969b\u60c5\u6cc1\u4e2d\uff0c\u540c\u6642\u6700\u4f73\u5316\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u975e\u5e38\u56f0\u96e3\u3002</p> <p>\u964d\u4f4e\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba\u4ee3\u50f9\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u5011\u5c07\u72a7\u7272\u8a18\u61b6\u9ad4\u7a7a\u9593\u4f86\u63d0\u5347\u6f14\u7b97\u6cd5\u57f7\u884c\u901f\u5ea6\u7684\u601d\u8def\u7a31\u70ba\u201c\u4ee5\u7a7a\u9593\u63db\u6642\u9593\u201d\uff1b\u53cd\u4e4b\uff0c\u5247\u7a31\u70ba\u201c\u4ee5\u6642\u9593\u63db\u7a7a\u9593\u201d\u3002</p> <p>\u9078\u64c7\u54ea\u7a2e\u601d\u8def\u53d6\u6c7a\u65bc\u6211\u5011\u66f4\u770b\u91cd\u54ea\u500b\u65b9\u9762\u3002\u5728\u5927\u591a\u6578\u60c5\u6cc1\u4e0b\uff0c\u6642\u9593\u6bd4\u7a7a\u9593\u66f4\u5bf6\u8cb4\uff0c\u56e0\u6b64\u201c\u4ee5\u7a7a\u9593\u63db\u6642\u9593\u201d\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u7576\u7136\uff0c\u5728\u8cc7\u6599\u91cf\u5f88\u5927\u7684\u60c5\u6cc1\u4e0b\uff0c\u63a7\u5236\u7a7a\u9593\u8907\u96dc\u5ea6\u4e5f\u975e\u5e38\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_computational_complexity/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<p>\u6f14\u7b97\u6cd5\u6548\u7387\u8a55\u4f30</p> <ul> <li>\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u662f\u8861\u91cf\u6f14\u7b97\u6cd5\u512a\u52a3\u7684\u5169\u500b\u4e3b\u8981\u8a55\u50f9\u6307\u6a19\u3002</li> <li>\u6211\u5011\u53ef\u4ee5\u900f\u904e\u5be6\u969b\u6e2c\u8a66\u4f86\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96e3\u4ee5\u6d88\u9664\u6e2c\u8a66\u74b0\u5883\u7684\u5f71\u97ff\uff0c\u4e14\u6703\u8017\u8cbb\u5927\u91cf\u8a08\u7b97\u8cc7\u6e90\u3002</li> <li>\u8907\u96dc\u5ea6\u5206\u6790\u53ef\u4ee5\u6d88\u9664\u5be6\u969b\u6e2c\u8a66\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7d50\u679c\u9069\u7528\u65bc\u6240\u6709\u57f7\u884c\u5e73\u81fa\uff0c\u4e26\u4e14\u80fd\u5920\u63ed\u793a\u6f14\u7b97\u6cd5\u5728\u4e0d\u540c\u8cc7\u6599\u898f\u6a21\u4e0b\u7684\u6548\u7387\u3002</li> </ul> <p>\u6642\u9593\u8907\u96dc\u5ea6</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u7528\u65bc\u8861\u91cf\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u96a8\u8cc7\u6599\u91cf\u589e\u9577\u7684\u8da8\u52e2\uff0c\u53ef\u4ee5\u6709\u6548\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u6cc1\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f38\u5165\u7684\u8cc7\u6599\u91cf\u8f03\u5c0f\u6216\u6642\u9593\u8907\u96dc\u5ea6\u76f8\u540c\u6642\uff0c\u7121\u6cd5\u7cbe\u78ba\u5c0d\u6bd4\u6f14\u7b97\u6cd5\u6548\u7387\u7684\u512a\u52a3\u3002</li> <li>\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u865f\u8868\u793a\uff0c\u5c0d\u61c9\u51fd\u5f0f\u6f38\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u7576 \\(n\\) \u8da8\u5411\u6b63\u7121\u7aae\u6642\uff0c\u64cd\u4f5c\u6578\u91cf \\(T(n)\\) \u7684\u589e\u9577\u7d1a\u5225\u3002</li> <li>\u63a8\u7b97\u6642\u9593\u8907\u96dc\u5ea6\u5206\u70ba\u5169\u6b65\uff0c\u9996\u5148\u7d71\u8a08\u64cd\u4f5c\u6578\u91cf\uff0c\u7136\u5f8c\u5224\u65b7\u6f38\u8fd1\u4e0a\u754c\u3002</li> <li>\u5e38\u898b\u6642\u9593\u8907\u96dc\u5ea6\u5f9e\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n \\log n)\\)\u3001\\(O(n^2)\\)\u3001\\(O(2^n)\\) \u548c \\(O(n!)\\) \u7b49\u3002</li> <li>\u67d0\u4e9b\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u8207\u8f38\u5165\u8cc7\u6599\u7684\u5206\u4f48\u6709\u95dc\u3002\u6642\u9593\u8907\u96dc\u5ea6\u5206\u70ba\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u5e7e\u4e4e\u4e0d\u7528\uff0c\u56e0\u70ba\u8f38\u5165\u8cc7\u6599\u4e00\u822c\u9700\u8981\u6eff\u8db3\u56b4\u683c\u689d\u4ef6\u624d\u80fd\u9054\u5230\u6700\u4f73\u60c5\u6cc1\u3002</li> <li>\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u53cd\u6620\u6f14\u7b97\u6cd5\u5728\u96a8\u6a5f\u8cc7\u6599\u8f38\u5165\u4e0b\u7684\u57f7\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5be6\u969b\u61c9\u7528\u4e2d\u7684\u6f14\u7b97\u6cd5\u6548\u80fd\u3002\u8a08\u7b97\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u9700\u8981\u7d71\u8a08\u8f38\u5165\u8cc7\u6599\u5206\u4f48\u4ee5\u53ca\u7d9c\u5408\u5f8c\u7684\u6578\u5b78\u671f\u671b\u3002</li> </ul> <p>\u7a7a\u9593\u8907\u96dc\u5ea6</p> <ul> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u7684\u4f5c\u7528\u985e\u4f3c\u65bc\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u7528\u65bc\u8861\u91cf\u6f14\u7b97\u6cd5\u4f54\u7528\u8a18\u61b6\u9ad4\u7a7a\u9593\u96a8\u8cc7\u6599\u91cf\u589e\u9577\u7684\u8da8\u52e2\u3002</li> <li>\u6f14\u7b97\u6cd5\u57f7\u884c\u904e\u7a0b\u4e2d\u7684\u76f8\u95dc\u8a18\u61b6\u9ad4\u7a7a\u9593\u53ef\u5206\u70ba\u8f38\u5165\u7a7a\u9593\u3001\u66ab\u5b58\u7a7a\u9593\u3001\u8f38\u51fa\u7a7a\u9593\u3002\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\u8f38\u5165\u7a7a\u9593\u4e0d\u7d0d\u5165\u7a7a\u9593\u8907\u96dc\u5ea6\u8a08\u7b97\u3002\u66ab\u5b58\u7a7a\u9593\u53ef\u5206\u70ba\u66ab\u5b58\u8cc7\u6599\u3001\u5806\u758a\u5e40\u7a7a\u9593\u548c\u6307\u4ee4\u7a7a\u9593\uff0c\u5176\u4e2d\u5806\u758a\u5e40\u7a7a\u9593\u901a\u5e38\u50c5\u5728\u905e\u8ff4\u51fd\u5f0f\u4e2d\u5f71\u97ff\u7a7a\u9593\u8907\u96dc\u5ea6\u3002</li> <li>\u6211\u5011\u901a\u5e38\u53ea\u95dc\u6ce8\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\uff0c\u5373\u7d71\u8a08\u6f14\u7b97\u6cd5\u5728\u6700\u5dee\u8f38\u5165\u8cc7\u6599\u548c\u6700\u5dee\u57f7\u884c\u6642\u523b\u4e0b\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u3002</li> <li>\u5e38\u898b\u7a7a\u9593\u8907\u96dc\u5ea6\u5f9e\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n^2)\\) \u548c \\(O(2^n)\\) \u7b49\u3002</li> </ul>"},{"location":"chapter_computational_complexity/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u5c3e\u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u662f \\(O(1)\\) \u55ce\uff1f</p> <p>\u7406\u8ad6\u4e0a\uff0c\u5c3e\u905e\u8ff4\u51fd\u5f0f\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u6700\u4f73\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u904e\u7d55\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\uff08\u4f8b\u5982 Java\u3001Python\u3001C++\u3001Go\u3001C# \u7b49\uff09\u4e0d\u652f\u6301\u81ea\u52d5\u6700\u4f73\u5316\u5c3e\u905e\u8ff4\uff0c\u56e0\u6b64\u901a\u5e38\u8a8d\u70ba\u7a7a\u9593\u8907\u96dc\u5ea6\u662f \\(O(n)\\) \u3002</p> <p>Q\uff1a\u51fd\u5f0f\u548c\u65b9\u6cd5\u9019\u5169\u500b\u8853\u8a9e\u7684\u5340\u5225\u662f\u4ec0\u9ebc\uff1f</p> <p>\u51fd\u5f0f\uff08function\uff09\u53ef\u4ee5\u88ab\u7368\u7acb\u57f7\u884c\uff0c\u6240\u6709\u53c3\u6578\u90fd\u4ee5\u986f\u5f0f\u50b3\u905e\u3002\u65b9\u6cd5\uff08method\uff09\u8207\u4e00\u500b\u7269\u4ef6\u95dc\u806f\uff0c\u88ab\u96b1\u5f0f\u50b3\u905e\u7d66\u547c\u53eb\u5b83\u7684\u7269\u4ef6\uff0c\u80fd\u5920\u5c0d\u985e\u5225\u7684\u4f8b\u9805\u4e2d\u5305\u542b\u7684\u8cc7\u6599\u9032\u884c\u64cd\u4f5c\u3002</p> <p>\u4e0b\u9762\u4ee5\u5e7e\u7a2e\u5e38\u898b\u7684\u7a0b\u5f0f\u8a9e\u8a00\u70ba\u4f8b\u4f86\u8aaa\u660e\u3002</p> <ul> <li>C \u8a9e\u8a00\u662f\u7a0b\u5e8f\u5f0f\u7a0b\u5f0f\u8a2d\u8a08\u8a9e\u8a00\uff0c\u6c92\u6709\u7269\u4ef6\u5c0e\u5411\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u53ea\u6709\u51fd\u5f0f\u3002\u4f46\u6211\u5011\u53ef\u4ee5\u900f\u904e\u5efa\u7acb\u7d50\u69cb\u9ad4\uff08struct\uff09\u4f86\u6a21\u64ec\u7269\u4ef6\u5c0e\u5411\u7a0b\u5f0f\u8a2d\u8a08\uff0c\u8207\u7d50\u69cb\u9ad4\u76f8\u95dc\u806f\u7684\u51fd\u5f0f\u5c31\u76f8\u7576\u65bc\u5176\u4ed6\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u7684\u65b9\u6cd5\u3002</li> <li>Java \u548c C# \u662f\u7269\u4ef6\u5c0e\u5411\u7684\u7a0b\u5f0f\u8a9e\u8a00\uff0c\u7a0b\u5f0f\u78bc\u584a\uff08\u65b9\u6cd5\uff09\u901a\u5e38\u4f5c\u70ba\u67d0\u500b\u985e\u5225\u7684\u4e00\u90e8\u5206\u3002\u975c\u614b\u65b9\u6cd5\u7684\u884c\u70ba\u985e\u4f3c\u65bc\u51fd\u5f0f\uff0c\u56e0\u70ba\u5b83\u88ab\u7e6b\u7d50\u5728\u985e\u5225\u4e0a\uff0c\u4e0d\u80fd\u8a2a\u554f\u7279\u5b9a\u7684\u4f8b\u9805\u8b8a\u6578\u3002</li> <li>C++ \u548c Python \u65e2\u652f\u6301\u7a0b\u5e8f\u5f0f\u7a0b\u5f0f\u8a2d\u8a08\uff08\u51fd\u5f0f\uff09\uff0c\u4e5f\u652f\u6301\u7269\u4ef6\u5c0e\u5411\u7a0b\u5f0f\u8a2d\u8a08\uff08\u65b9\u6cd5\uff09\u3002</li> </ul> <p>Q\uff1a\u5716\u89e3\u201c\u5e38\u898b\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u578b\u5225\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u4f54\u7528\u7a7a\u9593\u7684\u7d55\u5c0d\u5927\u5c0f\uff1f</p> <p>\u4e0d\u662f\uff0c\u8a72\u5716\u5c55\u793a\u7684\u662f\u7a7a\u9593\u8907\u96dc\u5ea6\uff0c\u5176\u53cd\u6620\u7684\u662f\u589e\u9577\u8da8\u52e2\uff0c\u800c\u4e0d\u662f\u4f54\u7528\u7a7a\u9593\u7684\u7d55\u5c0d\u5927\u5c0f\u3002</p> <p>\u5047\u8a2d\u53d6 \\(n = 8\\) \uff0c\u4f60\u53ef\u80fd\u6703\u767c\u73fe\u6bcf\u689d\u66f2\u7dda\u7684\u503c\u8207\u51fd\u5f0f\u5c0d\u61c9\u4e0d\u4e0a\u3002\u9019\u662f\u56e0\u70ba\u6bcf\u689d\u66f2\u7dda\u90fd\u5305\u542b\u4e00\u500b\u5e38\u6578\u9805\uff0c\u7528\u65bc\u5c07\u53d6\u503c\u7bc4\u570d\u58d3\u7e2e\u5230\u4e00\u500b\u8996\u89ba\u8212\u9069\u7684\u7bc4\u570d\u5167\u3002</p> <p>\u5728\u5be6\u969b\u4e2d\uff0c\u56e0\u70ba\u6211\u5011\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u500b\u65b9\u6cd5\u7684\u201c\u5e38\u6578\u9805\u201d\u8907\u96dc\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u7121\u6cd5\u50c5\u6191\u8907\u96dc\u5ea6\u4f86\u9078\u64c7 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u512a\u89e3\u6cd5\u3002\u4f46\u5c0d\u65bc \\(n = 8^5\\) \u5c31\u5f88\u597d\u9078\u4e86\uff0c\u9019\u6642\u589e\u9577\u8da8\u52e2\u5df2\u7d93\u4f54\u4e3b\u5c0e\u4e86\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 \u6642\u9593\u8907\u96dc\u5ea6","text":"<p>\u57f7\u884c\u6642\u9593\u53ef\u4ee5\u76f4\u89c0\u4e14\u6e96\u78ba\u5730\u53cd\u6620\u6f14\u7b97\u6cd5\u7684\u6548\u7387\u3002\u5982\u679c\u6211\u5011\u60f3\u6e96\u78ba\u9810\u4f30\u4e00\u6bb5\u7a0b\u5f0f\u78bc\u7684\u57f7\u884c\u6642\u9593\uff0c\u61c9\u8a72\u5982\u4f55\u64cd\u4f5c\u5462\uff1f</p> <ol> <li>\u78ba\u5b9a\u57f7\u884c\u5e73\u81fa\uff0c\u5305\u62ec\u786c\u9ad4\u914d\u7f6e\u3001\u7a0b\u5f0f\u8a9e\u8a00\u3001\u7cfb\u7d71\u74b0\u5883\u7b49\uff0c\u9019\u4e9b\u56e0\u7d20\u90fd\u6703\u5f71\u97ff\u7a0b\u5f0f\u78bc\u7684\u57f7\u884c\u6548\u7387\u3002</li> <li>\u8a55\u4f30\u5404\u7a2e\u8a08\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u57f7\u884c\u6642\u9593\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c <code>+</code> \u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c <code>*</code> \u9700\u8981 10 ns \uff0c\u5217\u5370\u64cd\u4f5c <code>print()</code> \u9700\u8981 5 ns \u7b49\u3002</li> <li>\u7d71\u8a08\u7a0b\u5f0f\u78bc\u4e2d\u6240\u6709\u7684\u8a08\u7b97\u64cd\u4f5c\uff0c\u4e26\u5c07\u6240\u6709\u64cd\u4f5c\u7684\u57f7\u884c\u6642\u9593\u6c42\u548c\uff0c\u5f9e\u800c\u5f97\u5230\u57f7\u884c\u6642\u9593\u3002</li> </ol> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # \u8ff4\u5708 n \u6b21\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // \u8ff4\u5708 n \u6b21\n  for (int i = 0; i &lt; n; i++) { // 1 ns\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for _ in 0..n {     // 1 ns\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // 1 ns\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfun algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (i in 0..&lt;n) {  // 1 ns\n        println(0)      // 5 ns\n    }\n}\n</code></pre> <pre><code># \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\ndef algorithm(n)\n    a = 2       # 1 ns\n    a = a + 1   # 1 ns\n    a = a * 2   # 10 ns\n    # \u8ff4\u5708 n \u6b21\n    (0...n).each do # 1 ns\n        puts 0      # 5 ns\n    end\nend\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfn algorithm(n: usize) void {\n    var a: i32 = 2; // 1 ns\n    a += 1; // 1 ns\n    a *= 2; // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (0..n) |_| { // 1 ns\n        std.debug.print(\"{}\\n\", .{0}); // 5 ns\n    }\n}\n</code></pre> <p>\u6839\u64da\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u6f14\u7b97\u6cd5\u7684\u57f7\u884c\u6642\u9593\u70ba \\((6n + 12)\\) ns \uff1a</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>\u4f46\u5be6\u969b\u4e0a\uff0c\u7d71\u8a08\u6f14\u7b97\u6cd5\u7684\u57f7\u884c\u6642\u9593\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73fe\u5be6\u3002\u9996\u5148\uff0c\u6211\u5011\u4e0d\u5e0c\u671b\u5c07\u9810\u4f30\u6642\u9593\u548c\u57f7\u884c\u5e73\u81fa\u7e6b\u7d50\uff0c\u56e0\u70ba\u6f14\u7b97\u6cd5\u9700\u8981\u5728\u5404\u7a2e\u4e0d\u540c\u7684\u5e73\u81fa\u4e0a\u57f7\u884c\u3002\u5176\u6b21\uff0c\u6211\u5011\u5f88\u96e3\u7372\u77e5\u6bcf\u7a2e\u64cd\u4f5c\u7684\u57f7\u884c\u6642\u9593\uff0c\u9019\u7d66\u9810\u4f30\u904e\u7a0b\u5e36\u4f86\u4e86\u6975\u5927\u7684\u96e3\u5ea6\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#231","title":"2.3.1 \u00a0 \u7d71\u8a08\u6642\u9593\u589e\u9577\u8da8\u52e2","text":"<p>\u6642\u9593\u8907\u96dc\u5ea6\u5206\u6790\u7d71\u8a08\u7684\u4e0d\u662f\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\uff0c\u800c\u662f\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u96a8\u8457\u8cc7\u6599\u91cf\u8b8a\u5927\u6642\u7684\u589e\u9577\u8da8\u52e2\u3002</p> <p>\u201c\u6642\u9593\u589e\u9577\u8da8\u52e2\u201d\u9019\u500b\u6982\u5ff5\u6bd4\u8f03\u62bd\u8c61\uff0c\u6211\u5011\u900f\u904e\u4e00\u500b\u4f8b\u5b50\u4f86\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8a2d\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff0c\u7d66\u5b9a\u4e09\u500b\u6f14\u7b97\u6cd5 <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\ndef algorithm_A(n: int):\n    print(0)\n# \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithmA(int n) {\n  print(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfun algoritm_A(n: Int) {\n    println(0)\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfun algorithm_B(n: Int) {\n    for (i in 0..&lt;n){\n        println(0)\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfun algorithm_C(n: Int) {\n    for (i in 0..&lt;1000000) {\n        println(0)\n    }\n}\n</code></pre> <pre><code># \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\ndef algorithm_A(n)\n    puts 0\nend\n\n# \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\ndef algorithm_B(n)\n    (0...n).each { puts 0 }\nend\n\n# \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\ndef algorithm_C(n)\n    (0...1_000_000).each { puts 0 }\nend\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfn algorithm_A(n: usize) void {\n    _ = n;\n    std.debug.print(\"{}\\n\", .{0});\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfn algorithm_B(n: i32) void {\n    for (0..n) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfn algorithm_C(n: i32) void {\n    _ = n;\n    for (0..1000000) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n</code></pre> <p>\u5716 2-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4e09\u500b\u6f14\u7b97\u6cd5\u51fd\u5f0f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <ul> <li>\u6f14\u7b97\u6cd5 <code>A</code> \u53ea\u6709 \\(1\\) \u500b\u5217\u5370\u64cd\u4f5c\uff0c\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u4e0d\u96a8\u8457 \\(n\\) \u589e\u5927\u800c\u589e\u9577\u3002\u6211\u5011\u7a31\u6b64\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba\u201c\u5e38\u6578\u968e\u201d\u3002</li> <li>\u6f14\u7b97\u6cd5 <code>B</code> \u4e2d\u7684\u5217\u5370\u64cd\u4f5c\u9700\u8981\u8ff4\u5708 \\(n\\) \u6b21\uff0c\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u96a8\u8457 \\(n\\) \u589e\u5927\u5448\u7dda\u6027\u589e\u9577\u3002\u6b64\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u88ab\u7a31\u70ba\u201c\u7dda\u6027\u968e\u201d\u3002</li> <li>\u6f14\u7b97\u6cd5 <code>C</code> \u4e2d\u7684\u5217\u5370\u64cd\u4f5c\u9700\u8981\u8ff4\u5708 \\(1000000\\) \u6b21\uff0c\u96d6\u7136\u57f7\u884c\u6642\u9593\u5f88\u9577\uff0c\u4f46\u5b83\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7121\u95dc\u3002\u56e0\u6b64 <code>C</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u548c <code>A</code> \u76f8\u540c\uff0c\u4ecd\u70ba\u201c\u5e38\u6578\u968e\u201d\u3002</li> </ul> <p></p> <p> \u5716 2-7 \u00a0 \u6f14\u7b97\u6cd5 A\u3001B \u548c C \u7684\u6642\u9593\u589e\u9577\u8da8\u52e2 </p> <p>\u76f8\u8f03\u65bc\u76f4\u63a5\u7d71\u8a08\u6f14\u7b97\u6cd5\u7684\u57f7\u884c\u6642\u9593\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u7279\u9ede\u5462\uff1f</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u80fd\u5920\u6709\u6548\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u6f14\u7b97\u6cd5 <code>B</code> \u7684\u57f7\u884c\u6642\u9593\u5448\u7dda\u6027\u589e\u9577\uff0c\u5728 \\(n &gt; 1\\) \u6642\u6bd4\u6f14\u7b97\u6cd5 <code>A</code> \u66f4\u6162\uff0c\u5728 \\(n &gt; 1000000\\) \u6642\u6bd4\u6f14\u7b97\u6cd5 <code>C</code> \u66f4\u6162\u3002\u4e8b\u5be6\u4e0a\uff0c\u53ea\u8981\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u8db3\u5920\u5927\uff0c\u8907\u96dc\u5ea6\u70ba\u201c\u5e38\u6578\u968e\u201d\u7684\u6f14\u7b97\u6cd5\u4e00\u5b9a\u512a\u65bc\u201c\u7dda\u6027\u968e\u201d\u7684\u6f14\u7b97\u6cd5\uff0c\u9019\u6b63\u662f\u6642\u9593\u589e\u9577\u8da8\u52e2\u7684\u542b\u7fa9\u3002</li> <li>\u6642\u9593\u8907\u96dc\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7c21\u4fbf\u3002\u986f\u7136\uff0c\u57f7\u884c\u5e73\u81fa\u548c\u8a08\u7b97\u64cd\u4f5c\u578b\u5225\u90fd\u8207\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u7684\u589e\u9577\u8da8\u52e2\u7121\u95dc\u3002\u56e0\u6b64\u5728\u6642\u9593\u8907\u96dc\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u5011\u53ef\u4ee5\u7c21\u55ae\u5730\u5c07\u6240\u6709\u8a08\u7b97\u64cd\u4f5c\u7684\u57f7\u884c\u6642\u9593\u8996\u70ba\u76f8\u540c\u7684\u201c\u55ae\u4f4d\u6642\u9593\u201d\uff0c\u5f9e\u800c\u5c07\u201c\u8a08\u7b97\u64cd\u4f5c\u57f7\u884c\u6642\u9593\u7d71\u8a08\u201d\u7c21\u5316\u70ba\u201c\u8a08\u7b97\u64cd\u4f5c\u6578\u91cf\u7d71\u8a08\u201d\uff0c\u9019\u6a23\u4e00\u4f86\u4f30\u7b97\u96e3\u5ea6\u5c31\u5927\u5927\u964d\u4f4e\u4e86\u3002</li> <li>\u6642\u9593\u8907\u96dc\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u4fb7\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5118\u7ba1\u6f14\u7b97\u6cd5 <code>A</code> \u548c <code>C</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u76f8\u540c\uff0c\u4f46\u5be6\u969b\u57f7\u884c\u6642\u9593\u5dee\u5225\u5f88\u5927\u3002\u540c\u6a23\uff0c\u5118\u7ba1\u6f14\u7b97\u6cd5 <code>B</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u6bd4 <code>C</code> \u9ad8\uff0c\u4f46\u5728\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u8f03\u5c0f\u6642\uff0c\u6f14\u7b97\u6cd5 <code>B</code> \u660e\u986f\u512a\u65bc\u6f14\u7b97\u6cd5 <code>C</code> \u3002\u5c0d\u65bc\u6b64\u985e\u60c5\u6cc1\uff0c\u6211\u5011\u6642\u5e38\u96e3\u4ee5\u50c5\u6191\u6642\u9593\u8907\u96dc\u5ea6\u5224\u65b7\u6f14\u7b97\u6cd5\u6548\u7387\u7684\u9ad8\u4f4e\u3002\u7576\u7136\uff0c\u5118\u7ba1\u5b58\u5728\u4e0a\u8ff0\u554f\u984c\uff0c\u8907\u96dc\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8a55\u5224\u6f14\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002</li> </ul>"},{"location":"chapter_computational_complexity/time_complexity/#232","title":"2.3.2 \u00a0 \u51fd\u5f0f\u6f38\u8fd1\u4e0a\u754c","text":"<p>\u7d66\u5b9a\u4e00\u500b\u8f38\u5165\u5927\u5c0f\u70ba \\(n\\) \u7684\u51fd\u5f0f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # \u8ff4\u5708 n \u6b21\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // \u8ff4\u5708 n \u6b21\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u8ff4\u5708 n \u6b21\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u8ff4\u5708 n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u8ff4\u5708 n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // \u8ff4\u5708 n \u6b21\n  for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // \u8ff4\u5708 n \u6b21\n    for _ in 0..n { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        printf(\"%d\", 0);            // +1\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u8ff4\u5708 n \u6b21\n    for (i in 0..&lt;n) { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        println(0) // +1\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +1\n    a = a + 1   # +1\n    a = a * 2   # +1\n    # \u8ff4\u5708 n \u6b21\n    (0...n).each do # +1\n        puts 0      # +1\n    end\nend\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u8ff4\u5708 n \u6b21\n    for (0..n) |_| { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        std.debug.print(\"{}\\n\", .{0}); // +1\n    }\n}\n</code></pre> <p>\u8a2d\u6f14\u7b97\u6cd5\u7684\u64cd\u4f5c\u6578\u91cf\u662f\u4e00\u500b\u95dc\u65bc\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7684\u51fd\u5f0f\uff0c\u8a18\u70ba \\(T(n)\\) \uff0c\u5247\u4ee5\u4e0a\u51fd\u5f0f\u7684\u64cd\u4f5c\u6578\u91cf\u70ba\uff1a</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u5f0f\uff0c\u8aaa\u660e\u5176\u57f7\u884c\u6642\u9593\u7684\u589e\u9577\u8da8\u52e2\u662f\u7dda\u6027\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f\u7dda\u6027\u968e\u3002</p> <p>\u6211\u5011\u5c07\u7dda\u6027\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6\u8a18\u70ba \\(O(n)\\) \uff0c\u9019\u500b\u6578\u5b78\u7b26\u865f\u7a31\u70ba\u5927 \\(O\\) \u8a18\u865f\uff08big-\\(O\\) notation\uff09\uff0c\u8868\u793a\u51fd\u5f0f \\(T(n)\\) \u7684\u6f38\u8fd1\u4e0a\u754c\uff08asymptotic upper bound\uff09\u3002</p> <p>\u6642\u9593\u8907\u96dc\u5ea6\u5206\u6790\u672c\u8cea\u4e0a\u662f\u8a08\u7b97\u201c\u64cd\u4f5c\u6578\u91cf \\(T(n)\\)\u201d\u7684\u6f38\u8fd1\u4e0a\u754c\uff0c\u5b83\u5177\u6709\u660e\u78ba\u7684\u6578\u5b78\u5b9a\u7fa9\u3002</p> <p>\u51fd\u5f0f\u6f38\u8fd1\u4e0a\u754c</p> <p>\u82e5\u5b58\u5728\u6b63\u5be6\u6578 \\(c\\) \u548c\u5be6\u6578 \\(n_0\\) \uff0c\u4f7f\u5f97\u5c0d\u65bc\u6240\u6709\u7684 \\(n &gt; n_0\\) \uff0c\u5747\u6709 \\(T(n) \\leq c \\cdot f(n)\\) \uff0c\u5247\u53ef\u8a8d\u70ba \\(f(n)\\) \u7d66\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u500b\u6f38\u8fd1\u4e0a\u754c\uff0c\u8a18\u70ba \\(T(n) = O(f(n))\\) \u3002</p> <p>\u5982\u5716 2-8 \u6240\u793a\uff0c\u8a08\u7b97\u6f38\u8fd1\u4e0a\u754c\u5c31\u662f\u5c0b\u627e\u4e00\u500b\u51fd\u5f0f \\(f(n)\\) \uff0c\u4f7f\u5f97\u7576 \\(n\\) \u8da8\u5411\u65bc\u7121\u7aae\u5927\u6642\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u8655\u65bc\u76f8\u540c\u7684\u589e\u9577\u7d1a\u5225\uff0c\u50c5\u76f8\u5dee\u4e00\u500b\u5e38\u6578\u9805 \\(c\\) \u7684\u500d\u6578\u3002</p> <p></p> <p> \u5716 2-8 \u00a0 \u51fd\u5f0f\u7684\u6f38\u8fd1\u4e0a\u754c </p>"},{"location":"chapter_computational_complexity/time_complexity/#233","title":"2.3.3 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u6f38\u8fd1\u4e0a\u754c\u7684\u6578\u5b78\u5473\u5152\u6709\u9ede\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89ba\u6c92\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u7121\u9808\u64d4\u5fc3\u3002\u6211\u5011\u53ef\u4ee5\u5148\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u5728\u4e0d\u65b7\u7684\u5be6\u8e10\u4e2d\uff0c\u5c31\u53ef\u4ee5\u9010\u6f38\u9818\u609f\u5176\u6578\u5b78\u610f\u7fa9\u3002</p> <p>\u6839\u64da\u5b9a\u7fa9\uff0c\u78ba\u5b9a \\(f(n)\\) \u4e4b\u5f8c\uff0c\u6211\u5011\u4fbf\u53ef\u5f97\u5230\u6642\u9593\u8907\u96dc\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u9ebc\u5982\u4f55\u78ba\u5b9a\u6f38\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u7e3d\u9ad4\u5206\u70ba\u5169\u6b65\uff1a\u9996\u5148\u7d71\u8a08\u64cd\u4f5c\u6578\u91cf\uff0c\u7136\u5f8c\u5224\u65b7\u6f38\u8fd1\u4e0a\u754c\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1. \u00a0 \u7b2c\u4e00\u6b65\uff1a\u7d71\u8a08\u64cd\u4f5c\u6578\u91cf","text":"<p>\u91dd\u5c0d\u7a0b\u5f0f\u78bc\uff0c\u9010\u884c\u5f9e\u4e0a\u5230\u4e0b\u8a08\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u65bc\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6578\u9805 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6578\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u7a2e\u4fc2\u6578\u3001\u5e38\u6578\u9805\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002\u6839\u64da\u6b64\u539f\u5247\uff0c\u53ef\u4ee5\u7e3d\u7d50\u51fa\u4ee5\u4e0b\u8a08\u6578\u7c21\u5316\u6280\u5de7\u3002</p> <ol> <li>\u5ffd\u7565 \\(T(n)\\) \u4e2d\u7684\u5e38\u6578\u9805\u3002\u56e0\u70ba\u5b83\u5011\u90fd\u8207 \\(n\\) \u7121\u95dc\uff0c\u6240\u4ee5\u5c0d\u6642\u9593\u8907\u96dc\u5ea6\u4e0d\u7522\u751f\u5f71\u97ff\u3002</li> <li>\u7701\u7565\u6240\u6709\u4fc2\u6578\u3002\u4f8b\u5982\uff0c\u8ff4\u5708 \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7c21\u5316\u8a18\u70ba \\(n\\) \u6b21\uff0c\u56e0\u70ba \\(n\\) \u524d\u9762\u7684\u4fc2\u6578\u5c0d\u6642\u9593\u8907\u96dc\u5ea6\u6c92\u6709\u5f71\u97ff\u3002</li> <li>\u8ff4\u5708\u5de2\u72c0\u6642\u4f7f\u7528\u4e58\u6cd5\u3002\u7e3d\u64cd\u4f5c\u6578\u91cf\u7b49\u65bc\u5916\u5c64\u8ff4\u5708\u548c\u5167\u5c64\u8ff4\u5708\u64cd\u4f5c\u6578\u91cf\u4e4b\u7a4d\uff0c\u6bcf\u4e00\u5c64\u8ff4\u5708\u4f9d\u7136\u53ef\u4ee5\u5206\u5225\u5957\u7528\u7b2c <code>1.</code> \u9ede\u548c\u7b2c <code>2.</code> \u9ede\u7684\u6280\u5de7\u3002</li> </ol> <p>\u7d66\u5b9a\u4e00\u500b\u51fd\u5f0f\uff0c\u6211\u5011\u53ef\u4ee5\u7528\u4e0a\u8ff0\u6280\u5de7\u4f86\u7d71\u8a08\u64cd\u4f5c\u6578\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 1      # +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  # +0\uff08\u6280\u5de7 1\uff09\n    # +n\uff08\u6280\u5de7 2\uff09\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n\uff08\u6280\u5de7 3\uff09\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n  a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n  // +n\uff08\u6280\u5de7 2\uff09\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n\uff08\u6280\u5de7 3\uff09\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1   // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n   // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (i in 0..&lt;5 * n + 1) {\n        println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (i in 0..&lt;2 * n) {\n        for (j in 0..&lt;n + 1) {\n            println(0)\n        }\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +0\uff08\u6280\u5de7 1\uff09\n    a = a + n   # +0\uff08\u6280\u5de7 1\uff09\n    # +n\uff08\u6280\u5de7 2\uff09\n    (0...(5 * n + 1)).each do { puts 0 }\n    # +n*n\uff08\u6280\u5de7 3\uff09\n    (0...(2 * n)).each do\n        (0...(n + 1)).each do { puts 0 }\n    end\nend\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + @as(i32, @intCast(n));        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for(0..(5 * n + 1)) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for(0..(2 * n)) |_| {\n        for(0..(n + 1)) |_| {\n            std.debug.print(\"{}\\n\", .{0});\n        }\n    }\n}\n</code></pre> <p>\u4ee5\u4e0b\u516c\u5f0f\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u5f8c\u7684\u7d71\u8a08\u7d50\u679c\uff0c\u5169\u8005\u63a8\u7b97\u51fa\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n^2)\\) \u3002</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{\u5b8c\u6574\u7d71\u8a08 (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{\u5077\u61f6\u7d71\u8a08 (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2. \u00a0 \u7b2c\u4e8c\u6b65\uff1a\u5224\u65b7\u6f38\u8fd1\u4e0a\u754c","text":"<p>\u6642\u9593\u8907\u96dc\u5ea6\u7531 \\(T(n)\\) \u4e2d\u6700\u9ad8\u968e\u7684\u9805\u4f86\u6c7a\u5b9a\u3002\u9019\u662f\u56e0\u70ba\u5728 \\(n\\) \u8da8\u65bc\u7121\u7aae\u5927\u6642\uff0c\u6700\u9ad8\u968e\u7684\u9805\u5c07\u767c\u63ee\u4e3b\u5c0e\u4f5c\u7528\uff0c\u5176\u4ed6\u9805\u7684\u5f71\u97ff\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002</p> <p>\u8868 2-2 \u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u8a87\u5f35\u7684\u503c\u662f\u70ba\u4e86\u5f37\u8abf\u201c\u4fc2\u6578\u7121\u6cd5\u64bc\u52d5\u968e\u6578\u201d\u9019\u4e00\u7d50\u8ad6\u3002\u7576 \\(n\\) \u8da8\u65bc\u7121\u7aae\u5927\u6642\uff0c\u9019\u4e9b\u5e38\u6578\u8b8a\u5f97\u7121\u8db3\u8f15\u91cd\u3002</p> <p> \u8868 2-2 \u00a0 \u4e0d\u540c\u64cd\u4f5c\u6578\u91cf\u5c0d\u61c9\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> \u64cd\u4f5c\u6578\u91cf \\(T(n)\\) \u6642\u9593\u8907\u96dc\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234","title":"2.3.4 \u00a0 \u5e38\u898b\u578b\u5225","text":"<p>\u8a2d\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff0c\u5e38\u898b\u7684\u6642\u9593\u8907\u96dc\u5ea6\u578b\u5225\u5982\u5716 2-9 \u6240\u793a\uff08\u6309\u7167\u5f9e\u4f4e\u5230\u9ad8\u7684\u9806\u5e8f\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline \\text{\u5e38\u6578\u968e} &lt; \\text{\u5c0d\u6578\u968e} &lt; \\text{\u7dda\u6027\u968e} &lt; \\text{\u7dda\u6027\u5c0d\u6578\u968e} &lt; \\text{\u5e73\u65b9\u968e} &lt; \\text{\u6307\u6578\u968e} &lt; \\text{\u968e\u4e58\u968e} \\end{aligned} \\] <p></p> <p> \u5716 2-9 \u00a0 \u5e38\u898b\u7684\u6642\u9593\u8907\u96dc\u5ea6\u578b\u5225 </p>"},{"location":"chapter_computational_complexity/time_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6578\u968e \\(O(1)\\)","text":"<p>\u5e38\u6578\u968e\u7684\u64cd\u4f5c\u6578\u91cf\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7121\u95dc\uff0c\u5373\u4e0d\u96a8\u8457 \\(n\\) \u7684\u8b8a\u5316\u800c\u8b8a\u5316\u3002</p> <p>\u5728\u4ee5\u4e0b\u51fd\u5f0f\u4e2d\uff0c\u5118\u7ba1\u64cd\u4f5c\u6578\u91cf <code>size</code> \u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u65bc\u5176\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7121\u95dc\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u4ecd\u70ba \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"\u5e38\u6578\u968e\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e38\u6578\u968e */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e38\u6578\u968e */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e38\u6578\u968e */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e38\u6578\u968e */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e38\u6578\u968e */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e38\u6578\u968e */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e38\u6578\u968e */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e38\u6578\u968e */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e38\u6578\u968e */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e38\u6578\u968e */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    int i = 0;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e38\u6578\u968e */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 100000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u5e38\u6578\u968e ###\ndef constant(n)\n  count = 0\n  size = 100000\n\n  (0...size).each { count += 1 }\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u5e38\u6578\u968e\nfn constant(n: i32) i32 {\n    _ = n;\n    var count: i32 = 0;\n    const size: i32 = 100_000;\n    var i: i32 = 0;\n    while(i&lt;size) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#2-on","title":"2. \u00a0 \u7dda\u6027\u968e \\(O(n)\\)","text":"<p>\u7dda\u6027\u968e\u7684\u64cd\u4f5c\u6578\u91cf\u76f8\u5c0d\u65bc\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u4ee5\u7dda\u6027\u7d1a\u5225\u589e\u9577\u3002\u7dda\u6027\u968e\u901a\u5e38\u51fa\u73fe\u5728\u55ae\u5c64\u8ff4\u5708\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"\u7dda\u6027\u968e\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7dda\u6027\u968e */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7dda\u6027\u968e */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7dda\u6027\u968e */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7dda\u6027\u968e */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7dda\u6027\u968e */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7dda\u6027\u968e */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7dda\u6027\u968e */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7dda\u6027\u968e */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7dda\u6027\u968e */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7dda\u6027\u968e */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7dda\u6027\u968e */\nfun linear(n: Int): Int {\n    var count = 0\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u7dda\u6027\u968e ###\ndef linear(n)\n  count = 0\n  (0...n).each { count += 1 }\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u7dda\u6027\u968e\nfn linear(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u8d70\u8a2a\u9663\u5217\u548c\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u7b49\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u70ba\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u7684\u9577\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"\u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09\"\"\"\n    count = 0\n    # \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09###\ndef array_traversal(nums)\n  count = 0\n\n  # \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n  for num in nums\n    count += 1\n  end\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09\nfn arrayTraversal(nums: []i32) i32 {\n    var count: i32 = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (nums) |_| {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u9700\u6839\u64da\u8f38\u5165\u8cc7\u6599\u7684\u578b\u5225\u4f86\u5177\u9ad4\u78ba\u5b9a\u3002\u6bd4\u5982\u5728\u7b2c\u4e00\u500b\u793a\u4f8b\u4e2d\uff0c\u8b8a\u6578 \\(n\\) \u70ba\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\uff1b\u5728\u7b2c\u4e8c\u500b\u793a\u4f8b\u4e2d\uff0c\u9663\u5217\u9577\u5ea6 \\(n\\) \u70ba\u8cc7\u6599\u5927\u5c0f\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u968e \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u968e\u7684\u64cd\u4f5c\u6578\u91cf\u76f8\u5c0d\u65bc\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u4ee5\u5e73\u65b9\u7d1a\u5225\u589e\u9577\u3002\u5e73\u65b9\u968e\u901a\u5e38\u51fa\u73fe\u5728\u5de2\u72c0\u8ff4\u5708\u4e2d\uff0c\u5916\u5c64\u8ff4\u5708\u548c\u5167\u5c64\u8ff4\u5708\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n)\\) \uff0c\u56e0\u6b64\u7e3d\u9ad4\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u968e\"\"\"\n    count = 0\n    # \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u968e */\nint quadratic(int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u968e */\nint quadratic(int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u968e */\nint Quadratic(int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u968e */\nfunc quadratic(n int) int {\n    count := 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u968e */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u968e */\nfunction quadratic(n) {\n    let count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u968e */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u968e */\nint quadratic(int n) {\n  int count = 0;\n  // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u968e */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u968e */\nint quadratic(int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u968e */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u5e73\u65b9\u968e ###\ndef quadratic(n)\n  count = 0\n\n  # \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n  for i in 0...n\n    for j in 0...n\n      count += 1\n    end\n  end\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u968e\nfn quadratic(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            count += 1;\n        }\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-10 \u5c0d\u6bd4\u4e86\u5e38\u6578\u968e\u3001\u7dda\u6027\u968e\u548c\u5e73\u65b9\u968e\u4e09\u7a2e\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <p></p> <p> \u5716 2-10 \u00a0 \u5e38\u6578\u968e\u3001\u7dda\u6027\u968e\u548c\u5e73\u65b9\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u4ee5\u6ce1\u6cab\u6392\u5e8f\u70ba\u4f8b\uff0c\u5916\u5c64\u8ff4\u5708\u57f7\u884c \\(n - 1\\) \u6b21\uff0c\u5167\u5c64\u8ff4\u5708\u57f7\u884c \\(n-1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u5e73\u5747\u70ba \\(n / 2\\) \u6b21\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O((n - 1) n / 2) = O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"\u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09\"\"\"\n    count = 0  # \u8a08\u6578\u5668\n    # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n    int count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n    int count = 0;  // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n    count := 0 // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n    let count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // \u8a08\u6578\u5668\n  // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // \u8a08\u6578\u5668\n\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0 // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                count += 3 // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09###\ndef bubble_sort(nums)\n  count = 0  # \u8a08\u6578\u5668\n\n  # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for i in (nums.length - 1).downto(0)\n    # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        tmp = nums[j]\n        nums[j] = nums[j + 1]\n        nums[j + 1] = tmp\n        count += 3 # \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n      end\n    end\n  end\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n    var count: i32 = 0;  // \u8a08\u6578\u5668 \n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3;  // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6578\u968e \\(O(2^n)\\)","text":"<p>\u751f\u7269\u5b78\u7684\u201c\u7d30\u80de\u5206\u88c2\u201d\u662f\u6307\u6578\u968e\u589e\u9577\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72c0\u614b\u70ba \\(1\\) \u500b\u7d30\u80de\uff0c\u5206\u88c2\u4e00\u8f2a\u5f8c\u8b8a\u70ba \\(2\\) \u500b\uff0c\u5206\u88c2\u5169\u8f2a\u5f8c\u8b8a\u70ba \\(4\\) \u500b\uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f2a\u5f8c\u6709 \\(2^n\\) \u500b\u7d30\u80de\u3002</p> <p>\u5716 2-11 \u548c\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6a21\u64ec\u4e86\u7d30\u80de\u5206\u88c2\u7684\u904e\u7a0b\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^n)\\) \u3002\u8acb\u6ce8\u610f\uff0c\u8f38\u5165 \\(n\\) \u8868\u793a\u5206\u88c2\u8f2a\u6578\uff0c\u8fd4\u56de\u503c <code>count</code> \u8868\u793a\u7e3d\u5206\u88c2\u6b21\u6578\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"\u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09\"\"\"\n    count = 0\n    base = 1\n    # \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfun exponential(n: Int): Int {\n    var count = 0\n    var base = 1\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09###\ndef exponential(n)\n  count, base = 0, 1\n\n  # \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  (0...n).each do\n    (0...base).each { count += 1 }\n    base *= 2\n  end\n\n  # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09\nfn exponential(n: i32) i32 {\n    var count: i32 = 0;\n    var bas: i32 = 1;\n    var i: i32 = 0;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; bas) : (j += 1) {\n            count += 1;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-11 \u00a0 \u6307\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u5728\u5be6\u969b\u6f14\u7b97\u6cd5\u4e2d\uff0c\u6307\u6578\u968e\u5e38\u51fa\u73fe\u65bc\u905e\u8ff4\u51fd\u5f0f\u4e2d\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u5176\u905e\u8ff4\u5730\u4e00\u5206\u70ba\u4e8c\uff0c\u7d93\u904e \\(n\\) \u6b21\u5206\u88c2\u5f8c\u505c\u6b62\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"\u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef exp_recur(n)\n  return 1 if n == 1\n  exp_recur(n - 1) + exp_recur(n - 1) + 1\nend\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn expRecur(n: i32) i32 {\n    if (n == 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6307\u6578\u968e\u589e\u9577\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u7aae\u8209\u6cd5\uff08\u66b4\u529b\u641c\u5c0b\u3001\u56de\u6eaf\u7b49\uff09\u4e2d\u6bd4\u8f03\u5e38\u898b\u3002\u5c0d\u65bc\u8cc7\u6599\u898f\u6a21\u8f03\u5927\u7684\u554f\u984c\uff0c\u6307\u6578\u968e\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff0c\u901a\u5e38\u9700\u8981\u4f7f\u7528\u52d5\u614b\u898f\u5283\u6216\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7b49\u4f86\u89e3\u6c7a\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","title":"5. \u00a0 \u5c0d\u6578\u968e \\(O(\\log n)\\)","text":"<p>\u8207\u6307\u6578\u968e\u76f8\u53cd\uff0c\u5c0d\u6578\u968e\u53cd\u6620\u4e86\u201c\u6bcf\u8f2a\u7e2e\u6e1b\u5230\u4e00\u534a\u201d\u7684\u60c5\u6cc1\u3002\u8a2d\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff0c\u7531\u65bc\u6bcf\u8f2a\u7e2e\u6e1b\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u8ff4\u5708\u6b21\u6578\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u5f0f\u3002</p> <p>\u5716 2-12 \u548c\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6a21\u64ec\u4e86\u201c\u6bcf\u8f2a\u7e2e\u6e1b\u5230\u4e00\u534a\u201d\u7684\u904e\u7a0b\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log_2 n)\\) \uff0c\u7c21\u8a18\u70ba \\(O(\\log n)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"\u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09###\ndef logarithmic(n)\n  count = 0\n\n  while n &gt; 1\n    n /= 2\n    count += 1\n  end\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09\nfn logarithmic(n: i32) i32 {\n    var count: i32 = 0;\n    var n_var = n;\n    while (n_var &gt; 1)\n    {\n        n_var = n_var / 2;\n        count +=1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-12 \u00a0 \u5c0d\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u8207\u6307\u6578\u968e\u985e\u4f3c\uff0c\u5c0d\u6578\u968e\u4e5f\u5e38\u51fa\u73fe\u65bc\u905e\u8ff4\u51fd\u5f0f\u4e2d\u3002\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u5f62\u6210\u4e86\u4e00\u68f5\u9ad8\u5ea6\u70ba \\(\\log_2 n\\) \u7684\u905e\u8ff4\u6a39\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"\u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef log_recur(n)\n  return 0 unless n &gt; 1\n  log_recur(n / 2) + 1\nend\n</code></pre> time_complexity.zig<pre><code>// \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn logRecur(n: i32) i32 {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5c0d\u6578\u968e\u5e38\u51fa\u73fe\u65bc\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6f14\u7b97\u6cd5\u4e2d\uff0c\u9ad4\u73fe\u4e86\u201c\u4e00\u5206\u70ba\u591a\u201d\u548c\u201c\u5316\u7e41\u70ba\u7c21\u201d\u7684\u6f14\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u589e\u9577\u7de9\u6162\uff0c\u662f\u50c5\u6b21\u65bc\u5e38\u6578\u968e\u7684\u7406\u60f3\u7684\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <p>\\(O(\\log n)\\) \u7684\u5e95\u6578\u662f\u591a\u5c11\uff1f</p> <p>\u6e96\u78ba\u4f86\u8aaa\uff0c\u201c\u4e00\u5206\u70ba \\(m\\)\u201d\u5c0d\u61c9\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(\\log_m n)\\) \u3002\u800c\u900f\u904e\u5c0d\u6578\u63db\u5e95\u516c\u5f0f\uff0c\u6211\u5011\u53ef\u4ee5\u5f97\u5230\u5177\u6709\u4e0d\u540c\u5e95\u6578\u3001\u76f8\u7b49\u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u5e95\u6578 \\(m\\) \u53ef\u4ee5\u5728\u4e0d\u5f71\u97ff\u8907\u96dc\u5ea6\u7684\u524d\u63d0\u4e0b\u8f49\u63db\u3002\u56e0\u6b64\u6211\u5011\u901a\u5e38\u6703\u7701\u7565\u5e95\u6578 \\(m\\) \uff0c\u5c07\u5c0d\u6578\u968e\u76f4\u63a5\u8a18\u70ba \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","title":"6. \u00a0 \u7dda\u6027\u5c0d\u6578\u968e \\(O(n \\log n)\\)","text":"<p>\u7dda\u6027\u5c0d\u6578\u968e\u5e38\u51fa\u73fe\u65bc\u5de2\u72c0\u8ff4\u5708\u4e2d\uff0c\u5169\u5c64\u8ff4\u5708\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5206\u5225\u70ba \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002\u76f8\u95dc\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"\u7dda\u6027\u5c0d\u6578\u968e\"\"\"\n    if n &lt;= 1:\n        return 1\n    # \u4e00\u5206\u70ba\u4e8c\uff0c\u5b50\u554f\u984c\u7684\u898f\u6a21\u6e1b\u5c0f\u4e00\u534a\n    count = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    # \u7576\u524d\u5b50\u554f\u984c\u5305\u542b n \u500b\u64cd\u4f5c\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u7dda\u6027\u5c0d\u6578\u968e ###\ndef linear_log_recur(n)\n  return 1 unless n &gt; 1\n\n  count = linear_log_recur(n / 2) + linear_log_recur(n / 2)\n  (0...n).each { count += 1 }\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u7dda\u6027\u5c0d\u6578\u968e\nfn linearLogRecur(n: i32) i32 {\n    if (n &lt;= 1) return 1;\n    var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-13 \u5c55\u793a\u4e86\u7dda\u6027\u5c0d\u6578\u968e\u7684\u751f\u6210\u65b9\u5f0f\u3002\u4e8c\u5143\u6a39\u7684\u6bcf\u4e00\u5c64\u7684\u64cd\u4f5c\u7e3d\u6578\u90fd\u70ba \\(n\\) \uff0c\u6a39\u5171\u6709 \\(\\log_2 n + 1\\) \u5c64\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</p> <p></p> <p> \u5716 2-13 \u00a0 \u7dda\u6027\u5c0d\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u4e3b\u6d41\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u70ba \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5408\u4f75\u6392\u5e8f\u3001\u5806\u7a4d\u6392\u5e8f\u7b49\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#7-on","title":"7. \u00a0 \u968e\u4e58\u968e \\(O(n!)\\)","text":"<p>\u968e\u4e58\u968e\u5c0d\u61c9\u6578\u5b78\u4e0a\u7684\u201c\u5168\u6392\u5217\u201d\u554f\u984c\u3002\u7d66\u5b9a \\(n\\) \u500b\u4e92\u4e0d\u91cd\u8907\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6578\u91cf\u70ba\uff1a</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>\u968e\u4e58\u901a\u5e38\u4f7f\u7528\u905e\u8ff4\u5be6\u73fe\u3002\u5982\u5716 2-14 \u548c\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\uff0c\u7b2c\u4e00\u5c64\u5206\u88c2\u51fa \\(n\\) \u500b\uff0c\u7b2c\u4e8c\u5c64\u5206\u88c2\u51fa \\(n - 1\\) \u500b\uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c64\u6642\u505c\u6b62\u5206\u88c2\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"\u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef factorial_recur(n)\n  return 1 if n == 0\n\n  count = 0\n  # \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n  (0...n).each { count += factorial_recur(n - 1) }\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn factorialRecur(n: i32) i32 {\n    if (n == 0) return 1;\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    while (i &lt; n) : (i += 1) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-14 \u00a0 \u968e\u4e58\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u8acb\u6ce8\u610f\uff0c\u56e0\u70ba\u7576 \\(n \\geq 4\\) \u6642\u6046\u6709 \\(n! &gt; 2^n\\) \uff0c\u6240\u4ee5\u968e\u4e58\u968e\u6bd4\u6307\u6578\u968e\u589e\u9577\u5f97\u66f4\u5feb\uff0c\u5728 \\(n\\) \u8f03\u5927\u6642\u4e5f\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#235","title":"2.3.5 \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6","text":"<p>\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u6548\u7387\u5f80\u5f80\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u8207\u8f38\u5165\u8cc7\u6599\u7684\u5206\u4f48\u6709\u95dc\u3002\u5047\u8a2d\u8f38\u5165\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217 <code>nums</code> \uff0c\u5176\u4e2d <code>nums</code> \u7531\u5f9e \\(1\\) \u81f3 \\(n\\) \u7684\u6578\u5b57\u7d44\u6210\uff0c\u6bcf\u500b\u6578\u5b57\u53ea\u51fa\u73fe\u4e00\u6b21\uff1b\u4f46\u5143\u7d20\u9806\u5e8f\u662f\u96a8\u6a5f\u6253\u4e82\u7684\uff0c\u4efb\u52d9\u76ee\u6a19\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u5011\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7d50\u8ad6\u3002</p> <ul> <li>\u7576 <code>nums = [?, ?, ..., 1]</code> \uff0c\u5373\u7576\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u6642\uff0c\u9700\u8981\u5b8c\u6574\u8d70\u8a2a\u9663\u5217\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\) \u3002</li> <li>\u7576 <code>nums = [1, ?, ?, ...]</code> \uff0c\u5373\u7576\u9996\u500b\u5143\u7d20\u70ba \\(1\\) \u6642\uff0c\u7121\u8ad6\u9663\u5217\u591a\u9577\u90fd\u4e0d\u9700\u8981\u7e7c\u7e8c\u8d70\u8a2a\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 \\(\\Omega(1)\\) \u3002</li> </ul> <p>\u201c\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u201d\u5c0d\u61c9\u51fd\u5f0f\u6f38\u8fd1\u4e0a\u754c\uff0c\u4f7f\u7528\u5927 \\(O\\) \u8a18\u865f\u8868\u793a\u3002\u76f8\u61c9\u5730\uff0c\u201c\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u201d\u5c0d\u61c9\u51fd\u5f0f\u6f38\u8fd1\u4e0b\u754c\uff0c\u7528 \\(\\Omega\\) \u8a18\u865f\u8868\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"\u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba: 1, 2, ..., n \uff0c\u9806\u5e8f\u88ab\u6253\u4e82\"\"\"\n    # \u751f\u6210\u9663\u5217 nums =: 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"\u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n    for i in range(len(nums)):\n        # \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        # \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u4f7f\u7528\u7cfb\u7d71\u6642\u9593\u751f\u6210\u96a8\u6a5f\u7a2e\u5b50\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    nums.shuffle()\n    return nums\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n  nums.shuffle();\n\n  return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n    // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nint *randomNumbers(int n) {\n    // \u5206\u914d\u5806\u7a4d\u5340\u8a18\u61b6\u9ad4\uff08\u5efa\u7acb\u4e00\u7dad\u53ef\u8b8a\u9577\u9663\u5217\uff1a\u9663\u5217\u4e2d\u5143\u7d20\u6578\u91cf\u70ba n \uff0c\u5143\u7d20\u578b\u5225\u70ba int \uff09\n    int *nums = (int *)malloc(n * sizeof(int));\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    nums.shuffle()\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = nums[i]\n    }\n    return res\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>### \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba: 1, 2, ..., n \uff0c\u9806\u5e8f\u88ab\u6253\u4e82 ###\ndef random_numbers(n)\n  # \u751f\u6210\u9663\u5217 nums =: 1, 2, 3, ..., n\n  nums = Array.new(n) { |i| i + 1 }\n  # \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n  nums.shuffle!\nend\n\n### \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 ###\ndef find_one(nums)\n  for i in 0...nums.length\n    # \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n    # \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n    return i if nums[i] == 1\n  end\n\n  -1\nend\n</code></pre> worst_best_time_complexity.zig<pre><code>// \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82\nfn randomNumbers(comptime n: usize) [n]i32 {\n    var nums: [n]i32 = undefined;\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (&amp;nums, 0..) |*num, i| {\n        num.* = @as(i32, @intCast(i)) + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    const rand = std.crypto.random;\n    rand.shuffle(i32, &amp;nums);\n    return nums;\n}\n\n// \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n    for (nums, 0..) |num, i| {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (num == 1) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u6211\u5011\u5728\u5be6\u969b\u4e2d\u5f88\u5c11\u4f7f\u7528\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u56e0\u70ba\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6a5f\u7387\u4e0b\u624d\u80fd\u9054\u5230\uff0c\u53ef\u80fd\u6703\u5e36\u4f86\u4e00\u5b9a\u7684\u8aa4\u5c0e\u6027\u3002\u800c\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u66f4\u70ba\u5be6\u7528\uff0c\u56e0\u70ba\u5b83\u7d66\u51fa\u4e86\u4e00\u500b\u6548\u7387\u5b89\u5168\u503c\uff0c\u8b93\u6211\u5011\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u6f14\u7b97\u6cd5\u3002</p> <p>\u5f9e\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u548c\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u53ea\u51fa\u73fe\u65bc\u201c\u7279\u6b8a\u7684\u8cc7\u6599\u5206\u4f48\u201d\uff0c\u9019\u4e9b\u60c5\u6cc1\u7684\u51fa\u73fe\u6a5f\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u4e26\u4e0d\u80fd\u771f\u5be6\u5730\u53cd\u6620\u6f14\u7b97\u6cd5\u57f7\u884c\u6548\u7387\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u9ad4\u73fe\u6f14\u7b97\u6cd5\u5728\u96a8\u6a5f\u8f38\u5165\u8cc7\u6599\u4e0b\u7684\u57f7\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8a18\u865f\u4f86\u8868\u793a\u3002</p> <p>\u5c0d\u65bc\u90e8\u5206\u6f14\u7b97\u6cd5\uff0c\u6211\u5011\u53ef\u4ee5\u7c21\u55ae\u5730\u63a8\u7b97\u51fa\u96a8\u6a5f\u8cc7\u6599\u5206\u4f48\u4e0b\u7684\u5e73\u5747\u60c5\u6cc1\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u65bc\u8f38\u5165\u9663\u5217\u662f\u88ab\u6253\u4e82\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73fe\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6a5f\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u9ebc\u6f14\u7b97\u6cd5\u7684\u5e73\u5747\u8ff4\u5708\u6b21\u6578\u5c31\u662f\u9663\u5217\u9577\u5ea6\u7684\u4e00\u534a \\(n / 2\\) \uff0c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(\\Theta(n / 2) = \\Theta(n)\\) \u3002</p> <p>\u4f46\u5c0d\u65bc\u8f03\u70ba\u8907\u96dc\u7684\u6f14\u7b97\u6cd5\uff0c\u8a08\u7b97\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u5f80\u5f80\u6bd4\u8f03\u56f0\u96e3\uff0c\u56e0\u70ba\u5f88\u96e3\u5206\u6790\u51fa\u5728\u8cc7\u6599\u5206\u4f48\u4e0b\u7684\u6574\u9ad4\u6578\u5b78\u671f\u671b\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u6211\u5011\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u4f5c\u70ba\u6f14\u7b97\u6cd5\u6548\u7387\u7684\u8a55\u5224\u6a19\u6e96\u3002</p> <p>\u70ba\u4ec0\u9ebc\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u865f\uff1f</p> <p>\u53ef\u80fd\u7531\u65bc \\(O\\) \u7b26\u865f\u904e\u65bc\u6717\u6717\u4e0a\u53e3\uff0c\u56e0\u6b64\u6211\u5011\u5e38\u5e38\u4f7f\u7528\u5b83\u4f86\u8868\u793a\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u3002\u4f46\u5f9e\u56b4\u683c\u610f\u7fa9\u4e0a\u8b1b\uff0c\u9019\u7a2e\u505a\u6cd5\u4e26\u4e0d\u898f\u7bc4\u3002\u5728\u672c\u66f8\u548c\u5176\u4ed6\u8cc7\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u985e\u4f3c\u201c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8acb\u5c07\u5176\u76f4\u63a5\u7406\u89e3\u70ba \\(\\Theta(n)\\) \u3002</p>"},{"location":"chapter_data_structure/","title":"\u7b2c 3 \u7ae0 \u00a0 \u8cc7\u6599\u7d50\u69cb","text":"<p>Abstract</p> <p>\u8cc7\u6599\u7d50\u69cb\u5982\u540c\u4e00\u526f\u7a69\u56fa\u800c\u591a\u6a23\u7684\u6846\u67b6\u3002</p> <p>\u5b83\u70ba\u8cc7\u6599\u7684\u6709\u5e8f\u7d44\u7e54\u63d0\u4f9b\u4e86\u85cd\u5716\uff0c\u6f14\u7b97\u6cd5\u5f97\u4ee5\u5728\u6b64\u57fa\u790e\u4e0a\u751f\u52d5\u8d77\u4f86\u3002</p>"},{"location":"chapter_data_structure/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>3.1 \u00a0 \u8cc7\u6599\u7d50\u69cb\u5206\u985e</li> <li>3.2 \u00a0 \u57fa\u672c\u8cc7\u6599\u578b\u5225</li> <li>3.3 \u00a0 \u6578\u5b57\u7de8\u78bc *</li> <li>3.4 \u00a0 \u5b57\u5143\u7de8\u78bc *</li> <li>3.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_data_structure/basic_data_types/","title":"3.2 \u00a0 \u57fa\u672c\u8cc7\u6599\u578b\u5225","text":"<p>\u7576\u8ac7\u53ca\u8a08\u7b97\u6a5f\u4e2d\u7684\u8cc7\u6599\u6642\uff0c\u6211\u5011\u6703\u60f3\u5230\u6587\u5b57\u3001\u5716\u7247\u3001\u5f71\u7247\u3001\u8a9e\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u7a2e\u5f62\u5f0f\u3002\u5118\u7ba1\u9019\u4e9b\u8cc7\u6599\u7684\u7d44\u7e54\u5f62\u5f0f\u5404\u7570\uff0c\u4f46\u5b83\u5011\u90fd\u7531\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u69cb\u6210\u3002</p> <p>\u57fa\u672c\u8cc7\u6599\u578b\u5225\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u9032\u884c\u904b\u7b97\u7684\u578b\u5225\uff0c\u5728\u6f14\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\uff0c\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u5e7e\u7a2e\u3002</p> <ul> <li>\u6574\u6578\u578b\u5225 <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \u3002</li> <li>\u6d6e\u9ede\u6578\u578b\u5225 <code>float</code>\u3001<code>double</code> \uff0c\u7528\u65bc\u8868\u793a\u5c0f\u6578\u3002</li> <li>\u5b57\u5143\u578b\u5225 <code>char</code> \uff0c\u7528\u65bc\u8868\u793a\u5404\u7a2e\u8a9e\u8a00\u7684\u5b57\u6bcd\u3001\u6a19\u9ede\u7b26\u865f\u751a\u81f3\u8868\u60c5\u7b26\u865f\u7b49\u3002</li> <li>\u5e03\u6797\u578b\u5225 <code>bool</code> \uff0c\u7528\u65bc\u8868\u793a\u201c\u662f\u201d\u8207\u201c\u5426\u201d\u5224\u65b7\u3002</li> </ul> <p>\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4ee5\u4e8c\u9032\u4f4d\u5236\u7684\u5f62\u5f0f\u5132\u5b58\u5728\u8a08\u7b97\u6a5f\u4e2d\u3002\u4e00\u500b\u4e8c\u9032\u4f4d\u5236\u4f4d\u5373\u70ba \\(1\\) \u4f4d\u5143\u3002\u5728\u7d55\u5927\u591a\u6578\u73fe\u4ee3\u4f5c\u696d\u7cfb\u7d71\u4e2d\uff0c\\(1\\) \u4f4d\u5143\u7d44\uff08byte\uff09\u7531 \\(8\\) \u4f4d\u5143\uff08bit\uff09\u7d44\u6210\u3002</p> <p>\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u53d6\u503c\u7bc4\u570d\u53d6\u6c7a\u65bc\u5176\u4f54\u7528\u7684\u7a7a\u9593\u5927\u5c0f\u3002\u4e0b\u9762\u4ee5 Java \u70ba\u4f8b\u3002</p> <ul> <li>\u6574\u6578\u578b\u5225 <code>byte</code> \u4f54\u7528 \\(1\\) \u4f4d\u5143\u7d44 = \\(8\\) \u4f4d\u5143 \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u500b\u6578\u5b57\u3002</li> <li>\u6574\u6578\u578b\u5225 <code>int</code> \u4f54\u7528 \\(4\\) \u4f4d\u5143\u7d44 = \\(32\\) \u4f4d\u5143 \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u500b\u6578\u5b57\u3002</li> </ul> <p>\u8868 3-1 \u5217\u8209\u4e86 Java \u4e2d\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u4f54\u7528\u7a7a\u9593\u3001\u53d6\u503c\u7bc4\u570d\u548c\u9810\u8a2d\u503c\u3002\u6b64\u8868\u683c\u7121\u9808\u6b7b\u8a18\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u6642\u53ef\u4ee5\u900f\u904e\u67e5\u8868\u4f86\u56de\u61b6\u3002</p> <p> \u8868 3-1 \u00a0 \u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u4f54\u7528\u7a7a\u9593\u548c\u53d6\u503c\u7bc4\u570d </p> \u578b\u5225 \u7b26\u865f \u4f54\u7528\u7a7a\u9593 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9810\u8a2d\u503c \u6574\u6578 <code>byte</code> 1 \u4f4d\u5143\u7d44 \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2 \u4f4d\u5143\u7d44 \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4 \u4f4d\u5143\u7d44 \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8 \u4f4d\u5143\u7d44 \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u9ede\u6578 <code>float</code> 4 \u4f4d\u5143\u7d44 \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 \u4f4d\u5143\u7d44 \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u5143 <code>char</code> 2 \u4f4d\u5143\u7d44 \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u6797 <code>bool</code> 1 \u4f4d\u5143\u7d44 \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>\u8acb\u6ce8\u610f\uff0c\u8868 3-1 \u91dd\u5c0d\u7684\u662f Java \u7684\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u60c5\u6cc1\u3002\u6bcf\u7a2e\u7a0b\u5f0f\u8a9e\u8a00\u90fd\u6709\u5404\u81ea\u7684\u8cc7\u6599\u578b\u5225\u5b9a\u7fa9\uff0c\u5b83\u5011\u7684\u4f54\u7528\u7a7a\u9593\u3001\u53d6\u503c\u7bc4\u570d\u548c\u9810\u8a2d\u503c\u53ef\u80fd\u6703\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u5728 Python \u4e2d\uff0c\u6574\u6578\u578b\u5225 <code>int</code> \u53ef\u4ee5\u662f\u4efb\u610f\u5927\u5c0f\uff0c\u53ea\u53d7\u9650\u65bc\u53ef\u7528\u8a18\u61b6\u9ad4\uff1b\u6d6e\u9ede\u6578 <code>float</code> \u662f\u96d9\u7cbe\u5ea6 64 \u4f4d\uff1b\u6c92\u6709 <code>char</code> \u578b\u5225\uff0c\u55ae\u500b\u5b57\u5143\u5be6\u969b\u4e0a\u662f\u9577\u5ea6\u70ba 1 \u7684\u5b57\u4e32 <code>str</code> \u3002</li> <li>C \u548c C++ \u672a\u660e\u78ba\u898f\u5b9a\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u5927\u5c0f\uff0c\u800c\u56e0\u5be6\u73fe\u548c\u5e73\u81fa\u5404\u7570\u3002\u8868 3-1 \u9075\u5faa LP64 \u8cc7\u6599\u6a21\u578b\uff0c\u5176\u7528\u65bc\u5305\u62ec Linux \u548c macOS \u5728\u5167\u7684 Unix 64 \u4f4d\u4f5c\u696d\u7cfb\u7d71\u3002</li> <li>\u5b57\u5143 <code>char</code> \u7684\u5927\u5c0f\u5728 C \u548c C++ \u4e2d\u70ba 1 \u4f4d\u5143\u7d44\uff0c\u5728\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u53d6\u6c7a\u65bc\u7279\u5b9a\u7684\u5b57\u5143\u7de8\u78bc\u65b9\u6cd5\uff0c\u8a73\u898b\u201c\u5b57\u5143\u7de8\u78bc\u201d\u7ae0\u7bc0\u3002</li> <li>\u5373\u4f7f\u8868\u793a\u5e03\u6797\u91cf\u50c5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u8a18\u61b6\u9ad4\u4e2d\u901a\u5e38\u4e5f\u5132\u5b58\u70ba 1 \u4f4d\u5143\u7d44\u3002\u9019\u662f\u56e0\u70ba\u73fe\u4ee3\u8a08\u7b97\u6a5f CPU \u901a\u5e38\u5c07 1 \u4f4d\u5143\u7d44\u4f5c\u70ba\u6700\u5c0f\u5b9a\u5740\u8a18\u61b6\u9ad4\u55ae\u5143\u3002</li> </ul> <p>\u90a3\u9ebc\uff0c\u57fa\u672c\u8cc7\u6599\u578b\u5225\u8207\u8cc7\u6599\u7d50\u69cb\u4e4b\u9593\u6709\u4ec0\u9ebc\u95dc\u806f\u5462\uff1f\u6211\u5011\u77e5\u9053\uff0c\u8cc7\u6599\u7d50\u69cb\u662f\u5728\u8a08\u7b97\u6a5f\u4e2d\u7d44\u7e54\u8207\u5132\u5b58\u8cc7\u6599\u7684\u65b9\u5f0f\u3002\u9019\u53e5\u8a71\u7684\u4e3b\u8a9e\u662f\u201c\u7d50\u69cb\u201d\u800c\u975e\u201c\u8cc7\u6599\u201d\u3002</p> <p>\u5982\u679c\u60f3\u8868\u793a\u201c\u4e00\u6392\u6578\u5b57\u201d\uff0c\u6211\u5011\u81ea\u7136\u6703\u60f3\u5230\u4f7f\u7528\u9663\u5217\u3002\u9019\u662f\u56e0\u70ba\u9663\u5217\u7684\u7dda\u6027\u7d50\u69cb\u53ef\u4ee5\u8868\u793a\u6578\u5b57\u7684\u76f8\u9130\u95dc\u4fc2\u548c\u9806\u5e8f\u95dc\u4fc2\uff0c\u4f46\u81f3\u65bc\u5132\u5b58\u7684\u5167\u5bb9\u662f\u6574\u6578 <code>int</code>\u3001\u5c0f\u6578 <code>float</code> \u9084\u662f\u5b57\u5143 <code>char</code> \uff0c\u5247\u8207\u201c\u8cc7\u6599\u7d50\u69cb\u201d\u7121\u95dc\u3002</p> <p>\u63db\u53e5\u8a71\u8aaa\uff0c\u57fa\u672c\u8cc7\u6599\u578b\u5225\u63d0\u4f9b\u4e86\u8cc7\u6599\u7684\u201c\u5167\u5bb9\u578b\u5225\u201d\uff0c\u800c\u8cc7\u6599\u7d50\u69cb\u63d0\u4f9b\u4e86\u8cc7\u6599\u7684\u201c\u7d44\u7e54\u65b9\u5f0f\u201d\u3002\u4f8b\u5982\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u7528\u76f8\u540c\u7684\u8cc7\u6599\u7d50\u69cb\uff08\u9663\u5217\uff09\u4f86\u5132\u5b58\u8207\u8868\u793a\u4e0d\u540c\u7684\u57fa\u672c\u8cc7\u6599\u578b\u5225\uff0c\u5305\u62ec <code>int</code>\u3001<code>float</code>\u3001<code>char</code>\u3001<code>bool</code> \u7b49\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python \u7684\u5b57\u5143\u5be6\u969b\u4e0a\u662f\u9577\u5ea6\u70ba 1 \u7684\u5b57\u4e32\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python \u7684\u4e32\u5217\u53ef\u4ee5\u81ea\u7531\u5132\u5b58\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u548c\u7269\u4ef6\u5f15\u7528\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript \u7684\u9663\u5217\u53ef\u4ee5\u81ea\u7531\u5132\u5b58\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u548c\u7269\u4ef6\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0; 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nval numbers = IntArray(5)\nval decinals = FloatArray(5)\nval characters = CharArray(5)\nval bools = BooleanArray(5)\n</code></pre> <pre><code># Ruby \u7684\u4e32\u5217\u53ef\u4ee5\u81ea\u7531\u5132\u5b58\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u548c\u7269\u4ef6\u5f15\u7528\ndata = [0, 0.0, 'a', false, ListNode(0)]\n</code></pre> <pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 \u5b57\u5143\u7de8\u78bc *","text":"<p>\u5728\u8a08\u7b97\u6a5f\u4e2d\uff0c\u6240\u6709\u8cc7\u6599\u90fd\u662f\u4ee5\u4e8c\u9032\u4f4d\u5236\u6578\u7684\u5f62\u5f0f\u5132\u5b58\u7684\uff0c\u5b57\u5143 <code>char</code> \u4e5f\u4e0d\u4f8b\u5916\u3002\u70ba\u4e86\u8868\u793a\u5b57\u5143\uff0c\u6211\u5011\u9700\u8981\u5efa\u7acb\u4e00\u5957\u201c\u5b57\u7b26\u96c6\u201d\uff0c\u898f\u5b9a\u6bcf\u500b\u5b57\u5143\u548c\u4e8c\u9032\u4f4d\u5236\u6578\u4e4b\u9593\u7684\u4e00\u4e00\u5c0d\u61c9\u95dc\u4fc2\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u5f8c\uff0c\u8a08\u7b97\u6a5f\u5c31\u53ef\u4ee5\u900f\u904e\u67e5\u8868\u5b8c\u6210\u4e8c\u9032\u4f4d\u5236\u6578\u5230\u5b57\u5143\u7684\u8f49\u63db\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1 \u00a0 ASCII \u5b57\u7b26\u96c6","text":"<p>ASCII \u78bc\u662f\u6700\u65e9\u51fa\u73fe\u7684\u5b57\u7b26\u96c6\uff0c\u5176\u5168\u7a31\u70ba American Standard Code for Information Interchange\uff08\u7f8e\u570b\u6a19\u6e96\u8cc7\u8a0a\u4ea4\u63db\u7a0b\u5f0f\u78bc\uff09\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u9032\u4f4d\u5236\u6578\uff08\u4e00\u500b\u4f4d\u5143\u7d44\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u500b\u5b57\u5143\uff0c\u6700\u591a\u80fd\u5920\u8868\u793a 128 \u500b\u4e0d\u540c\u7684\u5b57\u5143\u3002\u5982\u5716 3-6 \u6240\u793a\uff0cASCII \u78bc\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5beb\u3001\u6578\u5b57 0 ~ 9\u3001\u4e00\u4e9b\u6a19\u9ede\u7b26\u865f\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u5143\uff08\u5982\u63db\u884c\u7b26\u548c\u88fd\u8868\u7b26\uff09\u3002</p> <p></p> <p> \u5716 3-6 \u00a0 ASCII \u78bc </p> <p>\u7136\u800c\uff0cASCII \u78bc\u50c5\u80fd\u5920\u8868\u793a\u82f1\u6587\u3002\u96a8\u8457\u8a08\u7b97\u6a5f\u7684\u5168\u7403\u5316\uff0c\u8a95\u751f\u4e86\u4e00\u7a2e\u80fd\u5920\u8868\u793a\u66f4\u591a\u8a9e\u8a00\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u790e\u4e0a\u64f4\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u5920\u8868\u793a 256 \u500b\u4e0d\u540c\u7684\u5b57\u5143\u3002</p> <p>\u5728\u4e16\u754c\u7bc4\u570d\u5167\uff0c\u9678\u7e8c\u51fa\u73fe\u4e86\u4e00\u6279\u9069\u7528\u65bc\u4e0d\u540c\u5730\u5340\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u9019\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u500b\u5b57\u5143\u7d71\u4e00\u70ba ASCII \u78bc\uff0c\u5f8c 128 \u500b\u5b57\u5143\u5b9a\u7fa9\u4e0d\u540c\uff0c\u4ee5\u9069\u61c9\u4e0d\u540c\u8a9e\u8a00\u7684\u9700\u6c42\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2 \u00a0 GBK \u5b57\u7b26\u96c6","text":"<p>\u5f8c\u4f86\u4eba\u5011\u767c\u73fe\uff0cEASCII \u78bc\u4ecd\u7136\u7121\u6cd5\u6eff\u8db3\u8a31\u591a\u8a9e\u8a00\u7684\u5b57\u5143\u6578\u91cf\u8981\u6c42\u3002\u6bd4\u5982\u6f22\u5b57\u6709\u8fd1\u5341\u842c\u500b\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u5e7e\u5343\u500b\u3002\u4e2d\u570b\u570b\u5bb6\u6a19\u6e96\u7e3d\u5c40\u65bc 1980 \u5e74\u91cb\u51fa\u4e86 GB2312 \u5b57\u7b26\u96c6\uff0c\u5176\u6536\u9304\u4e86 6763 \u500b\u6f22\u5b57\uff0c\u57fa\u672c\u6eff\u8db3\u4e86\u6f22\u5b57\u7684\u8a08\u7b97\u6a5f\u8655\u7406\u9700\u8981\u3002</p> <p>\u7136\u800c\uff0cGB2312 \u7121\u6cd5\u8655\u7406\u90e8\u5206\u7f55\u898b\u5b57\u548c\u7e41\u9ad4\u5b57\u3002GBK \u5b57\u7b26\u96c6\u662f\u5728 GB2312 \u7684\u57fa\u790e\u4e0a\u64f4\u5c55\u5f97\u5230\u7684\uff0c\u5b83\u5171\u6536\u9304\u4e86 21886 \u500b\u6f22\u5b57\u3002\u5728 GBK \u7684\u7de8\u78bc\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u5143\u4f7f\u7528\u4e00\u500b\u4f4d\u5143\u7d44\u8868\u793a\uff0c\u6f22\u5b57\u4f7f\u7528\u5169\u500b\u4f4d\u5143\u7d44\u8868\u793a\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3 \u00a0 Unicode \u5b57\u7b26\u96c6","text":"<p>\u96a8\u8457\u8a08\u7b97\u6a5f\u6280\u8853\u7684\u84ec\u52c3\u767c\u5c55\uff0c\u5b57\u7b26\u96c6\u8207\u7de8\u78bc\u6a19\u6e96\u767e\u82b1\u9f4a\u653e\uff0c\u800c\u9019\u5e36\u4f86\u4e86\u8a31\u591a\u554f\u984c\u3002\u4e00\u65b9\u9762\uff0c\u9019\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u7fa9\u4e86\u7279\u5b9a\u8a9e\u8a00\u7684\u5b57\u5143\uff0c\u7121\u6cd5\u5728\u591a\u8a9e\u8a00\u74b0\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u7a2e\u8a9e\u8a00\u5b58\u5728\u591a\u7a2e\u5b57\u7b26\u96c6\u6a19\u6e96\uff0c\u5982\u679c\u5169\u81fa\u8a08\u7b97\u6a5f\u4f7f\u7528\u7684\u662f\u4e0d\u540c\u7684\u7de8\u78bc\u6a19\u6e96\uff0c\u5247\u5728\u8cc7\u8a0a\u50b3\u905e\u6642\u5c31\u6703\u51fa\u73fe\u4e82\u78bc\u3002</p> <p>\u90a3\u500b\u6642\u4ee3\u7684\u7814\u7a76\u4eba\u54e1\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u500b\u8db3\u5920\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c07\u4e16\u754c\u7bc4\u570d\u5167\u7684\u6240\u6709\u8a9e\u8a00\u548c\u7b26\u865f\u90fd\u6536\u9304\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u6c7a\u8de8\u8a9e\u8a00\u74b0\u5883\u548c\u4e82\u78bc\u554f\u984c\u4e86\u55ce\uff1f\u5728\u9019\u7a2e\u60f3\u6cd5\u7684\u9a45\u52d5\u4e0b\uff0c\u4e00\u500b\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u61c9\u904b\u800c\u751f\u3002</p> <p>Unicode \u7684\u4e2d\u6587\u540d\u7a31\u70ba\u201c\u7d71\u4e00\u78bc\u201d\uff0c\u7406\u8ad6\u4e0a\u80fd\u5bb9\u7d0d 100 \u591a\u842c\u500b\u5b57\u5143\u3002\u5b83\u81f4\u529b\u65bc\u5c07\u5168\u7403\u7bc4\u570d\u5167\u7684\u5b57\u5143\u7d0d\u5165\u7d71\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u7a2e\u901a\u7528\u7684\u5b57\u7b26\u96c6\u4f86\u8655\u7406\u548c\u986f\u793a\u5404\u7a2e\u8a9e\u8a00\u6587\u5b57\uff0c\u6e1b\u5c11\u56e0\u70ba\u7de8\u78bc\u6a19\u6e96\u4e0d\u540c\u800c\u7522\u751f\u7684\u4e82\u78bc\u554f\u984c\u3002</p> <p>\u81ea 1991 \u5e74\u91cb\u51fa\u4ee5\u4f86\uff0cUnicode \u4e0d\u65b7\u64f4\u5145\u65b0\u7684\u8a9e\u8a00\u8207\u5b57\u5143\u3002\u622a\u81f3 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7d93\u5305\u542b 149186 \u500b\u5b57\u5143\uff0c\u5305\u62ec\u5404\u7a2e\u8a9e\u8a00\u7684\u5b57\u5143\u3001\u7b26\u865f\u751a\u81f3\u8868\u60c5\u7b26\u865f\u7b49\u3002\u5728\u9f90\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u5143\u4f54\u7528 2 \u4f4d\u5143\u7d44\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u5143\u4f54\u7528 3 \u4f4d\u5143\u7d44\u751a\u81f3 4 \u4f4d\u5143\u7d44\u3002</p> <p>Unicode \u662f\u4e00\u7a2e\u901a\u7528\u5b57\u7b26\u96c6\uff0c\u672c\u8cea\u4e0a\u662f\u7d66\u6bcf\u500b\u5b57\u5143\u5206\u914d\u4e00\u500b\u7de8\u865f\uff08\u7a31\u70ba\u201c\u78bc\u9ede\u201d\uff09\uff0c\u4f46\u5b83\u4e26\u6c92\u6709\u898f\u5b9a\u5728\u8a08\u7b97\u6a5f\u4e2d\u5982\u4f55\u5132\u5b58\u9019\u4e9b\u5b57\u5143\u78bc\u9ede\u3002\u6211\u5011\u4e0d\u7981\u6703\u554f\uff1a\u7576\u591a\u7a2e\u9577\u5ea6\u7684 Unicode \u78bc\u9ede\u540c\u6642\u51fa\u73fe\u5728\u4e00\u500b\u6587\u5b57\u4e2d\u6642\uff0c\u7cfb\u7d71\u5982\u4f55\u89e3\u6790\u5b57\u5143\uff1f\u4f8b\u5982\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba 2 \u4f4d\u5143\u7d44\u7684\u7de8\u78bc\uff0c\u7cfb\u7d71\u5982\u4f55\u78ba\u8a8d\u5b83\u662f\u4e00\u500b 2 \u4f4d\u5143\u7d44\u7684\u5b57\u5143\u9084\u662f\u5169\u500b 1 \u4f4d\u5143\u7d44\u7684\u5b57\u5143\uff1f</p> <p>\u5c0d\u65bc\u4ee5\u4e0a\u554f\u984c\uff0c\u4e00\u7a2e\u76f4\u63a5\u7684\u89e3\u6c7a\u65b9\u6848\u662f\u5c07\u6240\u6709\u5b57\u5143\u5132\u5b58\u70ba\u7b49\u9577\u7684\u7de8\u78bc\u3002\u5982\u5716 3-7 \u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u500b\u5b57\u5143\u4f54\u7528 1 \u4f4d\u5143\u7d44\uff0c\u201c\u6f14\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u500b\u5b57\u5143\u4f54\u7528 2 \u4f4d\u5143\u7d44\u3002\u6211\u5011\u53ef\u4ee5\u900f\u904e\u9ad8\u4f4d\u586b 0 \u5c07\u201cHello \u6f14\u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u5143\u90fd\u7de8\u78bc\u70ba 2 \u4f4d\u5143\u7d44\u9577\u5ea6\u3002\u9019\u6a23\u7cfb\u7d71\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u4f4d\u5143\u7d44\u89e3\u6790\u4e00\u500b\u5b57\u5143\uff0c\u6062\u5fa9\u9019\u500b\u77ed\u8a9e\u7684\u5167\u5bb9\u4e86\u3002</p> <p></p> <p> \u5716 3-7 \u00a0 Unicode \u7de8\u78bc\u793a\u4f8b </p> <p>\u7136\u800c ASCII \u78bc\u5df2\u7d93\u5411\u6211\u5011\u8b49\u660e\uff0c\u7de8\u78bc\u82f1\u6587\u53ea\u9700 1 \u4f4d\u5143\u7d44\u3002\u82e5\u63a1\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u5b57\u4f54\u7528\u7a7a\u9593\u7684\u5927\u5c0f\u5c07\u6703\u662f ASCII \u7de8\u78bc\u4e0b\u7684\u5169\u500d\uff0c\u975e\u5e38\u6d6a\u8cbb\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002\u56e0\u6b64\uff0c\u6211\u5011\u9700\u8981\u4e00\u7a2e\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7de8\u78bc\u65b9\u6cd5\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4 \u00a0 UTF-8 \u7de8\u78bc","text":"<p>\u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u70ba\u570b\u969b\u4e0a\u4f7f\u7528\u6700\u5ee3\u6cdb\u7684 Unicode \u7de8\u78bc\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u7a2e\u53ef\u8b8a\u9577\u5ea6\u7684\u7de8\u78bc\uff0c\u4f7f\u7528 1 \u5230 4 \u4f4d\u5143\u7d44\u4f86\u8868\u793a\u4e00\u500b\u5b57\u5143\uff0c\u6839\u64da\u5b57\u5143\u7684\u8907\u96dc\u6027\u800c\u8b8a\u3002ASCII \u5b57\u5143\u53ea\u9700 1 \u4f4d\u5143\u7d44\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u81d8\u5b57\u6bcd\u9700\u8981 2 \u4f4d\u5143\u7d44\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u5143\u9700\u8981 3 \u4f4d\u5143\u7d44\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u5143\u9700\u8981 4 \u4f4d\u5143\u7d44\u3002</p> <p>UTF-8 \u7684\u7de8\u78bc\u898f\u5247\u4e26\u4e0d\u8907\u96dc\uff0c\u5206\u70ba\u4ee5\u4e0b\u5169\u7a2e\u60c5\u6cc1\u3002</p> <ul> <li>\u5c0d\u65bc\u9577\u5ea6\u70ba 1 \u4f4d\u5143\u7d44\u7684\u5b57\u5143\uff0c\u5c07\u6700\u9ad8\u4f4d\u8a2d\u5b9a\u70ba \\(0\\) \uff0c\u5176\u9918 7 \u4f4d\u8a2d\u5b9a\u70ba Unicode \u78bc\u9ede\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u5143\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u4f54\u64da\u4e86\u524d 128 \u500b\u78bc\u9ede\u3002\u4e5f\u5c31\u662f\u8aaa\uff0cUTF-8 \u7de8\u78bc\u53ef\u4ee5\u5411\u4e0b\u76f8\u5bb9 ASCII \u78bc\u3002\u9019\u610f\u5473\u8457\u6211\u5011\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u4f86\u89e3\u6790\u5e74\u4ee3\u4e45\u9060\u7684 ASCII \u78bc\u6587\u5b57\u3002</li> <li>\u5c0d\u65bc\u9577\u5ea6\u70ba \\(n\\) \u4f4d\u5143\u7d44\u7684\u5b57\u5143\uff08\u5176\u4e2d \\(n &gt; 1\\)\uff09\uff0c\u5c07\u9996\u500b\u4f4d\u5143\u7d44\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8a2d\u5b9a\u70ba \\(1\\) \uff0c\u7b2c \\(n + 1\\) \u4f4d\u8a2d\u5b9a\u70ba \\(0\\) \uff1b\u5f9e\u7b2c\u4e8c\u500b\u4f4d\u5143\u7d44\u958b\u59cb\uff0c\u5c07\u6bcf\u500b\u4f4d\u5143\u7d44\u7684\u9ad8 2 \u4f4d\u90fd\u8a2d\u5b9a\u70ba \\(10\\) \uff1b\u5176\u9918\u6240\u6709\u4f4d\u7528\u65bc\u586b\u5145\u5b57\u5143\u7684 Unicode \u78bc\u9ede\u3002</li> </ul> <p>\u5716 3-8 \u5c55\u793a\u4e86\u201cHello\u6f14\u7b97\u6cd5\u201d\u5c0d\u61c9\u7684 UTF-8 \u7de8\u78bc\u3002\u89c0\u5bdf\u767c\u73fe\uff0c\u7531\u65bc\u6700\u9ad8 \\(n\\) \u4f4d\u90fd\u8a2d\u5b9a\u70ba \\(1\\) \uff0c\u56e0\u6b64\u7cfb\u7d71\u53ef\u4ee5\u900f\u904e\u8b80\u53d6\u6700\u9ad8\u4f4d \\(1\\) \u7684\u500b\u6578\u4f86\u89e3\u6790\u51fa\u5b57\u5143\u7684\u9577\u5ea6\u70ba \\(n\\) \u3002</p> <p>\u4f46\u70ba\u4ec0\u9ebc\u8981\u5c07\u5176\u9918\u6240\u6709\u4f4d\u5143\u7d44\u7684\u9ad8 2 \u4f4d\u90fd\u8a2d\u5b9a\u70ba \\(10\\) \u5462\uff1f\u5be6\u969b\u4e0a\uff0c\u9019\u500b \\(10\\) \u80fd\u5920\u8d77\u5230\u6821\u9a57\u7b26\u7684\u4f5c\u7528\u3002\u5047\u8a2d\u7cfb\u7d71\u5f9e\u4e00\u500b\u932f\u8aa4\u7684\u4f4d\u5143\u7d44\u958b\u59cb\u89e3\u6790\u6587\u5b57\uff0c\u4f4d\u5143\u7d44\u982d\u90e8\u7684 \\(10\\) \u80fd\u5920\u5e6b\u52a9\u7cfb\u7d71\u5feb\u901f\u5224\u65b7\u51fa\u7570\u5e38\u3002</p> <p>\u4e4b\u6240\u4ee5\u5c07 \\(10\\) \u7576\u4f5c\u6821\u9a57\u7b26\uff0c\u662f\u56e0\u70ba\u5728 UTF-8 \u7de8\u78bc\u898f\u5247\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u5143\u7684\u6700\u9ad8\u5169\u4f4d\u662f \\(10\\) \u3002\u9019\u500b\u7d50\u8ad6\u53ef\u4ee5\u7528\u53cd\u8b49\u6cd5\u4f86\u8b49\u660e\uff1a\u5047\u8a2d\u4e00\u500b\u5b57\u5143\u7684\u6700\u9ad8\u5169\u4f4d\u662f \\(10\\) \uff0c\u8aaa\u660e\u8a72\u5b57\u5143\u7684\u9577\u5ea6\u70ba \\(1\\) \uff0c\u5c0d\u61c9 ASCII \u78bc\u3002\u800c ASCII \u78bc\u7684\u6700\u9ad8\u4f4d\u61c9\u8a72\u662f \\(0\\) \uff0c\u8207\u5047\u8a2d\u77db\u76fe\u3002</p> <p></p> <p> \u5716 3-8 \u00a0 UTF-8 \u7de8\u78bc\u793a\u4f8b </p> <p>\u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u898b\u7684\u7de8\u78bc\u65b9\u5f0f\u9084\u5305\u62ec\u4ee5\u4e0b\u5169\u7a2e\u3002</p> <ul> <li>UTF-16 \u7de8\u78bc\uff1a\u4f7f\u7528 2 \u6216 4 \u4f4d\u5143\u7d44\u4f86\u8868\u793a\u4e00\u500b\u5b57\u5143\u3002\u6240\u6709\u7684 ASCII \u5b57\u5143\u548c\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u5143\uff0c\u90fd\u7528 2 \u4f4d\u5143\u7d44\u8868\u793a\uff1b\u5c11\u6578\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u4f4d\u5143\u7d44\u8868\u793a\u3002\u5c0d\u65bc 2 \u4f4d\u5143\u7d44\u7684\u5b57\u5143\uff0cUTF-16 \u7de8\u78bc\u8207 Unicode \u78bc\u9ede\u76f8\u7b49\u3002</li> <li>UTF-32 \u7de8\u78bc\uff1a\u6bcf\u500b\u5b57\u5143\u90fd\u4f7f\u7528 4 \u4f4d\u5143\u7d44\u3002\u9019\u610f\u5473\u8457 UTF-32 \u6bd4 UTF-8 \u548c UTF-16 \u66f4\u4f54\u7528\u7a7a\u9593\uff0c\u7279\u5225\u662f\u5c0d\u65bc ASCII \u5b57\u5143\u4f54\u6bd4\u8f03\u9ad8\u7684\u6587\u5b57\u3002</li> </ul> <p>\u5f9e\u5132\u5b58\u7a7a\u9593\u4f54\u7528\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u5143\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u70ba\u5b83\u50c5\u9700 1 \u4f4d\u5143\u7d44\uff1b\u4f7f\u7528 UTF-16 \u7de8\u78bc\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u5143\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u6703\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u70ba\u5b83\u50c5\u9700 2 \u4f4d\u5143\u7d44\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u4f4d\u5143\u7d44\u3002</p> <p>\u5f9e\u76f8\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8a31\u591a\u5de5\u5177\u548c\u5eab\u512a\u5148\u652f\u6301 UTF-8 \u3002</p>"},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5 \u00a0 \u7a0b\u5f0f\u8a9e\u8a00\u7684\u5b57\u5143\u7de8\u78bc","text":"<p>\u5c0d\u65bc\u4ee5\u5f80\u7684\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\uff0c\u7a0b\u5f0f\u57f7\u884c\u4e2d\u7684\u5b57\u4e32\u90fd\u63a1\u7528 UTF-16 \u6216 UTF-32 \u9019\u985e\u7b49\u9577\u7de8\u78bc\u3002\u5728\u7b49\u9577\u7de8\u78bc\u4e0b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5b57\u4e32\u770b\u4f5c\u9663\u5217\u4f86\u8655\u7406\uff0c\u9019\u7a2e\u505a\u6cd5\u5177\u6709\u4ee5\u4e0b\u512a\u9ede\u3002</p> <ul> <li>\u96a8\u6a5f\u8a2a\u554f\uff1aUTF-16 \u7de8\u78bc\u7684\u5b57\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u9032\u884c\u96a8\u6a5f\u8a2a\u554f\u3002UTF-8 \u662f\u4e00\u7a2e\u8b8a\u9577\u7de8\u78bc\uff0c\u8981\u60f3\u627e\u5230\u7b2c \\(i\\) \u500b\u5b57\u5143\uff0c\u6211\u5011\u9700\u8981\u5f9e\u5b57\u4e32\u7684\u958b\u59cb\u8655\u8d70\u8a2a\u5230\u7b2c \\(i\\) \u500b\u5b57\u5143\uff0c\u9019\u9700\u8981 \\(O(n)\\) \u7684\u6642\u9593\u3002</li> <li>\u5b57\u5143\u8a08\u6578\uff1a\u8207\u96a8\u6a5f\u8a2a\u554f\u985e\u4f3c\uff0c\u8a08\u7b97 UTF-16 \u7de8\u78bc\u7684\u5b57\u4e32\u7684\u9577\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8a08\u7b97 UTF-8 \u7de8\u78bc\u7684\u5b57\u4e32\u7684\u9577\u5ea6\u9700\u8981\u8d70\u8a2a\u6574\u500b\u5b57\u4e32\u3002</li> <li>\u5b57\u4e32\u64cd\u4f5c\uff1a\u5728 UTF-16 \u7de8\u78bc\u7684\u5b57\u4e32\u4e0a\uff0c\u5f88\u591a\u5b57\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u9023\u7dda\u3001\u63d2\u5165\u3001\u522a\u9664\u7b49\uff09\u66f4\u5bb9\u6613\u9032\u884c\u3002\u5728 UTF-8 \u7de8\u78bc\u7684\u5b57\u4e32\u4e0a\uff0c\u9032\u884c\u9019\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u984d\u5916\u7684\u8a08\u7b97\uff0c\u4ee5\u78ba\u4fdd\u4e0d\u6703\u7522\u751f\u7121\u6548\u7684 UTF-8 \u7de8\u78bc\u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u7a0b\u5f0f\u8a9e\u8a00\u7684\u5b57\u5143\u7de8\u78bc\u65b9\u6848\u8a2d\u8a08\u662f\u4e00\u500b\u5f88\u6709\u8da3\u7684\u8a71\u984c\uff0c\u6d89\u53ca\u8a31\u591a\u56e0\u7d20\u3002</p> <ul> <li>Java \u7684 <code>String</code> \u578b\u5225\u4f7f\u7528 UTF-16 \u7de8\u78bc\uff0c\u6bcf\u500b\u5b57\u5143\u4f54\u7528 2 \u4f4d\u5143\u7d44\u3002\u9019\u662f\u56e0\u70ba Java \u8a9e\u8a00\u8a2d\u8a08\u4e4b\u521d\uff0c\u4eba\u5011\u8a8d\u70ba 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u5143\u3002\u7136\u800c\uff0c\u9019\u662f\u4e00\u500b\u4e0d\u6b63\u78ba\u7684\u5224\u65b7\u3002\u5f8c\u4f86 Unicode \u898f\u7bc4\u64f4\u5c55\u5230\u4e86\u8d85\u904e 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u5143\u73fe\u5728\u53ef\u80fd\u7531\u4e00\u5c0d 16 \u4f4d\u7684\u503c\uff08\u7a31\u70ba\u201c\u4ee3\u7406\u5c0d\u201d\uff09\u8868\u793a\u3002</li> <li>JavaScript \u548c TypeScript \u7684\u5b57\u4e32\u4f7f\u7528 UTF-16 \u7de8\u78bc\u7684\u539f\u56e0\u8207 Java \u985e\u4f3c\u3002\u7576 1995 \u5e74 Netscape \u516c\u53f8\u9996\u6b21\u63a8\u51fa JavaScript \u8a9e\u8a00\u6642\uff0cUnicode \u9084\u8655\u65bc\u767c\u5c55\u65e9\u671f\uff0c\u90a3\u6642\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7de8\u78bc\u5c31\u8db3\u4ee5\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u5143\u4e86\u3002</li> <li>C# \u4f7f\u7528 UTF-16 \u7de8\u78bc\uff0c\u4e3b\u8981\u662f\u56e0\u70ba .NET \u5e73\u81fa\u662f\u7531 Microsoft \u8a2d\u8a08\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u8853\uff08\u5305\u62ec Windows \u4f5c\u696d\u7cfb\u7d71\uff09\u90fd\u5ee3\u6cdb\u4f7f\u7528 UTF-16 \u7de8\u78bc\u3002</li> </ul> <p>\u7531\u65bc\u4ee5\u4e0a\u7a0b\u5f0f\u8a9e\u8a00\u5c0d\u5b57\u5143\u6578\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u5011\u4e0d\u5f97\u4e0d\u63a1\u53d6\u201c\u4ee3\u7406\u5c0d\u201d\u7684\u65b9\u5f0f\u4f86\u8868\u793a\u8d85\u904e 16 \u4f4d\u9577\u5ea6\u7684 Unicode \u5b57\u5143\u3002\u9019\u662f\u4e00\u500b\u4e0d\u5f97\u5df2\u70ba\u4e4b\u7684\u7121\u5948\u4e4b\u8209\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5c0d\u7684\u5b57\u4e32\u4e2d\uff0c\u4e00\u500b\u5b57\u5143\u53ef\u80fd\u4f54\u7528 2 \u4f4d\u5143\u7d44\u6216 4 \u4f4d\u5143\u7d44\uff0c\u5f9e\u800c\u55aa\u5931\u4e86\u7b49\u9577\u7de8\u78bc\u7684\u512a\u52e2\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u8655\u7406\u4ee3\u7406\u5c0d\u9700\u8981\u984d\u5916\u589e\u52a0\u7a0b\u5f0f\u78bc\uff0c\u9019\u63d0\u9ad8\u4e86\u7a0b\u5f0f\u8a2d\u8a08\u7684\u8907\u96dc\u6027\u548c\u9664\u932f\u96e3\u5ea6\u3002</p> <p>\u51fa\u65bc\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7a0b\u5f0f\u8a9e\u8a00\u63d0\u51fa\u4e86\u4e00\u4e9b\u4e0d\u540c\u7684\u7de8\u78bc\u65b9\u6848\u3002</p> <ul> <li>Python \u4e2d\u7684 <code>str</code> \u4f7f\u7528 Unicode \u7de8\u78bc\uff0c\u4e26\u63a1\u7528\u4e00\u7a2e\u9748\u6d3b\u7684\u5b57\u4e32\u8868\u793a\uff0c\u5132\u5b58\u7684\u5b57\u5143\u9577\u5ea6\u53d6\u6c7a\u65bc\u5b57\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u78bc\u9ede\u3002\u82e5\u5b57\u4e32\u4e2d\u5168\u90e8\u662f ASCII \u5b57\u5143\uff0c\u5247\u6bcf\u500b\u5b57\u5143\u4f54\u7528 1 \u4f4d\u5143\u7d44\uff1b\u5982\u679c\u6709\u5b57\u5143\u8d85\u51fa\u4e86 ASCII \u7bc4\u570d\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8a9e\u8a00\u5e73\u9762\uff08BMP\uff09\u5167\uff0c\u5247\u6bcf\u500b\u5b57\u5143\u4f54\u7528 2 \u4f4d\u5143\u7d44\uff1b\u5982\u679c\u6709\u8d85\u51fa BMP \u7684\u5b57\u5143\uff0c\u5247\u6bcf\u500b\u5b57\u5143\u4f54\u7528 4 \u4f4d\u5143\u7d44\u3002</li> <li>Go \u8a9e\u8a00\u7684 <code>string</code> \u578b\u5225\u5728\u5167\u90e8\u4f7f\u7528 UTF-8 \u7de8\u78bc\u3002Go \u8a9e\u8a00\u9084\u63d0\u4f9b\u4e86 <code>rune</code> \u578b\u5225\uff0c\u5b83\u7528\u65bc\u8868\u793a\u55ae\u500b Unicode \u78bc\u9ede\u3002</li> <li>Rust \u8a9e\u8a00\u7684 <code>str</code> \u548c <code>String</code> \u578b\u5225\u5728\u5167\u90e8\u4f7f\u7528 UTF-8 \u7de8\u78bc\u3002Rust \u4e5f\u63d0\u4f9b\u4e86 <code>char</code> \u578b\u5225\uff0c\u7528\u65bc\u8868\u793a\u55ae\u500b Unicode \u78bc\u9ede\u3002</li> </ul> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8a0e\u8ad6\u7684\u90fd\u662f\u5b57\u4e32\u5728\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u7684\u5132\u5b58\u65b9\u5f0f\uff0c\u9019\u548c\u5b57\u4e32\u5982\u4f55\u5728\u6a94\u6848\u4e2d\u5132\u5b58\u6216\u5728\u7db2\u8def\u4e2d\u50b3\u8f38\u662f\u4e0d\u540c\u7684\u554f\u984c\u3002\u5728\u6a94\u6848\u5132\u5b58\u6216\u7db2\u8def\u50b3\u8f38\u4e2d\uff0c\u6211\u5011\u901a\u5e38\u6703\u5c07\u5b57\u4e32\u7de8\u78bc\u70ba UTF-8 \u683c\u5f0f\uff0c\u4ee5\u9054\u5230\u6700\u512a\u7684\u76f8\u5bb9\u6027\u548c\u7a7a\u9593\u6548\u7387\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 \u8cc7\u6599\u7d50\u69cb\u5206\u985e","text":"<p>\u5e38\u898b\u7684\u8cc7\u6599\u7d50\u69cb\u5305\u62ec\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u6a39\u3001\u5806\u7a4d\u3001\u5716\uff0c\u5b83\u5011\u53ef\u4ee5\u5f9e\u201c\u908f\u8f2f\u7d50\u69cb\u201d\u548c\u201c\u7269\u7406\u7d50\u69cb\u201d\u5169\u500b\u7dad\u5ea6\u9032\u884c\u5206\u985e\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1 \u00a0 \u908f\u8f2f\u7d50\u69cb\uff1a\u7dda\u6027\u8207\u975e\u7dda\u6027","text":"<p>\u908f\u8f2f\u7d50\u69cb\u63ed\u793a\u4e86\u8cc7\u6599\u5143\u7d20\u4e4b\u9593\u7684\u908f\u8f2f\u95dc\u4fc2\u3002\u5728\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u4e2d\uff0c\u8cc7\u6599\u6309\u7167\u4e00\u5b9a\u9806\u5e8f\u6392\u5217\uff0c\u9ad4\u73fe\u4e86\u8cc7\u6599\u4e4b\u9593\u7684\u7dda\u6027\u95dc\u4fc2\uff1b\u800c\u5728\u6a39\u4e2d\uff0c\u8cc7\u6599\u5f9e\u9802\u90e8\u5411\u4e0b\u6309\u5c64\u6b21\u6392\u5217\uff0c\u8868\u73fe\u51fa\u201c\u7956\u5148\u201d\u8207\u201c\u5f8c\u4ee3\u201d\u4e4b\u9593\u7684\u6d3e\u751f\u95dc\u4fc2\uff1b\u5716\u5247\u7531\u7bc0\u9ede\u548c\u908a\u69cb\u6210\uff0c\u53cd\u6620\u4e86\u8907\u96dc\u7684\u7db2\u8def\u95dc\u4fc2\u3002</p> <p>\u5982\u5716 3-1 \u6240\u793a\uff0c\u908f\u8f2f\u7d50\u69cb\u53ef\u5206\u70ba\u201c\u7dda\u6027\u201d\u548c\u201c\u975e\u7dda\u6027\u201d\u5169\u5927\u985e\u3002\u7dda\u6027\u7d50\u69cb\u6bd4\u8f03\u76f4\u89c0\uff0c\u6307\u8cc7\u6599\u5728\u908f\u8f2f\u95dc\u4fc2\u4e0a\u5448\u7dda\u6027\u6392\u5217\uff1b\u975e\u7dda\u6027\u7d50\u69cb\u5247\u76f8\u53cd\uff0c\u5448\u975e\u7dda\u6027\u6392\u5217\u3002</p> <ul> <li>\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff1a\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\uff0c\u5143\u7d20\u4e4b\u9593\u662f\u4e00\u5c0d\u4e00\u7684\u9806\u5e8f\u95dc\u4fc2\u3002</li> <li>\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff1a\u6a39\u3001\u5806\u7a4d\u3001\u5716\u3001\u96dc\u6e4a\u8868\u3002</li> </ul> <p>\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u53ef\u4ee5\u9032\u4e00\u6b65\u5283\u5206\u70ba\u6a39\u5f62\u7d50\u69cb\u548c\u7db2\u72c0\u7d50\u69cb\u3002</p> <ul> <li>\u6a39\u5f62\u7d50\u69cb\uff1a\u6a39\u3001\u5806\u7a4d\u3001\u96dc\u6e4a\u8868\uff0c\u5143\u7d20\u4e4b\u9593\u662f\u4e00\u5c0d\u591a\u7684\u95dc\u4fc2\u3002</li> <li>\u7db2\u72c0\u7d50\u69cb\uff1a\u5716\uff0c\u5143\u7d20\u4e4b\u9593\u662f\u591a\u5c0d\u591a\u7684\u95dc\u4fc2\u3002</li> </ul> <p></p> <p> \u5716 3-1 \u00a0 \u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u8207\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb </p>"},{"location":"chapter_data_structure/classification_of_data_structure/#312","title":"3.1.2 \u00a0 \u7269\u7406\u7d50\u69cb\uff1a\u9023\u7e8c\u8207\u5206\u6563","text":"<p>\u7576\u6f14\u7b97\u6cd5\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u6b63\u5728\u8655\u7406\u7684\u8cc7\u6599\u4e3b\u8981\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u4e2d\u3002\u5716 3-2 \u5c55\u793a\u4e86\u4e00\u500b\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u689d\uff0c\u5176\u4e2d\u6bcf\u500b\u9ed1\u8272\u65b9\u584a\u90fd\u5305\u542b\u4e00\u584a\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002\u6211\u5011\u53ef\u4ee5\u5c07\u8a18\u61b6\u9ad4\u60f3\u8c61\u6210\u4e00\u500b\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u500b\u55ae\u5143\u683c\u90fd\u53ef\u4ee5\u5132\u5b58\u4e00\u5b9a\u5927\u5c0f\u7684\u8cc7\u6599\u3002</p> <p>\u7cfb\u7d71\u900f\u904e\u8a18\u61b6\u9ad4\u4f4d\u5740\u4f86\u8a2a\u554f\u76ee\u6a19\u4f4d\u7f6e\u7684\u8cc7\u6599\u3002\u5982\u5716 3-2 \u6240\u793a\uff0c\u8a08\u7b97\u6a5f\u6839\u64da\u7279\u5b9a\u898f\u5247\u70ba\u8868\u683c\u4e2d\u7684\u6bcf\u500b\u55ae\u5143\u683c\u5206\u914d\u7de8\u865f\uff0c\u78ba\u4fdd\u6bcf\u500b\u8a18\u61b6\u9ad4\u7a7a\u9593\u90fd\u6709\u552f\u4e00\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u3002\u6709\u4e86\u9019\u4e9b\u4f4d\u5740\uff0c\u7a0b\u5f0f\u4fbf\u53ef\u4ee5\u8a2a\u554f\u8a18\u61b6\u9ad4\u4e2d\u7684\u8cc7\u6599\u3002</p> <p></p> <p> \u5716 3-2 \u00a0 \u8a18\u61b6\u9ad4\u689d\u3001\u8a18\u61b6\u9ad4\u7a7a\u9593\u3001\u8a18\u61b6\u9ad4\u4f4d\u5740 </p> <p>Tip</p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u5c07\u8a18\u61b6\u9ad4\u6bd4\u4f5c Excel \u8868\u683c\u662f\u4e00\u500b\u7c21\u5316\u7684\u985e\u6bd4\uff0c\u5be6\u969b\u8a18\u61b6\u9ad4\u7684\u5de5\u4f5c\u6a5f\u5236\u6bd4\u8f03\u8907\u96dc\uff0c\u6d89\u53ca\u4f4d\u5740\u7a7a\u9593\u3001\u8a18\u61b6\u9ad4\u7ba1\u7406\u3001\u5feb\u53d6\u6a5f\u5236\u3001\u865b\u64ec\u8a18\u61b6\u9ad4\u548c\u7269\u7406\u8a18\u61b6\u9ad4\u7b49\u6982\u5ff5\u3002</p> <p>\u8a18\u61b6\u9ad4\u662f\u6240\u6709\u7a0b\u5f0f\u7684\u5171\u4eab\u8cc7\u6e90\uff0c\u7576\u67d0\u584a\u8a18\u61b6\u9ad4\u88ab\u67d0\u500b\u7a0b\u5f0f\u4f54\u7528\u6642\uff0c\u5247\u901a\u5e38\u7121\u6cd5\u88ab\u5176\u4ed6\u7a0b\u5f0f\u540c\u6642\u4f7f\u7528\u4e86\u3002\u56e0\u6b64\u5728\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u8a2d\u8a08\u4e2d\uff0c\u8a18\u61b6\u9ad4\u8cc7\u6e90\u662f\u4e00\u500b\u91cd\u8981\u7684\u8003\u616e\u56e0\u7d20\u3002\u6bd4\u5982\uff0c\u6f14\u7b97\u6cd5\u6240\u4f54\u7528\u7684\u8a18\u61b6\u9ad4\u5cf0\u503c\u4e0d\u61c9\u8d85\u904e\u7cfb\u7d71\u5269\u9918\u7a7a\u9592\u8a18\u61b6\u9ad4\uff1b\u5982\u679c\u7f3a\u5c11\u9023\u7e8c\u5927\u584a\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\uff0c\u90a3\u9ebc\u6240\u9078\u7528\u7684\u8cc7\u6599\u7d50\u69cb\u5fc5\u9808\u80fd\u5920\u5132\u5b58\u5728\u5206\u6563\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u5167\u3002</p> <p>\u5982\u5716 3-3 \u6240\u793a\uff0c\u7269\u7406\u7d50\u69cb\u53cd\u6620\u4e86\u8cc7\u6599\u5728\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u4e2d\u7684\u5132\u5b58\u65b9\u5f0f\uff0c\u53ef\u5206\u70ba\u9023\u7e8c\u7a7a\u9593\u5132\u5b58\uff08\u9663\u5217\uff09\u548c\u5206\u6563\u7a7a\u9593\u5132\u5b58\uff08\u93c8\u7d50\u4e32\u5217\uff09\u3002\u7269\u7406\u7d50\u69cb\u5f9e\u5e95\u5c64\u6c7a\u5b9a\u4e86\u8cc7\u6599\u7684\u8a2a\u554f\u3001\u66f4\u65b0\u3001\u589e\u522a\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u5169\u7a2e\u7269\u7406\u7d50\u69cb\u5728\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u65b9\u9762\u5448\u73fe\u51fa\u4e92\u88dc\u7684\u7279\u9ede\u3002</p> <p></p> <p> \u5716 3-3 \u00a0 \u9023\u7e8c\u7a7a\u9593\u5132\u5b58\u8207\u5206\u6563\u7a7a\u9593\u5132\u5b58 </p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u6240\u6709\u8cc7\u6599\u7d50\u69cb\u90fd\u662f\u57fa\u65bc\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u6216\u4e8c\u8005\u7684\u7d44\u5408\u5be6\u73fe\u7684\u3002\u4f8b\u5982\uff0c\u5806\u758a\u548c\u4f47\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u9663\u5217\u5be6\u73fe\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\uff1b\u800c\u96dc\u6e4a\u8868\u7684\u5be6\u73fe\u53ef\u80fd\u540c\u6642\u5305\u542b\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u3002</p> <ul> <li>\u57fa\u65bc\u9663\u5217\u53ef\u5be6\u73fe\uff1a\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u6a39\u3001\u5806\u7a4d\u3001\u5716\u3001\u77e9\u9663\u3001\u5f35\u91cf\uff08\u7dad\u5ea6 \\(\\geq 3\\) \u7684\u9663\u5217\uff09\u7b49\u3002</li> <li>\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u53ef\u5be6\u73fe\uff1a\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u6a39\u3001\u5806\u7a4d\u3001\u5716\u7b49\u3002</li> </ul> <p>\u93c8\u7d50\u4e32\u5217\u5728\u521d\u59cb\u5316\u5f8c\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5f0f\u57f7\u884c\u904e\u7a0b\u4e2d\u5c0d\u5176\u9577\u5ea6\u9032\u884c\u8abf\u6574\uff0c\u56e0\u6b64\u4e5f\u7a31\u201c\u52d5\u614b\u8cc7\u6599\u7d50\u69cb\u201d\u3002\u9663\u5217\u5728\u521d\u59cb\u5316\u5f8c\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u4e5f\u7a31\u201c\u975c\u614b\u8cc7\u6599\u7d50\u69cb\u201d\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u9663\u5217\u53ef\u900f\u904e\u91cd\u65b0\u5206\u914d\u8a18\u61b6\u9ad4\u5be6\u73fe\u9577\u5ea6\u8b8a\u5316\uff0c\u5f9e\u800c\u5177\u5099\u4e00\u5b9a\u7684\u201c\u52d5\u614b\u6027\u201d\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u611f\u89ba\u7269\u7406\u7d50\u69cb\u7406\u89e3\u8d77\u4f86\u6709\u56f0\u96e3\uff0c\u5efa\u8b70\u5148\u95b1\u8b80\u4e0b\u4e00\u7ae0\uff0c\u7136\u5f8c\u518d\u56de\u9867\u672c\u7bc0\u5167\u5bb9\u3002</p>"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 \u6578\u5b57\u7de8\u78bc *","text":"<p>Tip</p> <p>\u5728\u672c\u66f8\u4e2d\uff0c\u6a19\u984c\u5e36\u6709 * \u7b26\u865f\u7684\u662f\u9078\u8b80\u7ae0\u7bc0\u3002\u5982\u679c\u4f60\u6642\u9593\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96e3\uff0c\u53ef\u4ee5\u5148\u8df3\u904e\uff0c\u7b49\u5b78\u5b8c\u5fc5\u8b80\u7ae0\u7bc0\u5f8c\u518d\u55ae\u7368\u653b\u514b\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1 \u00a0 \u539f\u78bc\u3001\u4e00\u88dc\u6578\u548c\u4e8c\u88dc\u6578","text":"<p>\u5728\u4e0a\u4e00\u7bc0\u7684\u8868\u683c\u4e2d\u6211\u5011\u767c\u73fe\uff0c\u6240\u6709\u6574\u6578\u578b\u5225\u80fd\u5920\u8868\u793a\u7684\u8ca0\u6578\u90fd\u6bd4\u6b63\u6578\u591a\u4e00\u500b\uff0c\u4f8b\u5982 <code>byte</code> \u7684\u53d6\u503c\u7bc4\u570d\u662f \\([-128, 127]\\) \u3002\u9019\u500b\u73fe\u8c61\u6bd4\u8f03\u53cd\u76f4\u89ba\uff0c\u5b83\u7684\u5167\u5728\u539f\u56e0\u6d89\u53ca\u539f\u78bc\u3001\u4e00\u88dc\u6578\u3001\u4e8c\u88dc\u6578\u7684\u76f8\u95dc\u77e5\u8b58\u3002</p> <p>\u9996\u5148\u9700\u8981\u6307\u51fa\uff0c\u6578\u5b57\u662f\u4ee5\u201c\u4e8c\u88dc\u6578\u201d\u7684\u5f62\u5f0f\u5132\u5b58\u5728\u8a08\u7b97\u6a5f\u4e2d\u7684\u3002\u5728\u5206\u6790\u9019\u6a23\u505a\u7684\u539f\u56e0\u4e4b\u524d\uff0c\u9996\u5148\u7d66\u51fa\u4e09\u8005\u7684\u5b9a\u7fa9\u3002</p> <ul> <li>\u539f\u78bc\uff1a\u6211\u5011\u5c07\u6578\u5b57\u7684\u4e8c\u9032\u4f4d\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u8996\u70ba\u7b26\u865f\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6578\uff0c\\(1\\) \u8868\u793a\u8ca0\u6578\uff0c\u5176\u9918\u4f4d\u8868\u793a\u6578\u5b57\u7684\u503c\u3002</li> <li>\u4e00\u88dc\u6578\uff1a\u6b63\u6578\u7684\u4e00\u88dc\u6578\u8207\u5176\u539f\u78bc\u76f8\u540c\uff0c\u8ca0\u6578\u7684\u4e00\u88dc\u6578\u662f\u5c0d\u5176\u539f\u78bc\u9664\u7b26\u865f\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002</li> <li>\u4e8c\u88dc\u6578\uff1a\u6b63\u6578\u7684\u4e8c\u88dc\u6578\u8207\u5176\u539f\u78bc\u76f8\u540c\uff0c\u8ca0\u6578\u7684\u4e8c\u88dc\u6578\u662f\u5728\u5176\u4e00\u88dc\u6578\u7684\u57fa\u790e\u4e0a\u52a0 \\(1\\) \u3002</li> </ul> <p>\u5716 3-4 \u5c55\u793a\u4e86\u539f\u78bc\u3001\u4e00\u88dc\u6578\u548c\u4e8c\u88dc\u6578\u4e4b\u9593\u7684\u8f49\u63db\u65b9\u6cd5\u3002</p> <p></p> <p> \u5716 3-4 \u00a0 \u539f\u78bc\u3001\u4e00\u88dc\u6578\u8207\u4e8c\u88dc\u6578\u4e4b\u9593\u7684\u76f8\u4e92\u8f49\u63db </p> <p>\u539f\u78bc\uff08sign-magnitude\uff09\u96d6\u7136\u6700\u76f4\u89c0\uff0c\u4f46\u5b58\u5728\u4e00\u4e9b\u4fb7\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u8ca0\u6578\u7684\u539f\u78bc\u4e0d\u80fd\u76f4\u63a5\u7528\u65bc\u904b\u7b97\u3002\u4f8b\u5982\u5728\u539f\u78bc\u4e0b\u8a08\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7d50\u679c\u662f \\(-3\\) \uff0c\u9019\u986f\u7136\u662f\u4e0d\u5c0d\u7684\u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>\u70ba\u4e86\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u8a08\u7b97\u6a5f\u5f15\u5165\u4e86\u4e00\u88dc\u6578\uff081's complement\uff09\u3002\u5982\u679c\u6211\u5011\u5148\u5c07\u539f\u78bc\u8f49\u63db\u70ba\u4e00\u88dc\u6578\uff0c\u4e26\u5728\u4e00\u88dc\u6578\u4e0b\u8a08\u7b97 \\(1 + (-2)\\) \uff0c\u6700\u5f8c\u5c07\u7d50\u679c\u5f9e\u4e00\u88dc\u6578\u8f49\u63db\u56de\u539f\u78bc\uff0c\u5247\u53ef\u5f97\u5230\u6b63\u78ba\u7d50\u679c \\(-1\\) \u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(\u539f\u78bc)} + 1000 \\; 0010 \\; \\text{(\u539f\u78bc)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(\u4e00\u88dc\u6578)} + 1111  \\; 1101 \\; \\text{(\u4e00\u88dc\u6578)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(\u4e00\u88dc\u6578)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u539f\u78bc)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>\u53e6\u4e00\u65b9\u9762\uff0c\u6578\u5b57\u96f6\u7684\u539f\u78bc\u6709 \\(+0\\) \u548c \\(-0\\) \u5169\u7a2e\u8868\u793a\u65b9\u5f0f\u3002\u9019\u610f\u5473\u8457\u6578\u5b57\u96f6\u5c0d\u61c9\u5169\u500b\u4e0d\u540c\u7684\u4e8c\u9032\u4f4d\u5236\u7de8\u78bc\uff0c\u9019\u53ef\u80fd\u6703\u5e36\u4f86\u6b67\u7fa9\u3002\u6bd4\u5982\u5728\u689d\u4ef6\u5224\u65b7\u4e2d\uff0c\u5982\u679c\u6c92\u6709\u5340\u5206\u6b63\u96f6\u548c\u8ca0\u96f6\uff0c\u5247\u53ef\u80fd\u6703\u5c0e\u81f4\u5224\u65b7\u7d50\u679c\u51fa\u932f\u3002\u800c\u5982\u679c\u6211\u5011\u60f3\u8655\u7406\u6b63\u96f6\u548c\u8ca0\u96f6\u6b67\u7fa9\uff0c\u5247\u9700\u8981\u5f15\u5165\u984d\u5916\u7684\u5224\u65b7\u64cd\u4f5c\uff0c\u9019\u53ef\u80fd\u6703\u964d\u4f4e\u8a08\u7b97\u6a5f\u7684\u904b\u7b97\u6548\u7387\u3002</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>\u8207\u539f\u78bc\u4e00\u6a23\uff0c\u4e00\u88dc\u6578\u4e5f\u5b58\u5728\u6b63\u8ca0\u96f6\u6b67\u7fa9\u554f\u984c\uff0c\u56e0\u6b64\u8a08\u7b97\u6a5f\u9032\u4e00\u6b65\u5f15\u5165\u4e86\u4e8c\u88dc\u6578\uff082's complement\uff09\u3002\u6211\u5011\u5148\u4f86\u89c0\u5bdf\u4e00\u4e0b\u8ca0\u96f6\u7684\u539f\u78bc\u3001\u4e00\u88dc\u6578\u3001\u4e8c\u88dc\u6578\u7684\u8f49\u63db\u904e\u7a0b\uff1a</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(\u539f\u78bc)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(\u4e00\u88dc\u6578)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(\u4e8c\u88dc\u6578)} \\newline \\end{aligned} \\] <p>\u5728\u8ca0\u96f6\u7684\u4e00\u88dc\u6578\u57fa\u790e\u4e0a\u52a0 \\(1\\) \u6703\u7522\u751f\u9032\u4f4d\uff0c\u4f46 <code>byte</code> \u578b\u5225\u7684\u9577\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u4f4d\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u6703\u88ab\u6368\u68c4\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u8ca0\u96f6\u7684\u4e8c\u88dc\u6578\u70ba \\(0000 \\; 0000\\) \uff0c\u8207\u6b63\u96f6\u7684\u4e8c\u88dc\u6578\u76f8\u540c\u3002\u9019\u610f\u5473\u8457\u5728\u4e8c\u88dc\u6578\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u500b\u96f6\uff0c\u6b63\u8ca0\u96f6\u6b67\u7fa9\u5f9e\u800c\u5f97\u5230\u89e3\u6c7a\u3002</p> <p>\u9084\u5269\u6700\u5f8c\u4e00\u500b\u7591\u60d1\uff1a<code>byte</code> \u578b\u5225\u7684\u53d6\u503c\u7bc4\u570d\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u4f86\u7684\u4e00\u500b\u8ca0\u6578 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u5011\u6ce8\u610f\u5230\uff0c\u5340\u9593 \\([-127, +127]\\) \u5167\u7684\u6240\u6709\u6574\u6578\u90fd\u6709\u5c0d\u61c9\u7684\u539f\u78bc\u3001\u4e00\u88dc\u6578\u548c\u4e8c\u88dc\u6578\uff0c\u4e26\u4e14\u539f\u78bc\u548c\u4e8c\u88dc\u6578\u4e4b\u9593\u53ef\u4ee5\u4e92\u76f8\u8f49\u63db\u3002</p> <p>\u7136\u800c\uff0c\u4e8c\u88dc\u6578 \\(1000 \\; 0000\\) \u662f\u4e00\u500b\u4f8b\u5916\uff0c\u5b83\u4e26\u6c92\u6709\u5c0d\u61c9\u7684\u539f\u78bc\u3002\u6839\u64da\u8f49\u63db\u65b9\u6cd5\uff0c\u6211\u5011\u5f97\u5230\u8a72\u4e8c\u88dc\u6578\u7684\u539f\u78bc\u70ba \\(0000 \\; 0000\\) \u3002\u9019\u986f\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u70ba\u8a72\u539f\u78bc\u8868\u793a\u6578\u5b57 \\(0\\) \uff0c\u5b83\u7684\u4e8c\u88dc\u6578\u61c9\u8a72\u662f\u81ea\u8eab\u3002\u8a08\u7b97\u6a5f\u898f\u5b9a\u9019\u500b\u7279\u6b8a\u7684\u4e8c\u88dc\u6578 \\(1000 \\; 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5be6\u969b\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u4e8c\u88dc\u6578\u4e0b\u7684\u8a08\u7b97\u7d50\u679c\u5c31\u662f \\(-128\\) \u3002</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(\u539f\u78bc)} + 1000 \\; 0001 \\; \\text{(\u539f\u78bc)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u4e00\u88dc\u6578)} + 1111  \\; 1110 \\; \\text{(\u4e00\u88dc\u6578)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u4e8c\u88dc\u6578)} + 1111  \\; 1111 \\; \\text{(\u4e8c\u88dc\u6578)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u4e8c\u88dc\u6578)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>\u4f60\u53ef\u80fd\u5df2\u7d93\u767c\u73fe\u4e86\uff0c\u4e0a\u8ff0\u6240\u6709\u8a08\u7b97\u90fd\u662f\u52a0\u6cd5\u904b\u7b97\u3002\u9019\u6697\u793a\u8457\u4e00\u500b\u91cd\u8981\u4e8b\u5be6\uff1a\u8a08\u7b97\u6a5f\u5167\u90e8\u7684\u786c\u9ad4\u96fb\u8def\u4e3b\u8981\u662f\u57fa\u65bc\u52a0\u6cd5\u904b\u7b97\u8a2d\u8a08\u7684\u3002\u9019\u662f\u56e0\u70ba\u52a0\u6cd5\u904b\u7b97\u76f8\u5c0d\u65bc\u5176\u4ed6\u904b\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u6e1b\u6cd5\uff09\u4f86\u8aaa\uff0c\u786c\u9ad4\u5be6\u73fe\u8d77\u4f86\u66f4\u7c21\u55ae\uff0c\u66f4\u5bb9\u6613\u9032\u884c\u4e26\u884c\u5316\u8655\u7406\uff0c\u904b\u7b97\u901f\u5ea6\u66f4\u5feb\u3002</p> <p>\u8acb\u6ce8\u610f\uff0c\u9019\u4e26\u4e0d\u610f\u5473\u8457\u8a08\u7b97\u6a5f\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u900f\u904e\u5c07\u52a0\u6cd5\u8207\u4e00\u4e9b\u57fa\u672c\u908f\u8f2f\u904b\u7b97\u7d50\u5408\uff0c\u8a08\u7b97\u6a5f\u80fd\u5920\u5be6\u73fe\u5404\u7a2e\u5176\u4ed6\u7684\u6578\u5b78\u904b\u7b97\u3002\u4f8b\u5982\uff0c\u8a08\u7b97\u6e1b\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f49\u63db\u70ba\u8a08\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8a08\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f49\u63db\u70ba\u8a08\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u6e1b\u6cd5\u3002</p> <p>\u73fe\u5728\u6211\u5011\u53ef\u4ee5\u7e3d\u7d50\u51fa\u8a08\u7b97\u6a5f\u4f7f\u7528\u4e8c\u88dc\u6578\u7684\u539f\u56e0\uff1a\u57fa\u65bc\u4e8c\u88dc\u6578\u8868\u793a\uff0c\u8a08\u7b97\u6a5f\u53ef\u4ee5\u7528\u540c\u6a23\u7684\u96fb\u8def\u548c\u64cd\u4f5c\u4f86\u8655\u7406\u6b63\u6578\u548c\u8ca0\u6578\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8a2d\u8a08\u7279\u6b8a\u7684\u786c\u9ad4\u96fb\u8def\u4f86\u8655\u7406\u6e1b\u6cd5\uff0c\u4e26\u4e14\u7121\u9808\u7279\u5225\u8655\u7406\u6b63\u8ca0\u96f6\u7684\u6b67\u7fa9\u554f\u984c\u3002\u9019\u5927\u5927\u7c21\u5316\u4e86\u786c\u9ad4\u8a2d\u8a08\uff0c\u63d0\u9ad8\u4e86\u904b\u7b97\u6548\u7387\u3002</p> <p>\u4e8c\u88dc\u6578\u7684\u8a2d\u8a08\u975e\u5e38\u7cbe\u5999\uff0c\u56e0\u7bc7\u5e45\u95dc\u4fc2\u6211\u5011\u5c31\u5148\u4ecb\u7d39\u5230\u9019\u88e1\uff0c\u5efa\u8b70\u6709\u8208\u8da3\u7684\u8b80\u8005\u9032\u4e00\u6b65\u6df1\u5165\u77ad\u89e3\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2 \u00a0 \u6d6e\u9ede\u6578\u7de8\u78bc","text":"<p>\u7d30\u5fc3\u7684\u4f60\u53ef\u80fd\u6703\u767c\u73fe\uff1a<code>int</code> \u548c <code>float</code> \u9577\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 \u4f4d\u5143\u7d44 \uff0c\u4f46\u70ba\u4ec0\u9ebc <code>float</code> \u7684\u53d6\u503c\u7bc4\u570d\u9060\u5927\u65bc <code>int</code> \uff1f\u9019\u975e\u5e38\u53cd\u76f4\u89ba\uff0c\u56e0\u70ba\u6309\u7406\u8aaa <code>float</code> \u9700\u8981\u8868\u793a\u5c0f\u6578\uff0c\u53d6\u503c\u7bc4\u570d\u61c9\u8a72\u8b8a\u5c0f\u624d\u5c0d\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u9019\u662f\u56e0\u70ba\u6d6e\u9ede\u6578 <code>float</code> \u63a1\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u8a18\u4e00\u500b 32 \u4f4d\u5143\u9577\u5ea6\u7684\u4e8c\u9032\u4f4d\u5236\u6578\u70ba\uff1a</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>\u6839\u64da IEEE 754 \u6a19\u6e96\uff0c32-bit \u9577\u5ea6\u7684 <code>float</code> \u7531\u4ee5\u4e0b\u4e09\u500b\u90e8\u5206\u69cb\u6210\u3002</p> <ul> <li>\u7b26\u865f\u4f4d \\(\\mathrm{S}\\) \uff1a\u4f54 1 \u4f4d \uff0c\u5c0d\u61c9 \\(b_{31}\\) \u3002</li> <li>\u6307\u6578\u4f4d \\(\\mathrm{E}\\) \uff1a\u4f54 8 \u4f4d \uff0c\u5c0d\u61c9 \\(b_{30} b_{29} \\ldots b_{23}\\) \u3002</li> <li>\u5206\u6578\u4f4d \\(\\mathrm{N}\\) \uff1a\u4f54 23 \u4f4d \uff0c\u5c0d\u61c9 \\(b_{22} b_{21} \\ldots b_0\\) \u3002</li> </ul> <p>\u4e8c\u9032\u4f4d\u5236\u6578 <code>float</code> \u5c0d\u61c9\u503c\u7684\u8a08\u7b97\u65b9\u6cd5\u70ba\uff1a</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>\u8f49\u5316\u5230\u5341\u9032\u4f4d\u5236\u4e0b\u7684\u8a08\u7b97\u516c\u5f0f\u70ba\uff1a</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>\u5176\u4e2d\u5404\u9805\u7684\u53d6\u503c\u7bc4\u570d\u70ba\uff1a</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> \u5716 3-5 \u00a0 IEEE 754 \u6a19\u6e96\u4e0b\u7684 float \u7684\u8a08\u7b97\u793a\u4f8b </p> <p>\u89c0\u5bdf\u5716 3-5 \uff0c\u7d66\u5b9a\u4e00\u500b\u793a\u4f8b\u8cc7\u6599 \\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u5247\u6709\uff1a</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>\u73fe\u5728\u6211\u5011\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u554f\u984c\uff1a<code>float</code> \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6578\u4f4d\uff0c\u5c0e\u81f4\u5176\u53d6\u503c\u7bc4\u570d\u9060\u5927\u65bc <code>int</code> \u3002\u6839\u64da\u4ee5\u4e0a\u8a08\u7b97\uff0c<code>float</code> \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6578\u70ba \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u63db\u7b26\u865f\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8ca0\u6578\u3002</p> <p>\u5118\u7ba1\u6d6e\u9ede\u6578 <code>float</code> \u64f4\u5c55\u4e86\u53d6\u503c\u7bc4\u570d\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u72a7\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6578\u578b\u5225 <code>int</code> \u5c07\u5168\u90e8 32 \u4f4d\u5143\u7528\u65bc\u8868\u793a\u6578\u5b57\uff0c\u6578\u5b57\u662f\u5747\u52fb\u5206\u4f48\u7684\uff1b\u800c\u7531\u65bc\u6307\u6578\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u9ede\u6578 <code>float</code> \u7684\u6578\u503c\u8d8a\u5927\uff0c\u76f8\u9130\u5169\u500b\u6578\u5b57\u4e4b\u9593\u7684\u5dee\u503c\u5c31\u6703\u8da8\u5411\u8d8a\u5927\u3002</p> <p>\u5982\u8868 3-2 \u6240\u793a\uff0c\u6307\u6578\u4f4d \\(\\mathrm{E} = 0\\) \u548c \\(\\mathrm{E} = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u7fa9\uff0c\u7528\u65bc\u8868\u793a\u96f6\u3001\u7121\u7aae\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002</p> <p> \u8868 3-2 \u00a0 \u6307\u6578\u4f4d\u542b\u7fa9 </p> \u6307\u6578\u4f4d E \u5206\u6578\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6578\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8a08\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u898f\u6578 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u898f\u6578 \u6b63\u898f\u6578 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u6b21\u6b63\u898f\u6578\u986f\u8457\u63d0\u5347\u4e86\u6d6e\u9ede\u6578\u7684\u7cbe\u5ea6\u3002\u6700\u5c0f\u6b63\u6b63\u898f\u6578\u70ba \\(2^{-126}\\) \uff0c\u6700\u5c0f\u6b63\u6b21\u6b63\u898f\u6578\u70ba \\(2^{-126} \\times 2^{-23}\\) \u3002</p> <p>\u96d9\u7cbe\u5ea6 <code>double</code> \u4e5f\u63a1\u7528\u985e\u4f3c\u65bc <code>float</code> \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u505a\u8d05\u8ff0\u3002</p>"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_data_structure/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u8cc7\u6599\u7d50\u69cb\u53ef\u4ee5\u5f9e\u908f\u8f2f\u7d50\u69cb\u548c\u7269\u7406\u7d50\u69cb\u5169\u500b\u89d2\u5ea6\u9032\u884c\u5206\u985e\u3002\u908f\u8f2f\u7d50\u69cb\u63cf\u8ff0\u4e86\u8cc7\u6599\u5143\u7d20\u4e4b\u9593\u7684\u908f\u8f2f\u95dc\u4fc2\uff0c\u800c\u7269\u7406\u7d50\u69cb\u63cf\u8ff0\u4e86\u8cc7\u6599\u5728\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u4e2d\u7684\u5132\u5b58\u65b9\u5f0f\u3002</li> <li>\u5e38\u898b\u7684\u908f\u8f2f\u7d50\u69cb\u5305\u62ec\u7dda\u6027\u3001\u6a39\u72c0\u548c\u7db2\u72c0\u7b49\u3002\u901a\u5e38\u6211\u5011\u6839\u64da\u908f\u8f2f\u7d50\u69cb\u5c07\u8cc7\u6599\u7d50\u69cb\u5206\u70ba\u7dda\u6027\uff08\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\uff09\u548c\u975e\u7dda\u6027\uff08\u6a39\u3001\u5716\u3001\u5806\u7a4d\uff09\u5169\u7a2e\u3002\u96dc\u6e4a\u8868\u7684\u5be6\u73fe\u53ef\u80fd\u540c\u6642\u5305\u542b\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u548c\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u7576\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u8cc7\u6599\u88ab\u5132\u5b58\u5728\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u4e2d\u3002\u6bcf\u500b\u8a18\u61b6\u9ad4\u7a7a\u9593\u90fd\u64c1\u6709\u5c0d\u61c9\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\uff0c\u7a0b\u5f0f\u900f\u904e\u9019\u4e9b\u8a18\u61b6\u9ad4\u4f4d\u5740\u8a2a\u554f\u8cc7\u6599\u3002</li> <li>\u7269\u7406\u7d50\u69cb\u4e3b\u8981\u5206\u70ba\u9023\u7e8c\u7a7a\u9593\u5132\u5b58\uff08\u9663\u5217\uff09\u548c\u5206\u6563\u7a7a\u9593\u5132\u5b58\uff08\u93c8\u7d50\u4e32\u5217\uff09\u3002\u6240\u6709\u8cc7\u6599\u7d50\u69cb\u90fd\u662f\u7531\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u6216\u5169\u8005\u7684\u7d44\u5408\u5be6\u73fe\u7684\u3002</li> <li>\u8a08\u7b97\u6a5f\u4e2d\u7684\u57fa\u672c\u8cc7\u6599\u578b\u5225\u5305\u62ec\u6574\u6578 <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \uff0c\u6d6e\u9ede\u6578 <code>float</code>\u3001<code>double</code> \uff0c\u5b57\u5143 <code>char</code> \u548c\u5e03\u6797 <code>bool</code> \u3002\u5b83\u5011\u7684\u53d6\u503c\u7bc4\u570d\u53d6\u6c7a\u65bc\u4f54\u7528\u7a7a\u9593\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002</li> <li>\u539f\u78bc\u3001\u4e00\u88dc\u6578\u548c\u4e8c\u88dc\u6578\u662f\u5728\u8a08\u7b97\u6a5f\u4e2d\u7de8\u78bc\u6578\u5b57\u7684\u4e09\u7a2e\u65b9\u6cd5\uff0c\u5b83\u5011\u4e4b\u9593\u53ef\u4ee5\u76f8\u4e92\u8f49\u63db\u3002\u6574\u6578\u7684\u539f\u78bc\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u865f\u4f4d\uff0c\u5176\u9918\u4f4d\u662f\u6578\u5b57\u7684\u503c\u3002</li> <li>\u6574\u6578\u5728\u8a08\u7b97\u6a5f\u4e2d\u662f\u4ee5\u4e8c\u88dc\u6578\u7684\u5f62\u5f0f\u5132\u5b58\u7684\u3002\u5728\u4e8c\u88dc\u6578\u8868\u793a\u4e0b\uff0c\u8a08\u7b97\u6a5f\u53ef\u4ee5\u5c0d\u6b63\u6578\u548c\u8ca0\u6578\u7684\u52a0\u6cd5\u4e00\u8996\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u70ba\u6e1b\u6cd5\u64cd\u4f5c\u55ae\u7368\u8a2d\u8a08\u7279\u6b8a\u7684\u786c\u9ad4\u96fb\u8def\uff0c\u4e26\u4e14\u4e0d\u5b58\u5728\u6b63\u8ca0\u96f6\u6b67\u7fa9\u7684\u554f\u984c\u3002</li> <li>\u6d6e\u9ede\u6578\u7684\u7de8\u78bc\u7531 1 \u4f4d\u7b26\u865f\u4f4d\u30018 \u4f4d\u6307\u6578\u4f4d\u548c 23 \u4f4d\u5206\u6578\u4f4d\u69cb\u6210\u3002\u7531\u65bc\u5b58\u5728\u6307\u6578\u4f4d\uff0c\u56e0\u6b64\u6d6e\u9ede\u6578\u7684\u53d6\u503c\u7bc4\u570d\u9060\u5927\u65bc\u6574\u6578\uff0c\u4ee3\u50f9\u662f\u72a7\u7272\u4e86\u7cbe\u5ea6\u3002</li> <li>ASCII \u78bc\u662f\u6700\u65e9\u51fa\u73fe\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u9577\u5ea6\u70ba 1 \u4f4d\u5143\u7d44\uff0c\u5171\u6536\u9304 127 \u500b\u5b57\u5143\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u9304\u5169\u842c\u591a\u500b\u6f22\u5b57\u3002Unicode \u81f4\u529b\u65bc\u63d0\u4f9b\u4e00\u500b\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6a19\u6e96\uff0c\u6536\u9304\u4e16\u754c\u4e0a\u5404\u7a2e\u8a9e\u8a00\u7684\u5b57\u5143\uff0c\u5f9e\u800c\u89e3\u6c7a\u7531\u65bc\u5b57\u5143\u7de8\u78bc\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5c0e\u81f4\u7684\u4e82\u78bc\u554f\u984c\u3002</li> <li>UTF-8 \u662f\u6700\u53d7\u6b61\u8fce\u7684 Unicode \u7de8\u78bc\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u7a2e\u8b8a\u9577\u7684\u7de8\u78bc\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u64f4\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5132\u5b58\u7a7a\u9593\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u9577\u7684\u7de8\u78bc\u65b9\u6cd5\u3002\u5728\u7de8\u78bc\u4e2d\u6587\u6642\uff0cUTF-16 \u4f54\u7528\u7684\u7a7a\u9593\u6bd4 UTF-8 \u66f4\u5c0f\u3002Java \u548c C# \u7b49\u7a0b\u5f0f\u8a9e\u8a00\u9810\u8a2d\u4f7f\u7528 UTF-16 \u7de8\u78bc\u3002</li> </ul>"},{"location":"chapter_data_structure/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u70ba\u4ec0\u9ebc\u96dc\u6e4a\u8868\u540c\u6642\u5305\u542b\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u548c\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff1f</p> <p>\u96dc\u6e4a\u8868\u5e95\u5c64\u662f\u9663\u5217\uff0c\u800c\u70ba\u4e86\u89e3\u6c7a\u96dc\u6e4a\u885d\u7a81\uff0c\u6211\u5011\u53ef\u80fd\u6703\u4f7f\u7528\u201c\u93c8\u5f0f\u4f4d\u5740\u201d\uff08\u5f8c\u7e8c\u201c\u96dc\u6e4a\u885d\u7a81\u201d\u7ae0\u7bc0\u6703\u8b1b\uff09\uff1a\u9663\u5217\u4e2d\u6bcf\u500b\u6876\u6307\u5411\u4e00\u500b\u93c8\u7d50\u4e32\u5217\uff0c\u7576\u93c8\u7d50\u4e32\u5217\u9577\u5ea6\u8d85\u904e\u4e00\u5b9a\u95be\u503c\u6642\uff0c\u53c8\u53ef\u80fd\u88ab\u8f49\u5316\u70ba\u6a39\uff08\u901a\u5e38\u70ba\u7d05\u9ed1\u6a39\uff09\u3002</p> <p>\u5f9e\u5132\u5b58\u7684\u89d2\u5ea6\u4f86\u770b\uff0c\u96dc\u6e4a\u8868\u7684\u5e95\u5c64\u662f\u9663\u5217\uff0c\u5176\u4e2d\u6bcf\u4e00\u500b\u6876\u69fd\u4f4d\u53ef\u80fd\u5305\u542b\u4e00\u500b\u503c\uff0c\u4e5f\u53ef\u80fd\u5305\u542b\u4e00\u500b\u93c8\u7d50\u4e32\u5217\u6216\u4e00\u68f5\u6a39\u3002\u56e0\u6b64\uff0c\u96dc\u6e4a\u8868\u53ef\u80fd\u540c\u6642\u5305\u542b\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff08\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\uff09\u548c\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff08\u6a39\uff09\u3002</p> <p>Q\uff1a<code>char</code> \u578b\u5225\u7684\u9577\u5ea6\u662f 1 \u4f4d\u5143\u7d44\u55ce\uff1f</p> <p><code>char</code> \u578b\u5225\u7684\u9577\u5ea6\u7531\u7a0b\u5f0f\u8a9e\u8a00\u63a1\u7528\u7684\u7de8\u78bc\u65b9\u6cd5\u6c7a\u5b9a\u3002\u4f8b\u5982\uff0cJava\u3001JavaScript\u3001TypeScript\u3001C# \u90fd\u63a1\u7528 UTF-16 \u7de8\u78bc\uff08\u5132\u5b58 Unicode \u78bc\u9ede\uff09\uff0c\u56e0\u6b64 <code>char</code> \u578b\u5225\u7684\u9577\u5ea6\u70ba 2 \u4f4d\u5143\u7d44\u3002</p> <p>Q\uff1a\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u8cc7\u6599\u7d50\u69cb\u4e5f\u7a31\u201c\u975c\u614b\u8cc7\u6599\u7d50\u69cb\u201d \u662f\u5426\u6709\u6b67\u7fa9\uff1f\u5806\u758a\u4e5f\u53ef\u4ee5\u9032\u884c\u51fa\u5806\u758a\u548c\u5165\u5806\u758a\u7b49\u64cd\u4f5c\uff0c\u9019\u4e9b\u64cd\u4f5c\u90fd\u662f\u201c\u52d5\u614b\u201d\u7684\u3002</p> <p>\u5806\u758a\u78ba\u5be6\u53ef\u4ee5\u5be6\u73fe\u52d5\u614b\u7684\u8cc7\u6599\u64cd\u4f5c\uff0c\u4f46\u8cc7\u6599\u7d50\u69cb\u4ecd\u7136\u662f\u201c\u975c\u614b\u201d\uff08\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff09\u7684\u3002\u5118\u7ba1\u57fa\u65bc\u9663\u5217\u7684\u8cc7\u6599\u7d50\u69cb\u53ef\u4ee5\u52d5\u614b\u5730\u65b0\u589e\u6216\u522a\u9664\u5143\u7d20\uff0c\u4f46\u5b83\u5011\u7684\u5bb9\u91cf\u662f\u56fa\u5b9a\u7684\u3002\u5982\u679c\u8cc7\u6599\u91cf\u8d85\u51fa\u4e86\u9810\u5206\u914d\u7684\u5927\u5c0f\uff0c\u5c31\u9700\u8981\u5efa\u7acb\u4e00\u500b\u65b0\u7684\u66f4\u5927\u7684\u9663\u5217\uff0c\u4e26\u5c07\u820a\u9663\u5217\u7684\u5167\u5bb9\u8907\u88fd\u5230\u65b0\u9663\u5217\u4e2d\u3002</p> <p>Q\uff1a\u5728\u69cb\u5efa\u5806\u758a\uff08\u4f47\u5217\uff09\u7684\u6642\u5019\uff0c\u672a\u6307\u5b9a\u5b83\u7684\u5927\u5c0f\uff0c\u70ba\u4ec0\u9ebc\u5b83\u5011\u662f\u201c\u975c\u614b\u8cc7\u6599\u7d50\u69cb\u201d\u5462\uff1f</p> <p>\u5728\u9ad8\u968e\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\uff0c\u6211\u5011\u7121\u9808\u4eba\u5de5\u6307\u5b9a\u5806\u758a\uff08\u4f47\u5217\uff09\u7684\u521d\u59cb\u5bb9\u91cf\uff0c\u9019\u500b\u5de5\u4f5c\u7531\u985e\u5225\u5167\u90e8\u81ea\u52d5\u5b8c\u6210\u3002\u4f8b\u5982\uff0cJava \u7684 <code>ArrayList</code> \u7684\u521d\u59cb\u5bb9\u91cf\u901a\u5e38\u70ba 10\u3002\u53e6\u5916\uff0c\u64f4\u5bb9\u64cd\u4f5c\u4e5f\u662f\u81ea\u52d5\u5be6\u73fe\u7684\u3002\u8a73\u898b\u5f8c\u7e8c\u7684\u201c\u4e32\u5217\u201d\u7ae0\u7bc0\u3002</p> <p>Q\uff1a\u539f\u78bc\u8f49\u4e8c\u88dc\u6578\u7684\u65b9\u6cd5\u662f\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\uff0c\u90a3\u9ebc\u4e8c\u88dc\u6578\u8f49\u539f\u78bc\u61c9\u8a72\u662f\u9006\u904b\u7b97\u201c\u5148\u6e1b 1 \u5f8c\u53d6\u53cd\u201d\uff0c\u800c\u4e8c\u88dc\u6578\u8f49\u539f\u78bc\u4e5f\u4e00\u6a23\u53ef\u4ee5\u900f\u904e\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u5f97\u5230\uff0c\u9019\u662f\u70ba\u4ec0\u9ebc\u5462\uff1f</p> <p>\u9019\u662f\u56e0\u70ba\u539f\u78bc\u548c\u4e8c\u88dc\u6578\u7684\u76f8\u4e92\u8f49\u63db\u5be6\u969b\u4e0a\u662f\u8a08\u7b97\u201c\u88dc\u6578\u201d\u7684\u904e\u7a0b\u3002\u6211\u5011\u5148\u7d66\u51fa\u88dc\u6578\u7684\u5b9a\u7fa9\uff1a\u5047\u8a2d \\(a + b = c\\) \uff0c\u90a3\u9ebc\u6211\u5011\u7a31 \\(a\\) \u662f \\(b\\) \u5230 \\(c\\) \u7684\u88dc\u6578\uff0c\u53cd\u4e4b\u4e5f\u7a31 \\(b\\) \u662f \\(a\\) \u5230 \\(c\\) \u7684\u88dc\u6578\u3002</p> <p>\u7d66\u5b9a\u4e00\u500b \\(n = 4\\) \u4f4d\u9577\u5ea6\u7684\u4e8c\u9032\u4f4d\u5236\u6578 \\(0010\\) \uff0c\u5982\u679c\u5c07\u9019\u500b\u6578\u5b57\u770b\u4f5c\u539f\u78bc\uff08\u4e0d\u8003\u616e\u7b26\u865f\u4f4d\uff09\uff0c\u90a3\u9ebc\u5b83\u7684\u4e8c\u88dc\u6578\u9700\u900f\u904e\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u5f97\u5230\uff1a</p> \\[ 0010 \\rightarrow 1101 \\rightarrow 1110 \\] <p>\u6211\u5011\u6703\u767c\u73fe\uff0c\u539f\u78bc\u548c\u4e8c\u88dc\u6578\u7684\u548c\u662f \\(0010 + 1110 = 10000\\) \uff0c\u4e5f\u5c31\u662f\u8aaa\uff0c\u4e8c\u88dc\u6578 \\(1110\\) \u662f\u539f\u78bc \\(0010\\) \u5230 \\(10000\\) \u7684\u201c\u88dc\u6578\u201d\u3002\u9019\u610f\u5473\u8457\u4e0a\u8ff0\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u5be6\u969b\u4e0a\u662f\u8a08\u7b97\u5230 \\(10000\\) \u7684\u88dc\u6578\u7684\u904e\u7a0b\u3002</p> <p>\u90a3\u9ebc\uff0c\u4e8c\u88dc\u6578 \\(1110\\) \u5230 \\(10000\\) \u7684\u201c\u88dc\u6578\u201d\u662f\u591a\u5c11\u5462\uff1f\u6211\u5011\u4f9d\u7136\u53ef\u4ee5\u7528\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u5f97\u5230\u5b83\uff1a</p> \\[ 1110 \\rightarrow 0001 \\rightarrow 0010 \\] <p>\u63db\u53e5\u8a71\u8aaa\uff0c\u539f\u78bc\u548c\u4e8c\u88dc\u6578\u4e92\u70ba\u5c0d\u65b9\u5230 \\(10000\\) \u7684\u201c\u88dc\u6578\u201d\uff0c\u56e0\u6b64\u201c\u539f\u78bc\u8f49\u4e8c\u88dc\u6578\u201d\u548c\u201c\u4e8c\u88dc\u6578\u8f49\u539f\u78bc\u201d\u53ef\u4ee5\u7528\u76f8\u540c\u7684\u64cd\u4f5c\uff08\u5148\u53d6\u53cd\u5f8c\u52a0 1 \uff09\u5be6\u73fe\u3002</p> <p>\u7576\u7136\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u7528\u9006\u904b\u7b97\u4f86\u6c42\u4e8c\u88dc\u6578 \\(1110\\) \u7684\u539f\u78bc\uff0c\u5373\u201c\u5148\u6e1b 1 \u5f8c\u53d6\u53cd\u201d\uff1a</p> \\[ 1110 \\rightarrow 1101 \\rightarrow 0010 \\] <p>\u7e3d\u7d50\u4f86\u770b\uff0c\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u548c\u201c\u5148\u6e1b 1 \u5f8c\u53d6\u53cd\u201d\u9019\u5169\u7a2e\u904b\u7b97\u90fd\u662f\u5728\u8a08\u7b97\u5230 \\(10000\\) \u7684\u88dc\u6578\uff0c\u5b83\u5011\u662f\u7b49\u50f9\u7684\u3002</p> <p>\u672c\u8cea\u4e0a\u770b\uff0c\u201c\u53d6\u53cd\u201d\u64cd\u4f5c\u5be6\u969b\u4e0a\u662f\u6c42\u5230 \\(1111\\) \u7684\u88dc\u6578\uff08\u56e0\u70ba\u6046\u6709 <code>\u539f\u78bc + \u4e00\u88dc\u6578 = 1111</code>\uff09\uff1b\u800c\u5728\u4e00\u88dc\u6578\u57fa\u790e\u4e0a\u518d\u52a0 1 \u5f97\u5230\u7684\u4e8c\u88dc\u6578\uff0c\u5c31\u662f\u5230 \\(10000\\) \u7684\u88dc\u6578\u3002</p> <p>\u4e0a\u8ff0\u4ee5 \\(n = 4\\) \u70ba\u4f8b\uff0c\u5176\u53ef\u88ab\u63a8\u5ee3\u81f3\u4efb\u610f\u4f4d\u6578\u7684\u4e8c\u9032\u4f4d\u5236\u6578\u3002</p>"},{"location":"chapter_divide_and_conquer/","title":"\u7b2c 12 \u7ae0 \u00a0 \u5206\u6cbb","text":"<p>Abstract</p> <p>\u96e3\u984c\u88ab\u9010\u5c64\u62c6\u89e3\uff0c\u6bcf\u4e00\u6b21\u7684\u62c6\u89e3\u90fd\u4f7f\u5b83\u8b8a\u5f97\u66f4\u70ba\u7c21\u55ae\u3002</p> <p>\u5206\u800c\u6cbb\u4e4b\u63ed\u793a\u4e86\u4e00\u500b\u91cd\u8981\u7684\u4e8b\u5be6\uff1a\u5f9e\u7c21\u55ae\u505a\u8d77\uff0c\u4e00\u5207\u90fd\u4e0d\u518d\u8907\u96dc\u3002</p>"},{"location":"chapter_divide_and_conquer/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>12.1 \u00a0 \u5206\u6cbb\u6f14\u7b97\u6cd5</li> <li>12.2 \u00a0 \u5206\u6cbb\u641c\u5c0b\u7b56\u7565</li> <li>12.3 \u00a0 \u69cb\u5efa\u6a39\u554f\u984c</li> <li>12.4 \u00a0 \u6cb3\u5167\u5854\u554f\u984c</li> <li>12.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_divide_and_conquer/binary_search_recur/","title":"12.2 \u00a0 \u5206\u6cbb\u641c\u5c0b\u7b56\u7565","text":"<p>\u6211\u5011\u5df2\u7d93\u5b78\u904e\uff0c\u641c\u5c0b\u6f14\u7b97\u6cd5\u5206\u70ba\u5169\u5927\u985e\u3002</p> <ul> <li>\u66b4\u529b\u641c\u5c0b\uff1a\u5b83\u900f\u904e\u8d70\u8a2a\u8cc7\u6599\u7d50\u69cb\u5be6\u73fe\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</li> <li>\u81ea\u9069\u61c9\u641c\u5c0b\uff1a\u5b83\u5229\u7528\u7279\u6709\u7684\u8cc7\u6599\u7d44\u7e54\u5f62\u5f0f\u6216\u5148\u9a57\u8cc7\u8a0a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \u7684\u641c\u5c0b\u6f14\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u5be6\u73fe\u7684\uff0c\u4f8b\u5982\u4e8c\u5206\u641c\u5c0b\u548c\u6a39\u3002</p> <ul> <li>\u4e8c\u5206\u641c\u5c0b\u7684\u6bcf\u4e00\u6b65\u90fd\u5c07\u554f\u984c\uff08\u5728\u9663\u5217\u4e2d\u641c\u7d22\u76ee\u6a19\u5143\u7d20\uff09\u5206\u89e3\u70ba\u4e00\u500b\u5c0f\u554f\u984c\uff08\u5728\u9663\u5217\u7684\u4e00\u534a\u4e2d\u641c\u7d22\u76ee\u6a19\u5143\u7d20\uff09\uff0c\u9019\u500b\u904e\u7a0b\u4e00\u76f4\u6301\u7e8c\u5230\u9663\u5217\u70ba\u7a7a\u6216\u627e\u5230\u76ee\u6a19\u5143\u7d20\u70ba\u6b62\u3002</li> <li>\u6a39\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4ee3\u8868\uff0c\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u3001AVL \u6a39\u3001\u5806\u7a4d\u7b49\u8cc7\u6599\u7d50\u69cb\u4e2d\uff0c\u5404\u7a2e\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u7686\u70ba \\(O(\\log n)\\) \u3002</li> </ul> <p>\u4e8c\u5206\u641c\u5c0b\u7684\u5206\u6cbb\u7b56\u7565\u5982\u4e0b\u6240\u793a\u3002</p> <ul> <li>\u554f\u984c\u53ef\u4ee5\u5206\u89e3\uff1a\u4e8c\u5206\u641c\u5c0b\u905e\u8ff4\u5730\u5c07\u539f\u554f\u984c\uff08\u5728\u9663\u5217\u4e2d\u9032\u884c\u67e5\u8a62\uff09\u5206\u89e3\u70ba\u5b50\u554f\u984c\uff08\u5728\u9663\u5217\u7684\u4e00\u534a\u4e2d\u9032\u884c\u67e5\u8a62\uff09\uff0c\u9019\u662f\u900f\u904e\u6bd4\u8f03\u4e2d\u9593\u5143\u7d20\u548c\u76ee\u6a19\u5143\u7d20\u4f86\u5be6\u73fe\u7684\u3002</li> <li>\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff1a\u5728\u4e8c\u5206\u641c\u5c0b\u4e2d\uff0c\u6bcf\u8f2a\u53ea\u8655\u7406\u4e00\u500b\u5b50\u554f\u984c\uff0c\u5b83\u4e0d\u53d7\u5176\u4ed6\u5b50\u554f\u984c\u7684\u5f71\u97ff\u3002</li> <li>\u5b50\u554f\u984c\u7684\u89e3\u7121\u9808\u5408\u4f75\uff1a\u4e8c\u5206\u641c\u5c0b\u65e8\u5728\u67e5\u8a62\u4e00\u500b\u7279\u5b9a\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5c07\u5b50\u554f\u984c\u7684\u89e3\u9032\u884c\u5408\u4f75\u3002\u7576\u5b50\u554f\u984c\u5f97\u5230\u89e3\u6c7a\u6642\uff0c\u539f\u554f\u984c\u4e5f\u6703\u540c\u6642\u5f97\u5230\u89e3\u6c7a\u3002</li> </ul> <p>\u5206\u6cbb\u80fd\u5920\u63d0\u5347\u641c\u5c0b\u6548\u7387\uff0c\u672c\u8cea\u4e0a\u662f\u56e0\u70ba\u66b4\u529b\u641c\u5c0b\u6bcf\u8f2a\u53ea\u80fd\u6392\u9664\u4e00\u500b\u9078\u9805\uff0c\u800c\u5206\u6cbb\u641c\u5c0b\u6bcf\u8f2a\u53ef\u4ee5\u6392\u9664\u4e00\u534a\u9078\u9805\u3002</p>"},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","title":"1. \u00a0 \u57fa\u65bc\u5206\u6cbb\u5be6\u73fe\u4e8c\u5206\u641c\u5c0b","text":"<p>\u5728\u4e4b\u524d\u7684\u7ae0\u7bc0\u4e2d\uff0c\u4e8c\u5206\u641c\u5c0b\u662f\u57fa\u65bc\u905e\u63a8\uff08\u8fed\u4ee3\uff09\u5be6\u73fe\u7684\u3002\u73fe\u5728\u6211\u5011\u57fa\u65bc\u5206\u6cbb\uff08\u905e\u8ff4\uff09\u4f86\u5be6\u73fe\u5b83\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u6709\u5e8f\u9663\u5217 <code>nums</code> \uff0c\u5176\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\uff0c\u8acb\u67e5\u8a62\u5143\u7d20 <code>target</code> \u3002</p> <p>\u5f9e\u5206\u6cbb\u89d2\u5ea6\uff0c\u6211\u5011\u5c07\u641c\u5c0b\u5340\u9593 \\([i, j]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\u8a18\u70ba \\(f(i, j)\\) \u3002</p> <p>\u4ee5\u539f\u554f\u984c \\(f(0, n-1)\\) \u70ba\u8d77\u59cb\u9ede\uff0c\u900f\u904e\u4ee5\u4e0b\u6b65\u9a5f\u9032\u884c\u4e8c\u5206\u641c\u5c0b\u3002</p> <ol> <li>\u8a08\u7b97\u641c\u5c0b\u5340\u9593 \\([i, j]\\) \u7684\u4e2d\u9ede \\(m\\) \uff0c\u6839\u64da\u5b83\u6392\u9664\u4e00\u534a\u641c\u5c0b\u5340\u9593\u3002</li> <li>\u905e\u8ff4\u6c42\u89e3\u898f\u6a21\u6e1b\u5c0f\u4e00\u534a\u7684\u5b50\u554f\u984c\uff0c\u53ef\u80fd\u70ba \\(f(i, m-1)\\) \u6216 \\(f(m+1, j)\\) \u3002</li> <li>\u8ff4\u5708\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\uff0c\u76f4\u81f3\u627e\u5230 <code>target</code> \u6216\u5340\u9593\u70ba\u7a7a\u6642\u8fd4\u56de\u3002</li> </ol> <p>\u5716 12-4 \u5c55\u793a\u4e86\u5728\u9663\u5217\u4e2d\u4e8c\u5206\u641c\u5c0b\u5143\u7d20 \\(6\\) \u7684\u5206\u6cbb\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 12-4 \u00a0 \u4e8c\u5206\u641c\u5c0b\u7684\u5206\u6cbb\u904e\u7a0b </p> <p>\u5728\u5be6\u73fe\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u6211\u5011\u5ba3\u544a\u4e00\u500b\u905e\u8ff4\u51fd\u5f0f <code>dfs()</code> \u4f86\u6c42\u89e3\u554f\u984c \\(f(i, j)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j)\"\"\"\n    # \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if i &gt; j:\n        return -1\n    # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1)\n    else:\n        # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\"\"\"\n    n = len(nums)\n    # \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // \u5982\u679c\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u6c92\u6709\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    //    \u8a08\u7b97\u7d22\u5f15\u4e2d\u9ede\n    m := i + ((j - i) &gt;&gt; 1)\n    //\u5224\u65b7\u4e2d\u9ede\u8207\u76ee\u6a19\u5143\u7d20\u5927\u5c0f\n    if nums[m] &lt; target {\n        // \u5c0f\u65bc\u5247\u905e\u8ff4\u53f3\u534a\u9663\u5217\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // \u5927\u65bc\u5247\u905e\u8ff4\u5de6\u534a\u9663\u5217\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m-1)\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    dfs(nums: nums, target: target, i: nums.startIndex, j: nums.endIndex - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n  if (i &gt; j) {\n    return -1;\n  }\n  // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return m;\n  }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // \u6c42\u89e3\u554f\u984c f(0, n-1)\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if i &gt; j {\n        return -1;\n    }\n    let m: i32 = i + (j - i) / 2;\n    if nums[m as usize] &lt; target {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfun dfs(\n    nums: IntArray,\n    target: Int,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    val m = (i + j) / 2\n    return if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        dfs(nums, target, m + 1, j)\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        dfs(nums, target, i, m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        m\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    val n = nums.size\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) ###\ndef dfs(nums, target, i, j)\n  # \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n  return -1 if i &gt; j\n\n  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n  m = (i + j) / 2\n\n  if nums[m] &lt; target\n    # \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n    return dfs(nums, target, m + 1, j)\n  elsif nums[m] &gt; target\n    # \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n    return dfs(nums, target, i, m - 1)\n  else\n    # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return m\n  end\nend\n\n### \u4e8c\u5206\u641c\u5c0b ###\ndef binary_search(nums, target)\n  n = nums.length\n  # \u6c42\u89e3\u554f\u984c f(0, n-1)\n  dfs(nums, target, 0, n - 1)\nend\n</code></pre> binary_search_recur.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{binarySearch}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","title":"12.3 \u00a0 \u69cb\u5efa\u4e8c\u5143\u6a39\u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u68f5\u4e8c\u5143\u6a39\u7684\u524d\u5e8f\u8d70\u8a2a <code>preorder</code> \u548c\u4e2d\u5e8f\u8d70\u8a2a <code>inorder</code> \uff0c\u8acb\u5f9e\u4e2d\u69cb\u5efa\u4e8c\u5143\u6a39\uff0c\u8fd4\u56de\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede\u3002\u5047\u8a2d\u4e8c\u5143\u6a39\u4e2d\u6c92\u6709\u503c\u91cd\u8907\u7684\u7bc0\u9ede\uff08\u5982\u5716 12-5 \u6240\u793a\uff09\u3002</p> <p></p> <p> \u5716 12-5 \u00a0 \u69cb\u5efa\u4e8c\u5143\u6a39\u7684\u793a\u4f8b\u8cc7\u6599 </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","title":"1. \u00a0 \u5224\u65b7\u662f\u5426\u70ba\u5206\u6cbb\u554f\u984c","text":"<p>\u539f\u554f\u984c\u5b9a\u7fa9\u70ba\u5f9e <code>preorder</code> \u548c <code>inorder</code> \u69cb\u5efa\u4e8c\u5143\u6a39\uff0c\u662f\u4e00\u500b\u5178\u578b\u7684\u5206\u6cbb\u554f\u984c\u3002</p> <ul> <li>\u554f\u984c\u53ef\u4ee5\u5206\u89e3\uff1a\u5f9e\u5206\u6cbb\u7684\u89d2\u5ea6\u5207\u5165\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u539f\u554f\u984c\u5283\u5206\u70ba\u5169\u500b\u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\u3001\u69cb\u5efa\u53f3\u5b50\u6a39\uff0c\u52a0\u4e0a\u4e00\u6b65\u64cd\u4f5c\uff1a\u521d\u59cb\u5316\u6839\u7bc0\u9ede\u3002\u800c\u5c0d\u65bc\u6bcf\u68f5\u5b50\u6a39\uff08\u5b50\u554f\u984c\uff09\uff0c\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u8907\u7528\u4ee5\u4e0a\u5283\u5206\u65b9\u6cd5\uff0c\u5c07\u5176\u5283\u5206\u70ba\u66f4\u5c0f\u7684\u5b50\u6a39\uff08\u5b50\u554f\u984c\uff09\uff0c\u76f4\u81f3\u9054\u5230\u6700\u5c0f\u5b50\u554f\u984c\uff08\u7a7a\u5b50\u6a39\uff09\u6642\u7d42\u6b62\u3002</li> <li>\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff1a\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\u662f\u76f8\u4e92\u7368\u7acb\u7684\uff0c\u5b83\u5011\u4e4b\u9593\u6c92\u6709\u4ea4\u96c6\u3002\u5728\u69cb\u5efa\u5de6\u5b50\u6a39\u6642\uff0c\u6211\u5011\u53ea\u9700\u95dc\u6ce8\u4e2d\u5e8f\u8d70\u8a2a\u548c\u524d\u5e8f\u8d70\u8a2a\u4e2d\u8207\u5de6\u5b50\u6a39\u5c0d\u61c9\u7684\u90e8\u5206\u3002\u53f3\u5b50\u6a39\u540c\u7406\u3002</li> <li>\u5b50\u554f\u984c\u7684\u89e3\u53ef\u4ee5\u5408\u4f75\uff1a\u4e00\u65e6\u5f97\u5230\u4e86\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\uff08\u5b50\u554f\u984c\u7684\u89e3\uff09\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5c07\u5b83\u5011\u9023\u7d50\u5230\u6839\u7bc0\u9ede\u4e0a\uff0c\u5f97\u5230\u539f\u554f\u984c\u7684\u89e3\u3002</li> </ul>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2","title":"2. \u00a0 \u5982\u4f55\u5283\u5206\u5b50\u6a39","text":"<p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u9019\u9053\u984c\u53ef\u4ee5\u4f7f\u7528\u5206\u6cbb\u4f86\u6c42\u89e3\uff0c\u4f46\u5982\u4f55\u900f\u904e\u524d\u5e8f\u8d70\u8a2a <code>preorder</code> \u548c\u4e2d\u5e8f\u8d70\u8a2a <code>inorder</code> \u4f86\u5283\u5206\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\u5462\uff1f</p> <p>\u6839\u64da\u5b9a\u7fa9\uff0c<code>preorder</code> \u548c <code>inorder</code> \u90fd\u53ef\u4ee5\u5283\u5206\u70ba\u4e09\u500b\u90e8\u5206\u3002</p> <ul> <li>\u524d\u5e8f\u8d70\u8a2a\uff1a<code>[ \u6839\u7bc0\u9ede | \u5de6\u5b50\u6a39 | \u53f3\u5b50\u6a39 ]</code> \uff0c\u4f8b\u5982\u5716 12-5 \u7684\u6a39\u5c0d\u61c9 <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> <li>\u4e2d\u5e8f\u8d70\u8a2a\uff1a<code>[ \u5de6\u5b50\u6a39 | \u6839\u7bc0\u9ede \uff5c \u53f3\u5b50\u6a39 ]</code> \uff0c\u4f8b\u5982\u5716 12-5 \u7684\u6a39\u5c0d\u61c9 <code>[ 9 | 3 | 1 2 7 ]</code> \u3002</li> </ul> <p>\u4ee5\u4e0a\u5716\u8cc7\u6599\u70ba\u4f8b\uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u5716 12-6 \u6240\u793a\u7684\u6b65\u9a5f\u5f97\u5230\u5283\u5206\u7d50\u679c\u3002</p> <ol> <li>\u524d\u5e8f\u8d70\u8a2a\u7684\u9996\u5143\u7d20 3 \u662f\u6839\u7bc0\u9ede\u7684\u503c\u3002</li> <li>\u67e5\u8a62\u6839\u7bc0\u9ede 3 \u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u5229\u7528\u8a72\u7d22\u5f15\u53ef\u5c07 <code>inorder</code> \u5283\u5206\u70ba <code>[ 9 | 3 \uff5c 1 2 7 ]</code> \u3002</li> <li>\u6839\u64da <code>inorder</code> \u7684\u5283\u5206\u7d50\u679c\uff0c\u6613\u5f97\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u5206\u5225\u70ba 1 \u548c 3 \uff0c\u5f9e\u800c\u53ef\u5c07 <code>preorder</code> \u5283\u5206\u70ba <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> </ol> <p></p> <p> \u5716 12-6 \u00a0 \u5728\u524d\u5e8f\u8d70\u8a2a\u548c\u4e2d\u5e8f\u8d70\u8a2a\u4e2d\u5283\u5206\u5b50\u6a39 </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","title":"3. \u00a0 \u57fa\u65bc\u8b8a\u6578\u63cf\u8ff0\u5b50\u6a39\u5340\u9593","text":"<p>\u6839\u64da\u4ee5\u4e0a\u5283\u5206\u65b9\u6cd5\uff0c\u6211\u5011\u5df2\u7d93\u5f97\u5230\u6839\u7bc0\u9ede\u3001\u5de6\u5b50\u6a39\u3001\u53f3\u5b50\u6a39\u5728 <code>preorder</code> \u548c <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u5340\u9593\u3002\u800c\u70ba\u4e86\u63cf\u8ff0\u9019\u4e9b\u7d22\u5f15\u5340\u9593\uff0c\u6211\u5011\u9700\u8981\u85c9\u52a9\u5e7e\u500b\u6307\u6a19\u8b8a\u6578\u3002</p> <ul> <li>\u5c07\u7576\u524d\u6a39\u7684\u6839\u7bc0\u9ede\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\u8a18\u70ba \\(i\\) \u3002</li> <li>\u5c07\u7576\u524d\u6a39\u7684\u6839\u7bc0\u9ede\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u8a18\u70ba \\(m\\) \u3002</li> <li>\u5c07\u7576\u524d\u6a39\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u5340\u9593\u8a18\u70ba \\([l, r]\\) \u3002</li> </ul> <p>\u5982\u8868 12-1 \u6240\u793a\uff0c\u900f\u904e\u4ee5\u4e0a\u8b8a\u6578\u5373\u53ef\u8868\u793a\u6839\u7bc0\u9ede\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u4ee5\u53ca\u5b50\u6a39\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u5340\u9593\u3002</p> <p> \u8868 12-1 \u00a0 \u6839\u7bc0\u9ede\u548c\u5b50\u6a39\u5728\u524d\u5e8f\u8d70\u8a2a\u548c\u4e2d\u5e8f\u8d70\u8a2a\u4e2d\u7684\u7d22\u5f15 </p> \u6839\u7bc0\u9ede\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15 \u5b50\u6a39\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u5340\u9593 \u7576\u524d\u6a39 \\(i\\) \\([l, r]\\) \u5de6\u5b50\u6a39 \\(i + 1\\) \\([l, m-1]\\) \u53f3\u5b50\u6a39 \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>\u8acb\u6ce8\u610f\uff0c\u53f3\u5b50\u6a39\u6839\u7bc0\u9ede\u7d22\u5f15\u4e2d\u7684 \\((m-l)\\) \u7684\u542b\u7fa9\u662f\u201c\u5de6\u5b50\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u201d\uff0c\u5efa\u8b70\u7d50\u5408\u5716 12-7 \u7406\u89e3\u3002</p> <p></p> <p> \u5716 12-7 \u00a0 \u6839\u7bc0\u9ede\u548c\u5de6\u53f3\u5b50\u6a39\u7684\u7d22\u5f15\u5340\u9593\u8868\u793a </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","title":"4. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u70ba\u4e86\u63d0\u5347\u67e5\u8a62 \\(m\\) \u7684\u6548\u7387\uff0c\u6211\u5011\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u8868 <code>hmap</code> \u4f86\u5132\u5b58\u9663\u5217 <code>inorder</code> \u4e2d\u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"\u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb\"\"\"\n    # \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if r - l &lt; 0:\n        return None\n    # \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    root = TreeNode(preorder[i])\n    # \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    m = inorder_map[preorder[i]]\n    # \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"\u69cb\u5efa\u4e8c\u5143\u6a39\"\"\"\n    # \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    TreeNode *root = new TreeNode(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    TreeNode root = new TreeNode(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    int m = inorderMap.get(preorder[i]);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    TreeNode root = new(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if r-l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    root := NewTreeNode(preorder[i])\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    m := inorderMap[preorder[i]]\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if r - l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    let root = TreeNode(x: preorder[i])\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    let m = inorderMap[preorder[i]]!\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: inorder.startIndex, l: inorder.startIndex, r: inorder.endIndex - 1)\n}\n</code></pre> build_tree.js<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    const root = new TreeNode(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfunction buildTree(preorder, inorder) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n  TreeNode? root = TreeNode(preorder[i]);\n  // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n  int m = inorderMap[preorder[i]]!;\n  // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // \u8fd4\u56de\u6839\u7bc0\u9ede\n  return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfn dfs(\n    preorder: &amp;[i32],\n    inorder_map: &amp;HashMap&lt;i32, i32&gt;,\n    i: i32,\n    l: i32,\n    r: i32,\n) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if r - l &lt; 0 {\n        return None;\n    }\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    let root = TreeNode::new(preorder[i as usize]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    Some(root)\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.kt<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfun dfs(\n    preorder: IntArray,\n    inorderMap: Map&lt;Int?, Int?&gt;,\n    i: Int,\n    l: Int,\n    r: Int\n): TreeNode? {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0) return null\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    val root = TreeNode(preorder[i])\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    val m = inorderMap[preorder[i]]!!\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1)\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r)\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfun buildTree(preorder: IntArray, inorder: IntArray): TreeNode? {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    val inorderMap = HashMap&lt;Int?, Int?&gt;()\n    for (i in inorder.indices) {\n        inorderMap[inorder[i]] = i\n    }\n    val root = dfs(preorder, inorderMap, 0, 0, inorder.size - 1)\n    return root\n}\n</code></pre> build_tree.rb<pre><code>### \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb ###\ndef dfs(preorder, inorder_map, i, l, r)\n  # \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n  return if r - l &lt; 0\n\n  # \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n  root = TreeNode.new(preorder[i])\n  # \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n  m = inorder_map[preorder[i]]\n  # \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n  root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n  # \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n  root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n\n  # \u8fd4\u56de\u6839\u7bc0\u9ede\n  root\nend\n\n### \u69cb\u5efa\u4e8c\u5143\u6a39 ###\ndef build_tree(preorder, inorder)\n  # \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n  inorder_map = {}\n  inorder.each_with_index { |val, i| inorder_map[val] = i }\n  dfs(preorder, inorder_map, 0, 0, inorder.length - 1)\nend\n</code></pre> build_tree.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{buildTree}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 12-8 \u5c55\u793a\u4e86\u69cb\u5efa\u4e8c\u5143\u6a39\u7684\u905e\u8ff4\u904e\u7a0b\uff0c\u5404\u500b\u7bc0\u9ede\u662f\u5728\u5411\u4e0b\u201c\u905e\u201d\u7684\u904e\u7a0b\u4e2d\u5efa\u7acb\u7684\uff0c\u800c\u5404\u689d\u908a\uff08\u5f15\u7528\uff09\u662f\u5728\u5411\u4e0a\u201c\u8ff4\u201d\u7684\u904e\u7a0b\u4e2d\u5efa\u7acb\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 12-8 \u00a0 \u69cb\u5efa\u4e8c\u5143\u6a39\u7684\u905e\u8ff4\u904e\u7a0b </p> <p>\u6bcf\u500b\u905e\u8ff4\u51fd\u5f0f\u5167\u7684\u524d\u5e8f\u8d70\u8a2a <code>preorder</code> \u548c\u4e2d\u5e8f\u8d70\u8a2a <code>inorder</code> \u7684\u5283\u5206\u7d50\u679c\u5982\u5716 12-9 \u6240\u793a\u3002</p> <p></p> <p> \u5716 12-9 \u00a0 \u6bcf\u500b\u905e\u8ff4\u51fd\u5f0f\u4e2d\u7684\u5283\u5206\u7d50\u679c </p> <p>\u8a2d\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\(n\\) \uff0c\u521d\u59cb\u5316\u6bcf\u4e00\u500b\u7bc0\u9ede\uff08\u57f7\u884c\u4e00\u500b\u905e\u8ff4\u51fd\u5f0f <code>dfs()</code> \uff09\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002\u56e0\u6b64\u7e3d\u9ad4\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> <p>\u96dc\u6e4a\u8868\u5132\u5b58 <code>inorder</code> \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u5373\u4e8c\u5143\u6a39\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u6642\uff0c\u905e\u8ff4\u6df1\u5ea6\u9054\u5230 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002\u56e0\u6b64\u7e3d\u9ad4\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/","title":"12.1 \u00a0 \u5206\u6cbb\u6f14\u7b97\u6cd5","text":"<p>\u5206\u6cbb\uff08divide and conquer\uff09\uff0c\u5168\u7a31\u5206\u800c\u6cbb\u4e4b\uff0c\u662f\u4e00\u7a2e\u975e\u5e38\u91cd\u8981\u4e14\u5e38\u898b\u7684\u6f14\u7b97\u6cd5\u7b56\u7565\u3002\u5206\u6cbb\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u5be6\u73fe\uff0c\u5305\u62ec\u201c\u5206\u201d\u548c\u201c\u6cbb\u201d\u5169\u500b\u6b65\u9a5f\u3002</p> <ol> <li>\u5206\uff08\u5283\u5206\u968e\u6bb5\uff09\uff1a\u905e\u8ff4\u5730\u5c07\u539f\u554f\u984c\u5206\u89e3\u70ba\u5169\u500b\u6216\u591a\u500b\u5b50\u554f\u984c\uff0c\u76f4\u81f3\u5230\u9054\u6700\u5c0f\u5b50\u554f\u984c\u6642\u7d42\u6b62\u3002</li> <li>\u6cbb\uff08\u5408\u4f75\u968e\u6bb5\uff09\uff1a\u5f9e\u5df2\u77e5\u89e3\u7684\u6700\u5c0f\u5b50\u554f\u984c\u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5730\u5c07\u5b50\u554f\u984c\u7684\u89e3\u9032\u884c\u5408\u4f75\uff0c\u5f9e\u800c\u69cb\u5efa\u51fa\u539f\u554f\u984c\u7684\u89e3\u3002</li> </ol> <p>\u5982\u5716 12-1 \u6240\u793a\uff0c\u201c\u5408\u4f75\u6392\u5e8f\u201d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u61c9\u7528\u4e4b\u4e00\u3002</p> <ol> <li>\u5206\uff1a\u905e\u8ff4\u5730\u5c07\u539f\u9663\u5217\uff08\u539f\u554f\u984c\uff09\u5283\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff08\u5b50\u554f\u984c\uff09\uff0c\u76f4\u5230\u5b50\u9663\u5217\u53ea\u5269\u4e00\u500b\u5143\u7d20\uff08\u6700\u5c0f\u5b50\u554f\u984c\uff09\u3002</li> <li>\u6cbb\uff1a\u5f9e\u5e95\u81f3\u9802\u5730\u5c07\u6709\u5e8f\u7684\u5b50\u9663\u5217\uff08\u5b50\u554f\u984c\u7684\u89e3\uff09\u9032\u884c\u5408\u4f75\uff0c\u5f9e\u800c\u5f97\u5230\u6709\u5e8f\u7684\u539f\u9663\u5217\uff08\u539f\u554f\u984c\u7684\u89e3\uff09\u3002</li> </ol> <p></p> <p> \u5716 12-1 \u00a0 \u5408\u4f75\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","title":"12.1.1 \u00a0 \u5982\u4f55\u5224\u65b7\u5206\u6cbb\u554f\u984c","text":"<p>\u4e00\u500b\u554f\u984c\u662f\u5426\u9069\u5408\u4f7f\u7528\u5206\u6cbb\u89e3\u6c7a\uff0c\u901a\u5e38\u53ef\u4ee5\u53c3\u8003\u4ee5\u4e0b\u5e7e\u500b\u5224\u65b7\u4f9d\u64da\u3002</p> <ol> <li>\u554f\u984c\u53ef\u4ee5\u5206\u89e3\uff1a\u539f\u554f\u984c\u53ef\u4ee5\u5206\u89e3\u6210\u898f\u6a21\u66f4\u5c0f\u3001\u985e\u4f3c\u7684\u5b50\u554f\u984c\uff0c\u4ee5\u53ca\u80fd\u5920\u4ee5\u76f8\u540c\u65b9\u5f0f\u905e\u8ff4\u5730\u9032\u884c\u5283\u5206\u3002</li> <li>\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff1a\u5b50\u554f\u984c\u4e4b\u9593\u6c92\u6709\u91cd\u758a\uff0c\u4e92\u4e0d\u4f9d\u8cf4\uff0c\u53ef\u4ee5\u7368\u7acb\u89e3\u6c7a\u3002</li> <li>\u5b50\u554f\u984c\u7684\u89e3\u53ef\u4ee5\u5408\u4f75\uff1a\u539f\u554f\u984c\u7684\u89e3\u900f\u904e\u5408\u4f75\u5b50\u554f\u984c\u7684\u89e3\u5f97\u4f86\u3002</li> </ol> <p>\u986f\u7136\uff0c\u5408\u4f75\u6392\u5e8f\u6eff\u8db3\u4ee5\u4e0a\u4e09\u500b\u5224\u65b7\u4f9d\u64da\u3002</p> <ol> <li>\u554f\u984c\u53ef\u4ee5\u5206\u89e3\uff1a\u905e\u8ff4\u5730\u5c07\u9663\u5217\uff08\u539f\u554f\u984c\uff09\u5283\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff08\u5b50\u554f\u984c\uff09\u3002</li> <li>\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff1a\u6bcf\u500b\u5b50\u9663\u5217\u90fd\u53ef\u4ee5\u7368\u7acb\u5730\u9032\u884c\u6392\u5e8f\uff08\u5b50\u554f\u984c\u53ef\u4ee5\u7368\u7acb\u9032\u884c\u6c42\u89e3\uff09\u3002</li> <li>\u5b50\u554f\u984c\u7684\u89e3\u53ef\u4ee5\u5408\u4f75\uff1a\u5169\u500b\u6709\u5e8f\u5b50\u9663\u5217\uff08\u5b50\u554f\u984c\u7684\u89e3\uff09\u53ef\u4ee5\u5408\u4f75\u70ba\u4e00\u500b\u6709\u5e8f\u9663\u5217\uff08\u539f\u554f\u984c\u7684\u89e3\uff09\u3002</li> </ol>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212","title":"12.1.2 \u00a0 \u900f\u904e\u5206\u6cbb\u63d0\u5347\u6548\u7387","text":"<p>\u5206\u6cbb\u4e0d\u50c5\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u6c7a\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u5f80\u5f80\u9084\u53ef\u4ee5\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\u3002\u5728\u6392\u5e8f\u6f14\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u3001\u5408\u4f75\u6392\u5e8f\u3001\u5806\u7a4d\u6392\u5e8f\u76f8\u8f03\u65bc\u9078\u64c7\u3001\u5192\u6ce1\u3001\u63d2\u5165\u6392\u5e8f\u66f4\u5feb\uff0c\u5c31\u662f\u56e0\u70ba\u5b83\u5011\u61c9\u7528\u4e86\u5206\u6cbb\u7b56\u7565\u3002</p> <p>\u90a3\u9ebc\uff0c\u6211\u5011\u4e0d\u7981\u767c\u554f\uff1a\u70ba\u4ec0\u9ebc\u5206\u6cbb\u53ef\u4ee5\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u5176\u5e95\u5c64\u908f\u8f2f\u662f\u4ec0\u9ebc\uff1f\u63db\u53e5\u8a71\u8aaa\uff0c\u5c07\u5927\u554f\u984c\u5206\u89e3\u70ba\u591a\u500b\u5b50\u554f\u984c\u3001\u89e3\u6c7a\u5b50\u554f\u984c\u3001\u5c07\u5b50\u554f\u984c\u7684\u89e3\u5408\u4f75\u70ba\u539f\u554f\u984c\u7684\u89e3\uff0c\u9019\u5e7e\u6b65\u7684\u6548\u7387\u70ba\u4ec0\u9ebc\u6bd4\u76f4\u63a5\u89e3\u6c7a\u539f\u554f\u984c\u7684\u6548\u7387\u66f4\u9ad8\uff1f\u9019\u500b\u554f\u984c\u53ef\u4ee5\u5f9e\u64cd\u4f5c\u6578\u91cf\u548c\u5e73\u884c\u8a08\u7b97\u5169\u65b9\u9762\u4f86\u8a0e\u8ad6\u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","title":"1. \u00a0 \u64cd\u4f5c\u6578\u91cf\u6700\u4f73\u5316","text":"<p>\u4ee5\u201c\u6ce1\u6cab\u6392\u5e8f\u201d\u70ba\u4f8b\uff0c\u5176\u8655\u7406\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217\u9700\u8981 \\(O(n^2)\\) \u6642\u9593\u3002\u5047\u8a2d\u6211\u5011\u6309\u7167\u5716 12-2 \u6240\u793a\u7684\u65b9\u5f0f\uff0c\u5c07\u9663\u5217\u5f9e\u4e2d\u9ede\u8655\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff0c\u5247\u5283\u5206\u9700\u8981 \\(O(n)\\) \u6642\u9593\uff0c\u6392\u5e8f\u6bcf\u500b\u5b50\u9663\u5217\u9700\u8981 \\(O((n / 2)^2)\\) \u6642\u9593\uff0c\u5408\u4f75\u5169\u500b\u5b50\u9663\u5217\u9700\u8981 \\(O(n)\\) \u6642\u9593\uff0c\u7e3d\u9ad4\u6642\u9593\u8907\u96dc\u5ea6\u70ba\uff1a</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> \u5716 12-2 \u00a0 \u5283\u5206\u9663\u5217\u524d\u5f8c\u7684\u6ce1\u6cab\u6392\u5e8f </p> <p>\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u8a08\u7b97\u4ee5\u4e0b\u4e0d\u7b49\u5f0f\uff0c\u5176\u5de6\u908a\u548c\u53f3\u908a\u5206\u5225\u70ba\u5283\u5206\u524d\u548c\u5283\u5206\u5f8c\u7684\u64cd\u4f5c\u7e3d\u6578\uff1a</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>\u9019\u610f\u5473\u8457\u7576 \\(n &gt; 4\\) \u6642\uff0c\u5283\u5206\u5f8c\u7684\u64cd\u4f5c\u6578\u91cf\u66f4\u5c11\uff0c\u6392\u5e8f\u6548\u7387\u61c9\u8a72\u66f4\u9ad8\u3002\u8acb\u6ce8\u610f\uff0c\u5283\u5206\u5f8c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4ecd\u7136\u662f\u5e73\u65b9\u968e \\(O(n^2)\\) \uff0c\u53ea\u662f\u8907\u96dc\u5ea6\u4e2d\u7684\u5e38\u6578\u9805\u8b8a\u5c0f\u4e86\u3002</p> <p>\u9032\u4e00\u6b65\u60f3\uff0c\u5982\u679c\u6211\u5011\u628a\u5b50\u9663\u5217\u4e0d\u65b7\u5730\u518d\u5f9e\u4e2d\u9ede\u8655\u5283\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff0c\u76f4\u81f3\u5b50\u9663\u5217\u53ea\u5269\u4e00\u500b\u5143\u7d20\u6642\u505c\u6b62\u5283\u5206\u5462\uff1f\u9019\u7a2e\u601d\u8def\u5be6\u969b\u4e0a\u5c31\u662f\u201c\u5408\u4f75\u6392\u5e8f\u201d\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</p> <p>\u518d\u601d\u8003\uff0c\u5982\u679c\u6211\u5011\u591a\u8a2d\u5b9a\u5e7e\u500b\u5283\u5206\u9ede\uff0c\u5c07\u539f\u9663\u5217\u5e73\u5747\u5283\u5206\u70ba \\(k\\) \u500b\u5b50\u9663\u5217\u5462\uff1f\u9019\u7a2e\u60c5\u6cc1\u8207\u201c\u6876\u6392\u5e8f\u201d\u975e\u5e38\u985e\u4f3c\uff0c\u5b83\u975e\u5e38\u9069\u5408\u6392\u5e8f\u6d77\u91cf\u8cc7\u6599\uff0c\u7406\u8ad6\u4e0a\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u9054\u5230 \\(O(n + k)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","title":"2. \u00a0 \u5e73\u884c\u8a08\u7b97\u6700\u4f73\u5316","text":"<p>\u6211\u5011\u77e5\u9053\uff0c\u5206\u6cbb\u751f\u6210\u7684\u5b50\u554f\u984c\u662f\u76f8\u4e92\u7368\u7acb\u7684\uff0c\u56e0\u6b64\u901a\u5e38\u53ef\u4ee5\u4e26\u884c\u89e3\u6c7a\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u5206\u6cbb\u4e0d\u50c5\u53ef\u4ee5\u964d\u4f4e\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u9084\u6709\u5229\u65bc\u4f5c\u696d\u7cfb\u7d71\u7684\u4e26\u884c\u6700\u4f73\u5316\u3002</p> <p>\u4e26\u884c\u6700\u4f73\u5316\u5728\u591a\u6838\u6216\u591a\u8655\u7406\u5668\u7684\u74b0\u5883\u4e2d\u5c24\u5176\u6709\u6548\uff0c\u56e0\u70ba\u7cfb\u7d71\u53ef\u4ee5\u540c\u6642\u8655\u7406\u591a\u500b\u5b50\u554f\u984c\uff0c\u66f4\u52a0\u5145\u5206\u5730\u5229\u7528\u8a08\u7b97\u8cc7\u6e90\uff0c\u5f9e\u800c\u986f\u8457\u6e1b\u5c11\u7e3d\u9ad4\u7684\u57f7\u884c\u6642\u9593\u3002</p> <p>\u6bd4\u5982\u5728\u5716 12-3 \u6240\u793a\u7684\u201c\u6876\u6392\u5e8f\u201d\u4e2d\uff0c\u6211\u5011\u5c07\u6d77\u91cf\u7684\u8cc7\u6599\u5e73\u5747\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\uff0c\u5247\u53ef\u5c07\u6240\u6709\u6876\u7684\u6392\u5e8f\u4efb\u52d9\u5206\u6563\u5230\u5404\u500b\u8a08\u7b97\u55ae\u5143\uff0c\u5b8c\u6210\u5f8c\u518d\u5408\u4f75\u7d50\u679c\u3002</p> <p></p> <p> \u5716 12-3 \u00a0 \u6876\u6392\u5e8f\u7684\u5e73\u884c\u8a08\u7b97 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","title":"12.1.3 \u00a0 \u5206\u6cbb\u5e38\u898b\u61c9\u7528","text":"<p>\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u53ef\u4ee5\u7528\u4f86\u89e3\u6c7a\u8a31\u591a\u7d93\u5178\u6f14\u7b97\u6cd5\u554f\u984c\u3002</p> <ul> <li>\u5c0b\u627e\u6700\u8fd1\u9ede\u5c0d\uff1a\u8a72\u6f14\u7b97\u6cd5\u9996\u5148\u5c07\u9ede\u96c6\u5206\u6210\u5169\u90e8\u5206\uff0c\u7136\u5f8c\u5206\u5225\u627e\u51fa\u5169\u90e8\u5206\u4e2d\u7684\u6700\u8fd1\u9ede\u5c0d\uff0c\u6700\u5f8c\u627e\u51fa\u8de8\u8d8a\u5169\u90e8\u5206\u7684\u6700\u8fd1\u9ede\u5c0d\u3002</li> <li>\u5927\u6574\u6578\u4e58\u6cd5\uff1a\u4f8b\u5982 Karatsuba \u6f14\u7b97\u6cd5\uff0c\u5b83\u5c07\u5927\u6574\u6578\u4e58\u6cd5\u5206\u89e3\u70ba\u5e7e\u500b\u8f03\u5c0f\u7684\u6574\u6578\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u77e9\u9663\u4e58\u6cd5\uff1a\u4f8b\u5982 Strassen \u6f14\u7b97\u6cd5\uff0c\u5b83\u5c07\u5927\u77e9\u9663\u4e58\u6cd5\u5206\u89e3\u70ba\u591a\u500b\u5c0f\u77e9\u9663\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u6cb3\u5167\u5854\u554f\u984c\uff1a\u6cb3\u5167\u5854\u554f\u984c\u53ef\u4ee5\u900f\u904e\u905e\u8ff4\u89e3\u6c7a\uff0c\u9019\u662f\u5178\u578b\u7684\u5206\u6cbb\u7b56\u7565\u61c9\u7528\u3002</li> <li>\u6c42\u89e3\u9006\u5e8f\u5c0d\uff1a\u5728\u4e00\u500b\u5e8f\u5217\u4e2d\uff0c\u5982\u679c\u524d\u9762\u7684\u6578\u5b57\u5927\u65bc\u5f8c\u9762\u7684\u6578\u5b57\uff0c\u90a3\u9ebc\u9019\u5169\u500b\u6578\u5b57\u69cb\u6210\u4e00\u500b\u9006\u5e8f\u5c0d\u3002\u6c42\u89e3\u9006\u5e8f\u5c0d\u554f\u984c\u53ef\u4ee5\u5229\u7528\u5206\u6cbb\u7684\u601d\u60f3\uff0c\u85c9\u52a9\u5408\u4f75\u6392\u5e8f\u9032\u884c\u6c42\u89e3\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5728\u6f14\u7b97\u6cd5\u548c\u8cc7\u6599\u7d50\u69cb\u7684\u8a2d\u8a08\u4e2d\u61c9\u7528\u5f97\u975e\u5e38\u5ee3\u6cdb\u3002</p> <ul> <li>\u4e8c\u5206\u641c\u5c0b\uff1a\u4e8c\u5206\u641c\u5c0b\u662f\u5c07\u6709\u5e8f\u9663\u5217\u5f9e\u4e2d\u9ede\u7d22\u5f15\u8655\u5206\u70ba\u5169\u90e8\u5206\uff0c\u7136\u5f8c\u6839\u64da\u76ee\u6a19\u503c\u8207\u4e2d\u9593\u5143\u7d20\u503c\u6bd4\u8f03\u7d50\u679c\uff0c\u6c7a\u5b9a\u6392\u9664\u54ea\u4e00\u534a\u5340\u9593\uff0c\u4e26\u5728\u5269\u9918\u5340\u9593\u57f7\u884c\u76f8\u540c\u7684\u4e8c\u5206\u64cd\u4f5c\u3002</li> <li>\u5408\u4f75\u6392\u5e8f\uff1a\u672c\u7bc0\u958b\u982d\u5df2\u4ecb\u7d39\uff0c\u4e0d\u518d\u8d05\u8ff0\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\uff1a\u5feb\u901f\u6392\u5e8f\u662f\u9078\u53d6\u4e00\u500b\u57fa\u6e96\u503c\uff0c\u7136\u5f8c\u628a\u9663\u5217\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff0c\u4e00\u500b\u5b50\u9663\u5217\u7684\u5143\u7d20\u6bd4\u57fa\u6e96\u503c\u5c0f\uff0c\u53e6\u4e00\u5b50\u9663\u5217\u7684\u5143\u7d20\u6bd4\u57fa\u6e96\u503c\u5927\uff0c\u518d\u5c0d\u9019\u5169\u90e8\u5206\u9032\u884c\u76f8\u540c\u7684\u5283\u5206\u64cd\u4f5c\uff0c\u76f4\u81f3\u5b50\u9663\u5217\u53ea\u5269\u4e0b\u4e00\u500b\u5143\u7d20\u3002</li> <li>\u6876\u6392\u5e8f\uff1a\u6876\u6392\u5e8f\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c07\u8cc7\u6599\u5206\u6563\u5230\u591a\u500b\u6876\uff0c\u7136\u5f8c\u5c0d\u6bcf\u500b\u6876\u5167\u7684\u5143\u7d20\u9032\u884c\u6392\u5e8f\uff0c\u6700\u5f8c\u5c07\u5404\u500b\u6876\u7684\u5143\u7d20\u4f9d\u6b21\u53d6\u51fa\uff0c\u5f9e\u800c\u5f97\u5230\u4e00\u500b\u6709\u5e8f\u9663\u5217\u3002</li> <li>\u6a39\uff1a\u4f8b\u5982\u4e8c\u5143\u641c\u5c0b\u6a39\u3001AVL \u6a39\u3001\u7d05\u9ed1\u6a39\u3001B \u6a39\u3001B+ \u6a39\u7b49\uff0c\u5b83\u5011\u7684\u67e5\u8a62\u3001\u63d2\u5165\u548c\u522a\u9664\u7b49\u64cd\u4f5c\u90fd\u53ef\u4ee5\u8996\u70ba\u5206\u6cbb\u7b56\u7565\u7684\u61c9\u7528\u3002</li> <li>\u5806\u7a4d\uff1a\u5806\u7a4d\u662f\u4e00\u7a2e\u7279\u6b8a\u7684\u5b8c\u5168\u4e8c\u5143\u6a39\uff0c\u5176\u5404\u7a2e\u64cd\u4f5c\uff0c\u5982\u63d2\u5165\u3001\u522a\u9664\u548c\u5806\u7a4d\u5316\uff0c\u5be6\u969b\u4e0a\u90fd\u96b1\u542b\u4e86\u5206\u6cbb\u7684\u601d\u60f3\u3002</li> <li>\u96dc\u6e4a\u8868\uff1a\u96d6\u7136\u96dc\u6e4a\u8868\u4e26\u4e0d\u76f4\u63a5\u61c9\u7528\u5206\u6cbb\uff0c\u4f46\u67d0\u4e9b\u96dc\u6e4a\u885d\u7a81\u89e3\u6c7a\u65b9\u6848\u9593\u63a5\u61c9\u7528\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u4f8b\u5982\uff0c\u93c8\u5f0f\u4f4d\u5740\u4e2d\u7684\u9577\u93c8\u7d50\u4e32\u5217\u6703\u88ab\u8f49\u5316\u70ba\u7d05\u9ed1\u6a39\uff0c\u4ee5\u63d0\u5347\u67e5\u8a62\u6548\u7387\u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u51fa\uff0c\u5206\u6cbb\u662f\u4e00\u7a2e\u201c\u6f64\u7269\u7d30\u7121\u8072\u201d\u7684\u6f14\u7b97\u6cd5\u601d\u60f3\uff0c\u96b1\u542b\u5728\u5404\u7a2e\u6f14\u7b97\u6cd5\u8207\u8cc7\u6599\u7d50\u69cb\u4e4b\u4e2d\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/","title":"12.4 \u00a0 \u6cb3\u5167\u5854\u554f\u984c","text":"<p>\u5728\u5408\u4f75\u6392\u5e8f\u548c\u69cb\u5efa\u4e8c\u5143\u6a39\u4e2d\uff0c\u6211\u5011\u90fd\u662f\u5c07\u539f\u554f\u984c\u5206\u89e3\u70ba\u5169\u500b\u898f\u6a21\u70ba\u539f\u554f\u984c\u4e00\u534a\u7684\u5b50\u554f\u984c\u3002\u7136\u800c\u5c0d\u65bc\u6cb3\u5167\u5854\u554f\u984c\uff0c\u6211\u5011\u63a1\u7528\u4e0d\u540c\u7684\u5206\u89e3\u7b56\u7565\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e09\u6839\u67f1\u5b50\uff0c\u8a18\u70ba <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \u3002\u8d77\u59cb\u72c0\u614b\u4e0b\uff0c\u67f1\u5b50 <code>A</code> \u4e0a\u5957\u8457 \\(n\\) \u500b\u5713\u76e4\uff0c\u5b83\u5011\u5f9e\u4e0a\u5230\u4e0b\u6309\u7167\u5f9e\u5c0f\u5230\u5927\u7684\u9806\u5e8f\u6392\u5217\u3002\u6211\u5011\u7684\u4efb\u52d9\u662f\u8981\u628a\u9019 \\(n\\) \u500b\u5713\u76e4\u79fb\u5230\u67f1\u5b50 <code>C</code> \u4e0a\uff0c\u4e26\u4fdd\u6301\u5b83\u5011\u7684\u539f\u6709\u9806\u5e8f\u4e0d\u8b8a\uff08\u5982\u5716 12-10 \u6240\u793a\uff09\u3002\u5728\u79fb\u52d5\u5713\u76e4\u7684\u904e\u7a0b\u4e2d\uff0c\u9700\u8981\u9075\u5b88\u4ee5\u4e0b\u898f\u5247\u3002</p> <ol> <li>\u5713\u76e4\u53ea\u80fd\u5f9e\u4e00\u6839\u67f1\u5b50\u9802\u90e8\u62ff\u51fa\uff0c\u5f9e\u53e6\u4e00\u6839\u67f1\u5b50\u9802\u90e8\u653e\u5165\u3002</li> <li>\u6bcf\u6b21\u53ea\u80fd\u79fb\u52d5\u4e00\u500b\u5713\u76e4\u3002</li> <li>\u5c0f\u5713\u76e4\u5fc5\u9808\u6642\u523b\u4f4d\u65bc\u5927\u5713\u76e4\u4e4b\u4e0a\u3002</li> </ol> <p></p> <p> \u5716 12-10 \u00a0 \u6cb3\u5167\u5854\u554f\u984c\u793a\u4f8b </p> <p>\u6211\u5011\u5c07\u898f\u6a21\u70ba \\(i\\) \u7684\u6cb3\u5167\u5854\u554f\u984c\u8a18\u4f5c \\(f(i)\\) \u3002\u4f8b\u5982 \\(f(3)\\) \u4ee3\u8868\u5c07 \\(3\\) \u500b\u5713\u76e4\u5f9e <code>A</code> \u79fb\u52d5\u81f3 <code>C</code> \u7684\u6cb3\u5167\u5854\u554f\u984c\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#1","title":"1. \u00a0 \u8003\u616e\u57fa\u672c\u60c5\u6cc1","text":"<p>\u5982\u5716 12-11 \u6240\u793a\uff0c\u5c0d\u65bc\u554f\u984c \\(f(1)\\) \uff0c\u5373\u7576\u53ea\u6709\u4e00\u500b\u5713\u76e4\u6642\uff0c\u6211\u5011\u5c07\u5b83\u76f4\u63a5\u5f9e <code>A</code> \u79fb\u52d5\u81f3 <code>C</code> \u5373\u53ef\u3002</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> \u5716 12-11 \u00a0 \u898f\u6a21\u70ba 1 \u7684\u554f\u984c\u7684\u89e3 </p> <p>\u5982\u5716 12-12 \u6240\u793a\uff0c\u5c0d\u65bc\u554f\u984c \\(f(2)\\) \uff0c\u5373\u7576\u6709\u5169\u500b\u5713\u76e4\u6642\uff0c\u7531\u65bc\u8981\u6642\u523b\u6eff\u8db3\u5c0f\u5713\u76e4\u5728\u5927\u5713\u76e4\u4e4b\u4e0a\uff0c\u56e0\u6b64\u9700\u8981\u85c9\u52a9 <code>B</code> \u4f86\u5b8c\u6210\u79fb\u52d5\u3002</p> <ol> <li>\u5148\u5c07\u4e0a\u9762\u7684\u5c0f\u5713\u76e4\u5f9e <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u518d\u5c07\u5927\u5713\u76e4\u5f9e <code>A</code> \u79fb\u81f3 <code>C</code> \u3002</li> <li>\u6700\u5f8c\u5c07\u5c0f\u5713\u76e4\u5f9e <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 12-12 \u00a0 \u898f\u6a21\u70ba 2 \u7684\u554f\u984c\u7684\u89e3 </p> <p>\u89e3\u6c7a\u554f\u984c \\(f(2)\\) \u7684\u904e\u7a0b\u53ef\u7e3d\u7d50\u70ba\uff1a\u5c07\u5169\u500b\u5713\u76e4\u85c9\u52a9 <code>B</code> \u5f9e <code>A</code> \u79fb\u81f3 <code>C</code> \u3002\u5176\u4e2d\uff0c<code>C</code> \u7a31\u70ba\u76ee\u6a19\u67f1\u3001<code>B</code> \u7a31\u70ba\u7de9\u885d\u67f1\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#2","title":"2. \u00a0 \u5b50\u554f\u984c\u5206\u89e3","text":"<p>\u5c0d\u65bc\u554f\u984c \\(f(3)\\) \uff0c\u5373\u7576\u6709\u4e09\u500b\u5713\u76e4\u6642\uff0c\u60c5\u6cc1\u8b8a\u5f97\u7a0d\u5fae\u8907\u96dc\u4e86\u4e00\u4e9b\u3002</p> <p>\u56e0\u70ba\u5df2\u77e5 \\(f(1)\\) \u548c \\(f(2)\\) \u7684\u89e3\uff0c\u6240\u4ee5\u6211\u5011\u53ef\u5f9e\u5206\u6cbb\u89d2\u5ea6\u601d\u8003\uff0c\u5c07 <code>A</code> \u9802\u90e8\u7684\u5169\u500b\u5713\u76e4\u770b\u4f5c\u4e00\u500b\u6574\u9ad4\uff0c\u57f7\u884c\u5716 12-13 \u6240\u793a\u7684\u6b65\u9a5f\u3002\u9019\u6a23\u4e09\u500b\u5713\u76e4\u5c31\u88ab\u9806\u5229\u5730\u5f9e <code>A</code> \u79fb\u81f3 <code>C</code> \u4e86\u3002</p> <ol> <li>\u4ee4 <code>B</code> \u70ba\u76ee\u6a19\u67f1\u3001<code>C</code> \u70ba\u7de9\u885d\u67f1\uff0c\u5c07\u5169\u500b\u5713\u76e4\u5f9e <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c07 <code>A</code> \u4e2d\u5269\u9918\u7684\u4e00\u500b\u5713\u76e4\u5f9e <code>A</code> \u76f4\u63a5\u79fb\u52d5\u81f3 <code>C</code> \u3002</li> <li>\u4ee4 <code>C</code> \u70ba\u76ee\u6a19\u67f1\u3001<code>A</code> \u70ba\u7de9\u885d\u67f1\uff0c\u5c07\u5169\u500b\u5713\u76e4\u5f9e <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 12-13 \u00a0 \u898f\u6a21\u70ba 3 \u7684\u554f\u984c\u7684\u89e3 </p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u6211\u5011\u5c07\u554f\u984c \\(f(3)\\) \u5283\u5206\u70ba\u5169\u500b\u5b50\u554f\u984c \\(f(2)\\) \u548c\u4e00\u500b\u5b50\u554f\u984c \\(f(1)\\) \u3002\u6309\u9806\u5e8f\u89e3\u6c7a\u9019\u4e09\u500b\u5b50\u554f\u984c\u4e4b\u5f8c\uff0c\u539f\u554f\u984c\u96a8\u4e4b\u5f97\u5230\u89e3\u6c7a\u3002\u9019\u8aaa\u660e\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff0c\u800c\u4e14\u89e3\u53ef\u4ee5\u5408\u4f75\u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u5011\u53ef\u7e3d\u7d50\u51fa\u5716 12-14 \u6240\u793a\u7684\u89e3\u6c7a\u6cb3\u5167\u5854\u554f\u984c\u7684\u5206\u6cbb\u7b56\u7565\uff1a\u5c07\u539f\u554f\u984c \\(f(n)\\) \u5283\u5206\u70ba\u5169\u500b\u5b50\u554f\u984c \\(f(n-1)\\) \u548c\u4e00\u500b\u5b50\u554f\u984c \\(f(1)\\) \uff0c\u4e26\u6309\u7167\u4ee5\u4e0b\u9806\u5e8f\u89e3\u6c7a\u9019\u4e09\u500b\u5b50\u554f\u984c\u3002</p> <ol> <li>\u5c07 \\(n-1\\) \u500b\u5713\u76e4\u85c9\u52a9 <code>C</code> \u5f9e <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c07\u5269\u9918 \\(1\\) \u500b\u5713\u76e4\u5f9e <code>A</code> \u76f4\u63a5\u79fb\u81f3 <code>C</code> \u3002</li> <li>\u5c07 \\(n-1\\) \u500b\u5713\u76e4\u85c9\u52a9 <code>A</code> \u5f9e <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> <p>\u5c0d\u65bc\u9019\u5169\u500b\u5b50\u554f\u984c \\(f(n-1)\\) \uff0c\u53ef\u4ee5\u900f\u904e\u76f8\u540c\u7684\u65b9\u5f0f\u9032\u884c\u905e\u8ff4\u5283\u5206\uff0c\u76f4\u81f3\u9054\u5230\u6700\u5c0f\u5b50\u554f\u984c \\(f(1)\\) \u3002\u800c \\(f(1)\\) \u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u53ea\u9700\u4e00\u6b21\u79fb\u52d5\u64cd\u4f5c\u5373\u53ef\u3002</p> <p></p> <p> \u5716 12-14 \u00a0 \u89e3\u6c7a\u6cb3\u5167\u5854\u554f\u984c\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#3","title":"3. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5728\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u6211\u5011\u5ba3\u544a\u4e00\u500b\u905e\u8ff4\u51fd\u5f0f <code>dfs(i, src, buf, tar)</code> \uff0c\u5b83\u7684\u4f5c\u7528\u662f\u5c07\u67f1 <code>src</code> \u9802\u90e8\u7684 \\(i\\) \u500b\u5713\u76e4\u85c9\u52a9\u7de9\u885d\u67f1 <code>buf</code> \u79fb\u52d5\u81f3\u76ee\u6a19\u67f1 <code>tar</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"\u79fb\u52d5\u4e00\u500b\u5713\u76e4\"\"\"\n    # \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    pan = src.pop()\n    # \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"\u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i)\"\"\"\n    # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if i == 1:\n        move(src, tar)\n        return\n    # \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    # \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar)\n    # \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"\u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c\"\"\"\n    n = len(A)\n    # \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    int pan = src.back();\n    src.pop_back();\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.push_back(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    Integer pan = src.remove(src.size() - 1);\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.Add(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    DFS(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    Move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    DFS(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfunc move(src, tar *list.List) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    pan := src.Back()\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.PushBack(pan.Value)\n    // \u79fb\u9664 src \u9802\u90e8\u5713\u76e4\n    src.Remove(pan)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfsHanota(i-1, src, tar, buf)\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    let pan = src.popLast()!\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.append(pan)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src: &amp;src, tar: &amp;tar)\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // \u4e32\u5217\u5c3e\u90e8\u662f\u67f1\u5b50\u9802\u90e8\n    // \u5c07 src \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfunction move(src, tar) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    const pan = src.pop();\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfunction dfs(i, src, buf, tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfunction move(src: number[], tar: number[]): void {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    const pan = src.pop();\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n  int pan = src.removeLast();\n  // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n  tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n  dfs(i - 1, src, tar, buf);\n  // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n  move(src, tar);\n  // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n  dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    let pan = src.pop().unwrap();\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move_pan(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, srcSize, tar, tarSize);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.kt<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfun move(src: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    val pan = src.removeAt(src.size - 1)\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.add(pan)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfun dfs(i: Int, src: MutableList&lt;Int&gt;, buf: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfun solveHanota(A: MutableList&lt;Int&gt;, B: MutableList&lt;Int&gt;, C: MutableList&lt;Int&gt;) {\n    val n = A.size\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n}\n</code></pre> hanota.rb<pre><code>### \u79fb\u52d5\u4e00\u500b\u5713\u76e4 ###\ndef move(src, tar)\n  # \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n  pan = src.pop\n  # \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n  tar &lt;&lt; pan\nend\n\n### \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) ###\ndef dfs(i, src, buf, tar)\n  # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n  if i == 1\n    move(src, tar)\n    return\n  end\n\n  # \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n  dfs(i - 1, src, tar, buf)\n  # \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n  move(src, tar)\n  # \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n  dfs(i - 1, buf, src, tar)\nend\n\n### \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c ###\ndef solve_hanota(_A, _B, _C)\n  n = _A.length\n  # \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n  dfs(n, _A, _B, _C)\nend\n</code></pre> hanota.zig<pre><code>[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solveHanota}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 12-15 \u6240\u793a\uff0c\u6cb3\u5167\u5854\u554f\u984c\u5f62\u6210\u4e00\u68f5\u9ad8\u5ea6\u70ba \\(n\\) \u7684\u905e\u8ff4\u6a39\uff0c\u6bcf\u500b\u7bc0\u9ede\u4ee3\u8868\u4e00\u500b\u5b50\u554f\u984c\uff0c\u5c0d\u61c9\u4e00\u500b\u958b\u555f\u7684 <code>dfs()</code> \u51fd\u5f0f\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^n)\\) \uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> <p></p> <p> \u5716 12-15 \u00a0 \u6cb3\u5167\u5854\u554f\u984c\u7684\u905e\u8ff4\u6a39 </p> <p>Quote</p> <p>\u6cb3\u5167\u5854\u554f\u984c\u6e90\u81ea\u4e00\u500b\u53e4\u8001\u7684\u50b3\u8aaa\u3002\u5728\u53e4\u5370\u5ea6\u7684\u4e00\u500b\u5bfa\u5edf\u88e1\uff0c\u50e7\u4fb6\u5011\u6709\u4e09\u6839\u9ad8\u5927\u7684\u947d\u77f3\u67f1\u5b50\uff0c\u4ee5\u53ca \\(64\\) \u500b\u5927\u5c0f\u4e0d\u4e00\u7684\u91d1\u5713\u76e4\u3002\u50e7\u4fb6\u5011\u4e0d\u65b7\u5730\u79fb\u52d5\u5713\u76e4\uff0c\u4ed6\u5011\u76f8\u4fe1\u5728\u6700\u5f8c\u4e00\u500b\u5713\u76e4\u88ab\u6b63\u78ba\u653e\u7f6e\u7684\u90a3\u4e00\u523b\uff0c\u9019\u500b\u4e16\u754c\u5c31\u6703\u7d50\u675f\u3002</p> <p>\u7136\u800c\uff0c\u5373\u4f7f\u50e7\u4fb6\u5011\u6bcf\u79d2\u9418\u79fb\u52d5\u4e00\u6b21\uff0c\u7e3d\u5171\u9700\u8981\u5927\u7d04 \\(2^{64} \\approx 1.84\u00d710^{19}\\) \u79d2\uff0c\u5408\u7d04 \\(5850\\) \u5104\u5e74\uff0c\u9060\u9060\u8d85\u904e\u4e86\u73fe\u5728\u5c0d\u5b87\u5b99\u5e74\u9f61\u7684\u4f30\u8a08\u3002\u6240\u4ee5\uff0c\u5018\u82e5\u9019\u500b\u50b3\u8aaa\u662f\u771f\u7684\uff0c\u6211\u5011\u61c9\u8a72\u4e0d\u9700\u8981\u64d4\u5fc3\u4e16\u754c\u672b\u65e5\u7684\u5230\u4f86\u3002</p>"},{"location":"chapter_divide_and_conquer/summary/","title":"12.5 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u5206\u6cbb\u662f\u4e00\u7a2e\u5e38\u898b\u7684\u6f14\u7b97\u6cd5\u8a2d\u8a08\u7b56\u7565\uff0c\u5305\u62ec\u5206\uff08\u5283\u5206\uff09\u548c\u6cbb\uff08\u5408\u4f75\uff09\u5169\u500b\u968e\u6bb5\uff0c\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u5be6\u73fe\u3002</li> <li>\u5224\u65b7\u662f\u5426\u662f\u5206\u6cbb\u6f14\u7b97\u6cd5\u554f\u984c\u7684\u4f9d\u64da\u5305\u62ec\uff1a\u554f\u984c\u80fd\u5426\u5206\u89e3\u3001\u5b50\u554f\u984c\u662f\u5426\u7368\u7acb\u3001\u5b50\u554f\u984c\u80fd\u5426\u5408\u4f75\u3002</li> <li>\u5408\u4f75\u6392\u5e8f\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u61c9\u7528\uff0c\u5176\u905e\u8ff4\u5730\u5c07\u9663\u5217\u5283\u5206\u70ba\u7b49\u9577\u7684\u5169\u500b\u5b50\u9663\u5217\uff0c\u76f4\u5230\u53ea\u5269\u4e00\u500b\u5143\u7d20\u6642\u958b\u59cb\u9010\u5c64\u5408\u4f75\uff0c\u5f9e\u800c\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u5f15\u5165\u5206\u6cbb\u7b56\u7565\u5f80\u5f80\u53ef\u4ee5\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\u3002\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u7b56\u7565\u6e1b\u5c11\u4e86\u64cd\u4f5c\u6578\u91cf\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5f8c\u6709\u5229\u65bc\u7cfb\u7d71\u7684\u4e26\u884c\u6700\u4f73\u5316\u3002</li> <li>\u5206\u6cbb\u65e2\u53ef\u4ee5\u89e3\u6c7a\u8a31\u591a\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u4e5f\u5ee3\u6cdb\u61c9\u7528\u65bc\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u8a2d\u8a08\u4e2d\uff0c\u8655\u8655\u53ef\u898b\u5176\u8eab\u5f71\u3002</li> <li>\u76f8\u8f03\u65bc\u66b4\u529b\u641c\u5c0b\uff0c\u81ea\u9069\u61c9\u641c\u5c0b\u6548\u7387\u66f4\u9ad8\u3002\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \u7684\u641c\u5c0b\u6f14\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u5be6\u73fe\u7684\u3002</li> <li>\u4e8c\u5206\u641c\u5c0b\u662f\u5206\u6cbb\u7b56\u7565\u7684\u53e6\u4e00\u500b\u5178\u578b\u61c9\u7528\uff0c\u5b83\u4e0d\u5305\u542b\u5c07\u5b50\u554f\u984c\u7684\u89e3\u9032\u884c\u5408\u4f75\u7684\u6b65\u9a5f\u3002\u6211\u5011\u53ef\u4ee5\u900f\u904e\u905e\u8ff4\u5206\u6cbb\u5be6\u73fe\u4e8c\u5206\u641c\u5c0b\u3002</li> <li>\u5728\u69cb\u5efa\u4e8c\u5143\u6a39\u7684\u554f\u984c\u4e2d\uff0c\u69cb\u5efa\u6a39\uff08\u539f\u554f\u984c\uff09\u53ef\u4ee5\u5283\u5206\u70ba\u69cb\u5efa\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\uff08\u5b50\u554f\u984c\uff09\uff0c\u9019\u53ef\u4ee5\u900f\u904e\u5283\u5206\u524d\u5e8f\u8d70\u8a2a\u548c\u4e2d\u5e8f\u8d70\u8a2a\u7684\u7d22\u5f15\u5340\u9593\u4f86\u5be6\u73fe\u3002</li> <li>\u5728\u6cb3\u5167\u5854\u554f\u984c\u4e2d\uff0c\u4e00\u500b\u898f\u6a21\u70ba \\(n\\) \u7684\u554f\u984c\u53ef\u4ee5\u5283\u5206\u70ba\u5169\u500b\u898f\u6a21\u70ba \\(n-1\\) \u7684\u5b50\u554f\u984c\u548c\u4e00\u500b\u898f\u6a21\u70ba \\(1\\) \u7684\u5b50\u554f\u984c\u3002\u6309\u9806\u5e8f\u89e3\u6c7a\u9019\u4e09\u500b\u5b50\u554f\u984c\u5f8c\uff0c\u539f\u554f\u984c\u96a8\u4e4b\u5f97\u5230\u89e3\u6c7a\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/","title":"\u7b2c 14 \u7ae0 \u00a0 \u52d5\u614b\u898f\u5283","text":"<p>Abstract</p> <p>\u5c0f\u6eaa\u532f\u5165\u6cb3\u6d41\uff0c\u6c5f\u6cb3\u532f\u5165\u5927\u6d77\u3002</p> <p>\u52d5\u614b\u898f\u5283\u5c07\u5c0f\u554f\u984c\u7684\u89e3\u5f59\u96c6\u6210\u5927\u554f\u984c\u7684\u7b54\u6848\uff0c\u4e00\u6b65\u6b65\u5f15\u9818\u6211\u5011\u8d70\u5411\u89e3\u6c7a\u554f\u984c\u7684\u5f7c\u5cb8\u3002</p>"},{"location":"chapter_dynamic_programming/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>14.1 \u00a0 \u521d\u63a2\u52d5\u614b\u898f\u5283</li> <li>14.2 \u00a0 DP \u554f\u984c\u7279\u6027</li> <li>14.3 \u00a0 DP \u89e3\u984c\u601d\u8def</li> <li>14.4 \u00a0 0-1 \u80cc\u5305\u554f\u984c</li> <li>14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c</li> <li>14.6 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u554f\u984c</li> <li>14.7 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_dynamic_programming/dp_problem_features/","title":"14.2 \u00a0 \u52d5\u614b\u898f\u5283\u554f\u984c\u7279\u6027","text":"<p>\u5728\u4e0a\u4e00\u7bc0\u4e2d\uff0c\u6211\u5011\u5b78\u7fd2\u4e86\u52d5\u614b\u898f\u5283\u662f\u5982\u4f55\u900f\u904e\u5b50\u554f\u984c\u5206\u89e3\u4f86\u6c42\u89e3\u539f\u554f\u984c\u7684\u3002\u5be6\u969b\u4e0a\uff0c\u5b50\u554f\u984c\u5206\u89e3\u662f\u4e00\u7a2e\u901a\u7528\u7684\u6f14\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52d5\u614b\u898f\u5283\u3001\u56de\u6eaf\u4e2d\u7684\u5074\u91cd\u9ede\u4e0d\u540c\u3002</p> <ul> <li>\u5206\u6cbb\u6f14\u7b97\u6cd5\u905e\u8ff4\u5730\u5c07\u539f\u554f\u984c\u5283\u5206\u70ba\u591a\u500b\u76f8\u4e92\u7368\u7acb\u7684\u5b50\u554f\u984c\uff0c\u76f4\u81f3\u6700\u5c0f\u5b50\u554f\u984c\uff0c\u4e26\u5728\u56de\u6eaf\u4e2d\u5408\u4f75\u5b50\u554f\u984c\u7684\u89e3\uff0c\u6700\u7d42\u5f97\u5230\u539f\u554f\u984c\u7684\u89e3\u3002</li> <li>\u52d5\u614b\u898f\u5283\u4e5f\u5c0d\u554f\u984c\u9032\u884c\u905e\u8ff4\u5206\u89e3\uff0c\u4f46\u8207\u5206\u6cbb\u6f14\u7b97\u6cd5\u7684\u4e3b\u8981\u5340\u5225\u662f\uff0c\u52d5\u614b\u898f\u5283\u4e2d\u7684\u5b50\u554f\u984c\u662f\u76f8\u4e92\u4f9d\u8cf4\u7684\uff0c\u5728\u5206\u89e3\u904e\u7a0b\u4e2d\u6703\u51fa\u73fe\u8a31\u591a\u91cd\u758a\u5b50\u554f\u984c\u3002</li> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u5728\u5617\u8a66\u548c\u56de\u9000\u4e2d\u7aae\u8209\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u4e26\u900f\u904e\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u5c0b\u5206\u652f\u3002\u539f\u554f\u984c\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u6c7a\u7b56\u6b65\u9a5f\u69cb\u6210\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u6bcf\u500b\u6c7a\u7b56\u6b65\u9a5f\u4e4b\u524d\u7684\u5b50\u5e8f\u5217\u770b\u4f5c\u4e00\u500b\u5b50\u554f\u984c\u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u52d5\u614b\u898f\u5283\u5e38\u7528\u4f86\u6c42\u89e3\u6700\u6700\u4f73\u5316\u554f\u984c\uff0c\u5b83\u5011\u4e0d\u50c5\u5305\u542b\u91cd\u758a\u5b50\u554f\u984c\uff0c\u9084\u5177\u6709\u53e6\u5916\u5169\u5927\u7279\u6027\uff1a\u6700\u512a\u5b50\u7d50\u69cb\u3001\u7121\u5f8c\u6548\u6027\u3002</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","title":"14.2.1 \u00a0 \u6700\u512a\u5b50\u7d50\u69cb","text":"<p>\u6211\u5011\u5c0d\u722c\u6a13\u68af\u554f\u984c\u7a0d\u4f5c\u6539\u52d5\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9069\u5408\u5c55\u793a\u6700\u512a\u5b50\u7d50\u69cb\u6982\u5ff5\u3002</p> <p>\u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9</p> <p>\u7d66\u5b9a\u4e00\u500b\u6a13\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u968e\u6216\u8005 \\(2\\) \u968e\uff0c\u6bcf\u4e00\u968e\u6a13\u68af\u4e0a\u90fd\u8cbc\u6709\u4e00\u500b\u975e\u8ca0\u6574\u6578\uff0c\u8868\u793a\u4f60\u5728\u8a72\u81fa\u968e\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u50f9\u3002\u7d66\u5b9a\u4e00\u500b\u975e\u8ca0\u6574\u6578\u9663\u5217 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u500b\u81fa\u968e\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u50f9\uff0c\\(cost[0]\\) \u70ba\u5730\u9762\uff08\u8d77\u59cb\u9ede\uff09\u3002\u8acb\u8a08\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u50f9\u624d\u80fd\u5230\u9054\u9802\u90e8\uff1f</p> <p>\u5982\u5716 14-6 \u6240\u793a\uff0c\u82e5\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u968e\u7684\u4ee3\u50f9\u5206\u5225\u70ba \\(1\\)\u3001\\(10\\)\u3001\\(1\\) \uff0c\u5247\u5f9e\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u968e\u7684\u6700\u5c0f\u4ee3\u50f9\u70ba \\(2\\) \u3002</p> <p></p> <p> \u5716 14-6 \u00a0 \u722c\u5230\u7b2c 3 \u968e\u7684\u6700\u5c0f\u4ee3\u50f9 </p> <p>\u8a2d \\(dp[i]\\) \u70ba\u722c\u5230\u7b2c \\(i\\) \u968e\u7d2f\u8a08\u4ed8\u51fa\u7684\u4ee3\u50f9\uff0c\u7531\u65bc\u7b2c \\(i\\) \u968e\u53ea\u53ef\u80fd\u5f9e \\(i - 1\\) \u968e\u6216 \\(i - 2\\) \u968e\u8d70\u4f86\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u65bc \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u70ba\u4e86\u5118\u53ef\u80fd\u6e1b\u5c11\u4ee3\u50f9\uff0c\u6211\u5011\u61c9\u8a72\u9078\u64c7\u5169\u8005\u4e2d\u8f03\u5c0f\u7684\u90a3\u4e00\u500b\uff1a</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>\u9019\u4fbf\u53ef\u4ee5\u5f15\u51fa\u6700\u512a\u5b50\u7d50\u69cb\u7684\u542b\u7fa9\uff1a\u539f\u554f\u984c\u7684\u6700\u512a\u89e3\u662f\u5f9e\u5b50\u554f\u984c\u7684\u6700\u512a\u89e3\u69cb\u5efa\u5f97\u4f86\u7684\u3002</p> <p>\u672c\u984c\u986f\u7136\u5177\u6709\u6700\u512a\u5b50\u7d50\u69cb\uff1a\u6211\u5011\u5f9e\u5169\u500b\u5b50\u554f\u984c\u6700\u512a\u89e3 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u4e2d\u6311\u9078\u51fa\u8f03\u512a\u7684\u90a3\u4e00\u500b\uff0c\u4e26\u7528\u5b83\u69cb\u5efa\u51fa\u539f\u554f\u984c \\(dp[i]\\) \u7684\u6700\u512a\u89e3\u3002</p> <p>\u90a3\u9ebc\uff0c\u4e0a\u4e00\u7bc0\u7684\u722c\u6a13\u68af\u984c\u76ee\u6709\u6c92\u6709\u6700\u512a\u5b50\u7d50\u69cb\u5462\uff1f\u5b83\u7684\u76ee\u6a19\u662f\u6c42\u89e3\u65b9\u6848\u6578\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u500b\u8a08\u6578\u554f\u984c\uff0c\u4f46\u5982\u679c\u63db\u4e00\u7a2e\u554f\u6cd5\uff1a\u201c\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6578\u91cf\u201d\u3002\u6211\u5011\u610f\u5916\u5730\u767c\u73fe\uff0c\u96d6\u7136\u984c\u76ee\u4fee\u6539\u524d\u5f8c\u662f\u7b49\u50f9\u7684\uff0c\u4f46\u6700\u512a\u5b50\u7d50\u69cb\u6d6e\u73fe\u51fa\u4f86\u4e86\uff1a\u7b2c \\(n\\) \u968e\u6700\u5927\u65b9\u6848\u6578\u91cf\u7b49\u65bc\u7b2c \\(n-1\\) \u968e\u548c\u7b2c \\(n-2\\) \u968e\u6700\u5927\u65b9\u6848\u6578\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8aaa\uff0c\u6700\u512a\u5b50\u7d50\u69cb\u7684\u89e3\u91cb\u65b9\u5f0f\u6bd4\u8f03\u9748\u6d3b\uff0c\u5728\u4e0d\u540c\u554f\u984c\u4e2d\u6703\u6709\u4e0d\u540c\u7684\u542b\u7fa9\u3002</p> <p>\u6839\u64da\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72c0\u614b \\(dp[1] = cost[1]\\) \u548c \\(dp[2] = cost[2]\\) \uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5f97\u5230\u52d5\u614b\u898f\u5283\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1], dp[2] = cost[1], cost[2]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3 ... n {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int *dp = calloc(n + 1, sizeof(int));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfun minCostClimbingStairsDP(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    val dp = IntArray(n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (i in 3..n) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  dp = Array.new(n + 1, 0)\n  # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1], dp[2] = cost[1], cost[2]\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283\nfn minCostClimbingStairsDP(comptime cost: []i32) i32 {\n    comptime var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (3..n + 1) |i| {\n        dp[i] = @min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 14-7 \u00a0 \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p> <p>\u672c\u984c\u4e5f\u53ef\u4ee5\u9032\u884c\u7a7a\u9593\u6700\u4f73\u5316\uff0c\u5c07\u4e00\u7dad\u58d3\u7e2e\u81f3\u96f6\u7dad\uff0c\u4f7f\u5f97\u7a7a\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    a, b := cost[1], cost[2]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in 3 ... n {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun minCostClimbingStairsDPComp(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    var a = cost[1]\n    var b = cost[2]\n    for (i in 3..n) {\n        val tmp = b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  dp = Array.new(n + 1, 0)\n  # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1], dp[2] = cost[1], cost[2]\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n\n# \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\ndef min_cost_climbing_stairs_dp_comp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  a, b = cost[1], cost[2]\n  (3...(n + 1)).each { |i| a, b = b, [a, b].min + cost[i] }\n  b\nend\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn minCostClimbingStairsDPComp(cost: []i32) i32 {\n    var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    var a = cost[1];\n    var b = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (3..n + 1) |i| {\n        var tmp = b;\n        b = @min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","title":"14.2.2 \u00a0 \u7121\u5f8c\u6548\u6027","text":"<p>\u7121\u5f8c\u6548\u6027\u662f\u52d5\u614b\u898f\u5283\u80fd\u5920\u6709\u6548\u89e3\u6c7a\u554f\u984c\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5176\u5b9a\u7fa9\u70ba\uff1a\u7d66\u5b9a\u4e00\u500b\u78ba\u5b9a\u7684\u72c0\u614b\uff0c\u5b83\u7684\u672a\u4f86\u767c\u5c55\u53ea\u8207\u7576\u524d\u72c0\u614b\u6709\u95dc\uff0c\u800c\u8207\u904e\u53bb\u7d93\u6b77\u7684\u6240\u6709\u72c0\u614b\u7121\u95dc\u3002</p> <p>\u4ee5\u722c\u6a13\u68af\u554f\u984c\u70ba\u4f8b\uff0c\u7d66\u5b9a\u72c0\u614b \\(i\\) \uff0c\u5b83\u6703\u767c\u5c55\u51fa\u72c0\u614b \\(i+1\\) \u548c\u72c0\u614b \\(i+2\\) \uff0c\u5206\u5225\u5c0d\u61c9\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u9019\u5169\u7a2e\u9078\u64c7\u6642\uff0c\u6211\u5011\u7121\u9808\u8003\u616e\u72c0\u614b \\(i\\) \u4e4b\u524d\u7684\u72c0\u614b\uff0c\u5b83\u5011\u5c0d\u72c0\u614b \\(i\\) \u7684\u672a\u4f86\u6c92\u6709\u5f71\u97ff\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u5011\u7d66\u722c\u6a13\u68af\u554f\u984c\u65b0\u589e\u4e00\u500b\u7d04\u675f\uff0c\u60c5\u6cc1\u5c31\u4e0d\u4e00\u6a23\u4e86\u3002</p> <p>\u5e36\u7d04\u675f\u722c\u6a13\u68af</p> <p>\u7d66\u5b9a\u4e00\u500b\u5171\u6709 \\(n\\) \u968e\u7684\u6a13\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u968e\u6216\u8005 \\(2\\) \u968e\uff0c\u4f46\u4e0d\u80fd\u9023\u7e8c\u5169\u8f2a\u8df3 \\(1\\) \u968e\uff0c\u8acb\u554f\u6709\u591a\u5c11\u7a2e\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u6a13\u9802\uff1f</p> <p>\u5982\u5716 14-8 \u6240\u793a\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u968e\u50c5\u5269 \\(2\\) \u7a2e\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u9023\u7e8c\u4e09\u6b21\u8df3 \\(1\\) \u968e\u7684\u65b9\u6848\u4e0d\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\uff0c\u56e0\u6b64\u88ab\u6368\u68c4\u3002</p> <p></p> <p> \u5716 14-8 \u00a0 \u5e36\u7d04\u675f\u722c\u5230\u7b2c 3 \u968e\u7684\u65b9\u6848\u6578\u91cf </p> <p>\u5728\u8a72\u554f\u984c\u4e2d\uff0c\u5982\u679c\u4e0a\u4e00\u8f2a\u662f\u8df3 \\(1\\) \u968e\u4e0a\u4f86\u7684\uff0c\u90a3\u9ebc\u4e0b\u4e00\u8f2a\u5c31\u5fc5\u9808\u8df3 \\(2\\) \u968e\u3002\u9019\u610f\u5473\u8457\uff0c\u4e0b\u4e00\u6b65\u9078\u64c7\u4e0d\u80fd\u7531\u7576\u524d\u72c0\u614b\uff08\u7576\u524d\u6240\u5728\u6a13\u68af\u968e\u6578\uff09\u7368\u7acb\u6c7a\u5b9a\uff0c\u9084\u548c\u524d\u4e00\u500b\u72c0\u614b\uff08\u4e0a\u4e00\u8f2a\u6240\u5728\u6a13\u68af\u968e\u6578\uff09\u6709\u95dc\u3002</p> <p>\u4e0d\u96e3\u767c\u73fe\uff0c\u6b64\u554f\u984c\u5df2\u4e0d\u6eff\u8db3\u7121\u5f8c\u6548\u6027\uff0c\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u70ba \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f2a\u8df3 \\(1\\) \u968e\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8a31\u591a\u201c\u4e0a\u4e00\u8f2a\u662f\u8df3 \\(1\\) \u968e\u4e0a\u4f86\u7684\u201d\u65b9\u6848\uff0c\u800c\u70ba\u4e86\u6eff\u8db3\u7d04\u675f\uff0c\u6211\u5011\u5c31\u4e0d\u80fd\u5c07 \\(dp[i-1]\\) \u76f4\u63a5\u8a08\u5165 \\(dp[i]\\) \u4e2d\u3002</p> <p>\u70ba\u6b64\uff0c\u6211\u5011\u9700\u8981\u64f4\u5c55\u72c0\u614b\u5b9a\u7fa9\uff1a\u72c0\u614b \\([i, j]\\) \u8868\u793a\u8655\u5728\u7b2c \\(i\\) \u968e\u4e26\u4e14\u4e0a\u4e00\u8f2a\u8df3\u4e86 \\(j\\) \u968e\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72c0\u614b\u5b9a\u7fa9\u6709\u6548\u5730\u5340\u5206\u4e86\u4e0a\u4e00\u8f2a\u8df3\u4e86 \\(1\\) \u968e\u9084\u662f \\(2\\) \u968e\uff0c\u6211\u5011\u53ef\u4ee5\u64da\u6b64\u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5f9e\u4f55\u800c\u4f86\u7684\u3002</p> <ul> <li>\u7576\u4e0a\u4e00\u8f2a\u8df3\u4e86 \\(1\\) \u968e\u6642\uff0c\u4e0a\u4e0a\u4e00\u8f2a\u53ea\u80fd\u9078\u64c7\u8df3 \\(2\\) \u968e\uff0c\u5373 \\(dp[i, 1]\\) \u53ea\u80fd\u5f9e \\(dp[i-1, 2]\\) \u8f49\u79fb\u904e\u4f86\u3002</li> <li>\u7576\u4e0a\u4e00\u8f2a\u8df3\u4e86 \\(2\\) \u968e\u6642\uff0c\u4e0a\u4e0a\u4e00\u8f2a\u53ef\u9078\u64c7\u8df3 \\(1\\) \u968e\u6216\u8df3 \\(2\\) \u968e\uff0c\u5373 \\(dp[i, 2]\\) \u53ef\u4ee5\u5f9e \\(dp[i-2, 1]\\) \u6216 \\(dp[i-2, 2]\\) \u8f49\u79fb\u904e\u4f86\u3002</li> </ul> <p>\u5982\u5716 14-9 \u6240\u793a\uff0c\u5728\u8a72\u5b9a\u7fa9\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72c0\u614b \\([i, j]\\) \u5c0d\u61c9\u7684\u65b9\u6848\u6578\u3002\u6b64\u6642\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u70ba\uff1a</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> \u5716 14-9 \u00a0 \u8003\u616e\u7d04\u675f\u4e0b\u7684\u905e\u63a8\u95dc\u4fc2 </p> <p>\u6700\u7d42\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u5169\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"\u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[][] dp = new int[n + 1][3];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[,] dp = new int[n + 1, 3];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp := make([][3]int, n+1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3 ... n {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return 1;\n    };\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.kt<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfun climbingStairsConstraintDP(n: Int): Int {\n    if (n == 1 || n == 2) {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    val dp = Array(n + 1) { IntArray(3) }\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (i in 3..n) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.rb<pre><code>### \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 ###\ndef climbing_stairs_constraint_dp(n)\n  return 1 if n == 1 || n == 2\n\n  # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  dp = Array.new(n + 1) { Array.new(3, 0) }\n  # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1][1], dp[1][2] = 1, 0\n  dp[2][1], dp[2][2] = 0, 1\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  for i in 3...(n + 1)\n    dp[i][1] = dp[i - 1][2]\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n  end\n\n  dp[n][1] + dp[n][2]\nend\n</code></pre> climbing_stairs_constraint_dp.zig<pre><code>// \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283\nfn climbingStairsConstraintDP(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (3..n + 1) |i| {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u65bc\u50c5\u9700\u591a\u8003\u616e\u524d\u9762\u4e00\u500b\u72c0\u614b\uff0c\u56e0\u6b64\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u900f\u904e\u64f4\u5c55\u72c0\u614b\u5b9a\u7fa9\uff0c\u4f7f\u5f97\u554f\u984c\u91cd\u65b0\u6eff\u8db3\u7121\u5f8c\u6548\u6027\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u554f\u984c\u5177\u6709\u975e\u5e38\u56b4\u91cd\u7684\u201c\u6709\u5f8c\u6548\u6027\u201d\u3002</p> <p>\u722c\u6a13\u68af\u8207\u969c\u7919\u751f\u6210</p> <p>\u7d66\u5b9a\u4e00\u500b\u5171\u6709 \\(n\\) \u968e\u7684\u6a13\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u968e\u6216\u8005 \\(2\\) \u968e\u3002\u898f\u5b9a\u7576\u722c\u5230\u7b2c \\(i\\) \u968e\u6642\uff0c\u7cfb\u7d71\u81ea\u52d5\u6703\u5728\u7b2c \\(2i\\) \u968e\u4e0a\u653e\u4e0a\u969c\u7919\u7269\uff0c\u4e4b\u5f8c\u6240\u6709\u8f2a\u90fd\u4e0d\u5141\u8a31\u8df3\u5230\u7b2c \\(2i\\) \u968e\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u5169\u8f2a\u5206\u5225\u8df3\u5230\u4e86\u7b2c \\(2\\)\u3001\\(3\\) \u968e\u4e0a\uff0c\u5247\u4e4b\u5f8c\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4\\)\u3001\\(6\\) \u968e\u4e0a\u3002\u8acb\u554f\u6709\u591a\u5c11\u7a2e\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u6a13\u9802\uff1f</p> <p>\u5728\u9019\u500b\u554f\u984c\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8e8d\u4f9d\u8cf4\u904e\u53bb\u6240\u6709\u7684\u72c0\u614b\uff0c\u56e0\u70ba\u6bcf\u4e00\u6b21\u8df3\u8e8d\u90fd\u6703\u5728\u66f4\u9ad8\u7684\u968e\u68af\u4e0a\u8a2d\u5b9a\u969c\u7919\uff0c\u4e26\u5f71\u97ff\u672a\u4f86\u7684\u8df3\u8e8d\u3002\u5c0d\u65bc\u9019\u985e\u554f\u984c\uff0c\u52d5\u614b\u898f\u5283\u5f80\u5f80\u96e3\u4ee5\u89e3\u6c7a\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u8907\u96dc\u7684\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\uff08\u4f8b\u5982\u65c5\u884c\u5546\u554f\u984c\uff09\u4e0d\u6eff\u8db3\u7121\u5f8c\u6548\u6027\u3002\u5c0d\u65bc\u9019\u985e\u554f\u984c\uff0c\u6211\u5011\u901a\u5e38\u6703\u9078\u64c7\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u555f\u767c\u5f0f\u641c\u5c0b\u3001\u907a\u50b3\u6f14\u7b97\u6cd5\u3001\u5f37\u5316\u5b78\u7fd2\u7b49\uff0c\u5f9e\u800c\u5728\u6709\u9650\u6642\u9593\u5167\u5f97\u5230\u53ef\u7528\u7684\u5340\u57df\u6027\u6700\u512a\u89e3\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","title":"14.3 \u00a0 \u52d5\u614b\u898f\u5283\u89e3\u984c\u601d\u8def","text":"<p>\u4e0a\u5169\u7bc0\u4ecb\u7d39\u4e86\u52d5\u614b\u898f\u5283\u554f\u984c\u7684\u4e3b\u8981\u7279\u5fb5\uff0c\u63a5\u4e0b\u4f86\u6211\u5011\u4e00\u8d77\u63a2\u7a76\u5169\u500b\u66f4\u52a0\u5be6\u7528\u7684\u554f\u984c\u3002</p> <ol> <li>\u5982\u4f55\u5224\u65b7\u4e00\u500b\u554f\u984c\u662f\u4e0d\u662f\u52d5\u614b\u898f\u5283\u554f\u984c\uff1f</li> <li>\u6c42\u89e3\u52d5\u614b\u898f\u5283\u554f\u984c\u8a72\u5f9e\u4f55\u8655\u5165\u624b\uff0c\u5b8c\u6574\u6b65\u9a5f\u662f\u4ec0\u9ebc\uff1f</li> </ol>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431","title":"14.3.1 \u00a0 \u554f\u984c\u5224\u65b7","text":"<p>\u7e3d\u7684\u4f86\u8aaa\uff0c\u5982\u679c\u4e00\u500b\u554f\u984c\u5305\u542b\u91cd\u758a\u5b50\u554f\u984c\u3001\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u4e26\u6eff\u8db3\u7121\u5f8c\u6548\u6027\uff0c\u90a3\u9ebc\u5b83\u901a\u5e38\u9069\u5408\u7528\u52d5\u614b\u898f\u5283\u6c42\u89e3\u3002\u7136\u800c\uff0c\u6211\u5011\u5f88\u96e3\u5f9e\u554f\u984c\u63cf\u8ff0\u4e2d\u76f4\u63a5\u63d0\u53d6\u51fa\u9019\u4e9b\u7279\u6027\u3002\u56e0\u6b64\u6211\u5011\u901a\u5e38\u6703\u653e\u5bec\u689d\u4ef6\uff0c\u5148\u89c0\u5bdf\u554f\u984c\u662f\u5426\u9069\u5408\u4f7f\u7528\u56de\u6eaf\uff08\u7aae\u8209\uff09\u89e3\u6c7a\u3002</p> <p>\u9069\u5408\u7528\u56de\u6eaf\u89e3\u6c7a\u7684\u554f\u984c\u901a\u5e38\u6eff\u8db3\u201c\u6c7a\u7b56\u6a39\u6a21\u578b\u201d\uff0c\u9019\u7a2e\u554f\u984c\u53ef\u4ee5\u4f7f\u7528\u6a39\u5f62\u7d50\u69cb\u4f86\u63cf\u8ff0\uff0c\u5176\u4e2d\u6bcf\u4e00\u500b\u7bc0\u9ede\u4ee3\u8868\u4e00\u500b\u6c7a\u7b56\uff0c\u6bcf\u4e00\u689d\u8def\u5f91\u4ee3\u8868\u4e00\u500b\u6c7a\u7b56\u5e8f\u5217\u3002</p> <p>\u63db\u53e5\u8a71\u8aaa\uff0c\u5982\u679c\u554f\u984c\u5305\u542b\u660e\u78ba\u7684\u6c7a\u7b56\u6982\u5ff5\uff0c\u4e26\u4e14\u89e3\u662f\u900f\u904e\u4e00\u7cfb\u5217\u6c7a\u7b56\u7522\u751f\u7684\uff0c\u90a3\u9ebc\u5b83\u5c31\u6eff\u8db3\u6c7a\u7b56\u6a39\u6a21\u578b\uff0c\u901a\u5e38\u53ef\u4ee5\u4f7f\u7528\u56de\u6eaf\u4f86\u89e3\u6c7a\u3002</p> <p>\u5728\u6b64\u57fa\u790e\u4e0a\uff0c\u52d5\u614b\u898f\u5283\u554f\u984c\u9084\u6709\u4e00\u4e9b\u5224\u65b7\u7684\u201c\u52a0\u5206\u9805\u201d\u3002</p> <ul> <li>\u554f\u984c\u5305\u542b\u6700\u5927\uff08\u5c0f\uff09\u6216\u6700\u591a\uff08\u5c11\uff09\u7b49\u6700\u6700\u4f73\u5316\u63cf\u8ff0\u3002</li> <li>\u554f\u984c\u7684\u72c0\u614b\u80fd\u5920\u4f7f\u7528\u4e00\u500b\u4e32\u5217\u3001\u591a\u7dad\u77e9\u9663\u6216\u6a39\u4f86\u8868\u793a\uff0c\u4e26\u4e14\u4e00\u500b\u72c0\u614b\u8207\u5176\u5468\u570d\u7684\u72c0\u614b\u5b58\u5728\u905e\u63a8\u95dc\u4fc2\u3002</li> </ul> <p>\u76f8\u61c9\u5730\uff0c\u4e5f\u5b58\u5728\u4e00\u4e9b\u201c\u6e1b\u5206\u9805\u201d\u3002</p> <ul> <li>\u554f\u984c\u7684\u76ee\u6a19\u662f\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u89e3\u6c7a\u65b9\u6848\uff0c\u800c\u4e0d\u662f\u627e\u51fa\u6700\u512a\u89e3\u3002</li> <li>\u554f\u984c\u63cf\u8ff0\u4e2d\u6709\u660e\u986f\u7684\u6392\u5217\u7d44\u5408\u7684\u7279\u5fb5\uff0c\u9700\u8981\u8fd4\u56de\u5177\u9ad4\u7684\u591a\u500b\u65b9\u6848\u3002</li> </ul> <p>\u5982\u679c\u4e00\u500b\u554f\u984c\u6eff\u8db3\u6c7a\u7b56\u6a39\u6a21\u578b\uff0c\u4e26\u5177\u6709\u8f03\u70ba\u660e\u986f\u7684\u201c\u52a0\u5206\u9805\u201d\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5047\u8a2d\u5b83\u662f\u4e00\u500b\u52d5\u614b\u898f\u5283\u554f\u984c\uff0c\u4e26\u5728\u6c42\u89e3\u904e\u7a0b\u4e2d\u9a57\u8b49\u5b83\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","title":"14.3.2 \u00a0 \u554f\u984c\u6c42\u89e3\u6b65\u9a5f","text":"<p>\u52d5\u614b\u898f\u5283\u7684\u89e3\u984c\u6d41\u7a0b\u6703\u56e0\u554f\u984c\u7684\u6027\u8cea\u548c\u96e3\u5ea6\u800c\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u901a\u5e38\u9075\u5faa\u4ee5\u4e0b\u6b65\u9a5f\uff1a\u63cf\u8ff0\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5efa\u7acb \\(dp\\) \u8868\uff0c\u63a8\u5c0e\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff0c\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u7b49\u3002</p> <p>\u70ba\u4e86\u66f4\u5f62\u8c61\u5730\u5c55\u793a\u89e3\u984c\u6b65\u9a5f\uff0c\u6211\u5011\u4f7f\u7528\u4e00\u500b\u7d93\u5178\u554f\u984c\u201c\u6700\u5c0f\u8def\u5f91\u548c\u201d\u4f86\u8209\u4f8b\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b \\(n \\times m\\) \u7684\u4e8c\u7dad\u7db2\u683c <code>grid</code> \uff0c\u7db2\u683c\u4e2d\u7684\u6bcf\u500b\u55ae\u5143\u683c\u5305\u542b\u4e00\u500b\u975e\u8ca0\u6574\u6578\uff0c\u8868\u793a\u8a72\u55ae\u5143\u683c\u7684\u4ee3\u50f9\u3002\u6a5f\u5668\u4eba\u4ee5\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\u70ba\u8d77\u59cb\u9ede\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u8005\u5411\u53f3\u79fb\u52d5\u4e00\u6b65\uff0c\u76f4\u81f3\u5230\u9054\u53f3\u4e0b\u89d2\u55ae\u5143\u683c\u3002\u8acb\u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\u7684\u6700\u5c0f\u8def\u5f91\u548c\u3002</p> <p>\u5716 14-10 \u5c55\u793a\u4e86\u4e00\u500b\u4f8b\u5b50\uff0c\u7d66\u5b9a\u7db2\u683c\u7684\u6700\u5c0f\u8def\u5f91\u548c\u70ba \\(13\\) \u3002</p> <p></p> <p> \u5716 14-10 \u00a0 \u6700\u5c0f\u8def\u5f91\u548c\u793a\u4f8b\u8cc7\u6599 </p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f2a\u7684\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5f9e\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u672c\u984c\u7684\u6bcf\u4e00\u8f2a\u7684\u6c7a\u7b56\u5c31\u662f\u5f9e\u7576\u524d\u683c\u5b50\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u3002\u8a2d\u7576\u524d\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u70ba \\([i, j]\\) \uff0c\u5247\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u5f8c\uff0c\u7d22\u5f15\u8b8a\u70ba \\([i+1, j]\\) \u6216 \\([i, j+1]\\) \u3002\u56e0\u6b64\uff0c\u72c0\u614b\u61c9\u5305\u542b\u884c\u7d22\u5f15\u548c\u5217\u7d22\u5f15\u5169\u500b\u8b8a\u6578\uff0c\u8a18\u70ba \\([i, j]\\) \u3002</p> <p>\u72c0\u614b \\([i, j]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\u70ba\uff1a\u5f9e\u8d77\u59cb\u9ede \\([0, 0]\\) \u8d70\u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c\uff0c\u89e3\u8a18\u70ba \\(dp[i, j]\\) \u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u5011\u5c31\u5f97\u5230\u4e86\u5716 14-11 \u6240\u793a\u7684\u4e8c\u7dad \\(dp\\) \u77e9\u9663\uff0c\u5176\u5c3a\u5bf8\u8207\u8f38\u5165\u7db2\u683c \\(grid\\) \u76f8\u540c\u3002</p> <p></p> <p> \u5716 14-11 \u00a0 \u72c0\u614b\u5b9a\u7fa9\u8207 dp \u8868 </p> <p>Note</p> <p>\u52d5\u614b\u898f\u5283\u548c\u56de\u6eaf\u904e\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u70ba\u4e00\u500b\u6c7a\u7b56\u5e8f\u5217\uff0c\u800c\u72c0\u614b\u7531\u6240\u6709\u6c7a\u7b56\u8b8a\u6578\u69cb\u6210\u3002\u5b83\u61c9\u7576\u5305\u542b\u63cf\u8ff0\u89e3\u984c\u9032\u5ea6\u7684\u6240\u6709\u8b8a\u6578\uff0c\u5176\u5305\u542b\u4e86\u8db3\u5920\u7684\u8cc7\u8a0a\uff0c\u80fd\u5920\u7528\u4f86\u63a8\u5c0e\u51fa\u4e0b\u4e00\u500b\u72c0\u614b\u3002</p> <p>\u6bcf\u500b\u72c0\u614b\u90fd\u5c0d\u61c9\u4e00\u500b\u5b50\u554f\u984c\uff0c\u6211\u5011\u6703\u5b9a\u7fa9\u4e00\u500b \\(dp\\) \u8868\u4f86\u5132\u5b58\u6240\u6709\u5b50\u554f\u984c\u7684\u89e3\uff0c\u72c0\u614b\u7684\u6bcf\u500b\u7368\u7acb\u8b8a\u6578\u90fd\u662f \\(dp\\) \u8868\u7684\u4e00\u500b\u7dad\u5ea6\u3002\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\\(dp\\) \u8868\u662f\u72c0\u614b\u548c\u5b50\u554f\u984c\u7684\u89e3\u4e4b\u9593\u7684\u5c0d\u6620\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u9032\u800c\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b</p> <p>\u5c0d\u65bc\u72c0\u614b \\([i, j]\\) \uff0c\u5b83\u53ea\u80fd\u5f9e\u4e0a\u908a\u683c\u5b50 \\([i-1, j]\\) \u548c\u5de6\u908a\u683c\u5b50 \\([i, j-1]\\) \u8f49\u79fb\u800c\u4f86\u3002\u56e0\u6b64\u6700\u512a\u5b50\u7d50\u69cb\u70ba\uff1a\u5230\u9054 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c\u7531 \\([i, j-1]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c\u8207 \\([i-1, j]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c\u4e2d\u8f03\u5c0f\u7684\u90a3\u4e00\u500b\u6c7a\u5b9a\u3002</p> <p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u63a8\u51fa\u5716 14-12 \u6240\u793a\u7684\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> \u5716 14-12 \u00a0 \u6700\u512a\u5b50\u7d50\u69cb\u8207\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b </p> <p>Note</p> <p>\u6839\u64da\u5b9a\u7fa9\u597d\u7684 \\(dp\\) \u8868\uff0c\u601d\u8003\u539f\u554f\u984c\u548c\u5b50\u554f\u984c\u7684\u95dc\u4fc2\uff0c\u627e\u51fa\u900f\u904e\u5b50\u554f\u984c\u7684\u6700\u512a\u89e3\u4f86\u69cb\u9020\u539f\u554f\u984c\u7684\u6700\u512a\u89e3\u7684\u65b9\u6cd5\uff0c\u5373\u6700\u512a\u5b50\u7d50\u69cb\u3002</p> <p>\u4e00\u65e6\u6211\u5011\u627e\u5230\u4e86\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528\u5b83\u4f86\u69cb\u5efa\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u548c\u72c0\u614b\u8f49\u79fb\u9806\u5e8f</p> <p>\u5728\u672c\u984c\u4e2d\uff0c\u8655\u5728\u9996\u884c\u7684\u72c0\u614b\u53ea\u80fd\u5f9e\u5176\u5de6\u908a\u7684\u72c0\u614b\u5f97\u4f86\uff0c\u8655\u5728\u9996\u5217\u7684\u72c0\u614b\u53ea\u80fd\u5f9e\u5176\u4e0a\u908a\u7684\u72c0\u614b\u5f97\u4f86\uff0c\u56e0\u6b64\u9996\u884c \\(i = 0\\) \u548c\u9996\u5217 \\(j = 0\\) \u662f\u908a\u754c\u689d\u4ef6\u3002</p> <p>\u5982\u5716 14-13 \u6240\u793a\uff0c\u7531\u65bc\u6bcf\u500b\u683c\u5b50\u662f\u7531\u5176\u5de6\u65b9\u683c\u5b50\u548c\u4e0a\u65b9\u683c\u5b50\u8f49\u79fb\u800c\u4f86\uff0c\u56e0\u6b64\u6211\u5011\u4f7f\u7528\u8ff4\u5708\u4f86\u8d70\u8a2a\u77e9\u9663\uff0c\u5916\u8ff4\u5708\u8d70\u8a2a\u5404\u884c\uff0c\u5167\u8ff4\u5708\u8d70\u8a2a\u5404\u5217\u3002</p> <p></p> <p> \u5716 14-13 \u00a0 \u908a\u754c\u689d\u4ef6\u8207\u72c0\u614b\u8f49\u79fb\u9806\u5e8f </p> <p>Note</p> <p>\u908a\u754c\u689d\u4ef6\u5728\u52d5\u614b\u898f\u5283\u4e2d\u7528\u65bc\u521d\u59cb\u5316 \\(dp\\) \u8868\uff0c\u5728\u641c\u5c0b\u4e2d\u7528\u65bc\u526a\u679d\u3002</p> <p>\u72c0\u614b\u8f49\u79fb\u9806\u5e8f\u7684\u6838\u5fc3\u662f\u8981\u4fdd\u8b49\u5728\u8a08\u7b97\u7576\u524d\u554f\u984c\u7684\u89e3\u6642\uff0c\u6240\u6709\u5b83\u4f9d\u8cf4\u7684\u66f4\u5c0f\u5b50\u554f\u984c\u7684\u89e3\u90fd\u5df2\u7d93\u88ab\u6b63\u78ba\u5730\u8a08\u7b97\u51fa\u4f86\u3002</p> <p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u5011\u5df2\u7d93\u53ef\u4ee5\u76f4\u63a5\u5beb\u51fa\u52d5\u614b\u898f\u5283\u7a0b\u5f0f\u78bc\u3002\u7136\u800c\u5b50\u554f\u984c\u5206\u89e3\u662f\u4e00\u7a2e\u5f9e\u9802\u81f3\u5e95\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u6309\u7167\u201c\u66b4\u529b\u641c\u5c0b \\(\\rightarrow\\) \u8a18\u61b6\u5316\u641c\u5c0b \\(\\rightarrow\\) \u52d5\u614b\u898f\u5283\u201d\u7684\u9806\u5e8f\u5be6\u73fe\u66f4\u52a0\u7b26\u5408\u601d\u7dad\u7fd2\u6163\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u5c0b","text":"<p>\u5f9e\u72c0\u614b \\([i, j]\\) \u958b\u59cb\u641c\u5c0b\uff0c\u4e0d\u65b7\u5206\u89e3\u70ba\u66f4\u5c0f\u7684\u72c0\u614b \\([i-1, j]\\) \u548c \\([i, j-1]\\) \uff0c\u905e\u8ff4\u51fd\u5f0f\u5305\u62ec\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u905e\u8ff4\u53c3\u6578\uff1a\u72c0\u614b \\([i, j]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u5f9e \\([0, 0]\\) \u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c \\(dp[i, j]\\) \u3002</li> <li>\u7d42\u6b62\u689d\u4ef6\uff1a\u7576 \\(i = 0\\) \u4e14 \\(j = 0\\) \u6642\uff0c\u8fd4\u56de\u4ee3\u50f9 \\(grid[0, 0]\\) \u3002</li> <li>\u526a\u679d\uff1a\u7576 \\(i &lt; 0\\) \u6642\u6216 \\(j &lt; 0\\) \u6642\u7d22\u5f15\u8d8a\u754c\uff0c\u6b64\u6642\u8fd4\u56de\u4ee3\u50f9 \\(+\\infty\\) \uff0c\u4ee3\u8868\u4e0d\u53ef\u884c\u3002</li> </ul> <p>\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b\"\"\"\n    # \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfunction minPathSumDFS(grid, i, j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u578b\u5225\u662f\u56fa\u5b9a\u7bc4\u570d\u7684\u6574\u6578\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u7121\u7aae\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfun minPathSumDFS(grid: Array&lt;IntArray&gt;, i: Int, j: Int): Int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    val up = minPathSumDFS(grid, i - 1, j)\n    val left = minPathSumDFS(grid, i, j - 1)\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b ###\ndef min_path_sum_dfs(grid, i, j)\n  # \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n  return grid[i][j] if i == 0 &amp;&amp; j == 0\n  # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  up = min_path_sum_dfs(grid, i - 1, j)\n  left = min_path_sum_dfs(grid, i, j - 1)\n  # \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  [left, up].min + grid[i][j]\nend\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b\nfn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    var up = minPathSumDFS(grid, i - 1, j);\n    var left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-14 \u7d66\u51fa\u4e86\u4ee5 \\(dp[2, 1]\\) \u70ba\u6839\u7bc0\u9ede\u7684\u905e\u8ff4\u6a39\uff0c\u5176\u4e2d\u5305\u542b\u4e00\u4e9b\u91cd\u758a\u5b50\u554f\u984c\uff0c\u5176\u6578\u91cf\u6703\u96a8\u8457\u7db2\u683c <code>grid</code> \u7684\u5c3a\u5bf8\u8b8a\u5927\u800c\u6025\u5287\u589e\u591a\u3002</p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u9020\u6210\u91cd\u758a\u5b50\u554f\u984c\u7684\u539f\u56e0\u70ba\uff1a\u5b58\u5728\u591a\u689d\u8def\u5f91\u53ef\u4ee5\u5f9e\u5de6\u4e0a\u89d2\u5230\u9054\u67d0\u4e00\u55ae\u5143\u683c\u3002</p> <p></p> <p> \u5716 14-14 \u00a0 \u66b4\u529b\u641c\u5c0b\u905e\u8ff4\u6a39 </p> <p>\u6bcf\u500b\u72c0\u614b\u90fd\u6709\u5411\u4e0b\u548c\u5411\u53f3\u5169\u7a2e\u9078\u64c7\uff0c\u5f9e\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u7e3d\u5171\u9700\u8981 \\(m + n - 2\\) \u6b65\uff0c\u6240\u4ee5\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^{m + n})\\) \u3002\u8acb\u6ce8\u610f\uff0c\u9019\u7a2e\u8a08\u7b97\u65b9\u5f0f\u672a\u8003\u616e\u81e8\u8fd1\u7db2\u683c\u908a\u754c\u7684\u60c5\u6cc1\uff0c\u7576\u5230\u9054\u7db2\u8def\u908a\u754c\u6642\u53ea\u5269\u4e0b\u4e00\u7a2e\u9078\u64c7\uff0c\u56e0\u6b64\u5be6\u969b\u7684\u8def\u5f91\u6578\u91cf\u6703\u5c11\u4e00\u4e9b\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b","text":"<p>\u6211\u5011\u5f15\u5165\u4e00\u500b\u548c\u7db2\u683c <code>grid</code> \u76f8\u540c\u5c3a\u5bf8\u7684\u8a18\u61b6\u4e32\u5217 <code>mem</code> \uff0c\u7528\u65bc\u8a18\u9304\u5404\u500b\u5b50\u554f\u984c\u7684\u89e3\uff0c\u4e26\u5c07\u91cd\u758a\u5b50\u554f\u984c\u9032\u884c\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\"\"\"\n    # \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u578b\u5225\u662f\u56fa\u5b9a\u7bc4\u570d\u7684\u6574\u6578\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u7121\u7aae\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfun minPathSumDFSMem(\n    grid: Array&lt;IntArray&gt;,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j]\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    val up = minPathSumDFSMem(grid, mem, i - 1, j)\n    val left = minPathSumDFSMem(grid, mem, i, j - 1)\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b ###\ndef min_path_sum_dfs_mem(grid, mem, i, j)\n  # \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n  return grid[0][0] if i == 0 &amp;&amp; j == 0\n  # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  return mem[i][j] if mem[i][j] != -1\n  # \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n  left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n  # \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  mem[i][j] = [left, up].min + grid[i][j]\nend\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\nfn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {\n        return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    var up = minPathSumDFSMem(grid, mem, i - 1, j);\n    var left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 14-15 \u6240\u793a\uff0c\u5728\u5f15\u5165\u8a18\u61b6\u5316\u5f8c\uff0c\u6240\u6709\u5b50\u554f\u984c\u7684\u89e3\u53ea\u9700\u8a08\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u53d6\u6c7a\u65bc\u72c0\u614b\u7e3d\u6578\uff0c\u5373\u7db2\u683c\u5c3a\u5bf8 \\(O(nm)\\) \u3002</p> <p></p> <p> \u5716 14-15 \u00a0 \u8a18\u61b6\u5316\u641c\u5c0b\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52d5\u614b\u898f\u5283","text":"<p>\u57fa\u65bc\u8fed\u4ee3\u5be6\u73fe\u52d5\u614b\u898f\u5283\u89e3\u6cd5\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in 1 ..&lt; m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for i in 1 ..&lt; n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1 ..&lt; n {\n        for j in 1 ..&lt; m {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfun minPathSumDP(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n) { IntArray(m) }\n    dp[0][0] = grid[0][0]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (j in 1..&lt;m) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (i in 1..&lt;n) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (i in 1..&lt;n) {\n        for (j in 1..&lt;m) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 ###\ndef min_path_sum_dp(grid)\n  n, m = grid.length, grid.first.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n) { Array.new(m, 0) }\n  dp[0][0] = grid[0][0]\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  (1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n  (1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for i in 1...n\n    for j in 1...m\n      dp[i][j] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]\n    end\n  end\n  dp[n -1][m -1]\nend\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283\nfn minPathSumDP(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][m]i32{[_]i32{0} ** m} ** n;\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (1..m) |j| {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (1..n) |i| {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (1..n) |i| {\n        for (1..m) |j| {\n            dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-16 \u5c55\u793a\u4e86\u6700\u5c0f\u8def\u5f91\u548c\u7684\u72c0\u614b\u8f49\u79fb\u904e\u7a0b\uff0c\u5176\u8d70\u8a2a\u4e86\u6574\u500b\u7db2\u683c\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(nm)\\) \u3002</p> <p>\u9663\u5217 <code>dp</code> \u5927\u5c0f\u70ba \\(n \\times m\\) \uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(nm)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-16 \u00a0 \u6700\u5c0f\u8def\u5f91\u548c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","title":"4. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7531\u65bc\u6bcf\u500b\u683c\u5b50\u53ea\u8207\u5176\u5de6\u908a\u548c\u4e0a\u908a\u7684\u683c\u5b50\u6709\u95dc\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u53ea\u7528\u4e00\u500b\u55ae\u884c\u9663\u5217\u4f86\u5be6\u73fe \\(dp\\) \u8868\u3002</p> <p>\u8acb\u6ce8\u610f\uff0c\u56e0\u70ba\u9663\u5217 <code>dp</code> \u53ea\u80fd\u8868\u793a\u4e00\u884c\u7684\u72c0\u614b\uff0c\u6240\u4ee5\u6211\u5011\u7121\u6cd5\u63d0\u524d\u521d\u59cb\u5316\u9996\u5217\u72c0\u614b\uff0c\u800c\u662f\u5728\u8d70\u8a2a\u6bcf\u884c\u6642\u66f4\u65b0\u5b83\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * m\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in range(1, n):\n        # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(m);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, m)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: m)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in 1 ..&lt; m {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in 1 ..&lt; n {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in 1 ..&lt; m {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n  for (int i = 1; i &lt; n; i++) {\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    dp[0] = dp[0] + grid[i][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; m];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in 1..n {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(m, sizeof(int));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun minPathSumDPComp(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(m)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for (j in 1..&lt;m) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (i in 1..&lt;n) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (j in 1..&lt;m) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef min_path_sum_dp_comp(grid)\n  n, m = grid.length, grid.first.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(m, 0)\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  dp[0] = grid[0][0]\n  (1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n  for i in 1...n\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    dp[0] = dp[0] + grid[i][0]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n    (1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }\n  end\n  dp[m - 1]\nend\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn minPathSumDPComp(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** m;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (1..m) |j| {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (1..n) |i| {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        for (1..m) |j| {\n            dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/","title":"14.6 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u554f\u984c","text":"<p>\u7de8\u8f2f\u8ddd\u96e2\uff0c\u4e5f\u7a31 Levenshtein \u8ddd\u96e2\uff0c\u6307\u5169\u500b\u5b57\u4e32\u4e4b\u9593\u4e92\u76f8\u8f49\u63db\u7684\u6700\u5c11\u4fee\u6539\u6b21\u6578\uff0c\u901a\u5e38\u7528\u65bc\u5728\u8cc7\u8a0a\u6aa2\u7d22\u548c\u81ea\u7136\u8a9e\u8a00\u8655\u7406\u4e2d\u5ea6\u91cf\u5169\u500b\u5e8f\u5217\u7684\u76f8\u4f3c\u5ea6\u3002</p> <p>Question</p> <p>\u8f38\u5165\u5169\u500b\u5b57\u4e32 \\(s\\) \u548c \\(t\\) \uff0c\u8fd4\u56de\u5c07 \\(s\\) \u8f49\u63db\u70ba \\(t\\) \u6240\u9700\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u3002</p> <p>\u4f60\u53ef\u4ee5\u5728\u4e00\u500b\u5b57\u4e32\u4e2d\u9032\u884c\u4e09\u7a2e\u7de8\u8f2f\u64cd\u4f5c\uff1a\u63d2\u5165\u4e00\u500b\u5b57\u5143\u3001\u522a\u9664\u4e00\u500b\u5b57\u5143\u3001\u5c07\u5b57\u5143\u66ff\u63db\u70ba\u4efb\u610f\u4e00\u500b\u5b57\u5143\u3002</p> <p>\u5982\u5716 14-27 \u6240\u793a\uff0c\u5c07 <code>kitten</code> \u8f49\u63db\u70ba <code>sitting</code> \u9700\u8981\u7de8\u8f2f 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u63db\u64cd\u4f5c\u8207 1 \u6b21\u65b0\u589e\u64cd\u4f5c\uff1b\u5c07 <code>hello</code> \u8f49\u63db\u70ba <code>algo</code> \u9700\u8981 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u63db\u64cd\u4f5c\u548c 1 \u6b21\u522a\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u5716 14-27 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u7684\u793a\u4f8b\u8cc7\u6599 </p> <p>\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c\u53ef\u4ee5\u5f88\u81ea\u7136\u5730\u7528\u6c7a\u7b56\u6a39\u6a21\u578b\u4f86\u89e3\u91cb\u3002\u5b57\u4e32\u5c0d\u61c9\u6a39\u7bc0\u9ede\uff0c\u4e00\u8f2a\u6c7a\u7b56\uff08\u4e00\u6b21\u7de8\u8f2f\u64cd\u4f5c\uff09\u5c0d\u61c9\u6a39\u7684\u4e00\u689d\u908a\u3002</p> <p>\u5982\u5716 14-28 \u6240\u793a\uff0c\u5728\u4e0d\u9650\u5236\u64cd\u4f5c\u7684\u60c5\u6cc1\u4e0b\uff0c\u6bcf\u500b\u7bc0\u9ede\u90fd\u53ef\u4ee5\u6d3e\u751f\u51fa\u8a31\u591a\u689d\u908a\uff0c\u6bcf\u689d\u908a\u5c0d\u61c9\u4e00\u7a2e\u64cd\u4f5c\uff0c\u9019\u610f\u5473\u8457\u5f9e <code>hello</code> \u8f49\u63db\u5230 <code>algo</code> \u6709\u8a31\u591a\u7a2e\u53ef\u80fd\u7684\u8def\u5f91\u3002</p> <p>\u5f9e\u6c7a\u7b56\u6a39\u7684\u89d2\u5ea6\u770b\uff0c\u672c\u984c\u7684\u76ee\u6a19\u662f\u6c42\u89e3\u7bc0\u9ede <code>hello</code> \u548c\u7bc0\u9ede <code>algo</code> \u4e4b\u9593\u7684\u6700\u77ed\u8def\u5f91\u3002</p> <p></p> <p> \u5716 14-28 \u00a0 \u57fa\u65bc\u6c7a\u7b56\u6a39\u6a21\u578b\u8868\u793a\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","title":"1. \u00a0 \u52d5\u614b\u898f\u5283\u601d\u8def","text":"<p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f2a\u7684\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5f9e\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u6bcf\u4e00\u8f2a\u7684\u6c7a\u7b56\u662f\u5c0d\u5b57\u4e32 \\(s\\) \u9032\u884c\u4e00\u6b21\u7de8\u8f2f\u64cd\u4f5c\u3002</p> <p>\u6211\u5011\u5e0c\u671b\u5728\u7de8\u8f2f\u64cd\u4f5c\u7684\u904e\u7a0b\u4e2d\uff0c\u554f\u984c\u7684\u898f\u6a21\u9010\u6f38\u7e2e\u5c0f\uff0c\u9019\u6a23\u624d\u80fd\u69cb\u5efa\u5b50\u554f\u984c\u3002\u8a2d\u5b57\u4e32 \\(s\\) \u548c \\(t\\) \u7684\u9577\u5ea6\u5206\u5225\u70ba \\(n\\) \u548c \\(m\\) \uff0c\u6211\u5011\u5148\u8003\u616e\u5169\u5b57\u4e32\u5c3e\u90e8\u7684\u5b57\u5143 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u3002</p> <ul> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u76f8\u540c\uff0c\u6211\u5011\u53ef\u4ee5\u8df3\u904e\u5b83\u5011\uff0c\u76f4\u63a5\u8003\u616e \\(s[n-2]\\) \u548c \\(t[m-2]\\) \u3002</li> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u4e0d\u540c\uff0c\u6211\u5011\u9700\u8981\u5c0d \\(s\\) \u9032\u884c\u4e00\u6b21\u7de8\u8f2f\uff08\u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\uff09\uff0c\u4f7f\u5f97\u5169\u5b57\u4e32\u5c3e\u90e8\u7684\u5b57\u5143\u76f8\u540c\uff0c\u5f9e\u800c\u53ef\u4ee5\u8df3\u904e\u5b83\u5011\uff0c\u8003\u616e\u898f\u6a21\u66f4\u5c0f\u7684\u554f\u984c\u3002</li> </ul> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u6211\u5011\u5728\u5b57\u4e32 \\(s\\) \u4e2d\u9032\u884c\u7684\u6bcf\u4e00\u8f2a\u6c7a\u7b56\uff08\u7de8\u8f2f\u64cd\u4f5c\uff09\uff0c\u90fd\u6703\u4f7f\u5f97 \\(s\\) \u548c \\(t\\) \u4e2d\u5269\u9918\u7684\u5f85\u5339\u914d\u5b57\u5143\u767c\u751f\u8b8a\u5316\u3002\u56e0\u6b64\uff0c\u72c0\u614b\u70ba\u7576\u524d\u5728 \\(s\\) \u548c \\(t\\) \u4e2d\u8003\u616e\u7684\u7b2c \\(i\\) \u548c\u7b2c \\(j\\) \u500b\u5b57\u5143\uff0c\u8a18\u70ba \\([i, j]\\) \u3002</p> <p>\u72c0\u614b \\([i, j]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\uff1a\u5c07 \\(s\\) \u7684\u524d \\(i\\) \u500b\u5b57\u5143\u66f4\u6539\u70ba \\(t\\) \u7684\u524d \\(j\\) \u500b\u5b57\u5143\u6240\u9700\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u3002</p> <p>\u81f3\u6b64\uff0c\u5f97\u5230\u4e00\u500b\u5c3a\u5bf8\u70ba \\((i+1) \\times (j+1)\\) \u7684\u4e8c\u7dad \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u9032\u800c\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b</p> <p>\u8003\u616e\u5b50\u554f\u984c \\(dp[i, j]\\) \uff0c\u5176\u5c0d\u61c9\u7684\u5169\u500b\u5b57\u4e32\u7684\u5c3e\u90e8\u5b57\u5143\u70ba \\(s[i-1]\\) \u548c \\(t[j-1]\\) \uff0c\u53ef\u6839\u64da\u4e0d\u540c\u7de8\u8f2f\u64cd\u4f5c\u5206\u70ba\u5716 14-29 \u6240\u793a\u7684\u4e09\u7a2e\u60c5\u6cc1\u3002</p> <ol> <li>\u5728 \\(s[i-1]\\) \u4e4b\u5f8c\u65b0\u589e \\(t[j-1]\\) \uff0c\u5247\u5269\u9918\u5b50\u554f\u984c \\(dp[i, j-1]\\) \u3002</li> <li>\u522a\u9664 \\(s[i-1]\\) \uff0c\u5247\u5269\u9918\u5b50\u554f\u984c \\(dp[i-1, j]\\) \u3002</li> <li>\u5c07 \\(s[i-1]\\) \u66ff\u63db\u70ba \\(t[j-1]\\) \uff0c\u5247\u5269\u9918\u5b50\u554f\u984c \\(dp[i-1, j-1]\\) \u3002</li> </ol> <p></p> <p> \u5716 14-29 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u7684\u72c0\u614b\u8f49\u79fb </p> <p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u5f97\u6700\u512a\u5b50\u7d50\u69cb\uff1a\\(dp[i, j]\\) \u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u7b49\u65bc \\(dp[i, j-1]\\)\u3001\\(dp[i-1, j]\\)\u3001\\(dp[i-1, j-1]\\) \u4e09\u8005\u4e2d\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\uff0c\u518d\u52a0\u4e0a\u672c\u6b21\u7684\u7de8\u8f2f\u6b65\u6578 \\(1\\) \u3002\u5c0d\u61c9\u7684\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u70ba\uff1a</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>\u8acb\u6ce8\u610f\uff0c\u7576 \\(s[i-1]\\) \u548c \\(t[j-1]\\) \u76f8\u540c\u6642\uff0c\u7121\u9808\u7de8\u8f2f\u7576\u524d\u5b57\u5143\uff0c\u9019\u7a2e\u60c5\u6cc1\u4e0b\u7684\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u70ba\uff1a</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>\u7b2c\u4e09\u6b65\uff1a\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u548c\u72c0\u614b\u8f49\u79fb\u9806\u5e8f</p> <p>\u7576\u5169\u5b57\u4e32\u90fd\u70ba\u7a7a\u6642\uff0c\u7de8\u8f2f\u6b65\u6578\u70ba \\(0\\) \uff0c\u5373 \\(dp[0, 0] = 0\\) \u3002\u7576 \\(s\\) \u70ba\u7a7a\u4f46 \\(t\\) \u4e0d\u70ba\u7a7a\u6642\uff0c\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u7b49\u65bc \\(t\\) \u7684\u9577\u5ea6\uff0c\u5373\u9996\u884c \\(dp[0, j] = j\\) \u3002\u7576 \\(s\\) \u4e0d\u70ba\u7a7a\u4f46 \\(t\\) \u70ba\u7a7a\u6642\uff0c\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u7b49\u65bc \\(s\\) \u7684\u9577\u5ea6\uff0c\u5373\u9996\u5217 \\(dp[i, 0] = i\\) \u3002</p> <p>\u89c0\u5bdf\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff0c\u89e3 \\(dp[i, j]\\) \u4f9d\u8cf4\u5de6\u65b9\u3001\u4e0a\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u89e3\uff0c\u56e0\u6b64\u900f\u904e\u5169\u5c64\u8ff4\u5708\u6b63\u5e8f\u8d70\u8a2a\u6574\u500b \\(dp\\) \u8868\u5373\u53ef\u3002</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"\u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1 ... n {\n        dp[i][0] = i\n    }\n    for j in 1 ... m {\n        dp[0][j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1 ... n {\n        for j in 1 ... m {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1..=n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] =\n                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfun editDistanceDP(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = Array(n + 1) { IntArray(m + 1) }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (i in 1..n) {\n        dp[i][0] = i\n    }\n    for (j in 1..m) {\n        dp[0][j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (i in 1..n) {\n        for (j in 1..m) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.rb<pre><code>### \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 ###\ndef edit_distance_dp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(n + 1) { Array.new(m + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n  (1...(n + 1)).each { |i| dp[i][0] = i }\n  (1...(m + 1)).each { |j| dp[0][j] = j }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for i in 1...(n + 1)\n    for j in 1...(m +1)\n      if s[i - 1] == t[j - 1]\n        # \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n        dp[i][j] = dp[i - 1][j - 1]\n      else\n        # \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n        dp[i][j] = [dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]].min + 1\n      end\n    end\n  end\n  dp[n][m]\nend\n</code></pre> edit_distance.zig<pre><code>// \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283\nfn editDistanceDP(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..n + 1) |i| {\n        dp[i][0] = @intCast(i);\n    }\n    for (1..m + 1) |j| {\n        dp[0][j] = @intCast(j);\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..m + 1) |j| {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = @min(@min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 14-30 \u6240\u793a\uff0c\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c\u7684\u72c0\u614b\u8f49\u79fb\u904e\u7a0b\u8207\u80cc\u5305\u554f\u984c\u975e\u5e38\u985e\u4f3c\uff0c\u90fd\u53ef\u4ee5\u770b\u4f5c\u586b\u5beb\u4e00\u500b\u4e8c\u7dad\u7db2\u683c\u7684\u904e\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-30 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","title":"3. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7531\u65bc \\(dp[i,j]\\) \u662f\u7531\u4e0a\u65b9 \\(dp[i-1, j]\\)\u3001\u5de6\u65b9 \\(dp[i, j-1]\\)\u3001\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \u8f49\u79fb\u800c\u4f86\u7684\uff0c\u800c\u6b63\u5e8f\u8d70\u8a2a\u6703\u4e1f\u5931\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \uff0c\u5012\u5e8f\u8d70\u8a2a\u7121\u6cd5\u63d0\u524d\u69cb\u5efa \\(dp[i, j-1]\\) \uff0c\u56e0\u6b64\u5169\u7a2e\u8d70\u8a2a\u9806\u5e8f\u90fd\u4e0d\u53ef\u53d6\u3002</p> <p>\u70ba\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u4e00\u500b\u8b8a\u6578 <code>leftup</code> \u4f86\u66ab\u5b58\u5de6\u4e0a\u65b9\u7684\u89e3 \\(dp[i-1, j-1]\\) \uff0c\u5f9e\u800c\u53ea\u9700\u8003\u616e\u5de6\u65b9\u548c\u4e0a\u65b9\u7684\u89e3\u3002\u6b64\u6642\u7684\u60c5\u6cc1\u8207\u5b8c\u5168\u80cc\u5305\u554f\u984c\u76f8\u540c\uff0c\u53ef\u4f7f\u7528\u6b63\u5e8f\u8d70\u8a2a\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"\u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in range(1, m + 1):\n        dp[j] = j\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in range(1, n + 1):\n        # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        leftup = dp[0]  # \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] += 1\n        # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup\n            else:\n                # \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i := 1; i &lt;= n; i++ {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        leftUp := dp[0] // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftUp\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in 1 ... m {\n        dp[j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in 1 ... n {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in 1 ... m {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n  for (int i = 1; i &lt;= n; i++) {\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n    dp[0] = i;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n        dp[j] = leftup;\n      } else {\n        // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in 1..=n {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        let mut leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i as i32;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    int res = dp[m];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun editDistanceDPComp(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = IntArray(m + 1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (j in 1..m) {\n        dp[j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (i in 1..n) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (j in 1..m) {\n            val temp = dp[j]\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.rb<pre><code>### \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef edit_distance_dp_comp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(m + 1, 0)\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  (1...(m + 1)).each { |j| dp[j] = j }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n  for i in 1...(n + 1)\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    leftup = dp.first # \u66ab\u5b58 dp[i-1, j-1]\n    dp[0] += 1\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n    for j in 1...(m + 1)\n      temp = dp[j]\n      if s[i - 1] == t[j - 1]\n        # \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n        dp[j] = leftup\n      else\n        # \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n        dp[j] = [dp[j - 1], dp[j], leftup].min + 1\n      end\n      leftup = temp # \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n    end\n  end\n  dp[m]\nend\n</code></pre> edit_distance.zig<pre><code>// \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn editDistanceDPComp(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_]i32{0} ** (m + 1);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (1..m + 1) |j| {\n        dp[j] = @intCast(j);\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (1..n + 1) |i| {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = @intCast(i);\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (1..m + 1) |j| {\n            var temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = @min(@min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"14.1 \u00a0 \u521d\u63a2\u52d5\u614b\u898f\u5283","text":"<p>\u52d5\u614b\u898f\u5283\uff08dynamic programming\uff09\u662f\u4e00\u500b\u91cd\u8981\u7684\u6f14\u7b97\u6cd5\u7bc4\u5f0f\uff0c\u5b83\u5c07\u4e00\u500b\u554f\u984c\u5206\u89e3\u70ba\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u554f\u984c\uff0c\u4e26\u900f\u904e\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\u4f86\u907f\u514d\u91cd\u8907\u8a08\u7b97\uff0c\u5f9e\u800c\u5927\u5e45\u63d0\u5347\u6642\u9593\u6548\u7387\u3002</p> <p>\u5728\u672c\u7bc0\u4e2d\uff0c\u6211\u5011\u5f9e\u4e00\u500b\u7d93\u5178\u4f8b\u984c\u5165\u624b\uff0c\u5148\u7d66\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c0\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u758a\u5b50\u554f\u984c\uff0c\u518d\u9010\u6b65\u5c0e\u51fa\u66f4\u9ad8\u6548\u7684\u52d5\u614b\u898f\u5283\u89e3\u6cd5\u3002</p> <p>\u722c\u6a13\u68af</p> <p>\u7d66\u5b9a\u4e00\u500b\u5171\u6709 \\(n\\) \u968e\u7684\u6a13\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u968e\u6216\u8005 \\(2\\) \u968e\uff0c\u8acb\u554f\u6709\u591a\u5c11\u7a2e\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u6a13\u9802\uff1f</p> <p>\u5982\u5716 14-1 \u6240\u793a\uff0c\u5c0d\u65bc\u4e00\u500b \\(3\\) \u968e\u6a13\u68af\uff0c\u5171\u6709 \\(3\\) \u7a2e\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u6a13\u9802\u3002</p> <p></p> <p> \u5716 14-1 \u00a0 \u722c\u5230\u7b2c 3 \u968e\u7684\u65b9\u6848\u6578\u91cf </p> <p>\u672c\u984c\u7684\u76ee\u6a19\u662f\u6c42\u89e3\u65b9\u6848\u6578\u91cf\uff0c\u6211\u5011\u53ef\u4ee5\u8003\u616e\u900f\u904e\u56de\u6eaf\u4f86\u7aae\u8209\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u9ad4\u4f86\u8aaa\uff0c\u5c07\u722c\u6a13\u68af\u60f3\u8c61\u70ba\u4e00\u500b\u591a\u8f2a\u9078\u64c7\u7684\u904e\u7a0b\uff1a\u5f9e\u5730\u9762\u51fa\u767c\uff0c\u6bcf\u8f2a\u9078\u64c7\u4e0a \\(1\\) \u968e\u6216 \\(2\\) \u968e\uff0c\u6bcf\u7576\u5230\u9054\u6a13\u68af\u9802\u90e8\u6642\u5c31\u5c07\u65b9\u6848\u6578\u91cf\u52a0 \\(1\\) \uff0c\u7576\u8d8a\u904e\u6a13\u68af\u9802\u90e8\u6642\u5c31\u5c07\u5176\u526a\u679d\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"\u56de\u6eaf\"\"\"\n    # \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if state == n:\n        res[0] += 1\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices:\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if state + choice &gt; n:\n            continue\n        # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res)\n        # \u56de\u9000\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u56de\u6eaf\"\"\"\n    choices = [1, 2]  # \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    state = 0  # \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    res = [0]  # \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* \u56de\u6eaf */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (auto &amp;choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n)\n            continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    int state = 0;                // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    vector&lt;int&gt; res = {0};        // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (Integer choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n)\n            continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    int state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* \u56de\u6eaf */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    foreach (int choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n)\n            continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        Backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    int state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    List&lt;int&gt; res = [0]; // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if state+choice &gt; n {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state+choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n int) int {\n    // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    choices := []int{1, 2}\n    // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    state := 0\n    res := make([]int, 1)\n    // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if state == n {\n        res[0] += 1\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if state + choice &gt; n {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    let state = 0 // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    var res: [Int] = []\n    res.append(0) // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* \u56de\u6eaf */\nfunction backtrack(choices, state, n, res) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n) continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    const state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* \u56de\u6eaf */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n) continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    const state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n  if (state == n) {\n    res[0]++;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (int choice in choices) {\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n    if (state + choice &gt; n) continue;\n    // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n    backtrack(choices, state + choice, n, res);\n    // \u56de\u9000\n  }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n  int state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n  List&lt;int&gt; res = [];\n  res.add(0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* \u56de\u6eaf */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for &amp;choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if state + choice &gt; n {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    let state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    let mut res = Vec::new();\n    res.push(0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* \u56de\u6eaf */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n)\n            continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res, len);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    int state = 0;           // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.kt<pre><code>/* \u56de\u6eaf */\nfun backtrack(\n    choices: MutableList&lt;Int&gt;,\n    state: Int,\n    n: Int,\n    res: MutableList&lt;Int&gt;\n) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res[0] = res[0] + 1\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n) continue\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfun climbingStairsBacktrack(n: Int): Int {\n    val choices = mutableListOf(1, 2) // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    val state = 0 // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    val res = mutableListOf&lt;Int&gt;()\n    res.add(0) // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.rb<pre><code>### \u56de\u6eaf ###\ndef backtrack(choices, state, n, res)\n  # \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n  res[0] += 1 if state == n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for choice in choices\n    # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n    next if state + choice &gt; n\n\n    # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n    backtrack(choices, state + choice, n, res)\n  end\n  # \u56de\u9000\nend\n\n### \u722c\u6a13\u68af\uff1a\u56de\u6eaf ###\ndef climbing_stairs_backtrack(n)\n  choices = [1, 2] # \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n  state = 0 # \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n  res = [0] # \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n  backtrack(choices, state, n, res)\n  res.first\nend\n</code></pre> climbing_stairs_backtrack.zig<pre><code>// \u56de\u6eaf\nfn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n) {\n        res.items[0] = res.items[0] + 1;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (choices) |choice| {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n// \u722c\u6a13\u68af\uff1a\u56de\u6eaf\nfn climbingStairsBacktrack(n: usize) !i32 {\n    var choices = [_]i32{ 1, 2 }; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    var state: i32 = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    var res = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer res.deinit();\n    try res.append(0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(&amp;choices, state, @intCast(n), res);\n    return res.items[0];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411","title":"14.1.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u5c0b","text":"<p>\u56de\u6eaf\u6f14\u7b97\u6cd5\u901a\u5e38\u4e26\u4e0d\u986f\u5f0f\u5730\u5c0d\u554f\u984c\u9032\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c07\u6c42\u89e3\u554f\u984c\u770b\u4f5c\u4e00\u7cfb\u5217\u6c7a\u7b56\u6b65\u9a5f\uff0c\u900f\u904e\u8a66\u63a2\u548c\u526a\u679d\uff0c\u641c\u5c0b\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u5617\u8a66\u5f9e\u554f\u984c\u5206\u89e3\u7684\u89d2\u5ea6\u5206\u6790\u9019\u9053\u984c\u3002\u8a2d\u722c\u5230\u7b2c \\(i\\) \u968e\u5171\u6709 \\(dp[i]\\) \u7a2e\u65b9\u6848\uff0c\u90a3\u9ebc \\(dp[i]\\) \u5c31\u662f\u539f\u554f\u984c\uff0c\u5176\u5b50\u554f\u984c\u5305\u62ec\uff1a</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>\u7531\u65bc\u6bcf\u8f2a\u53ea\u80fd\u4e0a \\(1\\) \u968e\u6216 \\(2\\) \u968e\uff0c\u56e0\u6b64\u7576\u6211\u5011\u7ad9\u5728\u7b2c \\(i\\) \u968e\u6a13\u68af\u4e0a\u6642\uff0c\u4e0a\u4e00\u8f2a\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u968e\u6216\u7b2c \\(i - 2\\) \u968e\u4e0a\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u6211\u5011\u53ea\u80fd\u5f9e\u7b2c \\(i -1\\) \u968e\u6216\u7b2c \\(i - 2\\) \u968e\u9081\u5411\u7b2c \\(i\\) \u968e\u3002</p> <p>\u7531\u6b64\u4fbf\u53ef\u5f97\u51fa\u4e00\u500b\u91cd\u8981\u63a8\u8ad6\uff1a\u722c\u5230\u7b2c \\(i - 1\\) \u968e\u7684\u65b9\u6848\u6578\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u968e\u7684\u65b9\u6848\u6578\u5c31\u7b49\u65bc\u722c\u5230\u7b2c \\(i\\) \u968e\u7684\u65b9\u6848\u6578\u3002\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>\u9019\u610f\u5473\u8457\u5728\u722c\u6a13\u68af\u554f\u984c\u4e2d\uff0c\u5404\u500b\u5b50\u554f\u984c\u4e4b\u9593\u5b58\u5728\u905e\u63a8\u95dc\u4fc2\uff0c\u539f\u554f\u984c\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u554f\u984c\u7684\u89e3\u69cb\u5efa\u5f97\u4f86\u3002\u5716 14-2 \u5c55\u793a\u4e86\u8a72\u905e\u63a8\u95dc\u4fc2\u3002</p> <p></p> <p> \u5716 14-2 \u00a0 \u65b9\u6848\u6578\u91cf\u905e\u63a8\u95dc\u4fc2 </p> <p>\u6211\u5011\u53ef\u4ee5\u6839\u64da\u905e\u63a8\u516c\u5f0f\u5f97\u5230\u66b4\u529b\u641c\u5c0b\u89e3\u6cd5\u3002\u4ee5 \\(dp[n]\\) \u70ba\u8d77\u59cb\u9ede\uff0c\u905e\u8ff4\u5730\u5c07\u4e00\u500b\u8f03\u5927\u554f\u984c\u62c6\u89e3\u70ba\u5169\u500b\u8f03\u5c0f\u554f\u984c\u7684\u548c\uff0c\u76f4\u81f3\u5230\u9054\u6700\u5c0f\u5b50\u554f\u984c \\(dp[1]\\) \u548c \\(dp[2]\\) \u6642\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373 \\(dp[1] = 1\\)\u3001\\(dp[2] = 2\\) \uff0c\u8868\u793a\u722c\u5230\u7b2c \\(1\\)\u3001\\(2\\) \u968e\u5206\u5225\u6709 \\(1\\)\u3001\\(2\\) \u7a2e\u65b9\u6848\u3002</p> <p>\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u5b83\u548c\u6a19\u6e96\u56de\u6eaf\u7a0b\u5f0f\u78bc\u90fd\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\uff0c\u4f46\u66f4\u52a0\u7c21\u6f54\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"\u641c\u5c0b\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u641c\u5c0b\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* \u641c\u5c0b */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* \u641c\u5c0b */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* \u641c\u5c0b */\nint DFS(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* \u641c\u5c0b */\nfunc dfs(i int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* \u641c\u5c0b */\nfunc dfs(i: Int) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* \u641c\u5c0b */\nfunction dfs(i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* \u641c\u5c0b */\nfunction dfs(i: number): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* \u641c\u5c0b */\nint dfs(int i) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* \u641c\u5c0b */\nfn dfs(i: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n)\n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* \u641c\u5c0b */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.kt<pre><code>/* \u641c\u5c0b */\nfun dfs(i: Int): Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2) return i\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1) + dfs(i - 2)\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfun climbingStairsDFS(n: Int): Int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.rb<pre><code>### \u641c\u5c0b ###\ndef dfs(i)\n  # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  return i if i == 1 || i == 2\n  # dp[i] = dp[i-1] + dp[i-2]\n  dfs(i - 1) + dfs(i - 2)\nend\n\n### \u722c\u6a13\u68af\uff1a\u641c\u5c0b ###\ndef climbing_stairs_dfs(n)\n  dfs(n)\nend\n</code></pre> climbing_stairs_dfs.zig<pre><code>// \u641c\u5c0b\nfn dfs(i: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n// \u722c\u6a13\u68af\uff1a\u641c\u5c0b\nfn climbingStairsDFS(comptime n: usize) i32 {\n    return dfs(n);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-3 \u5c55\u793a\u4e86\u66b4\u529b\u641c\u5c0b\u5f62\u6210\u7684\u905e\u8ff4\u6a39\u3002\u5c0d\u65bc\u554f\u984c \\(dp[n]\\) \uff0c\u5176\u905e\u8ff4\u6a39\u7684\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^n)\\) \u3002\u6307\u6578\u968e\u5c6c\u65bc\u7206\u70b8\u5f0f\u589e\u9577\uff0c\u5982\u679c\u6211\u5011\u8f38\u5165\u4e00\u500b\u6bd4\u8f03\u5927\u7684 \\(n\\) \uff0c\u5247\u6703\u9677\u5165\u6f2b\u9577\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002</p> <p></p> <p> \u5716 14-3 \u00a0 \u722c\u6a13\u68af\u5c0d\u61c9\u905e\u8ff4\u6a39 </p> <p>\u89c0\u5bdf\u5716 14-3 \uff0c\u6307\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f\u201c\u91cd\u758a\u5b50\u554f\u984c\u201d\u5c0e\u81f4\u7684\u3002\u4f8b\u5982 \\(dp[9]\\) \u88ab\u5206\u89e3\u70ba \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\\(dp[8]\\) \u88ab\u5206\u89e3\u70ba \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u5169\u8005\u90fd\u5305\u542b\u5b50\u554f\u984c \\(dp[7]\\) \u3002</p> <p>\u4ee5\u6b64\u985e\u63a8\uff0c\u5b50\u554f\u984c\u4e2d\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u758a\u5b50\u554f\u984c\uff0c\u5b50\u5b50\u5b6b\u5b6b\u7121\u7aae\u76e1\u4e5f\u3002\u7d55\u5927\u90e8\u5206\u8a08\u7b97\u8cc7\u6e90\u90fd\u6d6a\u8cbb\u5728\u9019\u4e9b\u91cd\u758a\u7684\u5b50\u554f\u984c\u4e0a\u3002</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412","title":"14.1.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b","text":"<p>\u70ba\u4e86\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u5011\u5e0c\u671b\u6240\u6709\u7684\u91cd\u758a\u5b50\u554f\u984c\u90fd\u53ea\u88ab\u8a08\u7b97\u4e00\u6b21\u3002\u70ba\u6b64\uff0c\u6211\u5011\u5ba3\u544a\u4e00\u500b\u9663\u5217 <code>mem</code> \u4f86\u8a18\u9304\u6bcf\u500b\u5b50\u554f\u984c\u7684\u89e3\uff0c\u4e26\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\u5c07\u91cd\u758a\u5b50\u554f\u984c\u526a\u679d\u3002</p> <ol> <li>\u7576\u9996\u6b21\u8a08\u7b97 \\(dp[i]\\) \u6642\uff0c\u6211\u5011\u5c07\u5176\u8a18\u9304\u81f3 <code>mem[i]</code> \uff0c\u4ee5\u4fbf\u4e4b\u5f8c\u4f7f\u7528\u3002</li> <li>\u7576\u518d\u6b21\u9700\u8981\u8a08\u7b97 \\(dp[i]\\) \u6642\uff0c\u6211\u5011\u4fbf\u53ef\u76f4\u63a5\u5f9e <code>mem[i]</code> \u4e2d\u7372\u53d6\u7d50\u679c\uff0c\u5f9e\u800c\u907f\u514d\u91cd\u8907\u8a08\u7b97\u8a72\u5b50\u554f\u984c\u3002</li> </ol> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"\u8a18\u61b6\u5316\u641c\u5c0b\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # \u8a18\u9304 dp[i]\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\"\"\"\n    # mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint dfs(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint DFS(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfunc dfsMem(i int, mem []int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // \u8a18\u9304 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // \u8a18\u9304 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfunction dfs(i, mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction climbingStairsDFSMem(n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfunction dfs(i: number, mem: number[]): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint dfs(int i, List&lt;int&gt; mem) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // \u8a18\u9304 dp[i]\n  mem[i] = count;\n  return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint climbingStairsDFSMem(int n) {\n  // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint dfs(int i, int *mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.kt<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfun dfs(i: Int, mem: IntArray): Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2) return i\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i]\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    // \u8a18\u9304 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfun climbingStairsDFSMem(n: Int): Int {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    val mem = IntArray(n + 1)\n    mem.fill(-1)\n    return dfs(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.rb<pre><code>### \u8a18\u61b6\u5316\u641c\u5c0b ###\ndef dfs(i, mem)\n  # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  return i if i == 1 || i == 2\n  # \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n  return mem[i] if mem[i] != -1\n\n  # dp[i] = dp[i-1] + dp[i-2]\n  count = dfs(i - 1, mem) + dfs(i - 2, mem)\n  # \u8a18\u9304 dp[i]\n  mem[i] = count\nend\n\n### \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b ###\ndef climbing_stairs_dfs_mem(n)\n  # mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n  mem = Array.new(n + 1, -1)\n  dfs(n, mem)\nend\n</code></pre> climbing_stairs_dfs_mem.zig<pre><code>// \u8a18\u61b6\u5316\u641c\u5c0b\nfn dfs(i: usize, mem: []i32) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n// \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\nfn climbingStairsDFSMem(comptime n: usize) i32 {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    var mem = [_]i32{ -1 } ** (n + 1);\n    return dfs(n, &amp;mem);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u5716 14-4 \uff0c\u7d93\u904e\u8a18\u61b6\u5316\u8655\u7406\u5f8c\uff0c\u6240\u6709\u91cd\u758a\u5b50\u554f\u984c\u90fd\u53ea\u9700\u8a08\u7b97\u4e00\u6b21\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(n)\\) \uff0c\u9019\u662f\u4e00\u500b\u5de8\u5927\u7684\u98db\u8e8d\u3002</p> <p></p> <p> \u5716 14-4 \u00a0 \u8a18\u61b6\u5316\u641c\u5c0b\u5c0d\u61c9\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413","title":"14.1.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52d5\u614b\u898f\u5283","text":"<p>\u8a18\u61b6\u5316\u641c\u5c0b\u662f\u4e00\u7a2e\u201c\u5f9e\u9802\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u5011\u5f9e\u539f\u554f\u984c\uff08\u6839\u7bc0\u9ede\uff09\u958b\u59cb\uff0c\u905e\u8ff4\u5730\u5c07\u8f03\u5927\u5b50\u554f\u984c\u5206\u89e3\u70ba\u8f03\u5c0f\u5b50\u554f\u984c\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u554f\u984c\uff08\u8449\u7bc0\u9ede\uff09\u3002\u4e4b\u5f8c\uff0c\u900f\u904e\u56de\u6eaf\u9010\u5c64\u6536\u96c6\u5b50\u554f\u984c\u7684\u89e3\uff0c\u69cb\u5efa\u51fa\u539f\u554f\u984c\u7684\u89e3\u3002</p> <p>\u8207\u4e4b\u76f8\u53cd\uff0c\u52d5\u614b\u898f\u5283\u662f\u4e00\u7a2e\u201c\u5f9e\u5e95\u81f3\u9802\u201d\u7684\u65b9\u6cd5\uff1a\u5f9e\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\u958b\u59cb\uff0c\u8fed\u4ee3\u5730\u69cb\u5efa\u66f4\u5927\u5b50\u554f\u984c\u7684\u89e3\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u554f\u984c\u7684\u89e3\u3002</p> <p>\u7531\u65bc\u52d5\u614b\u898f\u5283\u4e0d\u5305\u542b\u56de\u6eaf\u904e\u7a0b\uff0c\u56e0\u6b64\u53ea\u9700\u4f7f\u7528\u8ff4\u5708\u8fed\u4ee3\u5be6\u73fe\uff0c\u7121\u9808\u4f7f\u7528\u905e\u8ff4\u3002\u5728\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u6211\u5011\u521d\u59cb\u5316\u4e00\u500b\u9663\u5217 <code>dp</code> \u4f86\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\uff0c\u5b83\u8d77\u5230\u4e86\u8207\u8a18\u61b6\u5316\u641c\u5c0b\u4e2d\u9663\u5217 <code>mem</code> \u76f8\u540c\u7684\u8a18\u9304\u4f5c\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1], dp[2] = 1, 2\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3 ... n {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1] = 1;\n  dp[2] = 2;\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfun climbingStairsDP(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    val dp = IntArray(n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (i in 3..n) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 ###\ndef climbing_stairs_dp(n)\n  return n  if n == 1 || n == 2\n\n  # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  dp = Array.new(n + 1, 0)\n  # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1], dp[2] = 1, 2\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  (3...(n + 1)).each { |i| dp[i] = dp[i - 1] + dp[i - 2] }\n\n  dp[n]\nend\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283\nfn climbingStairsDP(comptime n: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (3..n + 1) |i| {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-5 \u6a21\u64ec\u4e86\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u7684\u57f7\u884c\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 14-5 \u00a0 \u722c\u6a13\u68af\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p> <p>\u8207\u56de\u6eaf\u6f14\u7b97\u6cd5\u4e00\u6a23\uff0c\u52d5\u614b\u898f\u5283\u4e5f\u4f7f\u7528\u201c\u72c0\u614b\u201d\u6982\u5ff5\u4f86\u8868\u793a\u554f\u984c\u6c42\u89e3\u7684\u7279\u5b9a\u968e\u6bb5\uff0c\u6bcf\u500b\u72c0\u614b\u90fd\u5c0d\u61c9\u4e00\u500b\u5b50\u554f\u984c\u4ee5\u53ca\u76f8\u61c9\u7684\u5340\u57df\u6027\u6700\u512a\u89e3\u3002\u4f8b\u5982\uff0c\u722c\u6a13\u68af\u554f\u984c\u7684\u72c0\u614b\u5b9a\u7fa9\u70ba\u7576\u524d\u6240\u5728\u6a13\u68af\u968e\u6578 \\(i\\) \u3002</p> <p>\u6839\u64da\u4ee5\u4e0a\u5167\u5bb9\uff0c\u6211\u5011\u53ef\u4ee5\u7e3d\u7d50\u51fa\u52d5\u614b\u898f\u5283\u7684\u5e38\u7528\u8853\u8a9e\u3002</p> <ul> <li>\u5c07\u9663\u5217 <code>dp</code> \u7a31\u70ba dp \u8868\uff0c\\(dp[i]\\) \u8868\u793a\u72c0\u614b \\(i\\) \u5c0d\u61c9\u5b50\u554f\u984c\u7684\u89e3\u3002</li> <li>\u5c07\u6700\u5c0f\u5b50\u554f\u984c\u5c0d\u61c9\u7684\u72c0\u614b\uff08\u7b2c \\(1\\) \u968e\u548c\u7b2c \\(2\\) \u968e\u6a13\u68af\uff09\u7a31\u70ba\u521d\u59cb\u72c0\u614b\u3002</li> <li>\u5c07\u905e\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u7a31\u70ba\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414","title":"14.1.4 \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7d30\u5fc3\u7684\u8b80\u8005\u53ef\u80fd\u767c\u73fe\u4e86\uff0c\u7531\u65bc \\(dp[i]\\) \u53ea\u8207 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u95dc\uff0c\u56e0\u6b64\u6211\u5011\u7121\u9808\u4f7f\u7528\u4e00\u500b\u9663\u5217 <code>dp</code> \u4f86\u5132\u5b58\u6240\u6709\u5b50\u554f\u984c\u7684\u89e3\uff0c\u800c\u53ea\u9700\u5169\u500b\u8b8a\u6578\u6efe\u52d5\u524d\u9032\u5373\u53ef\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in 3 ... n {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun climbingStairsDPComp(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    var a = 1\n    var b = 2\n    for (i in 3..n) {\n        val temp = b\n        b += a\n        a = temp\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef climbing_stairs_dp_comp(n)\n  return n if n == 1 || n == 2\n\n  a, b = 1, 2\n  (3...(n + 1)).each { a, b = b, a + b }\n\n  b\nend\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn climbingStairsDPComp(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    var a: i32 = 1;\n    var b: i32 = 2;\n    for (3..n + 1) |_| {\n        var tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\uff0c\u7531\u65bc\u7701\u53bb\u4e86\u9663\u5217 <code>dp</code> \u4f54\u7528\u7684\u7a7a\u9593\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</p> <p>\u5728\u52d5\u614b\u898f\u5283\u554f\u984c\u4e2d\uff0c\u7576\u524d\u72c0\u614b\u5f80\u5f80\u50c5\u8207\u524d\u9762\u6709\u9650\u500b\u72c0\u614b\u6709\u95dc\uff0c\u9019\u6642\u6211\u5011\u53ef\u4ee5\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72c0\u614b\uff0c\u900f\u904e\u201c\u964d\u7dad\u201d\u4f86\u7bc0\u7701\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002\u9019\u7a2e\u7a7a\u9593\u6700\u4f73\u5316\u6280\u5de7\u88ab\u7a31\u70ba\u201c\u6efe\u52d5\u8b8a\u6578\u201d\u6216\u201c\u6efe\u52d5\u9663\u5217\u201d\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"14.4 \u00a0 0-1 \u80cc\u5305\u554f\u984c","text":"<p>\u80cc\u5305\u554f\u984c\u662f\u4e00\u500b\u975e\u5e38\u597d\u7684\u52d5\u614b\u898f\u5283\u5165\u9580\u984c\u76ee\uff0c\u662f\u52d5\u614b\u898f\u5283\u4e2d\u6700\u5e38\u898b\u7684\u554f\u984c\u5f62\u5f0f\u3002\u5176\u5177\u6709\u5f88\u591a\u8b8a\u7a2e\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u554f\u984c\u3001\u5b8c\u5168\u80cc\u5305\u554f\u984c\u3001\u591a\u91cd\u80cc\u5305\u554f\u984c\u7b49\u3002</p> <p>\u5728\u672c\u7bc0\u4e2d\uff0c\u6211\u5011\u5148\u4f86\u6c42\u89e3\u6700\u5e38\u898b\u7684 0-1 \u80cc\u5305\u554f\u984c\u3002</p> <p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u500b\u7269\u54c1\uff0c\u7b2c \\(i\\) \u500b\u7269\u54c1\u7684\u91cd\u91cf\u70ba \\(wgt[i-1]\\)\u3001\u50f9\u503c\u70ba \\(val[i-1]\\) \uff0c\u548c\u4e00\u500b\u5bb9\u91cf\u70ba \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u500b\u7269\u54c1\u53ea\u80fd\u9078\u64c7\u4e00\u6b21\uff0c\u554f\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u50f9\u503c\u3002</p> <p>\u89c0\u5bdf\u5716 14-17 \uff0c\u7531\u65bc\u7269\u54c1\u7de8\u865f \\(i\\) \u5f9e \\(1\\) \u958b\u59cb\u8a08\u6578\uff0c\u9663\u5217\u7d22\u5f15\u5f9e \\(0\\) \u958b\u59cb\u8a08\u6578\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5c0d\u61c9\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u50f9\u503c \\(val[i-1]\\) \u3002</p> <p></p> <p> \u5716 14-17 \u00a0 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u8cc7\u6599 </p> <p>\u6211\u5011\u53ef\u4ee5\u5c07 0-1 \u80cc\u5305\u554f\u984c\u770b\u4f5c\u4e00\u500b\u7531 \\(n\\) \u8f2a\u6c7a\u7b56\u7d44\u6210\u7684\u904e\u7a0b\uff0c\u5c0d\u65bc\u6bcf\u500b\u7269\u9ad4\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u5169\u7a2e\u6c7a\u7b56\uff0c\u56e0\u6b64\u8a72\u554f\u984c\u6eff\u8db3\u6c7a\u7b56\u6a39\u6a21\u578b\u3002</p> <p>\u8a72\u554f\u984c\u7684\u76ee\u6a19\u662f\u6c42\u89e3\u201c\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u50f9\u503c\u201d\uff0c\u56e0\u6b64\u8f03\u5927\u6a5f\u7387\u662f\u4e00\u500b\u52d5\u614b\u898f\u5283\u554f\u984c\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f2a\u7684\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5f9e\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u5c0d\u65bc\u6bcf\u500b\u7269\u54c1\u4f86\u8aaa\uff0c\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u8b8a\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u6e1b\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\u72c0\u614b\u5b9a\u7fa9\uff1a\u7576\u524d\u7269\u54c1\u7de8\u865f \\(i\\) \u548c\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u8a18\u70ba \\([i, c]\\) \u3002</p> <p>\u72c0\u614b \\([i, c]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\u70ba\uff1a\u524d \\(i\\) \u500b\u7269\u54c1\u5728\u5bb9\u91cf\u70ba \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u50f9\u503c\uff0c\u8a18\u70ba \\(dp[i, c]\\) \u3002</p> <p>\u5f85\u6c42\u89e3\u7684\u662f \\(dp[n, cap]\\) \uff0c\u56e0\u6b64\u9700\u8981\u4e00\u500b\u5c3a\u5bf8\u70ba \\((n+1) \\times (cap+1)\\) \u7684\u4e8c\u7dad \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u9032\u800c\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b</p> <p>\u7576\u6211\u5011\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u6c7a\u7b56\u5f8c\uff0c\u5269\u9918\u7684\u662f\u524d \\(i-1\\) \u500b\u7269\u54c1\u6c7a\u7b56\u7684\u5b50\u554f\u984c\uff0c\u53ef\u5206\u70ba\u4ee5\u4e0b\u5169\u7a2e\u60c5\u6cc1\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u8b8a\uff0c\u72c0\u614b\u8b8a\u5316\u70ba \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u6e1b\u5c11 \\(wgt[i-1]\\) \uff0c\u50f9\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72c0\u614b\u8b8a\u5316\u70ba \\([i-1, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4e0a\u8ff0\u5206\u6790\u5411\u6211\u5011\u63ed\u793a\u4e86\u672c\u984c\u7684\u6700\u512a\u5b50\u7d50\u69cb\uff1a\u6700\u5927\u50f9\u503c \\(dp[i, c]\\) \u7b49\u65bc\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\u3002\u7531\u6b64\u53ef\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u7576\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u9918\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u548c\u72c0\u614b\u8f49\u79fb\u9806\u5e8f</p> <p>\u7576\u7121\u7269\u54c1\u6216\u80cc\u5305\u5bb9\u91cf\u70ba \\(0\\) \u6642\u6700\u5927\u50f9\u503c\u70ba \\(0\\) \uff0c\u5373\u9996\u5217 \\(dp[i, 0]\\) \u548c\u9996\u884c \\(dp[0, c]\\) \u90fd\u7b49\u65bc \\(0\\) \u3002</p> <p>\u7576\u524d\u72c0\u614b \\([i, c]\\) \u5f9e\u4e0a\u65b9\u7684\u72c0\u614b \\([i-1, c]\\) \u548c\u5de6\u4e0a\u65b9\u7684\u72c0\u614b \\([i-1, c-wgt[i-1]]\\) \u8f49\u79fb\u800c\u4f86\uff0c\u56e0\u6b64\u900f\u904e\u5169\u5c64\u8ff4\u5708\u6b63\u5e8f\u8d70\u8a2a\u6574\u500b \\(dp\\) \u8868\u5373\u53ef\u3002</p> <p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u5011\u63a5\u4e0b\u4f86\u6309\u9806\u5e8f\u5be6\u73fe\u66b4\u529b\u641c\u5c0b\u3001\u8a18\u61b6\u5316\u641c\u5c0b\u3001\u52d5\u614b\u898f\u5283\u89e3\u6cd5\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u5c0b","text":"<p>\u641c\u5c0b\u7a0b\u5f0f\u78bc\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u905e\u8ff4\u53c3\u6578\uff1a\u72c0\u614b \\([i, c]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u5b50\u554f\u984c\u7684\u89e3 \\(dp[i, c]\\) \u3002</li> <li>\u7d42\u6b62\u689d\u4ef6\uff1a\u7576\u7269\u54c1\u7de8\u865f\u8d8a\u754c \\(i = 0\\) \u6216\u80cc\u5305\u5269\u9918\u5bb9\u91cf\u70ba \\(0\\) \u6642\uff0c\u7d42\u6b62\u905e\u8ff4\u4e26\u8fd4\u56de\u50f9\u503c \\(0\\) \u3002</li> <li>\u526a\u679d\uff1a\u82e5\u7576\u524d\u7269\u54c1\u91cd\u91cf\u8d85\u51fa\u80cc\u5305\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b\"\"\"\n    # \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfunction knapsackDFS(wgt, val, i, c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return myMax(no, yes);\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfun knapsackDFS(\n    wgt: IntArray,\n    _val: IntArray,\n    i: Int,\n    c: Int\n): Int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, _val, i - 1, c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    val no = knapsackDFS(wgt, _val, i - 1, c)\n    val yes = knapsackDFS(wgt, _val, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return max(no, yes)\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b ###\ndef knapsack_dfs(wgt, val, i, c)\n  # \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n  return 0 if i == 0 || c == 0\n  # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n  return knapsack_dfs(wgt, val, i - 1, c) if wgt[i - 1] &gt; c\n  # \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n  no = knapsack_dfs(wgt, val, i - 1, c)\n  yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n  [no, yes].max\nend\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b\nfn knapsackDFS(wgt: []i32, val: []i32, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    var no = knapsackDFS(wgt, val, i - 1, c);\n    var yes = knapsackDFS(wgt, val, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return @max(no, yes);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 14-18 \u6240\u793a\uff0c\u7531\u65bc\u6bcf\u500b\u7269\u54c1\u90fd\u6703\u7522\u751f\u4e0d\u9078\u548c\u9078\u5169\u689d\u641c\u5c0b\u5206\u652f\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^n)\\) \u3002</p> <p>\u89c0\u5bdf\u905e\u8ff4\u6a39\uff0c\u5bb9\u6613\u767c\u73fe\u5176\u4e2d\u5b58\u5728\u91cd\u758a\u5b50\u554f\u984c\uff0c\u4f8b\u5982 \\(dp[1, 10]\\) \u7b49\u3002\u800c\u7576\u7269\u54c1\u8f03\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f03\u5927\uff0c\u5c24\u5176\u662f\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f03\u591a\u6642\uff0c\u91cd\u758a\u5b50\u554f\u984c\u7684\u6578\u91cf\u5c07\u6703\u5927\u5e45\u589e\u591a\u3002</p> <p></p> <p> \u5716 14-18 \u00a0 0-1 \u80cc\u5305\u554f\u984c\u7684\u66b4\u529b\u641c\u5c0b\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b","text":"<p>\u70ba\u4e86\u4fdd\u8b49\u91cd\u758a\u5b50\u554f\u984c\u53ea\u88ab\u8a08\u7b97\u4e00\u6b21\uff0c\u6211\u5011\u85c9\u52a9\u8a18\u61b6\u4e32\u5217 <code>mem</code> \u4f86\u8a18\u9304\u5b50\u554f\u984c\u7684\u89e3\uff0c\u5176\u4e2d <code>mem[i][c]</code> \u5c0d\u61c9 \\(dp[i, c]\\) \u3002</p> <p>\u5f15\u5165\u8a18\u61b6\u5316\u4e4b\u5f8c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53d6\u6c7a\u65bc\u5b50\u554f\u984c\u6578\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\"\"\"\n    # \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfun knapsackDFSMem(\n    wgt: IntArray,\n    _val: IntArray,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    c: Int\n): Int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    val no = knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    val yes = knapsackDFSMem(wgt, _val, mem, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b ###\ndef knapsack_dfs_mem(wgt, val, mem, i, c)\n  # \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n  return 0 if i == 0 || c == 0\n  # \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  return mem[i][c] if mem[i][c] != -1\n  # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n  return knapsack_dfs_mem(wgt, val, mem, i - 1, c) if wgt[i - 1] &gt; c\n  # \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n  no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n  yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n  mem[i][c] = [no, yes].max\nend\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\nfn knapsackDFSMem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    var no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    var yes = knapsackDFSMem(wgt, val, mem, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = @max(no, yes);\n    return mem[i][c];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-19 \u5c55\u793a\u4e86\u5728\u8a18\u61b6\u5316\u641c\u5c0b\u4e2d\u88ab\u526a\u6389\u7684\u641c\u5c0b\u5206\u652f\u3002</p> <p></p> <p> \u5716 14-19 \u00a0 0-1 \u80cc\u5305\u554f\u984c\u7684\u8a18\u61b6\u5316\u641c\u5c0b\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52d5\u614b\u898f\u5283","text":"<p>\u52d5\u614b\u898f\u5283\u5be6\u8cea\u4e0a\u5c31\u662f\u5728\u72c0\u614b\u8f49\u79fb\u4e2d\u586b\u5145 \\(dp\\) \u8868\u7684\u904e\u7a0b\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = std::cmp::max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],\n                );\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfun knapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 ###\ndef knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = [dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283\nfn knapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i - 1][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 14-20 \u6240\u793a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u90fd\u7531\u9663\u5217 <code>dp</code> \u5927\u5c0f\u6c7a\u5b9a\uff0c\u5373 \\(O(n \\times cap)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-20 \u00a0 0-1 \u80cc\u5305\u554f\u984c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#4","title":"4. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7531\u65bc\u6bcf\u500b\u72c0\u614b\u90fd\u53ea\u8207\u5176\u4e0a\u4e00\u884c\u7684\u72c0\u614b\u6709\u95dc\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u5169\u500b\u9663\u5217\u6efe\u52d5\u524d\u9032\uff0c\u5c07\u7a7a\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \u3002</p> <p>\u9032\u4e00\u6b65\u601d\u8003\uff0c\u6211\u5011\u80fd\u5426\u50c5\u7528\u4e00\u500b\u9663\u5217\u5be6\u73fe\u7a7a\u9593\u6700\u4f73\u5316\u5462\uff1f\u89c0\u5bdf\u53ef\u77e5\uff0c\u6bcf\u500b\u72c0\u614b\u90fd\u662f\u7531\u6b63\u4e0a\u65b9\u6216\u5de6\u4e0a\u65b9\u7684\u683c\u5b50\u8f49\u79fb\u904e\u4f86\u7684\u3002\u5047\u8a2d\u53ea\u6709\u4e00\u500b\u9663\u5217\uff0c\u7576\u958b\u59cb\u8d70\u8a2a\u7b2c \\(i\\) \u884c\u6642\uff0c\u8a72\u9663\u5217\u5132\u5b58\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72c0\u614b\u3002</p> <ul> <li>\u5982\u679c\u63a1\u53d6\u6b63\u5e8f\u8d70\u8a2a\uff0c\u90a3\u9ebc\u8d70\u8a2a\u5230 \\(dp[i, j]\\) \u6642\uff0c\u5de6\u4e0a\u65b9 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) \u503c\u53ef\u80fd\u5df2\u7d93\u88ab\u8986\u84cb\uff0c\u6b64\u6642\u5c31\u7121\u6cd5\u5f97\u5230\u6b63\u78ba\u7684\u72c0\u614b\u8f49\u79fb\u7d50\u679c\u3002</li> <li>\u5982\u679c\u63a1\u53d6\u5012\u5e8f\u8d70\u8a2a\uff0c\u5247\u4e0d\u6703\u767c\u751f\u8986\u84cb\u554f\u984c\uff0c\u72c0\u614b\u8f49\u79fb\u53ef\u4ee5\u6b63\u78ba\u9032\u884c\u3002</li> </ul> <p>\u5716 14-21 \u5c55\u793a\u4e86\u5728\u55ae\u500b\u9663\u5217\u4e0b\u5f9e\u7b2c \\(i = 1\\) \u884c\u8f49\u63db\u81f3\u7b2c \\(i = 2\\) \u884c\u7684\u904e\u7a0b\u3002\u8acb\u601d\u8003\u6b63\u5e8f\u8d70\u8a2a\u548c\u5012\u5e8f\u8d70\u8a2a\u7684\u5340\u5225\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-21 \u00a0 0-1 \u80cc\u5305\u7684\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p> <p>\u5728\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e2d\uff0c\u6211\u5011\u50c5\u9700\u5c07\u9663\u5217 <code>dp</code> \u7684\u7b2c\u4e00\u7dad \\(i\\) \u76f4\u63a5\u522a\u9664\uff0c\u4e26\u4e14\u628a\u5167\u8ff4\u5708\u66f4\u6539\u70ba\u5012\u5e8f\u8d70\u8a2a\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        # \u5012\u5e8f\u8d70\u8a2a\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for c in (1 ... cap).reversed() {\n            if wgt[i - 1] &lt;= c {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5012\u5e8f\u8d70\u8a2a\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun knapsackDPComp(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(cap + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (c in cap downTo 1) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(cap + 1, 0)\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    # \u5012\u5e8f\u8d70\u8a2a\n    for c in cap.downto(1)\n      if wgt[i - 1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[c] = dp[c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn knapsackDPComp(wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        // \u5012\u5e8f\u8d70\u8a2a\n        var c = cap;\n        while (c &gt; 0) : (c -= 1) {\n            if (wgt[i - 1] &lt; c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/summary/","title":"14.7 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u52d5\u614b\u898f\u5283\u5c0d\u554f\u984c\u9032\u884c\u5206\u89e3\uff0c\u4e26\u900f\u904e\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\u4f86\u898f\u907f\u91cd\u8907\u8a08\u7b97\uff0c\u63d0\u9ad8\u8a08\u7b97\u6548\u7387\u3002</li> <li>\u4e0d\u8003\u616e\u6642\u9593\u7684\u524d\u63d0\u4e0b\uff0c\u6240\u6709\u52d5\u614b\u898f\u5283\u554f\u984c\u90fd\u53ef\u4ee5\u7528\u56de\u6eaf\uff08\u66b4\u529b\u641c\u5c0b\uff09\u9032\u884c\u6c42\u89e3\uff0c\u4f46\u905e\u8ff4\u6a39\u4e2d\u5b58\u5728\u5927\u91cf\u7684\u91cd\u758a\u5b50\u554f\u984c\uff0c\u6548\u7387\u6975\u4f4e\u3002\u900f\u904e\u5f15\u5165\u8a18\u61b6\u5316\u4e32\u5217\uff0c\u53ef\u4ee5\u5132\u5b58\u6240\u6709\u8a08\u7b97\u904e\u7684\u5b50\u554f\u984c\u7684\u89e3\uff0c\u5f9e\u800c\u4fdd\u8b49\u91cd\u758a\u5b50\u554f\u984c\u53ea\u88ab\u8a08\u7b97\u4e00\u6b21\u3002</li> <li>\u8a18\u61b6\u5316\u641c\u5c0b\u662f\u4e00\u7a2e\u5f9e\u9802\u81f3\u5e95\u7684\u905e\u8ff4\u5f0f\u89e3\u6cd5\uff0c\u800c\u8207\u4e4b\u5c0d\u61c9\u7684\u52d5\u614b\u898f\u5283\u662f\u4e00\u7a2e\u5f9e\u5e95\u81f3\u9802\u7684\u905e\u63a8\u5f0f\u89e3\u6cd5\uff0c\u5176\u5982\u540c\u201c\u586b\u5beb\u8868\u683c\u201d\u4e00\u6a23\u3002\u7531\u65bc\u7576\u524d\u72c0\u614b\u50c5\u4f9d\u8cf4\u67d0\u4e9b\u5340\u57df\u6027\u72c0\u614b\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u6d88\u9664 \\(dp\\) \u8868\u7684\u4e00\u500b\u7dad\u5ea6\uff0c\u5f9e\u800c\u964d\u4f4e\u7a7a\u9593\u8907\u96dc\u5ea6\u3002</li> <li>\u5b50\u554f\u984c\u5206\u89e3\u662f\u4e00\u7a2e\u901a\u7528\u7684\u6f14\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52d5\u614b\u898f\u5283\u3001\u56de\u6eaf\u4e2d\u5177\u6709\u4e0d\u540c\u7684\u6027\u8cea\u3002</li> <li>\u52d5\u614b\u898f\u5283\u554f\u984c\u6709\u4e09\u5927\u7279\u6027\uff1a\u91cd\u758a\u5b50\u554f\u984c\u3001\u6700\u512a\u5b50\u7d50\u69cb\u3001\u7121\u5f8c\u6548\u6027\u3002</li> <li>\u5982\u679c\u539f\u554f\u984c\u7684\u6700\u512a\u89e3\u53ef\u4ee5\u5f9e\u5b50\u554f\u984c\u7684\u6700\u512a\u89e3\u69cb\u5efa\u5f97\u4f86\uff0c\u5247\u5b83\u5c31\u5177\u6709\u6700\u512a\u5b50\u7d50\u69cb\u3002</li> <li>\u7121\u5f8c\u6548\u6027\u6307\u5c0d\u65bc\u4e00\u500b\u72c0\u614b\uff0c\u5176\u672a\u4f86\u767c\u5c55\u53ea\u8207\u8a72\u72c0\u614b\u6709\u95dc\uff0c\u800c\u8207\u904e\u53bb\u7d93\u6b77\u7684\u6240\u6709\u72c0\u614b\u7121\u95dc\u3002\u8a31\u591a\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\u4e0d\u5177\u6709\u7121\u5f8c\u6548\u6027\uff0c\u7121\u6cd5\u4f7f\u7528\u52d5\u614b\u898f\u5283\u5feb\u901f\u6c42\u89e3\u3002</li> </ul> <p>\u80cc\u5305\u554f\u984c</p> <ul> <li>\u80cc\u5305\u554f\u984c\u662f\u6700\u5178\u578b\u7684\u52d5\u614b\u898f\u5283\u554f\u984c\u4e4b\u4e00\uff0c\u5177\u6709 0-1 \u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305\u7b49\u8b8a\u7a2e\u3002</li> <li>0-1 \u80cc\u5305\u7684\u72c0\u614b\u5b9a\u7fa9\u70ba\u524d \\(i\\) \u500b\u7269\u54c1\u5728\u5bb9\u91cf\u70ba \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u50f9\u503c\u3002\u6839\u64da\u4e0d\u653e\u5165\u80cc\u5305\u548c\u653e\u5165\u80cc\u5305\u5169\u7a2e\u6c7a\u7b56\uff0c\u53ef\u5f97\u5230\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u4e26\u69cb\u5efa\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u3002\u5728\u7a7a\u9593\u6700\u4f73\u5316\u4e2d\uff0c\u7531\u65bc\u6bcf\u500b\u72c0\u614b\u4f9d\u8cf4\u6b63\u4e0a\u65b9\u548c\u5de6\u4e0a\u65b9\u7684\u72c0\u614b\uff0c\u56e0\u6b64\u9700\u8981\u5012\u5e8f\u8d70\u8a2a\u4e32\u5217\uff0c\u907f\u514d\u5de6\u4e0a\u65b9\u72c0\u614b\u88ab\u8986\u84cb\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u6bcf\u7a2e\u7269\u54c1\u7684\u9078\u53d6\u6578\u91cf\u7121\u9650\u5236\uff0c\u56e0\u6b64\u9078\u64c7\u653e\u5165\u7269\u54c1\u7684\u72c0\u614b\u8f49\u79fb\u8207 0-1 \u80cc\u5305\u554f\u984c\u4e0d\u540c\u3002\u7531\u65bc\u72c0\u614b\u4f9d\u8cf4\u6b63\u4e0a\u65b9\u548c\u6b63\u5de6\u65b9\u7684\u72c0\u614b\uff0c\u56e0\u6b64\u5728\u7a7a\u9593\u6700\u4f73\u5316\u4e2d\u61c9\u7576\u6b63\u5e8f\u8d70\u8a2a\u3002</li> <li>\u96f6\u9322\u514c\u63db\u554f\u984c\u662f\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u4e00\u500b\u8b8a\u7a2e\u3002\u5b83\u5f9e\u6c42\u201c\u6700\u5927\u201d\u50f9\u503c\u8b8a\u70ba\u6c42\u201c\u6700\u5c0f\u201d\u786c\u5e63\u6578\u91cf\uff0c\u56e0\u6b64\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\max()\\) \u61c9\u6539\u70ba \\(\\min()\\) \u3002\u5f9e\u8ffd\u6c42\u201c\u4e0d\u8d85\u904e\u201d\u80cc\u5305\u5bb9\u91cf\u5230\u8ffd\u6c42\u201c\u6070\u597d\u201d\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\uff0c\u56e0\u6b64\u4f7f\u7528 \\(amt + 1\\) \u4f86\u8868\u793a\u201c\u7121\u6cd5\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u201d\u7684\u7121\u6548\u89e3\u3002</li> <li>\u96f6\u9322\u514c\u63db\u554f\u984c II \u5f9e\u6c42\u201c\u6700\u5c11\u786c\u5e63\u6578\u91cf\u201d\u6539\u70ba\u6c42\u201c\u786c\u5e63\u7d44\u5408\u6578\u91cf\u201d\uff0c\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u76f8\u61c9\u5730\u5f9e \\(\\min()\\) \u6539\u70ba\u6c42\u548c\u904b\u7b97\u5b50\u3002</li> </ul> <p>\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c</p> <ul> <li>\u7de8\u8f2f\u8ddd\u96e2\uff08Levenshtein \u8ddd\u96e2\uff09\u7528\u65bc\u8861\u91cf\u5169\u500b\u5b57\u4e32\u4e4b\u9593\u7684\u76f8\u4f3c\u5ea6\uff0c\u5176\u5b9a\u7fa9\u70ba\u5f9e\u4e00\u500b\u5b57\u4e32\u5230\u53e6\u4e00\u500b\u5b57\u4e32\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\uff0c\u7de8\u8f2f\u64cd\u4f5c\u5305\u62ec\u65b0\u589e\u3001\u522a\u9664\u3001\u66ff\u63db\u3002</li> <li>\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c\u7684\u72c0\u614b\u5b9a\u7fa9\u70ba\u5c07 \\(s\\) \u7684\u524d \\(i\\) \u500b\u5b57\u5143\u66f4\u6539\u70ba \\(t\\) \u7684\u524d \\(j\\) \u500b\u5b57\u5143\u6240\u9700\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u3002\u7576 \\(s[i] \\ne t[j]\\) \u6642\uff0c\u5177\u6709\u4e09\u7a2e\u6c7a\u7b56\uff1a\u65b0\u589e\u3001\u522a\u9664\u3001\u66ff\u63db\uff0c\u5b83\u5011\u90fd\u6709\u76f8\u61c9\u7684\u5269\u9918\u5b50\u554f\u984c\u3002\u64da\u6b64\u4fbf\u53ef\u4ee5\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\u8207\u69cb\u5efa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u3002\u800c\u7576 \\(s[i] = t[j]\\) \u6642\uff0c\u7121\u9808\u7de8\u8f2f\u7576\u524d\u5b57\u5143\u3002</li> <li>\u5728\u7de8\u8f2f\u8ddd\u96e2\u4e2d\uff0c\u72c0\u614b\u4f9d\u8cf4\u5176\u6b63\u4e0a\u65b9\u3001\u6b63\u5de6\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u72c0\u614b\uff0c\u56e0\u6b64\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u6b63\u5e8f\u6216\u5012\u5e8f\u8d70\u8a2a\u90fd\u7121\u6cd5\u6b63\u78ba\u5730\u9032\u884c\u72c0\u614b\u8f49\u79fb\u3002\u70ba\u6b64\uff0c\u6211\u5011\u5229\u7528\u4e00\u500b\u8b8a\u6578\u66ab\u5b58\u5de6\u4e0a\u65b9\u72c0\u614b\uff0c\u5f9e\u800c\u8f49\u5316\u5230\u8207\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7b49\u50f9\u7684\u60c5\u6cc1\uff0c\u53ef\u4ee5\u5728\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u9032\u884c\u6b63\u5e8f\u8d70\u8a2a\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","title":"14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c","text":"<p>\u5728\u672c\u7bc0\u4e2d\uff0c\u6211\u5011\u5148\u6c42\u89e3\u53e6\u4e00\u500b\u5e38\u898b\u7684\u80cc\u5305\u554f\u984c\uff1a\u5b8c\u5168\u80cc\u5305\uff0c\u518d\u77ad\u89e3\u5b83\u7684\u4e00\u7a2e\u7279\u4f8b\uff1a\u96f6\u9322\u514c\u63db\u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","title":"14.5.1 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u500b\u7269\u54c1\uff0c\u7b2c \\(i\\) \u500b\u7269\u54c1\u7684\u91cd\u91cf\u70ba \\(wgt[i-1]\\)\u3001\u50f9\u503c\u70ba \\(val[i-1]\\) \uff0c\u548c\u4e00\u500b\u5bb9\u91cf\u70ba \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u500b\u7269\u54c1\u53ef\u4ee5\u91cd\u8907\u9078\u53d6\uff0c\u554f\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u50f9\u503c\u3002\u793a\u4f8b\u5982\u5716 14-22 \u6240\u793a\u3002</p> <p></p> <p> \u5716 14-22 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u793a\u4f8b\u8cc7\u6599 </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","title":"1. \u00a0 \u52d5\u614b\u898f\u5283\u601d\u8def","text":"<p>\u5b8c\u5168\u80cc\u5305\u554f\u984c\u548c 0-1 \u80cc\u5305\u554f\u984c\u975e\u5e38\u76f8\u4f3c\uff0c\u5340\u5225\u50c5\u5728\u65bc\u4e0d\u9650\u5236\u7269\u54c1\u7684\u9078\u64c7\u6b21\u6578\u3002</p> <ul> <li>\u5728 0-1 \u80cc\u5305\u554f\u984c\u4e2d\uff0c\u6bcf\u7a2e\u7269\u54c1\u53ea\u6709\u4e00\u500b\uff0c\u56e0\u6b64\u5c07\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u5f8c\uff0c\u53ea\u80fd\u5f9e\u524d \\(i-1\\) \u500b\u7269\u54c1\u4e2d\u9078\u64c7\u3002</li> <li>\u5728\u5b8c\u5168\u80cc\u5305\u554f\u984c\u4e2d\uff0c\u6bcf\u7a2e\u7269\u54c1\u7684\u6578\u91cf\u662f\u7121\u9650\u7684\uff0c\u56e0\u6b64\u5c07\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u5f8c\uff0c\u4ecd\u53ef\u4ee5\u5f9e\u524d \\(i\\) \u500b\u7269\u54c1\u4e2d\u9078\u64c7\u3002</li> </ul> <p>\u5728\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u898f\u5b9a\u4e0b\uff0c\u72c0\u614b \\([i, c]\\) \u7684\u8b8a\u5316\u5206\u70ba\u5169\u7a2e\u60c5\u6cc1\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u8207 0-1 \u80cc\u5305\u554f\u984c\u76f8\u540c\uff0c\u8f49\u79fb\u81f3 \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u8207 0-1 \u80cc\u5305\u554f\u984c\u4e0d\u540c\uff0c\u8f49\u79fb\u81f3 \\([i, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u5f9e\u800c\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u8b8a\u70ba\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5c0d\u6bd4\u5169\u9053\u984c\u76ee\u7684\u7a0b\u5f0f\u78bc\uff0c\u72c0\u614b\u8f49\u79fb\u4e2d\u6709\u4e00\u8655\u5f9e \\(i-1\\) \u8b8a\u70ba \\(i\\) \uff0c\u5176\u9918\u5b8c\u5168\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfun unboundedKnapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283\nfn unboundedKnapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","title":"3. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7531\u65bc\u7576\u524d\u72c0\u614b\u662f\u5f9e\u5de6\u908a\u548c\u4e0a\u908a\u7684\u72c0\u614b\u8f49\u79fb\u800c\u4f86\u7684\uff0c\u56e0\u6b64\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u61c9\u8a72\u5c0d \\(dp\\) \u8868\u4e2d\u7684\u6bcf\u4e00\u884c\u9032\u884c\u6b63\u5e8f\u8d70\u8a2a\u3002</p> <p>\u9019\u500b\u8d70\u8a2a\u9806\u5e8f\u8207 0-1 \u80cc\u5305\u6b63\u597d\u76f8\u53cd\u3002\u8acb\u85c9\u52a9\u5716 14-23 \u4f86\u7406\u89e3\u5169\u8005\u7684\u5340\u5225\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-23 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c\u5728\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p> <p>\u7a0b\u5f0f\u78bc\u5be6\u73fe\u6bd4\u8f03\u7c21\u55ae\uff0c\u50c5\u9700\u5c07\u9663\u5217 <code>dp</code> \u7684\u7b2c\u4e00\u7dad\u522a\u9664\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u8d70\u8a2a\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[c] = dp[c];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun unboundedKnapsackDPComp(\n    wgt: IntArray,\n    _val: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(cap + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n\n### \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ##3\ndef unbounded_knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(cap + 1, 0)\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    # \u6b63\u5e8f\u8d70\u8a2a\n    for c in 1...(cap + 1)\n      if wgt[i -1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[c] = dp[c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn unboundedKnapsackDPComp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","title":"14.5.2 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c","text":"<p>\u80cc\u5305\u554f\u984c\u662f\u4e00\u5927\u985e\u52d5\u614b\u898f\u5283\u554f\u984c\u7684\u4ee3\u8868\uff0c\u5176\u64c1\u6709\u5f88\u591a\u8b8a\u7a2e\uff0c\u4f8b\u5982\u96f6\u9322\u514c\u63db\u554f\u984c\u3002</p> <p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u7a2e\u786c\u5e63\uff0c\u7b2c \\(i\\) \u7a2e\u786c\u5e63\u7684\u9762\u503c\u70ba \\(coins[i - 1]\\) \uff0c\u76ee\u6a19\u91d1\u984d\u70ba \\(amt\\) \uff0c\u6bcf\u7a2e\u786c\u5e63\u53ef\u4ee5\u91cd\u8907\u9078\u53d6\uff0c\u554f\u80fd\u5920\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u7684\u6700\u5c11\u786c\u5e63\u6578\u91cf\u3002\u5982\u679c\u7121\u6cd5\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u5716 14-24 \u6240\u793a\u3002</p> <p></p> <p> \u5716 14-24 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c\u7684\u793a\u4f8b\u8cc7\u6599 </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","title":"1. \u00a0 \u52d5\u614b\u898f\u5283\u601d\u8def","text":"<p>\u96f6\u9322\u514c\u63db\u53ef\u4ee5\u770b\u4f5c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u4e00\u7a2e\u7279\u6b8a\u60c5\u6cc1\uff0c\u5169\u8005\u5177\u6709\u4ee5\u4e0b\u95dc\u806f\u8207\u4e0d\u540c\u9ede\u3002</p> <ul> <li>\u5169\u9053\u984c\u53ef\u4ee5\u76f8\u4e92\u8f49\u63db\uff0c\u201c\u7269\u54c1\u201d\u5c0d\u61c9\u201c\u786c\u5e63\u201d\u3001\u201c\u7269\u54c1\u91cd\u91cf\u201d\u5c0d\u61c9\u201c\u786c\u5e63\u9762\u503c\u201d\u3001\u201c\u80cc\u5305\u5bb9\u91cf\u201d\u5c0d\u61c9\u201c\u76ee\u6a19\u91d1\u984d\u201d\u3002</li> <li>\u6700\u4f73\u5316\u76ee\u6a19\u76f8\u53cd\uff0c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u662f\u8981\u6700\u5927\u5316\u7269\u54c1\u50f9\u503c\uff0c\u96f6\u9322\u514c\u63db\u554f\u984c\u662f\u8981\u6700\u5c0f\u5316\u786c\u5e63\u6578\u91cf\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u554f\u984c\u662f\u6c42\u201c\u4e0d\u8d85\u904e\u201d\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u89e3\uff0c\u96f6\u9322\u514c\u63db\u662f\u6c42\u201c\u6070\u597d\u201d\u6e4a\u5230\u76ee\u6a19\u91d1\u984d\u7684\u89e3\u3002</li> </ul> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f2a\u7684\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5f9e\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u72c0\u614b \\([i, a]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\u70ba\uff1a\u524d \\(i\\) \u7a2e\u786c\u5e63\u80fd\u5920\u6e4a\u51fa\u91d1\u984d \\(a\\) \u7684\u6700\u5c11\u786c\u5e63\u6578\u91cf\uff0c\u8a18\u70ba \\(dp[i, a]\\) \u3002</p> <p>\u4e8c\u7dad \\(dp\\) \u8868\u7684\u5c3a\u5bf8\u70ba \\((n+1) \\times (amt+1)\\) \u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u9032\u800c\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b</p> <p>\u672c\u984c\u8207\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u5b58\u5728\u4ee5\u4e0b\u5169\u9ede\u5dee\u7570\u3002</p> <ul> <li>\u672c\u984c\u8981\u6c42\u6700\u5c0f\u503c\uff0c\u56e0\u6b64\u9700\u5c07\u904b\u7b97\u5b50 \\(\\max()\\) \u66f4\u6539\u70ba \\(\\min()\\) \u3002</li> <li>\u6700\u4f73\u5316\u4e3b\u9ad4\u662f\u786c\u5e63\u6578\u91cf\u800c\u975e\u5546\u54c1\u50f9\u503c\uff0c\u56e0\u6b64\u5728\u9078\u4e2d\u786c\u5e63\u6642\u57f7\u884c \\(+1\\) \u5373\u53ef\u3002</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>\u7b2c\u4e09\u6b65\uff1a\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u548c\u72c0\u614b\u8f49\u79fb\u9806\u5e8f</p> <p>\u7576\u76ee\u6a19\u91d1\u984d\u70ba \\(0\\) \u6642\uff0c\u6e4a\u51fa\u5b83\u7684\u6700\u5c11\u786c\u5e63\u6578\u91cf\u70ba \\(0\\) \uff0c\u5373\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u7b49\u65bc \\(0\\) \u3002</p> <p>\u7576\u7121\u786c\u5e63\u6642\uff0c\u7121\u6cd5\u6e4a\u51fa\u4efb\u610f \\(&gt; 0\\) \u7684\u76ee\u6a19\u91d1\u984d\uff0c\u5373\u662f\u7121\u6548\u89e3\u3002\u70ba\u4f7f\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\min()\\) \u51fd\u5f0f\u80fd\u5920\u8b58\u5225\u4e26\u904e\u6ffe\u7121\u6548\u89e3\uff0c\u6211\u5011\u8003\u616e\u4f7f\u7528 \\(+ \\infty\\) \u4f86\u8868\u793a\u5b83\u5011\uff0c\u5373\u4ee4\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u65bc \\(+ \\infty\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u4e26\u672a\u63d0\u4f9b \\(+ \\infty\\) \u8b8a\u6578\uff0c\u53ea\u80fd\u4f7f\u7528\u6574\u578b <code>int</code> \u7684\u6700\u5927\u503c\u4f86\u4ee3\u66ff\u3002\u800c\u9019\u53c8\u6703\u5c0e\u81f4\u5927\u6578\u8d8a\u754c\uff1a\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(+ 1\\) \u64cd\u4f5c\u53ef\u80fd\u767c\u751f\u6ea2\u4f4d\u3002</p> <p>\u70ba\u6b64\uff0c\u6211\u5011\u63a1\u7528\u6578\u5b57 \\(amt + 1\\) \u4f86\u8868\u793a\u7121\u6548\u89e3\uff0c\u56e0\u70ba\u6e4a\u51fa \\(amt\\) \u7684\u786c\u5e63\u6578\u91cf\u6700\u591a\u70ba \\(amt\\) \u3002\u6700\u5f8c\u8fd4\u56de\u524d\uff0c\u5224\u65b7 \\(dp[n, amt]\\) \u662f\u5426\u7b49\u65bc \\(amt + 1\\) \uff0c\u82e5\u662f\u5247\u8fd4\u56de \\(-1\\) \uff0c\u4ee3\u8868\u7121\u6cd5\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1 ... amt {\n        dp[0][a] = MAX\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1..=amt {\n        dp[0][a] = max;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfun coinChangeDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (a in 1..amt) {\n        dp[0][a] = MAX\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[n][amt] != MAX) dp[n][amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 ###\ndef coin_change_dp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n  (1...(amt + 1)).each { |a| dp[0][a] = _MAX }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[i][a] = dp[i - 1][a]\n      else\n        # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n        dp[i][a] = [dp[i - 1][a], dp[i][a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[n][amt] != _MAX ? dp[n][amt] : -1\nend\n</code></pre> coin_change.zig<pre><code>// \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283\nfn coinChangeDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..amt + 1) |a| {\n        dp[0][a] = max;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = @min(dp[i - 1][a], dp[i][a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[n][amt] != max) {\n        return @intCast(dp[n][amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-25 \u5c55\u793a\u4e86\u96f6\u9322\u514c\u63db\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b\uff0c\u548c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u975e\u5e38\u76f8\u4f3c\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-25 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","title":"3. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u96f6\u9322\u514c\u63db\u7684\u7a7a\u9593\u6700\u4f73\u5316\u7684\u8655\u7406\u65b9\u5f0f\u548c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u8d70\u8a2a\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u6b63\u5e8f\u8d70\u8a2a\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = malloc((amt + 1) * sizeof(int));\n    for (int j = 1; j &lt;= amt; j++) {\n        dp[j] = MAX;\n    } \n    dp[0] = 0;\n\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun coinChangeDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(amt + 1)\n    dp.fill(MAX)\n    dp[0] = 0\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[amt] != MAX) dp[amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef coin_change_dp_comp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(amt + 1, _MAX)\n  dp[0] = 0\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    # \u6b63\u5e8f\u8d70\u8a2a\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[a] = dp[a]\n      else\n        # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n        dp[a] = [dp[a], dp[a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[amt] != _MAX ? dp[amt] : -1\nend\n</code></pre> coin_change.zig<pre><code>// \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn coinChangeDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    @memset(&amp;dp, max);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = @min(dp[a], dp[a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[amt] != max) {\n        return @intCast(dp[amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","title":"14.5.3 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c II","text":"<p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u7a2e\u786c\u5e63\uff0c\u7b2c \\(i\\) \u7a2e\u786c\u5e63\u7684\u9762\u503c\u70ba \\(coins[i - 1]\\) \uff0c\u76ee\u6a19\u91d1\u984d\u70ba \\(amt\\) \uff0c\u6bcf\u7a2e\u786c\u5e63\u53ef\u4ee5\u91cd\u8907\u9078\u53d6\uff0c\u554f\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u7684\u786c\u5e63\u7d44\u5408\u6578\u91cf\u3002\u793a\u4f8b\u5982\u5716 14-26 \u6240\u793a\u3002</p> <p></p> <p> \u5716 14-26 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c II \u7684\u793a\u4f8b\u8cc7\u6599 </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","title":"1. \u00a0 \u52d5\u614b\u898f\u5283\u601d\u8def","text":"<p>\u76f8\u6bd4\u65bc\u4e0a\u4e00\u984c\uff0c\u672c\u984c\u76ee\u6a19\u662f\u6c42\u7d44\u5408\u6578\u91cf\uff0c\u56e0\u6b64\u5b50\u554f\u984c\u8b8a\u70ba\uff1a\u524d \\(i\\) \u7a2e\u786c\u5e63\u80fd\u5920\u6e4a\u51fa\u91d1\u984d \\(a\\) \u7684\u7d44\u5408\u6578\u91cf\u3002\u800c \\(dp\\) \u8868\u4ecd\u7136\u662f\u5c3a\u5bf8\u70ba \\((n+1) \\times (amt + 1)\\) \u7684\u4e8c\u7dad\u77e9\u9663\u3002</p> <p>\u7576\u524d\u72c0\u614b\u7684\u7d44\u5408\u6578\u91cf\u7b49\u65bc\u4e0d\u9078\u7576\u524d\u786c\u5e63\u8207\u9078\u7576\u524d\u786c\u5e63\u9019\u5169\u7a2e\u6c7a\u7b56\u7684\u7d44\u5408\u6578\u91cf\u4e4b\u548c\u3002\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u70ba\uff1a</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>\u7576\u76ee\u6a19\u91d1\u984d\u70ba \\(0\\) \u6642\uff0c\u7121\u9808\u9078\u64c7\u4efb\u4f55\u786c\u5e63\u5373\u53ef\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\uff0c\u56e0\u6b64\u61c9\u5c07\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u521d\u59cb\u5316\u70ba \\(1\\) \u3002\u7576\u7121\u786c\u5e63\u6642\uff0c\u7121\u6cd5\u6e4a\u51fa\u4efb\u4f55 \\(&gt;0\\) \u7684\u76ee\u6a19\u91d1\u984d\uff0c\u56e0\u6b64\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u65bc \\(0\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u521d\u59cb\u5316\u9996\u5217\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0 ... n {\n        dp[i][0] = 1\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u521d\u59cb\u5316\u9996\u5217\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0..=n {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfun coinChangeIIDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (i in 0..n) {\n        dp[i][0] = 1\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 ###\ndef coin_change_ii_dp(coins, amt)\n  n = coins.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # \u521d\u59cb\u5316\u9996\u5217\n  (0...(n + 1)).each { |i| dp[i][0] = 1 }\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[i][a] = dp[i - 1][a]\n      else\n        # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n      end\n    end\n  end\n  dp[n][amt]\nend\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283\nfn coinChangeIIDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (0..n + 1) |i| {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","title":"3. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7a7a\u9593\u6700\u4f73\u5316\u8655\u7406\u65b9\u5f0f\u76f8\u540c\uff0c\u522a\u9664\u786c\u5e63\u7dad\u5ea6\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u8d70\u8a2a\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u6b63\u5e8f\u8d70\u8a2a\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(amt + 1)\n    dp[0] = 1\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef coin_change_ii_dp_comp(coins, amt)\n  n = coins.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(amt + 1, 0)\n  dp[0] = 1\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    # \u6b63\u5e8f\u8d70\u8a2a\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[a] = dp[a]\n      else\n        # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n        dp[a] = dp[a] + dp[a - coins[i - 1]]\n      end\n    end\n  end\n  dp[amt]\nend\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn coinChangeIIDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_graph/","title":"\u7b2c 9 \u7ae0 \u00a0 \u5716","text":"<p>Abstract</p> <p>\u5728\u751f\u547d\u65c5\u9014\u4e2d\uff0c\u6211\u5011\u5c31\u50cf\u662f\u4e00\u500b\u500b\u7bc0\u9ede\uff0c\u88ab\u7121\u6578\u770b\u4e0d\u898b\u7684\u908a\u76f8\u9023\u3002</p> <p>\u6bcf\u4e00\u6b21\u7684\u76f8\u8b58\u8207\u76f8\u96e2\uff0c\u90fd\u5728\u9019\u5f35\u5de8\u5927\u7684\u7db2\u8def\u5716\u4e2d\u7559\u4e0b\u7368\u7279\u7684\u5370\u8a18\u3002</p>"},{"location":"chapter_graph/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>9.1 \u00a0 \u5716</li> <li>9.2 \u00a0 \u5716\u57fa\u790e\u64cd\u4f5c</li> <li>9.3 \u00a0 \u5716\u7684\u8d70\u8a2a</li> <li>9.4 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_graph/graph/","title":"9.1 \u00a0 \u5716","text":"<p>\u5716\uff08graph\uff09\u662f\u4e00\u7a2e\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u7531\u9802\u9ede\uff08vertex\uff09\u548c\u908a\uff08edge\uff09\u7d44\u6210\u3002\u6211\u5011\u53ef\u4ee5\u5c07\u5716 \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u70ba\u4e00\u7d44\u9802\u9ede \\(V\\) \u548c\u4e00\u7d44\u908a \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u500b\u5305\u542b 5 \u500b\u9802\u9ede\u548c 7 \u689d\u908a\u7684\u5716\u3002</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>\u5982\u679c\u5c07\u9802\u9ede\u770b\u4f5c\u7bc0\u9ede\uff0c\u5c07\u908a\u770b\u4f5c\u9023\u7dda\u5404\u500b\u7bc0\u9ede\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5c07\u5716\u770b\u4f5c\u4e00\u7a2e\u5f9e\u93c8\u7d50\u4e32\u5217\u62d3\u5c55\u800c\u4f86\u7684\u8cc7\u6599\u7d50\u69cb\u3002\u5982\u5716 9-1 \u6240\u793a\uff0c\u76f8\u8f03\u65bc\u7dda\u6027\u95dc\u4fc2\uff08\u93c8\u7d50\u4e32\u5217\uff09\u548c\u5206\u6cbb\u95dc\u4fc2\uff08\u6a39\uff09\uff0c\u7db2\u8def\u95dc\u4fc2\uff08\u5716\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u56e0\u800c\u66f4\u70ba\u8907\u96dc\u3002</p> <p></p> <p> \u5716 9-1 \u00a0 \u93c8\u7d50\u4e32\u5217\u3001\u6a39\u3001\u5716\u4e4b\u9593\u7684\u95dc\u4fc2 </p>"},{"location":"chapter_graph/graph/#911","title":"9.1.1 \u00a0 \u5716\u7684\u5e38\u898b\u578b\u5225\u8207\u8853\u8a9e","text":"<p>\u6839\u64da\u908a\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u70ba\u7121\u5411\u5716\uff08undirected graph\uff09\u548c\u6709\u5411\u5716\uff08directed graph\uff09\uff0c\u5982\u5716 9-2 \u6240\u793a\u3002</p> <ul> <li>\u5728\u7121\u5411\u5716\u4e2d\uff0c\u908a\u8868\u793a\u5169\u9802\u9ede\u4e4b\u9593\u7684\u201c\u96d9\u5411\u201d\u9023\u7dda\u95dc\u4fc2\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u95dc\u4fc2\u201d\u3002</li> <li>\u5728\u6709\u5411\u5716\u4e2d\uff0c\u908a\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u5169\u500b\u65b9\u5411\u7684\u908a\u662f\u76f8\u4e92\u7368\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u95dc\u6ce8\u201d\u8207\u201c\u88ab\u95dc\u6ce8\u201d\u95dc\u4fc2\u3002</li> </ul> <p></p> <p> \u5716 9-2 \u00a0 \u6709\u5411\u5716\u8207\u7121\u5411\u5716 </p> <p>\u6839\u64da\u6240\u6709\u9802\u9ede\u662f\u5426\u9023\u901a\uff0c\u53ef\u5206\u70ba\u9023\u901a\u5716\uff08connected graph\uff09\u548c\u975e\u9023\u901a\u5716\uff08disconnected graph\uff09\uff0c\u5982\u5716 9-3 \u6240\u793a\u3002</p> <ul> <li>\u5c0d\u65bc\u9023\u901a\u5716\uff0c\u5f9e\u67d0\u500b\u9802\u9ede\u51fa\u767c\uff0c\u53ef\u4ee5\u5230\u9054\u5176\u9918\u4efb\u610f\u9802\u9ede\u3002</li> <li>\u5c0d\u65bc\u975e\u9023\u901a\u5716\uff0c\u5f9e\u67d0\u500b\u9802\u9ede\u51fa\u767c\uff0c\u81f3\u5c11\u6709\u4e00\u500b\u9802\u9ede\u7121\u6cd5\u5230\u9054\u3002</li> </ul> <p></p> <p> \u5716 9-3 \u00a0 \u9023\u901a\u5716\u8207\u975e\u9023\u901a\u5716 </p> <p>\u6211\u5011\u9084\u53ef\u4ee5\u70ba\u908a\u65b0\u589e\u201c\u6b0a\u91cd\u201d\u8b8a\u6578\uff0c\u5f9e\u800c\u5f97\u5230\u5982\u5716 9-4 \u6240\u793a\u7684\u6709\u6b0a\u5716\uff08weighted graph\uff09\u3002\u4f8b\u5982\u5728\u300a\u738b\u8005\u69ae\u8000\u300b\u7b49\u624b\u904a\u4e2d\uff0c\u7cfb\u7d71\u6703\u6839\u64da\u5171\u540c\u904a\u6232\u6642\u9593\u4f86\u8a08\u7b97\u73a9\u5bb6\u4e4b\u9593\u7684\u201c\u89aa\u5bc6\u5ea6\u201d\uff0c\u9019\u7a2e\u89aa\u5bc6\u5ea6\u7db2\u8def\u5c31\u53ef\u4ee5\u7528\u6709\u6b0a\u5716\u4f86\u8868\u793a\u3002</p> <p></p> <p> \u5716 9-4 \u00a0 \u6709\u6b0a\u5716\u8207\u7121\u6b0a\u5716 </p> <p>\u5716\u8cc7\u6599\u7d50\u69cb\u5305\u542b\u4ee5\u4e0b\u5e38\u7528\u8853\u8a9e\u3002</p> <ul> <li>\u9130\u63a5\uff08adjacency\uff09\uff1a\u7576\u5169\u9802\u9ede\u4e4b\u9593\u5b58\u5728\u908a\u76f8\u9023\u6642\uff0c\u7a31\u9019\u5169\u9802\u9ede\u201c\u9130\u63a5\u201d\u3002\u5728\u5716 9-4 \u4e2d\uff0c\u9802\u9ede 1 \u7684\u9130\u63a5\u9802\u9ede\u70ba\u9802\u9ede 2\u30013\u30015\u3002</li> <li>\u8def\u5f91\uff08path\uff09\uff1a\u5f9e\u9802\u9ede A \u5230\u9802\u9ede B \u7d93\u904e\u7684\u908a\u69cb\u6210\u7684\u5e8f\u5217\u88ab\u7a31\u70ba\u5f9e A \u5230 B \u7684\u201c\u8def\u5f91\u201d\u3002\u5728\u5716 9-4 \u4e2d\uff0c\u908a\u5e8f\u5217 1-5-2-4 \u662f\u9802\u9ede 1 \u5230\u9802\u9ede 4 \u7684\u4e00\u689d\u8def\u5f91\u3002</li> <li>\u5ea6\uff08degree\uff09\uff1a\u4e00\u500b\u9802\u9ede\u64c1\u6709\u7684\u908a\u6578\u3002\u5c0d\u65bc\u6709\u5411\u5716\uff0c\u5165\u5ea6\uff08in-degree\uff09\u8868\u793a\u6709\u591a\u5c11\u689d\u908a\u6307\u5411\u8a72\u9802\u9ede\uff0c\u51fa\u5ea6\uff08out-degree\uff09\u8868\u793a\u6709\u591a\u5c11\u689d\u908a\u5f9e\u8a72\u9802\u9ede\u6307\u51fa\u3002</li> </ul>"},{"location":"chapter_graph/graph/#912","title":"9.1.2 \u00a0 \u5716\u7684\u8868\u793a","text":"<p>\u5716\u7684\u5e38\u7528\u8868\u793a\u65b9\u5f0f\u5305\u62ec\u201c\u9130\u63a5\u77e9\u9663\u201d\u548c\u201c\u9130\u63a5\u8868\u201d\u3002\u4ee5\u4e0b\u4f7f\u7528\u7121\u5411\u5716\u9032\u884c\u8209\u4f8b\u3002</p>"},{"location":"chapter_graph/graph/#1","title":"1. \u00a0 \u9130\u63a5\u77e9\u9663","text":"<p>\u8a2d\u5716\u7684\u9802\u9ede\u6578\u91cf\u70ba \\(n\\) \uff0c\u9130\u63a5\u77e9\u9663\uff08adjacency matrix\uff09\u4f7f\u7528\u4e00\u500b \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9663\u4f86\u8868\u793a\u5716\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u500b\u9802\u9ede\uff0c\u77e9\u9663\u5143\u7d20\u4ee3\u8868\u908a\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u5169\u500b\u9802\u9ede\u4e4b\u9593\u662f\u5426\u5b58\u5728\u908a\u3002</p> <p>\u5982\u5716 9-5 \u6240\u793a\uff0c\u8a2d\u9130\u63a5\u77e9\u9663\u70ba \\(M\\)\u3001\u9802\u9ede\u4e32\u5217\u70ba \\(V\\) \uff0c\u90a3\u9ebc\u77e9\u9663\u5143\u7d20 \\(M[i, j] = 1\\) \u8868\u793a\u9802\u9ede \\(V[i]\\) \u5230\u9802\u9ede \\(V[j]\\) \u4e4b\u9593\u5b58\u5728\u908a\uff0c\u53cd\u4e4b \\(M[i, j] = 0\\) \u8868\u793a\u5169\u9802\u9ede\u4e4b\u9593\u7121\u908a\u3002</p> <p></p> <p> \u5716 9-5 \u00a0 \u5716\u7684\u9130\u63a5\u77e9\u9663\u8868\u793a </p> <p>\u9130\u63a5\u77e9\u9663\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u5728\u7c21\u55ae\u5716\u4e2d\uff0c\u9802\u9ede\u4e0d\u80fd\u8207\u81ea\u8eab\u76f8\u9023\uff0c\u6b64\u6642\u9130\u63a5\u77e9\u9663\u4e3b\u5c0d\u89d2\u7dda\u5143\u7d20\u6c92\u6709\u610f\u7fa9\u3002</li> <li>\u5c0d\u65bc\u7121\u5411\u5716\uff0c\u5169\u500b\u65b9\u5411\u7684\u908a\u7b49\u50f9\uff0c\u6b64\u6642\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\u3002</li> <li>\u5c07\u9130\u63a5\u77e9\u9663\u7684\u5143\u7d20\u5f9e \\(1\\) \u548c \\(0\\) \u66ff\u63db\u70ba\u6b0a\u91cd\uff0c\u5247\u53ef\u8868\u793a\u6709\u6b0a\u5716\u3002</li> </ul> <p>\u4f7f\u7528\u9130\u63a5\u77e9\u9663\u8868\u793a\u5716\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u8a2a\u554f\u77e9\u9663\u5143\u7d20\u4ee5\u7372\u53d6\u908a\uff0c\u56e0\u6b64\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9663\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u8a18\u61b6\u9ad4\u4f54\u7528\u8f03\u591a\u3002</p>"},{"location":"chapter_graph/graph/#2","title":"2. \u00a0 \u9130\u63a5\u8868","text":"<p>\u9130\u63a5\u8868\uff08adjacency list\uff09\u4f7f\u7528 \\(n\\) \u500b\u93c8\u7d50\u4e32\u5217\u4f86\u8868\u793a\u5716\uff0c\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u8868\u793a\u9802\u9ede\u3002\u7b2c \\(i\\) \u500b\u93c8\u7d50\u4e32\u5217\u5c0d\u61c9\u9802\u9ede \\(i\\) \uff0c\u5176\u4e2d\u5132\u5b58\u4e86\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\uff08\u8207\u8a72\u9802\u9ede\u76f8\u9023\u7684\u9802\u9ede\uff09\u3002\u5716 9-6 \u5c55\u793a\u4e86\u4e00\u500b\u4f7f\u7528\u9130\u63a5\u8868\u5132\u5b58\u7684\u5716\u7684\u793a\u4f8b\u3002</p> <p></p> <p> \u5716 9-6 \u00a0 \u5716\u7684\u9130\u63a5\u8868\u8868\u793a </p> <p>\u9130\u63a5\u8868\u50c5\u5132\u5b58\u5be6\u969b\u5b58\u5728\u7684\u908a\uff0c\u800c\u908a\u7684\u7e3d\u6578\u901a\u5e38\u9060\u5c0f\u65bc \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u7bc0\u7701\u7a7a\u9593\u3002\u7136\u800c\uff0c\u5728\u9130\u63a5\u8868\u4e2d\u9700\u8981\u900f\u904e\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4f86\u67e5\u8a62\u908a\uff0c\u56e0\u6b64\u5176\u6642\u9593\u6548\u7387\u4e0d\u5982\u9130\u63a5\u77e9\u9663\u3002</p> <p>\u89c0\u5bdf\u5716 9-6 \uff0c\u9130\u63a5\u8868\u7d50\u69cb\u8207\u96dc\u6e4a\u8868\u4e2d\u7684\u201c\u93c8\u5f0f\u4f4d\u5740\u201d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u5011\u4e5f\u53ef\u4ee5\u63a1\u7528\u985e\u4f3c\u7684\u65b9\u6cd5\u4f86\u6700\u4f73\u5316\u6548\u7387\u3002\u6bd4\u5982\u7576\u93c8\u7d50\u4e32\u5217\u8f03\u9577\u6642\uff0c\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba AVL \u6a39\u6216\u7d05\u9ed1\u6a39\uff0c\u5f9e\u800c\u5c07\u6642\u9593\u6548\u7387\u5f9e \\(O(n)\\) \u6700\u4f73\u5316\u81f3 \\(O(\\log n)\\) \uff1b\u9084\u53ef\u4ee5\u628a\u93c8\u7d50\u4e32\u5217\u8f49\u63db\u70ba\u96dc\u6e4a\u8868\uff0c\u5f9e\u800c\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u964d\u81f3 \\(O(1)\\) \u3002</p>"},{"location":"chapter_graph/graph/#913","title":"9.1.3 \u00a0 \u5716\u7684\u5e38\u898b\u61c9\u7528","text":"<p>\u5982\u8868 9-1 \u6240\u793a\uff0c\u8a31\u591a\u73fe\u5be6\u7cfb\u7d71\u53ef\u4ee5\u7528\u5716\u4f86\u5efa\u6a21\uff0c\u76f8\u61c9\u7684\u554f\u984c\u4e5f\u53ef\u4ee5\u7d04\u5316\u70ba\u5716\u8a08\u7b97\u554f\u984c\u3002</p> <p> \u8868 9-1 \u00a0 \u73fe\u5be6\u751f\u6d3b\u4e2d\u5e38\u898b\u7684\u5716 </p> \u9802\u9ede \u908a \u5716\u8a08\u7b97\u554f\u984c \u793e\u4ea4\u7db2\u8def \u4f7f\u7528\u8005 \u597d\u53cb\u95dc\u4fc2 \u6f5b\u5728\u597d\u53cb\u63a8\u85a6 \u5730\u9435\u7dda\u8def \u7ad9\u9ede \u7ad9\u9ede\u9593\u7684\u9023\u901a\u6027 \u6700\u77ed\u8def\u7dda\u63a8\u85a6 \u592a\u967d\u7cfb \u661f\u9ad4 \u661f\u9ad4\u9593\u7684\u842c\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8ecc\u9053\u8a08\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 \u5716\u7684\u57fa\u790e\u64cd\u4f5c","text":"<p>\u5716\u7684\u57fa\u790e\u64cd\u4f5c\u53ef\u5206\u70ba\u5c0d\u201c\u908a\u201d\u7684\u64cd\u4f5c\u548c\u5c0d\u201c\u9802\u9ede\u201d\u7684\u64cd\u4f5c\u3002\u5728\u201c\u9130\u63a5\u77e9\u9663\u201d\u548c\u201c\u9130\u63a5\u8868\u201d\u5169\u7a2e\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5be6\u73fe\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002</p>"},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1 \u00a0 \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u7684\u5be6\u73fe","text":"<p>\u7d66\u5b9a\u4e00\u500b\u9802\u9ede\u6578\u91cf\u70ba \\(n\\) \u7684\u7121\u5411\u5716\uff0c\u5247\u5404\u7a2e\u64cd\u4f5c\u7684\u5be6\u73fe\u65b9\u5f0f\u5982\u5716 9-7 \u6240\u793a\u3002</p> <ul> <li>\u65b0\u589e\u6216\u522a\u9664\u908a\uff1a\u76f4\u63a5\u5728\u9130\u63a5\u77e9\u9663\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u908a\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002\u800c\u7531\u65bc\u662f\u7121\u5411\u5716\uff0c\u56e0\u6b64\u9700\u8981\u540c\u6642\u66f4\u65b0\u5169\u500b\u65b9\u5411\u7684\u908a\u3002</li> <li>\u65b0\u589e\u9802\u9ede\uff1a\u5728\u9130\u63a5\u77e9\u9663\u7684\u5c3e\u90e8\u65b0\u589e\u4e00\u884c\u4e00\u5217\uff0c\u4e26\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002</li> <li>\u522a\u9664\u9802\u9ede\uff1a\u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u4e00\u884c\u4e00\u5217\u3002\u7576\u522a\u9664\u9996\u884c\u9996\u5217\u6642\u9054\u5230\u6700\u5dee\u60c5\u6cc1\uff0c\u9700\u8981\u5c07 \\((n-1)^2\\) \u500b\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52d5\u201d\uff0c\u5f9e\u800c\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u50b3\u5165 \\(n\\) \u500b\u9802\u9ede\uff0c\u521d\u59cb\u5316\u9577\u5ea6\u70ba \\(n\\) \u7684\u9802\u9ede\u4e32\u5217 <code>vertices</code> \uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u9130\u63a5\u77e9\u9663 <code>adjMat</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002</li> </ul> \u521d\u59cb\u5316\u9130\u63a5\u77e9\u9663\u65b0\u589e\u908a\u522a\u9664\u908a\u65b0\u589e\u9802\u9ede\u522a\u9664\u9802\u9ede <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 9-7 \u00a0 \u9130\u63a5\u77e9\u9663\u7684\u521d\u59cb\u5316\u3001\u589e\u522a\u908a\u3001\u589e\u522a\u9802\u9ede </p> <p>\u4ee5\u4e0b\u662f\u57fa\u65bc\u9130\u63a5\u77e9\u9663\u8868\u793a\u5716\u7684\u5be6\u73fe\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"\u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        # \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n        self.vertices: list[int] = []\n        # \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n        self.adj_mat: list[list[int]] = []\n        # \u65b0\u589e\u9802\u9ede\n        for val in vertices:\n            self.add_vertex(val)\n        # \u65b0\u589e\u908a\n        # \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u9802\u9ede\u6578\u91cf\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"\u65b0\u589e\u9802\u9ede\"\"\"\n        n = self.size()\n        # \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        self.vertices.append(val)\n        # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"\u522a\u9664\u9802\u9ede\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        self.vertices.pop(index)\n        # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.pop(index)\n        # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"\u65b0\u589e\u908a\"\"\"\n        # \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"\u522a\u9664\u908a\"\"\"\n        # \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"\u5217\u5370\u9130\u63a5\u77e9\u9663\"\"\"\n        print(\"\u9802\u9ede\u4e32\u5217 =\", self.vertices)\n        print(\"\u9130\u63a5\u77e9\u9663 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // \u65b0\u589e\u9802\u9ede\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    void addVertex(int val) {\n        int n = size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.push_back(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"\u9802\u9ede\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.erase(vertices.begin() + index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.erase(adjMat.begin() + index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9802\u9ede\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9802\u9ede\u4e0d\u5b58\u5728\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    void print() {\n        cout &lt;&lt; \"\u9802\u9ede\u4e32\u5217 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"\u9130\u63a5\u77e9\u9663 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // \u65b0\u589e\u9802\u9ede\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public void addVertex(int val) {\n        int n = size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.add(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.remove(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    public void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    public void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    public void print() {\n        System.out.print(\"\u9802\u9ede\u4e32\u5217 = \");\n        System.out.println(vertices);\n        System.out.println(\"\u9130\u63a5\u77e9\u9663 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    List&lt;List&lt;int&gt;&gt; adjMat; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u65b0\u589e\u9802\u9ede\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public void AddVertex(int val) {\n        int n = Size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.Add(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.RemoveAt(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.RemoveAt(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    public void AddEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    public void RemoveEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    public void Print() {\n        Console.Write(\"\u9802\u9ede\u4e32\u5217 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"\u9130\u63a5\u77e9\u9663 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\ntype graphAdjMat struct {\n    // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    vertices []int\n    // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    adjMat [][]int\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // \u65b0\u589e\u9802\u9ede\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // \u521d\u59cb\u5316\u5716\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // \u65b0\u589e\u908a\n    // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* \u65b0\u589e\u9802\u9ede */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n    g.vertices = append(g.vertices, val)\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* \u522a\u9664\u9802\u9ede */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* \u65b0\u589e\u908a */\n// \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* \u522a\u9664\u908a */\n// \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t\u9802\u9ede\u4e32\u5217 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t\u9130\u63a5\u77e9\u9663 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    private var vertices: [Int] // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    private var adjMat: [[Int]] // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // \u65b0\u589e\u9802\u9ede\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    func addVertex(val: Int) {\n        let n = size()\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.append(val)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.remove(at: index)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(at: index)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    func addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    func removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    func print() {\n        Swift.print(\"\u9802\u9ede\u4e32\u5217 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"\u9130\u63a5\u77e9\u9663 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    vertices; // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    adjMat; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u65b0\u589e\u9802\u9ede\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    addVertex(val) {\n        const n = this.size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        this.vertices.splice(index, 1);\n\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    print() {\n        console.log('\u9802\u9ede\u4e32\u5217 = ', this.vertices);\n        console.log('\u9130\u63a5\u77e9\u9663 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    vertices: number[]; // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    adjMat: number[][]; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u65b0\u589e\u9802\u9ede\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        this.vertices.splice(index, 1);\n\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    print(): void {\n        console.log('\u9802\u9ede\u4e32\u5217 = ', this.vertices);\n        console.log('\u9130\u63a5\u77e9\u9663 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // \u9802\u9ede\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //\u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n  /* \u5efa\u69cb\u5b50 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // \u65b0\u589e\u9802\u9ede\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // \u65b0\u589e\u908a\n    // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n  int size() {\n    return vertices.length;\n  }\n\n  /* \u65b0\u589e\u9802\u9ede */\n  void addVertex(int val) {\n    int n = size();\n    // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n    vertices.add(val);\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* \u522a\u9664\u9802\u9ede */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n    vertices.removeAt(index);\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n    adjMat.removeAt(index);\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* \u65b0\u589e\u908a */\n  // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n  void addEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* \u522a\u9664\u908a */\n  // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n  void removeEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n  void printAdjMat() {\n    print(\"\u9802\u9ede\u4e32\u5217 = $vertices\");\n    print(\"\u9130\u63a5\u77e9\u9663 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u578b\u5225 */\npub struct GraphAdjMat {\n    // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    pub vertices: Vec&lt;i32&gt;,\n    // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* \u5efa\u69cb\u5b50 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // \u65b0\u589e\u9802\u9ede\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        self.vertices.push(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        self.adj_mat.push(vec![0; n]);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for row in &amp;mut self.adj_mat {\n            row.push(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"index error\")\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        self.vertices.remove(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.remove(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in &amp;mut self.adj_mat {\n            row.remove(index);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    pub fn print(&amp;self) {\n        println!(\"\u9802\u9ede\u4e32\u5217 = {:?}\", self.vertices);\n        println!(\"\u9130\u63a5\u77e9\u9663 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u7d50\u69cb\u9ad4 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* \u5efa\u69cb\u5b50 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* \u65b0\u589e\u9802\u9ede */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"\u5716\u7684\u9802\u9ede\u6578\u91cf\u5df2\u9054\u6700\u5927\u503c\\n\");\n        return;\n    }\n    // \u65b0\u589e\u7b2c n \u500b\u9802\u9ede\uff0c\u4e26\u5c07\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* \u522a\u9664\u9802\u9ede */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"\u9802\u9ede\u7d22\u5f15\u8d8a\u754c\\n\");\n        return;\n    }\n    // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* \u65b0\u589e\u908a */\n// \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u908a\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* \u522a\u9664\u908a */\n// \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u908a\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"\u9802\u9ede\u4e32\u5217 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"\u9130\u63a5\u77e9\u9663 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices = mutableListOf&lt;Int&gt;() // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    val adjMat = mutableListOf&lt;MutableList&lt;Int&gt;&gt;() // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        // \u65b0\u589e\u9802\u9ede\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    fun addVertex(_val: Int) {\n        val n = size()\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.add(_val)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        val newRow = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size())\n            throw IndexOutOfBoundsException()\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.removeAt(index)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.removeAt(index)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    fun addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    fun removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    fun print() {\n        print(\"\u9802\u9ede\u4e32\u5217 = \")\n        println(vertices)\n        println(\"\u9130\u63a5\u77e9\u9663 =\")\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>### \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 ###\nclass GraphAdjMat\n  def initialize(vertices, edges)\n    ### \u5efa\u69cb\u5b50 ###\n    # \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    @vertices = []\n    # \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    @adj_mat = []\n    # \u65b0\u589e\u9802\u9ede\n    vertices.each { |val| add_vertex(val) }\n    # \u65b0\u589e\u908a\n    # \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    edges.each { |e| add_edge(e[0], e[1]) }\n  end\n\n  ### \u7372\u53d6\u9802\u9ede\u6578\u91cf ###\n  def size\n    @vertices.length\n  end\n\n  ### \u65b0\u589e\u9802\u9ede ###\n  def add_vertex(val)\n    n = size\n    # \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n    @vertices &lt;&lt; val\n    # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n    new_row = Array.new(n, 0)\n    @adj_mat &lt;&lt; new_row\n    # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n    @adj_mat.each { |row| row &lt;&lt; 0 }\n  end\n\n  ### \u522a\u9664\u9802\u9ede ###\n  def remove_vertex(index)\n    raise IndexError if index &gt;= size\n\n    # \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n    @vertices.delete_at(index)\n    # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n    @adj_mat.delete_at(index)\n    # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n    @adj_mat.each { |row| row.delete_at(index) }\n  end\n\n  ### \u65b0\u589e\u908a ###\n  def add_edge(i, j)\n    # \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    # \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    # \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n    @adj_mat[i][j] = 1\n    @adj_mat[j][i] = 1\n  end\n\n  ### \u522a\u9664\u908a ###\n  def remove_edge(i, j)\n    # \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    # \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    @adj_mat[i][j] = 0\n    @adj_mat[j][i] = 0\n  end\n\n  ### \u5217\u5370\u9130\u63a5\u77e9\u9663 ###\n  def __print__\n    puts \"\u9802\u9ede\u4e32\u5217 = #{@vertices}\"\n    puts '\u9130\u63a5\u77e9\u9663 ='\n    print_matrix(@adj_mat)\n  end\nend\n</code></pre> graph_adjacency_matrix.zig<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2 \u00a0 \u57fa\u65bc\u9130\u63a5\u8868\u7684\u5be6\u73fe","text":"<p>\u8a2d\u7121\u5411\u5716\u7684\u9802\u9ede\u7e3d\u6578\u70ba \\(n\\)\u3001\u908a\u7e3d\u6578\u70ba \\(m\\) \uff0c\u5247\u53ef\u6839\u64da\u5716 9-8 \u6240\u793a\u7684\u65b9\u6cd5\u5be6\u73fe\u5404\u7a2e\u64cd\u4f5c\u3002</p> <ul> <li>\u65b0\u589e\u908a\uff1a\u5728\u9802\u9ede\u5c0d\u61c9\u93c8\u7d50\u4e32\u5217\u7684\u672b\u5c3e\u65b0\u589e\u908a\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002\u56e0\u70ba\u662f\u7121\u5411\u5716\uff0c\u6240\u4ee5\u9700\u8981\u540c\u6642\u65b0\u589e\u5169\u500b\u65b9\u5411\u7684\u908a\u3002</li> <li>\u522a\u9664\u908a\uff1a\u5728\u9802\u9ede\u5c0d\u61c9\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u4e26\u522a\u9664\u6307\u5b9a\u908a\uff0c\u4f7f\u7528 \\(O(m)\\) \u6642\u9593\u3002\u5728\u7121\u5411\u5716\u4e2d\uff0c\u9700\u8981\u540c\u6642\u522a\u9664\u5169\u500b\u65b9\u5411\u7684\u908a\u3002</li> <li>\u65b0\u589e\u9802\u9ede\uff1a\u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u93c8\u7d50\u4e32\u5217\uff0c\u4e26\u5c07\u65b0\u589e\u9802\u9ede\u4f5c\u70ba\u93c8\u7d50\u4e32\u5217\u982d\u7bc0\u9ede\uff0c\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002</li> <li>\u522a\u9664\u9802\u9ede\uff1a\u9700\u8d70\u8a2a\u6574\u500b\u9130\u63a5\u8868\uff0c\u522a\u9664\u5305\u542b\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u908a\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u6642\u9593\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u5728\u9130\u63a5\u8868\u4e2d\u5efa\u7acb \\(n\\) \u500b\u9802\u9ede\u548c \\(2m\\) \u689d\u908a\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u6642\u9593\u3002</li> </ul> \u521d\u59cb\u5316\u9130\u63a5\u8868\u65b0\u589e\u908a\u522a\u9664\u908a\u65b0\u589e\u9802\u9ede\u522a\u9664\u9802\u9ede <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 9-8 \u00a0 \u9130\u63a5\u8868\u7684\u521d\u59cb\u5316\u3001\u589e\u522a\u908a\u3001\u589e\u522a\u9802\u9ede </p> <p>\u4ee5\u4e0b\u662f\u9130\u63a5\u8868\u7684\u7a0b\u5f0f\u78bc\u5be6\u73fe\u3002\u5c0d\u6bd4\u5716 9-8 \uff0c\u5be6\u969b\u7a0b\u5f0f\u78bc\u6709\u4ee5\u4e0b\u4e0d\u540c\u3002</p> <ul> <li>\u70ba\u4e86\u65b9\u4fbf\u65b0\u589e\u8207\u522a\u9664\u9802\u9ede\uff0c\u4ee5\u53ca\u7c21\u5316\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u4f7f\u7528\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\u4f86\u4ee3\u66ff\u93c8\u7d50\u4e32\u5217\u3002</li> <li>\u4f7f\u7528\u96dc\u6e4a\u8868\u4f86\u5132\u5b58\u9130\u63a5\u8868\uff0c<code>key</code> \u70ba\u9802\u9ede\u4f8b\u9805\uff0c<code>value</code> \u70ba\u8a72\u9802\u9ede\u7684\u9130\u63a5\u9802\u9ede\u4e32\u5217\uff08\u93c8\u7d50\u4e32\u5217\uff09\u3002</li> </ul> <p>\u53e6\u5916\uff0c\u6211\u5011\u5728\u9130\u63a5\u8868\u4e2d\u4f7f\u7528 <code>Vertex</code> \u985e\u5225\u4f86\u8868\u793a\u9802\u9ede\uff0c\u9019\u6a23\u505a\u7684\u539f\u56e0\u662f\uff1a\u5982\u679c\u8207\u9130\u63a5\u77e9\u9663\u4e00\u6a23\uff0c\u7528\u4e32\u5217\u7d22\u5f15\u4f86\u5340\u5206\u4e0d\u540c\u9802\u9ede\uff0c\u90a3\u9ebc\u5047\u8a2d\u8981\u522a\u9664\u7d22\u5f15\u70ba \\(i\\) \u7684\u9802\u9ede\uff0c\u5247\u9700\u8d70\u8a2a\u6574\u500b\u9130\u63a5\u8868\uff0c\u5c07\u6240\u6709\u5927\u65bc \\(i\\) \u7684\u7d22\u5f15\u5168\u90e8\u6e1b \\(1\\) \uff0c\u6548\u7387\u5f88\u4f4e\u3002\u800c\u5982\u679c\u6bcf\u500b\u9802\u9ede\u90fd\u662f\u552f\u4e00\u7684 <code>Vertex</code> \u4f8b\u9805\uff0c\u522a\u9664\u67d0\u4e00\u9802\u9ede\u4e4b\u5f8c\u5c31\u7121\u9808\u6539\u52d5\u5176\u4ed6\u9802\u9ede\u4e86\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"\u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        # \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u9802\u9ede\u6578\u91cf\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u65b0\u589e\u908a\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u65b0\u589e\u908a vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u522a\u9664\u908a\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u522a\u9664\u908a vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"\u65b0\u589e\u9802\u9ede\"\"\"\n        if vet in self.adj_list:\n            return\n        # \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"\u522a\u9664\u9802\u9ede\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        self.adj_list.pop(vet)\n        # \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"\u5217\u5370\u9130\u63a5\u8868\"\"\"\n        print(\"\u9130\u63a5\u8868 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n  public:\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* \u5728 vector \u4e2d\u522a\u9664\u6307\u5b9a\u7bc0\u9ede */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* \u5efa\u69cb\u5b50 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    int size() {\n        return adjList.size();\n    }\n\n    /* \u65b0\u589e\u908a */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9802\u9ede\");\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9802\u9ede\");\n        // \u522a\u9664\u908a vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9802\u9ede\");\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.erase(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    void print() {\n        cout &lt;&lt; \"\u9130\u63a5\u8868 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u5efa\u69cb\u5b50 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* \u65b0\u589e\u908a */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u522a\u9664\u908a vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.remove(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    public void print() {\n        System.out.println(\"\u9130\u63a5\u8868 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u5efa\u69cb\u5b50 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* \u65b0\u589e\u908a */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u522a\u9664\u908a vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList.Add(vet, []);\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.Remove(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    public void Print() {\n        Console.WriteLine(\"\u9130\u63a5\u8868 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\ntype graphAdjList struct {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    adjList map[Vertex][]Vertex\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* \u65b0\u589e\u908a */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u65b0\u589e\u908a vet1 - vet2, \u65b0\u589e\u533f\u540d struct{},\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u522a\u9664\u908a */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u522a\u9664\u908a vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u65b0\u589e\u9802\u9ede */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u522a\u9664\u9802\u9ede */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n    delete(g.adjList, vet)\n    // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* \u5217\u5370\u9130\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"\u9130\u63a5\u8868 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* \u5efa\u69cb\u5b50 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* \u65b0\u589e\u908a */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c3\u6578\u932f\u8aa4\")\n        }\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* \u522a\u9664\u908a */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c3\u6578\u932f\u8aa4\")\n        }\n        // \u522a\u9664\u908a vet1 - vet2\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList[vet] = []\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"\u53c3\u6578\u932f\u8aa4\")\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.removeValue(forKey: vet)\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    public func print() {\n        Swift.print(\"\u9130\u63a5\u8868 =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    adjList;\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* \u65b0\u589e\u908a */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u65b0\u589e\u908a vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u522a\u9664\u908a vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        this.adjList.set(vet, []);\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        this.adjList.delete(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    print() {\n        console.log('\u9130\u63a5\u8868 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* \u65b0\u589e\u908a */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u65b0\u589e\u908a vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u522a\u9664\u908a vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        this.adjList.set(vet, []);\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        this.adjList.delete(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    print(): void {\n        console.log('\u9130\u63a5\u8868 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n  // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* \u5efa\u69cb\u5b50 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n  int size() {\n    return adjList.length;\n  }\n\n  /* \u65b0\u589e\u908a */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u65b0\u589e\u908a vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* \u522a\u9664\u908a */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u522a\u9664\u908a vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* \u65b0\u589e\u9802\u9ede */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n    adjList[vet] = [];\n  }\n\n  /* \u522a\u9664\u9802\u9ede */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n    adjList.remove(vet);\n    // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* \u5217\u5370\u9130\u63a5\u8868 */\n  void printAdjList() {\n    print(\"\u9130\u63a5\u8868 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u578b\u5225 */\npub struct GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;,\n}\n\nimpl GraphAdjList {\n    /* \u5efa\u69cb\u5b50 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* \u65b0\u589e\u908a */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u65b0\u589e\u908a vet1 - vet2\n        self.adj_list.get_mut(&amp;vet1).unwrap().push(vet2);\n        self.adj_list.get_mut(&amp;vet2).unwrap().push(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u522a\u9664\u908a vet1 - vet2\n        self.adj_list\n            .get_mut(&amp;vet1)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet2);\n        self.adj_list\n            .get_mut(&amp;vet2)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet) {\n            panic!(\"value error\");\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        self.adj_list.remove(&amp;vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    pub fn print(&amp;self) {\n        println!(\"\u9130\u63a5\u8868 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* \u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // \u9802\u9ede\n    struct AdjListNode *next; // \u5f8c\u7e7c\u7bc0\u9ede\n} AdjListNode;\n\n/* \u67e5\u8a62\u9802\u9ede\u5c0d\u61c9\u7684\u7bc0\u9ede */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u65b0\u589e\u908a\u8f14\u52a9\u51fd\u5f0f */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // \u982d\u63d2\u6cd5\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* \u522a\u9664\u908a\u8f14\u52a9\u51fd\u5f0f */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u641c\u7d22 vet \u5c0d\u61c9\u7bc0\u9ede\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // \u5c07 vet \u5c0d\u61c9\u7bc0\u9ede\u5f9e\u93c8\u7d50\u4e32\u5217\u4e2d\u522a\u9664\n    pre-&gt;next = cur-&gt;next;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(cur);\n}\n\n/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // \u7bc0\u9ede\u9663\u5217\n    int size;                     // \u7bc0\u9ede\u6578\u91cf\n} GraphAdjList;\n\n/* \u5efa\u69cb\u5b50 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* \u67e5\u8a62\u9802\u9ede\u5c0d\u61c9\u7684\u7bc0\u9ede */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u65b0\u589e\u908a */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // \u65b0\u589e\u908a vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* \u522a\u9664\u908a */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // \u522a\u9664\u908a vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* \u65b0\u589e\u9802\u9ede */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* \u522a\u9664\u9802\u9ede */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // \u5c07\u8a72\u9802\u9ede\u4e4b\u5f8c\u7684\u9802\u9ede\u5411\u524d\u79fb\u52d5\uff0c\u4ee5\u586b\u88dc\u7a7a\u7f3a\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    val adjList = HashMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt;()\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (edge in edges) {\n            addVertex(edge[0]!!)\n            addVertex(edge[1]!!)\n            addEdge(edge[0]!!, edge[1]!!)\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* \u65b0\u589e\u908a */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1)\n    }\n\n    /* \u522a\u9664\u908a */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u522a\u9664\u908a vet1 - vet2\n        adjList[vet1]?.remove(vet2)\n        adjList[vet2]?.remove(vet1)\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList[vet] = mutableListOf()\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.remove(vet)\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    fun print() {\n        println(\"\u9130\u63a5\u8868 =\")\n        for (pair in adjList.entries) {\n            val tmp = mutableListOf&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex._val)\n            }\n            println(\"${pair.key._val}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>### \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 ###\nclass GraphAdjList\n  attr_reader :adj_list\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(edges)\n    # \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    @adj_list = {}\n    # \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n    for edge in edges\n      add_vertex(edge[0])\n      add_vertex(edge[1])\n      add_edge(edge[0], edge[1])\n    end\n  end\n\n  ### \u7372\u53d6\u9802\u9ede\u6578\u91cf ###\n  def size\n    @adj_list.length\n  end\n\n  ### \u65b0\u589e\u908a ###\n  def add_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    @adj_list[vet1] &lt;&lt; vet2\n    @adj_list[vet2] &lt;&lt; vet1\n  end\n\n  ### \u522a\u9664\u908a ###\n  def remove_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    # \u522a\u9664\u908a vet1 - vet2\n    @adj_list[vet1].delete(vet2)\n    @adj_list[vet2].delete(vet1)\n  end\n\n  ### \u65b0\u589e\u9802\u9ede ###\n  def add_vertex(vet)\n    return if @adj_list.include?(vet)\n\n    # \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n    @adj_list[vet] = []\n  end\n\n  ### \u522a\u9664\u9802\u9ede ###\n  def remove_vertex(vet)\n    raise ArgumentError unless @adj_list.include?(vet)\n\n    # \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n    @adj_list.delete(vet)\n    # \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n    for vertex in @adj_list\n      @adj_list[vertex.first].delete(vet) if @adj_list[vertex.first].include?(vet)\n    end\n  end\n\n  ### \u5217\u5370\u9130\u63a5\u8868 ###\n  def __print__\n    puts '\u9130\u63a5\u8868 ='\n    for vertex in @adj_list\n      tmp = @adj_list[vertex.first].map { |v| v.val }\n      puts \"#{vertex.first.val}: #{tmp},\"\n    end\n  end\nend\n</code></pre> graph_adjacency_list.zig<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3 \u00a0 \u6548\u7387\u5c0d\u6bd4","text":"<p>\u8a2d\u5716\u4e2d\u5171\u6709 \\(n\\) \u500b\u9802\u9ede\u548c \\(m\\) \u689d\u908a\uff0c\u8868 9-2 \u5c0d\u6bd4\u4e86\u9130\u63a5\u77e9\u9663\u548c\u9130\u63a5\u8868\u7684\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u3002</p> <p> \u8868 9-2 \u00a0 \u9130\u63a5\u77e9\u9663\u8207\u9130\u63a5\u8868\u5c0d\u6bd4 </p> \u9130\u63a5\u77e9\u9663 \u9130\u63a5\u8868\uff08\u93c8\u7d50\u4e32\u5217\uff09 \u9130\u63a5\u8868\uff08\u96dc\u6e4a\u8868\uff09 \u5224\u65b7\u662f\u5426\u9130\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u65b0\u589e\u908a \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u522a\u9664\u908a \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u65b0\u589e\u9802\u9ede \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u522a\u9664\u9802\u9ede \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u8a18\u61b6\u9ad4\u7a7a\u9593\u4f54\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>\u89c0\u5bdf\u8868 9-2 \uff0c\u4f3c\u4e4e\u9130\u63a5\u8868\uff08\u96dc\u6e4a\u8868\uff09\u7684\u6642\u9593\u6548\u7387\u8207\u7a7a\u9593\u6548\u7387\u6700\u512a\u3002\u4f46\u5be6\u969b\u4e0a\uff0c\u5728\u9130\u63a5\u77e9\u9663\u4e2d\u64cd\u4f5c\u908a\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u4e00\u6b21\u9663\u5217\u8a2a\u554f\u6216\u8ce6\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7d9c\u5408\u4f86\u770b\uff0c\u9130\u63a5\u77e9\u9663\u9ad4\u73fe\u4e86\u201c\u4ee5\u7a7a\u9593\u63db\u6642\u9593\u201d\u7684\u539f\u5247\uff0c\u800c\u9130\u63a5\u8868\u9ad4\u73fe\u4e86\u201c\u4ee5\u6642\u9593\u63db\u7a7a\u9593\u201d\u7684\u539f\u5247\u3002</p>"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 \u5716\u7684\u8d70\u8a2a","text":"<p>\u6a39\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5c0d\u591a\u201d\u7684\u95dc\u4fc2\uff0c\u800c\u5716\u5247\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5c0d\u591a\u201d\u95dc\u4fc2\u3002\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u6a39\u770b\u4f5c\u5716\u7684\u4e00\u7a2e\u7279\u4f8b\u3002\u986f\u7136\uff0c\u6a39\u7684\u8d70\u8a2a\u64cd\u4f5c\u4e5f\u662f\u5716\u7684\u8d70\u8a2a\u64cd\u4f5c\u7684\u4e00\u7a2e\u7279\u4f8b\u3002</p> <p>\u5716\u548c\u6a39\u90fd\u9700\u8981\u61c9\u7528\u641c\u7d22\u6f14\u7b97\u6cd5\u4f86\u5be6\u73fe\u8d70\u8a2a\u64cd\u4f5c\u3002\u5716\u7684\u8d70\u8a2a\u65b9\u5f0f\u4e5f\u53ef\u5206\u70ba\u5169\u7a2e\uff1a\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u548c\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u3002</p>"},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1 \u00a0 \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a","text":"<p>\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u662f\u4e00\u7a2e\u7531\u8fd1\u53ca\u9060\u7684\u8d70\u8a2a\u65b9\u5f0f\uff0c\u5f9e\u67d0\u500b\u7bc0\u9ede\u51fa\u767c\uff0c\u59cb\u7d42\u512a\u5148\u8a2a\u554f\u8ddd\u96e2\u6700\u8fd1\u7684\u9802\u9ede\uff0c\u4e26\u4e00\u5c64\u5c64\u5411\u5916\u64f4\u5f35\u3002\u5982\u5716 9-9 \u6240\u793a\uff0c\u5f9e\u5de6\u4e0a\u89d2\u9802\u9ede\u51fa\u767c\uff0c\u9996\u5148\u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\uff0c\u7136\u5f8c\u8d70\u8a2a\u4e0b\u4e00\u500b\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9802\u9ede\u8a2a\u554f\u5b8c\u7562\u3002</p> <p></p> <p> \u5716 9-9 \u00a0 \u5716\u7684\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a </p>"},{"location":"chapter_graph/graph_traversal/#1","title":"1. \u00a0 \u6f14\u7b97\u6cd5\u5be6\u73fe","text":"<p>BFS \u901a\u5e38\u85c9\u52a9\u4f47\u5217\u4f86\u5be6\u73fe\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\u3002\u4f47\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8cea\uff0c\u9019\u8207 BFS \u7684\u201c\u7531\u8fd1\u53ca\u9060\u201d\u7684\u601d\u60f3\u7570\u66f2\u540c\u5de5\u3002</p> <ol> <li>\u5c07\u8d70\u8a2a\u8d77\u59cb\u9802\u9ede <code>startVet</code> \u52a0\u5165\u5217\u5217\uff0c\u4e26\u958b\u555f\u8ff4\u5708\u3002</li> <li>\u5728\u8ff4\u5708\u7684\u6bcf\u8f2a\u8fed\u4ee3\u4e2d\uff0c\u5f48\u51fa\u4f47\u5217\u9996\u9802\u9ede\u4e26\u8a18\u9304\u8a2a\u554f\uff0c\u7136\u5f8c\u5c07\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\u52a0\u5165\u5230\u4f47\u5217\u5c3e\u90e8\u3002</li> <li>\u8ff4\u5708\u6b65\u9a5f <code>2.</code> \uff0c\u76f4\u5230\u6240\u6709\u9802\u9ede\u88ab\u8a2a\u554f\u5b8c\u7562\u5f8c\u7d50\u675f\u3002</li> </ol> <p>\u70ba\u4e86\u9632\u6b62\u91cd\u8907\u8d70\u8a2a\u9802\u9ede\uff0c\u6211\u5011\u9700\u8981\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u96c6\u5408 <code>visited</code> \u4f86\u8a18\u9304\u54ea\u4e9b\u7bc0\u9ede\u5df2\u88ab\u8a2a\u554f\u3002</p> <p>Tip</p> <p>\u96dc\u6e4a\u96c6\u5408\u53ef\u4ee5\u770b\u4f5c\u4e00\u500b\u53ea\u5132\u5b58 <code>key</code> \u800c\u4e0d\u5132\u5b58 <code>value</code> \u7684\u96dc\u6e4a\u8868\uff0c\u5b83\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u8907\u96dc\u5ea6\u4e0b\u9032\u884c <code>key</code> \u7684\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u3002\u6839\u64da <code>key</code> \u7684\u552f\u4e00\u6027\uff0c\u96dc\u6e4a\u96c6\u5408\u901a\u5e38\u7528\u65bc\u8cc7\u6599\u53bb\u91cd\u7b49\u5834\u666f\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\"\"\"\n    # \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    # \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res = []\n    # \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    visited = set[Vertex]([start_vet])\n    # \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    que = deque[Vertex]([start_vet])\n    # \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while len(que) &gt; 0:\n        vet = que.popleft()  # \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.append(vet)  # \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        # \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            que.append(adj_vet)  # \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.add(adj_vet)  # \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    # \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.push_back(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            que.push(adjVet);        // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.emplace(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.add(vet);            // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            que.offer(adjVet);   // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.add(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.Add(vet);               // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            que.Enqueue(adjVet);   // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.Add(adjVet);   // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS, \u4f7f\u7528\u5207\u7247\u6a21\u64ec\u4f47\u5217\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    for len(queue) &gt; 0 {\n        // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        vet := queue[0]\n        queue = queue[1:]\n        // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        res = append(res, vet)\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    var que: [Vertex] = [startVet]\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while !que.isEmpty {\n        let vet = que.removeFirst() // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.append(vet) // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            que.append(adjVet) // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.insert(adjVet) // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction graphBFS(graph, startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    const res = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    const visited = new Set();\n    visited.add(startVet);\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    const que = [startVet];\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (que.length) {\n        const vet = que.shift(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            que.push(adjVet); // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.add(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    const que = [startVet];\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (que.length) {\n        const vet = que.shift(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            que.push(adjVet); // \u53ea\u5165\u5217\u672a\u8a2a\u554f\n            visited.add(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n    res.add(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n      }\n      que.add(adjVet); // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n      visited.add(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    }\n  }\n  // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    let mut res = vec![];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while !que.is_empty() {\n        let vet = que.pop_front().unwrap(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n                }\n                que.push_back(adj_vet); // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n                visited.insert(adj_vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n            }\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* \u7bc0\u9ede\u4f47\u5217\u7d50\u69cb\u9ad4 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* \u5efa\u69cb\u5b50 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* \u5165\u5217\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* \u51fa\u5217\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* \u6aa2\u67e5\u9802\u9ede\u662f\u5426\u5df2\u88ab\u8a2a\u554f */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // \u8d70\u8a2a\u67e5\u8a62\u7bc0\u9ede\uff0c\u4f7f\u7528 O(n) \u6642\u9593\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res[(*resSize)++] = vet;      // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n                visited[(*visitedSize)++] = node-&gt;vertex; // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n            }\n            node = node-&gt;next;\n        }\n    }\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): MutableList&lt;Vertex?&gt; {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    val visited = HashSet&lt;Vertex&gt;()\n    visited.add(startVet)\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    val que = LinkedList&lt;Vertex&gt;()\n    que.offer(startVet)\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (!que.isEmpty()) {\n        val vet = que.poll() // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.add(vet)         // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet))\n                continue        // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            que.offer(adjVet)   // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.add(adjVet) // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>### \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a ###\ndef graph_bfs(graph, start_vet)\n  # \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  # \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  res = []\n  # \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n  visited = Set.new([start_vet])\n  # \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n  que = [start_vet]\n  # \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n  while que.length &gt; 0\n    vet = que.shift # \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n    res &lt;&lt; vet # \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    # \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for adj_vet in graph.adj_list[vet]\n      next if visited.include?(adj_vet) # \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n      que &lt;&lt; adj_vet # \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n      visited.add(adj_vet) # \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    end\n  end\n  # \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  res\nend\n</code></pre> graph_bfs.zig<pre><code>[class]{}-[func]{graphBFS}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u7a0b\u5f0f\u78bc\u76f8\u5c0d\u62bd\u8c61\uff0c\u5efa\u8b70\u5c0d\u7167\u5716 9-10 \u4f86\u52a0\u6df1\u7406\u89e3\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 9-10 \u00a0 \u5716\u7684\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u6b65\u9a5f </p> <p>\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0d\u552f\u4e00\u3002\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u9060\u201d\u7684\u9806\u5e8f\u8d70\u8a2a\uff0c\u800c\u591a\u500b\u76f8\u540c\u8ddd\u96e2\u7684\u9802\u9ede\u7684\u8d70\u8a2a\u9806\u5e8f\u5141\u8a31\u88ab\u4efb\u610f\u6253\u4e82\u3002\u4ee5\u5716 9-10 \u70ba\u4f8b\uff0c\u9802\u9ede \\(1\\)\u3001\\(3\\) \u7684\u8a2a\u554f\u9806\u5e8f\u53ef\u4ee5\u4ea4\u63db\uff0c\u9802\u9ede \\(2\\)\u3001\\(4\\)\u3001\\(6\\) \u7684\u8a2a\u554f\u9806\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u63db\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2","title":"2. \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<p>\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u6240\u6709\u9802\u9ede\u90fd\u6703\u5165\u5217\u4e26\u51fa\u968a\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u6642\u9593\uff1b\u5728\u8d70\u8a2a\u9130\u63a5\u9802\u9ede\u7684\u904e\u7a0b\u4e2d\uff0c\u7531\u65bc\u662f\u7121\u5411\u5716\uff0c\u56e0\u6b64\u6240\u6709\u908a\u90fd\u6703\u88ab\u8a2a\u554f \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u6642\u9593\uff1b\u7e3d\u9ad4\u4f7f\u7528 \\(O(|V| + |E|)\\) \u6642\u9593\u3002</p> <p>\u7a7a\u9593\u8907\u96dc\u5ea6\uff1a\u4e32\u5217 <code>res</code> \uff0c\u96dc\u6e4a\u96c6\u5408 <code>visited</code> \uff0c\u4f47\u5217 <code>que</code> \u4e2d\u7684\u9802\u9ede\u6578\u91cf\u6700\u591a\u70ba \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u9593\u3002</p>"},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2 \u00a0 \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a","text":"<p>\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u662f\u4e00\u7a2e\u512a\u5148\u8d70\u5230\u5e95\u3001\u7121\u8def\u53ef\u8d70\u518d\u56de\u982d\u7684\u8d70\u8a2a\u65b9\u5f0f\u3002\u5982\u5716 9-11 \u6240\u793a\uff0c\u5f9e\u5de6\u4e0a\u89d2\u9802\u9ede\u51fa\u767c\uff0c\u8a2a\u554f\u7576\u524d\u9802\u9ede\u7684\u67d0\u500b\u9130\u63a5\u9802\u9ede\uff0c\u76f4\u5230\u8d70\u5230\u76e1\u982d\u6642\u8fd4\u56de\uff0c\u518d\u7e7c\u7e8c\u8d70\u5230\u76e1\u982d\u4e26\u8fd4\u56de\uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9802\u9ede\u8d70\u8a2a\u5b8c\u6210\u3002</p> <p></p> <p> \u5716 9-11 \u00a0 \u5716\u7684\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a </p>"},{"location":"chapter_graph/graph_traversal/#1_1","title":"1. \u00a0 \u6f14\u7b97\u6cd5\u5be6\u73fe","text":"<p>\u9019\u7a2e\u201c\u8d70\u5230\u76e1\u982d\u518d\u8fd4\u56de\u201d\u7684\u6f14\u7b97\u6cd5\u7bc4\u5f0f\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u4f86\u5be6\u73fe\u3002\u8207\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u985e\u4f3c\uff0c\u5728\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u4e2d\uff0c\u6211\u5011\u4e5f\u9700\u8981\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u96c6\u5408 <code>visited</code> \u4f86\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\uff0c\u4ee5\u907f\u514d\u91cd\u8907\u8a2a\u554f\u9802\u9ede\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f\"\"\"\n    res.append(vet)  # \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet)  # \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    # \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        # \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\"\"\"\n    # \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    # \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res = []\n    # \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.emplace(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.Add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede                             \n        }\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append \u64cd\u4f5c\u6703\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u9808\u8b93\u539f\u5f15\u7528\u91cd\u65b0\u8ce6\u503c\u70ba\u65b0slice\u7684\u5f15\u7528\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.insert(vet) // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        }\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        }\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction graphDFS(graph, startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    const res = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        }\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n  visited.add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n  // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n    }\n    // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.insert(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n                         // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    let mut res = vec![];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* \u6aa2\u67e5\u9802\u9ede\u662f\u5426\u5df2\u88ab\u8a2a\u554f */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // \u8d70\u8a2a\u67e5\u8a62\u7bc0\u9ede\uff0c\u4f7f\u7528 O(n) \u6642\u9593\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    res[(*resSize)++] = vet;\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet)     // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet) // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet))\n            continue  // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfun graphDFS(graph: GraphAdjList, startVet: Vertex?): MutableList&lt;Vertex?&gt; {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    val visited = HashSet&lt;Vertex?&gt;()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>### \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f ###\ndef dfs(graph, visited, res, vet)\n  res &lt;&lt; vet # \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n  visited.add(vet) # \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n  # \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  for adj_vet in graph.adj_list[vet]\n    next if visited.include?(adj_vet) # \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n    # \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n    dfs(graph, visited, res, adj_vet)\n  end\nend\n\n### \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a ###\ndef graph_dfs(graph, start_vet)\n  # \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  # \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  res = []\n  # \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n  visited = Set.new\n  dfs(graph, visited, res, start_vet)\n  res\nend\n</code></pre> graph_dfs.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u7684\u6f14\u7b97\u6cd5\u6d41\u7a0b\u5982\u5716 9-12 \u6240\u793a\u3002</p> <ul> <li>\u76f4\u865b\u7dda\u4ee3\u8868\u5411\u4e0b\u905e\u63a8\uff0c\u8868\u793a\u958b\u555f\u4e86\u4e00\u500b\u65b0\u7684\u905e\u8ff4\u65b9\u6cd5\u4f86\u8a2a\u554f\u65b0\u9802\u9ede\u3002</li> <li>\u66f2\u865b\u7dda\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u905e\u8ff4\u65b9\u6cd5\u5df2\u7d93\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u958b\u555f\u6b64\u65b9\u6cd5\u7684\u4f4d\u7f6e\u3002</li> </ul> <p>\u70ba\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8b70\u5c07\u5716 9-12 \u8207\u7a0b\u5f0f\u78bc\u7d50\u5408\u8d77\u4f86\uff0c\u5728\u8166\u4e2d\u6a21\u64ec\uff08\u6216\u8005\u7528\u7b46\u756b\u4e0b\u4f86\uff09\u6574\u500b DFS \u904e\u7a0b\uff0c\u5305\u62ec\u6bcf\u500b\u905e\u8ff4\u65b9\u6cd5\u4f55\u6642\u958b\u555f\u3001\u4f55\u6642\u8fd4\u56de\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 9-12 \u00a0 \u5716\u7684\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u6b65\u9a5f </p> <p>\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u8207\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u985e\u4f3c\uff0c\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u5e8f\u5217\u7684\u9806\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7d66\u5b9a\u67d0\u9802\u9ede\uff0c\u5148\u5f80\u54ea\u500b\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u9130\u63a5\u9802\u9ede\u7684\u9806\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e82\uff0c\u90fd\u662f\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u3002</p> <p>\u4ee5\u6a39\u7684\u8d70\u8a2a\u70ba\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u5225\u5c0d\u61c9\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a\uff0c\u5b83\u5011\u5c55\u793a\u4e86\u4e09\u7a2e\u8d70\u8a2a\u512a\u5148\u9806\u5e8f\uff0c\u7136\u800c\u9019\u4e09\u8005\u90fd\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2_1","title":"2. \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<p>\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u6240\u6709\u9802\u9ede\u90fd\u6703\u88ab\u8a2a\u554f \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u6642\u9593\uff1b\u6240\u6709\u908a\u90fd\u6703\u88ab\u8a2a\u554f \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u6642\u9593\uff1b\u7e3d\u9ad4\u4f7f\u7528 \\(O(|V| + |E|)\\) \u6642\u9593\u3002</p> <p>\u7a7a\u9593\u8907\u96dc\u5ea6\uff1a\u4e32\u5217 <code>res</code> \uff0c\u96dc\u6e4a\u96c6\u5408 <code>visited</code> \u9802\u9ede\u6578\u91cf\u6700\u591a\u70ba \\(|V|\\) \uff0c\u905e\u8ff4\u6df1\u5ea6\u6700\u5927\u70ba \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u9593\u3002</p>"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_graph/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u5716\u7531\u9802\u9ede\u548c\u908a\u7d44\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u70ba\u4e00\u7d44\u9802\u9ede\u548c\u4e00\u7d44\u908a\u69cb\u6210\u7684\u96c6\u5408\u3002</li> <li>\u76f8\u8f03\u65bc\u7dda\u6027\u95dc\u4fc2\uff08\u93c8\u7d50\u4e32\u5217\uff09\u548c\u5206\u6cbb\u95dc\u4fc2\uff08\u6a39\uff09\uff0c\u7db2\u8def\u95dc\u4fc2\uff08\u5716\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u70ba\u8907\u96dc\u3002</li> <li>\u6709\u5411\u5716\u7684\u908a\u5177\u6709\u65b9\u5411\u6027\uff0c\u9023\u901a\u5716\u4e2d\u7684\u4efb\u610f\u9802\u9ede\u5747\u53ef\u9054\uff0c\u6709\u6b0a\u5716\u7684\u6bcf\u689d\u908a\u90fd\u5305\u542b\u6b0a\u91cd\u8b8a\u6578\u3002</li> <li>\u9130\u63a5\u77e9\u9663\u5229\u7528\u77e9\u9663\u4f86\u8868\u793a\u5716\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u500b\u9802\u9ede\uff0c\u77e9\u9663\u5143\u7d20\u4ee3\u8868\u908a\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u5169\u500b\u9802\u9ede\u4e4b\u9593\u6709\u908a\u6216\u7121\u908a\u3002\u9130\u63a5\u77e9\u9663\u5728\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u9593\u4f54\u7528\u8f03\u591a\u3002</li> <li>\u9130\u63a5\u8868\u4f7f\u7528\u591a\u500b\u93c8\u7d50\u4e32\u5217\u4f86\u8868\u793a\u5716\uff0c\u7b2c \\(i\\) \u500b\u93c8\u7d50\u4e32\u5217\u5c0d\u61c9\u9802\u9ede \\(i\\) \uff0c\u5176\u4e2d\u5132\u5b58\u4e86\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\u3002\u9130\u63a5\u8868\u76f8\u5c0d\u65bc\u9130\u63a5\u77e9\u9663\u66f4\u52a0\u7bc0\u7701\u7a7a\u9593\uff0c\u4f46\u7531\u65bc\u9700\u8981\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4f86\u67e5\u8a62\u908a\uff0c\u56e0\u6b64\u6642\u9593\u6548\u7387\u8f03\u4f4e\u3002</li> <li>\u7576\u9130\u63a5\u8868\u4e2d\u7684\u93c8\u7d50\u4e32\u5217\u904e\u9577\u6642\uff0c\u53ef\u4ee5\u5c07\u5176\u8f49\u63db\u70ba\u7d05\u9ed1\u6a39\u6216\u96dc\u6e4a\u8868\uff0c\u5f9e\u800c\u63d0\u5347\u67e5\u8a62\u6548\u7387\u3002</li> <li>\u5f9e\u6f14\u7b97\u6cd5\u601d\u60f3\u7684\u89d2\u5ea6\u5206\u6790\uff0c\u9130\u63a5\u77e9\u9663\u9ad4\u73fe\u4e86\u201c\u4ee5\u7a7a\u9593\u63db\u6642\u9593\u201d\uff0c\u9130\u63a5\u8868\u9ad4\u73fe\u4e86\u201c\u4ee5\u6642\u9593\u63db\u7a7a\u9593\u201d\u3002</li> <li>\u5716\u53ef\u7528\u65bc\u5efa\u6a21\u5404\u985e\u73fe\u5be6\u7cfb\u7d71\uff0c\u5982\u793e\u4ea4\u7db2\u8def\u3001\u5730\u9435\u7dda\u8def\u7b49\u3002</li> <li>\u6a39\u662f\u5716\u7684\u4e00\u7a2e\u7279\u4f8b\uff0c\u6a39\u7684\u8d70\u8a2a\u4e5f\u662f\u5716\u7684\u8d70\u8a2a\u7684\u4e00\u7a2e\u7279\u4f8b\u3002</li> <li>\u5716\u7684\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u662f\u4e00\u7a2e\u7531\u8fd1\u53ca\u9060\u3001\u5c64\u5c64\u64f4\u5f35\u7684\u641c\u5c0b\u65b9\u5f0f\uff0c\u901a\u5e38\u85c9\u52a9\u4f47\u5217\u5be6\u73fe\u3002</li> <li>\u5716\u7684\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u662f\u4e00\u7a2e\u512a\u5148\u8d70\u5230\u5e95\u3001\u7121\u8def\u53ef\u8d70\u6642\u518d\u56de\u6eaf\u7684\u641c\u5c0b\u65b9\u5f0f\uff0c\u5e38\u57fa\u65bc\u905e\u8ff4\u4f86\u5be6\u73fe\u3002</li> </ul>"},{"location":"chapter_graph/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u8def\u5f91\u7684\u5b9a\u7fa9\u662f\u9802\u9ede\u5e8f\u5217\u9084\u662f\u908a\u5e8f\u5217\uff1f</p> <p>\u7dad\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8a9e\u8a00\u7248\u672c\u7684\u5b9a\u7fa9\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f91\u662f\u4e00\u500b\u908a\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f91\u662f\u4e00\u500b\u9802\u9ede\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p> <p>\u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f91\u88ab\u8996\u70ba\u4e00\u500b\u908a\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u500b\u9802\u9ede\u5e8f\u5217\u3002\u9019\u662f\u56e0\u70ba\u5169\u500b\u9802\u9ede\u4e4b\u9593\u53ef\u80fd\u5b58\u5728\u591a\u689d\u908a\u9023\u7dda\uff0c\u6b64\u6642\u6bcf\u689d\u908a\u90fd\u5c0d\u61c9\u4e00\u689d\u8def\u5f91\u3002</p> <p>Q\uff1a\u975e\u9023\u901a\u5716\u4e2d\u662f\u5426\u6703\u6709\u7121\u6cd5\u8d70\u8a2a\u5230\u7684\u9ede\uff1f</p> <p>\u5728\u975e\u9023\u901a\u5716\u4e2d\uff0c\u5f9e\u67d0\u500b\u9802\u9ede\u51fa\u767c\uff0c\u81f3\u5c11\u6709\u4e00\u500b\u9802\u9ede\u7121\u6cd5\u5230\u9054\u3002\u8d70\u8a2a\u975e\u9023\u901a\u5716\u9700\u8981\u8a2d\u5b9a\u591a\u500b\u8d77\u9ede\uff0c\u4ee5\u8d70\u8a2a\u5230\u5716\u7684\u6240\u6709\u9023\u901a\u5206\u91cf\u3002</p> <p>Q\uff1a\u5728\u9130\u63a5\u8868\u4e2d\uff0c\u201c\u8207\u8a72\u9802\u9ede\u76f8\u9023\u7684\u6240\u6709\u9802\u9ede\u201d\u7684\u9802\u9ede\u9806\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f</p> <p>\u53ef\u4ee5\u662f\u4efb\u610f\u9806\u5e8f\u3002\u4f46\u5728\u5be6\u969b\u61c9\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u6309\u7167\u6307\u5b9a\u898f\u5247\u4f86\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9802\u9ede\u65b0\u589e\u7684\u6b21\u5e8f\uff0c\u6216\u8005\u6309\u7167\u9802\u9ede\u503c\u5927\u5c0f\u7684\u9806\u5e8f\u7b49\uff0c\u9019\u6a23\u6709\u52a9\u65bc\u5feb\u901f\u67e5\u8a62\u201c\u5e36\u6709\u67d0\u7a2e\u6975\u503c\u201d\u7684\u9802\u9ede\u3002</p>"},{"location":"chapter_greedy/","title":"\u7b2c 15 \u7ae0 \u00a0 \u8caa\u5a6a","text":"<p>Abstract</p> <p>\u5411\u65e5\u8475\u671d\u8457\u592a\u967d\u8f49\u52d5\uff0c\u6642\u523b\u8ffd\u6c42\u81ea\u8eab\u6210\u9577\u7684\u6700\u5927\u53ef\u80fd\u3002</p> <p>\u8caa\u5a6a\u7b56\u7565\u5728\u4e00\u8f2a\u8f2a\u7684\u7c21\u55ae\u9078\u64c7\u4e2d\uff0c\u9010\u6b65\u5c0e\u5411\u6700\u4f73\u7b54\u6848\u3002</p>"},{"location":"chapter_greedy/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>15.1 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5</li> <li>15.2 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c</li> <li>15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u554f\u984c</li> <li>15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u7a4d\u554f\u984c</li> <li>15.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_greedy/fractional_knapsack_problem/","title":"15.2 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u500b\u7269\u54c1\uff0c\u7b2c \\(i\\) \u500b\u7269\u54c1\u7684\u91cd\u91cf\u70ba \\(wgt[i-1]\\)\u3001\u50f9\u503c\u70ba \\(val[i-1]\\) \uff0c\u548c\u4e00\u500b\u5bb9\u91cf\u70ba \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u500b\u7269\u54c1\u53ea\u80fd\u9078\u64c7\u4e00\u6b21\uff0c\u4f46\u53ef\u4ee5\u9078\u64c7\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u50f9\u503c\u6839\u64da\u9078\u64c7\u7684\u91cd\u91cf\u6bd4\u4f8b\u8a08\u7b97\uff0c\u554f\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u50f9\u503c\u3002\u793a\u4f8b\u5982\u5716 15-3 \u6240\u793a\u3002</p> <p></p> <p> \u5716 15-3 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c\u7684\u793a\u4f8b\u8cc7\u6599 </p> <p>\u5206\u6578\u80cc\u5305\u554f\u984c\u548c 0-1 \u80cc\u5305\u554f\u984c\u6574\u9ad4\u4e0a\u975e\u5e38\u76f8\u4f3c\uff0c\u72c0\u614b\u5305\u542b\u7576\u524d\u7269\u54c1 \\(i\\) \u548c\u5bb9\u91cf \\(c\\) \uff0c\u76ee\u6a19\u662f\u6c42\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u6700\u5927\u50f9\u503c\u3002</p> <p>\u4e0d\u540c\u9ede\u5728\u65bc\uff0c\u672c\u984c\u5141\u8a31\u53ea\u9078\u64c7\u7269\u54c1\u7684\u4e00\u90e8\u5206\u3002\u5982\u5716 15-4 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u5c0d\u7269\u54c1\u4efb\u610f\u5730\u9032\u884c\u5207\u5206\uff0c\u4e26\u6309\u7167\u91cd\u91cf\u6bd4\u4f8b\u4f86\u8a08\u7b97\u76f8\u61c9\u50f9\u503c\u3002</p> <ol> <li>\u5c0d\u65bc\u7269\u54c1 \\(i\\) \uff0c\u5b83\u5728\u55ae\u4f4d\u91cd\u91cf\u4e0b\u7684\u50f9\u503c\u70ba \\(val[i-1] / wgt[i-1]\\) \uff0c\u7c21\u7a31\u55ae\u4f4d\u50f9\u503c\u3002</li> <li>\u5047\u8a2d\u653e\u5165\u4e00\u90e8\u5206\u7269\u54c1 \\(i\\) \uff0c\u91cd\u91cf\u70ba \\(w\\) \uff0c\u5247\u80cc\u5305\u589e\u52a0\u7684\u50f9\u503c\u70ba \\(w \\times val[i-1] / wgt[i-1]\\) \u3002</li> </ol> <p></p> <p> \u5716 15-4 \u00a0 \u7269\u54c1\u5728\u55ae\u4f4d\u91cd\u91cf\u4e0b\u7684\u50f9\u503c </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#1","title":"1. \u00a0 \u8caa\u5a6a\u7b56\u7565\u78ba\u5b9a","text":"<p>\u6700\u5927\u5316\u80cc\u5305\u5167\u7269\u54c1\u7e3d\u50f9\u503c\uff0c\u672c\u8cea\u4e0a\u662f\u6700\u5927\u5316\u55ae\u4f4d\u91cd\u91cf\u4e0b\u7684\u7269\u54c1\u50f9\u503c\u3002\u7531\u6b64\u4fbf\u53ef\u63a8\u7406\u51fa\u5716 15-5 \u6240\u793a\u7684\u8caa\u5a6a\u7b56\u7565\u3002</p> <ol> <li>\u5c07\u7269\u54c1\u6309\u7167\u55ae\u4f4d\u50f9\u503c\u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\u3002</li> <li>\u8d70\u8a2a\u6240\u6709\u7269\u54c1\uff0c\u6bcf\u8f2a\u8caa\u5a6a\u5730\u9078\u64c7\u55ae\u4f4d\u50f9\u503c\u6700\u9ad8\u7684\u7269\u54c1\u3002</li> <li>\u82e5\u5269\u9918\u80cc\u5305\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u4f7f\u7528\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u586b\u6eff\u80cc\u5305\u3002</li> </ol> <p></p> <p> \u5716 15-5 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c\u7684\u8caa\u5a6a\u7b56\u7565 </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u6211\u5011\u5efa\u7acb\u4e86\u4e00\u500b\u7269\u54c1\u985e\u5225 <code>Item</code> \uff0c\u4ee5\u4fbf\u5c07\u7269\u54c1\u6309\u7167\u55ae\u4f4d\u50f9\u503c\u9032\u884c\u6392\u5e8f\u3002\u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u7576\u80cc\u5305\u5df2\u6eff\u6642\u8df3\u51fa\u4e26\u8fd4\u56de\u89e3\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig fractional_knapsack.py<pre><code>class Item:\n    \"\"\"\u7269\u54c1\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # \u7269\u54c1\u91cd\u91cf\n        self.v = v  # \u7269\u54c1\u50f9\u503c\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a\"\"\"\n    # \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v\n            cap -= item.w\n        else:\n            # \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (item.v / item.w) * cap\n            # \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* \u7269\u54c1 */\nclass Item {\n  public:\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u50f9\u503c\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* \u7269\u54c1 */\nclass Item {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u50f9\u503c\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (double) item.v / item.w * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* \u7269\u54c1 */\nclass Item(int w, int v) {\n    public int w = w; // \u7269\u54c1\u91cd\u91cf\n    public int v = v; // \u7269\u54c1\u50f9\u503c\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* \u7269\u54c1 */\ntype Item struct {\n    w int // \u7269\u54c1\u91cd\u91cf\n    v int // \u7269\u54c1\u50f9\u503c\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* \u7269\u54c1 */\nclass Item {\n    var w: Int // \u7269\u54c1\u91cd\u91cf\n    var v: Int // \u7269\u54c1\u50f9\u503c\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) }\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* \u7269\u54c1 */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // \u7269\u54c1\u91cd\u91cf\n        this.v = v; // \u7269\u54c1\u50f9\u503c\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* \u7269\u54c1 */\nclass Item {\n    w: number; // \u7269\u54c1\u91cd\u91cf\n    v: number; // \u7269\u54c1\u50f9\u503c\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* \u7269\u54c1 */\nclass Item {\n  int w; // \u7269\u54c1\u91cd\u91cf\n  int v; // \u7269\u54c1\u50f9\u503c\n\n  Item(this.w, this.v);\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n      res += item.v / item.w * cap;\n      // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* \u7269\u54c1 */\nstruct Item {\n    w: i32, // \u7269\u54c1\u91cd\u91cf\n    v: i32, // \u7269\u54c1\u50f9\u503c\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* \u7269\u54c1 */\ntypedef struct {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u50f9\u503c\n} Item;\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.kt<pre><code>/* \u7269\u54c1 */\nclass Item(\n    val w: Int, // \u7269\u54c1\n    val v: Int  // \u7269\u54c1\u50f9\u503c\n)\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfun fractionalKnapsack(wgt: IntArray, _val: IntArray, c: Int): Double {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], _val[i])\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sortBy { item: Item? -&gt; -(item!!.v.toDouble() / item.w) }\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += item.v.toDouble() / item.w * cap\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.rb<pre><code>### \u7269\u54c1 ###\nclass Item\n  attr_accessor :w # \u7269\u54c1\u91cd\u91cf\n  attr_accessor :v # \u7269\u54c1\u50f9\u503c\n\n  def initialize(w, v)\n    @w = w\n    @v = v\n  end\nend\n\n### \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a ###\ndef fractional_knapsack(wgt, val, cap)\n  # \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\uff0c\u50f9\u503c\n  items = wgt.each_with_index.map { |w, i| Item.new(w, val[i]) }\n  # \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n  items.sort! { |a, b| (b.v.to_f / b.w) &lt;=&gt; (a.v.to_f / a.w) }\n  # \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n  res = 0\n  for item in items\n    if item.w &lt;= cap\n      # \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n      res += item.v\n      cap -= item.w\n    else\n      # \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n      res += (item.v.to_f / item.w) * cap\n      # \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n      break\n    end\n  end\n  res\nend\n</code></pre> fractional_knapsack.zig<pre><code>[class]{Item}-[func]{}\n\n[class]{}-[func]{fractionalKnapsack}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u9664\u6392\u5e8f\u4e4b\u5916\uff0c\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u9700\u8981\u8d70\u8a2a\u6574\u500b\u7269\u54c1\u4e32\u5217\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u70ba\u7269\u54c1\u6578\u91cf\u3002</p> <p>\u7531\u65bc\u521d\u59cb\u5316\u4e86\u4e00\u500b <code>Item</code> \u7269\u4ef6\u4e32\u5217\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#3","title":"3. \u00a0 \u6b63\u78ba\u6027\u8b49\u660e","text":"<p>\u63a1\u7528\u53cd\u8b49\u6cd5\u3002\u5047\u8a2d\u7269\u54c1 \\(x\\) \u662f\u55ae\u4f4d\u50f9\u503c\u6700\u9ad8\u7684\u7269\u54c1\uff0c\u4f7f\u7528\u67d0\u6f14\u7b97\u6cd5\u6c42\u5f97\u6700\u5927\u50f9\u503c\u70ba <code>res</code> \uff0c\u4f46\u8a72\u89e3\u4e2d\u4e0d\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u73fe\u5728\u5f9e\u80cc\u5305\u4e2d\u62ff\u51fa\u55ae\u4f4d\u91cd\u91cf\u7684\u4efb\u610f\u7269\u54c1\uff0c\u4e26\u66ff\u63db\u70ba\u55ae\u4f4d\u91cd\u91cf\u7684\u7269\u54c1 \\(x\\) \u3002\u7531\u65bc\u7269\u54c1 \\(x\\) \u7684\u55ae\u4f4d\u50f9\u503c\u6700\u9ad8\uff0c\u56e0\u6b64\u66ff\u63db\u5f8c\u7684\u7e3d\u50f9\u503c\u4e00\u5b9a\u5927\u65bc <code>res</code> \u3002\u9019\u8207 <code>res</code> \u662f\u6700\u512a\u89e3\u77db\u76fe\uff0c\u8aaa\u660e\u6700\u512a\u89e3\u4e2d\u5fc5\u9808\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u5c0d\u65bc\u8a72\u89e3\u4e2d\u7684\u5176\u4ed6\u7269\u54c1\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u69cb\u5efa\u51fa\u4e0a\u8ff0\u77db\u76fe\u3002\u7e3d\u800c\u8a00\u4e4b\uff0c\u55ae\u4f4d\u50f9\u503c\u66f4\u5927\u7684\u7269\u54c1\u7e3d\u662f\u66f4\u512a\u9078\u64c7\uff0c\u9019\u8aaa\u660e\u8caa\u5a6a\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p> <p>\u5982\u5716 15-6 \u6240\u793a\uff0c\u5982\u679c\u5c07\u7269\u54c1\u91cd\u91cf\u548c\u7269\u54c1\u55ae\u4f4d\u50f9\u503c\u5206\u5225\u770b\u4f5c\u4e00\u5f35\u4e8c\u7dad\u5716\u8868\u7684\u6a6b\u8ef8\u548c\u7e31\u8ef8\uff0c\u5247\u5206\u6578\u80cc\u5305\u554f\u984c\u53ef\u8f49\u5316\u70ba\u201c\u6c42\u5728\u6709\u9650\u6a6b\u8ef8\u5340\u9593\u4e0b\u570d\u6210\u7684\u6700\u5927\u9762\u7a4d\u201d\u3002\u9019\u500b\u985e\u6bd4\u53ef\u4ee5\u5e6b\u52a9\u6211\u5011\u5f9e\u5e7e\u4f55\u89d2\u5ea6\u7406\u89e3\u8caa\u5a6a\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</p> <p></p> <p> \u5716 15-6 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c\u7684\u5e7e\u4f55\u8868\u793a </p>"},{"location":"chapter_greedy/greedy_algorithm/","title":"15.1 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5","text":"<p>\u8caa\u5a6a\u6f14\u7b97\u6cd5\uff08greedy algorithm\uff09\u662f\u4e00\u7a2e\u5e38\u898b\u7684\u89e3\u6c7a\u6700\u4f73\u5316\u554f\u984c\u7684\u6f14\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u554f\u984c\u7684\u6bcf\u500b\u6c7a\u7b56\u968e\u6bb5\uff0c\u90fd\u9078\u64c7\u7576\u524d\u770b\u8d77\u4f86\u6700\u512a\u7684\u9078\u64c7\uff0c\u5373\u8caa\u5a6a\u5730\u505a\u51fa\u5340\u57df\u6027\u6700\u512a\u7684\u6c7a\u7b56\uff0c\u4ee5\u671f\u7372\u5f97\u5168\u57df\u6027\u6700\u512a\u89e3\u3002\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7c21\u6f54\u4e14\u9ad8\u6548\uff0c\u5728\u8a31\u591a\u5be6\u969b\u554f\u984c\u4e2d\u6709\u8457\u5ee3\u6cdb\u7684\u61c9\u7528\u3002</p> <p>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u548c\u52d5\u614b\u898f\u5283\u90fd\u5e38\u7528\u65bc\u89e3\u6c7a\u6700\u4f73\u5316\u554f\u984c\u3002\u5b83\u5011\u4e4b\u9593\u5b58\u5728\u4e00\u4e9b\u76f8\u4f3c\u4e4b\u8655\uff0c\u6bd4\u5982\u90fd\u4f9d\u8cf4\u6700\u512a\u5b50\u7d50\u69cb\u6027\u8cea\uff0c\u4f46\u5de5\u4f5c\u539f\u7406\u4e0d\u540c\u3002</p> <ul> <li>\u52d5\u614b\u898f\u5283\u6703\u6839\u64da\u4e4b\u524d\u968e\u6bb5\u7684\u6240\u6709\u6c7a\u7b56\u4f86\u8003\u616e\u7576\u524d\u6c7a\u7b56\uff0c\u4e26\u4f7f\u7528\u904e\u53bb\u5b50\u554f\u984c\u7684\u89e3\u4f86\u69cb\u5efa\u7576\u524d\u5b50\u554f\u984c\u7684\u89e3\u3002</li> <li>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u4e0d\u6703\u8003\u616e\u904e\u53bb\u7684\u6c7a\u7b56\uff0c\u800c\u662f\u4e00\u8def\u5411\u524d\u5730\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u4e0d\u65b7\u7e2e\u5c0f\u554f\u984c\u7bc4\u570d\uff0c\u76f4\u81f3\u554f\u984c\u88ab\u89e3\u6c7a\u3002</li> </ul> <p>\u6211\u5011\u5148\u900f\u904e\u4f8b\u984c\u201c\u96f6\u9322\u514c\u63db\u201d\u77ad\u89e3\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002\u9019\u9053\u984c\u5df2\u7d93\u5728\u201c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u201d\u7ae0\u7bc0\u4e2d\u4ecb\u7d39\u904e\uff0c\u76f8\u4fe1\u4f60\u5c0d\u5b83\u4e26\u4e0d\u964c\u751f\u3002</p> <p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u7a2e\u786c\u5e63\uff0c\u7b2c \\(i\\) \u7a2e\u786c\u5e63\u7684\u9762\u503c\u70ba \\(coins[i - 1]\\) \uff0c\u76ee\u6a19\u91d1\u984d\u70ba \\(amt\\) \uff0c\u6bcf\u7a2e\u786c\u5e63\u53ef\u4ee5\u91cd\u8907\u9078\u53d6\uff0c\u554f\u80fd\u5920\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u7684\u6700\u5c11\u786c\u5e63\u6578\u91cf\u3002\u5982\u679c\u7121\u6cd5\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u672c\u984c\u63a1\u53d6\u7684\u8caa\u5a6a\u7b56\u7565\u5982\u5716 15-1 \u6240\u793a\u3002\u7d66\u5b9a\u76ee\u6a19\u91d1\u984d\uff0c\u6211\u5011\u8caa\u5a6a\u5730\u9078\u64c7\u4e0d\u5927\u65bc\u4e14\u6700\u63a5\u8fd1\u5b83\u7684\u786c\u5e63\uff0c\u4e0d\u65b7\u8ff4\u5708\u8a72\u6b65\u9a5f\uff0c\u76f4\u81f3\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u70ba\u6b62\u3002</p> <p></p> <p> \u5716 15-1 \u00a0 \u96f6\u9322\u514c\u63db\u7684\u8caa\u5a6a\u7b56\u7565 </p> <p>\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a\"\"\"\n    # \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    i = len(coins) - 1\n    count = 0\n    # \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while amt &gt; 0:\n        # \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # \u9078\u64c7 coins[i]\n        amt -= coins[i]\n        count += 1\n    # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    int i = coins.size() - 1;\n    int count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint coinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    int i = coins.length - 1;\n    int count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    int i = coins.Length - 1;\n    int count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    i := len(coins) - 1\n    count := 0\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    for amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i]\n        count += 1\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfunction coinChangeGreedy(coins, amt) {\n    // \u5047\u8a2d coins \u9663\u5217\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // \u5047\u8a2d coins \u9663\u5217\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n  int i = coins.length - 1;\n  int count = 0;\n  // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n  while (amt &gt; 0) {\n    // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // \u9078\u64c7 coins[i]\n    amt -= coins[i];\n    count++;\n  }\n  // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count += 1;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    int i = size - 1;\n    int count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.kt<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfun coinChangeGreedy(coins: IntArray, amt: Int): Int {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    var am = amt\n    var i = coins.size - 1\n    var count = 0\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (am &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; am) {\n            i--\n        }\n        // \u9078\u64c7 coins[i]\n        am -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return if (am == 0) count else -1\n}\n</code></pre> coin_change_greedy.rb<pre><code>### \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a ###\ndef coin_change_greedy(coins, amt)\n  # \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n  i = coins.length - 1\n  count = 0\n  # \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n  while amt &gt; 0\n    # \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n    while i &gt; 0 &amp;&amp; coins[i] &gt; amt\n      i -= 1\n    end\n    # \u9078\u64c7 coins[i]\n    amt -= coins[i]\n    count += 1\n  end\n  # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c \u5247\u8fd4\u56de -1\n  amt == 0 ? count : -1\nend\n</code></pre> coin_change_greedy.zig<pre><code>[class]{}-[func]{coinChangeGreedy}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u4f60\u53ef\u80fd\u6703\u4e0d\u7531\u5730\u767c\u51fa\u611f\u5606\uff1aSo clean \uff01\u8caa\u5a6a\u6f14\u7b97\u6cd5\u50c5\u7528\u7d04\u5341\u884c\u7a0b\u5f0f\u78bc\u5c31\u89e3\u6c7a\u4e86\u96f6\u9322\u514c\u63db\u554f\u984c\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1511","title":"15.1.1 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u4e0d\u50c5\u64cd\u4f5c\u76f4\u63a5\u3001\u5be6\u73fe\u7c21\u55ae\uff0c\u800c\u4e14\u901a\u5e38\u6548\u7387\u4e5f\u5f88\u9ad8\u3002\u5728\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u8a18\u786c\u5e63\u6700\u5c0f\u9762\u503c\u70ba \\(\\min(coins)\\) \uff0c\u5247\u8caa\u5a6a\u9078\u64c7\u6700\u591a\u8ff4\u5708 \\(amt / \\min(coins)\\) \u6b21\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(amt / \\min(coins))\\) \u3002\u9019\u6bd4\u52d5\u614b\u898f\u5283\u89e3\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n \\times amt)\\) \u5c0f\u4e86\u4e00\u500b\u6578\u91cf\u7d1a\u3002</p> <p>\u7136\u800c\uff0c\u5c0d\u65bc\u67d0\u4e9b\u786c\u5e63\u9762\u503c\u7d44\u5408\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u4e26\u4e0d\u80fd\u627e\u5230\u6700\u512a\u89e3\u3002\u5716 15-2 \u7d66\u51fa\u4e86\u5169\u500b\u793a\u4f8b\u3002</p> <ul> <li>\u6b63\u4f8b \\(coins = [1, 5, 10, 20, 50, 100]\\)\uff1a\u5728\u8a72\u786c\u5e63\u7d44\u5408\u4e0b\uff0c\u7d66\u5b9a\u4efb\u610f \\(amt\\) \uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u90fd\u53ef\u4ee5\u627e\u5230\u6700\u512a\u89e3\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 20, 50]\\)\uff1a\u5047\u8a2d \\(amt = 60\\) \uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 10\\) \u7684\u514c\u63db\u7d44\u5408\uff0c\u5171\u8a08 \\(11\\) \u679a\u786c\u5e63\uff0c\u4f46\u52d5\u614b\u898f\u5283\u53ef\u4ee5\u627e\u5230\u6700\u512a\u89e3 \\(20 + 20 + 20\\) \uff0c\u50c5\u9700 \\(3\\) \u679a\u786c\u5e63\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 49, 50]\\)\uff1a\u5047\u8a2d \\(amt = 98\\) \uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 48\\) \u7684\u514c\u63db\u7d44\u5408\uff0c\u5171\u8a08 \\(49\\) \u679a\u786c\u5e63\uff0c\u4f46\u52d5\u614b\u898f\u5283\u53ef\u4ee5\u627e\u5230\u6700\u512a\u89e3 \\(49 + 49\\) \uff0c\u50c5\u9700 \\(2\\) \u679a\u786c\u5e63\u3002</li> </ul> <p></p> <p> \u5716 15-2 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u7121\u6cd5\u627e\u51fa\u6700\u512a\u89e3\u7684\u793a\u4f8b </p> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u5c0d\u65bc\u96f6\u9322\u514c\u63db\u554f\u984c\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7121\u6cd5\u4fdd\u8b49\u627e\u5230\u5168\u57df\u6027\u6700\u512a\u89e3\uff0c\u4e26\u4e14\u6709\u53ef\u80fd\u627e\u5230\u975e\u5e38\u5dee\u7684\u89e3\u3002\u5b83\u66f4\u9069\u5408\u7528\u52d5\u614b\u898f\u5283\u89e3\u6c7a\u3002</p> <p>\u4e00\u822c\u60c5\u6cc1\u4e0b\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u9069\u7528\u60c5\u6cc1\u5206\u4ee5\u4e0b\u5169\u7a2e\u3002</p> <ol> <li>\u53ef\u4ee5\u4fdd\u8b49\u627e\u5230\u6700\u512a\u89e3\uff1a\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\u5f80\u5f80\u662f\u6700\u512a\u9078\u64c7\uff0c\u56e0\u70ba\u5b83\u5f80\u5f80\u6bd4\u56de\u6eaf\u3001\u52d5\u614b\u898f\u5283\u66f4\u9ad8\u6548\u3002</li> <li>\u53ef\u4ee5\u627e\u5230\u8fd1\u4f3c\u6700\u512a\u89e3\uff1a\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\u4e5f\u662f\u53ef\u7528\u7684\u3002\u5c0d\u65bc\u5f88\u591a\u8907\u96dc\u554f\u984c\u4f86\u8aaa\uff0c\u5c0b\u627e\u5168\u57df\u6027\u6700\u512a\u89e3\u975e\u5e38\u56f0\u96e3\uff0c\u80fd\u4ee5\u8f03\u9ad8\u6548\u7387\u627e\u5230\u6b21\u512a\u89e3\u4e5f\u662f\u975e\u5e38\u4e0d\u932f\u7684\u3002</li> </ol>"},{"location":"chapter_greedy/greedy_algorithm/#1512","title":"15.1.2 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u7279\u6027","text":"<p>\u90a3\u9ebc\u554f\u984c\u4f86\u4e86\uff0c\u4ec0\u9ebc\u6a23\u7684\u554f\u984c\u9069\u5408\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6c42\u89e3\u5462\uff1f\u6216\u8005\u8aaa\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5728\u4ec0\u9ebc\u60c5\u6cc1\u4e0b\u53ef\u4ee5\u4fdd\u8b49\u627e\u5230\u6700\u512a\u89e3\uff1f</p> <p>\u76f8\u8f03\u65bc\u52d5\u614b\u898f\u5283\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u4f7f\u7528\u689d\u4ef6\u66f4\u52a0\u82db\u523b\uff0c\u5176\u4e3b\u8981\u95dc\u6ce8\u554f\u984c\u7684\u5169\u500b\u6027\u8cea\u3002</p> <ul> <li>\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\uff1a\u53ea\u6709\u7576\u5c40\u90e8\u6700\u512a\u9078\u64c7\u59cb\u7d42\u53ef\u4ee5\u5c0e\u81f4\u5168\u57df\u6027\u6700\u512a\u89e3\u6642\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u624d\u80fd\u4fdd\u8b49\u5f97\u5230\u6700\u512a\u89e3\u3002</li> <li>\u6700\u512a\u5b50\u7d50\u69cb\uff1a\u539f\u554f\u984c\u7684\u6700\u512a\u89e3\u5305\u542b\u5b50\u554f\u984c\u7684\u6700\u512a\u89e3\u3002</li> </ul> <p>\u6700\u512a\u5b50\u7d50\u69cb\u5df2\u7d93\u5728\u201c\u52d5\u614b\u898f\u5283\u201d\u7ae0\u7bc0\u4e2d\u4ecb\u7d39\u904e\uff0c\u9019\u88e1\u4e0d\u518d\u8d05\u8ff0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u554f\u984c\u7684\u6700\u512a\u5b50\u7d50\u69cb\u4e26\u4e0d\u660e\u986f\uff0c\u4f46\u4ecd\u7136\u53ef\u4f7f\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u89e3\u6c7a\u3002</p> <p>\u6211\u5011\u4e3b\u8981\u63a2\u7a76\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u7684\u5224\u65b7\u65b9\u6cd5\u3002\u96d6\u7136\u5b83\u7684\u63cf\u8ff0\u770b\u4e0a\u53bb\u6bd4\u8f03\u7c21\u55ae\uff0c\u4f46\u5be6\u969b\u4e0a\u5c0d\u65bc\u8a31\u591a\u554f\u984c\uff0c\u8b49\u660e\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u4e26\u975e\u6613\u4e8b\u3002</p> <p>\u4f8b\u5982\u96f6\u9322\u514c\u63db\u554f\u984c\uff0c\u6211\u5011\u96d6\u7136\u80fd\u5920\u5bb9\u6613\u5730\u8209\u51fa\u53cd\u4f8b\uff0c\u5c0d\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u9032\u884c\u8b49\u507d\uff0c\u4f46\u8b49\u5be6\u7684\u96e3\u5ea6\u8f03\u5927\u3002\u5982\u679c\u554f\uff1a\u6eff\u8db3\u4ec0\u9ebc\u689d\u4ef6\u7684\u786c\u5e63\u7d44\u5408\u53ef\u4ee5\u4f7f\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6c42\u89e3\uff1f\u6211\u5011\u5f80\u5f80\u53ea\u80fd\u6191\u85c9\u76f4\u89ba\u6216\u8209\u4f8b\u5b50\u4f86\u7d66\u51fa\u4e00\u500b\u6a21\u7a1c\u5169\u53ef\u7684\u7b54\u6848\uff0c\u800c\u96e3\u4ee5\u7d66\u51fa\u56b4\u8b39\u7684\u6578\u5b78\u8b49\u660e\u3002</p> <p>Quote</p> <p>\u6709\u4e00\u7bc7\u8ad6\u6587\u7d66\u51fa\u4e86\u4e00\u500b \\(O(n^3)\\) \u6642\u9593\u8907\u96dc\u5ea6\u7684\u6f14\u7b97\u6cd5\uff0c\u7528\u65bc\u5224\u65b7\u4e00\u500b\u786c\u5e63\u7d44\u5408\u80fd\u5426\u4f7f\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u627e\u51fa\u4efb\u610f\u91d1\u984d\u7684\u6700\u512a\u89e3\u3002</p> <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>"},{"location":"chapter_greedy/greedy_algorithm/#1513","title":"15.1.3 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u89e3\u984c\u6b65\u9a5f","text":"<p>\u8caa\u5a6a\u554f\u984c\u7684\u89e3\u6c7a\u6d41\u7a0b\u5927\u9ad4\u53ef\u5206\u70ba\u4ee5\u4e0b\u4e09\u6b65\u3002</p> <ol> <li>\u554f\u984c\u5206\u6790\uff1a\u68b3\u7406\u8207\u7406\u89e3\u554f\u984c\u7279\u6027\uff0c\u5305\u62ec\u72c0\u614b\u5b9a\u7fa9\u3001\u6700\u4f73\u5316\u76ee\u6a19\u548c\u7d04\u675f\u689d\u4ef6\u7b49\u3002\u9019\u4e00\u6b65\u5728\u56de\u6eaf\u548c\u52d5\u614b\u898f\u5283\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</li> <li>\u78ba\u5b9a\u8caa\u5a6a\u7b56\u7565\uff1a\u78ba\u5b9a\u5982\u4f55\u5728\u6bcf\u4e00\u6b65\u4e2d\u505a\u51fa\u8caa\u5a6a\u9078\u64c7\u3002\u9019\u500b\u7b56\u7565\u80fd\u5920\u5728\u6bcf\u4e00\u6b65\u6e1b\u5c0f\u554f\u984c\u7684\u898f\u6a21\uff0c\u4e26\u6700\u7d42\u89e3\u6c7a\u6574\u500b\u554f\u984c\u3002</li> <li>\u6b63\u78ba\u6027\u8b49\u660e\uff1a\u901a\u5e38\u9700\u8981\u8b49\u660e\u554f\u984c\u5177\u6709\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u548c\u6700\u512a\u5b50\u7d50\u69cb\u3002\u9019\u500b\u6b65\u9a5f\u53ef\u80fd\u9700\u8981\u7528\u5230\u6578\u5b78\u8b49\u660e\uff0c\u4f8b\u5982\u6b78\u7d0d\u6cd5\u6216\u53cd\u8b49\u6cd5\u7b49\u3002</li> </ol> <p>\u78ba\u5b9a\u8caa\u5a6a\u7b56\u7565\u662f\u6c42\u89e3\u554f\u984c\u7684\u6838\u5fc3\u6b65\u9a5f\uff0c\u4f46\u5be6\u65bd\u8d77\u4f86\u53ef\u80fd\u4e26\u4e0d\u5bb9\u6613\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e0d\u540c\u554f\u984c\u7684\u8caa\u5a6a\u7b56\u7565\u7684\u5dee\u7570\u8f03\u5927\u3002\u5c0d\u65bc\u8a31\u591a\u554f\u984c\u4f86\u8aaa\uff0c\u8caa\u5a6a\u7b56\u7565\u6bd4\u8f03\u6dfa\u986f\uff0c\u6211\u5011\u900f\u904e\u4e00\u4e9b\u5927\u6982\u7684\u601d\u8003\u8207\u5617\u8a66\u5c31\u80fd\u5f97\u51fa\u3002\u800c\u5c0d\u65bc\u4e00\u4e9b\u8907\u96dc\u554f\u984c\uff0c\u8caa\u5a6a\u7b56\u7565\u53ef\u80fd\u975e\u5e38\u96b1\u853d\uff0c\u9019\u7a2e\u60c5\u6cc1\u5c31\u975e\u5e38\u8003\u9a57\u500b\u4eba\u7684\u89e3\u984c\u7d93\u9a57\u8207\u6f14\u7b97\u6cd5\u80fd\u529b\u4e86\u3002</li> <li>\u67d0\u4e9b\u8caa\u5a6a\u7b56\u7565\u5177\u6709\u8f03\u5f37\u7684\u8ff7\u60d1\u6027\u3002\u7576\u6211\u5011\u6eff\u61f7\u4fe1\u5fc3\u8a2d\u8a08\u597d\u8caa\u5a6a\u7b56\u7565\uff0c\u5beb\u51fa\u89e3\u984c\u7a0b\u5f0f\u78bc\u4e26\u63d0\u4ea4\u57f7\u884c\uff0c\u5f88\u53ef\u80fd\u767c\u73fe\u90e8\u5206\u6e2c\u8a66\u6a23\u4f8b\u7121\u6cd5\u900f\u904e\u3002\u9019\u662f\u56e0\u70ba\u8a2d\u8a08\u7684\u8caa\u5a6a\u7b56\u7565\u53ea\u662f\u201c\u90e8\u5206\u6b63\u78ba\u201d\u7684\uff0c\u4e0a\u6587\u4ecb\u7d39\u7684\u96f6\u9322\u514c\u63db\u5c31\u662f\u4e00\u500b\u5178\u578b\u6848\u4f8b\u3002</li> </ul> <p>\u70ba\u4e86\u4fdd\u8b49\u6b63\u78ba\u6027\uff0c\u6211\u5011\u61c9\u8a72\u5c0d\u8caa\u5a6a\u7b56\u7565\u9032\u884c\u56b4\u8b39\u7684\u6578\u5b78\u8b49\u660e\uff0c\u901a\u5e38\u9700\u8981\u7528\u5230\u53cd\u8b49\u6cd5\u6216\u6578\u5b78\u6b78\u7d0d\u6cd5\u3002</p> <p>\u7136\u800c\uff0c\u6b63\u78ba\u6027\u8b49\u660e\u4e5f\u5f88\u53ef\u80fd\u4e0d\u662f\u4e00\u4ef6\u6613\u4e8b\u3002\u5982\u82e5\u6c92\u6709\u982d\u7dd2\uff0c\u6211\u5011\u901a\u5e38\u6703\u9078\u64c7\u9762\u5411\u6e2c\u8a66\u7528\u4f8b\u9032\u884c\u7a0b\u5f0f\u78bc\u9664\u932f\uff0c\u4e00\u6b65\u6b65\u4fee\u6539\u8207\u9a57\u8b49\u8caa\u5a6a\u7b56\u7565\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1514","title":"15.1.4 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u5178\u578b\u4f8b\u984c","text":"<p>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5e38\u5e38\u61c9\u7528\u5728\u6eff\u8db3\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u548c\u6700\u512a\u5b50\u7d50\u69cb\u7684\u6700\u4f73\u5316\u554f\u984c\u4e2d\uff0c\u4ee5\u4e0b\u5217\u8209\u4e86\u4e00\u4e9b\u5178\u578b\u7684\u8caa\u5a6a\u6f14\u7b97\u6cd5\u554f\u984c\u3002</p> <ul> <li>\u786c\u5e63\u627e\u96f6\u554f\u984c\uff1a\u5728\u67d0\u4e9b\u786c\u5e63\u7d44\u5408\u4e0b\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7e3d\u662f\u53ef\u4ee5\u5f97\u5230\u6700\u512a\u89e3\u3002</li> <li>\u5340\u9593\u6392\u7a0b\u554f\u984c\uff1a\u5047\u8a2d\u4f60\u6709\u4e00\u4e9b\u4efb\u52d9\uff0c\u6bcf\u500b\u4efb\u52d9\u5728\u4e00\u6bb5\u6642\u9593\u5167\u9032\u884c\uff0c\u4f60\u7684\u76ee\u6a19\u662f\u5b8c\u6210\u5118\u53ef\u80fd\u591a\u7684\u4efb\u52d9\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9078\u64c7\u7d50\u675f\u6642\u9593\u6700\u65e9\u7684\u4efb\u52d9\uff0c\u90a3\u9ebc\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u512a\u89e3\u3002</li> <li>\u5206\u6578\u80cc\u5305\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u7d44\u7269\u54c1\u548c\u4e00\u500b\u8f09\u91cd\u91cf\uff0c\u4f60\u7684\u76ee\u6a19\u662f\u9078\u64c7\u4e00\u7d44\u7269\u54c1\uff0c\u4f7f\u5f97\u7e3d\u91cd\u91cf\u4e0d\u8d85\u904e\u8f09\u91cd\u91cf\uff0c\u4e14\u7e3d\u50f9\u503c\u6700\u5927\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9078\u64c7\u50f9\u6548\u6bd4\u6700\u9ad8\uff08\u50f9\u503c / \u91cd\u91cf\uff09\u7684\u7269\u54c1\uff0c\u90a3\u9ebc\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5728\u4e00\u4e9b\u60c5\u6cc1\u4e0b\u53ef\u4ee5\u5f97\u5230\u6700\u512a\u89e3\u3002</li> <li>\u80a1\u7968\u8cb7\u8ce3\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u7d44\u80a1\u7968\u7684\u6b77\u53f2\u50f9\u683c\uff0c\u4f60\u53ef\u4ee5\u9032\u884c\u591a\u6b21\u8cb7\u8ce3\uff0c\u4f46\u5982\u679c\u4f60\u5df2\u7d93\u6301\u6709\u80a1\u7968\uff0c\u90a3\u9ebc\u5728\u8ce3\u51fa\u4e4b\u524d\u4e0d\u80fd\u518d\u8cb7\uff0c\u76ee\u6a19\u662f\u7372\u53d6\u6700\u5927\u5229\u6f64\u3002</li> <li>\u970d\u592b\u66fc\u7de8\u78bc\uff1a\u970d\u592b\u66fc\u7de8\u78bc\u662f\u4e00\u7a2e\u7528\u65bc\u7121\u640d\u8cc7\u6599\u58d3\u7e2e\u7684\u8caa\u5a6a\u6f14\u7b97\u6cd5\u3002\u900f\u904e\u69cb\u5efa\u970d\u592b\u66fc\u6a39\uff0c\u6bcf\u6b21\u9078\u64c7\u51fa\u73fe\u983b\u7387\u6700\u4f4e\u7684\u5169\u500b\u7bc0\u9ede\u5408\u4f75\uff0c\u6700\u5f8c\u5f97\u5230\u7684\u970d\u592b\u66fc\u6a39\u7684\u5e36\u6b0a\u8def\u5f91\u9577\u5ea6\uff08\u7de8\u78bc\u9577\u5ea6\uff09\u6700\u5c0f\u3002</li> <li>Dijkstra \u6f14\u7b97\u6cd5\uff1a\u5b83\u662f\u4e00\u7a2e\u89e3\u6c7a\u7d66\u5b9a\u6e90\u9802\u9ede\u5230\u5176\u9918\u5404\u9802\u9ede\u7684\u6700\u77ed\u8def\u5f91\u554f\u984c\u7684\u8caa\u5a6a\u6f14\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_greedy/max_capacity_problem/","title":"15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u554f\u984c","text":"<p>Question</p> <p>\u8f38\u5165\u4e00\u500b\u9663\u5217 \\(ht\\) \uff0c\u5176\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\u4ee3\u8868\u4e00\u500b\u5782\u76f4\u9694\u677f\u7684\u9ad8\u5ea6\u3002\u9663\u5217\u4e2d\u7684\u4efb\u610f\u5169\u500b\u9694\u677f\uff0c\u4ee5\u53ca\u5b83\u5011\u4e4b\u9593\u7684\u7a7a\u9593\u53ef\u4ee5\u7d44\u6210\u4e00\u500b\u5bb9\u5668\u3002</p> <p>\u5bb9\u5668\u7684\u5bb9\u91cf\u7b49\u65bc\u9ad8\u5ea6\u548c\u5bec\u5ea6\u7684\u4e58\u7a4d\uff08\u9762\u7a4d\uff09\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u8f03\u77ed\u7684\u9694\u677f\u6c7a\u5b9a\uff0c\u5bec\u5ea6\u662f\u5169\u500b\u9694\u677f\u7684\u9663\u5217\u7d22\u5f15\u4e4b\u5dee\u3002</p> <p>\u8acb\u5728\u9663\u5217\u4e2d\u9078\u64c7\u5169\u500b\u9694\u677f\uff0c\u4f7f\u5f97\u7d44\u6210\u7684\u5bb9\u5668\u7684\u5bb9\u91cf\u6700\u5927\uff0c\u8fd4\u56de\u6700\u5927\u5bb9\u91cf\u3002\u793a\u4f8b\u5982\u5716 15-7 \u6240\u793a\u3002</p> <p></p> <p> \u5716 15-7 \u00a0 \u6700\u5927\u5bb9\u91cf\u554f\u984c\u7684\u793a\u4f8b\u8cc7\u6599 </p> <p>\u5bb9\u5668\u7531\u4efb\u610f\u5169\u500b\u9694\u677f\u570d\u6210\uff0c\u56e0\u6b64\u672c\u984c\u7684\u72c0\u614b\u70ba\u5169\u500b\u9694\u677f\u7684\u7d22\u5f15\uff0c\u8a18\u70ba \\([i, j]\\) \u3002</p> <p>\u6839\u64da\u984c\u610f\uff0c\u5bb9\u91cf\u7b49\u65bc\u9ad8\u5ea6\u4e58\u4ee5\u5bec\u5ea6\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u77ed\u677f\u6c7a\u5b9a\uff0c\u5bec\u5ea6\u662f\u5169\u9694\u677f\u7684\u9663\u5217\u7d22\u5f15\u4e4b\u5dee\u3002\u8a2d\u5bb9\u91cf\u70ba \\(cap[i, j]\\) \uff0c\u5247\u53ef\u5f97\u8a08\u7b97\u516c\u5f0f\uff1a</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>\u8a2d\u9663\u5217\u9577\u5ea6\u70ba \\(n\\) \uff0c\u5169\u500b\u9694\u677f\u7684\u7d44\u5408\u6578\u91cf\uff08\u72c0\u614b\u7e3d\u6578\uff09\u70ba \\(C_n^2 = \\frac{n(n - 1)}{2}\\) \u500b\u3002\u6700\u76f4\u63a5\u5730\uff0c\u6211\u5011\u53ef\u4ee5\u7aae\u8209\u6240\u6709\u72c0\u614b\uff0c\u5f9e\u800c\u6c42\u5f97\u6700\u5927\u5bb9\u91cf\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_greedy/max_capacity_problem/#1","title":"1. \u00a0 \u8caa\u5a6a\u7b56\u7565\u78ba\u5b9a","text":"<p>\u9019\u9053\u984c\u9084\u6709\u66f4\u9ad8\u6548\u7387\u7684\u89e3\u6cd5\u3002\u5982\u5716 15-8 \u6240\u793a\uff0c\u73fe\u9078\u53d6\u4e00\u500b\u72c0\u614b \\([i, j]\\) \uff0c\u5176\u6eff\u8db3\u7d22\u5f15 \\(i &lt; j\\) \u4e14\u9ad8\u5ea6 \\(ht[i] &lt; ht[j]\\) \uff0c\u5373 \\(i\\) \u70ba\u77ed\u677f\u3001\\(j\\) \u70ba\u9577\u677f\u3002</p> <p></p> <p> \u5716 15-8 \u00a0 \u521d\u59cb\u72c0\u614b </p> <p>\u5982\u5716 15-9 \u6240\u793a\uff0c\u82e5\u6b64\u6642\u5c07\u9577\u677f \\(j\\) \u5411\u77ed\u677f \\(i\\) \u9760\u8fd1\uff0c\u5247\u5bb9\u91cf\u4e00\u5b9a\u8b8a\u5c0f\u3002</p> <p>\u9019\u662f\u56e0\u70ba\u5728\u79fb\u52d5\u9577\u677f \\(j\\) \u5f8c\uff0c\u5bec\u5ea6 \\(j-i\\) \u80af\u5b9a\u8b8a\u5c0f\uff1b\u800c\u9ad8\u5ea6\u7531\u77ed\u677f\u6c7a\u5b9a\uff0c\u56e0\u6b64\u9ad8\u5ea6\u53ea\u53ef\u80fd\u4e0d\u8b8a\uff08 \\(i\\) \u4ecd\u70ba\u77ed\u677f\uff09\u6216\u8b8a\u5c0f\uff08\u79fb\u52d5\u5f8c\u7684 \\(j\\) \u6210\u70ba\u77ed\u677f\uff09\u3002</p> <p></p> <p> \u5716 15-9 \u00a0 \u5411\u5167\u79fb\u52d5\u9577\u677f\u5f8c\u7684\u72c0\u614b </p> <p>\u53cd\u5411\u601d\u8003\uff0c\u6211\u5011\u53ea\u6709\u5411\u5167\u6536\u7e2e\u77ed\u677f \\(i\\) \uff0c\u624d\u6709\u53ef\u80fd\u4f7f\u5bb9\u91cf\u8b8a\u5927\u3002\u56e0\u70ba\u96d6\u7136\u5bec\u5ea6\u4e00\u5b9a\u8b8a\u5c0f\uff0c\u4f46\u9ad8\u5ea6\u53ef\u80fd\u6703\u8b8a\u5927\uff08\u79fb\u52d5\u5f8c\u7684\u77ed\u677f \\(i\\) \u53ef\u80fd\u6703\u8b8a\u9577\uff09\u3002\u4f8b\u5982\u5728\u5716 15-10 \u4e2d\uff0c\u79fb\u52d5\u77ed\u677f\u5f8c\u9762\u7a4d\u8b8a\u5927\u3002</p> <p></p> <p> \u5716 15-10 \u00a0 \u5411\u5167\u79fb\u52d5\u77ed\u677f\u5f8c\u7684\u72c0\u614b </p> <p>\u7531\u6b64\u4fbf\u53ef\u63a8\u51fa\u672c\u984c\u7684\u8caa\u5a6a\u7b56\u7565\uff1a\u521d\u59cb\u5316\u5169\u6307\u6a19\uff0c\u4f7f\u5176\u5206\u5217\u5bb9\u5668\u5169\u7aef\uff0c\u6bcf\u8f2a\u5411\u5167\u6536\u7e2e\u77ed\u677f\u5c0d\u61c9\u7684\u6307\u6a19\uff0c\u76f4\u81f3\u5169\u6307\u6a19\u76f8\u9047\u3002</p> <p>\u5716 15-11 \u5c55\u793a\u4e86\u8caa\u5a6a\u7b56\u7565\u7684\u57f7\u884c\u904e\u7a0b\u3002</p> <ol> <li>\u521d\u59cb\u72c0\u614b\u4e0b\uff0c\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u5206\u5217\u9663\u5217\u5169\u7aef\u3002</li> <li>\u8a08\u7b97\u7576\u524d\u72c0\u614b\u7684\u5bb9\u91cf \\(cap[i, j]\\) \uff0c\u4e26\u66f4\u65b0\u6700\u5927\u5bb9\u91cf\u3002</li> <li>\u6bd4\u8f03\u677f \\(i\\) \u548c\u677f \\(j\\) \u7684\u9ad8\u5ea6\uff0c\u4e26\u5c07\u77ed\u677f\u5411\u5167\u79fb\u52d5\u4e00\u683c\u3002</li> <li>\u8ff4\u5708\u57f7\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u76f4\u81f3 \\(i\\) \u548c \\(j\\) \u76f8\u9047\u6642\u7d50\u675f\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 15-11 \u00a0 \u6700\u5927\u5bb9\u91cf\u554f\u984c\u7684\u8caa\u5a6a\u904e\u7a0b </p>"},{"location":"chapter_greedy/max_capacity_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u7a0b\u5f0f\u78bc\u8ff4\u5708\u6700\u591a \\(n\\) \u8f2a\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> <p>\u8b8a\u6578 \\(i\\)\u3001\\(j\\)\u3001\\(res\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"\u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a\"\"\"\n    # \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    i, j = 0, len(ht) - 1\n    # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    res = 0\n    # \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while i &lt; j:\n        # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    int i = 0, j = ht.size() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    int res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint maxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    int i = 0, j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    int res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint MaxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    int i = 0, j = ht.Length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    int res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfunc maxCapacity(ht []int) int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    i, j := 0, len(ht)-1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    res := 0\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    for i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    var i = ht.startIndex, j = ht.endIndex - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    var res = 0\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfunction maxCapacity(ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    let res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfunction maxCapacity(ht: number[]): number {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    let res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint maxCapacity(List&lt;int&gt; ht) {\n  // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n  int i = 0, j = ht.length - 1;\n  // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n  int res = 0;\n  // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n  while (i &lt; j) {\n    // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    let mut res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint maxCapacity(int ht[], int htLength) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    int i = 0;\n    int j = htLength - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    int res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.kt<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfun maxCapacity(ht: IntArray): Int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    var i = 0\n    var j = ht.size - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    var res = 0\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        val cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.rb<pre><code>### \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a ###\ndef max_capacity(ht)\n  # \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n  i, j = 0, ht.length - 1\n  # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n  res = 0\n\n  # \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n  while i &lt; j\n    # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n    cap = [ht[i], ht[j]].min * (j - i)\n    res = [res, cap].max\n    # \u5411\u5167\u79fb\u52d5\u77ed\u677f\n    if ht[i] &lt; ht[j]\n      i += 1\n    else\n      j -= 1\n    end\n  end\n\n  res\nend\n</code></pre> max_capacity.zig<pre><code>[class]{}-[func]{maxCapacity}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_greedy/max_capacity_problem/#3","title":"3. \u00a0 \u6b63\u78ba\u6027\u8b49\u660e","text":"<p>\u4e4b\u6240\u4ee5\u8caa\u5a6a\u6bd4\u7aae\u8209\u66f4\u5feb\uff0c\u662f\u56e0\u70ba\u6bcf\u8f2a\u7684\u8caa\u5a6a\u9078\u64c7\u90fd\u6703\u201c\u8df3\u904e\u201d\u4e00\u4e9b\u72c0\u614b\u3002</p> <p>\u6bd4\u5982\u5728\u72c0\u614b \\(cap[i, j]\\) \u4e0b\uff0c\\(i\\) \u70ba\u77ed\u677f\u3001\\(j\\) \u70ba\u9577\u677f\u3002\u82e5\u8caa\u5a6a\u5730\u5c07\u77ed\u677f \\(i\\) \u5411\u5167\u79fb\u52d5\u4e00\u683c\uff0c\u6703\u5c0e\u81f4\u5716 15-12 \u6240\u793a\u7684\u72c0\u614b\u88ab\u201c\u8df3\u904e\u201d\u3002\u9019\u610f\u5473\u8457\u4e4b\u5f8c\u7121\u6cd5\u9a57\u8b49\u9019\u4e9b\u72c0\u614b\u7684\u5bb9\u91cf\u5927\u5c0f\u3002</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> \u5716 15-12 \u00a0 \u79fb\u52d5\u77ed\u677f\u5c0e\u81f4\u88ab\u8df3\u904e\u7684\u72c0\u614b </p> <p>\u89c0\u5bdf\u767c\u73fe\uff0c\u9019\u4e9b\u88ab\u8df3\u904e\u7684\u72c0\u614b\u5be6\u969b\u4e0a\u5c31\u662f\u5c07\u9577\u677f \\(j\\) \u5411\u5167\u79fb\u52d5\u7684\u6240\u6709\u72c0\u614b\u3002\u524d\u9762\u6211\u5011\u5df2\u7d93\u8b49\u660e\u5167\u79fb\u9577\u677f\u4e00\u5b9a\u6703\u5c0e\u81f4\u5bb9\u91cf\u8b8a\u5c0f\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u88ab\u8df3\u904e\u7684\u72c0\u614b\u90fd\u4e0d\u53ef\u80fd\u662f\u6700\u512a\u89e3\uff0c\u8df3\u904e\u5b83\u5011\u4e0d\u6703\u5c0e\u81f4\u932f\u904e\u6700\u512a\u89e3\u3002</p> <p>\u4ee5\u4e0a\u5206\u6790\u8aaa\u660e\uff0c\u79fb\u52d5\u77ed\u677f\u7684\u64cd\u4f5c\u662f\u201c\u5b89\u5168\u201d\u7684\uff0c\u8caa\u5a6a\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/","title":"15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u7a4d\u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u6b63\u6574\u6578 \\(n\\) \uff0c\u5c07\u5176\u5207\u5206\u70ba\u81f3\u5c11\u5169\u500b\u6b63\u6574\u6578\u7684\u548c\uff0c\u6c42\u5207\u5206\u5f8c\u6240\u6709\u6574\u6578\u7684\u4e58\u7a4d\u6700\u5927\u662f\u591a\u5c11\uff0c\u5982\u5716 15-13 \u6240\u793a\u3002</p> <p></p> <p> \u5716 15-13 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u7a4d\u7684\u554f\u984c\u5b9a\u7fa9 </p> <p>\u5047\u8a2d\u6211\u5011\u5c07 \\(n\\) \u5207\u5206\u70ba \\(m\\) \u500b\u6574\u6578\u56e0\u5b50\uff0c\u5176\u4e2d\u7b2c \\(i\\) \u500b\u56e0\u5b50\u8a18\u70ba \\(n_i\\) \uff0c\u5373</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>\u672c\u984c\u7684\u76ee\u6a19\u662f\u6c42\u5f97\u6240\u6709\u6574\u6578\u56e0\u5b50\u7684\u6700\u5927\u4e58\u7a4d\uff0c\u5373</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>\u6211\u5011\u9700\u8981\u601d\u8003\u7684\u662f\uff1a\u5207\u5206\u6578\u91cf \\(m\\) \u61c9\u8a72\u591a\u5927\uff0c\u6bcf\u500b \\(n_i\\) \u61c9\u8a72\u662f\u591a\u5c11\uff1f</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#1","title":"1. \u00a0 \u8caa\u5a6a\u7b56\u7565\u78ba\u5b9a","text":"<p>\u6839\u64da\u7d93\u9a57\uff0c\u5169\u500b\u6574\u6578\u7684\u4e58\u7a4d\u5f80\u5f80\u6bd4\u5b83\u5011\u7684\u52a0\u548c\u66f4\u5927\u3002\u5047\u8a2d\u5f9e \\(n\\) \u4e2d\u5206\u51fa\u4e00\u500b\u56e0\u5b50 \\(2\\) \uff0c\u5247\u5b83\u5011\u7684\u4e58\u7a4d\u70ba \\(2(n-2)\\) \u3002\u6211\u5011\u5c07\u8a72\u4e58\u7a4d\u8207 \\(n\\) \u4f5c\u6bd4\u8f03\uff1a</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>\u5982\u5716 15-14 \u6240\u793a\uff0c\u7576 \\(n \\geq 4\\) \u6642\uff0c\u5207\u5206\u51fa\u4e00\u500b \\(2\\) \u5f8c\u4e58\u7a4d\u6703\u8b8a\u5927\uff0c\u9019\u8aaa\u660e\u5927\u65bc\u7b49\u65bc \\(4\\) \u7684\u6574\u6578\u90fd\u61c9\u8a72\u88ab\u5207\u5206\u3002</p> <p>\u8caa\u5a6a\u7b56\u7565\u4e00\uff1a\u5982\u679c\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b \\(\\geq 4\\) \u7684\u56e0\u5b50\uff0c\u90a3\u9ebc\u5b83\u5c31\u61c9\u8a72\u88ab\u7e7c\u7e8c\u5207\u5206\u3002\u6700\u7d42\u7684\u5207\u5206\u65b9\u6848\u53ea\u61c9\u51fa\u73fe \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9019\u4e09\u7a2e\u56e0\u5b50\u3002</p> <p></p> <p> \u5716 15-14 \u00a0 \u5207\u5206\u5c0e\u81f4\u4e58\u7a4d\u8b8a\u5927 </p> <p>\u63a5\u4e0b\u4f86\u601d\u8003\u54ea\u500b\u56e0\u5b50\u662f\u6700\u512a\u7684\u3002\u5728 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9019\u4e09\u500b\u56e0\u5b50\u4e2d\uff0c\u986f\u7136 \\(1\\) \u662f\u6700\u5dee\u7684\uff0c\u56e0\u70ba \\(1 \\times (n-1) &lt; n\\) \u6046\u6210\u7acb\uff0c\u5373\u5207\u5206\u51fa \\(1\\) \u53cd\u800c\u6703\u5c0e\u81f4\u4e58\u7a4d\u6e1b\u5c0f\u3002</p> <p>\u5982\u5716 15-15 \u6240\u793a\uff0c\u7576 \\(n = 6\\) \u6642\uff0c\u6709 \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\) \u3002\u9019\u610f\u5473\u8457\u5207\u5206\u51fa \\(3\\) \u6bd4\u5207\u5206\u51fa \\(2\\) \u66f4\u512a\u3002</p> <p>\u8caa\u5a6a\u7b56\u7565\u4e8c\uff1a\u5728\u5207\u5206\u65b9\u6848\u4e2d\uff0c\u6700\u591a\u96bb\u61c9\u5b58\u5728\u5169\u500b \\(2\\) \u3002\u56e0\u70ba\u4e09\u500b \\(2\\) \u7e3d\u662f\u53ef\u4ee5\u66ff\u63db\u70ba\u5169\u500b \\(3\\) \uff0c\u5f9e\u800c\u7372\u5f97\u66f4\u5927\u7684\u4e58\u7a4d\u3002</p> <p></p> <p> \u5716 15-15 \u00a0 \u6700\u512a\u5207\u5206\u56e0\u5b50 </p> <p>\u7d9c\u4e0a\u6240\u8ff0\uff0c\u53ef\u63a8\u7406\u51fa\u4ee5\u4e0b\u8caa\u5a6a\u7b56\u7565\u3002</p> <ol> <li>\u8f38\u5165\u6574\u6578 \\(n\\) \uff0c\u5f9e\u5176\u4e0d\u65b7\u5730\u5207\u5206\u51fa\u56e0\u5b50 \\(3\\) \uff0c\u76f4\u81f3\u9918\u6578\u70ba \\(0\\)\u3001\\(1\\)\u3001\\(2\\) \u3002</li> <li>\u7576\u9918\u6578\u70ba \\(0\\) \u6642\uff0c\u4ee3\u8868 \\(n\\) \u662f \\(3\\) \u7684\u500d\u6578\uff0c\u56e0\u6b64\u4e0d\u505a\u4efb\u4f55\u8655\u7406\u3002</li> <li>\u7576\u9918\u6578\u70ba \\(2\\) \u6642\uff0c\u4e0d\u7e7c\u7e8c\u5283\u5206\uff0c\u4fdd\u7559\u3002</li> <li>\u7576\u9918\u6578\u70ba \\(1\\) \u6642\uff0c\u7531\u65bc \\(2 \\times 2 &gt; 1 \\times 3\\) \uff0c\u56e0\u6b64\u61c9\u5c07\u6700\u5f8c\u4e00\u500b \\(3\\) \u66ff\u63db\u70ba \\(2\\) \u3002</li> </ol>"},{"location":"chapter_greedy/max_product_cutting_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5982\u5716 15-16 \u6240\u793a\uff0c\u6211\u5011\u7121\u9808\u900f\u904e\u8ff4\u5708\u4f86\u5207\u5206\u6574\u6578\uff0c\u800c\u53ef\u4ee5\u5229\u7528\u5411\u4e0b\u6574\u9664\u904b\u7b97\u5f97\u5230 \\(3\\) \u7684\u500b\u6578 \\(a\\) \uff0c\u7528\u53d6\u6a21\u904b\u7b97\u5f97\u5230\u9918\u6578 \\(b\\) \uff0c\u6b64\u6642\u6709\uff1a</p> \\[ n = 3 a + b \\] <p>\u8acb\u6ce8\u610f\uff0c\u5c0d\u65bc \\(n \\leq 3\\) \u7684\u908a\u754c\u60c5\u6cc1\uff0c\u5fc5\u9808\u62c6\u5206\u51fa\u4e00\u500b \\(1\\) \uff0c\u4e58\u7a4d\u70ba \\(1 \\times (n - 1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"\u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a\"\"\"\n    # \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    a, b = n // 3, n % 3\n    if b == 1:\n        # \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return int(math.pow(3, a)) * 2\n    # \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint maxProductCutting(int n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return (int)pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint maxProductCutting(int n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return (int) Math.pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint MaxProductCutting(int n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfunc maxProductCutting(n int) int {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return pow(3, a) * 2\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfunction maxProductCutting(n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfunction maxProductCutting(n: number): number {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint maxProductCutting(int n) {\n  // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return (pow(3, a) * 2).toInt();\n  }\n  // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        3_i32.pow(a as u32) * 2\n    } else {\n        // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint maxProductCutting(int n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.kt<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfun maxProductCutting(n: Int): Int {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1)\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    val a = n / 3\n    val b = n % 3\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return 3.0.pow((a - 1)).toInt() * 2 * 2\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return 3.0.pow(a).toInt() * 2 * 2\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return 3.0.pow(a).toInt()\n}\n</code></pre> max_product_cutting.rb<pre><code>### \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a ###\ndef max_product_cutting(n)\n  # \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n  return 1 * (n - 1) if n &lt;= 3\n  # \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n  a, b = n / 3, n % 3\n  # \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n  return (3.pow(a - 1) * 2 * 2).to_i if b == 1\n  # \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n  return (3.pow(a) * 2).to_i if b == 2\n  # \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n  3.pow(a).to_i\nend\n</code></pre> max_product_cutting.zig<pre><code>[class]{}-[func]{maxProductCutting}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 15-16 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u7a4d\u7684\u8a08\u7b97\u65b9\u6cd5 </p> <p>\u6642\u9593\u8907\u96dc\u5ea6\u53d6\u6c7a\u65bc\u7a0b\u5f0f\u8a9e\u8a00\u7684\u51aa\u904b\u7b97\u7684\u5be6\u73fe\u65b9\u6cd5\u3002\u4ee5 Python \u70ba\u4f8b\uff0c\u5e38\u7528\u7684\u51aa\u8a08\u7b97\u51fd\u5f0f\u6709\u4e09\u7a2e\u3002</p> <ul> <li>\u904b\u7b97\u5b50 <code>**</code> \u548c\u51fd\u5f0f <code>pow()</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(\\log\u2061 a)\\) \u3002</li> <li>\u51fd\u5f0f <code>math.pow()</code> \u5167\u90e8\u547c\u53eb C \u8a9e\u8a00\u5eab\u7684 <code>pow()</code> \u51fd\u5f0f\uff0c\u5176\u57f7\u884c\u6d6e\u9ede\u53d6\u51aa\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</li> </ul> <p>\u8b8a\u6578 \\(a\\) \u548c \\(b\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#3","title":"3. \u00a0 \u6b63\u78ba\u6027\u8b49\u660e","text":"<p>\u4f7f\u7528\u53cd\u8b49\u6cd5\uff0c\u53ea\u5206\u6790 \\(n \\geq 3\\) \u7684\u60c5\u6cc1\u3002</p> <ol> <li>\u6240\u6709\u56e0\u5b50 \\(\\leq 3\\) \uff1a\u5047\u8a2d\u6700\u512a\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728 \\(\\geq 4\\) \u7684\u56e0\u5b50 \\(x\\) \uff0c\u90a3\u9ebc\u4e00\u5b9a\u53ef\u4ee5\u5c07\u5176\u7e7c\u7e8c\u5283\u5206\u70ba \\(2(x-2)\\) \uff0c\u5f9e\u800c\u7372\u5f97\u66f4\u5927\u7684\u4e58\u7a4d\u3002\u9019\u8207\u5047\u8a2d\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u4e0d\u5305\u542b \\(1\\) \uff1a\u5047\u8a2d\u6700\u512a\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728\u4e00\u500b\u56e0\u5b50 \\(1\\) \uff0c\u90a3\u9ebc\u5b83\u4e00\u5b9a\u53ef\u4ee5\u5408\u4f75\u5165\u53e6\u5916\u4e00\u500b\u56e0\u5b50\u4e2d\uff0c\u4ee5\u7372\u5f97\u66f4\u5927\u7684\u4e58\u7a4d\u3002\u9019\u8207\u5047\u8a2d\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u6700\u591a\u5305\u542b\u5169\u500b \\(2\\) \uff1a\u5047\u8a2d\u6700\u512a\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b\u4e09\u500b \\(2\\) \uff0c\u90a3\u9ebc\u4e00\u5b9a\u53ef\u4ee5\u66ff\u63db\u70ba\u5169\u500b \\(3\\) \uff0c\u4e58\u7a4d\u66f4\u5927\u3002\u9019\u8207\u5047\u8a2d\u77db\u76fe\u3002</li> </ol>"},{"location":"chapter_greedy/summary/","title":"15.5 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u901a\u5e38\u7528\u65bc\u89e3\u6c7a\u6700\u6700\u4f73\u5316\u554f\u984c\uff0c\u5176\u539f\u7406\u662f\u5728\u6bcf\u500b\u6c7a\u7b56\u968e\u6bb5\u90fd\u505a\u51fa\u5340\u57df\u6027\u6700\u512a\u7684\u6c7a\u7b56\uff0c\u4ee5\u671f\u7372\u5f97\u5168\u57df\u6027\u6700\u512a\u89e3\u3002</li> <li>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6703\u8fed\u4ee3\u5730\u505a\u51fa\u4e00\u500b\u53c8\u4e00\u500b\u7684\u8caa\u5a6a\u9078\u64c7\uff0c\u6bcf\u8f2a\u90fd\u5c07\u554f\u984c\u8f49\u5316\u6210\u4e00\u500b\u898f\u6a21\u66f4\u5c0f\u7684\u5b50\u554f\u984c\uff0c\u76f4\u5230\u554f\u984c\u88ab\u89e3\u6c7a\u3002</li> <li>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u4e0d\u50c5\u5be6\u73fe\u7c21\u55ae\uff0c\u9084\u5177\u6709\u5f88\u9ad8\u7684\u89e3\u984c\u6548\u7387\u3002\u76f8\u6bd4\u65bc\u52d5\u614b\u898f\u5283\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u66f4\u4f4e\u3002</li> <li>\u5728\u96f6\u9322\u514c\u63db\u554f\u984c\u4e2d\uff0c\u5c0d\u65bc\u67d0\u4e9b\u786c\u5e63\u7d44\u5408\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8b49\u627e\u5230\u6700\u512a\u89e3\uff1b\u5c0d\u65bc\u53e6\u5916\u4e00\u4e9b\u786c\u5e63\u7d44\u5408\u5247\u4e0d\u7136\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u53ef\u80fd\u627e\u5230\u5f88\u5dee\u7684\u89e3\u3002</li> <li>\u9069\u5408\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6c42\u89e3\u7684\u554f\u984c\u5177\u6709\u5169\u5927\u6027\u8cea\uff1a\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u548c\u6700\u512a\u5b50\u7d50\u69cb\u3002\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u4ee3\u8868\u8caa\u5a6a\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</li> <li>\u5c0d\u65bc\u67d0\u4e9b\u8907\u96dc\u554f\u984c\uff0c\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u7684\u8b49\u660e\u4e26\u4e0d\u7c21\u55ae\u3002\u76f8\u5c0d\u4f86\u8aaa\uff0c\u8b49\u507d\u66f4\u52a0\u5bb9\u6613\uff0c\u4f8b\u5982\u96f6\u9322\u514c\u63db\u554f\u984c\u3002</li> <li>\u6c42\u89e3\u8caa\u5a6a\u554f\u984c\u4e3b\u8981\u5206\u70ba\u4e09\u6b65\uff1a\u554f\u984c\u5206\u6790\u3001\u78ba\u5b9a\u8caa\u5a6a\u7b56\u7565\u3001\u6b63\u78ba\u6027\u8b49\u660e\u3002\u5176\u4e2d\uff0c\u78ba\u5b9a\u8caa\u5a6a\u7b56\u7565\u662f\u6838\u5fc3\u6b65\u9a5f\uff0c\u6b63\u78ba\u6027\u8b49\u660e\u5f80\u5f80\u662f\u96e3\u9ede\u3002</li> <li>\u5206\u6578\u80cc\u5305\u554f\u984c\u5728 0-1 \u80cc\u5305\u7684\u57fa\u790e\u4e0a\uff0c\u5141\u8a31\u9078\u64c7\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6c42\u89e3\u3002\u8caa\u5a6a\u7b56\u7565\u7684\u6b63\u78ba\u6027\u53ef\u4ee5\u4f7f\u7528\u53cd\u8b49\u6cd5\u4f86\u8b49\u660e\u3002</li> <li>\u6700\u5927\u5bb9\u91cf\u554f\u984c\u53ef\u4f7f\u7528\u7aae\u8209\u6cd5\u6c42\u89e3\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002\u900f\u904e\u8a2d\u8a08\u8caa\u5a6a\u7b56\u7565\uff0c\u6bcf\u8f2a\u5411\u5167\u79fb\u52d5\u77ed\u677f\uff0c\u53ef\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u5728\u6700\u5927\u5207\u5206\u4e58\u7a4d\u554f\u984c\u4e2d\uff0c\u6211\u5011\u5148\u5f8c\u63a8\u7406\u51fa\u5169\u500b\u8caa\u5a6a\u7b56\u7565\uff1a\\(\\geq 4\\) \u7684\u6574\u6578\u90fd\u61c9\u8a72\u7e7c\u7e8c\u5207\u5206\uff0c\u6700\u512a\u5207\u5206\u56e0\u5b50\u70ba \\(3\\) \u3002\u7a0b\u5f0f\u78bc\u4e2d\u5305\u542b\u51aa\u904b\u7b97\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53d6\u6c7a\u65bc\u51aa\u904b\u7b97\u5be6\u73fe\u65b9\u6cd5\uff0c\u901a\u5e38\u70ba \\(O(1)\\) \u6216 \\(O(\\log n)\\) \u3002</li> </ul>"},{"location":"chapter_hashing/","title":"\u7b2c 6 \u7ae0 \u00a0 \u96dc\u6e4a\u8868","text":"<p>Abstract</p> <p>\u5728\u8a08\u7b97\u6a5f\u4e16\u754c\u4e2d\uff0c\u96dc\u6e4a\u8868\u5982\u540c\u4e00\u4f4d\u8070\u6167\u7684\u5716\u66f8\u7ba1\u7406\u54e1\u3002</p> <p>\u4ed6\u77e5\u9053\u5982\u4f55\u8a08\u7b97\u7d22\u66f8\u865f\uff0c\u5f9e\u800c\u53ef\u4ee5\u5feb\u901f\u627e\u5230\u76ee\u6a19\u5716\u66f8\u3002</p>"},{"location":"chapter_hashing/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>6.1 \u00a0 \u96dc\u6e4a\u8868</li> <li>6.2 \u00a0 \u96dc\u6e4a\u885d\u7a81</li> <li>6.3 \u00a0 \u96dc\u6e4a\u6f14\u7b97\u6cd5</li> <li>6.4 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_hashing/hash_algorithm/","title":"6.3 \u00a0 \u96dc\u6e4a\u6f14\u7b97\u6cd5","text":"<p>\u524d\u5169\u7bc0\u4ecb\u7d39\u4e86\u96dc\u6e4a\u8868\u7684\u5de5\u4f5c\u539f\u7406\u548c\u96dc\u6e4a\u885d\u7a81\u7684\u8655\u7406\u65b9\u6cd5\u3002\u7136\u800c\u7121\u8ad6\u662f\u958b\u653e\u5b9a\u5740\u9084\u662f\u93c8\u5f0f\u4f4d\u5740\uff0c\u5b83\u5011\u53ea\u80fd\u4fdd\u8b49\u96dc\u6e4a\u8868\u53ef\u4ee5\u5728\u767c\u751f\u885d\u7a81\u6642\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u7121\u6cd5\u6e1b\u5c11\u96dc\u6e4a\u885d\u7a81\u7684\u767c\u751f\u3002</p> <p>\u5982\u679c\u96dc\u6e4a\u885d\u7a81\u904e\u65bc\u983b\u7e41\uff0c\u96dc\u6e4a\u8868\u7684\u6548\u80fd\u5247\u6703\u6025\u5287\u52a3\u5316\u3002\u5982\u5716 6-8 \u6240\u793a\uff0c\u5c0d\u65bc\u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868\uff0c\u7406\u60f3\u60c5\u6cc1\u4e0b\u9375\u503c\u5c0d\u5747\u52fb\u5206\u4f48\u5728\u5404\u500b\u6876\u4e2d\uff0c\u9054\u5230\u6700\u4f73\u67e5\u8a62\u6548\u7387\uff1b\u6700\u5dee\u60c5\u6cc1\u4e0b\u6240\u6709\u9375\u503c\u5c0d\u90fd\u5132\u5b58\u5230\u540c\u4e00\u500b\u6876\u4e2d\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</p> <p></p> <p> \u5716 6-8 \u00a0 \u96dc\u6e4a\u885d\u7a81\u7684\u6700\u4f73\u60c5\u6cc1\u8207\u6700\u5dee\u60c5\u6cc1 </p> <p>\u9375\u503c\u5c0d\u7684\u5206\u4f48\u60c5\u6cc1\u7531\u96dc\u6e4a\u51fd\u5f0f\u6c7a\u5b9a\u3002\u56de\u61b6\u96dc\u6e4a\u51fd\u5f0f\u7684\u8a08\u7b97\u6b65\u9a5f\uff0c\u5148\u8a08\u7b97\u96dc\u6e4a\u503c\uff0c\u518d\u5c0d\u9663\u5217\u9577\u5ea6\u53d6\u6a21\uff1a</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u89c0\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u7576\u96dc\u6e4a\u8868\u5bb9\u91cf <code>capacity</code> \u56fa\u5b9a\u6642\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5 <code>hash()</code> \u6c7a\u5b9a\u4e86\u8f38\u51fa\u503c\uff0c\u9032\u800c\u6c7a\u5b9a\u4e86\u9375\u503c\u5c0d\u5728\u96dc\u6e4a\u8868\u4e2d\u7684\u5206\u4f48\u60c5\u6cc1\u3002</p> <p>\u9019\u610f\u5473\u8457\uff0c\u70ba\u4e86\u964d\u4f4e\u96dc\u6e4a\u885d\u7a81\u7684\u767c\u751f\u6a5f\u7387\uff0c\u6211\u5011\u61c9\u7576\u5c07\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u96dc\u6e4a\u6f14\u7b97\u6cd5 <code>hash()</code> \u7684\u8a2d\u8a08\u4e0a\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1 \u00a0 \u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u76ee\u6a19","text":"<p>\u70ba\u4e86\u5be6\u73fe\u201c\u65e2\u5feb\u53c8\u7a69\u201d\u7684\u96dc\u6e4a\u8868\u8cc7\u6599\u7d50\u69cb\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u61c9\u5177\u5099\u4ee5\u4e0b\u7279\u9ede\u3002</p> <ul> <li>\u78ba\u5b9a\u6027\uff1a\u5c0d\u65bc\u76f8\u540c\u7684\u8f38\u5165\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u61c9\u59cb\u7d42\u7522\u751f\u76f8\u540c\u7684\u8f38\u51fa\u3002\u9019\u6a23\u624d\u80fd\u78ba\u4fdd\u96dc\u6e4a\u8868\u662f\u53ef\u9760\u7684\u3002</li> <li>\u6548\u7387\u9ad8\uff1a\u8a08\u7b97\u96dc\u6e4a\u503c\u7684\u904e\u7a0b\u61c9\u8a72\u8db3\u5920\u5feb\u3002\u8a08\u7b97\u958b\u92b7\u8d8a\u5c0f\uff0c\u96dc\u6e4a\u8868\u7684\u5be6\u7528\u6027\u8d8a\u9ad8\u3002</li> <li>\u5747\u52fb\u5206\u4f48\uff1a\u96dc\u6e4a\u6f14\u7b97\u6cd5\u61c9\u4f7f\u5f97\u9375\u503c\u5c0d\u5747\u52fb\u5206\u4f48\u5728\u96dc\u6e4a\u8868\u4e2d\u3002\u5206\u4f48\u8d8a\u5747\u52fb\uff0c\u96dc\u6e4a\u885d\u7a81\u7684\u6a5f\u7387\u5c31\u8d8a\u4f4e\u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u65bc\u5be6\u73fe\u96dc\u6e4a\u8868\uff0c\u9084\u5ee3\u6cdb\u61c9\u7528\u65bc\u5176\u4ed6\u9818\u57df\u4e2d\u3002</p> <ul> <li>\u5bc6\u78bc\u5132\u5b58\uff1a\u70ba\u4e86\u4fdd\u8b77\u4f7f\u7528\u8005\u5bc6\u78bc\u7684\u5b89\u5168\uff0c\u7cfb\u7d71\u901a\u5e38\u4e0d\u6703\u76f4\u63a5\u5132\u5b58\u4f7f\u7528\u8005\u7684\u660e\u6587\u5bc6\u78bc\uff0c\u800c\u662f\u5132\u5b58\u5bc6\u78bc\u7684\u96dc\u6e4a\u503c\u3002\u7576\u7528\u6236\u8f38\u5165\u5bc6\u78bc\u6642\uff0c\u7cfb\u7d71\u6703\u5c0d\u8f38\u5165\u7684\u5bc6\u78bc\u8a08\u7b97\u96dc\u6e4a\u503c\uff0c\u7136\u5f8c\u8207\u5132\u5b58\u7684\u96dc\u6e4a\u503c\u9032\u884c\u6bd4\u8f03\u3002\u5982\u679c\u5169\u8005\u5339\u914d\uff0c\u90a3\u9ebc\u5bc6\u78bc\u5c31\u88ab\u8996\u70ba\u6b63\u78ba\u3002</li> <li>\u8cc7\u6599\u5b8c\u6574\u6027\u6aa2\u67e5\uff1a\u8cc7\u6599\u50b3\u9001\u65b9\u53ef\u4ee5\u8a08\u7b97\u8cc7\u6599\u7684\u96dc\u6e4a\u503c\u4e26\u5c07\u5176\u4e00\u540c\u50b3\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8a08\u7b97\u63a5\u6536\u5230\u7684\u8cc7\u6599\u7684\u96dc\u6e4a\u503c\uff0c\u4e26\u8207\u63a5\u6536\u5230\u7684\u96dc\u6e4a\u503c\u9032\u884c\u6bd4\u8f03\u3002\u5982\u679c\u5169\u8005\u5339\u914d\uff0c\u90a3\u9ebc\u8cc7\u6599\u5c31\u88ab\u8996\u70ba\u5b8c\u6574\u3002</li> </ul> <p>\u5c0d\u65bc\u5bc6\u78bc\u5b78\u7684\u76f8\u95dc\u61c9\u7528\uff0c\u70ba\u4e86\u9632\u6b62\u5f9e\u96dc\u6e4a\u503c\u63a8\u5c0e\u51fa\u539f\u59cb\u5bc6\u78bc\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u9700\u8981\u5177\u5099\u66f4\u9ad8\u7b49\u7d1a\u7684\u5b89\u5168\u7279\u6027\u3002</p> <ul> <li>\u55ae\u5411\u6027\uff1a\u7121\u6cd5\u900f\u904e\u96dc\u6e4a\u503c\u53cd\u63a8\u51fa\u95dc\u65bc\u8f38\u5165\u8cc7\u6599\u7684\u4efb\u4f55\u8cc7\u8a0a\u3002</li> <li>\u6297\u78b0\u649e\u6027\uff1a\u61c9\u7576\u6975\u96e3\u627e\u5230\u5169\u500b\u4e0d\u540c\u7684\u8f38\u5165\uff0c\u4f7f\u5f97\u5b83\u5011\u7684\u96dc\u6e4a\u503c\u76f8\u540c\u3002</li> <li>\u96ea\u5d29\u6548\u61c9\uff1a\u8f38\u5165\u7684\u5fae\u5c0f\u8b8a\u5316\u61c9\u7576\u5c0e\u81f4\u8f38\u51fa\u7684\u986f\u8457\u4e14\u4e0d\u53ef\u9810\u6e2c\u7684\u8b8a\u5316\u3002</li> </ul> <p>\u8acb\u6ce8\u610f\uff0c\u201c\u5747\u52fb\u5206\u4f48\u201d\u8207\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u5169\u500b\u7368\u7acb\u7684\u6982\u5ff5\uff0c\u6eff\u8db3\u5747\u52fb\u5206\u4f48\u4e0d\u4e00\u5b9a\u6eff\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u96a8\u6a5f\u8f38\u5165 <code>key</code> \u4e0b\uff0c\u96dc\u6e4a\u51fd\u5f0f <code>key % 100</code> \u53ef\u4ee5\u7522\u751f\u5747\u52fb\u5206\u4f48\u7684\u8f38\u51fa\u3002\u7136\u800c\u8a72\u96dc\u6e4a\u6f14\u7b97\u6cd5\u904e\u65bc\u7c21\u55ae\uff0c\u6240\u6709\u5f8c\u5169\u4f4d\u76f8\u7b49\u7684 <code>key</code> \u7684\u8f38\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u5f9e\u96dc\u6e4a\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684 <code>key</code> \uff0c\u5f9e\u800c\u7834\u89e3\u5bc6\u78bc\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2 \u00a0 \u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u8a2d\u8a08","text":"<p>\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u8a2d\u8a08\u662f\u4e00\u500b\u9700\u8981\u8003\u616e\u8a31\u591a\u56e0\u7d20\u7684\u8907\u96dc\u554f\u984c\u3002\u7136\u800c\u5c0d\u65bc\u67d0\u4e9b\u8981\u6c42\u4e0d\u9ad8\u7684\u5834\u666f\uff0c\u6211\u5011\u4e5f\u80fd\u8a2d\u8a08\u4e00\u4e9b\u7c21\u55ae\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\u3002</p> <ul> <li>\u52a0\u6cd5\u96dc\u6e4a\uff1a\u5c0d\u8f38\u5165\u7684\u6bcf\u500b\u5b57\u5143\u7684 ASCII \u78bc\u9032\u884c\u76f8\u52a0\uff0c\u5c07\u5f97\u5230\u7684\u7e3d\u548c\u4f5c\u70ba\u96dc\u6e4a\u503c\u3002</li> <li>\u4e58\u6cd5\u96dc\u6e4a\uff1a\u5229\u7528\u4e58\u6cd5\u7684\u4e0d\u76f8\u95dc\u6027\uff0c\u6bcf\u8f2a\u4e58\u4ee5\u4e00\u500b\u5e38\u6578\uff0c\u5c07\u5404\u500b\u5b57\u5143\u7684 ASCII \u78bc\u7d2f\u7a4d\u5230\u96dc\u6e4a\u503c\u4e2d\u3002</li> <li>\u4e92\u65a5\u6216\u96dc\u6e4a\uff1a\u5c07\u8f38\u5165\u8cc7\u6599\u7684\u6bcf\u500b\u5143\u7d20\u900f\u904e\u4e92\u65a5\u6216\u64cd\u4f5c\u7d2f\u7a4d\u5230\u4e00\u500b\u96dc\u6e4a\u503c\u4e2d\u3002</li> <li>\u65cb\u8f49\u96dc\u6e4a\uff1a\u5c07\u6bcf\u500b\u5b57\u5143\u7684 ASCII \u78bc\u7d2f\u7a4d\u5230\u4e00\u500b\u96dc\u6e4a\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u7a4d\u4e4b\u524d\u90fd\u6703\u5c0d\u96dc\u6e4a\u503c\u9032\u884c\u65cb\u8f49\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"\u52a0\u6cd5\u96dc\u6e4a\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"\u4e58\u6cd5\u96dc\u6e4a\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"\u4e92\u65a5\u6216\u96dc\u6e4a\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"\u65cb\u8f49\u96dc\u6e4a\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfun addHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfun xorHash(key: String): Int {\n    var hash = 0\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>### \u52a0\u6cd5\u96dc\u6e4a ###\ndef add_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash += c.ord }\n\n  hash % modulus\nend\n\n### \u4e58\u6cd5\u96dc\u6e4a ###\ndef mul_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = 31 * hash + c.ord }\n\n  hash % modulus\nend\n\n### \u4e92\u65a5\u6216\u96dc\u6e4a ###\ndef xor_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash ^= c.ord }\n\n  hash % modulus\nend\n\n### \u65cb\u8f49\u96dc\u6e4a ###\ndef rot_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.ord }\n\n  hash % modulus\nend\n</code></pre> simple_hash.zig<pre><code>[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u767c\u73fe\uff0c\u6bcf\u7a2e\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u6700\u5f8c\u4e00\u6b65\u90fd\u662f\u5c0d\u5927\u8cea\u6578 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u78ba\u4fdd\u96dc\u6e4a\u503c\u5728\u5408\u9069\u7684\u7bc4\u570d\u5167\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u70ba\u4ec0\u9ebc\u8981\u5f37\u8abf\u5c0d\u8cea\u6578\u53d6\u6a21\uff0c\u6216\u8005\u8aaa\u5c0d\u5408\u6578\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u9ebc\uff1f\u9019\u662f\u4e00\u500b\u6709\u8da3\u7684\u554f\u984c\u3002</p> <p>\u5148\u4e1f\u64f2\u7d50\u8ad6\uff1a\u4f7f\u7528\u5927\u8cea\u6578\u4f5c\u70ba\u6a21\u6578\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8b49\u96dc\u6e4a\u503c\u7684\u5747\u52fb\u5206\u4f48\u3002\u56e0\u70ba\u8cea\u6578\u4e0d\u8207\u5176\u4ed6\u6578\u5b57\u5b58\u5728\u516c\u7d04\u6578\uff0c\u53ef\u4ee5\u6e1b\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u7522\u751f\u7684\u9031\u671f\u6027\u6a21\u5f0f\uff0c\u5f9e\u800c\u907f\u514d\u96dc\u6e4a\u885d\u7a81\u3002</p> <p>\u8209\u500b\u4f8b\u5b50\uff0c\u5047\u8a2d\u6211\u5011\u9078\u64c7\u5408\u6578 \\(9\\) \u4f5c\u70ba\u6a21\u6578\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\uff0c\u90a3\u9ebc\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684 <code>key</code> \u90fd\u6703\u88ab\u5c0d\u6620\u5230 \\(0\\)\u3001\\(3\\)\u3001\\(6\\) \u9019\u4e09\u500b\u96dc\u6e4a\u503c\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>\u5982\u679c\u8f38\u5165 <code>key</code> \u6070\u597d\u6eff\u8db3\u9019\u7a2e\u7b49\u5dee\u6578\u5217\u7684\u8cc7\u6599\u5206\u4f48\uff0c\u90a3\u9ebc\u96dc\u6e4a\u503c\u5c31\u6703\u51fa\u73fe\u805a\u5806\u7a4d\uff0c\u5f9e\u800c\u52a0\u91cd\u96dc\u6e4a\u885d\u7a81\u3002\u73fe\u5728\uff0c\u5047\u8a2d\u5c07 <code>modulus</code> \u66ff\u63db\u70ba\u8cea\u6578 \\(13\\) \uff0c\u7531\u65bc <code>key</code> \u548c <code>modulus</code> \u4e4b\u9593\u4e0d\u5b58\u5728\u516c\u7d04\u6578\uff0c\u56e0\u6b64\u8f38\u51fa\u7684\u96dc\u6e4a\u503c\u7684\u5747\u52fb\u6027\u6703\u660e\u986f\u63d0\u5347\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u5982\u679c\u80fd\u5920\u4fdd\u8b49 <code>key</code> \u662f\u96a8\u6a5f\u5747\u52fb\u5206\u4f48\u7684\uff0c\u90a3\u9ebc\u9078\u64c7\u8cea\u6578\u6216\u8005\u5408\u6578\u4f5c\u70ba\u6a21\u6578\u90fd\u53ef\u4ee5\uff0c\u5b83\u5011\u90fd\u80fd\u8f38\u51fa\u5747\u52fb\u5206\u4f48\u7684\u96dc\u6e4a\u503c\u3002\u800c\u7576 <code>key</code> \u7684\u5206\u4f48\u5b58\u5728\u67d0\u7a2e\u9031\u671f\u6027\u6642\uff0c\u5c0d\u5408\u6578\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73fe\u805a\u96c6\u73fe\u8c61\u3002</p> <p>\u7e3d\u800c\u8a00\u4e4b\uff0c\u6211\u5011\u901a\u5e38\u9078\u53d6\u8cea\u6578\u4f5c\u70ba\u6a21\u6578\uff0c\u4e26\u4e14\u9019\u500b\u8cea\u6578\u6700\u597d\u8db3\u5920\u5927\uff0c\u4ee5\u5118\u53ef\u80fd\u6d88\u9664\u9031\u671f\u6027\u6a21\u5f0f\uff0c\u63d0\u5347\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u7a69\u5065\u6027\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3 \u00a0 \u5e38\u898b\u96dc\u6e4a\u6f14\u7b97\u6cd5","text":"<p>\u4e0d\u96e3\u767c\u73fe\uff0c\u4ee5\u4e0a\u4ecb\u7d39\u7684\u7c21\u55ae\u96dc\u6e4a\u6f14\u7b97\u6cd5\u90fd\u6bd4\u8f03\u201c\u8106\u5f31\u201d\uff0c\u9060\u9060\u6c92\u6709\u9054\u5230\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u8a2d\u8a08\u76ee\u6a19\u3002\u4f8b\u5982\uff0c\u7531\u65bc\u52a0\u6cd5\u548c\u4e92\u65a5\u6216\u6eff\u8db3\u4ea4\u63db\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u96dc\u6e4a\u548c\u4e92\u65a5\u6216\u96dc\u6e4a\u7121\u6cd5\u5340\u5206\u5167\u5bb9\u76f8\u540c\u4f46\u9806\u5e8f\u4e0d\u540c\u7684\u5b57\u4e32\uff0c\u9019\u53ef\u80fd\u6703\u52a0\u5287\u96dc\u6e4a\u885d\u7a81\uff0c\u4e26\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u554f\u984c\u3002</p> <p>\u5728\u5be6\u969b\u4e2d\uff0c\u6211\u5011\u901a\u5e38\u6703\u7528\u4e00\u4e9b\u6a19\u6e96\u96dc\u6e4a\u6f14\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002\u5b83\u5011\u53ef\u4ee5\u5c07\u4efb\u610f\u9577\u5ea6\u7684\u8f38\u5165\u8cc7\u6599\u5c0d\u6620\u5230\u6046\u5b9a\u9577\u5ea6\u7684\u96dc\u6e4a\u503c\u3002</p> <p>\u8fd1\u4e00\u500b\u4e16\u7d00\u4ee5\u4f86\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u8655\u5728\u4e0d\u65b7\u5347\u7d1a\u8207\u6700\u4f73\u5316\u7684\u904e\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u54e1\u52aa\u529b\u63d0\u5347\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u6548\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u54e1\u548c\u99ed\u5ba2\u5247\u81f4\u529b\u65bc\u5c0b\u627e\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u554f\u984c\u3002\u8868 6-2 \u5c55\u793a\u4e86\u5728\u5be6\u969b\u61c9\u7528\u4e2d\u5e38\u898b\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\u3002</p> <ul> <li>MD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u64ca\uff0c\u56e0\u6b64\u5b83\u5011\u88ab\u5404\u985e\u5b89\u5168\u61c9\u7528\u68c4\u7528\u3002</li> <li>SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73fe\u6210\u529f\u7684\u653b\u64ca\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u7528\u5728\u5404\u985e\u5b89\u5168\u61c9\u7528\u8207\u5354\u8b70\u4e2d\u3002</li> <li>SHA-3 \u76f8\u8f03 SHA-2 \u7684\u5be6\u73fe\u958b\u92b7\u66f4\u4f4e\u3001\u8a08\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u84cb\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002</li> </ul> <p> \u8868 6-2 \u00a0 \u5e38\u898b\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5 </p> MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u6642\u9593 1992 1995 2002 2008 \u8f38\u51fa\u9577\u5ea6 128 bit 160 bit 256/512 bit 224/256/384/512 bit \u96dc\u6e4a\u885d\u7a81 \u8f03\u591a \u8f03\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7d1a \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u64ca \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u64ca \u9ad8 \u9ad8 \u61c9\u7528 \u5df2\u88ab\u68c4\u7528\uff0c\u4ecd\u7528\u65bc\u8cc7\u6599\u5b8c\u6574\u6027\u6aa2\u67e5 \u5df2\u88ab\u68c4\u7528 \u52a0\u5bc6\u8ca8\u5e63\u4ea4\u6613\u9a57\u8b49\u3001\u6578\u5b57\u7c3d\u540d\u7b49 \u53ef\u7528\u65bc\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4 \u00a0 \u8cc7\u6599\u7d50\u69cb\u7684\u96dc\u6e4a\u503c","text":"<p>\u6211\u5011\u77e5\u9053\uff0c\u96dc\u6e4a\u8868\u7684 <code>key</code> \u53ef\u4ee5\u662f\u6574\u6578\u3001\u5c0f\u6578\u6216\u5b57\u4e32\u7b49\u8cc7\u6599\u578b\u5225\u3002\u7a0b\u5f0f\u8a9e\u8a00\u901a\u5e38\u6703\u70ba\u9019\u4e9b\u8cc7\u6599\u578b\u5225\u63d0\u4f9b\u5167\u5efa\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\uff0c\u7528\u65bc\u8a08\u7b97\u96dc\u6e4a\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u70ba\u4f8b\uff0c\u6211\u5011\u53ef\u4ee5\u547c\u53eb <code>hash()</code> \u51fd\u5f0f\u4f86\u8a08\u7b97\u5404\u7a2e\u8cc7\u6599\u578b\u5225\u7684\u96dc\u6e4a\u503c\u3002</p> <ul> <li>\u6574\u6578\u548c\u5e03\u6797\u91cf\u7684\u96dc\u6e4a\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002</li> <li>\u6d6e\u9ede\u6578\u548c\u5b57\u4e32\u7684\u96dc\u6e4a\u503c\u8a08\u7b97\u8f03\u70ba\u8907\u96dc\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u8acb\u81ea\u884c\u5b78\u7fd2\u3002</li> <li>\u5143\u7d44\u7684\u96dc\u6e4a\u503c\u662f\u5c0d\u5176\u4e2d\u6bcf\u4e00\u500b\u5143\u7d20\u9032\u884c\u96dc\u6e4a\uff0c\u7136\u5f8c\u5c07\u9019\u4e9b\u96dc\u6e4a\u503c\u7d44\u5408\u8d77\u4f86\uff0c\u5f97\u5230\u55ae\u4e00\u7684\u96dc\u6e4a\u503c\u3002</li> <li>\u7269\u4ef6\u7684\u96dc\u6e4a\u503c\u57fa\u65bc\u5176\u8a18\u61b6\u9ad4\u4f4d\u5740\u751f\u6210\u3002\u900f\u904e\u91cd\u5beb\u7269\u4ef6\u7684\u96dc\u6e4a\u65b9\u6cd5\uff0c\u53ef\u5be6\u73fe\u57fa\u65bc\u5167\u5bb9\u751f\u6210\u96dc\u6e4a\u503c\u3002</li> </ul> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u4e0d\u540c\u7a0b\u5f0f\u8a9e\u8a00\u7684\u5167\u5efa\u96dc\u6e4a\u503c\u8a08\u7b97\u51fd\u5f0f\u7684\u5b9a\u7fa9\u548c\u65b9\u6cd5\u4e0d\u540c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3\n\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u6797\u91cf True \u7684\u96dc\u6e4a\u503c\u70ba 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba 326484311674566659\n\nstr = \"Hello \u6f14\u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7d44 (12836, '\u5c0f\u54c8') \u7684\u96dc\u6e4a\u503c\u70ba 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u7bc0\u9ede\u7269\u4ef6 &lt;ListNode object at 0x1058fd810&gt; \u7684\u96dc\u6e4a\u503c\u70ba 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// \u5e03\u6797\u91cf 1 \u7684\u96dc\u6e4a\u503c\u70ba 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba 4614256650576692846\n\nstring str = \"Hello \u6f14\u7b97\u6cd5\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba 15466937326284535026\n\n// \u5728 C++ \u4e2d\uff0c\u5167\u5efa std:hash() \u50c5\u63d0\u4f9b\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u96dc\u6e4a\u503c\u8a08\u7b97\n// \u9663\u5217\u3001\u7269\u4ef6\u7684\u96dc\u6e4a\u503c\u8a08\u7b97\u9700\u8981\u81ea\u884c\u5be6\u73fe\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -1340954729\n\nString str = \"Hello \u6f14\u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u9663\u5217 [12836, \u5c0f\u54c8] \u7684\u96dc\u6e4a\u503c\u70ba 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u7bc0\u9ede\u7269\u4ef6 utils.ListNode@7dc5e7b4 \u7684\u96dc\u6e4a\u503c\u70ba 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -1340954729;\n\nstring str = \"Hello \u6f14\u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// \u9663\u5217 [12836, \u5c0f\u54c8] \u7684\u96dc\u6e4a\u503c\u70ba 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// \u7bc0\u9ede\u7269\u4ef6 0 \u7684\u96dc\u6e4a\u503c\u70ba 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go \u672a\u63d0\u4f9b\u5167\u5efa hash code \u51fd\u5f0f\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -2465384235396674631\n\nlet str = \"Hello \u6f14\u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u9663\u5217 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u96dc\u6e4a\u503c\u70ba -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u7bc0\u9ede\u7269\u4ef6 utils.ListNode \u7684\u96dc\u6e4a\u503c\u70ba -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5167\u5efa hash code \u51fd\u5f0f\n</code></pre> built_in_hash.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5167\u5efa hash code \u51fd\u5f0f\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u6797\u503c true \u7684\u96dc\u6e4a\u503c\u70ba 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba 2570631074981783\n\nString str = \"Hello \u6f14\u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u9663\u5217 [12836, \u5c0f\u54c8] \u7684\u96dc\u6e4a\u503c\u70ba 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u7bc0\u9ede\u7269\u4ef6 Instance of 'ListNode' \u7684\u96dc\u6e4a\u503c\u70ba 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba 2566941990314602357\n\nlet str = \"Hello \u6f14\u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// \u5143\u7d44 (12836, \"\u5c0f\u54c8\") \u7684\u96dc\u6e4a\u503c\u70ba 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// \u7bc0\u9ede\u7269\u4ef6 RefCell { value: ListNode { val: 42, next: None } } \u7684\u96dc\u6e4a\u503c\u70ba15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa hash code \u51fd\u5f0f\n</code></pre> built_in_hash.kt<pre><code>val num = 3\nval hashNum = num.hashCode()\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3\n\nval bol = true\nval hashBol = bol.hashCode()\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba 1231\n\nval dec = 3.14159\nval hashDec = dec.hashCode()\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -1340954729\n\nval str = \"Hello \u6f14\u7b97\u6cd5\"\nval hashStr = str.hashCode()\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -727081396\n\nval arr = arrayOf&lt;Any&gt;(12836, \"\u5c0f\u54c8\")\nval hashTup = arr.hashCode()\n// \u9663\u5217 [12836, \u5c0f\u54c8] \u7684\u96dc\u6e4a\u503c\u70ba 189568618\n\nval obj = ListNode(0)\nval hashObj = obj.hashCode()\n// \u7bc0\u9ede\u7269\u4ef6 utils.ListNode@1d81eb93 \u7684\u96dc\u6e4a\u503c\u70ba 495053715\n</code></pre> built_in_hash.rb<pre><code>num = 3\nhash_num = num.hash\n# \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba -4385856518450339636\n\nbol = true\nhash_bol = bol.hash\n# \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba -1617938112149317027\n\ndec = 3.14159\nhash_dec = dec.hash\n# \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -1479186995943067893\n\nstr = \"Hello \u6f14\u7b97\u6cd5\"\nhash_str = str.hash\n# \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -4075943250025831763\n\ntup = [12836, '\u5c0f\u54c8']\nhash_tup = tup.hash\n# \u5143\u7d44 (12836, '\u5c0f\u54c8') \u7684\u96dc\u6e4a\u503c\u70ba 1999544809202288822\n\nobj = ListNode.new(0)\nhash_obj = obj.hash\n# \u7bc0\u9ede\u7269\u4ef6 #&lt;ListNode:0x000078133140ab70&gt; \u7684\u96dc\u6e4a\u503c\u70ba 4302940560806366381\n</code></pre> built_in_hash.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5728\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u8b8a\u7269\u4ef6\u624d\u53ef\u4f5c\u70ba\u96dc\u6e4a\u8868\u7684 <code>key</code> \u3002\u5047\u5982\u6211\u5011\u5c07\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\u4f5c\u70ba <code>key</code> \uff0c\u7576\u4e32\u5217\u7684\u5167\u5bb9\u767c\u751f\u8b8a\u5316\u6642\uff0c\u5b83\u7684\u96dc\u6e4a\u503c\u4e5f\u96a8\u4e4b\u6539\u8b8a\uff0c\u6211\u5011\u5c31\u7121\u6cd5\u5728\u96dc\u6e4a\u8868\u4e2d\u67e5\u8a62\u5230\u539f\u5148\u7684 <code>value</code> \u4e86\u3002</p> <p>\u96d6\u7136\u81ea\u5b9a\u7fa9\u7269\u4ef6\uff08\u6bd4\u5982\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\uff09\u7684\u6210\u54e1\u8b8a\u6578\u662f\u53ef\u8b8a\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u96dc\u6e4a\u7684\u3002\u9019\u662f\u56e0\u70ba\u7269\u4ef6\u7684\u96dc\u6e4a\u503c\u901a\u5e38\u662f\u57fa\u65bc\u8a18\u61b6\u9ad4\u4f4d\u5740\u751f\u6210\u7684\uff0c\u5373\u4f7f\u7269\u4ef6\u7684\u5167\u5bb9\u767c\u751f\u4e86\u8b8a\u5316\uff0c\u4f46\u5b83\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u4e0d\u8b8a\uff0c\u96dc\u6e4a\u503c\u4ecd\u7136\u662f\u4e0d\u8b8a\u7684\u3002</p> <p>\u7d30\u5fc3\u7684\u4f60\u53ef\u80fd\u767c\u73fe\u5728\u4e0d\u540c\u63a7\u5236\u6aaf\u4e2d\u57f7\u884c\u7a0b\u5f0f\u6642\uff0c\u8f38\u51fa\u7684\u96dc\u6e4a\u503c\u662f\u4e0d\u540c\u7684\u3002\u9019\u662f\u56e0\u70ba Python \u76f4\u8b6f\u5668\u5728\u6bcf\u6b21\u555f\u52d5\u6642\uff0c\u90fd\u6703\u70ba\u5b57\u4e32\u96dc\u6e4a\u51fd\u5f0f\u52a0\u5165\u4e00\u500b\u96a8\u6a5f\u7684\u9e7d\uff08salt\uff09\u503c\u3002\u9019\u7a2e\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u64ca\uff0c\u63d0\u5347\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002</p>"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 \u96dc\u6e4a\u885d\u7a81","text":"<p>\u4e0a\u4e00\u7bc0\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u6cc1\u4e0b\u96dc\u6e4a\u51fd\u5f0f\u7684\u8f38\u5165\u7a7a\u9593\u9060\u5927\u65bc\u8f38\u51fa\u7a7a\u9593\uff0c\u56e0\u6b64\u7406\u8ad6\u4e0a\u96dc\u6e4a\u885d\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u6bd4\u5982\uff0c\u8f38\u5165\u7a7a\u9593\u70ba\u5168\u9ad4\u6574\u6578\uff0c\u8f38\u51fa\u7a7a\u9593\u70ba\u9663\u5217\u5bb9\u91cf\u5927\u5c0f\uff0c\u5247\u5fc5\u7136\u6709\u591a\u500b\u6574\u6578\u5c0d\u6620\u81f3\u540c\u4e00\u6876\u7d22\u5f15\u3002</p> <p>\u96dc\u6e4a\u885d\u7a81\u6703\u5c0e\u81f4\u67e5\u8a62\u7d50\u679c\u932f\u8aa4\uff0c\u56b4\u91cd\u5f71\u97ff\u96dc\u6e4a\u8868\u7684\u53ef\u7528\u6027\u3002\u70ba\u4e86\u89e3\u6c7a\u8a72\u554f\u984c\uff0c\u6bcf\u7576\u9047\u5230\u96dc\u6e4a\u885d\u7a81\u6642\uff0c\u6211\u5011\u5c31\u9032\u884c\u96dc\u6e4a\u8868\u64f4\u5bb9\uff0c\u76f4\u81f3\u885d\u7a81\u6d88\u5931\u70ba\u6b62\u3002\u6b64\u65b9\u6cd5\u7c21\u55ae\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u70ba\u96dc\u6e4a\u8868\u64f4\u5bb9\u9700\u8981\u9032\u884c\u5927\u91cf\u7684\u8cc7\u6599\u642c\u904b\u8207\u96dc\u6e4a\u503c\u8a08\u7b97\u3002\u70ba\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u5011\u53ef\u4ee5\u63a1\u7528\u4ee5\u4e0b\u7b56\u7565\u3002</p> <ol> <li>\u6539\u826f\u96dc\u6e4a\u8868\u8cc7\u6599\u7d50\u69cb\uff0c\u4f7f\u5f97\u96dc\u6e4a\u8868\u53ef\u4ee5\u5728\u51fa\u73fe\u96dc\u6e4a\u885d\u7a81\u6642\u6b63\u5e38\u5de5\u4f5c\u3002</li> <li>\u50c5\u5728\u5fc5\u8981\u6642\uff0c\u5373\u7576\u96dc\u6e4a\u885d\u7a81\u6bd4\u8f03\u56b4\u91cd\u6642\uff0c\u624d\u57f7\u884c\u64f4\u5bb9\u64cd\u4f5c\u3002</li> </ol> <p>\u96dc\u6e4a\u8868\u7684\u7d50\u69cb\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u201c\u93c8\u5f0f\u4f4d\u5740\u201d\u548c\u201c\u958b\u653e\u5b9a\u5740\u201d\u3002</p>"},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1 \u00a0 \u93c8\u5f0f\u4f4d\u5740","text":"<p>\u5728\u539f\u59cb\u96dc\u6e4a\u8868\u4e2d\uff0c\u6bcf\u500b\u6876\u50c5\u80fd\u5132\u5b58\u4e00\u500b\u9375\u503c\u5c0d\u3002\u93c8\u5f0f\u4f4d\u5740\uff08separate chaining\uff09\u5c07\u55ae\u500b\u5143\u7d20\u8f49\u63db\u70ba\u93c8\u7d50\u4e32\u5217\uff0c\u5c07\u9375\u503c\u5c0d\u4f5c\u70ba\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\uff0c\u5c07\u6240\u6709\u767c\u751f\u885d\u7a81\u7684\u9375\u503c\u5c0d\u90fd\u5132\u5b58\u5728\u540c\u4e00\u93c8\u7d50\u4e32\u5217\u4e2d\u3002\u5716 6-5 \u5c55\u793a\u4e86\u4e00\u500b\u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868\u7684\u4f8b\u5b50\u3002</p> <p></p> <p> \u5716 6-5 \u00a0 \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 </p> <p>\u57fa\u65bc\u93c8\u5f0f\u4f4d\u5740\u5be6\u73fe\u7684\u96dc\u6e4a\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u767c\u751f\u4e86\u4ee5\u4e0b\u8b8a\u5316\u3002</p> <ul> <li>\u67e5\u8a62\u5143\u7d20\uff1a\u8f38\u5165 <code>key</code> \uff0c\u7d93\u904e\u96dc\u6e4a\u51fd\u5f0f\u5f97\u5230\u6876\u7d22\u5f15\uff0c\u5373\u53ef\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u982d\u7bc0\u9ede\uff0c\u7136\u5f8c\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4e26\u5c0d\u6bd4 <code>key</code> \u4ee5\u67e5\u8a62\u76ee\u6a19\u9375\u503c\u5c0d\u3002</li> <li>\u65b0\u589e\u5143\u7d20\uff1a\u9996\u5148\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u982d\u7bc0\u9ede\uff0c\u7136\u5f8c\u5c07\u7bc0\u9ede\uff08\u9375\u503c\u5c0d\uff09\u65b0\u589e\u5230\u93c8\u7d50\u4e32\u5217\u4e2d\u3002</li> <li>\u522a\u9664\u5143\u7d20\uff1a\u6839\u64da\u96dc\u6e4a\u51fd\u5f0f\u7684\u7d50\u679c\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u982d\u90e8\uff0c\u63a5\u8457\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4ee5\u67e5\u8a62\u76ee\u6a19\u7bc0\u9ede\u4e26\u5c07\u5176\u522a\u9664\u3002</li> </ul> <p>\u93c8\u5f0f\u4f4d\u5740\u5b58\u5728\u4ee5\u4e0b\u4fb7\u9650\u6027\u3002</p> <ul> <li>\u4f54\u7528\u7a7a\u9593\u589e\u5927\uff1a\u93c8\u7d50\u4e32\u5217\u5305\u542b\u7bc0\u9ede\u6307\u6a19\uff0c\u5b83\u76f8\u6bd4\u9663\u5217\u66f4\u52a0\u8017\u8cbb\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> <li>\u67e5\u8a62\u6548\u7387\u964d\u4f4e\uff1a\u56e0\u70ba\u9700\u8981\u7dda\u6027\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4f86\u67e5\u8a62\u5c0d\u61c9\u5143\u7d20\u3002</li> </ul> <p>\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u7d66\u51fa\u4e86\u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868\u7684\u7c21\u55ae\u5be6\u73fe\uff0c\u9700\u8981\u6ce8\u610f\u5169\u9ede\u3002</p> <ul> <li>\u4f7f\u7528\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\u4ee3\u66ff\u93c8\u7d50\u4e32\u5217\uff0c\u5f9e\u800c\u7c21\u5316\u7a0b\u5f0f\u78bc\u3002\u5728\u9019\u7a2e\u8a2d\u5b9a\u4e0b\uff0c\u96dc\u6e4a\u8868\uff08\u9663\u5217\uff09\u5305\u542b\u591a\u500b\u6876\uff0c\u6bcf\u500b\u6876\u90fd\u662f\u4e00\u500b\u4e32\u5217\u3002</li> <li>\u4ee5\u4e0b\u5be6\u73fe\u5305\u542b\u96dc\u6e4a\u8868\u64f4\u5bb9\u65b9\u6cd5\u3002\u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e \\(\\frac{2}{3}\\) \u6642\uff0c\u6211\u5011\u5c07\u96dc\u6e4a\u8868\u64f4\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"\u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self.size = 0  # \u9375\u503c\u5c0d\u6578\u91cf\n        self.capacity = 4  # \u96dc\u6e4a\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n        self.extend_ratio = 2  # \u64f4\u5bb9\u500d\u6578\n        self.buckets = [[] for _ in range(self.capacity)]  # \u6876\u9663\u5217\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u96dc\u6e4a\u51fd\u5f0f\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8ca0\u8f09\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"\u67e5\u8a62\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u65b0\u589e\u64cd\u4f5c\"\"\"\n        # \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u522a\u9664\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"\u64f4\u5bb9\u96dc\u6e4a\u8868\"\"\"\n        # \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        buckets = self.buckets\n        # \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u5217\u5370\u96dc\u6e4a\u8868\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n  private:\n    int size;                       // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity;                   // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres;               // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio;                // \u64f4\u5bb9\u500d\u6578\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // \u6876\u9663\u5217\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* \u6790\u69cb\u65b9\u6cd5 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // \u91cb\u653e\u8a18\u61b6\u9ad4\n                delete pair;\n            }\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n        return \"\";\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // \u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n                delete tmp;                       // \u91cb\u653e\u8a18\u61b6\u9ad4\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // \u91cb\u653e\u8a18\u61b6\u9ad4\n                delete pair;\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    int size; // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio; // \u64f4\u5bb9\u500d\u6578\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    void put(int key, String val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    int size; // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio; // \u64f4\u5bb9\u500d\u6578\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void Extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\ntype hashMapChaining struct {\n    size        int      // \u9375\u503c\u5c0d\u6578\u91cf\n    capacity    int      // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    loadThres   float64  // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    extendRatio int      // \u64f4\u5bb9\u500d\u6578\n    buckets     [][]pair // \u6876\u9663\u5217\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* \u8ca0\u8f09\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n    return \"\"\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // \u5207\u7247\u522a\u9664\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\nfunc (m *hashMapChaining) extend() {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    var size: Int // \u9375\u503c\u5c0d\u6578\u91cf\n    var capacity: Int // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    var extendRatio: Int // \u64f4\u5bb9\u500d\u6578\n    var buckets: [[Pair]] // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de nil\n        return nil\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    func extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    #size; // \u9375\u503c\u5c0d\u6578\u91cf\n    #capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    #loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    #extendRatio; // \u64f4\u5bb9\u500d\u6578\n    #buckets; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    put(key, val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    #extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    #size: number; // \u9375\u503c\u5c0d\u6578\u91cf\n    #capacity: number; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    #loadThres: number; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    #extendRatio: number; // \u64f4\u5bb9\u500d\u6578\n    #buckets: Pair[][]; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    #extend(): void {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n  late int size; // \u9375\u503c\u5c0d\u6578\u91cf\n  late int capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n  late double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n  late int extendRatio; // \u64f4\u5bb9\u500d\u6578\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u9663\u5217\n\n  /* \u5efa\u69cb\u5b50 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* \u96dc\u6e4a\u51fd\u5f0f */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* \u8ca0\u8f09\u56e0\u5b50 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* \u67e5\u8a62\u64cd\u4f5c */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u65b0\u589e\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* \u522a\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n  void extend() {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u5217\u5370\u96dc\u6e4a\u8868 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nstruct HashMapChaining {\n    size: i32,\n    capacity: i32,\n    load_thres: f32,\n    extend_ratio: i32,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* \u5efa\u69cb\u5b50 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity as usize\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for i in 0..bucket.len() {\n            if bucket[i].key == key {\n                let pair = bucket.remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de None\n        None\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    fn extend(&amp;mut self) {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        let buckets_tmp = std::mem::replace(&amp;mut self.buckets, vec![]);\n\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val;\n                return;\n            }\n        }\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        let pair = Pair { key, val };\n        bucket.push(pair);\n        self.size += 1;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;self.buckets[index];\n\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for pair in bucket {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de None\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\ntypedef struct {\n    int size;         // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity;     // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio;  // \u64f4\u5bb9\u500d\u6578\n    Node **buckets;   // \u6876\u9663\u5217\n} HashMapChaining;\n\n/* \u5efa\u69cb\u5b50 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8ca0\u8f09\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // \u91cb\u653e\u8a18\u61b6\u9ad4\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // \u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // \u91cb\u653e\u8a18\u61b6\u9ad4\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    var size: Int // \u9375\u503c\u5c0d\u6578\u91cf\n    var capacity: Int // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    val loadThres: Double // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    val extendRatio: Int // \u64f4\u5bb9\u500d\u6578\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (pair in bucket) {\n            if (pair.key == key) return pair._val\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fun put(key: Int, _val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair._val = _val\n                return\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        val pair = Pair(key, _val)\n        bucket.add(pair)\n        size++\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    fun extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio\n        // mutablelist \u7121\u56fa\u5b9a\u5927\u5c0f\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair._val\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>### \u9375\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 ###\nclass HashMapChaining\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @size = 0 # \u9375\u503c\u5c0d\u6578\u91cf\n    @capacity = 4 # \u96dc\u6e4a\u8868\u5bb9\u91cf\n    @load_thres = 2.0 / 3.0 # \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    @extend_ratio = 2 # \u64f4\u5bb9\u500d\u6578\n    @buckets = Array.new(@capacity) { [] } # \u6876\u9663\u5217\n  end\n\n  ### \u96dc\u6e4a\u51fd\u5f0f ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### \u8ca0\u8f09\u56e0\u5b50 ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### \u67e5\u8a62\u64cd\u4f5c ###\n  def get(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    for pair in bucket\n      return pair.val if pair.key == key\n    end\n    # \u82e5\u672a\u627e\u5230 key , \u5247\u8fd4\u56de nil\n    nil\n  end\n\n  ### \u65b0\u589e\u64cd\u4f5c ###\n  def put(key, val)\n    # \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    extend if load_factor &gt; @load_thres\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n    for pair in bucket\n      if pair.key == key\n        pair.val = val\n        return\n      end\n    end\n    # \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n    pair = Pair.new(key, val)\n    bucket &lt;&lt; pair\n    @size += 1\n  end\n\n  ### \u522a\u9664\u64cd\u4f5c ###\n  def remove(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n    for pair in bucket\n      if pair.key == key\n        bucket.delete(pair)\n        @size -= 1\n        break\n      end\n    end\n  end\n\n  ### \u64f4\u5bb9\u96dc\u6e4a\u8868 ###\n  def extend\n    # \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    buckets = @buckets\n    # \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity) { [] }\n    @size = 0\n    # \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for bucket in buckets\n      for pair in bucket\n        put(pair.key, pair.val)\n      end\n    end\n  end\n\n  ### \u5217\u5370\u96dc\u6e4a\u8868 ###\n  def print\n    for bucket in @buckets\n      res = []\n      for pair in bucket\n        res &lt;&lt; \"#{pair.key} -&gt; #{pair.val}\"\n      end\n      pp res\n    end\n  end\nend\n</code></pre> hash_map_chaining.zig<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7576\u93c8\u7d50\u4e32\u5217\u5f88\u9577\u6642\uff0c\u67e5\u8a62\u6548\u7387 \\(O(n)\\) \u5f88\u5dee\u3002\u6b64\u6642\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u63db\u70ba\u201cAVL \u6a39\u201d\u6216\u201c\u7d05\u9ed1\u6a39\u201d\uff0c\u5f9e\u800c\u5c07\u67e5\u8a62\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2 \u00a0 \u958b\u653e\u5b9a\u5740","text":"<p>\u958b\u653e\u5b9a\u5740\uff08open addressing\uff09\u4e0d\u5f15\u5165\u984d\u5916\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u800c\u662f\u900f\u904e\u201c\u591a\u6b21\u63a2\u6e2c\u201d\u4f86\u8655\u7406\u96dc\u6e4a\u885d\u7a81\uff0c\u63a2\u6e2c\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7dda\u6027\u63a2\u67e5\u3001\u5e73\u65b9\u63a2\u6e2c\u548c\u591a\u6b21\u96dc\u6e4a\u7b49\u3002</p> <p>\u4e0b\u9762\u4ee5\u7dda\u6027\u63a2\u67e5\u70ba\u4f8b\uff0c\u4ecb\u7d39\u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868\u7684\u5de5\u4f5c\u6a5f\u5236\u3002</p>"},{"location":"chapter_hashing/hash_collision/#1","title":"1. \u00a0 \u7dda\u6027\u63a2\u67e5","text":"<p>\u7dda\u6027\u63a2\u67e5\u63a1\u7528\u56fa\u5b9a\u6b65\u9577\u7684\u7dda\u6027\u641c\u5c0b\u4f86\u9032\u884c\u63a2\u6e2c\uff0c\u5176\u64cd\u4f5c\u65b9\u6cd5\u8207\u666e\u901a\u96dc\u6e4a\u8868\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u82e5\u767c\u73fe\u6876\u5167\u5df2\u6709\u5143\u7d20\uff0c\u5247\u5f9e\u885d\u7a81\u4f4d\u7f6e\u5411\u5f8c\u7dda\u6027\u8d70\u8a2a\uff08\u6b65\u9577\u901a\u5e38\u70ba \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u6876\uff0c\u5c07\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002</li> <li>\u67e5\u8a62\u5143\u7d20\uff1a\u82e5\u767c\u73fe\u96dc\u6e4a\u885d\u7a81\uff0c\u5247\u4f7f\u7528\u76f8\u540c\u6b65\u9577\u5411\u5f8c\u9032\u884c\u7dda\u6027\u8d70\u8a2a\uff0c\u76f4\u5230\u627e\u5230\u5c0d\u61c9\u5143\u7d20\uff0c\u8fd4\u56de <code>value</code> \u5373\u53ef\uff1b\u5982\u679c\u9047\u5230\u7a7a\u6876\uff0c\u8aaa\u660e\u76ee\u6a19\u5143\u7d20\u4e0d\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u8fd4\u56de <code>None</code> \u3002</li> </ul> <p>\u5716 6-6 \u5c55\u793a\u4e86\u958b\u653e\u5b9a\u5740\uff08\u7dda\u6027\u63a2\u67e5\uff09\u96dc\u6e4a\u8868\u7684\u9375\u503c\u5c0d\u5206\u4f48\u3002\u6839\u64da\u6b64\u96dc\u6e4a\u51fd\u5f0f\uff0c\u6700\u5f8c\u5169\u4f4d\u76f8\u540c\u7684 <code>key</code> \u90fd\u6703\u88ab\u5c0d\u6620\u5230\u76f8\u540c\u7684\u6876\u3002\u800c\u900f\u904e\u7dda\u6027\u63a2\u67e5\uff0c\u5b83\u5011\u88ab\u4f9d\u6b21\u5132\u5b58\u5728\u8a72\u6876\u4ee5\u53ca\u4e4b\u4e0b\u7684\u6876\u4e2d\u3002</p> <p></p> <p> \u5716 6-6 \u00a0 \u958b\u653e\u5b9a\u5740\uff08\u7dda\u6027\u63a2\u67e5\uff09\u96dc\u6e4a\u8868\u7684\u9375\u503c\u5c0d\u5206\u4f48 </p> <p>\u7136\u800c\uff0c\u7dda\u6027\u63a2\u67e5\u5bb9\u6613\u7522\u751f\u201c\u805a\u96c6\u73fe\u8c61\u201d\u3002\u5177\u9ad4\u4f86\u8aaa\uff0c\u9663\u5217\u4e2d\u9023\u7e8c\u88ab\u4f54\u7528\u7684\u4f4d\u7f6e\u8d8a\u9577\uff0c\u9019\u4e9b\u9023\u7e8c\u4f4d\u7f6e\u767c\u751f\u96dc\u6e4a\u885d\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u5f9e\u800c\u9032\u4e00\u6b65\u4fc3\u4f7f\u8a72\u4f4d\u7f6e\u7684\u805a\u5806\u7a4d\u751f\u9577\uff0c\u5f62\u6210\u60e1\u6027\u8ff4\u5708\uff0c\u6700\u7d42\u5c0e\u81f4\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u52a3\u5316\u3002</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6211\u5011\u4e0d\u80fd\u5728\u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868\u4e2d\u76f4\u63a5\u522a\u9664\u5143\u7d20\u3002\u9019\u662f\u56e0\u70ba\u522a\u9664\u5143\u7d20\u6703\u5728\u9663\u5217\u5167\u7522\u751f\u4e00\u500b\u7a7a\u6876 <code>None</code> \uff0c\u800c\u7576\u67e5\u8a62\u5143\u7d20\u6642\uff0c\u7dda\u6027\u63a2\u67e5\u5230\u8a72\u7a7a\u6876\u5c31\u6703\u8fd4\u56de\uff0c\u56e0\u6b64\u5728\u8a72\u7a7a\u6876\u4e4b\u4e0b\u7684\u5143\u7d20\u90fd\u7121\u6cd5\u518d\u88ab\u8a2a\u554f\u5230\uff0c\u7a0b\u5f0f\u53ef\u80fd\u8aa4\u5224\u9019\u4e9b\u5143\u7d20\u4e0d\u5b58\u5728\uff0c\u5982\u5716 6-7 \u6240\u793a\u3002</p> <p></p> <p> \u5716 6-7 \u00a0 \u5728\u958b\u653e\u5b9a\u5740\u4e2d\u522a\u9664\u5143\u7d20\u5c0e\u81f4\u7684\u67e5\u8a62\u554f\u984c </p> <p>\u70ba\u4e86\u89e3\u6c7a\u8a72\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u63a1\u7528\u61f6\u522a\u9664\uff08lazy deletion\uff09\u6a5f\u5236\uff1a\u5b83\u4e0d\u76f4\u63a5\u5f9e\u96dc\u6e4a\u8868\u4e2d\u79fb\u9664\u5143\u7d20\uff0c\u800c\u662f\u5229\u7528\u4e00\u500b\u5e38\u6578 <code>TOMBSTONE</code> \u4f86\u6a19\u8a18\u9019\u500b\u6876\u3002\u5728\u8a72\u6a5f\u5236\u4e0b\uff0c<code>None</code> \u548c <code>TOMBSTONE</code> \u90fd\u4ee3\u8868\u7a7a\u6876\uff0c\u90fd\u53ef\u4ee5\u653e\u7f6e\u9375\u503c\u5c0d\u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u7dda\u6027\u63a2\u67e5\u5230 <code>TOMBSTONE</code> \u6642\u61c9\u8a72\u7e7c\u7e8c\u8d70\u8a2a\uff0c\u56e0\u70ba\u5176\u4e4b\u4e0b\u53ef\u80fd\u9084\u5b58\u5728\u9375\u503c\u5c0d\u3002</p> <p>\u7136\u800c\uff0c\u61f6\u522a\u9664\u53ef\u80fd\u6703\u52a0\u901f\u96dc\u6e4a\u8868\u7684\u6548\u80fd\u9000\u5316\u3002\u9019\u662f\u56e0\u70ba\u6bcf\u6b21\u522a\u9664\u64cd\u4f5c\u90fd\u6703\u7522\u751f\u4e00\u500b\u522a\u9664\u6a19\u8a18\uff0c\u96a8\u8457 <code>TOMBSTONE</code> \u7684\u589e\u52a0\uff0c\u641c\u5c0b\u6642\u9593\u4e5f\u6703\u589e\u52a0\uff0c\u56e0\u70ba\u7dda\u6027\u63a2\u67e5\u53ef\u80fd\u9700\u8981\u8df3\u904e\u591a\u500b <code>TOMBSTONE</code> \u624d\u80fd\u627e\u5230\u76ee\u6a19\u5143\u7d20\u3002</p> <p>\u70ba\u6b64\uff0c\u8003\u616e\u7dda\u4e0a\u6027\u63a2\u67e5\u4e2d\u8a18\u9304\u9047\u5230\u7684\u9996\u500b <code>TOMBSTONE</code> \u7684\u7d22\u5f15\uff0c\u4e26\u5c07\u641c\u5c0b\u5230\u7684\u76ee\u6a19\u5143\u7d20\u8207\u8a72 <code>TOMBSTONE</code> \u4ea4\u63db\u4f4d\u7f6e\u3002\u9019\u6a23\u505a\u7684\u597d\u8655\u662f\u7576\u6bcf\u6b21\u67e5\u8a62\u6216\u65b0\u589e\u5143\u7d20\u6642\uff0c\u5143\u7d20\u6703\u88ab\u79fb\u52d5\u81f3\u8ddd\u96e2\u7406\u60f3\u4f4d\u7f6e\uff08\u63a2\u6e2c\u8d77\u59cb\u9ede\uff09\u66f4\u8fd1\u7684\u6876\uff0c\u5f9e\u800c\u6700\u4f73\u5316\u67e5\u8a62\u6548\u7387\u3002</p> <p>\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e86\u4e00\u500b\u5305\u542b\u61f6\u522a\u9664\u7684\u958b\u653e\u5b9a\u5740\uff08\u7dda\u6027\u63a2\u67e5\uff09\u96dc\u6e4a\u8868\u3002\u70ba\u4e86\u66f4\u52a0\u5145\u5206\u5730\u4f7f\u7528\u96dc\u6e4a\u8868\u7684\u7a7a\u9593\uff0c\u6211\u5011\u5c07\u96dc\u6e4a\u8868\u770b\u4f5c\u4e00\u500b\u201c\u74b0\u5f62\u9663\u5217\u201d\uff0c\u7576\u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u56de\u5230\u982d\u90e8\u7e7c\u7e8c\u8d70\u8a2a\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"\u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self.size = 0  # \u9375\u503c\u5c0d\u6578\u91cf\n        self.capacity = 4  # \u96dc\u6e4a\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n        self.extend_ratio = 2  # \u64f4\u5bb9\u500d\u6578\n        self.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u9663\u5217\n        self.TOMBSTONE = Pair(-1, \"-1\")  # \u522a\u9664\u6a19\u8a18\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u96dc\u6e4a\u51fd\u5f0f\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8ca0\u8f09\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"\u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while self.buckets[index] is not None:\n            # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].key == key:\n                # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                return index  # \u8fd4\u56de\u6876\u7d22\u5f15\n            # \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % self.capacity\n        # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8a62\u64cd\u4f5c\"\"\"\n        # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u65b0\u589e\u64cd\u4f5c\"\"\"\n        # \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u522a\u9664\u64cd\u4f5c\"\"\"\n        # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"\u64f4\u5bb9\u96dc\u6e4a\u8868\"\"\"\n        # \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        buckets_tmp = self.buckets\n        # \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u5217\u5370\u96dc\u6e4a\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity = 4;                     // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    const double loadThres = 2.0 / 3.0;     // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    const int extendRatio = 2;            // \u64f4\u5bb9\u500d\u6578\n    vector&lt;Pair *&gt; buckets;               // \u6876\u9663\u5217\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u522a\u9664\u6a19\u8a18\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* \u6790\u69cb\u65b9\u6cd5 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (buckets[index] != nullptr) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (buckets[index]-&gt;key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    string get(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n        return \"\";\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    private int size; // \u9375\u503c\u5c0d\u6578\u91cf\n    private int capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    private final double loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    private final int extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n    private Pair[] buckets; // \u6876\u9663\u5217\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public String get(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void put(int key, String val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    private void extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    int size; // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n    Pair[] buckets; // \u6876\u9663\u5217\n    Pair TOMBSTONE = new(-1, \"-1\"); // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public string? Get(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void Extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\ntype hashMapOpenAddressing struct {\n    size        int     // \u9375\u503c\u5c0d\u6578\u91cf\n    capacity    int     // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    loadThres   float64 // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    extendRatio int     // \u64f4\u5bb9\u500d\u6578\n    buckets     []*pair // \u6876\u9663\u5217\n    TOMBSTONE   *pair   // \u522a\u9664\u6a19\u8a18\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // \u6839\u64da\u9375\u8a08\u7b97\u96dc\u6e4a\u503c\n}\n\n/* \u8ca0\u8f09\u56e0\u5b50 */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // \u8a08\u7b97\u7576\u524d\u8ca0\u8f09\u56e0\u5b50\n}\n\n/* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // \u7372\u53d6\u521d\u59cb\u7d22\u5f15\n    firstTombstone := -1     // \u8a18\u9304\u9047\u5230\u7684\u7b2c\u4e00\u500bTOMBSTONE\u7684\u4f4d\u7f6e\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n            }\n            return index // \u8fd4\u56de\u627e\u5230\u7684\u7d22\u5f15\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\u7684\u4f4d\u7f6e\n        }\n        index = (index + 1) % h.capacity // \u7dda\u6027\u63a2\u67e5\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    }\n    return \"\" // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de \"\"\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    }\n    index := h.findBucket(key) // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        h.size++\n    } else {\n        h.buckets[index].val = val // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val\n    }\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        h.size--\n    }\n}\n\n/* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    h.capacity *= h.extendRatio           // \u66f4\u65b0\u5bb9\u91cf\n    h.buckets = make([]*pair, h.capacity) // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    h.size = 0                            // \u91cd\u7f6e\u5927\u5c0f\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    var size: Int // \u9375\u503c\u5c0d\u6578\u91cf\n    var capacity: Int // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    var extendRatio: Int // \u64f4\u5bb9\u500d\u6578\n    var buckets: [Pair?] // \u6876\u9663\u5217\n    var TOMBSTONE: Pair // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while buckets[index] != nil {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if buckets[index]!.key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return nil\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    func extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    #size; // \u9375\u503c\u5c0d\u6578\u91cf\n    #capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    #loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    #extendRatio; // \u64f4\u5bb9\u500d\u6578\n    #buckets; // \u6876\u9663\u5217\n    #TOMBSTONE; // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.#size = 0; // \u9375\u503c\u5c0d\u6578\u91cf\n        this.#capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n        this.#loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n        this.#extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n        this.#buckets = Array(this.#capacity).fill(null); // \u6876\u9663\u5217\n        this.#TOMBSTONE = new Pair(-1, '-1'); // \u522a\u9664\u6a19\u8a18\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (this.#buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (this.#buckets[index].key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % this.#capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    put(key, val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    remove(key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    #extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    private size: number; // \u9375\u503c\u5c0d\u6578\u91cf\n    private capacity: number; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    private loadThres: number; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    private extendRatio: number; // \u64f4\u5bb9\u500d\u6578\n    private buckets: Array&lt;Pair | null&gt;; // \u6876\u9663\u5217\n    private TOMBSTONE: Pair; // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.size = 0; // \u9375\u503c\u5c0d\u6578\u91cf\n        this.capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n        this.loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n        this.extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n        this.buckets = Array(this.capacity).fill(null); // \u6876\u9663\u5217\n        this.TOMBSTONE = new Pair(-1, '-1'); // \u522a\u9664\u6a19\u8a18\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (this.buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (this.buckets[index]!.key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % this.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key: number): string | null {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    private extend(): void {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        const bucketsTmp = this.buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n  late int _size; // \u9375\u503c\u5c0d\u6578\u91cf\n  int _capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n  double _loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n  int _extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n  late List&lt;Pair?&gt; _buckets; // \u6876\u9663\u5217\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u522a\u9664\u6a19\u8a18\n\n  /* \u5efa\u69cb\u5b50 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* \u96dc\u6e4a\u51fd\u5f0f */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* \u8ca0\u8f09\u56e0\u5b50 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n    while (_buckets[index] != null) {\n      // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n      if (_buckets[index]!.key == key) {\n        // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n        }\n        return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n      }\n      // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n      index = (index + 1) % _capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* \u67e5\u8a62\u64cd\u4f5c */\n  String? get(int key) {\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u65b0\u589e\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* \u522a\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n  void extend() {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u5217\u5370\u96dc\u6e4a\u8868 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nstruct HashMapOpenAddressing {\n    size: usize,                // \u9375\u503c\u5c0d\u6578\u91cf\n    capacity: usize,            // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    load_thres: f64,            // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    extend_ratio: usize,        // \u64f4\u5bb9\u500d\u6578\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // \u6876\u9663\u5217\n    TOMBSTONE: Option&lt;Pair&gt;,    // \u522a\u9664\u6a19\u8a18\n}\n\nimpl HashMapOpenAddressing {\n    /* \u5efa\u69cb\u5b50 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while self.buckets[index].is_some() {\n            // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u5efa\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % self.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        None\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    fn extend(&amp;mut self) {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        let buckets_tmp = self.buckets.clone();\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\ntypedef struct {\n    int size;         // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity;     // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio;  // \u64f4\u5bb9\u500d\u6578\n    Pair **buckets;   // \u6876\u9663\u5217\n    Pair *TOMBSTONE;  // \u522a\u9664\u6a19\u8a18\n} HashMapOpenAddressing;\n\n/* \u5efa\u69cb\u5b50 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap-&gt;TOMBSTONE);\n    free(hashMap);\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8ca0\u8f09\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n            }\n            return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n        }\n        // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n    return \"\";\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;size = 0;\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    private var size: Int               // \u9375\u503c\u5c0d\u6578\u91cf\n    private var capacity: Int           // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    private val loadThres: Double       // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    private val extendRatio: Int        // \u64f4\u5bb9\u500d\u6578\n    private var buckets: Array&lt;Pair?&gt;   // \u6876\u9663\u5217\n    private val TOMBSTONE: Pair         // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = arrayOfNulls(capacity)\n        TOMBSTONE = Pair(-1, \"-1\")\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (buckets[index]?.key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?._val\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fun put(key: Int, _val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!._val = _val\n            return\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = Pair(key, _val)\n        size++\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    fun extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair._val}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>### \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 ###\nclass HashMapOpenAddressing\n  TOMBSTONE = Pair.new(-1, '-1') # \u522a\u9664\u6a19\u8a18\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @size = 0 # \u9375\u503c\u5c0d\u6578\u91cf\n    @capacity = 4 # \u96dc\u6e4a\u8868\u5bb9\u91cf\n    @load_thres = 2.0 / 3.0 # \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    @extend_ratio = 2 # \u64f4\u5bb9\u500d\u6578\n    @buckets = Array.new(@capacity) # \u6876\u9663\u5217\n  end\n\n  ### \u96dc\u6e4a\u51fd\u5f0f ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### \u8ca0\u8f09\u56e0\u5b50 ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 ###\n  def find_bucket(key)\n    index = hash_func(key)\n    first_tombstone = -1\n    # \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n    while !@buckets[index].nil?\n      # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n      if @buckets[index].key == key\n        # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n        if first_tombstone != -1\n          @buckets[first_tombstone] = @buckets[index]\n          @buckets[index] = TOMBSTONE\n          return first_tombstone # \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n        end\n        return index # \u8fd4\u56de\u6876\u7d22\u5f15\n      end\n      # \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n      first_tombstone = index if first_tombstone == -1 &amp;&amp; @buckets[index] == TOMBSTONE\n      # \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n      index = (index + 1) % @capacity\n    end\n    # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n    first_tombstone == -1 ? index : first_tombstone\n  end\n\n  ### \u67e5\u8a62\u64cd\u4f5c ###\n  def get(key)\n    # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    index = find_bucket(key)\n    # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    return @buckets[index].val unless [nil, TOMBSTONE].include?(@buckets[index])\n    # \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de nil\n    nil\n  end\n\n  ### \u65b0\u589e\u64cd\u4f5c ###\n  def put(key, val)\n    # \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    extend if load_factor &gt; @load_thres\n    # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    index = find_bucket(key)\n    # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u958b\u8fd4\u56de\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index].val = val\n      return\n    end\n    # \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n    @buckets[index] = Pair.new(key, val)\n    @size += 1\n  end\n\n  ### \u522a\u9664\u64cd\u4f5c ###\n  def remove(key)\n    # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    index = find_bucket(key)\n    # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index] = TOMBSTONE\n      @size -= 1\n    end\n  end\n\n  ### \u64f4\u5bb9\u96dc\u6e4a\u8868 ###\n  def extend\n    # \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    buckets_tmp = @buckets\n    # \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity)\n    @size = 0\n    # \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for pair in buckets_tmp\n      put(pair.key, pair.val) unless [nil, TOMBSTONE].include?(pair)\n    end\n  end\n\n  ### \u5217\u5370\u96dc\u6e4a\u8868 ###\n  def print\n    for pair in @buckets\n      if pair.nil?\n        puts \"Nil\"\n      elsif pair == TOMBSTONE\n        puts \"TOMBSTONE\"\n      else\n        puts \"#{pair.key} -&gt; #{pair.val}\"\n      end\n    end\n  end\nend\n</code></pre> hash_map_open_addressing.zig<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre>"},{"location":"chapter_hashing/hash_collision/#2","title":"2. \u00a0 \u5e73\u65b9\u63a2\u6e2c","text":"<p>\u5e73\u65b9\u63a2\u6e2c\u8207\u7dda\u6027\u63a2\u67e5\u985e\u4f3c\uff0c\u90fd\u662f\u958b\u653e\u5b9a\u5740\u7684\u5e38\u898b\u7b56\u7565\u4e4b\u4e00\u3002\u7576\u767c\u751f\u885d\u7a81\u6642\uff0c\u5e73\u65b9\u63a2\u6e2c\u4e0d\u662f\u7c21\u55ae\u5730\u8df3\u904e\u4e00\u500b\u56fa\u5b9a\u7684\u6b65\u6578\uff0c\u800c\u662f\u8df3\u904e\u201c\u63a2\u6e2c\u6b21\u6578\u7684\u5e73\u65b9\u201d\u7684\u6b65\u6578\uff0c\u5373 \\(1, 4, 9, \\dots\\) \u6b65\u3002</p> <p>\u5e73\u65b9\u63a2\u6e2c\u4e3b\u8981\u5177\u6709\u4ee5\u4e0b\u512a\u52e2\u3002</p> <ul> <li>\u5e73\u65b9\u63a2\u6e2c\u900f\u904e\u8df3\u904e\u63a2\u6e2c\u6b21\u6578\u5e73\u65b9\u7684\u8ddd\u96e2\uff0c\u8a66\u5716\u7de9\u89e3\u7dda\u6027\u63a2\u67e5\u7684\u805a\u96c6\u6548\u61c9\u3002</li> <li>\u5e73\u65b9\u63a2\u6e2c\u6703\u8df3\u904e\u66f4\u5927\u7684\u8ddd\u96e2\u4f86\u5c0b\u627e\u7a7a\u4f4d\u7f6e\uff0c\u6709\u52a9\u65bc\u8cc7\u6599\u5206\u4f48\u5f97\u66f4\u52a0\u5747\u52fb\u3002</li> </ul> <p>\u7136\u800c\uff0c\u5e73\u65b9\u63a2\u6e2c\u4e26\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002</p> <ul> <li>\u4ecd\u7136\u5b58\u5728\u805a\u96c6\u73fe\u8c61\uff0c\u5373\u67d0\u4e9b\u4f4d\u7f6e\u6bd4\u5176\u4ed6\u4f4d\u7f6e\u66f4\u5bb9\u6613\u88ab\u4f54\u7528\u3002</li> <li>\u7531\u65bc\u5e73\u65b9\u7684\u589e\u9577\uff0c\u5e73\u65b9\u63a2\u6e2c\u53ef\u80fd\u4e0d\u6703\u63a2\u6e2c\u6574\u500b\u96dc\u6e4a\u8868\uff0c\u9019\u610f\u5473\u8457\u5373\u4f7f\u96dc\u6e4a\u8868\u4e2d\u6709\u7a7a\u6876\uff0c\u5e73\u65b9\u63a2\u6e2c\u4e5f\u53ef\u80fd\u7121\u6cd5\u8a2a\u554f\u5230\u5b83\u3002</li> </ul>"},{"location":"chapter_hashing/hash_collision/#3","title":"3. \u00a0 \u591a\u6b21\u96dc\u6e4a","text":"<p>\u9867\u540d\u601d\u7fa9\uff0c\u591a\u6b21\u96dc\u6e4a\u65b9\u6cd5\u4f7f\u7528\u591a\u500b\u96dc\u6e4a\u51fd\u5f0f \\(f_1(x)\\)\u3001\\(f_2(x)\\)\u3001\\(f_3(x)\\)\u3001\\(\\dots\\) \u9032\u884c\u63a2\u6e2c\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u82e5\u96dc\u6e4a\u51fd\u5f0f \\(f_1(x)\\) \u51fa\u73fe\u885d\u7a81\uff0c\u5247\u5617\u8a66 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u4f4d\u5f8c\u63d2\u5165\u5143\u7d20\u3002</li> <li>\u67e5\u8a62\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u96dc\u6e4a\u51fd\u5f0f\u9806\u5e8f\u4e0b\u9032\u884c\u67e5\u8a62\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6a19\u5143\u7d20\u6642\u8fd4\u56de\uff1b\u82e5\u9047\u5230\u7a7a\u4f4d\u6216\u5df2\u5617\u8a66\u6240\u6709\u96dc\u6e4a\u51fd\u5f0f\uff0c\u8aaa\u660e\u96dc\u6e4a\u8868\u4e2d\u4e0d\u5b58\u5728\u8a72\u5143\u7d20\uff0c\u5247\u8fd4\u56de <code>None</code> \u3002</li> </ul> <p>\u8207\u7dda\u6027\u63a2\u67e5\u76f8\u6bd4\uff0c\u591a\u6b21\u96dc\u6e4a\u65b9\u6cd5\u4e0d\u6613\u7522\u751f\u805a\u96c6\uff0c\u4f46\u591a\u500b\u96dc\u6e4a\u51fd\u5f0f\u6703\u5e36\u4f86\u984d\u5916\u7684\u8a08\u7b97\u91cf\u3002</p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u958b\u653e\u5b9a\u5740\uff08\u7dda\u6027\u63a2\u67e5\u3001\u5e73\u65b9\u63a2\u6e2c\u548c\u591a\u6b21\u96dc\u6e4a\uff09\u96dc\u6e4a\u8868\u90fd\u5b58\u5728\u201c\u4e0d\u80fd\u76f4\u63a5\u522a\u9664\u5143\u7d20\u201d\u7684\u554f\u984c\u3002</p>"},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3 \u00a0 \u7a0b\u5f0f\u8a9e\u8a00\u7684\u9078\u64c7","text":"<p>\u5404\u7a2e\u7a0b\u5f0f\u8a9e\u8a00\u63a1\u53d6\u4e86\u4e0d\u540c\u7684\u96dc\u6e4a\u8868\u5be6\u73fe\u7b56\u7565\uff0c\u4e0b\u9762\u8209\u5e7e\u500b\u4f8b\u5b50\u3002</p> <ul> <li>Python \u63a1\u7528\u958b\u653e\u5b9a\u5740\u3002\u5b57\u5178 <code>dict</code> \u4f7f\u7528\u507d\u96a8\u6a5f\u6578\u9032\u884c\u63a2\u6e2c\u3002</li> <li>Java \u63a1\u7528\u93c8\u5f0f\u4f4d\u5740\u3002\u81ea JDK 1.8 \u4ee5\u4f86\uff0c\u7576 <code>HashMap</code> \u5167\u9663\u5217\u9577\u5ea6\u9054\u5230 64 \u4e14\u93c8\u7d50\u4e32\u5217\u9577\u5ea6\u9054\u5230 8 \u6642\uff0c\u93c8\u7d50\u4e32\u5217\u6703\u8f49\u63db\u70ba\u7d05\u9ed1\u6a39\u4ee5\u63d0\u5347\u67e5\u8a62\u6548\u80fd\u3002</li> <li>Go \u63a1\u7528\u93c8\u5f0f\u4f4d\u5740\u3002Go \u898f\u5b9a\u6bcf\u500b\u6876\u6700\u591a\u5132\u5b58 8 \u500b\u9375\u503c\u5c0d\uff0c\u8d85\u51fa\u5bb9\u91cf\u5247\u9023\u7dda\u4e00\u500b\u6ea2\u4f4d\u6876\uff1b\u7576\u6ea2\u4f4d\u6876\u904e\u591a\u6642\uff0c\u6703\u57f7\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u64f4\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u78ba\u4fdd\u6548\u80fd\u3002</li> </ul>"},{"location":"chapter_hashing/hash_map/","title":"6.1 \u00a0 \u96dc\u6e4a\u8868","text":"<p>\u96dc\u6e4a\u8868\uff08hash table\uff09\uff0c\u53c8\u7a31\u6563\u5217\u8868\uff0c\u5b83\u900f\u904e\u5efa\u7acb\u9375 <code>key</code> \u8207\u503c <code>value</code> \u4e4b\u9593\u7684\u5c0d\u6620\uff0c\u5be6\u73fe\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8a62\u3002\u5177\u9ad4\u800c\u8a00\uff0c\u6211\u5011\u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u4e00\u500b\u9375 <code>key</code> \uff0c\u5247\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u5167\u7372\u53d6\u5c0d\u61c9\u7684\u503c <code>value</code> \u3002</p> <p>\u5982\u5716 6-1 \u6240\u793a\uff0c\u7d66\u5b9a \\(n\\) \u500b\u5b78\u751f\uff0c\u6bcf\u500b\u5b78\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b78\u865f\u201d\u5169\u9805\u8cc7\u6599\u3002\u5047\u5982\u6211\u5011\u5e0c\u671b\u5be6\u73fe\u201c\u8f38\u5165\u4e00\u500b\u5b78\u865f\uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u59d3\u540d\u201d\u7684\u67e5\u8a62\u529f\u80fd\uff0c\u5247\u53ef\u4ee5\u63a1\u7528\u5716 6-1 \u6240\u793a\u7684\u96dc\u6e4a\u8868\u4f86\u5be6\u73fe\u3002</p> <p></p> <p> \u5716 6-1 \u00a0 \u96dc\u6e4a\u8868\u7684\u62bd\u8c61\u8868\u793a </p> <p>\u9664\u96dc\u6e4a\u8868\u5916\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u4e5f\u53ef\u4ee5\u5be6\u73fe\u67e5\u8a62\u529f\u80fd\uff0c\u5b83\u5011\u7684\u6548\u7387\u5c0d\u6bd4\u5982\u8868 6-1 \u6240\u793a\u3002</p> <ul> <li>\u65b0\u589e\u5143\u7d20\uff1a\u50c5\u9700\u5c07\u5143\u7d20\u65b0\u589e\u81f3\u9663\u5217\uff08\u93c8\u7d50\u4e32\u5217\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002</li> <li>\u67e5\u8a62\u5143\u7d20\uff1a\u7531\u65bc\u9663\u5217\uff08\u93c8\u7d50\u4e32\u5217\uff09\u662f\u4e82\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u8d70\u8a2a\u5176\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002</li> <li>\u522a\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8a62\u5230\u5143\u7d20\uff0c\u518d\u5f9e\u9663\u5217\uff08\u93c8\u7d50\u4e32\u5217\uff09\u4e2d\u522a\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002</li> </ul> <p> \u8868 6-1 \u00a0 \u5143\u7d20\u67e5\u8a62\u6548\u7387\u5c0d\u6bd4 </p> \u9663\u5217 \u93c8\u7d50\u4e32\u5217 \u96dc\u6e4a\u8868 \u67e5\u8a62\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u65b0\u589e\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u522a\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>\u89c0\u5bdf\u767c\u73fe\uff0c\u5728\u96dc\u6e4a\u8868\u4e2d\u9032\u884c\u589e\u522a\u67e5\u6539\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1 \u00a0 \u96dc\u6e4a\u8868\u5e38\u7528\u64cd\u4f5c","text":"<p>\u96dc\u6e4a\u8868\u7684\u5e38\u898b\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8a62\u64cd\u4f5c\u3001\u65b0\u589e\u9375\u503c\u5c0d\u548c\u522a\u9664\u9375\u503c\u5c0d\u7b49\uff0c\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map.py<pre><code># \u521d\u59cb\u5316\u96dc\u6e4a\u8868\nhmap: dict = {}\n\n# \u65b0\u589e\u64cd\u4f5c\n# \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u56c9\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9d28\"\n\n# \u67e5\u8a62\u64cd\u4f5c\n# \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nname: str = hmap[15937]\n\n# \u522a\u9664\u64cd\u4f5c\n# \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nunordered_map&lt;int, string&gt; map;\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u56c9\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9d28\";\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");\nmap.put(15937, \"\u5c0f\u56c9\");\nmap.put(16750, \"\u5c0f\u7b97\");\nmap.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9d28\");\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nDictionary&lt;int, string&gt; map = new() {\n    /* \u65b0\u589e\u64cd\u4f5c */\n    // \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\n    { 12836, \"\u5c0f\u54c8\" },\n    { 15937, \"\u5c0f\u56c9\" },\n    { 16750, \"\u5c0f\u7b97\" },\n    { 13276, \"\u5c0f\u6cd5\" },\n    { 10583, \"\u5c0f\u9d28\" }\n};\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nhmap := make(map[int]string)\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u56c9\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9d28\"\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nname := hmap[15937]\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nvar map: [Int: String] = [:]\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u56c9\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9d28\"\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nconst map = new Map();\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u56c9');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9d28');\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nconst map = new Map&lt;number, string&gt;();\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u56c9');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9d28');\nconsole.info('\\n\u65b0\u589e\u5b8c\u6210\u5f8c\uff0c\u96dc\u6e4a\u8868\u70ba\\nKey -&gt; Value');\nconsole.info(map);\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f38\u5165\u5b78\u865f 15937 \uff0c\u67e5\u8a62\u5230\u59d3\u540d ' + name);\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.delete(10583);\nconsole.info('\\n\u522a\u9664 10583 \u5f8c\uff0c\u96dc\u6e4a\u8868\u70ba\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nMap&lt;int, String&gt; map = {};\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u56c9\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9d28\";\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap.insert(12836, \"\u5c0f\u54c8\".to_string());\nmap.insert(15937, \"\u5c0f\u56c9\".to_string());\nmap.insert(16750, \"\u5c0f\u7b97\".to_string());\nmap.insert(13279, \"\u5c0f\u6cd5\".to_string());\nmap.insert(10583, \"\u5c0f\u9d28\".to_string());\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u96dc\u6e4a\u8868\n</code></pre> hash_map.kt<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nval map = HashMap&lt;Int,String&gt;()\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u56c9\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9d28\"\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nval name = map[15937]\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.remove(10583)\n</code></pre> hash_map.rb<pre><code># \u521d\u59cb\u5316\u96dc\u6e4a\u8868\nhmap = {}\n\n# \u65b0\u589e\u64cd\u4f5c\n# \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u56c9\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9d28\"\n\n# \u67e5\u8a62\u64cd\u4f5c\n# \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nname = hmap[15937]\n\n# \u522a\u9664\u64cd\u4f5c\n# \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nhmap.delete(10583)\n</code></pre> hash_map.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u96dc\u6e4a\u8868\u6709\u4e09\u7a2e\u5e38\u7528\u7684\u8d70\u8a2a\u65b9\u5f0f\uff1a\u8d70\u8a2a\u9375\u503c\u5c0d\u3001\u8d70\u8a2a\u9375\u548c\u8d70\u8a2a\u503c\u3002\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map.py<pre><code># \u8d70\u8a2a\u96dc\u6e4a\u8868\n# \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# \u55ae\u7368\u8d70\u8a2a\u9375 key\nfor key in hmap.keys():\n    print(key)\n# \u55ae\u7368\u8d70\u8a2a\u503c value\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u5668\u8d70\u8a2a key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 key\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c value\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 key\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c value\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 key\nfor key := range hmap {\n    fmt.Println(key)\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c value\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 Key\nfor key in map.keys {\n    print(key)\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c Value\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\nconsole.info('\\n\u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u55ae\u7368\u8d70\u8a2a\u9375 Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u55ae\u7368\u8d70\u8a2a\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\nconsole.info('\\n\u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u55ae\u7368\u8d70\u8a2a\u9375 Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u55ae\u7368\u8d70\u8a2a\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// \u55ae\u7368\u8d70\u8a2a\u9375 Key\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// \u55ae\u7368\u8d70\u8a2a\u503c Value\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// \u55ae\u7368\u8d70\u8a2a\u9375 Key\nfor key in map.keys() {\n    println!(\"{key}\");\n}\n\n// \u55ae\u7368\u8d70\u8a2a\u503c Value\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u96dc\u6e4a\u8868\n</code></pre> hash_map.kt<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor ((key, value) in map) {\n    println(\"$key -&gt; $value\")\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 key\nfor (key in map.keys) {\n    println(key)\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c value\nfor (_val in map.values) {\n    println(_val)\n}\n</code></pre> hash_map.rb<pre><code># \u8d70\u8a2a\u96dc\u6e4a\u8868\n# \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nhmap.entries.each { |key, value| puts \"#{key} -&gt; #{value}\" }\n\n# \u55ae\u7368\u8d70\u8a2a\u9375 key\nhmap.keys.each { |key| puts key }\n\n# \u55ae\u7368\u8d70\u8a2a\u503c value\nhmap.values.each { |val| puts val }\n</code></pre> hash_map.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2 \u00a0 \u96dc\u6e4a\u8868\u7c21\u55ae\u5be6\u73fe","text":"<p>\u6211\u5011\u5148\u8003\u616e\u6700\u7c21\u55ae\u7684\u60c5\u6cc1\uff0c\u50c5\u7528\u4e00\u500b\u9663\u5217\u4f86\u5be6\u73fe\u96dc\u6e4a\u8868\u3002\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u6211\u5011\u5c07\u9663\u5217\u4e2d\u7684\u6bcf\u500b\u7a7a\u4f4d\u7a31\u70ba\u6876\uff08bucket\uff09\uff0c\u6bcf\u500b\u6876\u53ef\u5132\u5b58\u4e00\u500b\u9375\u503c\u5c0d\u3002\u56e0\u6b64\uff0c\u67e5\u8a62\u64cd\u4f5c\u5c31\u662f\u627e\u5230 <code>key</code> \u5c0d\u61c9\u7684\u6876\uff0c\u4e26\u5728\u6876\u4e2d\u7372\u53d6 <code>value</code> \u3002</p> <p>\u90a3\u9ebc\uff0c\u5982\u4f55\u57fa\u65bc <code>key</code> \u5b9a\u4f4d\u5c0d\u61c9\u7684\u6876\u5462\uff1f\u9019\u662f\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\uff08hash function\uff09\u5be6\u73fe\u7684\u3002\u96dc\u6e4a\u51fd\u5f0f\u7684\u4f5c\u7528\u662f\u5c07\u4e00\u500b\u8f03\u5927\u7684\u8f38\u5165\u7a7a\u9593\u5c0d\u6620\u5230\u4e00\u500b\u8f03\u5c0f\u7684\u8f38\u51fa\u7a7a\u9593\u3002\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u8f38\u5165\u7a7a\u9593\u662f\u6240\u6709 <code>key</code> \uff0c\u8f38\u51fa\u7a7a\u9593\u662f\u6240\u6709\u6876\uff08\u9663\u5217\u7d22\u5f15\uff09\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u8f38\u5165\u4e00\u500b <code>key</code> \uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u5f97\u5230\u8a72 <code>key</code> \u5c0d\u61c9\u7684\u9375\u503c\u5c0d\u5728\u9663\u5217\u4e2d\u7684\u5132\u5b58\u4f4d\u7f6e\u3002</p> <p>\u8f38\u5165\u4e00\u500b <code>key</code> \uff0c\u96dc\u6e4a\u51fd\u5f0f\u7684\u8a08\u7b97\u904e\u7a0b\u5206\u70ba\u4ee5\u4e0b\u5169\u6b65\u3002</p> <ol> <li>\u900f\u904e\u67d0\u7a2e\u96dc\u6e4a\u6f14\u7b97\u6cd5 <code>hash()</code> \u8a08\u7b97\u5f97\u5230\u96dc\u6e4a\u503c\u3002</li> <li>\u5c07\u96dc\u6e4a\u503c\u5c0d\u6876\u6578\u91cf\uff08\u9663\u5217\u9577\u5ea6\uff09<code>capacity</code> \u53d6\u6a21\uff0c\u5f9e\u800c\u7372\u53d6\u8a72 <code>key</code> \u5c0d\u61c9\u7684\u9663\u5217\u7d22\u5f15 <code>index</code> \u3002</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u96a8\u5f8c\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5229\u7528 <code>index</code> \u5728\u96dc\u6e4a\u8868\u4e2d\u8a2a\u554f\u5c0d\u61c9\u7684\u6876\uff0c\u5f9e\u800c\u7372\u53d6 <code>value</code> \u3002</p> <p>\u8a2d\u9663\u5217\u9577\u5ea6 <code>capacity = 100</code>\u3001\u96dc\u6e4a\u6f14\u7b97\u6cd5 <code>hash(key) = key</code> \uff0c\u6613\u5f97\u96dc\u6e4a\u51fd\u5f0f\u70ba <code>key % 100</code> \u3002\u5716 6-2 \u4ee5 <code>key</code> \u5b78\u865f\u548c <code>value</code> \u59d3\u540d\u70ba\u4f8b\uff0c\u5c55\u793a\u4e86\u96dc\u6e4a\u51fd\u5f0f\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p></p> <p> \u5716 6-2 \u00a0 \u96dc\u6e4a\u51fd\u5f0f\u5de5\u4f5c\u539f\u7406 </p> <p>\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e86\u4e00\u500b\u7c21\u55ae\u96dc\u6e4a\u8868\u3002\u5176\u4e2d\uff0c\u6211\u5011\u5c07 <code>key</code> \u548c <code>value</code> \u5c01\u88dd\u6210\u4e00\u500b\u985e\u5225 <code>Pair</code> \uff0c\u4ee5\u8868\u793a\u9375\u503c\u5c0d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_hash_map.py<pre><code>class Pair:\n    \"\"\"\u9375\u503c\u5c0d\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        # \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u96dc\u6e4a\u51fd\u5f0f\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8a62\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"\u65b0\u589e\u64cd\u4f5c\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"\u522a\u9664\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        # \u7f6e\u70ba None \uff0c\u4ee3\u8868\u522a\u9664\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"\u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"\u7372\u53d6\u6240\u6709\u9375\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"\u7372\u53d6\u6240\u6709\u503c\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"\u5217\u5370\u96dc\u6e4a\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* \u9375\u503c\u5c0d */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\u4e26\u7f6e\u70ba nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* \u9375\u503c\u5c0d */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        buckets.set(index, null);\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* \u9375\u503c\u5c0d int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        buckets[index] = null;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* \u9375\u503c\u5c0d */\ntype pair struct {\n    key int\n    val string\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n    // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // \u7f6e\u70ba nil \uff0c\u4ee3\u8868\u522a\u9664\n    a.buckets[index] = nil\n}\n\n/* \u7372\u53d6\u6240\u6709\u9375\u5c0d */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* \u7372\u53d6\u6240\u6709\u9375 */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* \u7372\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* \u9375\u503c\u5c0d */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // \u7f6e\u70ba nil \uff0c\u4ee3\u8868\u522a\u9664\n        buckets[index] = nil\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* \u9375\u503c\u5c0d Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        this.#buckets[index] = null;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* \u9375\u503c\u5c0d Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        this.buckets[index] = null;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* \u9375\u503c\u5c0d */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n    _buckets = List.filled(100, null);\n  }\n\n  /* \u96dc\u6e4a\u51fd\u5f0f */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* \u67e5\u8a62\u64cd\u4f5c */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* \u65b0\u589e\u64cd\u4f5c */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* \u522a\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* \u7372\u53d6\u6240\u6709\u9375 */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* \u7372\u53d6\u6240\u6709\u503c */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* \u5217\u5370\u96dc\u6e4a\u8868 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* \u9375\u503c\u5c0d */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // \u7f6e\u70ba None \uff0c\u4ee3\u8868\u522a\u9664\n        self.buckets[index] = None;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* \u9375\u503c\u5c0d int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\ntypedef struct {\n    Pair *buckets[MAX_SIZE];\n} ArrayHashMap;\n\n/* \u5efa\u69cb\u5b50 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    for (int i=0; i &lt; MAX_SIZE; i++) {\n        hmap-&gt;buckets[i] = NULL;\n    }\n    return hmap;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7d71\u8a08\u6709\u6548\u9375\u503c\u5c0d\u6578\u91cf */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* \u7372\u53d6\u6240\u6709\u9375 */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7d71\u8a08\u6709\u6548\u9375\u503c\u5c0d\u6578\u91cf */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* \u7372\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7d71\u8a08\u6709\u6548\u9375\u503c\u5c0d\u6578\u91cf */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* \u9375\u503c\u5c0d */\nclass Pair(\n    var key: Int,\n    var _val: String\n)\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair._val\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fun put(key: Int, _val: String) {\n        val pair = Pair(key, _val)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        buckets[index] = null\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = mutableListOf&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = mutableListOf&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = mutableListOf&lt;String&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                valueSet.add(pair._val)\n        }\n        return valueSet\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val _val = kv._val\n            println(\"$key -&gt; $_val\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>### \u9375\u503c\u5c0d ###\nclass Pair\n  attr_accessor :key, :val\n\n  def initialize(key, val)\n    @key = key\n    @val = val\n  end\nend\n\n### \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 ###\nclass ArrayHashMap\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    # \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n    @buckets = Array.new(100)\n  end\n\n  ### \u96dc\u6e4a\u51fd\u5f0f ###\n  def hash_func(key)\n    index = key % 100\n  end\n\n  ### \u67e5\u8a62\u64cd\u4f5c ###\n  def get(key)\n    index = hash_func(key)\n    pair = @buckets[index]\n\n    return if pair.nil?\n    pair.val\n  end\n\n  ### \u65b0\u589e\u64cd\u4f5c ###\n  def put(key, val)\n    pair = Pair.new(key, val)\n    index = hash_func(key)\n    @buckets[index] = pair\n  end\n\n  ### \u522a\u9664\u64cd\u4f5c ###\n  def remove(key)\n    index = hash_func(key)\n    # \u7f6e\u70ba nil \uff0c\u4ee3\u8868\u522a\u9664\n    @buckets[index] = nil\n  end\n\n  ### \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d ###\n  def entry_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair unless pair.nil? }\n    result\n  end\n\n  ### \u7372\u53d6\u6240\u6709\u9375 ###\n  def key_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.key unless pair.nil? }\n    result\n  end\n\n  ### \u7372\u53d6\u6240\u6709\u503c ###\n  def value_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.val unless pair.nil? }\n    result\n  end\n\n  ### \u5217\u5370\u96dc\u6e4a\u8868 ###\n  def print\n    @buckets.each { |pair| puts \"#{pair.key} -&gt; #{pair.val}\" unless pair.nil? }\n  end\nend\n</code></pre> array_hash_map.zig<pre><code>// \u9375\u503c\u5c0d\nconst Pair = struct {\n    key: usize = undefined,\n    val: []const u8 = undefined,\n\n   pub fn init(key: usize, val: []const u8) Pair {\n        return Pair {\n            .key = key,\n            .val = val,\n        };\n    }\n};\n\n// \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868\nfn ArrayHashMap(comptime T: type) type {\n    return struct {\n        bucket: ?std.ArrayList(?T) = null,\n        mem_allocator: std.mem.Allocator = undefined,\n\n        const Self = @This();\n\n        // \u5efa\u69cb\u5b50\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            self.mem_allocator = allocator;\n            // \u521d\u59cb\u5316\u4e00\u500b\u9577\u5ea6\u70ba 100 \u7684\u6876\uff08\u9663\u5217\uff09\n            self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n            var i: i32 = 0;\n            while (i &lt; 100) : (i += 1) {\n                try self.bucket.?.append(null);\n            }\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\n        pub fn deinit(self: *Self) void {\n            if (self.bucket != null) self.bucket.?.deinit();\n        }\n\n        // \u96dc\u6e4a\u51fd\u5f0f\n        fn hashFunc(key: usize) usize {\n            var index = key % 100;\n            return index;\n        }\n\n        // \u67e5\u8a62\u64cd\u4f5c\n        pub fn get(self: *Self, key: usize) []const u8 {\n            var index = hashFunc(key);\n            var pair = self.bucket.?.items[index];\n            return pair.?.val;\n        }\n\n        // \u65b0\u589e\u64cd\u4f5c\n        pub fn put(self: *Self, key: usize, val: []const u8) !void {\n            var pair = Pair.init(key, val);\n            var index = hashFunc(key);\n            self.bucket.?.items[index] = pair;\n        }\n\n        // \u522a\u9664\u64cd\u4f5c\n        pub fn remove(self: *Self, key: usize) !void {\n            var index = hashFunc(key);\n            // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n            self.bucket.?.items[index] = null;\n        }       \n\n        // \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d\n        pub fn pairSet(self: *Self) !std.ArrayList(T) {\n            var entry_set = std.ArrayList(T).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try entry_set.append(item.?);\n            }\n            return entry_set;\n        }  \n\n        // \u7372\u53d6\u6240\u6709\u9375\n        pub fn keySet(self: *Self) !std.ArrayList(usize) {\n            var key_set = std.ArrayList(usize).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try key_set.append(item.?.key);\n            }\n            return key_set;\n        }  \n\n        // \u7372\u53d6\u6240\u6709\u503c\n        pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n            var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try value_set.append(item.?.val);\n            }\n            return value_set;\n        }\n\n        // \u5217\u5370\u96dc\u6e4a\u8868\n        pub fn print(self: *Self) !void {\n            var entry_set = try self.pairSet();\n            defer entry_set.deinit();\n            for (entry_set.items) |item| {\n                std.debug.print(\"{} -&gt; {s}\\n\", .{item.key, item.val});\n            }\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3 \u00a0 \u96dc\u6e4a\u885d\u7a81\u8207\u64f4\u5bb9","text":"<p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u96dc\u6e4a\u51fd\u5f0f\u7684\u4f5c\u7528\u662f\u5c07\u6240\u6709 <code>key</code> \u69cb\u6210\u7684\u8f38\u5165\u7a7a\u9593\u5c0d\u6620\u5230\u9663\u5217\u6240\u6709\u7d22\u5f15\u69cb\u6210\u7684\u8f38\u51fa\u7a7a\u9593\uff0c\u800c\u8f38\u5165\u7a7a\u9593\u5f80\u5f80\u9060\u5927\u65bc\u8f38\u51fa\u7a7a\u9593\u3002\u56e0\u6b64\uff0c\u7406\u8ad6\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u500b\u8f38\u5165\u5c0d\u61c9\u76f8\u540c\u8f38\u51fa\u201d\u7684\u60c5\u6cc1\u3002</p> <p>\u5c0d\u65bc\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u96dc\u6e4a\u51fd\u5f0f\uff0c\u7576\u8f38\u5165\u7684 <code>key</code> \u5f8c\u5169\u4f4d\u76f8\u540c\u6642\uff0c\u96dc\u6e4a\u51fd\u5f0f\u7684\u8f38\u51fa\u7d50\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8a62\u5b78\u865f\u70ba 12836 \u548c 20336 \u7684\u5169\u500b\u5b78\u751f\u6642\uff0c\u6211\u5011\u5f97\u5230\uff1a</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>\u5982\u5716 6-3 \u6240\u793a\uff0c\u5169\u500b\u5b78\u865f\u6307\u5411\u4e86\u540c\u4e00\u500b\u59d3\u540d\uff0c\u9019\u986f\u7136\u662f\u4e0d\u5c0d\u7684\u3002\u6211\u5011\u5c07\u9019\u7a2e\u591a\u500b\u8f38\u5165\u5c0d\u61c9\u540c\u4e00\u8f38\u51fa\u7684\u60c5\u6cc1\u7a31\u70ba\u96dc\u6e4a\u885d\u7a81\uff08hash collision\uff09\u3002</p> <p></p> <p> \u5716 6-3 \u00a0 \u96dc\u6e4a\u885d\u7a81\u793a\u4f8b </p> <p>\u5bb9\u6613\u60f3\u5230\uff0c\u96dc\u6e4a\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u500b <code>key</code> \u88ab\u5206\u914d\u5230\u540c\u4e00\u500b\u6876\u4e2d\u7684\u6a5f\u7387\u5c31\u8d8a\u4f4e\uff0c\u885d\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u64f4\u5bb9\u96dc\u6e4a\u8868\u4f86\u6e1b\u5c11\u96dc\u6e4a\u885d\u7a81\u3002</p> <p>\u5982\u5716 6-4 \u6240\u793a\uff0c\u64f4\u5bb9\u524d\u9375\u503c\u5c0d <code>(136, A)</code> \u548c <code>(236, D)</code> \u767c\u751f\u885d\u7a81\uff0c\u64f4\u5bb9\u5f8c\u885d\u7a81\u6d88\u5931\u3002</p> <p></p> <p> \u5716 6-4 \u00a0 \u96dc\u6e4a\u8868\u64f4\u5bb9 </p> <p>\u985e\u4f3c\u65bc\u9663\u5217\u64f4\u5bb9\uff0c\u96dc\u6e4a\u8868\u64f4\u5bb9\u9700\u5c07\u6240\u6709\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u9077\u79fb\u81f3\u65b0\u96dc\u6e4a\u8868\uff0c\u975e\u5e38\u8017\u6642\uff1b\u4e26\u4e14\u7531\u65bc\u96dc\u6e4a\u8868\u5bb9\u91cf <code>capacity</code> \u6539\u8b8a\uff0c\u6211\u5011\u9700\u8981\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u4f86\u91cd\u65b0\u8a08\u7b97\u6240\u6709\u9375\u503c\u5c0d\u7684\u5132\u5b58\u4f4d\u7f6e\uff0c\u9019\u9032\u4e00\u6b65\u589e\u52a0\u4e86\u64f4\u5bb9\u904e\u7a0b\u7684\u8a08\u7b97\u958b\u92b7\u3002\u70ba\u6b64\uff0c\u7a0b\u5f0f\u8a9e\u8a00\u901a\u5e38\u6703\u9810\u7559\u8db3\u5920\u5927\u7684\u96dc\u6e4a\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u983b\u7e41\u64f4\u5bb9\u3002</p> <p>\u8ca0\u8f09\u56e0\u5b50\uff08load factor\uff09\u662f\u96dc\u6e4a\u8868\u7684\u4e00\u500b\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u7fa9\u70ba\u96dc\u6e4a\u8868\u7684\u5143\u7d20\u6578\u91cf\u9664\u4ee5\u6876\u6578\u91cf\uff0c\u7528\u65bc\u8861\u91cf\u96dc\u6e4a\u885d\u7a81\u7684\u56b4\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u4f5c\u70ba\u96dc\u6e4a\u8868\u64f4\u5bb9\u7684\u89f8\u767c\u689d\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e \\(0.75\\) \u6642\uff0c\u7cfb\u7d71\u6703\u5c07\u96dc\u6e4a\u8868\u64f4\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</p>"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_hashing/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u8f38\u5165 <code>key</code> \uff0c\u96dc\u6e4a\u8868\u80fd\u5920\u5728 \\(O(1)\\) \u6642\u9593\u5167\u67e5\u8a62\u5230 <code>value</code> \uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002</li> <li>\u5e38\u898b\u7684\u96dc\u6e4a\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8a62\u3001\u65b0\u589e\u9375\u503c\u5c0d\u3001\u522a\u9664\u9375\u503c\u5c0d\u548c\u8d70\u8a2a\u96dc\u6e4a\u8868\u7b49\u3002</li> <li>\u96dc\u6e4a\u51fd\u5f0f\u5c07 <code>key</code> \u5c0d\u6620\u70ba\u9663\u5217\u7d22\u5f15\uff0c\u5f9e\u800c\u8a2a\u554f\u5c0d\u61c9\u6876\u4e26\u7372\u53d6 <code>value</code> \u3002</li> <li>\u5169\u500b\u4e0d\u540c\u7684 <code>key</code> \u53ef\u80fd\u5728\u7d93\u904e\u96dc\u6e4a\u51fd\u5f0f\u5f8c\u5f97\u5230\u76f8\u540c\u7684\u9663\u5217\u7d22\u5f15\uff0c\u5c0e\u81f4\u67e5\u8a62\u7d50\u679c\u51fa\u932f\uff0c\u9019\u7a2e\u73fe\u8c61\u88ab\u7a31\u70ba\u96dc\u6e4a\u885d\u7a81\u3002</li> <li>\u96dc\u6e4a\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u96dc\u6e4a\u885d\u7a81\u7684\u6a5f\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u900f\u904e\u64f4\u5bb9\u96dc\u6e4a\u8868\u4f86\u7de9\u89e3\u96dc\u6e4a\u885d\u7a81\u3002\u8207\u9663\u5217\u64f4\u5bb9\u985e\u4f3c\uff0c\u96dc\u6e4a\u8868\u64f4\u5bb9\u64cd\u4f5c\u7684\u958b\u92b7\u5f88\u5927\u3002</li> <li>\u8ca0\u8f09\u56e0\u5b50\u5b9a\u7fa9\u70ba\u96dc\u6e4a\u8868\u4e2d\u5143\u7d20\u6578\u91cf\u9664\u4ee5\u6876\u6578\u91cf\uff0c\u53cd\u6620\u4e86\u96dc\u6e4a\u885d\u7a81\u7684\u56b4\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89f8\u767c\u96dc\u6e4a\u8868\u64f4\u5bb9\u7684\u689d\u4ef6\u3002</li> <li>\u93c8\u5f0f\u4f4d\u5740\u900f\u904e\u5c07\u55ae\u500b\u5143\u7d20\u8f49\u5316\u70ba\u93c8\u7d50\u4e32\u5217\uff0c\u5c07\u6240\u6709\u885d\u7a81\u5143\u7d20\u5132\u5b58\u5728\u540c\u4e00\u500b\u93c8\u7d50\u4e32\u5217\u4e2d\u3002\u7136\u800c\uff0c\u93c8\u7d50\u4e32\u5217\u904e\u9577\u6703\u964d\u4f4e\u67e5\u8a62\u6548\u7387\uff0c\u53ef\u4ee5\u900f\u904e\u9032\u4e00\u6b65\u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u63db\u70ba\u7d05\u9ed1\u6a39\u4f86\u63d0\u9ad8\u6548\u7387\u3002</li> <li>\u958b\u653e\u5b9a\u5740\u900f\u904e\u591a\u6b21\u63a2\u6e2c\u4f86\u8655\u7406\u96dc\u6e4a\u885d\u7a81\u3002\u7dda\u6027\u63a2\u67e5\u4f7f\u7528\u56fa\u5b9a\u6b65\u9577\uff0c\u7f3a\u9ede\u662f\u4e0d\u80fd\u522a\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u7522\u751f\u805a\u96c6\u3002\u591a\u6b21\u96dc\u6e4a\u4f7f\u7528\u591a\u500b\u96dc\u6e4a\u51fd\u5f0f\u9032\u884c\u63a2\u6e2c\uff0c\u76f8\u8f03\u7dda\u6027\u63a2\u67e5\u66f4\u4e0d\u6613\u7522\u751f\u805a\u96c6\uff0c\u4f46\u591a\u500b\u96dc\u6e4a\u51fd\u5f0f\u589e\u52a0\u4e86\u8a08\u7b97\u91cf\u3002</li> <li>\u4e0d\u540c\u7a0b\u5f0f\u8a9e\u8a00\u63a1\u53d6\u4e86\u4e0d\u540c\u7684\u96dc\u6e4a\u8868\u5be6\u73fe\u3002\u4f8b\u5982\uff0cJava \u7684 <code>HashMap</code> \u4f7f\u7528\u93c8\u5f0f\u4f4d\u5740\uff0c\u800c Python \u7684 <code>Dict</code> \u63a1\u7528\u958b\u653e\u5b9a\u5740\u3002</li> <li>\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u6211\u5011\u5e0c\u671b\u96dc\u6e4a\u6f14\u7b97\u6cd5\u5177\u6709\u78ba\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u52fb\u5206\u4f48\u7684\u7279\u9ede\u3002\u5728\u5bc6\u78bc\u5b78\u4e2d\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u9084\u61c9\u8a72\u5177\u5099\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u61c9\u3002</li> <li>\u96dc\u6e4a\u6f14\u7b97\u6cd5\u901a\u5e38\u63a1\u7528\u5927\u8cea\u6578\u4f5c\u70ba\u6a21\u6578\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8b49\u96dc\u6e4a\u503c\u5747\u52fb\u5206\u4f48\uff0c\u6e1b\u5c11\u96dc\u6e4a\u885d\u7a81\u3002</li> <li>\u5e38\u898b\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\u5305\u62ec MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002MD5 \u5e38\u7528\u65bc\u6821\u9a57\u6a94\u6848\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u65bc\u5b89\u5168\u61c9\u7528\u8207\u5354\u8b70\u3002</li> <li>\u7a0b\u5f0f\u8a9e\u8a00\u901a\u5e38\u6703\u70ba\u8cc7\u6599\u578b\u5225\u63d0\u4f9b\u5167\u5efa\u96dc\u6e4a\u6f14\u7b97\u6cd5\uff0c\u7528\u65bc\u8a08\u7b97\u96dc\u6e4a\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u8b8a\u7269\u4ef6\u662f\u53ef\u96dc\u6e4a\u7684\u3002</li> </ul>"},{"location":"chapter_hashing/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u96dc\u6e4a\u8868\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5728\u4ec0\u9ebc\u60c5\u6cc1\u4e0b\u662f \\(O(n)\\) \uff1f</p> <p>\u7576\u96dc\u6e4a\u885d\u7a81\u6bd4\u8f03\u56b4\u91cd\u6642\uff0c\u96dc\u6e4a\u8868\u7684\u6642\u9593\u8907\u96dc\u5ea6\u6703\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u7576\u96dc\u6e4a\u51fd\u5f0f\u8a2d\u8a08\u5f97\u6bd4\u8f03\u597d\u3001\u5bb9\u91cf\u8a2d\u5b9a\u6bd4\u8f03\u5408\u7406\u3001\u885d\u7a81\u6bd4\u8f03\u5e73\u5747\u6642\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u5011\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u5167\u5efa\u7684\u96dc\u6e4a\u8868\u6642\uff0c\u901a\u5e38\u8a8d\u70ba\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(1)\\) \u3002</p> <p>Q\uff1a\u70ba\u4ec0\u9ebc\u4e0d\u4f7f\u7528\u96dc\u6e4a\u51fd\u5f0f \\(f(x) = x\\) \u5462\uff1f\u9019\u6a23\u5c31\u4e0d\u6703\u6709\u885d\u7a81\u4e86\u3002</p> <p>\u5728 \\(f(x) = x\\) \u96dc\u6e4a\u51fd\u5f0f\u4e0b\uff0c\u6bcf\u500b\u5143\u7d20\u5c0d\u61c9\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u9019\u8207\u9663\u5217\u7b49\u50f9\u3002\u7136\u800c\uff0c\u8f38\u5165\u7a7a\u9593\u901a\u5e38\u9060\u5927\u65bc\u8f38\u51fa\u7a7a\u9593\uff08\u9663\u5217\u9577\u5ea6\uff09\uff0c\u56e0\u6b64\u96dc\u6e4a\u51fd\u5f0f\u7684\u6700\u5f8c\u4e00\u6b65\u5f80\u5f80\u662f\u5c0d\u9663\u5217\u9577\u5ea6\u53d6\u6a21\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u96dc\u6e4a\u8868\u7684\u76ee\u6a19\u662f\u5c07\u4e00\u500b\u8f03\u5927\u7684\u72c0\u614b\u7a7a\u9593\u5c0d\u6620\u5230\u4e00\u500b\u8f03\u5c0f\u7684\u7a7a\u9593\uff0c\u4e26\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8a62\u6548\u7387\u3002</p> <p>Q\uff1a\u96dc\u6e4a\u8868\u5e95\u5c64\u5be6\u73fe\u662f\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u4e8c\u5143\u6a39\uff0c\u4f46\u70ba\u4ec0\u9ebc\u6548\u7387\u53ef\u4ee5\u6bd4\u5b83\u5011\u66f4\u9ad8\u5462\uff1f</p> <p>\u9996\u5148\uff0c\u96dc\u6e4a\u8868\u7684\u6642\u9593\u6548\u7387\u8b8a\u9ad8\uff0c\u4f46\u7a7a\u9593\u6548\u7387\u8b8a\u4f4e\u4e86\u3002\u96dc\u6e4a\u8868\u6709\u76f8\u7576\u4e00\u90e8\u5206\u8a18\u61b6\u9ad4\u672a\u4f7f\u7528\u3002</p> <p>\u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u5834\u666f\u4e0b\u6642\u9593\u6548\u7387\u8b8a\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u500b\u529f\u80fd\u80fd\u5920\u5728\u76f8\u540c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e0b\u4f7f\u7528\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\uff0c\u90a3\u9ebc\u901a\u5e38\u6bd4\u96dc\u6e4a\u8868\u66f4\u5feb\u3002\u9019\u662f\u56e0\u70ba\u96dc\u6e4a\u51fd\u5f0f\u8a08\u7b97\u9700\u8981\u958b\u92b7\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u7684\u5e38\u6578\u9805\u66f4\u5927\u3002</p> <p>\u6700\u5f8c\uff0c\u96dc\u6e4a\u8868\u7684\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u80fd\u767c\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u93c8\u5f0f\u4f4d\u5740\u4e2d\uff0c\u6211\u5011\u63a1\u53d6\u5728\u93c8\u7d50\u4e32\u5217\u6216\u7d05\u9ed1\u6a39\u4e2d\u57f7\u884c\u67e5\u8a62\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u6642\u9593\u7684\u98a8\u96aa\u3002</p> <p>Q\uff1a\u591a\u6b21\u96dc\u6e4a\u6709\u4e0d\u80fd\u76f4\u63a5\u522a\u9664\u5143\u7d20\u7684\u7f3a\u9677\u55ce\uff1f\u6a19\u8a18\u70ba\u5df2\u522a\u9664\u7684\u7a7a\u9593\u9084\u80fd\u518d\u6b21\u4f7f\u7528\u55ce\uff1f</p> <p>\u591a\u6b21\u96dc\u6e4a\u662f\u958b\u653e\u5b9a\u5740\u7684\u4e00\u7a2e\uff0c\u958b\u653e\u5b9a\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u522a\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u900f\u904e\u6a19\u8a18\u522a\u9664\u3002\u6a19\u8a18\u70ba\u5df2\u522a\u9664\u7684\u7a7a\u9593\u53ef\u4ee5\u518d\u6b21\u4f7f\u7528\u3002\u7576\u5c07\u65b0\u5143\u7d20\u63d2\u5165\u96dc\u6e4a\u8868\uff0c\u4e26\u4e14\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u627e\u5230\u6a19\u8a18\u70ba\u5df2\u522a\u9664\u7684\u4f4d\u7f6e\u6642\uff0c\u8a72\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u5143\u7d20\u4f7f\u7528\u3002\u9019\u6a23\u505a\u65e2\u80fd\u4fdd\u6301\u96dc\u6e4a\u8868\u7684\u63a2\u6e2c\u5e8f\u5217\u4e0d\u8b8a\uff0c\u53c8\u80fd\u4fdd\u8b49\u96dc\u6e4a\u8868\u7684\u7a7a\u9593\u4f7f\u7528\u7387\u3002</p> <p>Q\uff1a\u70ba\u4ec0\u9ebc\u7dda\u4e0a\u6027\u63a2\u67e5\u4e2d\uff0c\u67e5\u8a62\u5143\u7d20\u7684\u6642\u5019\u6703\u51fa\u73fe\u96dc\u6e4a\u885d\u7a81\u5462\uff1f</p> <p>\u67e5\u8a62\u7684\u6642\u5019\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u627e\u5230\u5c0d\u61c9\u7684\u6876\u548c\u9375\u503c\u5c0d\uff0c\u767c\u73fe <code>key</code> \u4e0d\u5339\u914d\uff0c\u9019\u5c31\u4ee3\u8868\u6709\u96dc\u6e4a\u885d\u7a81\u3002\u56e0\u6b64\uff0c\u7dda\u6027\u63a2\u67e5\u6cd5\u6703\u6839\u64da\u9810\u5148\u8a2d\u5b9a\u7684\u6b65\u9577\u4f9d\u6b21\u5411\u4e0b\u67e5\u8a62\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u78ba\u7684\u9375\u503c\u5c0d\u6216\u7121\u6cd5\u627e\u5230\u8df3\u51fa\u70ba\u6b62\u3002</p> <p>Q\uff1a\u70ba\u4ec0\u9ebc\u96dc\u6e4a\u8868\u64f4\u5bb9\u80fd\u5920\u7de9\u89e3\u96dc\u6e4a\u885d\u7a81\uff1f</p> <p>\u96dc\u6e4a\u51fd\u5f0f\u7684\u6700\u5f8c\u4e00\u6b65\u5f80\u5f80\u662f\u5c0d\u9663\u5217\u9577\u5ea6 \\(n\\) \u53d6\u6a21\uff08\u53d6\u9918\uff09\uff0c\u8b93\u8f38\u51fa\u503c\u843d\u5728\u9663\u5217\u7d22\u5f15\u7bc4\u570d\u5167\uff1b\u5728\u64f4\u5bb9\u5f8c\uff0c\u9663\u5217\u9577\u5ea6 \\(n\\) \u767c\u751f\u8b8a\u5316\uff0c\u800c <code>key</code> \u5c0d\u61c9\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u767c\u751f\u8b8a\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u500b\u6876\u7684\u591a\u500b <code>key</code> \uff0c\u5728\u64f4\u5bb9\u5f8c\u53ef\u80fd\u6703\u88ab\u5206\u914d\u5230\u591a\u500b\u6876\u4e2d\uff0c\u5f9e\u800c\u5be6\u73fe\u96dc\u6e4a\u885d\u7a81\u7684\u7de9\u89e3\u3002</p>"},{"location":"chapter_heap/","title":"\u7b2c 8 \u7ae0 \u00a0 \u5806\u7a4d","text":"<p>Abstract</p> <p>\u5806\u7a4d\u5c31\u50cf\u662f\u5c71\u5dbd\u5cf0\u5dd2\uff0c\u5c64\u758a\u8d77\u4f0f\u3001\u5f62\u614b\u5404\u7570\u3002</p> <p>\u5ea7\u5ea7\u5c71\u5cf0\u9ad8\u4f4e\u932f\u843d\uff0c\u800c\u6700\u9ad8\u7684\u5c71\u5cf0\u7e3d\u662f\u6700\u5148\u6620\u5165\u773c\u7c3e\u3002</p>"},{"location":"chapter_heap/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>8.1 \u00a0 \u5806\u7a4d</li> <li>8.2 \u00a0 \u5efa\u5806\u7a4d\u64cd\u4f5c</li> <li>8.3 \u00a0 Top-k \u554f\u984c</li> <li>8.4 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_heap/build_heap/","title":"8.2 \u00a0 \u5efa\u5806\u7a4d\u64cd\u4f5c","text":"<p>\u5728\u67d0\u4e9b\u60c5\u6cc1\u4e0b\uff0c\u6211\u5011\u5e0c\u671b\u4f7f\u7528\u4e00\u500b\u4e32\u5217\u7684\u6240\u6709\u5143\u7d20\u4f86\u69cb\u5efa\u4e00\u500b\u5806\u7a4d\uff0c\u9019\u500b\u904e\u7a0b\u88ab\u7a31\u70ba\u201c\u5efa\u5806\u7a4d\u64cd\u4f5c\u201d\u3002</p>"},{"location":"chapter_heap/build_heap/#821","title":"8.2.1 \u00a0 \u85c9\u52a9\u5165\u5806\u7a4d\u64cd\u4f5c\u5be6\u73fe","text":"<p>\u6211\u5011\u9996\u5148\u5efa\u7acb\u4e00\u500b\u7a7a\u5806\u7a4d\uff0c\u7136\u5f8c\u8d70\u8a2a\u4e32\u5217\uff0c\u4f9d\u6b21\u5c0d\u6bcf\u500b\u5143\u7d20\u57f7\u884c\u201c\u5165\u5806\u7a4d\u64cd\u4f5c\u201d\uff0c\u5373\u5148\u5c07\u5143\u7d20\u65b0\u589e\u81f3\u5806\u7a4d\u7684\u5c3e\u90e8\uff0c\u518d\u5c0d\u8a72\u5143\u7d20\u57f7\u884c\u201c\u5f9e\u5e95\u81f3\u9802\u201d\u5806\u7a4d\u5316\u3002</p> <p>\u6bcf\u7576\u4e00\u500b\u5143\u7d20\u5165\u5806\u7a4d\uff0c\u5806\u7a4d\u7684\u9577\u5ea6\u5c31\u52a0\u4e00\u3002\u7531\u65bc\u7bc0\u9ede\u662f\u5f9e\u9802\u5230\u5e95\u4f9d\u6b21\u88ab\u65b0\u589e\u9032\u4e8c\u5143\u6a39\u7684\uff0c\u56e0\u6b64\u5806\u7a4d\u662f\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u69cb\u5efa\u7684\u3002</p> <p>\u8a2d\u5143\u7d20\u6578\u91cf\u70ba \\(n\\) \uff0c\u6bcf\u500b\u5143\u7d20\u7684\u5165\u5806\u7a4d\u64cd\u4f5c\u4f7f\u7528 \\(O(\\log{n})\\) \u6642\u9593\uff0c\u56e0\u6b64\u8a72\u5efa\u5806\u7a4d\u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</p>"},{"location":"chapter_heap/build_heap/#822","title":"8.2.2 \u00a0 \u900f\u904e\u8d70\u8a2a\u5806\u7a4d\u5316\u5be6\u73fe","text":"<p>\u5be6\u969b\u4e0a\uff0c\u6211\u5011\u53ef\u4ee5\u5be6\u73fe\u4e00\u7a2e\u66f4\u70ba\u9ad8\u6548\u7684\u5efa\u5806\u7a4d\u65b9\u6cd5\uff0c\u5171\u5206\u70ba\u5169\u6b65\u3002</p> <ol> <li>\u5c07\u4e32\u5217\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u5730\u65b0\u589e\u5230\u5806\u7a4d\u4e2d\uff0c\u6b64\u6642\u5806\u7a4d\u7684\u6027\u8cea\u5c1a\u672a\u5f97\u5230\u6eff\u8db3\u3002</li> <li>\u5012\u5e8f\u8d70\u8a2a\u5806\u7a4d\uff08\u5c64\u5e8f\u8d70\u8a2a\u7684\u5012\u5e8f\uff09\uff0c\u4f9d\u6b21\u5c0d\u6bcf\u500b\u975e\u8449\u7bc0\u9ede\u57f7\u884c\u201c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\u201d\u3002</li> </ol> <p>\u6bcf\u7576\u5806\u7a4d\u5316\u4e00\u500b\u7bc0\u9ede\u5f8c\uff0c\u4ee5\u8a72\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\u7684\u5b50\u6a39\u5c31\u5f62\u6210\u4e00\u500b\u5408\u6cd5\u7684\u5b50\u5806\u7a4d\u3002\u800c\u7531\u65bc\u662f\u5012\u5e8f\u8d70\u8a2a\uff0c\u56e0\u6b64\u5806\u7a4d\u662f\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u69cb\u5efa\u7684\u3002</p> <p>\u4e4b\u6240\u4ee5\u9078\u64c7\u5012\u5e8f\u8d70\u8a2a\uff0c\u662f\u56e0\u70ba\u9019\u6a23\u80fd\u5920\u4fdd\u8b49\u7576\u524d\u7bc0\u9ede\u4e4b\u4e0b\u7684\u5b50\u6a39\u5df2\u7d93\u662f\u5408\u6cd5\u7684\u5b50\u5806\u7a4d\uff0c\u9019\u6a23\u5806\u7a4d\u5316\u7576\u524d\u7bc0\u9ede\u624d\u662f\u6709\u6548\u7684\u3002</p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u7531\u65bc\u8449\u7bc0\u9ede\u6c92\u6709\u5b50\u7bc0\u9ede\uff0c\u56e0\u6b64\u5b83\u5011\u5929\u7136\u5c31\u662f\u5408\u6cd5\u7684\u5b50\u5806\u7a4d\uff0c\u7121\u9808\u5806\u7a4d\u5316\u3002\u5982\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\uff0c\u6700\u5f8c\u4e00\u500b\u975e\u8449\u7bc0\u9ede\u662f\u6700\u5f8c\u4e00\u500b\u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\uff0c\u6211\u5011\u5f9e\u5b83\u958b\u59cb\u5012\u5e8f\u8d70\u8a2a\u4e26\u57f7\u884c\u5806\u7a4d\u5316\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"\u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d\"\"\"\n    # \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    self.max_heap = nums\n    # \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nMaxHeap(vector&lt;int&gt; nums) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    maxHeap = nums;\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    maxHeap = new List&lt;int&gt;(nums);\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u5207\u7247\u5efa\u5806\u7a4d */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\ninit(nums: [Int]) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    maxHeap = nums\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u5efa\u7acb\u7a7a\u5806\u7a4d\u6216\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nconstructor(nums) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u5efa\u7acb\u7a7a\u5806\u7a4d\u6216\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nconstructor(nums?: number[]) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nMaxHeap(List&lt;int&gt; nums) {\n  // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n  _maxHeap = nums;\n  // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    let mut heap = MaxHeap { max_heap: nums };\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u5207\u7247\u5efa\u5806\u7a4d */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // \u6240\u6709\u5143\u7d20\u5165\u5806\u7a4d\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* \u5927\u9802\u5806\u7a4d */\nclass MaxHeap(nums: MutableList&lt;Int&gt;?) {\n    // \u4f7f\u7528\u4e32\u5217\u800c\u975e\u9663\u5217\uff0c\u9019\u6a23\u7121\u9808\u8003\u616e\u64f4\u5bb9\u554f\u984c\n    private val maxHeap = mutableListOf&lt;Int&gt;()\n\n    /* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\n    init {\n        // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n        maxHeap.addAll(nums!!)\n        // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u4ea4\u63db\u5143\u7d20 */\n    private fun swap(i: Int, j: Int) {\n        val temp = maxHeap[i]\n        maxHeap[i] = maxHeap[j]\n        maxHeap[j] = temp\n    }\n\n    /* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        /* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\n        return size() == 0\n    }\n\n    /* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* \u5143\u7d20\u5165\u5806\u7a4d */\n    fun push(_val: Int) {\n        // \u65b0\u589e\u7bc0\u9ede\n        maxHeap.add(_val)\n        // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n        siftUp(size() - 1)\n    }\n\n    /* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\n    private fun siftUp(it: Int) {\n        // Kotlin\u7684\u51fd\u5f0f\u53c3\u6578\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5efa\u7acb\u81e8\u6642\u8b8a\u6578\n        var i = it\n        while (true) {\n            // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n            val p = parent(i)\n            // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // \u4ea4\u63db\u5169\u7bc0\u9ede\n            swap(i, p)\n            // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n            i = p\n        }\n    }\n\n    /* \u5143\u7d20\u51fa\u5806\u7a4d */\n    fun pop(): Int {\n        // \u5224\u7a7a\u8655\u7406\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        swap(0, size() - 1)\n        // \u522a\u9664\u7bc0\u9ede\n        val _val = maxHeap.removeAt(size() - 1)\n        // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n        siftDown(0)\n        // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n        return _val\n    }\n\n    /* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\n    private fun siftDown(it: Int) {\n        // Kotlin\u7684\u51fd\u5f0f\u53c3\u6578\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5efa\u7acb\u81e8\u6642\u8b8a\u6578\n        var i = it\n        while (true) {\n            // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n            if (ma == i) break\n            // \u4ea4\u63db\u5169\u7bc0\u9ede\n            swap(i, ma)\n            // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n            i = ma\n        }\n    }\n\n    /* \u5217\u5370\u5806\u7a4d\uff08\u4e8c\u5143\u6a39\uff09 */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>### \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d ###\ndef initialize(nums)\n  # \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n  @max_heap = nums\n  # \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n  parent(size - 1).downto(0) do |i|\n    sift_down(i)\n  end\nend\n</code></pre> my_heap.zig<pre><code>// \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n    if (self.max_heap != null) return;\n    self.max_heap = std.ArrayList(T).init(allocator);\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    try self.max_heap.?.appendSlice(nums);\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    var i: usize = parent(self.size() - 1) + 1;\n    while (i &gt; 0) : (i -= 1) {\n        try self.siftDown(i - 1);\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/build_heap/#823","title":"8.2.3 \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<p>\u4e0b\u9762\uff0c\u6211\u5011\u4f86\u5617\u8a66\u63a8\u7b97\u7b2c\u4e8c\u7a2e\u5efa\u5806\u7a4d\u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <ul> <li>\u5047\u8a2d\u5b8c\u5168\u4e8c\u5143\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\(n\\) \uff0c\u5247\u8449\u7bc0\u9ede\u6578\u91cf\u70ba \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u70ba\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u9700\u8981\u5806\u7a4d\u5316\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\((n - 1) / 2\\) \u3002</li> <li>\u5728\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\u7684\u904e\u7a0b\u4e2d\uff0c\u6bcf\u500b\u7bc0\u9ede\u6700\u591a\u5806\u7a4d\u5316\u5230\u8449\u7bc0\u9ede\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6578\u70ba\u4e8c\u5143\u6a39\u9ad8\u5ea6 \\(\\log n\\) \u3002</li> </ul> <p>\u5c07\u4e0a\u8ff0\u5169\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u7a4d\u904e\u7a0b\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002\u4f46\u9019\u500b\u4f30\u7b97\u7d50\u679c\u4e26\u4e0d\u6e96\u78ba\uff0c\u56e0\u70ba\u6211\u5011\u6c92\u6709\u8003\u616e\u5230\u4e8c\u5143\u6a39\u5e95\u5c64\u7bc0\u9ede\u6578\u91cf\u9060\u591a\u65bc\u9802\u5c64\u7bc0\u9ede\u7684\u6027\u8cea\u3002</p> <p>\u63a5\u4e0b\u4f86\u6211\u5011\u4f86\u9032\u884c\u66f4\u70ba\u6e96\u78ba\u7684\u8a08\u7b97\u3002\u70ba\u4e86\u964d\u4f4e\u8a08\u7b97\u96e3\u5ea6\uff0c\u5047\u8a2d\u7d66\u5b9a\u4e00\u500b\u7bc0\u9ede\u6578\u91cf\u70ba \\(n\\) \u3001\u9ad8\u5ea6\u70ba \\(h\\) \u7684\u201c\u5b8c\u7f8e\u4e8c\u5143\u6a39\u201d\uff0c\u8a72\u5047\u8a2d\u4e0d\u6703\u5f71\u97ff\u8a08\u7b97\u7d50\u679c\u7684\u6b63\u78ba\u6027\u3002</p> <p></p> <p> \u5716 8-5 \u00a0 \u5b8c\u7f8e\u4e8c\u5143\u6a39\u7684\u5404\u5c64\u7bc0\u9ede\u6578\u91cf </p> <p>\u5982\u5716 8-5 \u6240\u793a\uff0c\u7bc0\u9ede\u201c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\u201d\u7684\u6700\u5927\u8fed\u4ee3\u6b21\u6578\u7b49\u65bc\u8a72\u7bc0\u9ede\u5230\u8449\u7bc0\u9ede\u7684\u8ddd\u96e2\uff0c\u800c\u8a72\u8ddd\u96e2\u6b63\u662f\u201c\u7bc0\u9ede\u9ad8\u5ea6\u201d\u3002\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u5c0d\u5404\u5c64\u7684\u201c\u7bc0\u9ede\u6578\u91cf \\(\\times\\) \u7bc0\u9ede\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5f97\u5230\u6240\u6709\u7bc0\u9ede\u7684\u5806\u7a4d\u5316\u8fed\u4ee3\u6b21\u6578\u7684\u7e3d\u548c\u3002</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>\u5316\u7c21\u4e0a\u5f0f\u9700\u8981\u85c9\u52a9\u4e2d\u5b78\u7684\u6578\u5217\u77e5\u8b58\uff0c\u5148\u5c07 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>\u4f7f\u7528\u932f\u4f4d\u76f8\u6e1b\u6cd5\uff0c\u7528\u4e0b\u5f0f \\(2 T(h)\\) \u6e1b\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97\uff1a</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>\u89c0\u5bdf\u4e0a\u5f0f\uff0c\u767c\u73fe \\(T(h)\\) \u662f\u4e00\u500b\u7b49\u6bd4\u6578\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u6642\u9593\u8907\u96dc\u5ea6\u70ba\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>\u9032\u4e00\u6b65\uff0c\u9ad8\u5ea6\u70ba \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u5143\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u8907\u96dc\u5ea6\u70ba \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 \u5806\u7a4d","text":"<p>\u5806\u7a4d\uff08heap\uff09\u662f\u4e00\u7a2e\u6eff\u8db3\u7279\u5b9a\u689d\u4ef6\u7684\u5b8c\u5168\u4e8c\u5143\u6a39\uff0c\u4e3b\u8981\u53ef\u5206\u70ba\u5169\u7a2e\u578b\u5225\uff0c\u5982\u5716 8-1 \u6240\u793a\u3002</p> <ul> <li>\u5c0f\u9802\u5806\u7a4d\uff08min heap\uff09\uff1a\u4efb\u610f\u7bc0\u9ede\u7684\u503c \\(\\leq\\) \u5176\u5b50\u7bc0\u9ede\u7684\u503c\u3002</li> <li>\u5927\u9802\u5806\u7a4d\uff08max heap\uff09\uff1a\u4efb\u610f\u7bc0\u9ede\u7684\u503c \\(\\geq\\) \u5176\u5b50\u7bc0\u9ede\u7684\u503c\u3002</li> </ul> <p></p> <p> \u5716 8-1 \u00a0 \u5c0f\u9802\u5806\u7a4d\u8207\u5927\u9802\u5806\u7a4d </p> <p>\u5806\u7a4d\u4f5c\u70ba\u5b8c\u5168\u4e8c\u5143\u6a39\u7684\u4e00\u500b\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u6700\u5e95\u5c64\u7bc0\u9ede\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c64\u7684\u7bc0\u9ede\u90fd\u88ab\u586b\u6eff\u3002</li> <li>\u6211\u5011\u5c07\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede\u7a31\u70ba\u201c\u5806\u7a4d\u9802\u201d\uff0c\u5c07\u5e95\u5c64\u6700\u9760\u53f3\u7684\u7bc0\u9ede\u7a31\u70ba\u201c\u5806\u7a4d\u5e95\u201d\u3002</li> <li>\u5c0d\u65bc\u5927\u9802\u5806\u7a4d\uff08\u5c0f\u9802\u5806\u7a4d\uff09\uff0c\u5806\u7a4d\u9802\u5143\u7d20\uff08\u6839\u7bc0\u9ede\uff09\u7684\u503c\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002</li> </ul>"},{"location":"chapter_heap/heap/#811","title":"8.1.1 \u00a0 \u5806\u7a4d\u7684\u5e38\u7528\u64cd\u4f5c","text":"<p>\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u63d0\u4f9b\u7684\u662f\u512a\u5148\u4f47\u5217\uff08priority queue\uff09\uff0c\u9019\u662f\u4e00\u7a2e\u62bd\u8c61\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u5b9a\u7fa9\u70ba\u5177\u6709\u512a\u5148\u9806\u5e8f\u6392\u5e8f\u7684\u4f47\u5217\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u5806\u7a4d\u901a\u5e38\u7528\u65bc\u5be6\u73fe\u512a\u5148\u4f47\u5217\uff0c\u5927\u9802\u5806\u7a4d\u76f8\u7576\u65bc\u5143\u7d20\u6309\u5f9e\u5927\u5230\u5c0f\u7684\u9806\u5e8f\u51fa\u5217\u7684\u512a\u5148\u4f47\u5217\u3002\u5f9e\u4f7f\u7528\u89d2\u5ea6\u4f86\u770b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u201c\u512a\u5148\u4f47\u5217\u201d\u548c\u201c\u5806\u7a4d\u201d\u770b\u4f5c\u7b49\u50f9\u7684\u8cc7\u6599\u7d50\u69cb\u3002\u56e0\u6b64\uff0c\u672c\u66f8\u5c0d\u5169\u8005\u4e0d\u505a\u7279\u5225\u5340\u5206\uff0c\u7d71\u4e00\u7a31\u4f5c\u201c\u5806\u7a4d\u201d\u3002</p> <p>\u5806\u7a4d\u7684\u5e38\u7528\u64cd\u4f5c\u898b\u8868 8-1 \uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u64da\u7a0b\u5f0f\u8a9e\u8a00\u4f86\u78ba\u5b9a\u3002</p> <p> \u8868 8-1 \u00a0 \u5806\u7a4d\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u6642\u9593\u8907\u96dc\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u5806\u7a4d \\(O(\\log n)\\) <code>pop()</code> \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d \\(O(\\log n)\\) <code>peek()</code> \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20\uff08\u5c0d\u65bc\u5927 / \u5c0f\u9802\u5806\u7a4d\u5206\u5225\u70ba\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) <code>size()</code> \u7372\u53d6\u5806\u7a4d\u7684\u5143\u7d20\u6578\u91cf \\(O(1)\\) <code>isEmpty()</code> \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a \\(O(1)\\) <p>\u5728\u5be6\u969b\u61c9\u7528\u4e2d\uff0c\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u63d0\u4f9b\u7684\u5806\u7a4d\u985e\u5225\uff08\u6216\u512a\u5148\u4f47\u5217\u985e\u5225\uff09\u3002</p> <p>\u985e\u4f3c\u65bc\u6392\u5e8f\u6f14\u7b97\u6cd5\u4e2d\u7684\u201c\u5f9e\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u5f9e\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u8a2d\u5b9a\u4e00\u500b <code>flag</code> \u6216\u4fee\u6539 <code>Comparator</code> \u5be6\u73fe\u201c\u5c0f\u9802\u5806\u7a4d\u201d\u8207\u201c\u5927\u9802\u5806\u7a4d\u201d\u4e4b\u9593\u7684\u8f49\u63db\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap.py<pre><code># \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u7d44\u9810\u8a2d\u5be6\u73fe\u5c0f\u9802\u5806\u7a4d\n# \u8003\u616e\u5c07\u201c\u5143\u7d20\u53d6\u8ca0\u201d\u5f8c\u518d\u5165\u5806\u7a4d\uff0c\u9019\u6a23\u5c31\u53ef\u4ee5\u5c07\u5927\u5c0f\u95dc\u4fc2\u985b\u5012\uff0c\u5f9e\u800c\u5be6\u73fe\u5927\u9802\u5806\u7a4d\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u6642\u5c0d\u61c9\u5c0f\u9802\u5806\u7a4d\uff0cflag = -1 \u6642\u5c0d\u61c9\u5927\u9802\u5806\u7a4d\n\n# \u5143\u7d20\u5165\u5806\u7a4d\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\n# \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u7372\u53d6\u5806\u7a4d\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\uff08\u4f7f\u7528 lambda \u8868\u793a\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\uff08\u4f7f\u7528 lambda \u8868\u793a\u5f0f\u4fee\u6539 Comparer \u5373\u53ef\uff09\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y.CompareTo(x)));\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go \u8a9e\u8a00\u4e2d\u53ef\u4ee5\u900f\u904e\u5be6\u73fe heap.Interface \u4f86\u69cb\u5efa\u6574\u6578\u5927\u9802\u5806\u7a4d\n// \u5be6\u73fe heap.Interface \u9700\u8981\u540c\u6642\u5be6\u73fe sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5be6\u73fe\u63a8\u5165\u5143\u7d20\u5230\u5806\u7a4d\nfunc (h *intHeap) Push(x any) {\n    // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u70ba\u53c3\u6578\n    // \u56e0\u70ba\u5b83\u5011\u4e0d\u50c5\u6703\u5c0d\u5207\u7247\u7684\u5167\u5bb9\u9032\u884c\u8abf\u6574\uff0c\u9084\u6703\u4fee\u6539\u5207\u7247\u7684\u9577\u5ea6\u3002\n    *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5be6\u73fe\u5f48\u51fa\u5806\u7a4d\u9802\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n    // \u5f85\u51fa\u5806\u7a4d\u5143\u7d20\u5b58\u653e\u5728\u6700\u5f8c\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n    // \u5982\u679c\u5be6\u73fe\u5c0f\u9802\u5806\u7a4d\uff0c\u5247\u9700\u8981\u8abf\u6574\u70ba\u5c0f\u65bc\u865f\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* \u521d\u59cb\u5316\u5806\u7a4d */\n    // \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* \u5143\u7d20\u5165\u5806\u7a4d */\n    // \u547c\u53eb heap.Interface \u7684\u65b9\u6cd5\uff0c\u4f86\u65b0\u589e\u5143\u7d20\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\n    top := maxHeap.Top()\n    fmt.Printf(\"\u5806\u7a4d\u9802\u5143\u7d20\u70ba %d\\n\", top)\n\n    /* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n    // \u547c\u53eb heap.Interface \u7684\u65b9\u6cd5\uff0c\u4f86\u79fb\u9664\u5143\u7d20\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\n    size := len(*maxHeap)\n    fmt.Printf(\"\u5806\u7a4d\u5143\u7d20\u6578\u91cf\u70ba %d\\n\", size)\n\n    /* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"\u5806\u7a4d\u662f\u5426\u70ba\u7a7a %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// Swift \u7684 Heap \u578b\u5225\u540c\u6642\u652f\u6301\u6700\u5927\u5806\u7a4d\u548c\u6700\u5c0f\u5806\u7a4d\uff0c\u4e14\u9700\u8981\u5f15\u5165 swift-collections\nvar heap = Heap&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nvar peek = heap.max()!\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nlet size = heap.count\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nlet isEmpty = heap.isEmpty\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.dart<pre><code>// Dart \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.kt<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\uff08\u4f7f\u7528 lambda \u8868\u793a\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nvar peek = maxHeap.peek() // 5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nval size = maxHeap.size\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nval isEmpty = maxHeap.isEmpty()\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code># Ruby \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/heap/#812","title":"8.1.2 \u00a0 \u5806\u7a4d\u7684\u5be6\u73fe","text":"<p>\u4e0b\u6587\u5be6\u73fe\u7684\u662f\u5927\u9802\u5806\u7a4d\u3002\u82e5\u8981\u5c07\u5176\u8f49\u63db\u70ba\u5c0f\u9802\u5806\u7a4d\uff0c\u53ea\u9700\u5c07\u6240\u6709\u5927\u5c0f\u908f\u8f2f\u5224\u65b7\u9032\u884c\u9006\u8f49\uff08\u4f8b\u5982\uff0c\u5c07 \\(\\geq\\) \u66ff\u63db\u70ba \\(\\leq\\) \uff09\u3002\u611f\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u81ea\u884c\u5be6\u73fe\u3002</p>"},{"location":"chapter_heap/heap/#1","title":"1. \u00a0 \u5806\u7a4d\u7684\u5132\u5b58\u8207\u8868\u793a","text":"<p>\u201c\u4e8c\u5143\u6a39\u201d\u7ae0\u7bc0\u8b1b\u904e\uff0c\u5b8c\u5168\u4e8c\u5143\u6a39\u975e\u5e38\u9069\u5408\u7528\u9663\u5217\u4f86\u8868\u793a\u3002\u7531\u65bc\u5806\u7a4d\u6b63\u662f\u4e00\u7a2e\u5b8c\u5168\u4e8c\u5143\u6a39\uff0c\u56e0\u6b64\u6211\u5011\u5c07\u63a1\u7528\u9663\u5217\u4f86\u5132\u5b58\u5806\u7a4d\u3002</p> <p>\u7576\u4f7f\u7528\u9663\u5217\u8868\u793a\u4e8c\u5143\u6a39\u6642\uff0c\u5143\u7d20\u4ee3\u8868\u7bc0\u9ede\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u7bc0\u9ede\u5728\u4e8c\u5143\u6a39\u4e2d\u7684\u4f4d\u7f6e\u3002\u7bc0\u9ede\u6307\u6a19\u900f\u904e\u7d22\u5f15\u5c0d\u6620\u516c\u5f0f\u4f86\u5be6\u73fe\u3002</p> <p>\u5982\u5716 8-2 \u6240\u793a\uff0c\u7d66\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\u70ba \\(2i + 1\\) \uff0c\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\u70ba \\(2i + 2\\) \uff0c\u7236\u7bc0\u9ede\u7684\u7d22\u5f15\u70ba \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u7576\u7d22\u5f15\u8d8a\u754c\u6642\uff0c\u8868\u793a\u7a7a\u7bc0\u9ede\u6216\u7bc0\u9ede\u4e0d\u5b58\u5728\u3002</p> <p></p> <p> \u5716 8-2 \u00a0 \u5806\u7a4d\u7684\u8868\u793a\u8207\u5132\u5b58 </p> <p>\u6211\u5011\u53ef\u4ee5\u5c07\u7d22\u5f15\u5c0d\u6620\u516c\u5f0f\u5c01\u88dd\u6210\u51fd\u5f0f\uff0c\u65b9\u4fbf\u5f8c\u7e8c\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"\u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"\u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"\u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n    return (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n</code></pre> my_heap.cpp<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.java<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.cs<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint Parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.go<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n    // \u5411\u4e0b\u6574\u9664\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.js<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.ts<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.dart<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rs<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.c<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u53d6\u6574\n}\n</code></pre> my_heap.kt<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rb<pre><code>### \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 ###\ndef left(i)\n  2 * i + 1\nend\n\n### \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 ###\ndef right(i)\n  2 * i + 2\nend\n\n### \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 ###\ndef parent(i)\n  (i - 1) / 2     # \u5411\u4e0b\u6574\u9664\nend\n</code></pre> my_heap.zig<pre><code>// \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\nfn left(i: usize) usize {\n    return 2 * i + 1;\n}\n\n// \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\nfn right(i: usize) usize {\n    return 2 * i + 2;\n}\n\n// \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15\nfn parent(i: usize) usize {\n    // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n    return @divFloor(i - 1, 2);\n}\n</code></pre>"},{"location":"chapter_heap/heap/#2","title":"2. \u00a0 \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20","text":"<p>\u5806\u7a4d\u9802\u5143\u7d20\u5373\u70ba\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede\uff0c\u4e5f\u5c31\u662f\u4e32\u5217\u7684\u9996\u500b\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"\u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>### \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 ###\ndef peek\n  @max_heap[0]\nend\n</code></pre> my_heap.zig<pre><code>// \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20\nfn peek(self: *Self) T {\n    return self.max_heap.?.items[0];\n}  \n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/heap/#3","title":"3. \u00a0 \u5143\u7d20\u5165\u5806\u7a4d","text":"<p>\u7d66\u5b9a\u5143\u7d20 <code>val</code> \uff0c\u6211\u5011\u9996\u5148\u5c07\u5176\u65b0\u589e\u5230\u5806\u7a4d\u5e95\u3002\u65b0\u589e\u4e4b\u5f8c\uff0c\u7531\u65bc <code>val</code> \u53ef\u80fd\u5927\u65bc\u5806\u7a4d\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7a4d\u7684\u6210\u7acb\u689d\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u58de\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u5fa9\u5f9e\u63d2\u5165\u7bc0\u9ede\u5230\u6839\u7bc0\u9ede\u7684\u8def\u5f91\u4e0a\u7684\u5404\u500b\u7bc0\u9ede\uff0c\u9019\u500b\u64cd\u4f5c\u88ab\u7a31\u70ba\u5806\u7a4d\u5316\uff08heapify\uff09\u3002</p> <p>\u8003\u616e\u5f9e\u5165\u5806\u7a4d\u7bc0\u9ede\u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u57f7\u884c\u5806\u7a4d\u5316\u3002\u5982\u5716 8-3 \u6240\u793a\uff0c\u6211\u5011\u6bd4\u8f03\u63d2\u5165\u7bc0\u9ede\u8207\u5176\u7236\u7bc0\u9ede\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u7bc0\u9ede\u66f4\u5927\uff0c\u5247\u5c07\u5b83\u5011\u4ea4\u63db\u3002\u7136\u5f8c\u7e7c\u7e8c\u57f7\u884c\u6b64\u64cd\u4f5c\uff0c\u5f9e\u5e95\u81f3\u9802\u4fee\u5fa9\u5806\u7a4d\u4e2d\u7684\u5404\u500b\u7bc0\u9ede\uff0c\u76f4\u81f3\u8d8a\u904e\u6839\u7bc0\u9ede\u6216\u9047\u5230\u7121\u9808\u4ea4\u63db\u7684\u7bc0\u9ede\u6642\u7d50\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 8-3 \u00a0 \u5143\u7d20\u5165\u5806\u7a4d\u6b65\u9a5f </p> <p>\u8a2d\u7bc0\u9ede\u7e3d\u6578\u70ba \\(n\\) \uff0c\u5247\u6a39\u7684\u9ad8\u5ea6\u70ba \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u7a4d\u5316\u64cd\u4f5c\u7684\u8ff4\u5708\u8f2a\u6578\u6700\u591a\u70ba \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u7a4d\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"\u5143\u7d20\u5165\u5806\u7a4d\"\"\"\n    # \u65b0\u589e\u7bc0\u9ede\n    self.max_heap.append(val)\n    # \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\"\"\"\n    while True:\n        # \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        p = self.parent(i)\n        # \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # \u4ea4\u63db\u5169\u7bc0\u9ede\n        self.swap(i, p)\n        # \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid push(int val) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.push_back(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        int p = parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(maxHeap[i], maxHeap[p]);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid push(int val) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.add(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        int p = parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid Push(int val) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.Add(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    SiftUp(Size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid SiftUp(int i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        int p = Parent(i);\n        // \u82e5\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\uff0c\u5247\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        Swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfunc (h *maxHeap) push(val any) {\n    // \u65b0\u589e\u7bc0\u9ede\n    h.data = append(h.data, val)\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    h.siftUp(len(h.data) - 1)\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        p := h.parent(i)\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        h.swap(i, p)\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfunc push(val: Int) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.append(val)\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(i: size() - 1)\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        let p = parent(i: i)\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i: i, j: p)\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\npush(val) {\n    // \u65b0\u589e\u7bc0\u9ede\n    this.#maxHeap.push(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    this.#siftUp(this.size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\n#siftUp(i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        const p = this.#parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        this.#swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\npush(val: number): void {\n    // \u65b0\u589e\u7bc0\u9ede\n    this.maxHeap.push(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    this.siftUp(this.size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nsiftUp(i: number): void {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        const p = this.parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        this.swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid push(int val) {\n  // \u65b0\u589e\u7bc0\u9ede\n  _maxHeap.add(val);\n  // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n  siftUp(size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid siftUp(int i) {\n  while (true) {\n    // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n    int p = _parent(i);\n    // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // \u4ea4\u63db\u5169\u7bc0\u9ede\n    _swap(i, p);\n    // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfn push(&amp;mut self, val: i32) {\n    // \u65b0\u589e\u7bc0\u9ede\n    self.max_heap.push(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    self.sift_up(self.size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // \u7bc0\u9ede i \u5df2\u7d93\u662f\u5806\u7a4d\u9802\u7bc0\u9ede\u4e86\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if i == 0 {\n            break;\n        }\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        let p = Self::parent(i);\n        // \u7576\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        self.swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid push(MaxHeap *maxHeap, int val) {\n    // \u9810\u8a2d\u60c5\u6cc1\u4e0b\uff0c\u4e0d\u61c9\u8a72\u65b0\u589e\u9019\u9ebc\u591a\u7bc0\u9ede\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"heap is full!\");\n        return;\n    }\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        int p = parent(maxHeap, i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(maxHeap, i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfun push(_val: Int) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.add(_val)\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(size() - 1)\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nfun siftUp(it: Int) {\n    // Kotlin\u7684\u51fd\u5f0f\u53c3\u6578\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5efa\u7acb\u81e8\u6642\u8b8a\u6578\n    var i = it\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        val p = parent(i)\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i, p)\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>### \u5143\u7d20\u5165\u5806\u7a4d ###\ndef push(val)\n  # \u65b0\u589e\u7bc0\u9ede\n  @max_heap &lt;&lt; val\n  # \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n  sift_up(size - 1)\nend\n\n### \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 ###\ndef sift_up(i)\n  loop do\n    # \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n    p = parent(i)\n    # \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n    break if p &lt; 0 || @max_heap[i] &lt;= @max_heap[p]\n    # \u4ea4\u63db\u5169\u7bc0\u9ede\n    swap(i, p)\n    # \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n    i = p\n  end\nend\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u5165\u5806\u7a4d\nfn push(self: *Self, val: T) !void {\n    // \u65b0\u589e\u7bc0\u9ede\n    try self.max_heap.?.append(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    try self.siftUp(self.size() - 1);\n}  \n\n// \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        var p = parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 or self.max_heap.?.items[i] &lt;= self.max_heap.?.items[p]) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        try self.swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/heap/#4","title":"4. \u00a0 \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d","text":"<p>\u5806\u7a4d\u9802\u5143\u7d20\u662f\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede\uff0c\u5373\u4e32\u5217\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u5011\u76f4\u63a5\u5f9e\u4e32\u5217\u4e2d\u522a\u9664\u9996\u5143\u7d20\uff0c\u90a3\u9ebc\u4e8c\u5143\u6a39\u4e2d\u6240\u6709\u7bc0\u9ede\u7684\u7d22\u5f15\u90fd\u6703\u767c\u751f\u8b8a\u5316\uff0c\u9019\u5c07\u4f7f\u5f97\u5f8c\u7e8c\u4f7f\u7528\u5806\u7a4d\u5316\u9032\u884c\u4fee\u5fa9\u8b8a\u5f97\u56f0\u96e3\u3002\u70ba\u4e86\u5118\u91cf\u6e1b\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u8b8a\u52d5\uff0c\u6211\u5011\u63a1\u7528\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9a5f\u3002</p> <ol> <li>\u4ea4\u63db\u5806\u7a4d\u9802\u5143\u7d20\u8207\u5806\u7a4d\u5e95\u5143\u7d20\uff08\u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff09\u3002</li> <li>\u4ea4\u63db\u5b8c\u6210\u5f8c\uff0c\u5c07\u5806\u7a4d\u5e95\u5f9e\u4e32\u5217\u4e2d\u522a\u9664\uff08\u6ce8\u610f\uff0c\u7531\u65bc\u5df2\u7d93\u4ea4\u63db\uff0c\u56e0\u6b64\u5be6\u969b\u4e0a\u522a\u9664\u7684\u662f\u539f\u4f86\u7684\u5806\u7a4d\u9802\u5143\u7d20\uff09\u3002</li> <li>\u5f9e\u6839\u7bc0\u9ede\u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u57f7\u884c\u5806\u7a4d\u5316\u3002</li> </ol> <p>\u5982\u5716 8-4 \u6240\u793a\uff0c\u201c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\u201d\u7684\u64cd\u4f5c\u65b9\u5411\u8207\u201c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\u201d\u76f8\u53cd\uff0c\u6211\u5011\u5c07\u6839\u7bc0\u9ede\u7684\u503c\u8207\u5176\u5169\u500b\u5b50\u7bc0\u9ede\u7684\u503c\u9032\u884c\u6bd4\u8f03\uff0c\u5c07\u6700\u5927\u7684\u5b50\u7bc0\u9ede\u8207\u6839\u7bc0\u9ede\u4ea4\u63db\u3002\u7136\u5f8c\u8ff4\u5708\u57f7\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u904e\u8449\u7bc0\u9ede\u6216\u9047\u5230\u7121\u9808\u4ea4\u63db\u7684\u7bc0\u9ede\u6642\u7d50\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 8-4 \u00a0 \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u6b65\u9a5f </p> <p>\u8207\u5143\u7d20\u5165\u5806\u7a4d\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e5f\u70ba \\(O(\\log n)\\) \u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"\u5143\u7d20\u51fa\u5806\u7a4d\"\"\"\n    # \u5224\u7a7a\u8655\u7406\n    if self.is_empty():\n        raise IndexError(\"\u5806\u7a4d\u70ba\u7a7a\")\n    # \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1)\n    # \u522a\u9664\u7bc0\u9ede\n    val = self.max_heap.pop()\n    # \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    self.sift_down(0)\n    # \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\"\"\"\n    while True:\n        # \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u63db\u5169\u7bc0\u9ede\n        self.swap(i, ma)\n        # \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nvoid pop() {\n    // \u5224\u7a7a\u8655\u7406\n    if (isEmpty()) {\n        throw out_of_range(\"\u5806\u7a4d\u70ba\u7a7a\");\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // \u522a\u9664\u7bc0\u9ede\n    maxHeap.pop_back();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(0);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint pop() {\n    // \u5224\u7a7a\u8655\u7406\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    int val = maxHeap.remove(size() - 1);\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint Pop() {\n    // \u5224\u7a7a\u8655\u7406\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    Swap(0, Size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    SiftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid SiftDown(int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u201c\u7bc0\u9ede i \u6700\u5927\u201d\u6216\u201c\u8d8a\u904e\u8449\u7bc0\u9ede\u201d\uff0c\u5247\u7d50\u675f\u5806\u7a4d\u5316\n        if (ma == i) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        Swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfunc (h *maxHeap) pop() any {\n    // \u5224\u7a7a\u8655\u7406\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    h.swap(0, h.size()-1)\n    // \u522a\u9664\u7bc0\u9ede\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    h.siftDown(0)\n\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if max == i {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        h.swap(i, max)\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfunc pop() -&gt; Int {\n    // \u5224\u7a7a\u8655\u7406\n    if isEmpty() {\n        fatalError(\"\u5806\u7a4d\u70ba\u7a7a\")\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(i: 0, j: size() - 1)\n    // \u522a\u9664\u7bc0\u9ede\n    let val = maxHeap.remove(at: size() - 1)\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(i: 0)\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i: i, j: ma)\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\npop() {\n    // \u5224\u7a7a\u8655\u7406\n    if (this.isEmpty()) throw new Error('\u5806\u7a4d\u70ba\u7a7a');\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    this.#swap(0, this.size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    const val = this.#maxHeap.pop();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    this.#siftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\n#siftDown(i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        this.#swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\npop(): number {\n    // \u5224\u7a7a\u8655\u7406\n    if (this.isEmpty()) throw new RangeError('Heap is empty.');\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    this.swap(0, this.size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    const val = this.maxHeap.pop();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    this.siftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nsiftDown(i: number): void {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        this.swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint pop() {\n  // \u5224\u7a7a\u8655\u7406\n  if (isEmpty()) throw Exception('\u5806\u7a4d\u70ba\u7a7a');\n  // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n  _swap(0, size() - 1);\n  // \u522a\u9664\u7bc0\u9ede\n  int val = _maxHeap.removeLast();\n  // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n  siftDown(0);\n  // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n  return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int i) {\n  while (true) {\n    // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u63db\u5169\u7bc0\u9ede\n    _swap(i, ma);\n    // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfn pop(&amp;mut self) -&gt; i32 {\n    // \u5224\u7a7a\u8655\u7406\n    if self.is_empty() {\n        panic!(\"index out of bounds\");\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    let val = self.max_heap.pop().unwrap();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    self.sift_down(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    val\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        self.swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint pop(MaxHeap *maxHeap) {\n    // \u5224\u7a7a\u8655\u7406\n    if (isEmpty(maxHeap)) {\n        printf(\"heap is empty!\");\n        return INT_MAX;\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(maxHeap, 0);\n\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (max == i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(maxHeap, i, max);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfun pop(): Int {\n    // \u5224\u7a7a\u8655\u7406\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1)\n    // \u522a\u9664\u7bc0\u9ede\n    val _val = maxHeap.removeAt(size() - 1)\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(0)\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return _val\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfun siftDown(it: Int) {\n    // Kotlin\u7684\u51fd\u5f0f\u53c3\u6578\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5efa\u7acb\u81e8\u6642\u8b8a\u6578\n    var i = it\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i, ma)\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>### \u5143\u7d20\u51fa\u5806\u7a4d ###\ndef pop\n  # \u5224\u7a7a\u8655\u7406\n  raise IndexError, \"\u5806\u7a4d\u70ba\u7a7a\" if is_empty?\n  # \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n  swap(0, size - 1)\n  # \u522a\u9664\u7bc0\u9ede\n  val = @max_heap.pop\n  # \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n  sift_down(0)\n  # \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n  val\nend\n\n### \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 ###\ndef sift_down(i)\n  loop do\n    # \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n    l, r, ma = left(i), right(i), i\n    ma = l if l &lt; size &amp;&amp; @max_heap[l] &gt; @max_heap[ma]\n    ma = r if r &lt; size &amp;&amp; @max_heap[r] &gt; @max_heap[ma]\n\n    # \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n    break if ma == i\n\n    # \u4ea4\u63db\u5169\u7bc0\u9ede\n    swap(i, ma)\n    # \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n    i = ma\n  end\nend\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u51fa\u5806\u7a4d\nfn pop(self: *Self) !T {\n    // \u5224\u65b7\u8655\u7406\n    if (self.isEmpty()) unreachable;\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    try self.swap(0, self.size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    var val = self.max_heap.?.pop();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    try self.siftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n} \n\n// \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        var l = left(i);\n        var r = right(i);\n        var ma = i;\n        if (l &lt; self.size() and self.max_heap.?.items[l] &gt; self.max_heap.?.items[ma]) ma = l;\n        if (r &lt; self.size() and self.max_heap.?.items[r] &gt; self.max_heap.?.items[ma]) ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        try self.swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/heap/#813","title":"8.1.3 \u00a0 \u5806\u7a4d\u7684\u5e38\u898b\u61c9\u7528","text":"<ul> <li>\u512a\u5148\u4f47\u5217\uff1a\u5806\u7a4d\u901a\u5e38\u4f5c\u70ba\u5be6\u73fe\u512a\u5148\u4f47\u5217\u7684\u9996\u9078\u8cc7\u6599\u7d50\u69cb\uff0c\u5176\u5165\u5217\u548c\u51fa\u5217\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(\\log n)\\) \uff0c\u800c\u5efa\u5806\u7a4d\u64cd\u4f5c\u70ba \\(O(n)\\) \uff0c\u9019\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002</li> <li>\u5806\u7a4d\u6392\u5e8f\uff1a\u7d66\u5b9a\u4e00\u7d44\u8cc7\u6599\uff0c\u6211\u5011\u53ef\u4ee5\u7528\u5b83\u5011\u5efa\u7acb\u4e00\u500b\u5806\u7a4d\uff0c\u7136\u5f8c\u4e0d\u65b7\u5730\u57f7\u884c\u5143\u7d20\u51fa\u5806\u7a4d\u64cd\u4f5c\uff0c\u5f9e\u800c\u5f97\u5230\u6709\u5e8f\u8cc7\u6599\u3002\u7136\u800c\uff0c\u6211\u5011\u901a\u5e38\u6703\u4f7f\u7528\u4e00\u7a2e\u66f4\u512a\u96c5\u7684\u65b9\u5f0f\u5be6\u73fe\u5806\u7a4d\u6392\u5e8f\uff0c\u8a73\u898b\u201c\u5806\u7a4d\u6392\u5e8f\u201d\u7ae0\u7bc0\u3002</li> <li>\u7372\u53d6\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\uff1a\u9019\u662f\u4e00\u500b\u7d93\u5178\u7684\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u540c\u6642\u4e5f\u662f\u4e00\u7a2e\u5178\u578b\u61c9\u7528\uff0c\u4f8b\u5982\u9078\u64c7\u71b1\u5ea6\u524d 10 \u7684\u65b0\u805e\u4f5c\u70ba\u5fae\u535a\u71b1\u641c\uff0c\u9078\u53d6\u92b7\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002</li> </ul>"},{"location":"chapter_heap/summary/","title":"8.4 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_heap/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u5806\u7a4d\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u5143\u6a39\uff0c\u6839\u64da\u6210\u7acb\u689d\u4ef6\u53ef\u5206\u70ba\u5927\u9802\u5806\u7a4d\u548c\u5c0f\u9802\u5806\u7a4d\u3002\u5927\uff08\u5c0f\uff09\u9802\u5806\u7a4d\u7684\u5806\u7a4d\u9802\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002</li> <li>\u512a\u5148\u4f47\u5217\u7684\u5b9a\u7fa9\u662f\u5177\u6709\u51fa\u5217\u512a\u5148\u9806\u5e8f\u7684\u4f47\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u7a4d\u4f86\u5be6\u73fe\u3002</li> <li>\u5806\u7a4d\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5c0d\u61c9\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806\u7a4d \\(O(\\log n)\\)\u3001\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d \\(O(\\log n)\\) \u548c\u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 \\(O(1)\\) \u7b49\u3002</li> <li>\u5b8c\u5168\u4e8c\u5143\u6a39\u975e\u5e38\u9069\u5408\u7528\u9663\u5217\u8868\u793a\uff0c\u56e0\u6b64\u6211\u5011\u901a\u5e38\u4f7f\u7528\u9663\u5217\u4f86\u5132\u5b58\u5806\u7a4d\u3002</li> <li>\u5806\u7a4d\u5316\u64cd\u4f5c\u7528\u65bc\u7dad\u8b77\u5806\u7a4d\u7684\u6027\u8cea\uff0c\u5728\u5165\u5806\u7a4d\u548c\u51fa\u5806\u7a4d\u64cd\u4f5c\u4e2d\u90fd\u6703\u7528\u5230\u3002</li> <li>\u8f38\u5165 \\(n\\) \u500b\u5143\u7d20\u4e26\u5efa\u5806\u7a4d\u7684\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u6700\u4f73\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</li> <li>Top-k \u662f\u4e00\u500b\u7d93\u5178\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u7a4d\u8cc7\u6599\u7d50\u69cb\u9ad8\u6548\u89e3\u6c7a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log k)\\) \u3002</li> </ul>"},{"location":"chapter_heap/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u8cc7\u6599\u7d50\u69cb\u7684\u201c\u5806\u7a4d\u201d\u8207\u8a18\u61b6\u9ad4\u7ba1\u7406\u7684\u201c\u5806\u7a4d\u201d\u662f\u540c\u4e00\u500b\u6982\u5ff5\u55ce\uff1f</p> <p>\u5169\u8005\u4e0d\u662f\u540c\u4e00\u500b\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u201c\u5806\u7a4d\u201d\u3002\u8a08\u7b97\u6a5f\u7cfb\u7d71\u8a18\u61b6\u9ad4\u4e2d\u7684\u5806\u7a4d\u662f\u52d5\u614b\u8a18\u61b6\u9ad4\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5f0f\u5728\u57f7\u884c\u6642\u53ef\u4ee5\u4f7f\u7528\u5b83\u4f86\u5132\u5b58\u8cc7\u6599\u3002\u7a0b\u5f0f\u53ef\u4ee5\u8acb\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u7a4d\u8a18\u61b6\u9ad4\uff0c\u7528\u65bc\u5132\u5b58\u5982\u7269\u4ef6\u548c\u9663\u5217\u7b49\u8907\u96dc\u7d50\u69cb\u3002\u7576\u9019\u4e9b\u8cc7\u6599\u4e0d\u518d\u9700\u8981\u6642\uff0c\u7a0b\u5f0f\u9700\u8981\u91cb\u653e\u9019\u4e9b\u8a18\u61b6\u9ad4\uff0c\u4ee5\u9632\u6b62\u8a18\u61b6\u9ad4\u6d41\u5931\u3002\u76f8\u8f03\u65bc\u5806\u758a\u8a18\u61b6\u9ad4\uff0c\u5806\u7a4d\u8a18\u61b6\u9ad4\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8b39\u614e\uff0c\u4f7f\u7528\u4e0d\u7576\u53ef\u80fd\u6703\u5c0e\u81f4\u8a18\u61b6\u9ad4\u6d41\u5931\u548c\u91ce\u6307\u6a19\u7b49\u554f\u984c\u3002</p>"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-k \u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u7121\u5e8f\u9663\u5217 <code>nums</code> \uff0c\u8acb\u8fd4\u56de\u9663\u5217\u4e2d\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\u3002</p> <p>\u5c0d\u65bc\u8a72\u554f\u984c\uff0c\u6211\u5011\u5148\u4ecb\u7d39\u5169\u7a2e\u601d\u8def\u6bd4\u8f03\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7d39\u6548\u7387\u66f4\u9ad8\u7684\u5806\u7a4d\u89e3\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#831","title":"8.3.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u8d70\u8a2a\u9078\u64c7","text":"<p>\u6211\u5011\u53ef\u4ee5\u9032\u884c\u5716 8-6 \u6240\u793a\u7684 \\(k\\) \u8f2a\u8d70\u8a2a\uff0c\u5206\u5225\u5728\u6bcf\u8f2a\u4e2d\u63d0\u53d6\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(\\dots\\)\u3001\\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(nk)\\) \u3002</p> <p>\u6b64\u65b9\u6cd5\u53ea\u9069\u7528\u65bc \\(k \\ll n\\) \u7684\u60c5\u6cc1\uff0c\u56e0\u70ba\u7576 \\(k\\) \u8207 \\(n\\) \u6bd4\u8f03\u63a5\u8fd1\u6642\uff0c\u5176\u6642\u9593\u8907\u96dc\u5ea6\u8da8\u5411\u65bc \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u6642\u3002</p> <p></p> <p> \u5716 8-6 \u00a0 \u8d70\u8a2a\u5c0b\u627e\u6700\u5927\u7684 k \u500b\u5143\u7d20 </p> <p>Tip</p> <p>\u7576 \\(k = n\\) \u6642\uff0c\u6211\u5011\u53ef\u4ee5\u5f97\u5230\u5b8c\u6574\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6b64\u6642\u7b49\u50f9\u65bc\u201c\u9078\u64c7\u6392\u5e8f\u201d\u6f14\u7b97\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#832","title":"8.3.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"<p>\u5982\u5716 8-7 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u5148\u5c0d\u9663\u5217 <code>nums</code> \u9032\u884c\u6392\u5e8f\uff0c\u518d\u8fd4\u56de\u6700\u53f3\u908a\u7684 \\(k\\) \u500b\u5143\u7d20\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</p> <p>\u986f\u7136\uff0c\u8a72\u65b9\u6cd5\u201c\u8d85\u984d\u201d\u5b8c\u6210\u4efb\u52d9\u4e86\uff0c\u56e0\u70ba\u6211\u5011\u53ea\u9700\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002</p> <p></p> <p> \u5716 8-7 \u00a0 \u6392\u5e8f\u5c0b\u627e\u6700\u5927\u7684 k \u500b\u5143\u7d20 </p>"},{"location":"chapter_heap/top_k/#833","title":"8.3.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806\u7a4d","text":"<p>\u6211\u5011\u53ef\u4ee5\u57fa\u65bc\u5806\u7a4d\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u6c7a Top-k \u554f\u984c\uff0c\u6d41\u7a0b\u5982\u5716 8-8 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4e00\u500b\u5c0f\u9802\u5806\u7a4d\uff0c\u5176\u5806\u7a4d\u9802\u5143\u7d20\u6700\u5c0f\u3002</li> <li>\u5148\u5c07\u9663\u5217\u7684\u524d \\(k\\) \u500b\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u7a4d\u3002</li> <li>\u5f9e\u7b2c \\(k + 1\\) \u500b\u5143\u7d20\u958b\u59cb\uff0c\u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\uff0c\u4e26\u5c07\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\u3002</li> <li>\u8d70\u8a2a\u5b8c\u6210\u5f8c\uff0c\u5806\u7a4d\u4e2d\u5132\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 8-8 \u00a0 \u57fa\u65bc\u5806\u7a4d\u5c0b\u627e\u6700\u5927\u7684 k \u500b\u5143\u7d20 </p> <p>\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"\u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20\"\"\"\n    # \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    heap = []\n    # \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for i in range(k, len(nums)):\n        # \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (int i = k; i &lt; nums.length; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for i := k; i &lt; len(nums); i++ {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u500b\u5c0f\u9802\u5806\u7a4d\uff0c\u4e26\u5c07\u524d k \u500b\u5143\u7d20\u5efa\u5806\u7a4d\n    var heap = Heap(nums.prefix(k))\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for i in nums.indices.dropFirst(k) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfunction pushMinHeap(maxHeap, val) {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfunction popMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u7a4d\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    // \u8acb\u6ce8\u610f\uff1a\u6211\u5011\u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u7528\u5927\u9802\u5806\u7a4d\u4f86\u6a21\u64ec\u5c0f\u9802\u5806\u7a4d\n    const maxHeap = new MaxHeap([]);\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u7a4d\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u7a4d\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    // \u8acb\u6ce8\u610f\uff1a\u6211\u5011\u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u7528\u5927\u9802\u5806\u7a4d\u4f86\u6a21\u64ec\u5c0f\u9802\u5806\u7a4d\n    const maxHeap = new MaxHeap([]);\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u7a4d\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\uff0c\u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n  for (int i = k; i &lt; nums.length; i++) {\n    // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap \u662f\u5927\u9802\u5806\u7a4d\uff0c\u4f7f\u7528 Reverse \u5c07\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u5be6\u73fe\u5c0f\u9802\u5806\u7a4d\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for &amp;num in nums.iter().skip(k) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // \u5143\u7d20\u53d6\u53cd\n    push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint popMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -pop(maxHeap);\n}\n\n/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u7a4d\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u4e26\u5b58\u5165 res \u9663\u5217\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* \u53d6\u51fa\u5806\u7a4d\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u4e26\u5b58\u5165 res \u9663\u5217\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20\u7684\u51fd\u5f0f\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    // \u8acb\u6ce8\u610f\uff1a\u6211\u5011\u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u7528\u5927\u9802\u5806\u7a4d\u4f86\u6a21\u64ec\u5c0f\u9802\u5806\u7a4d\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    val heap = PriorityQueue&lt;Int&gt;()\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (i in k..&lt;nums.size) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>### \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 ###\ndef top_k_heap(nums, k)\n  # \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n  # \u8acb\u6ce8\u610f\uff1a\u6211\u5011\u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u7528\u5927\u9802\u5806\u7a4d\u4f86\u6a21\u64ec\u5c0f\u9802\u5806\u7a4d\n  max_heap = MaxHeap.new([])\n\n  # \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n  for i in 0...k\n    push_min_heap(max_heap, nums[i])\n  end\n\n  # \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n  for i in k...nums.length\n    # \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n    if nums[i] &gt; peek_min_heap(max_heap)\n      pop_min_heap(max_heap)\n      push_min_heap(max_heap, nums[i])\n    end\n  end\n\n  get_min_heap(max_heap)\nend\n</code></pre> top_k.zig<pre><code>[class]{}-[func]{topKHeap}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u7e3d\u5171\u57f7\u884c\u4e86 \\(n\\) \u8f2a\u5165\u5806\u7a4d\u548c\u51fa\u5806\u7a4d\uff0c\u5806\u7a4d\u7684\u6700\u5927\u9577\u5ea6\u70ba \\(k\\) \uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log k)\\) \u3002\u8a72\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u7576 \\(k\\) \u8f03\u5c0f\u6642\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u8da8\u5411 \\(O(n)\\) \uff1b\u7576 \\(k\\) \u8f03\u5927\u6642\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u4e0d\u6703\u8d85\u904e \\(O(n \\log n)\\) \u3002</p> <p>\u53e6\u5916\uff0c\u8a72\u65b9\u6cd5\u9069\u7528\u65bc\u52d5\u614b\u8cc7\u6599\u6d41\u7684\u4f7f\u7528\u5834\u666f\u3002\u5728\u4e0d\u65b7\u52a0\u5165\u8cc7\u6599\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u6301\u7e8c\u7dad\u8b77\u5806\u7a4d\u5167\u7684\u5143\u7d20\uff0c\u5f9e\u800c\u5be6\u73fe\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\u7684\u52d5\u614b\u66f4\u65b0\u3002</p>"},{"location":"chapter_hello_algo/","title":"\u5e8f","text":"<p>\u5e7e\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u201c\u528d\u6307 Offer\u201d\u7cfb\u5217\u984c\u89e3\uff0c\u53d7\u5230\u4e86\u8a31\u591a\u8b80\u8005\u7684\u9f13\u52f5\u548c\u652f\u6301\u3002\u5728\u8207\u8b80\u8005\u4ea4\u6d41\u671f\u9593\uff0c\u6211\u6700\u5e38\u88ab\u554f\u7684\u4e00\u500b\u554f\u984c\u662f\u201c\u5982\u4f55\u5165\u9580\u6f14\u7b97\u6cd5\u201d\u3002\u9010\u6f38\u5730\uff0c\u6211\u5c0d\u9019\u500b\u554f\u984c\u7522\u751f\u4e86\u6fc3\u539a\u7684\u8208\u8da3\u3002</p> <p>\u5169\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u984c\u4f3c\u4e4e\u662f\u6700\u53d7\u6b61\u8fce\u7684\u65b9\u6cd5\uff0c\u7c21\u55ae\u3001\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\u5237\u984c\u5c31\u5982\u540c\u73a9\u201c\u6383\u96f7\u201d\u904a\u6232\uff0c\u81ea\u5b78\u80fd\u529b\u5f37\u7684\u4eba\u80fd\u5920\u9806\u5229\u5c07\u5730\u96f7\u9010\u500b\u6392\u6389\uff0c\u800c\u57fa\u790e\u4e0d\u8db3\u7684\u4eba\u5f88\u53ef\u80fd\u88ab\u70b8\u5f97\u6eff\u982d\u662f\u5305\uff0c\u4e26\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7e2e\u3002\u901a\u8b80\u6559\u6750\u4e5f\u662f\u4e00\u7a2e\u5e38\u898b\u505a\u6cd5\uff0c\u4f46\u5c0d\u65bc\u9762\u5411\u6c42\u8077\u7684\u4eba\u4f86\u8aaa\uff0c\u7562\u696d\u8ad6\u6587\u3001\u6295\u905e\u7c21\u6b77\u3001\u6e96\u5099\u7b46\u8a66\u548c\u9762\u8a66\u5df2\u7d93\u6d88\u8017\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u5543\u539a\u91cd\u7684\u66f8\u5f80\u5f80\u8b8a\u6210\u4e86\u4e00\u9805\u8271\u9245\u7684\u6311\u6230\u3002</p> <p>\u5982\u679c\u4f60\u4e5f\u9762\u81e8\u985e\u4f3c\u7684\u56f0\u64fe\uff0c\u90a3\u9ebc\u5f88\u5e78\u904b\u9019\u672c\u66f8\u201c\u627e\u201d\u5230\u4e86\u4f60\u3002\u672c\u66f8\u662f\u6211\u5c0d\u9019\u500b\u554f\u984c\u7d66\u51fa\u7684\u7b54\u6848\uff0c\u5373\u4f7f\u4e0d\u662f\u6700\u512a\u89e3\uff0c\u4e5f\u81f3\u5c11\u662f\u4e00\u6b21\u7a4d\u6975\u7684\u5617\u8a66\u3002\u672c\u66f8\u96d6\u7136\u4e0d\u8db3\u4ee5\u8b93\u4f60\u76f4\u63a5\u62ff\u5230 Offer\uff0c\u4f46\u6703\u5f15\u5c0e\u4f60\u63a2\u7d22\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u201c\u77e5\u8b58\u5730\u5716\u201d\uff0c\u5e36\u4f60\u77ad\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72c0\u3001\u5927\u5c0f\u548c\u5206\u4f48\u4f4d\u7f6e\uff0c\u8b93\u4f60\u638c\u63e1\u5404\u7a2e\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u9019\u4e9b\u672c\u9818\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u81ea\u5982\u5730\u5237\u984c\u548c\u95b1\u8b80\u6587\u737b\uff0c\u9010\u6b65\u69cb\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8b58\u9ad4\u7cfb\u3002</p> <p>\u6211\u6df1\u6df1\u8d0a\u540c\u8cbb\u66fc\u6559\u6388\u6240\u8a00\uff1a\u201cKnowledge isn't free. You have to pay attention.\u201d\u5f9e\u9019\u500b\u610f\u7fa9\u4e0a\u770b\uff0c\u9019\u672c\u66f8\u4e26\u975e\u5b8c\u5168\u201c\u514d\u8cbb\u201d\u3002\u70ba\u4e86\u4e0d\u8f9c\u8ca0\u4f60\u70ba\u672c\u66f8\u6240\u4ed8\u51fa\u7684\u5bf6\u8cb4\u201c\u6ce8\u610f\u529b\u201d\uff0c\u6211\u6703\u7aed\u76e1\u6240\u80fd\uff0c\u6295\u5165\u6700\u5927\u7684\u201c\u6ce8\u610f\u529b\u201d\u4f86\u5b8c\u6210\u672c\u66f8\u7684\u5275\u4f5c\u3002</p> <p>\u672c\u4eba\u81ea\u77e5\u5b78\u758f\u624d\u6dfa\uff0c\u66f8\u4e2d\u5167\u5bb9\u96d6\u7136\u5df2\u7d93\u904e\u4e00\u6bb5\u6642\u9593\u7684\u6253\u78e8\uff0c\u4f46\u4e00\u5b9a\u4ecd\u6709\u8a31\u591a\u932f\u8aa4\uff0c\u61c7\u8acb\u5404\u4f4d\u8001\u5e2b\u548c\u540c\u5b78\u6279\u8a55\u6307\u6b63\u3002</p> <p></p> Hello\uff0c\u6f14\u7b97\u6cd5\uff01 <p>\u8a08\u7b97\u6a5f\u7684\u51fa\u73fe\u7d66\u4e16\u754c\u5e36\u4f86\u4e86\u5de8\u5927\u8b8a\u9769\uff0c\u5b83\u6191\u85c9\u9ad8\u901f\u7684\u8a08\u7b97\u80fd\u529b\u548c\u51fa\u8272\u7684\u53ef\u7a0b\u5f0f\u8a2d\u8a08\u6027\uff0c\u6210\u70ba\u4e86\u57f7\u884c\u6f14\u7b97\u6cd5\u8207\u8655\u7406\u8cc7\u6599\u7684\u7406\u60f3\u5a92\u4ecb\u3002\u7121\u8ad6\u662f\u96fb\u5b50\u904a\u6232\u7684\u903c\u771f\u756b\u9762\u3001\u81ea\u52d5\u99d5\u99db\u7684\u667a\u6167\u6c7a\u7b56\uff0c\u9084\u662f AlphaGo \u7684\u7cbe\u5f69\u68cb\u5c40\u3001ChatGPT \u7684\u81ea\u7136\u4e92\u52d5\uff0c\u9019\u4e9b\u61c9\u7528\u90fd\u662f\u6f14\u7b97\u6cd5\u5728\u8a08\u7b97\u6a5f\u4e0a\u7684\u7cbe\u5999\u6f14\u7e79\u3002</p> <p>\u4e8b\u5be6\u4e0a\uff0c\u5728\u8a08\u7b97\u6a5f\u554f\u4e16\u4e4b\u524d\uff0c\u6f14\u7b97\u6cd5\u548c\u8cc7\u6599\u7d50\u69cb\u5c31\u5df2\u7d93\u5b58\u5728\u65bc\u4e16\u754c\u7684\u5404\u500b\u89d2\u843d\u3002\u65e9\u671f\u7684\u6f14\u7b97\u6cd5\u76f8\u5c0d\u7c21\u55ae\uff0c\u4f8b\u5982\u53e4\u4ee3\u7684\u8a08\u6578\u65b9\u6cd5\u548c\u5de5\u5177\u88fd\u4f5c\u6b65\u9a5f\u7b49\u3002\u96a8\u8457\u6587\u660e\u7684\u9032\u6b65\uff0c\u6f14\u7b97\u6cd5\u9010\u6f38\u8b8a\u5f97\u66f4\u52a0\u7cbe\u7d30\u548c\u8907\u96dc\u3002\u5f9e\u5de7\u596a\u5929\u5de5\u7684\u5320\u4eba\u6280\u85dd\u3001\u5230\u89e3\u653e\u751f\u7522\u529b\u7684\u5de5\u696d\u7522\u54c1\u3001\u518d\u5230\u5b87\u5b99\u57f7\u884c\u7684\u79d1\u5b78\u898f\u5f8b\uff0c\u5e7e\u4e4e\u6bcf\u4e00\u4ef6\u5e73\u51e1\u6216\u4ee4\u4eba\u9a5a\u6b4e\u7684\u4e8b\u7269\u80cc\u5f8c\uff0c\u90fd\u96b1\u85cf\u8457\u7cbe\u5999\u7684\u6f14\u7b97\u6cd5\u601d\u60f3\u3002</p> <p>\u540c\u6a23\uff0c\u8cc7\u6599\u7d50\u69cb\u7121\u8655\u4e0d\u5728\uff1a\u5927\u5230\u793e\u6703\u7db2\u8def\uff0c\u5c0f\u5230\u5730\u9435\u7dda\u8def\uff0c\u8a31\u591a\u7cfb\u7d71\u90fd\u53ef\u4ee5\u5efa\u6a21\u70ba\u201c\u5716\u201d\uff1b\u5927\u5230\u4e00\u500b\u570b\u5bb6\uff0c\u5c0f\u5230\u4e00\u500b\u5bb6\u5ead\uff0c\u793e\u6703\u7684\u4e3b\u8981\u7d44\u7e54\u5f62\u5f0f\u5448\u73fe\u51fa\u201c\u6a39\u201d\u7684\u7279\u5fb5\uff1b\u51ac\u5929\u7684\u8863\u670d\u5c31\u50cf\u201c\u5806\u758a\u201d\uff0c\u6700\u5148\u7a7f\u4e0a\u7684\u6700\u5f8c\u624d\u80fd\u812b\u4e0b\uff1b\u7fbd\u6bdb\u7403\u7b52\u5247\u5982\u540c\u201c\u4f47\u5217\u201d\uff0c\u4e00\u7aef\u653e\u5165\u3001\u53e6\u4e00\u7aef\u53d6\u51fa\uff1b\u5b57\u5178\u5c31\u50cf\u4e00\u500b\u201c\u96dc\u6e4a\u8868\u201d\uff0c\u80fd\u5920\u5feb\u901f\u67e5\u8a62\u76ee\u6a19\u8a5e\u689d\u3002</p> <p>\u672c\u66f8\u65e8\u5728\u900f\u904e\u6e05\u6670\u6613\u61c2\u7684\u52d5\u756b\u5716\u89e3\u548c\u53ef\u57f7\u884c\u7684\u7a0b\u5f0f\u78bc\u793a\u4f8b\uff0c\u4f7f\u8b80\u8005\u7406\u89e3\u6f14\u7b97\u6cd5\u548c\u8cc7\u6599\u7d50\u69cb\u7684\u6838\u5fc3\u6982\u5ff5\uff0c\u4e26\u80fd\u5920\u900f\u904e\u7a0b\u5f0f\u8a2d\u8a08\u4f86\u5be6\u73fe\u5b83\u5011\u3002\u5728\u6b64\u57fa\u790e\u4e0a\uff0c\u672c\u66f8\u81f4\u529b\u65bc\u63ed\u793a\u6f14\u7b97\u6cd5\u5728\u8907\u96dc\u4e16\u754c\u4e2d\u7684\u751f\u52d5\u9ad4\u73fe\uff0c\u5c55\u73fe\u6f14\u7b97\u6cd5\u4e4b\u7f8e\u3002\u5e0c\u671b\u672c\u66f8\u80fd\u5920\u5e6b\u52a9\u5230\u4f60\uff01</p>"},{"location":"chapter_introduction/","title":"\u7b2c 1 \u7ae0 \u00a0 \u521d\u8b58\u6f14\u7b97\u6cd5","text":"<p>Abstract</p> <p>\u4e00\u4f4d\u5c11\u5973\u7fe9\u7fe9\u8d77\u821e\uff0c\u8207\u8cc7\u6599\u4ea4\u7e54\u5728\u4e00\u8d77\uff0c\u88d9\u896c\u4e0a\u98c4\u63da\u8457\u6f14\u7b97\u6cd5\u7684\u65cb\u5f8b\u3002</p> <p>\u5979\u9080\u8acb\u4f60\u5171\u821e\uff0c\u8acb\u7dca\u8ddf\u5979\u7684\u6b65\u4f10\uff0c\u8e0f\u5165\u5145\u6eff\u908f\u8f2f\u8207\u7f8e\u611f\u7684\u6f14\u7b97\u6cd5\u4e16\u754c\u3002</p>"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>1.1 \u00a0 \u6f14\u7b97\u6cd5\u7121\u8655\u4e0d\u5728</li> <li>1.2 \u00a0 \u6f14\u7b97\u6cd5\u662f\u4ec0\u9ebc</li> <li>1.3 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1 \u00a0 \u6f14\u7b97\u6cd5\u7121\u8655\u4e0d\u5728","text":"<p>\u7576\u6211\u5011\u807d\u5230\u201c\u6f14\u7b97\u6cd5\u201d\u9019\u500b\u8a5e\u6642\uff0c\u5f88\u81ea\u7136\u5730\u6703\u60f3\u5230\u6578\u5b78\u3002\u7136\u800c\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u6f14\u7b97\u6cd5\u4e26\u4e0d\u6d89\u53ca\u8907\u96dc\u6578\u5b78\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8cf4\u57fa\u672c\u908f\u8f2f\uff0c\u9019\u4e9b\u908f\u8f2f\u5728\u6211\u5011\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u8655\u8655\u53ef\u898b\u3002</p> <p>\u5728\u6b63\u5f0f\u63a2\u8a0e\u6f14\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u500b\u6709\u8da3\u7684\u4e8b\u5be6\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7d93\u5728\u4e0d\u77e5\u4e0d\u89ba\u4e2d\u5b78\u6703\u4e86\u8a31\u591a\u6f14\u7b97\u6cd5\uff0c\u4e26\u7fd2\u6163\u5c07\u5b83\u5011\u61c9\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\u6211\u5c07\u8209\u5e7e\u500b\u5177\u9ad4\u7684\u4f8b\u5b50\u4f86\u8b49\u5be6\u9019\u4e00\u9ede\u3002</p> <p>\u4f8b\u4e00\uff1a\u67e5\u5b57\u5178\u3002\u5728\u5b57\u5178\u88e1\uff0c\u6bcf\u500b\u6f22\u5b57\u90fd\u5c0d\u61c9\u4e00\u500b\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u5b57\u6bcd\u9806\u5e8f\u6392\u5217\u7684\u3002\u5047\u8a2d\u6211\u5011\u9700\u8981\u67e5\u8a62\u4e00\u500b\u62fc\u97f3\u9996\u5b57\u6bcd\u70ba \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u6703\u6309\u7167\u5716 1-1 \u6240\u793a\u7684\u65b9\u5f0f\u5be6\u73fe\u3002</p> <ol> <li>\u7ffb\u958b\u5b57\u5178\u7d04\u4e00\u534a\u7684\u9801\u6578\uff0c\u6aa2\u8996\u8a72\u9801\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u9ebc\uff0c\u5047\u8a2d\u9996\u5b57\u6bcd\u70ba \\(m\\) \u3002</li> <li>\u7531\u65bc\u5728\u62fc\u97f3\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u65bc \\(m\\) \u4e4b\u5f8c\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u8a62\u7bc4\u570d\u7e2e\u5c0f\u5230\u5f8c\u534a\u90e8\u5206\u3002</li> <li>\u4e0d\u65b7\u91cd\u8907\u6b65\u9a5f <code>1.</code> \u548c\u6b65\u9a5f <code>2.</code> \uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u70ba \\(r\\) \u7684\u9801\u78bc\u70ba\u6b62\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 1-1 \u00a0 \u67e5\u5b57\u5178\u6b65\u9a5f </p> <p>\u67e5\u5b57\u5178\u9019\u500b\u5c0f\u5b78\u751f\u5fc5\u5099\u6280\u80fd\uff0c\u5be6\u969b\u4e0a\u5c31\u662f\u8457\u540d\u7684\u201c\u4e8c\u5206\u641c\u5c0b\u201d\u6f14\u7b97\u6cd5\u3002\u5f9e\u8cc7\u6599\u7d50\u69cb\u7684\u89d2\u5ea6\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u5b57\u5178\u8996\u70ba\u4e00\u500b\u5df2\u6392\u5e8f\u7684\u201c\u9663\u5217\u201d\uff1b\u5f9e\u6f14\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u201c\u4e8c\u5206\u641c\u5c0b\u201d\u3002</p> <p>\u4f8b\u4e8c\uff1a\u6574\u7406\u64b2\u514b\u3002\u6211\u5011\u5728\u6253\u724c\u6642\uff0c\u6bcf\u5c40\u90fd\u9700\u8981\u6574\u7406\u624b\u4e2d\u7684\u64b2\u514b\u724c\uff0c\u4f7f\u5176\u5f9e\u5c0f\u5230\u5927\u6392\u5217\uff0c\u5be6\u73fe\u6d41\u7a0b\u5982\u5716 1-2 \u6240\u793a\u3002</p> <ol> <li>\u5c07\u64b2\u514b\u724c\u5283\u5206\u70ba\u201c\u6709\u5e8f\u201d\u548c\u201c\u7121\u5e8f\u201d\u5169\u90e8\u5206\uff0c\u4e26\u5047\u8a2d\u521d\u59cb\u72c0\u614b\u4e0b\u6700\u5de6 1 \u5f35\u64b2\u514b\u724c\u5df2\u7d93\u6709\u5e8f\u3002</li> <li>\u5728\u7121\u5e8f\u90e8\u5206\u62bd\u51fa\u4e00\u5f35\u64b2\u514b\u724c\uff0c\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\u7684\u6b63\u78ba\u4f4d\u7f6e\uff1b\u5b8c\u6210\u5f8c\u6700\u5de6 2 \u5f35\u64b2\u514b\u5df2\u7d93\u6709\u5e8f\u3002</li> <li>\u4e0d\u65b7\u8ff4\u5708\u6b65\u9a5f <code>2.</code> \uff0c\u6bcf\u4e00\u8f2a\u5c07\u4e00\u5f35\u64b2\u514b\u724c\u5f9e\u7121\u5e8f\u90e8\u5206\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\uff0c\u76f4\u81f3\u6240\u6709\u64b2\u514b\u724c\u90fd\u6709\u5e8f\u3002</li> </ol> <p></p> <p> \u5716 1-2 \u00a0 \u64b2\u514b\u6392\u5e8f\u6b65\u9a5f </p> <p>\u4e0a\u8ff0\u6574\u7406\u64b2\u514b\u724c\u7684\u65b9\u6cd5\u672c\u8cea\u4e0a\u662f\u201c\u63d2\u5165\u6392\u5e8f\u201d\u6f14\u7b97\u6cd5\uff0c\u5b83\u5728\u8655\u7406\u5c0f\u578b\u8cc7\u6599\u96c6\u6642\u975e\u5e38\u9ad8\u6548\u3002\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u7684\u6392\u5e8f\u5eab\u51fd\u5f0f\u4e2d\u90fd\u6709\u63d2\u5165\u6392\u5e8f\u7684\u8eab\u5f71\u3002</p> <p>\u4f8b\u4e09\uff1a\u8ca8\u5e63\u627e\u96f6\u3002\u5047\u8a2d\u6211\u5011\u5728\u8d85\u5e02\u8cfc\u8cb7\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7d66\u4e86\u6536\u9280\u54e1 \\(100\\) \u5143\uff0c\u5247\u6536\u9280\u54e1\u9700\u8981\u627e\u6211\u5011 \\(31\\) \u5143\u3002\u4ed6\u6703\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u5982\u5716 1-3 \u6240\u793a\u7684\u601d\u8003\u3002</p> <ol> <li>\u53ef\u9078\u9805\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8ca8\u5e63\uff0c\u5305\u62ec \\(1\\) \u5143\u3001\\(5\\) \u5143\u3001\\(10\\) \u5143\u3001\\(20\\) \u5143\u3002</li> <li>\u5f9e\u53ef\u9078\u9805\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u9918 \\(31 - 20 = 11\\) \u5143\u3002</li> <li>\u5f9e\u5269\u9918\u53ef\u9078\u9805\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u9918 \\(11 - 10 = 1\\) \u5143\u3002</li> <li>\u5f9e\u5269\u9918\u53ef\u9078\u9805\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u9918 \\(1 - 1 = 0\\) \u5143\u3002</li> <li>\u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u70ba \\(20 + 10 + 1 = 31\\) \u5143\u3002</li> </ol> <p></p> <p> \u5716 1-3 \u00a0 \u8ca8\u5e63\u627e\u96f6\u904e\u7a0b </p> <p>\u5728\u4ee5\u4e0a\u6b65\u9a5f\u4e2d\uff0c\u6211\u5011\u6bcf\u4e00\u6b65\u90fd\u63a1\u53d6\u7576\u524d\u770b\u4f86\u6700\u597d\u7684\u9078\u64c7\uff08\u5118\u53ef\u80fd\u7528\u5927\u9762\u984d\u7684\u8ca8\u5e63\uff09\uff0c\u6700\u7d42\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u5f9e\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u9019\u7a2e\u65b9\u6cd5\u672c\u8cea\u4e0a\u662f\u201c\u8caa\u5a6a\u201d\u6f14\u7b97\u6cd5\u3002</p> <p>\u5c0f\u5230\u70f9\u98ea\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u969b\u822a\u884c\uff0c\u5e7e\u4e4e\u6240\u6709\u554f\u984c\u7684\u89e3\u6c7a\u90fd\u96e2\u4e0d\u958b\u6f14\u7b97\u6cd5\u3002\u8a08\u7b97\u6a5f\u7684\u51fa\u73fe\u4f7f\u5f97\u6211\u5011\u80fd\u5920\u900f\u904e\u7a0b\u5f0f\u8a2d\u8a08\u5c07\u8cc7\u6599\u7d50\u69cb\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u4e2d\uff0c\u540c\u6642\u7de8\u5beb\u7a0b\u5f0f\u78bc\u547c\u53eb CPU \u548c GPU \u57f7\u884c\u6f14\u7b97\u6cd5\u3002\u9019\u6a23\u4e00\u4f86\uff0c\u6211\u5011\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u554f\u984c\u8f49\u79fb\u5230\u8a08\u7b97\u6a5f\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u6c7a\u5404\u7a2e\u8907\u96dc\u554f\u984c\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u5c0d\u8cc7\u6599\u7d50\u69cb\u3001\u6f14\u7b97\u6cd5\u3001\u9663\u5217\u548c\u4e8c\u5206\u641c\u5c0b\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u8acb\u7e7c\u7e8c\u5f80\u4e0b\u95b1\u8b80\uff0c\u672c\u66f8\u5c07\u5f15\u5c0e\u4f60\u9081\u5165\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u77e5\u8b58\u6bbf\u5802\u3002</p>"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u6f14\u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u7121\u8655\u4e0d\u5728\uff0c\u4e26\u4e0d\u662f\u9059\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8b58\u3002\u5be6\u969b\u4e0a\uff0c\u6211\u5011\u5df2\u7d93\u5728\u4e0d\u77e5\u4e0d\u89ba\u4e2d\u5b78\u6703\u4e86\u8a31\u591a\u6f14\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u6c7a\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u554f\u984c\u3002</li> <li>\u67e5\u5b57\u5178\u7684\u539f\u7406\u8207\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\u9ad4\u73fe\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u6f14\u7b97\u6cd5\u601d\u60f3\u3002</li> <li>\u6574\u7406\u64b2\u514b\u7684\u904e\u7a0b\u8207\u63d2\u5165\u6392\u5e8f\u6f14\u7b97\u6cd5\u975e\u5e38\u985e\u4f3c\u3002\u63d2\u5165\u6392\u5e8f\u6f14\u7b97\u6cd5\u9069\u5408\u6392\u5e8f\u5c0f\u578b\u8cc7\u6599\u96c6\u3002</li> <li>\u8ca8\u5e63\u627e\u96f6\u7684\u6b65\u9a5f\u672c\u8cea\u4e0a\u662f\u8caa\u5a6a\u6f14\u7b97\u6cd5\uff0c\u6bcf\u4e00\u6b65\u90fd\u63a1\u53d6\u7576\u524d\u770b\u4f86\u6700\u597d\u7684\u9078\u64c7\u3002</li> <li>\u6f14\u7b97\u6cd5\u662f\u5728\u6709\u9650\u6642\u9593\u5167\u89e3\u6c7a\u7279\u5b9a\u554f\u984c\u7684\u4e00\u7d44\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9a5f\uff0c\u800c\u8cc7\u6599\u7d50\u69cb\u662f\u8a08\u7b97\u6a5f\u4e2d\u7d44\u7e54\u548c\u5132\u5b58\u8cc7\u6599\u7684\u65b9\u5f0f\u3002</li> <li>\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7dca\u5bc6\u76f8\u9023\u3002\u8cc7\u6599\u7d50\u69cb\u662f\u6f14\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u6f14\u7b97\u6cd5\u70ba\u8cc7\u6599\u7d50\u69cb\u6ce8\u5165\u751f\u547d\u529b\u3002</li> <li>\u6211\u5011\u53ef\u4ee5\u5c07\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u985e\u6bd4\u70ba\u62fc\u88dd\u7a4d\u6728\uff0c\u7a4d\u6728\u4ee3\u8868\u8cc7\u6599\uff0c\u7a4d\u6728\u7684\u5f62\u72c0\u548c\u9023\u7dda\u65b9\u5f0f\u7b49\u4ee3\u8868\u8cc7\u6599\u7d50\u69cb\uff0c\u62fc\u88dd\u7a4d\u6728\u7684\u6b65\u9a5f\u5247\u5c0d\u61c9\u6f14\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_introduction/summary/#1-q-a","title":"1. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u4f5c\u70ba\u4e00\u540d\u7a0b\u5f0f\u8a2d\u8a08\u5e2b\uff0c\u6211\u5728\u65e5\u5e38\u5de5\u4f5c\u4e2d\u5f9e\u672a\u7528\u6f14\u7b97\u6cd5\u89e3\u6c7a\u904e\u554f\u984c\uff0c\u5e38\u7528\u6f14\u7b97\u6cd5\u90fd\u88ab\u7a0b\u5f0f\u8a9e\u8a00\u5c01\u88dd\u597d\u4e86\uff0c\u76f4\u63a5\u7528\u5c31\u53ef\u4ee5\u4e86\uff1b\u9019\u662f\u5426\u610f\u5473\u8457\u6211\u5011\u5de5\u4f5c\u4e2d\u7684\u554f\u984c\u9084\u6c92\u6709\u5230\u9054\u9700\u8981\u6f14\u7b97\u6cd5\u7684\u7a0b\u5ea6\uff1f</p> <p>\u5982\u679c\u628a\u5177\u9ad4\u7684\u5de5\u4f5c\u6280\u80fd\u6bd4\u4f5c\u662f\u6b66\u529f\u7684\u201c\u62db\u5f0f\u201d\u7684\u8a71\uff0c\u90a3\u9ebc\u57fa\u790e\u79d1\u76ee\u61c9\u8a72\u66f4\u50cf\u662f\u201c\u5167\u529f\u201d\u3002</p> <p>\u6211\u8a8d\u70ba\u5b78\u6f14\u7b97\u6cd5\uff08\u4ee5\u53ca\u5176\u4ed6\u57fa\u790e\u79d1\u76ee\uff09\u7684\u610f\u7fa9\u4e0d\u662f\u5728\u65bc\u5728\u5de5\u4f5c\u4e2d\u5f9e\u96f6\u5be6\u73fe\u5b83\uff0c\u800c\u662f\u57fa\u65bc\u5b78\u5230\u7684\u77e5\u8b58\uff0c\u5728\u89e3\u6c7a\u554f\u984c\u6642\u80fd\u5920\u4f5c\u51fa\u5c08\u696d\u7684\u53cd\u61c9\u548c\u5224\u65b7\uff0c\u5f9e\u800c\u63d0\u5347\u5de5\u4f5c\u7684\u6574\u9ad4\u8cea\u91cf\u3002\u8209\u4e00\u500b\u7c21\u55ae\u4f8b\u5b50\uff0c\u6bcf\u7a2e\u7a0b\u5f0f\u8a9e\u8a00\u90fd\u5167\u5efa\u4e86\u6392\u5e8f\u51fd\u5f0f\uff1a</p> <ul> <li>\u5982\u679c\u6211\u5011\u6c92\u6709\u5b78\u904e\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\uff0c\u90a3\u9ebc\u7d66\u5b9a\u4efb\u4f55\u8cc7\u6599\uff0c\u6211\u5011\u53ef\u80fd\u90fd\u585e\u7d66\u9019\u500b\u6392\u5e8f\u51fd\u5f0f\u53bb\u505a\u4e86\u3002\u57f7\u884c\u9806\u66a2\u3001\u6548\u80fd\u4e0d\u932f\uff0c\u770b\u4e0a\u53bb\u4e26\u6c92\u6709\u4ec0\u9ebc\u554f\u984c\u3002</li> <li>\u4f46\u5982\u679c\u5b78\u904e\u6f14\u7b97\u6cd5\uff0c\u6211\u5011\u5c31\u6703\u77e5\u9053\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(n \\log n)\\) \uff1b\u800c\u5982\u679c\u7d66\u5b9a\u7684\u8cc7\u6599\u662f\u56fa\u5b9a\u4f4d\u6578\u7684\u6574\u6578\uff08\u4f8b\u5982\u5b78\u865f\uff09\uff0c\u90a3\u9ebc\u6211\u5011\u5c31\u53ef\u4ee5\u7528\u6548\u7387\u66f4\u9ad8\u7684\u201c\u57fa\u6578\u6392\u5e8f\u201d\u4f86\u505a\uff0c\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u964d\u70ba \\(O(nk)\\) \uff0c\u5176\u4e2d \\(k\\) \u70ba\u4f4d\u6578\u3002\u7576\u8cc7\u6599\u9ad4\u91cf\u5f88\u5927\u6642\uff0c\u7bc0\u7701\u51fa\u4f86\u7684\u57f7\u884c\u6642\u9593\u5c31\u80fd\u5275\u9020\u8f03\u5927\u50f9\u503c\uff08\u6210\u672c\u964d\u4f4e\u3001\u9ad4\u9a57\u8b8a\u597d\u7b49\uff09\u3002</li> </ul> <p>\u5728\u5de5\u7a0b\u9818\u57df\u4e2d\uff0c\u5927\u91cf\u554f\u984c\u662f\u96e3\u4ee5\u9054\u5230\u6700\u512a\u89e3\u7684\uff0c\u8a31\u591a\u554f\u984c\u53ea\u662f\u88ab\u201c\u5dee\u4e0d\u591a\u201d\u5730\u89e3\u6c7a\u4e86\u3002\u554f\u984c\u7684\u96e3\u6613\u7a0b\u5ea6\u4e00\u65b9\u9762\u53d6\u6c7a\u65bc\u554f\u984c\u672c\u8eab\u7684\u6027\u8cea\uff0c\u53e6\u4e00\u65b9\u9762\u4e5f\u53d6\u6c7a\u65bc\u89c0\u6e2c\u554f\u984c\u7684\u4eba\u7684\u77e5\u8b58\u5132\u5099\u3002\u4eba\u7684\u77e5\u8b58\u8d8a\u5b8c\u5099\u3001\u7d93\u9a57\u8d8a\u591a\uff0c\u5206\u6790\u554f\u984c\u5c31\u6703\u8d8a\u6df1\u5165\uff0c\u554f\u984c\u5c31\u80fd\u88ab\u89e3\u6c7a\u5f97\u66f4\u512a\u96c5\u3002</p>"},{"location":"chapter_introduction/what_is_dsa/","title":"1.2 \u00a0 \u6f14\u7b97\u6cd5\u662f\u4ec0\u9ebc","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1 \u00a0 \u6f14\u7b97\u6cd5\u5b9a\u7fa9","text":"<p>\u6f14\u7b97\u6cd5\uff08algorithm\uff09\u662f\u5728\u6709\u9650\u6642\u9593\u5167\u89e3\u6c7a\u7279\u5b9a\u554f\u984c\u7684\u4e00\u7d44\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9a5f\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u554f\u984c\u662f\u660e\u78ba\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f38\u5165\u548c\u8f38\u51fa\u5b9a\u7fa9\u3002</li> <li>\u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u5920\u5728\u6709\u9650\u6b65\u9a5f\u3001\u6642\u9593\u548c\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e0b\u5b8c\u6210\u3002</li> <li>\u5404\u6b65\u9a5f\u90fd\u6709\u78ba\u5b9a\u7684\u542b\u7fa9\uff0c\u5728\u76f8\u540c\u7684\u8f38\u5165\u548c\u57f7\u884c\u689d\u4ef6\u4e0b\uff0c\u8f38\u51fa\u59cb\u7d42\u76f8\u540c\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2 \u00a0 \u8cc7\u6599\u7d50\u69cb\u5b9a\u7fa9","text":"<p>\u8cc7\u6599\u7d50\u69cb\uff08data structure\uff09\u662f\u7d44\u7e54\u548c\u5132\u5b58\u8cc7\u6599\u7684\u65b9\u5f0f\uff0c\u6db5\u84cb\u8cc7\u6599\u5167\u5bb9\u3001\u8cc7\u6599\u4e4b\u9593\u95dc\u4fc2\u548c\u8cc7\u6599\u64cd\u4f5c\u65b9\u6cd5\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u8a2d\u8a08\u76ee\u6a19\u3002</p> <ul> <li>\u7a7a\u9593\u4f54\u7528\u5118\u91cf\u5c11\uff0c\u4ee5\u7bc0\u7701\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u3002</li> <li>\u8cc7\u6599\u64cd\u4f5c\u5118\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u84cb\u8cc7\u6599\u8a2a\u554f\u3001\u65b0\u589e\u3001\u522a\u9664\u3001\u66f4\u65b0\u7b49\u3002</li> <li>\u63d0\u4f9b\u7c21\u6f54\u7684\u8cc7\u6599\u8868\u793a\u548c\u908f\u8f2f\u8cc7\u8a0a\uff0c\u4ee5\u4fbf\u6f14\u7b97\u6cd5\u9ad8\u6548\u57f7\u884c\u3002</li> </ul> <p>\u8cc7\u6599\u7d50\u69cb\u8a2d\u8a08\u662f\u4e00\u500b\u5145\u6eff\u6b0a\u8861\u7684\u904e\u7a0b\u3002\u5982\u679c\u60f3\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u63d0\u5347\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u5354\u3002\u4e0b\u9762\u8209\u5169\u500b\u4f8b\u5b50\u3002</p> <ul> <li>\u93c8\u7d50\u4e32\u5217\u76f8\u8f03\u65bc\u9663\u5217\uff0c\u5728\u8cc7\u6599\u65b0\u589e\u548c\u522a\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u72a7\u7272\u4e86\u8cc7\u6599\u8a2a\u554f\u901f\u5ea6\u3002</li> <li>\u5716\u76f8\u8f03\u65bc\u93c8\u7d50\u4e32\u5217\uff0c\u63d0\u4f9b\u4e86\u66f4\u8c50\u5bcc\u7684\u908f\u8f2f\u8cc7\u8a0a\uff0c\u4f46\u9700\u8981\u4f54\u7528\u66f4\u5927\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3 \u00a0 \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u95dc\u4fc2","text":"<p>\u5982\u5716 1-4 \u6240\u793a\uff0c\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u9ad8\u5ea6\u76f8\u95dc\u3001\u7dca\u5bc6\u7d50\u5408\uff0c\u5177\u9ad4\u8868\u73fe\u5728\u4ee5\u4e0b\u4e09\u500b\u65b9\u9762\u3002</p> <ul> <li>\u8cc7\u6599\u7d50\u69cb\u662f\u6f14\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u8cc7\u6599\u7d50\u69cb\u70ba\u6f14\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7d50\u69cb\u5316\u5132\u5b58\u7684\u8cc7\u6599\uff0c\u4ee5\u53ca\u64cd\u4f5c\u8cc7\u6599\u7684\u65b9\u6cd5\u3002</li> <li>\u6f14\u7b97\u6cd5\u70ba\u8cc7\u6599\u7d50\u69cb\u6ce8\u5165\u751f\u547d\u529b\u3002\u8cc7\u6599\u7d50\u69cb\u672c\u8eab\u50c5\u5132\u5b58\u8cc7\u6599\u8cc7\u8a0a\uff0c\u7d50\u5408\u6f14\u7b97\u6cd5\u624d\u80fd\u89e3\u6c7a\u7279\u5b9a\u554f\u984c\u3002</li> <li>\u6f14\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u65bc\u4e0d\u540c\u7684\u8cc7\u6599\u7d50\u69cb\u5be6\u73fe\uff0c\u4f46\u57f7\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\uff0c\u9078\u64c7\u5408\u9069\u7684\u8cc7\u6599\u7d50\u69cb\u662f\u95dc\u9375\u3002</li> </ul> <p></p> <p> \u5716 1-4 \u00a0 \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u95dc\u4fc2 </p> <p>\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7336\u5982\u5716 1-5 \u6240\u793a\u7684\u62fc\u88dd\u7a4d\u6728\u3002\u4e00\u5957\u7a4d\u6728\uff0c\u9664\u4e86\u5305\u542b\u8a31\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u9084\u9644\u6709\u8a73\u7d30\u7684\u7d44\u88dd\u8aaa\u660e\u66f8\u3002\u6211\u5011\u6309\u7167\u8aaa\u660e\u66f8\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7d44\u88dd\u51fa\u7cbe\u7f8e\u7684\u7a4d\u6728\u6a21\u578b\u3002</p> <p></p> <p> \u5716 1-5 \u00a0 \u62fc\u88dd\u7a4d\u6728 </p> <p>\u5169\u8005\u7684\u8a73\u7d30\u5c0d\u61c9\u95dc\u4fc2\u5982\u8868 1-1 \u6240\u793a\u3002</p> <p> \u8868 1-1 \u00a0 \u5c07\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u985e\u6bd4\u70ba\u62fc\u88dd\u7a4d\u6728 </p> \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5 \u62fc\u88dd\u7a4d\u6728 \u8f38\u5165\u8cc7\u6599 \u672a\u62fc\u88dd\u7684\u7a4d\u6728 \u8cc7\u6599\u7d50\u69cb \u7a4d\u6728\u7d44\u7e54\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72c0\u3001\u5927\u5c0f\u3001\u9023\u7dda\u65b9\u5f0f\u7b49 \u6f14\u7b97\u6cd5 \u628a\u7a4d\u6728\u62fc\u6210\u76ee\u6a19\u5f62\u614b\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9a5f \u8f38\u51fa\u8cc7\u6599 \u7a4d\u6728\u6a21\u578b <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u662f\u7368\u7acb\u65bc\u7a0b\u5f0f\u8a9e\u8a00\u7684\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u66f8\u5f97\u4ee5\u63d0\u4f9b\u57fa\u65bc\u591a\u7a2e\u7a0b\u5f0f\u8a9e\u8a00\u7684\u5be6\u73fe\u3002</p> <p>\u7d04\u5b9a\u4fd7\u6210\u7684\u7c21\u7a31</p> <p>\u5728\u5be6\u969b\u8a0e\u8ad6\u6642\uff0c\u6211\u5011\u901a\u5e38\u6703\u5c07\u201c\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u201d\u7c21\u7a31\u70ba\u201c\u6f14\u7b97\u6cd5\u201d\u3002\u6bd4\u5982\u773e\u6240\u5468\u77e5\u7684 LeetCode \u6f14\u7b97\u6cd5\u984c\u76ee\uff0c\u5be6\u969b\u4e0a\u540c\u6642\u8003\u67e5\u8cc7\u6599\u7d50\u69cb\u548c\u6f14\u7b97\u6cd5\u5169\u65b9\u9762\u7684\u77e5\u8b58\u3002</p>"},{"location":"chapter_preface/","title":"\u7b2c 0 \u7ae0 \u00a0 \u524d\u8a00","text":"<p>Abstract</p> <p>\u6f14\u7b97\u6cd5\u7336\u5982\u7f8e\u5999\u7684\u4ea4\u97ff\u6a02\uff0c\u6bcf\u4e00\u884c\u7a0b\u5f0f\u78bc\u90fd\u50cf\u97fb\u5f8b\u822c\u6d41\u6dcc\u3002</p> <p>\u9858\u9019\u672c\u66f8\u5728\u4f60\u7684\u8166\u6d77\u4e2d\u8f15\u8f15\u97ff\u8d77\uff0c\u7559\u4e0b\u7368\u7279\u800c\u6df1\u523b\u7684\u65cb\u5f8b\u3002</p>"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>0.1 \u00a0 \u95dc\u65bc\u672c\u66f8</li> <li>0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u66f8</li> <li>0.3 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_preface/about_the_book/","title":"0.1 \u00a0 \u95dc\u65bc\u672c\u66f8","text":"<p>\u672c\u5c08\u6848\u65e8\u5728\u5efa\u7acb\u4e00\u672c\u958b\u6e90\u3001\u514d\u8cbb\u3001\u5c0d\u65b0\u624b\u53cb\u597d\u7684\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u5165\u9580\u6559\u7a0b\u3002</p> <ul> <li>\u5168\u66f8\u63a1\u7528\u52d5\u756b\u5716\u89e3\uff0c\u5167\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b78\u7fd2\u66f2\u7dda\u5e73\u6ed1\uff0c\u5f15\u5c0e\u521d\u5b78\u8005\u63a2\u7d22\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u77e5\u8b58\u5730\u5716\u3002</li> <li>\u6e90\u7a0b\u5f0f\u78bc\u53ef\u4e00\u9375\u57f7\u884c\uff0c\u5e6b\u52a9\u8b80\u8005\u5728\u7df4\u7fd2\u4e2d\u63d0\u5347\u7a0b\u5f0f\u8a2d\u8a08\u6280\u80fd\uff0c\u77ad\u89e3\u6f14\u7b97\u6cd5\u5de5\u4f5c\u539f\u7406\u548c\u8cc7\u6599\u7d50\u69cb\u5e95\u5c64\u5be6\u73fe\u3002</li> <li>\u63d0\u5021\u8b80\u8005\u4e92\u52a9\u5b78\u7fd2\uff0c\u6b61\u8fce\u5927\u5bb6\u5728\u8a55\u8ad6\u5340\u63d0\u51fa\u554f\u984c\u8207\u5206\u4eab\u898b\u89e3\uff0c\u5728\u4ea4\u6d41\u8a0e\u8ad6\u4e2d\u5171\u540c\u9032\u6b65\u3002</li> </ul>"},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1 \u00a0 \u8b80\u8005\u7269\u4ef6","text":"<p>\u82e5\u4f60\u662f\u6f14\u7b97\u6cd5\u521d\u5b78\u8005\uff0c\u5f9e\u672a\u63a5\u89f8\u904e\u6f14\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7d93\u6709\u4e00\u4e9b\u5237\u984c\u7d93\u9a57\uff0c\u5c0d\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8a8d\u8b58\uff0c\u5728\u6703\u8207\u4e0d\u6703\u4e4b\u9593\u53cd\u8986\u6a6b\u8df3\uff0c\u90a3\u9ebc\u672c\u66f8\u6b63\u662f\u70ba\u4f60\u91cf\u8eab\u5b9a\u88fd\u7684\uff01</p> <p>\u5982\u679c\u4f60\u5df2\u7d93\u7a4d\u7d2f\u4e00\u5b9a\u7684\u5237\u984c\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u984c\u578b\uff0c\u90a3\u9ebc\u672c\u66f8\u53ef\u52a9\u4f60\u56de\u9867\u8207\u68b3\u7406\u6f14\u7b97\u6cd5\u77e5\u8b58\u9ad4\u7cfb\uff0c\u5009\u5eab\u6e90\u7a0b\u5f0f\u78bc\u53ef\u4ee5\u7576\u4f5c\u201c\u5237\u984c\u5de5\u5177\u5eab\u201d\u6216\u201c\u6f14\u7b97\u6cd5\u5b57\u5178\u201d\u4f86\u4f7f\u7528\u3002</p> <p>\u82e5\u4f60\u662f\u6f14\u7b97\u6cd5\u201c\u5927\u795e\u201d\uff0c\u6211\u5011\u671f\u5f85\u6536\u5230\u4f60\u7684\u5bf6\u8cb4\u5efa\u8b70\uff0c\u6216\u8005\u4e00\u8d77\u53c3\u8207\u5275\u4f5c\u3002</p> <p>\u524d\u7f6e\u689d\u4ef6</p> <p>\u4f60\u9700\u8981\u81f3\u5c11\u5177\u5099\u4efb\u4e00\u8a9e\u8a00\u7684\u7a0b\u5f0f\u8a2d\u8a08\u57fa\u790e\uff0c\u80fd\u5920\u95b1\u8b80\u548c\u7de8\u5beb\u7c21\u55ae\u7a0b\u5f0f\u78bc\u3002</p>"},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2 \u00a0 \u5167\u5bb9\u7d50\u69cb","text":"<p>\u672c\u66f8\u7684\u4e3b\u8981\u5167\u5bb9\u5982\u5716 0-1 \u6240\u793a\u3002</p> <ul> <li>\u8907\u96dc\u5ea6\u5206\u6790\uff1a\u8cc7\u6599\u7d50\u69cb\u548c\u6f14\u7b97\u6cd5\u7684\u8a55\u50f9\u7dad\u5ea6\u8207\u65b9\u6cd5\u3002\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u898b\u578b\u5225\u3001\u793a\u4f8b\u7b49\u3002</li> <li>\u8cc7\u6599\u7d50\u69cb\uff1a\u57fa\u672c\u8cc7\u6599\u578b\u5225\u548c\u8cc7\u6599\u7d50\u69cb\u7684\u5206\u985e\u65b9\u6cd5\u3002\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u6a39\u3001\u5806\u7a4d\u3001\u5716\u7b49\u8cc7\u6599\u7d50\u69cb\u7684\u5b9a\u7fa9\u3001\u512a\u7f3a\u9ede\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u898b\u578b\u5225\u3001\u5178\u578b\u61c9\u7528\u3001\u5be6\u73fe\u65b9\u6cd5\u7b49\u3002</li> <li>\u6f14\u7b97\u6cd5\uff1a\u641c\u5c0b\u3001\u6392\u5e8f\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52d5\u614b\u898f\u5283\u3001\u8caa\u5a6a\u7b49\u6f14\u7b97\u6cd5\u7684\u5b9a\u7fa9\u3001\u512a\u7f3a\u9ede\u3001\u6548\u7387\u3001\u61c9\u7528\u5834\u666f\u3001\u89e3\u984c\u6b65\u9a5f\u548c\u793a\u4f8b\u554f\u984c\u7b49\u3002</li> </ul> <p></p> <p> \u5716 0-1 \u00a0 \u672c\u66f8\u4e3b\u8981\u5167\u5bb9 </p>"},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3 \u00a0 \u81f4\u8b1d","text":"<p>\u672c\u66f8\u5728\u958b\u6e90\u793e\u7fa4\u773e\u591a\u8ca2\u737b\u8005\u7684\u5171\u540c\u52aa\u529b\u4e0b\u4e0d\u65b7\u5b8c\u5584\u3002\u611f\u8b1d\u6bcf\u4e00\u4f4d\u6295\u5165\u6642\u9593\u8207\u7cbe\u529b\u7684\u64b0\u7a3f\u4eba\uff0c\u4ed6\u5011\u662f\uff08\u6309\u7167 GitHub \u81ea\u52d5\u751f\u6210\u7684\u9806\u5e8f\uff09\uff1akrahets\u3001coderonion\u3001Gonglja\u3001nuomi1\u3001Reanon\u3001justin-tse\u3001hpstory\u3001danielsss\u3001curtishd\u3001night-cruise\u3001S-N-O-R-L-A-X\u3001msk397\u3001gvenusleo\u3001khoaxuantu\u3001RiverTwilight\u3001rongyi\u3001gyt95\u3001zhuoqinyue\u3001K3v123\u3001Zuoxun\u3001mingXta\u3001hello-ikun\u3001FangYuan33\u3001GN-Yu\u3001yuelinxin\u3001longsizhuo\u3001Cathay-Chen\u3001guowei-gong\u3001xBLACKICEx\u3001IsChristina\u3001JoseHung\u3001qualifier1024\u3001QiLOL\u3001pengchzn\u3001Guanngxu\u3001L-Super\u3001WSL0809\u3001Slone123c\u3001lhxsm\u3001yuan0221\u3001what-is-me\u3001theNefelibatas\u3001longranger2\u3001cy-by-side\u3001xiongsp\u3001JeffersonHuang\u3001Transmigration-zhou\u3001magentaqin\u3001Wonderdch\u3001malone6\u3001xiaomiusa87\u3001gaofer\u3001bluebean-cloud\u3001a16su\u3001Shyam-Chen\u3001nanlei\u3001hongyun-robot\u3001Phoenix0415\u3001MolDuM\u3001Nigh\u3001he-weilai\u3001junminhong\u3001mgisr\u3001iron-irax\u3001yd-j\u3001XiaChuerwu\u3001XC-Zero\u3001seven1240\u3001SamJin98\u3001wodray\u3001reeswell\u3001NI-SW\u3001Horbin-Magician\u3001Enlightenus\u3001xjr7670\u3001YangXuanyi\u3001DullSword\u3001boloboloda\u3001iStig\u3001qq909244296\u3001jiaxianhua\u3001wenjianmin\u3001keshida\u3001kilikilikid\u3001lclc6\u3001lwbaptx\u3001liuxjerry\u3001lucaswangdev\u3001lyl625760\u3001hts0000\u3001gledfish\u3001fbigm\u3001echo1937\u3001szu17dmy\u3001dshlstarr\u3001Yucao-cy\u3001coderlef\u3001czruby\u3001bongbongbakudan\u3001beintentional\u3001ZongYangL\u3001ZhongYuuu\u3001luluxia\u3001xb534\u3001bitsmi\u3001ElaBosak233\u3001baagod\u3001zhouLion\u3001yishangzhang\u3001yi427\u3001yabo083\u3001weibk\u3001wangwang105\u3001th1nk3r-ing\u3001tao363\u30014yDX3906\u3001syd168\u3001steventimes\u3001sslmj2020\u3001smilelsb\u3001siqyka\u3001selear\u3001sdshaoda\u3001Xi-Row\u3001popozhu\u3001nuquist19\u3001noobcodemaker\u3001XiaoK29\u3001chadyi\u3001ZhongGuanbin\u3001shanghai-Jerry\u3001JackYang-hellobobo\u3001Javesun99\u3001lipusheng\u3001BlindTerran\u3001ShiMaRing\u3001FreddieLi\u3001FloranceYeh\u3001iFleey\u3001fanchenggang\u3001gltianwen\u3001goerll\u3001Dr-XYZ\u3001nedchu\u3001curly210102\u3001CuB3y0nd\u3001KraHsu\u3001CarrotDLaw\u3001youshaoXG\u3001bubble9um\u3001fanenr\u3001eagleanurag\u3001LifeGoesOnionOnionOnion\u300152coder\u3001foursevenlove\u3001KorsChen\u3001hezhizhen\u3001linzeyan\u3001ZJKung\u3001GaochaoZhu\u3001hopkings2008\u3001yang-le\u3001Evilrabbit520\u3001Turing-1024-Lee\u3001thomasq0\u3001Suremotoo\u3001Allen-Scai\u3001Risuntsy\u3001Richard-Zhang1019\u3001qingpeng9802\u3001primexiao\u3001nidhoggfgg\u30011ch0\u3001MwumLi\u3001martinx\u3001ZnYang2018\u3001hugtyftg\u3001logan-qiu\u3001psychelzh\u3001Keynman\u3001KeiichiKasai\u30010130w \u3002</p> <p>\u672c\u66f8\u7684\u7a0b\u5f0f\u78bc\u5be9\u95b1\u5de5\u4f5c\u7531 coderonion\u3001curtishd\u3001Gonglja\u3001gvenusleo\u3001hpstory\u3001justin-tse\u3001khoaxuantu\u3001krahets\u3001night-cruise\u3001nuomi1\u3001Reanon \u548c rongyi \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u9806\u5e8f\u6392\u5217\uff09\u3002\u611f\u8b1d\u4ed6\u5011\u4ed8\u51fa\u7684\u6642\u9593\u8207\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u5011\u78ba\u4fdd\u4e86\u5404\u8a9e\u8a00\u7a0b\u5f0f\u78bc\u7684\u898f\u7bc4\u8207\u7d71\u4e00\u3002</p> <p>\u5728\u672c\u66f8\u7684\u5275\u4f5c\u904e\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8a31\u591a\u4eba\u7684\u5e6b\u52a9\u3002</p> <ul> <li>\u611f\u8b1d\u6211\u5728\u516c\u53f8\u7684\u5c0e\u5e2b\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u66a2\u8ac7\u4e2d\u4f60\u9f13\u52f5\u6211\u201c\u5feb\u884c\u52d5\u8d77\u4f86\u201d\uff0c\u5805\u5b9a\u4e86\u6211\u5beb\u9019\u672c\u66f8\u7684\u6c7a\u5fc3\uff1b</li> <li>\u611f\u8b1d\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u70ba\u672c\u66f8\u7684\u9996\u4f4d\u8b80\u8005\uff0c\u5f9e\u6f14\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8a31\u591a\u5bf6\u8cb4\u5efa\u8b70\uff0c\u4f7f\u5f97\u672c\u66f8\u66f4\u9069\u5408\u65b0\u624b\u95b1\u8b80\uff1b</li> <li>\u611f\u8b1d\u9a30\u5bf6\u3001\u7426\u5bf6\u3001\u98db\u5bf6\u70ba\u672c\u66f8\u8d77\u4e86\u4e00\u500b\u5bcc\u6709\u5275\u610f\u7684\u540d\u5b57\uff0c\u559a\u8d77\u5927\u5bb6\u5beb\u4e0b\u7b2c\u4e00\u884c\u7a0b\u5f0f\u78bc\u201cHello World!\u201d\u7684\u7f8e\u597d\u56de\u61b6\uff1b</li> <li>\u611f\u8b1d\u6821\u9293\u5728\u667a\u6167\u8ca1\u7522\u6b0a\u65b9\u9762\u63d0\u4f9b\u7684\u5c08\u696d\u5e6b\u52a9\uff0c\u9019\u5c0d\u672c\u958b\u6e90\u66f8\u7684\u5b8c\u5584\u8d77\u5230\u4e86\u91cd\u8981\u4f5c\u7528\uff1b</li> <li>\u611f\u8b1d\u8607\u6f7c\u70ba\u672c\u66f8\u8a2d\u8a08\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c logo \uff0c\u4e26\u5728\u6211\u7684\u5f37\u8feb\u75c7\u7684\u9a45\u4f7f\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\uff1b</li> <li>\u611f\u8b1d @squidfunk \u63d0\u4f9b\u7684\u6392\u7248\u5efa\u8b70\uff0c\u4ee5\u53ca\u4ed6\u958b\u767c\u7684\u958b\u6e90\u6587\u4ef6\u4e3b\u984c Material-for-MkDocs \u3002</li> </ul> <p>\u5728\u5beb\u4f5c\u904e\u7a0b\u4e2d\uff0c\u6211\u95b1\u8b80\u4e86\u8a31\u591a\u95dc\u65bc\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u9019\u4e9b\u4f5c\u54c1\u70ba\u672c\u66f8\u63d0\u4f9b\u4e86\u512a\u79c0\u7684\u7bc4\u672c\uff0c\u78ba\u4fdd\u4e86\u672c\u66f8\u5167\u5bb9\u7684\u6e96\u78ba\u6027\u8207\u54c1\u8cea\u3002\u5728\u6b64\u611f\u8b1d\u6240\u6709\u8001\u5e2b\u548c\u524d\u8f29\u7684\u5091\u51fa\u8ca2\u737b\uff01</p> <p>\u672c\u66f8\u5021\u5c0e\u624b\u8166\u4e26\u7528\u7684\u5b78\u7fd2\u65b9\u5f0f\uff0c\u5728\u9019\u4e00\u9ede\u4e0a\u6211\u6df1\u53d7\u300a\u52d5\u624b\u5b78\u6df1\u5ea6\u5b78\u7fd2\u300b\u7684\u555f\u767c\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8b80\u8005\u5f37\u70c8\u63a8\u85a6\u9019\u672c\u512a\u79c0\u7684\u8457\u4f5c\u3002</p> <p>\u8877\u5fc3\u611f\u8b1d\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u5011\u4e00\u76f4\u4ee5\u4f86\u7684\u652f\u6301\u8207\u9f13\u52f5\uff0c\u8b93\u6211\u6709\u6a5f\u6703\u505a\u9019\u4ef6\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002</p>"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u66f8","text":"<p>Tip</p> <p>\u70ba\u4e86\u7372\u5f97\u6700\u4f73\u7684\u95b1\u8b80\u9ad4\u9a57\uff0c\u5efa\u8b70\u4f60\u901a\u8b80\u672c\u7bc0\u5167\u5bb9\u3002</p>"},{"location":"chapter_preface/suggestions/#021","title":"0.2.1 \u00a0 \u884c\u6587\u98a8\u683c\u7d04\u5b9a","text":"<ul> <li>\u6a19\u984c\u5f8c\u6a19\u8a3b <code>*</code> \u7684\u662f\u9078\u8b80\u7ae0\u7bc0\uff0c\u5167\u5bb9\u76f8\u5c0d\u56f0\u96e3\u3002\u5982\u679c\u4f60\u7684\u6642\u9593\u6709\u9650\uff0c\u53ef\u4ee5\u5148\u8df3\u904e\u3002</li> <li>\u5c08\u696d\u8853\u8a9e\u6703\u4f7f\u7528\u9ed1\u9ad4\uff08\u7d19\u8cea\u7248\u548c PDF \u7248\uff09\u6216\u65b0\u589e\u4e0b\u5283\u7dda\uff08\u7db2\u9801\u7248\uff09\uff0c\u4f8b\u5982\u9663\u5217\uff08array\uff09\u3002\u5efa\u8b70\u8a18\u4f4f\u5b83\u5011\uff0c\u4ee5\u4fbf\u95b1\u8b80\u6587\u737b\u3002</li> <li>\u91cd\u9ede\u5167\u5bb9\u548c\u7e3d\u7d50\u6027\u8a9e\u53e5\u6703 \u52a0\u7c97\uff0c\u9019\u985e\u6587\u5b57\u503c\u5f97\u7279\u5225\u95dc\u6ce8\u3002</li> <li>\u6709\u7279\u6307\u542b\u7fa9\u7684\u8a5e\u53e5\u6703\u4f7f\u7528\u201c\u5f15\u865f\u201d\u6a19\u8a3b\uff0c\u4ee5\u907f\u514d\u6b67\u7fa9\u3002</li> <li>\u7576\u6d89\u53ca\u7a0b\u5f0f\u8a9e\u8a00\u4e4b\u9593\u4e0d\u4e00\u81f4\u7684\u540d\u8a5e\u6642\uff0c\u672c\u66f8\u5747\u4ee5 Python \u70ba\u6e96\uff0c\u4f8b\u5982\u4f7f\u7528 <code>None</code> \u4f86\u8868\u793a\u201c\u7a7a\u201d\u3002</li> <li>\u672c\u66f8\u90e8\u5206\u653e\u68c4\u4e86\u7a0b\u5f0f\u8a9e\u8a00\u7684\u8a3b\u91cb\u898f\u7bc4\uff0c\u4ee5\u63db\u53d6\u66f4\u52a0\u7dca\u6e4a\u7684\u5167\u5bb9\u6392\u7248\u3002\u8a3b\u91cb\u4e3b\u8981\u5206\u70ba\u4e09\u7a2e\u985e\u578b\uff1a\u6a19\u984c\u8a3b\u91cb\u3001\u5167\u5bb9\u8a3b\u91cb\u3001\u591a\u884c\u8a3b\u91cb\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>\"\"\"\u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49\"\"\"\n\n# \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n\"\"\"\n\u591a\u884c\n\u8a3b\u91cb\n\"\"\"\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>### \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 ###\n\n# \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n# \u591a\u884c\n# \u8a3b\u91cb\n</code></pre> <pre><code>// \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n// \u591a\u884c\n// \u8a3b\u91cb\n</code></pre>"},{"location":"chapter_preface/suggestions/#022","title":"0.2.2 \u00a0 \u5728\u52d5\u756b\u5716\u89e3\u4e2d\u9ad8\u6548\u5b78\u7fd2","text":"<p>\u76f8\u8f03\u65bc\u6587\u5b57\uff0c\u5f71\u7247\u548c\u5716\u7247\u5177\u6709\u66f4\u9ad8\u7684\u8cc7\u8a0a\u5bc6\u5ea6\u548c\u7d50\u69cb\u5316\u7a0b\u5ea6\uff0c\u66f4\u6613\u65bc\u7406\u89e3\u3002\u5728\u672c\u66f8\u4e2d\uff0c\u91cd\u9ede\u548c\u96e3\u9ede\u77e5\u8b58\u5c07\u4e3b\u8981\u900f\u904e\u52d5\u756b\u4ee5\u5716\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5247\u4f5c\u70ba\u89e3\u91cb\u8207\u88dc\u5145\u3002</p> <p>\u5982\u679c\u4f60\u5728\u95b1\u8b80\u672c\u66f8\u6642\uff0c\u767c\u73fe\u67d0\u6bb5\u5167\u5bb9\u63d0\u4f9b\u77ad\u5982\u5716 0-2 \u6240\u793a\u7684\u52d5\u756b\u5716\u89e3\uff0c\u8acb\u4ee5\u5716\u70ba\u4e3b\u3001\u4ee5\u6587\u5b57\u70ba\u8f14\uff0c\u7d9c\u5408\u5169\u8005\u4f86\u7406\u89e3\u5167\u5bb9\u3002</p> <p></p> <p> \u5716 0-2 \u00a0 \u52d5\u756b\u5716\u89e3\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#023","title":"0.2.3 \u00a0 \u5728\u7a0b\u5f0f\u78bc\u5be6\u8e10\u4e2d\u52a0\u6df1\u7406\u89e3","text":"<p>\u672c\u66f8\u7684\u914d\u5957\u7a0b\u5f0f\u78bc\u8a17\u7ba1\u5728 GitHub \u5009\u5eab\u3002\u5982\u5716 0-3 \u6240\u793a\uff0c\u6e90\u7a0b\u5f0f\u78bc\u9644\u6709\u6e2c\u8a66\u6a23\u4f8b\uff0c\u53ef\u4e00\u9375\u57f7\u884c\u3002</p> <p>\u5982\u679c\u6642\u9593\u5141\u8a31\uff0c\u5efa\u8b70\u4f60\u53c3\u7167\u7a0b\u5f0f\u78bc\u81ea\u884c\u6572\u4e00\u904d\u3002\u5982\u679c\u5b78\u7fd2\u6642\u9593\u6709\u9650\uff0c\u8acb\u81f3\u5c11\u901a\u8b80\u4e26\u57f7\u884c\u6240\u6709\u7a0b\u5f0f\u78bc\u3002</p> <p>\u8207\u95b1\u8b80\u7a0b\u5f0f\u78bc\u76f8\u6bd4\uff0c\u7de8\u5beb\u7a0b\u5f0f\u78bc\u7684\u904e\u7a0b\u5f80\u5f80\u80fd\u5e36\u4f86\u66f4\u591a\u6536\u7a6b\u3002\u52d5\u624b\u5b78\uff0c\u624d\u662f\u771f\u7684\u5b78\u3002</p> <p></p> <p> \u5716 0-3 \u00a0 \u57f7\u884c\u7a0b\u5f0f\u78bc\u793a\u4f8b </p> <p>\u57f7\u884c\u7a0b\u5f0f\u78bc\u7684\u524d\u7f6e\u5de5\u4f5c\u4e3b\u8981\u5206\u70ba\u4e09\u6b65\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u5b89\u88dd\u672c\u5730\u7a0b\u5f0f\u8a2d\u8a08\u74b0\u5883\u3002\u8acb\u53c3\u7167\u9644\u9304\u6240\u793a\u7684\u6559\u7a0b\u9032\u884c\u5b89\u88dd\uff0c\u5982\u679c\u5df2\u5b89\u88dd\uff0c\u5247\u53ef\u8df3\u904e\u6b64\u6b65\u9a5f\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u514b\u9686\u6216\u4e0b\u8f09\u7a0b\u5f0f\u78bc\u5009\u5eab\u3002\u524d\u5f80 GitHub \u5009\u5eab\u3002\u5982\u679c\u5df2\u7d93\u5b89\u88dd Git \uff0c\u53ef\u4ee5\u900f\u904e\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u5009\u5eab\uff1a</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>\u7576\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u5728\u5716 0-4 \u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u9ede\u9078\u201cDownload ZIP\u201d\u6309\u9215\u76f4\u63a5\u4e0b\u8f09\u7a0b\u5f0f\u78bc\u58d3\u7e2e\u5305\uff0c\u7136\u5f8c\u5728\u672c\u5730\u89e3\u58d3\u5373\u53ef\u3002</p> <p></p> <p> \u5716 0-4 \u00a0 \u514b\u9686\u5009\u5eab\u8207\u4e0b\u8f09\u7a0b\u5f0f\u78bc </p> <p>\u7b2c\u4e09\u6b65\uff1a\u57f7\u884c\u6e90\u7a0b\u5f0f\u78bc\u3002\u5982\u5716 0-5 \u6240\u793a\uff0c\u5c0d\u65bc\u9802\u90e8\u6a19\u6709\u6a94\u6848\u540d\u7a31\u7684\u7a0b\u5f0f\u78bc\u584a\uff0c\u6211\u5011\u53ef\u4ee5\u5728\u5009\u5eab\u7684 <code>codes</code> \u6a94\u6848\u593e\u5167\u627e\u5230\u5c0d\u61c9\u7684\u6e90\u7a0b\u5f0f\u78bc\u6a94\u6848\u3002\u6e90\u7a0b\u5f0f\u78bc\u6a94\u6848\u53ef\u4e00\u9375\u57f7\u884c\uff0c\u5c07\u5e6b\u52a9\u4f60\u7bc0\u7701\u4e0d\u5fc5\u8981\u7684\u9664\u932f\u6642\u9593\uff0c\u8b93\u4f60\u80fd\u5920\u5c08\u6ce8\u65bc\u5b78\u7fd2\u5167\u5bb9\u3002</p> <p></p> <p> \u5716 0-5 \u00a0 \u7a0b\u5f0f\u78bc\u584a\u8207\u5c0d\u61c9\u7684\u6e90\u7a0b\u5f0f\u78bc\u6a94\u6848 </p> <p>\u9664\u4e86\u672c\u5730\u57f7\u884c\u7a0b\u5f0f\u78bc\uff0c\u7db2\u9801\u7248\u9084\u652f\u6301 Python \u7a0b\u5f0f\u78bc\u7684\u8996\u89ba\u5316\u57f7\u884c\uff08\u57fa\u65bc pythontutor \u5be6\u73fe\uff09\u3002\u5982\u5716 0-6 \u6240\u793a\uff0c\u4f60\u53ef\u4ee5\u9ede\u9078\u7a0b\u5f0f\u78bc\u584a\u4e0b\u65b9\u7684\u201c\u8996\u89ba\u5316\u57f7\u884c\u201d\u4f86\u5c55\u958b\u6aa2\u8996\uff0c\u89c0\u5bdf\u6f14\u7b97\u6cd5\u7a0b\u5f0f\u78bc\u7684\u57f7\u884c\u904e\u7a0b\uff1b\u4e5f\u53ef\u4ee5\u9ede\u9078\u201c\u5168\u5c4f\u89c0\u770b\u201d\uff0c\u4ee5\u7372\u5f97\u66f4\u597d\u7684\u95b1\u89bd\u9ad4\u9a57\u3002</p> <p></p> <p> \u5716 0-6 \u00a0 Python \u7a0b\u5f0f\u78bc\u7684\u8996\u89ba\u5316\u57f7\u884c </p>"},{"location":"chapter_preface/suggestions/#024","title":"0.2.4 \u00a0 \u5728\u63d0\u554f\u8a0e\u8ad6\u4e2d\u5171\u540c\u6210\u9577","text":"<p>\u5728\u95b1\u8b80\u672c\u66f8\u6642\uff0c\u8acb\u4e0d\u8981\u8f15\u6613\u8df3\u904e\u90a3\u4e9b\u6c92\u5b78\u660e\u767d\u7684\u77e5\u8b58\u9ede\u3002\u6b61\u8fce\u5728\u8a55\u8ad6\u5340\u63d0\u51fa\u4f60\u7684\u554f\u984c\uff0c\u6211\u548c\u5c0f\u5925\u4f34\u5011\u5c07\u7aed\u8aa0\u70ba\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u6cc1\u4e0b\u53ef\u5728\u5169\u5929\u5167\u56de\u8986\u3002</p> <p>\u5982\u5716 0-7 \u6240\u793a\uff0c\u7db2\u9801\u7248\u6bcf\u500b\u7ae0\u7bc0\u7684\u5e95\u90e8\u90fd\u914d\u6709\u8a55\u8ad6\u5340\u3002\u5e0c\u671b\u4f60\u80fd\u591a\u95dc\u6ce8\u8a55\u8ad6\u5340\u7684\u5167\u5bb9\u3002\u4e00\u65b9\u9762\uff0c\u4f60\u53ef\u4ee5\u77ad\u89e3\u5927\u5bb6\u9047\u5230\u7684\u554f\u984c\uff0c\u5f9e\u800c\u67e5\u6f0f\u88dc\u7f3a\uff0c\u6fc0\u767c\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u671f\u5f85\u4f60\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u5925\u4f34\u7684\u554f\u984c\uff0c\u5206\u4eab\u4f60\u7684\u898b\u89e3\uff0c\u5e6b\u52a9\u4ed6\u4eba\u9032\u6b65\u3002</p> <p></p> <p> \u5716 0-7 \u00a0 \u8a55\u8ad6\u5340\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#025","title":"0.2.5 \u00a0 \u6f14\u7b97\u6cd5\u5b78\u7fd2\u8def\u7dda","text":"<p>\u5f9e\u7e3d\u9ad4\u4e0a\u770b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5b78\u7fd2\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u904e\u7a0b\u5283\u5206\u70ba\u4e09\u500b\u968e\u6bb5\u3002</p> <ol> <li>\u968e\u6bb5\u4e00\uff1a\u6f14\u7b97\u6cd5\u5165\u9580\u3002\u6211\u5011\u9700\u8981\u719f\u6089\u5404\u7a2e\u8cc7\u6599\u7d50\u69cb\u7684\u7279\u9ede\u548c\u7528\u6cd5\uff0c\u5b78\u7fd2\u4e0d\u540c\u6f14\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u7684\u5167\u5bb9\u3002</li> <li>\u968e\u6bb5\u4e8c\uff1a\u5237\u6f14\u7b97\u6cd5\u984c\u3002\u5efa\u8b70\u5f9e\u71b1\u9580\u984c\u76ee\u958b\u5237\uff0c\u5148\u7a4d\u7d2f\u81f3\u5c11 100 \u9053\u984c\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u6f14\u7b97\u6cd5\u554f\u984c\u3002\u521d\u6b21\u5237\u984c\u6642\uff0c\u201c\u77e5\u8b58\u907a\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u500b\u6311\u6230\uff0c\u4f46\u8acb\u653e\u5fc3\uff0c\u9019\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u5011\u53ef\u4ee5\u6309\u7167\u201c\u827e\u8cd3\u6d69\u65af\u907a\u5fd8\u66f2\u7dda\u201d\u4f86\u8907\u7fd2\u984c\u76ee\uff0c\u901a\u5e38\u5728\u9032\u884c 3\uff5e5 \u8f2a\u7684\u91cd\u8907\u5f8c\uff0c\u5c31\u80fd\u5c07\u5176\u7262\u8a18\u5728\u5fc3\u3002\u63a8\u85a6\u7684\u984c\u55ae\u548c\u5237\u984c\u8a08\u5283\u8acb\u898b\u6b64 GitHub \u5009\u5eab\u3002</li> <li>\u968e\u6bb5\u4e09\uff1a\u642d\u5efa\u77e5\u8b58\u9ad4\u7cfb\u3002\u5728\u5b78\u7fd2\u65b9\u9762\uff0c\u6211\u5011\u53ef\u4ee5\u95b1\u8b80\u6f14\u7b97\u6cd5\u5c08\u6b04\u6587\u7ae0\u3001\u89e3\u984c\u6846\u67b6\u548c\u6f14\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65b7\u8c50\u5bcc\u77e5\u8b58\u9ad4\u7cfb\u3002\u5728\u5237\u984c\u65b9\u9762\uff0c\u53ef\u4ee5\u5617\u8a66\u63a1\u7528\u9032\u968e\u5237\u984c\u7b56\u7565\uff0c\u5982\u6309\u5c08\u984c\u5206\u985e\u3001\u4e00\u984c\u591a\u89e3\u3001\u4e00\u89e3\u591a\u984c\u7b49\uff0c\u76f8\u95dc\u7684\u5237\u984c\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u500b\u793e\u7fa4\u627e\u5230\u3002</li> </ol> <p>\u5982\u5716 0-8 \u6240\u793a\uff0c\u672c\u66f8\u5167\u5bb9\u4e3b\u8981\u6db5\u84cb\u201c\u968e\u6bb5\u4e00\u201d\uff0c\u65e8\u5728\u5e6b\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u958b\u968e\u6bb5\u4e8c\u548c\u968e\u6bb5\u4e09\u7684\u5b78\u7fd2\u3002</p> <p></p> <p> \u5716 0-8 \u00a0 \u6f14\u7b97\u6cd5\u5b78\u7fd2\u8def\u7dda </p>"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u672c\u66f8\u7684\u4e3b\u8981\u53d7\u773e\u662f\u6f14\u7b97\u6cd5\u521d\u5b78\u8005\u3002\u5982\u679c\u4f60\u5df2\u6709\u4e00\u5b9a\u57fa\u790e\uff0c\u672c\u66f8\u80fd\u5e6b\u52a9\u4f60\u7cfb\u7d71\u56de\u9867\u6f14\u7b97\u6cd5\u77e5\u8b58\uff0c\u66f8\u4e2d\u6e90\u7a0b\u5f0f\u78bc\u4e5f\u53ef\u4f5c\u70ba\u201c\u5237\u984c\u5de5\u5177\u5eab\u201d\u4f7f\u7528\u3002</li> <li>\u66f8\u4e2d\u5167\u5bb9\u4e3b\u8981\u5305\u62ec\u8907\u96dc\u5ea6\u5206\u6790\u3001\u8cc7\u6599\u7d50\u69cb\u548c\u6f14\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u84cb\u4e86\u8a72\u9818\u57df\u7684\u5927\u90e8\u5206\u4e3b\u984c\u3002</li> <li>\u5c0d\u65bc\u6f14\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b78\u968e\u6bb5\u95b1\u8b80\u4e00\u672c\u5165\u9580\u66f8\u81f3\u95dc\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8a31\u591a\u5f4e\u8def\u3002</li> <li>\u66f8\u4e2d\u7684\u52d5\u756b\u5716\u89e3\u901a\u5e38\u7528\u65bc\u4ecb\u7d39\u91cd\u9ede\u548c\u96e3\u9ede\u77e5\u8b58\u3002\u95b1\u8b80\u672c\u66f8\u6642\uff0c\u61c9\u7d66\u4e88\u9019\u4e9b\u5167\u5bb9\u66f4\u591a\u95dc\u6ce8\u3002</li> <li>\u5be6\u8e10\u4e43\u5b78\u7fd2\u7a0b\u5f0f\u8a2d\u8a08\u4e4b\u6700\u4f73\u9014\u5f91\u3002\u5f37\u70c8\u5efa\u8b70\u57f7\u884c\u6e90\u7a0b\u5f0f\u78bc\u4e26\u89aa\u81ea\u6572\u7a0b\u5f0f\u78bc\u3002</li> <li>\u672c\u66f8\u7db2\u9801\u7248\u7684\u6bcf\u500b\u7ae0\u7bc0\u90fd\u8a2d\u6709\u8a55\u8ad6\u5340\uff0c\u6b61\u8fce\u96a8\u6642\u5206\u4eab\u4f60\u7684\u7591\u60d1\u8207\u898b\u89e3\u3002</li> </ul>"},{"location":"chapter_reference/","title":"\u53c3\u8003\u6587\u737b","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p> <p>[4] \u56b4\u851a\u654f. \u8cc7\u6599\u7d50\u69cb\uff08C \u8a9e\u8a00\u7248\uff09.</p> <p>[5] \u9127\u4fca\u8f1d. \u8cc7\u6599\u7d50\u69cb\uff08C++ \u8a9e\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.</p> <p>[6] \u99ac\u514b \u827e\u502b \u7dad\u65af\u8457\uff0c\u9673\u8d8a\u8b6f. \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u5206\u6790\uff1aJava\u8a9e\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.</p> <p>[7] \u7a0b\u5091. \u5927\u8a71\u8cc7\u6599\u7d50\u69cb.</p> <p>[8] \u738b\u722d. \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u4e4b\u7f8e.</p> <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p> <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>"},{"location":"chapter_searching/","title":"\u7b2c 10 \u7ae0 \u00a0 \u641c\u5c0b","text":"<p>Abstract</p> <p>\u641c\u5c0b\u662f\u4e00\u5834\u672a\u77e5\u7684\u5192\u96aa\uff0c\u6211\u5011\u6216\u8a31\u9700\u8981\u8d70\u904d\u795e\u79d8\u7a7a\u9593\u7684\u6bcf\u500b\u89d2\u843d\uff0c\u53c8\u6216\u8a31\u53ef\u4ee5\u5feb\u901f\u9396\u5b9a\u76ee\u6a19\u3002</p> <p>\u5728\u9019\u5834\u5c0b\u8993\u4e4b\u65c5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u63a2\u7d22\u90fd\u53ef\u80fd\u5f97\u5230\u4e00\u500b\u672a\u66fe\u6599\u60f3\u7684\u7b54\u6848\u3002</p>"},{"location":"chapter_searching/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>10.1 \u00a0 \u4e8c\u5206\u641c\u5c0b</li> <li>10.2 \u00a0 \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede</li> <li>10.3 \u00a0 \u4e8c\u5206\u641c\u5c0b\u908a\u754c</li> <li>10.4 \u00a0 \u96dc\u6e4a\u6700\u4f73\u5316\u7b56\u7565</li> <li>10.5 \u00a0 \u91cd\u8b58\u641c\u5c0b\u6f14\u7b97\u6cd5</li> <li>10.6 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_searching/binary_search/","title":"10.1 \u00a0 \u4e8c\u5206\u641c\u5c0b","text":"<p>\u4e8c\u5206\u641c\u5c0b\uff08binary search\uff09\u662f\u4e00\u7a2e\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u9ad8\u6548\u641c\u5c0b\u6f14\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u8cc7\u6599\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f2a\u7e2e\u5c0f\u4e00\u534a\u641c\u5c0b\u7bc4\u570d\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6a19\u5143\u7d20\u6216\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u70ba\u6b62\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217 <code>nums</code> \uff0c\u5143\u7d20\u6309\u5f9e\u5c0f\u5230\u5927\u7684\u9806\u5e8f\u6392\u5217\u4e14\u4e0d\u91cd\u8907\u3002\u8acb\u67e5\u8a62\u4e26\u8fd4\u56de\u5143\u7d20 <code>target</code> \u5728\u8a72\u9663\u5217\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u9663\u5217\u4e0d\u5305\u542b\u8a72\u5143\u7d20\uff0c\u5247\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u5716 10-1 \u6240\u793a\u3002</p> <p></p> <p> \u5716 10-1 \u00a0 \u4e8c\u5206\u641c\u5c0b\u793a\u4f8b\u8cc7\u6599 </p> <p>\u5982\u5716 10-2 \u6240\u793a\uff0c\u6211\u5011\u5148\u521d\u59cb\u5316\u6307\u6a19 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u5c0b\u5340\u9593 \\([0, n - 1]\\) \u3002\u8acb\u6ce8\u610f\uff0c\u4e2d\u62ec\u865f\u8868\u793a\u9589\u5340\u9593\uff0c\u5176\u5305\u542b\u908a\u754c\u503c\u672c\u8eab\u3002</p> <p>\u63a5\u4e0b\u4f86\uff0c\u8ff4\u5708\u57f7\u884c\u4ee5\u4e0b\u5169\u6b65\u3002</p> <ol> <li>\u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\: \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002</li> <li>\u5224\u65b7 <code>nums[m]</code> \u548c <code>target</code> \u7684\u5927\u5c0f\u95dc\u4fc2\uff0c\u5206\u70ba\u4ee5\u4e0b\u4e09\u7a2e\u60c5\u6cc1\u3002<ol> <li>\u7576 <code>nums[m] &lt; target</code> \u6642\uff0c\u8aaa\u660e <code>target</code> \u5728\u5340\u9593 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u57f7\u884c \\(i = m + 1\\) \u3002</li> <li>\u7576 <code>nums[m] &gt; target</code> \u6642\uff0c\u8aaa\u660e <code>target</code> \u5728\u5340\u9593 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u57f7\u884c \\(j = m - 1\\) \u3002</li> <li>\u7576 <code>nums[m] = target</code> \u6642\uff0c\u8aaa\u660e\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \u3002</li> </ol> </li> </ol> <p>\u82e5\u9663\u5217\u4e0d\u5305\u542b\u76ee\u6a19\u5143\u7d20\uff0c\u641c\u5c0b\u5340\u9593\u6700\u7d42\u6703\u7e2e\u5c0f\u70ba\u7a7a\u3002\u6b64\u6642\u8fd4\u56de \\(-1\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 10-2 \u00a0 \u4e8c\u5206\u641c\u5c0b\u6d41\u7a0b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u65bc \\(i\\) \u548c \\(j\\) \u90fd\u662f <code>int</code> \u578b\u5225\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u6703\u8d85\u51fa <code>int</code> \u578b\u5225\u7684\u53d6\u503c\u7bc4\u570d\u3002\u70ba\u4e86\u907f\u514d\u5927\u6578\u8d8a\u754c\uff0c\u6211\u5011\u901a\u5e38\u63a1\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u4f86\u8a08\u7b97\u4e2d\u9ede\u3002</p> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09\"\"\"\n    # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j = 0, len(nums) - 1\n    # \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while i &lt;= j:\n        # \u7406\u8ad6\u4e0a Python \u7684\u6578\u5b57\u53ef\u4ee5\u7121\u9650\u5927\uff08\u53d6\u6c7a\u65bc\u8a18\u61b6\u9ad4\u5927\u5c0f\uff09\uff0c\u7121\u9808\u8003\u616e\u5927\u6578\u8d8a\u754c\u554f\u984c\n        m = (i + j) // 2  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.size() - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint binarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.length - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint BinarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.Length - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else                       // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfunc binarySearch(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j := 0, len(nums)-1\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    for i &lt;= j {\n        m := i + (j-i)/2      // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfunction binarySearch(nums, target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else return m; // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n  int i = 0, j = nums.length - 1;\n  // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n      j = m - 1;\n    } else {\n      // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint binarySearch(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = len - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = 0\n    var j = nums.size - 1\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        else  // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 ###\ndef binary_search(nums, target)\n  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n  i, j = 0, nums.length - 1\n\n  # \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n  while i &lt;= j\n    # \u7406\u8ad6\u4e0a Ruby \u7684\u6578\u5b57\u53ef\u4ee5\u7121\u9650\u5927\uff08\u53d6\u6c7a\u65bc\u8a18\u61b6\u9ad4\u5927\u5c0f\uff09\uff0c\u7121\u9808\u8003\u616e\u5927\u6578\u8d8a\u754c\u554f\u984c\n    m = (i + j) / 2   # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n\n    if nums[m] &lt; target\n      i = m + 1 # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    elsif nums[m] &gt; target\n      j = m - 1 # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    else\n      return m  # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    end\n  end\n\n  -1  # \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\nend\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i: usize = 0;\n    var j: usize = nums.items.len - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {                                // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \uff1a\u5728\u4e8c\u5206\u8ff4\u5708\u4e2d\uff0c\u5340\u9593\u6bcf\u8f2a\u7e2e\u5c0f\u4e00\u534a\uff0c\u56e0\u6b64\u8ff4\u5708\u6b21\u6578\u70ba \\(\\log_2 n\\) \u3002</p> <p>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \uff1a\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7a7a\u9593\u3002</p>"},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1 \u00a0 \u5340\u9593\u8868\u793a\u65b9\u6cd5","text":"<p>\u9664\u4e86\u4e0a\u8ff0\u96d9\u9589\u5340\u9593\u5916\uff0c\u5e38\u898b\u7684\u5340\u9593\u8868\u793a\u9084\u6709\u201c\u5de6\u9589\u53f3\u958b\u201d\u5340\u9593\uff0c\u5b9a\u7fa9\u70ba \\([0, n)\\) \uff0c\u5373\u5de6\u908a\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u908a\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8a72\u8868\u793a\u4e0b\uff0c\u5340\u9593 \\([i, j)\\) \u5728 \\(i = j\\) \u6642\u70ba\u7a7a\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u57fa\u65bc\u8a72\u8868\u793a\u5be6\u73fe\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09\"\"\"\n    # \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j = 0, len(nums)\n    # \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while i &lt; j:\n        m = (i + j) // 2  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n        elif nums[m] &gt; target:\n            j = m  # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.size();\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.length;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint BinarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.Length;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        else                       // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j := 0, len(nums)\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    for i &lt; j {\n        m := i + (j-i)/2      // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = nums.startIndex\n    var j = nums.endIndex\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfunction binarySearchLCRO(nums, target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        else return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n  int i = 0, j = nums.length;\n  // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n      j = m;\n    } else {\n      // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = len;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfun binarySearchLCRO(nums: IntArray, target: Int): Int {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = 0\n    var j = nums.size\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        val m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m\n        else  // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 ###\ndef binary_search_lcro(nums, target)\n  # \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n  i, j = 0, nums.length\n\n  # \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n  while i &lt; j\n    # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n    elsif nums[m] &gt; target\n      j = m - 1 # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n    else\n      return m  # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    end\n  end\n\n  -1  # \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\nend\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i: usize = 0;\n    var j: usize = nums.items.len;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        } else {                                // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 10-3 \u6240\u793a\uff0c\u5728\u5169\u7a2e\u5340\u9593\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u8ff4\u5708\u689d\u4ef6\u548c\u7e2e\u5c0f\u5340\u9593\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002</p> <p>\u7531\u65bc\u201c\u96d9\u9589\u5340\u9593\u201d\u8868\u793a\u4e2d\u7684\u5de6\u53f3\u908a\u754c\u90fd\u88ab\u5b9a\u7fa9\u70ba\u9589\u5340\u9593\uff0c\u56e0\u6b64\u900f\u904e\u6307\u6a19 \\(i\\) \u548c\u6307\u6a19 \\(j\\) \u7e2e\u5c0f\u5340\u9593\u7684\u64cd\u4f5c\u4e5f\u662f\u5c0d\u7a31\u7684\u3002\u9019\u6a23\u66f4\u4e0d\u5bb9\u6613\u51fa\u932f\uff0c\u56e0\u6b64\u4e00\u822c\u5efa\u8b70\u63a1\u7528\u201c\u96d9\u9589\u5340\u9593\u201d\u7684\u5beb\u6cd5\u3002</p> <p></p> <p> \u5716 10-3 \u00a0 \u5169\u7a2e\u5340\u9593\u5b9a\u7fa9 </p>"},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2 \u00a0 \u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u4e8c\u5206\u641c\u5c0b\u5728\u6642\u9593\u548c\u7a7a\u9593\u65b9\u9762\u90fd\u6709\u8f03\u597d\u7684\u6548\u80fd\u3002</p> <ul> <li>\u4e8c\u5206\u641c\u5c0b\u7684\u6642\u9593\u6548\u7387\u9ad8\u3002\u5728\u5927\u8cc7\u6599\u91cf\u4e0b\uff0c\u5c0d\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5177\u6709\u986f\u8457\u512a\u52e2\u3002\u4f8b\u5982\uff0c\u7576\u8cc7\u6599\u5927\u5c0f \\(n = 2^{20}\\) \u6642\uff0c\u7dda\u6027\u67e5\u8a62\u9700\u8981 \\(2^{20} = 1048576\\) \u8f2a\u8ff4\u5708\uff0c\u800c\u4e8c\u5206\u641c\u5c0b\u50c5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f2a\u8ff4\u5708\u3002</li> <li>\u4e8c\u5206\u641c\u5c0b\u7121\u9808\u984d\u5916\u7a7a\u9593\u3002\u76f8\u8f03\u65bc\u9700\u8981\u85c9\u52a9\u984d\u5916\u7a7a\u9593\u7684\u641c\u5c0b\u6f14\u7b97\u6cd5\uff08\u4f8b\u5982\u96dc\u6e4a\u67e5\u8a62\uff09\uff0c\u4e8c\u5206\u641c\u5c0b\u66f4\u52a0\u7bc0\u7701\u7a7a\u9593\u3002</li> </ul> <p>\u7136\u800c\uff0c\u4e8c\u5206\u641c\u5c0b\u4e26\u975e\u9069\u7528\u65bc\u6240\u6709\u60c5\u6cc1\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e8c\u5206\u641c\u5c0b\u50c5\u9069\u7528\u65bc\u6709\u5e8f\u8cc7\u6599\u3002\u82e5\u8f38\u5165\u8cc7\u6599\u7121\u5e8f\uff0c\u70ba\u4e86\u4f7f\u7528\u4e8c\u5206\u641c\u5c0b\u800c\u5c08\u9580\u9032\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u511f\u5931\u3002\u56e0\u70ba\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u70ba \\(O(n \\log n)\\) \uff0c\u6bd4\u7dda\u6027\u67e5\u8a62\u548c\u4e8c\u5206\u641c\u5c0b\u90fd\u66f4\u9ad8\u3002\u5c0d\u65bc\u983b\u7e41\u63d2\u5165\u5143\u7d20\u7684\u5834\u666f\uff0c\u70ba\u4fdd\u6301\u9663\u5217\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c07\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8cb4\u7684\u3002</li> <li>\u4e8c\u5206\u641c\u5c0b\u50c5\u9069\u7528\u65bc\u9663\u5217\u3002\u4e8c\u5206\u641c\u5c0b\u9700\u8981\u8df3\u8e8d\u5f0f\uff08\u975e\u9023\u7e8c\u5730\uff09\u8a2a\u554f\u5143\u7d20\uff0c\u800c\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u57f7\u884c\u8df3\u8e8d\u5f0f\u8a2a\u554f\u7684\u6548\u7387\u8f03\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9069\u5408\u61c9\u7528\u5728\u93c8\u7d50\u4e32\u5217\u6216\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u5c0f\u8cc7\u6599\u91cf\u4e0b\uff0c\u7dda\u6027\u67e5\u8a62\u6548\u80fd\u66f4\u4f73\u3002\u7dda\u4e0a\u6027\u67e5\u8a62\u4e2d\uff0c\u6bcf\u8f2a\u53ea\u9700 1 \u6b21\u5224\u65b7\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u641c\u5c0b\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65b7\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u6e1b\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u500b\u55ae\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u7576\u8cc7\u6599\u91cf \\(n\\) \u8f03\u5c0f\u6642\uff0c\u7dda\u6027\u67e5\u8a62\u53cd\u800c\u6bd4\u4e8c\u5206\u641c\u5c0b\u66f4\u5feb\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_edge/","title":"10.3 \u00a0 \u4e8c\u5206\u641c\u5c0b\u908a\u754c","text":""},{"location":"chapter_searching/binary_search_edge/#1031","title":"10.3.1 \u00a0 \u67e5\u8a62\u5de6\u908a\u754c","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u6709\u5e8f\u9663\u5217 <code>nums</code> \uff0c\u5176\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\u3002\u8acb\u8fd4\u56de\u9663\u5217\u4e2d\u6700\u5de6\u4e00\u500b\u5143\u7d20 <code>target</code> \u7684\u7d22\u5f15\u3002\u82e5\u9663\u5217\u4e2d\u4e0d\u5305\u542b\u8a72\u5143\u7d20\uff0c\u5247\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u56de\u61b6\u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\u7684\u65b9\u6cd5\uff0c\u641c\u5c0b\u5b8c\u6210\u5f8c \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u500b <code>target</code> \uff0c\u56e0\u6b64\u67e5\u8a62\u63d2\u5165\u9ede\u672c\u8cea\u4e0a\u662f\u5728\u67e5\u8a62\u6700\u5de6\u4e00\u500b <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u8003\u616e\u900f\u904e\u67e5\u8a62\u63d2\u5165\u9ede\u7684\u51fd\u5f0f\u5be6\u73fe\u67e5\u8a62\u5de6\u908a\u754c\u3002\u8acb\u6ce8\u610f\uff0c\u9663\u5217\u4e2d\u53ef\u80fd\u4e0d\u5305\u542b <code>target</code> \uff0c\u9019\u7a2e\u60c5\u6cc1\u53ef\u80fd\u5c0e\u81f4\u4ee5\u4e0b\u5169\u7a2e\u7d50\u679c\u3002</p> <ul> <li>\u63d2\u5165\u9ede\u7684\u7d22\u5f15 \\(i\\) \u8d8a\u754c\u3002</li> <li>\u5143\u7d20 <code>nums[i]</code> \u8207 <code>target</code> \u4e0d\u76f8\u7b49\u3002</li> </ul> <p>\u7576\u9047\u5230\u4ee5\u4e0a\u5169\u7a2e\u60c5\u6cc1\u6642\uff0c\u76f4\u63a5\u8fd4\u56de \\(-1\\) \u5373\u53ef\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target\"\"\"\n    # \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    i = binary_search_insertion(nums, target)\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    int i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    i := binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.endIndex || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfunction binarySearchLeftEdge(nums, target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n  int i = binarySearchInsertion(nums, target);\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    let i = binary_search_insertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    int i = binarySearchInsertion(nums, numSize, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfun binarySearchLeftEdge(nums: IntArray, target: Int): Int {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    val i = binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size || nums[i] != target) {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target ###\ndef binary_search_left_edge(nums, target)\n  # \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n  i = binary_search_insertion(nums, target)\n\n  # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  return -1 if i == nums.length || nums[i] != target\n\n  i # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nend\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchLeftEdge}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_edge/#1032","title":"10.3.2 \u00a0 \u67e5\u8a62\u53f3\u908a\u754c","text":"<p>\u90a3\u9ebc\u5982\u4f55\u67e5\u8a62\u6700\u53f3\u4e00\u500b <code>target</code> \u5462\uff1f\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\u662f\u4fee\u6539\u7a0b\u5f0f\u78bc\uff0c\u66ff\u63db\u5728 <code>nums[m] == target</code> \u60c5\u6cc1\u4e0b\u7684\u6307\u6a19\u6536\u7e2e\u64cd\u4f5c\u3002\u7a0b\u5f0f\u78bc\u5728\u6b64\u7701\u7565\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u81ea\u884c\u5be6\u73fe\u3002</p> <p>\u4e0b\u9762\u6211\u5011\u4ecb\u7d39\u5169\u7a2e\u66f4\u52a0\u53d6\u5de7\u7684\u65b9\u6cd5\u3002</p>"},{"location":"chapter_searching/binary_search_edge/#1","title":"1. \u00a0 \u8907\u7528\u67e5\u8a62\u5de6\u908a\u754c","text":"<p>\u5be6\u969b\u4e0a\uff0c\u6211\u5011\u53ef\u4ee5\u5229\u7528\u67e5\u8a62\u6700\u5de6\u5143\u7d20\u7684\u51fd\u5f0f\u4f86\u67e5\u8a62\u6700\u53f3\u5143\u7d20\uff0c\u5177\u9ad4\u65b9\u6cd5\u70ba\uff1a\u5c07\u67e5\u8a62\u6700\u53f3\u4e00\u500b <code>target</code> \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b <code>target + 1</code>\u3002</p> <p>\u5982\u5716 10-7 \u6240\u793a\uff0c\u67e5\u8a62\u5b8c\u6210\u5f8c\uff0c\u6307\u6a19 \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u500b <code>target + 1</code>\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u800c \\(j\\) \u6307\u5411\u6700\u53f3\u4e00\u500b <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de \\(j\\) \u5373\u53ef\u3002</p> <p></p> <p> \u5716 10-7 \u00a0 \u5c07\u67e5\u8a62\u53f3\u908a\u754c\u8f49\u5316\u70ba\u67e5\u8a62\u5de6\u908a\u754c </p> <p>\u8acb\u6ce8\u610f\uff0c\u8fd4\u56de\u7684\u63d2\u5165\u9ede\u662f \\(i\\) \uff0c\u56e0\u6b64\u9700\u8981\u5c07\u5176\u6e1b \\(1\\) \uff0c\u5f9e\u800c\u7372\u5f97 \\(j\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target\"\"\"\n    # \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    i = binary_search_insertion(nums, target + 1)\n    # j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    j = i - 1\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 or nums[j] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    int i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint binarySearchRightEdge(int[] nums, int target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    i := binarySearchInsertion(nums, target+1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    j := i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    let j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfunction binarySearchRightEdge(nums, target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n  int i = binarySearchInsertion(nums, target + 1);\n  // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n  int j = i - 1;\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    let i = binary_search_insertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    let j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfun binarySearchRightEdge(nums: IntArray, target: Int): Int {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    val i = binarySearchInsertion(nums, target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    val j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target ###\ndef binary_search_right_edge(nums, target)\n  # \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n  i = binary_search_insertion(nums, target + 1)\n\n  # j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n  j = i - 1\n\n  # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  return -1 if j == -1 || nums[j] != target\n\n  j # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nend\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchRightEdge}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_edge/#2","title":"2. \u00a0 \u8f49\u5316\u70ba\u67e5\u8a62\u5143\u7d20","text":"<p>\u6211\u5011\u77e5\u9053\uff0c\u7576\u9663\u5217\u4e0d\u5305\u542b <code>target</code> \u6642\uff0c\u6700\u7d42 \\(i\\) \u548c \\(j\\) \u6703\u5206\u5225\u6307\u5411\u9996\u500b\u5927\u65bc\u3001\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\u3002</p> <p>\u56e0\u6b64\uff0c\u5982\u5716 10-8 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u69cb\u9020\u4e00\u500b\u9663\u5217\u4e2d\u4e0d\u5b58\u5728\u7684\u5143\u7d20\uff0c\u7528\u65bc\u67e5\u8a62\u5de6\u53f3\u908a\u754c\u3002</p> <ul> <li>\u67e5\u8a62\u6700\u5de6\u4e00\u500b <code>target</code> \uff1a\u53ef\u4ee5\u8f49\u5316\u70ba\u67e5\u8a62 <code>target - 0.5</code> \uff0c\u4e26\u8fd4\u56de\u6307\u6a19 \\(i\\) \u3002</li> <li>\u67e5\u8a62\u6700\u53f3\u4e00\u500b <code>target</code> \uff1a\u53ef\u4ee5\u8f49\u5316\u70ba\u67e5\u8a62 <code>target + 0.5</code> \uff0c\u4e26\u8fd4\u56de\u6307\u6a19 \\(j\\) \u3002</li> </ul> <p></p> <p> \u5716 10-8 \u00a0 \u5c07\u67e5\u8a62\u908a\u754c\u8f49\u5316\u70ba\u67e5\u8a62\u5143\u7d20 </p> <p>\u7a0b\u5f0f\u78bc\u5728\u6b64\u7701\u7565\uff0c\u4ee5\u4e0b\u5169\u9ede\u503c\u5f97\u6ce8\u610f\u3002</p> <ul> <li>\u7d66\u5b9a\u9663\u5217\u4e0d\u5305\u542b\u5c0f\u6578\uff0c\u9019\u610f\u5473\u8457\u6211\u5011\u7121\u9808\u95dc\u5fc3\u5982\u4f55\u8655\u7406\u76f8\u7b49\u7684\u60c5\u6cc1\u3002</li> <li>\u56e0\u70ba\u8a72\u65b9\u6cd5\u5f15\u5165\u4e86\u5c0f\u6578\uff0c\u6240\u4ee5\u9700\u8981\u5c07\u51fd\u5f0f\u4e2d\u7684\u8b8a\u6578 <code>target</code> \u6539\u70ba\u6d6e\u9ede\u6578\u578b\u5225\uff08Python \u7121\u9808\u6539\u52d5\uff09\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_insertion/","title":"10.2 \u00a0 \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede","text":"<p>\u4e8c\u5206\u641c\u5c0b\u4e0d\u50c5\u53ef\u7528\u65bc\u641c\u5c0b\u76ee\u6a19\u5143\u7d20\uff0c\u9084\u53ef\u7528\u65bc\u89e3\u6c7a\u8a31\u591a\u8b8a\u7a2e\u554f\u984c\uff0c\u6bd4\u5982\u641c\u5c0b\u76ee\u6a19\u5143\u7d20\u7684\u63d2\u5165\u4f4d\u7f6e\u3002</p>"},{"location":"chapter_searching/binary_search_insertion/#1021","title":"10.2.1 \u00a0 \u7121\u91cd\u8907\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u6709\u5e8f\u9663\u5217 <code>nums</code> \u548c\u4e00\u500b\u5143\u7d20 <code>target</code> \uff0c\u9663\u5217\u4e0d\u5b58\u5728\u91cd\u8907\u5143\u7d20\u3002\u73fe\u5c07 <code>target</code> \u63d2\u5165\u9663\u5217 <code>nums</code> \u4e2d\uff0c\u4e26\u4fdd\u6301\u5176\u6709\u5e8f\u6027\u3002\u82e5\u9663\u5217\u4e2d\u5df2\u5b58\u5728\u5143\u7d20 <code>target</code> \uff0c\u5247\u63d2\u5165\u5230\u5176\u5de6\u65b9\u3002\u8acb\u8fd4\u56de\u63d2\u5165\u5f8c <code>target</code> \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15\u3002\u793a\u4f8b\u5982\u5716 10-4 \u6240\u793a\u3002</p> <p></p> <p> \u5716 10-4 \u00a0 \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\u793a\u4f8b\u8cc7\u6599 </p> <p>\u5982\u679c\u60f3\u8907\u7528\u4e0a\u4e00\u7bc0\u7684\u4e8c\u5206\u641c\u5c0b\u7a0b\u5f0f\u78bc\uff0c\u5247\u9700\u8981\u56de\u7b54\u4ee5\u4e0b\u5169\u500b\u554f\u984c\u3002</p> <p>\u554f\u984c\u4e00\uff1a\u7576\u9663\u5217\u4e2d\u5305\u542b <code>target</code> \u6642\uff0c\u63d2\u5165\u9ede\u7684\u7d22\u5f15\u662f\u5426\u662f\u8a72\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u984c\u76ee\u8981\u6c42\u5c07 <code>target</code> \u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u5de6\u908a\uff0c\u9019\u610f\u5473\u8457\u65b0\u63d2\u5165\u7684 <code>target</code> \u66ff\u63db\u4e86\u539f\u4f86 <code>target</code> \u7684\u4f4d\u7f6e\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u7576\u9663\u5217\u5305\u542b <code>target</code> \u6642\uff0c\u63d2\u5165\u9ede\u7684\u7d22\u5f15\u5c31\u662f\u8a72 <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u554f\u984c\u4e8c\uff1a\u7576\u9663\u5217\u4e2d\u4e0d\u5b58\u5728 <code>target</code> \u6642\uff0c\u63d2\u5165\u9ede\u662f\u54ea\u500b\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u9032\u4e00\u6b65\u601d\u8003\u4e8c\u5206\u641c\u5c0b\u904e\u7a0b\uff1a\u7576 <code>nums[m] &lt; target</code> \u6642 \\(i\\) \u79fb\u52d5\uff0c\u9019\u610f\u5473\u8457\u6307\u6a19 \\(i\\) \u5728\u5411\u5927\u65bc\u7b49\u65bc <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002\u540c\u7406\uff0c\u6307\u6a19 \\(j\\) \u59cb\u7d42\u5728\u5411\u5c0f\u65bc\u7b49\u65bc <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</p> <p>\u56e0\u6b64\u4e8c\u5206\u7d50\u675f\u6642\u4e00\u5b9a\u6709\uff1a\\(i\\) \u6307\u5411\u9996\u500b\u5927\u65bc <code>target</code> \u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u9996\u500b\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\u3002\u6613\u5f97\u7576\u9663\u5217\u4e0d\u5305\u542b <code>target</code> \u6642\uff0c\u63d2\u5165\u7d22\u5f15\u70ba \\(i\\) \u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n            return m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    } else {\n      return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n    }\n  }\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfn binary_search_insertion_simple(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 ###\ndef binary_search_insertion_simple(nums, target)\n  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    elsif nums[m] &gt; target\n      j = m - 1 # target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    else\n      return m  # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n    end\n  end\n\n  i # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\nend\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertionSimple}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_insertion/#1022","title":"10.2.2 \u00a0 \u5b58\u5728\u91cd\u8907\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u5728\u4e0a\u4e00\u984c\u7684\u57fa\u790e\u4e0a\uff0c\u898f\u5b9a\u9663\u5217\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u5176\u9918\u4e0d\u8b8a\u3002</p> <p>\u5047\u8a2d\u9663\u5217\u4e2d\u5b58\u5728\u591a\u500b <code>target</code> \uff0c\u5247\u666e\u901a\u4e8c\u5206\u641c\u5c0b\u53ea\u80fd\u8fd4\u56de\u5176\u4e2d\u4e00\u500b <code>target</code> \u7684\u7d22\u5f15\uff0c\u800c\u7121\u6cd5\u78ba\u5b9a\u8a72\u5143\u7d20\u7684\u5de6\u908a\u548c\u53f3\u908a\u9084\u6709\u591a\u5c11 <code>target</code>\u3002</p> <p>\u984c\u76ee\u8981\u6c42\u5c07\u76ee\u6a19\u5143\u7d20\u63d2\u5165\u5230\u6700\u5de6\u908a\uff0c\u6240\u4ee5\u6211\u5011\u9700\u8981\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5de6\u4e00\u500b <code>target</code> \u7684\u7d22\u5f15\u3002\u521d\u6b65\u8003\u616e\u900f\u904e\u5716 10-5 \u6240\u793a\u7684\u6b65\u9a5f\u5be6\u73fe\u3002</p> <ol> <li>\u57f7\u884c\u4e8c\u5206\u641c\u5c0b\uff0c\u5f97\u5230\u4efb\u610f\u4e00\u500b <code>target</code> \u7684\u7d22\u5f15\uff0c\u8a18\u70ba \\(k\\) \u3002</li> <li>\u5f9e\u7d22\u5f15 \\(k\\) \u958b\u59cb\uff0c\u5411\u5de6\u9032\u884c\u7dda\u6027\u8d70\u8a2a\uff0c\u7576\u627e\u5230\u6700\u5de6\u908a\u7684 <code>target</code> \u6642\u8fd4\u56de\u3002</li> </ol> <p></p> <p> \u5716 10-5 \u00a0 \u7dda\u6027\u67e5\u8a62\u91cd\u8907\u5143\u7d20\u7684\u63d2\u5165\u9ede </p> <p>\u6b64\u65b9\u6cd5\u96d6\u7136\u53ef\u7528\uff0c\u4f46\u5176\u5305\u542b\u7dda\u6027\u67e5\u8a62\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002\u7576\u9663\u5217\u4e2d\u5b58\u5728\u5f88\u591a\u91cd\u8907\u7684 <code>target</code> \u6642\uff0c\u8a72\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\u3002</p> <p>\u73fe\u8003\u616e\u62d3\u5c55\u4e8c\u5206\u641c\u5c0b\u7a0b\u5f0f\u78bc\u3002\u5982\u5716 10-6 \u6240\u793a\uff0c\u6574\u9ad4\u6d41\u7a0b\u4fdd\u6301\u4e0d\u8b8a\uff0c\u6bcf\u8f2a\u5148\u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65b7 <code>target</code> \u548c <code>nums[m]</code> \u7684\u5927\u5c0f\u95dc\u4fc2\uff0c\u5206\u70ba\u4ee5\u4e0b\u5e7e\u7a2e\u60c5\u6cc1\u3002</p> <ul> <li>\u7576 <code>nums[m] &lt; target</code> \u6216 <code>nums[m] &gt; target</code> \u6642\uff0c\u8aaa\u660e\u9084\u6c92\u6709\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u63a1\u7528\u666e\u901a\u4e8c\u5206\u641c\u5c0b\u7684\u7e2e\u5c0f\u5340\u9593\u64cd\u4f5c\uff0c\u5f9e\u800c\u4f7f\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u5411 <code>target</code> \u9760\u8fd1\u3002</li> <li>\u7576 <code>nums[m] == target</code> \u6642\uff0c\u8aaa\u660e\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\u5728\u5340\u9593 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u63a1\u7528 \\(j = m - 1\\) \u4f86\u7e2e\u5c0f\u5340\u9593\uff0c\u5f9e\u800c\u4f7f\u6307\u6a19 \\(j\\) \u5411\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</li> </ul> <p>\u8ff4\u5708\u5b8c\u6210\u5f8c\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u908a\u7684 <code>target</code> \uff0c\\(j\\) \u6307\u5411\u9996\u500b\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\uff0c\u56e0\u6b64\u7d22\u5f15 \\(i\\) \u5c31\u662f\u63d2\u5165\u9ede\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 10-6 \u00a0 \u4e8c\u5206\u641c\u5c0b\u91cd\u8907\u5143\u7d20\u7684\u63d2\u5165\u9ede\u7684\u6b65\u9a5f </p> <p>\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u5224\u65b7\u5206\u652f <code>nums[m] &gt; target</code> \u548c <code>nums[m] == target</code> \u7684\u64cd\u4f5c\u76f8\u540c\uff0c\u56e0\u6b64\u5169\u8005\u53ef\u4ee5\u5408\u4f75\u3002</p> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u5c07\u5224\u65b7\u689d\u4ef6\u4fdd\u6301\u5c55\u958b\uff0c\u56e0\u70ba\u5176\u908f\u8f2f\u66f4\u52a0\u6e05\u6670\u3001\u53ef\u8b80\u6027\u66f4\u597d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        else:\n            j = m - 1  # \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n    # \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    } else {\n      j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n    }\n  }\n  // \u8fd4\u56de\u63d2\u5165\u9ede i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfun binarySearchInsertion(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 ###\ndef binary_search_insertion(nums, target)\n  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    elsif nums[m] &gt; target\n      j = m - 1 # target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    else\n      j = m - 1 # \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n    end\n  end\n\n  i # \u8fd4\u56de\u63d2\u5165\u9ede i\nend\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertion}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>Tip</p> <p>\u672c\u7bc0\u7684\u7a0b\u5f0f\u78bc\u90fd\u662f\u201c\u96d9\u9589\u5340\u9593\u201d\u5beb\u6cd5\u3002\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u81ea\u884c\u5be6\u73fe\u201c\u5de6\u9589\u53f3\u958b\u201d\u5beb\u6cd5\u3002</p> <p>\u7e3d\u7684\u4f86\u770b\uff0c\u4e8c\u5206\u641c\u5c0b\u7121\u975e\u5c31\u662f\u7d66\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u5206\u5225\u8a2d\u5b9a\u641c\u5c0b\u76ee\u6a19\uff0c\u76ee\u6a19\u53ef\u80fd\u662f\u4e00\u500b\u5177\u9ad4\u7684\u5143\u7d20\uff08\u4f8b\u5982 <code>target</code> \uff09\uff0c\u4e5f\u53ef\u80fd\u662f\u4e00\u500b\u5143\u7d20\u7bc4\u570d\uff08\u4f8b\u5982\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\uff09\u3002</p> <p>\u5728\u4e0d\u65b7\u7684\u8ff4\u5708\u4e8c\u5206\u4e2d\uff0c\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u90fd\u9010\u6f38\u903c\u8fd1\u9810\u5148\u8a2d\u5b9a\u7684\u76ee\u6a19\u3002\u6700\u7d42\uff0c\u5b83\u5011\u6216\u662f\u6210\u529f\u627e\u5230\u7b54\u6848\uff0c\u6216\u662f\u8d8a\u904e\u908a\u754c\u5f8c\u505c\u6b62\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.4 \u00a0 \u96dc\u6e4a\u6700\u4f73\u5316\u7b56\u7565","text":"<p>\u5728\u6f14\u7b97\u6cd5\u984c\u4e2d\uff0c\u6211\u5011\u5e38\u900f\u904e\u5c07\u7dda\u6027\u67e5\u8a62\u66ff\u63db\u70ba\u96dc\u6e4a\u67e5\u8a62\u4f86\u964d\u4f4e\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u3002\u6211\u5011\u85c9\u52a9\u4e00\u500b\u6f14\u7b97\u6cd5\u984c\u4f86\u52a0\u6df1\u7406\u89e3\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u6574\u6578\u9663\u5217 <code>nums</code> \u548c\u4e00\u500b\u76ee\u6a19\u5143\u7d20 <code>target</code> \uff0c\u8acb\u5728\u9663\u5217\u4e2d\u641c\u7d22\u201c\u548c\u201d\u70ba <code>target</code> \u7684\u5169\u500b\u5143\u7d20\uff0c\u4e26\u8fd4\u56de\u5b83\u5011\u7684\u9663\u5217\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u500b\u89e3\u5373\u53ef\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1041","title":"10.4.1 \u00a0 \u7dda\u6027\u67e5\u8a62\uff1a\u4ee5\u6642\u9593\u63db\u7a7a\u9593","text":"<p>\u8003\u616e\u76f4\u63a5\u8d70\u8a2a\u6240\u6709\u53ef\u80fd\u7684\u7d44\u5408\u3002\u5982\u5716 10-9 \u6240\u793a\uff0c\u6211\u5011\u958b\u555f\u4e00\u500b\u5169\u5c64\u8ff4\u5708\uff0c\u5728\u6bcf\u8f2a\u4e2d\u5224\u65b7\u5169\u500b\u6574\u6578\u7684\u548c\u662f\u5426\u70ba <code>target</code> \uff0c\u82e5\u662f\uff0c\u5247\u8fd4\u56de\u5b83\u5011\u7684\u7d22\u5f15\u3002</p> <p></p> <p> \u5716 10-9 \u00a0 \u7dda\u6027\u67e5\u8a62\u6c42\u89e3\u5169\u6578\u4e4b\u548c </p> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209\"\"\"\n    # \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; j &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u5217\u8209 */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfun twoSumBruteForce(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (i in 0..&lt;size - 1) {\n        for (j in i + 1..&lt;size) {\n            if (nums[i] + nums[j] == target) return intArrayOf(i, j)\n        }\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 ###\ndef two_sum_brute_force(nums, target)\n  # \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n  for i in 0...(nums.length - 1)\n    for j in (i + 1)...nums.length\n      return [i, j] if nums[i] + nums[j] == target\n    end\n  end\n\n  []\nend\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\n    var size: usize = nums.len;\n    var i: usize = 0;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    while (i &lt; size - 1) : (i += 1) {\n        var j = i + 1;\n        while (j &lt; size) : (j += 1) {\n            if (nums[i] + nums[j] == target) {\n                return [_]i32{@intCast(i), @intCast(j)};\n            }\n        }\n    }\n    return null;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6b64\u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \uff0c\u5728\u5927\u8cc7\u6599\u91cf\u4e0b\u975e\u5e38\u8017\u6642\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1042","title":"10.4.2 \u00a0 \u96dc\u6e4a\u67e5\u8a62\uff1a\u4ee5\u7a7a\u9593\u63db\u6642\u9593","text":"<p>\u8003\u616e\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u8868\uff0c\u9375\u503c\u5c0d\u5206\u5225\u70ba\u9663\u5217\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u8ff4\u5708\u8d70\u8a2a\u9663\u5217\uff0c\u6bcf\u8f2a\u57f7\u884c\u5716 10-10 \u6240\u793a\u7684\u6b65\u9a5f\u3002</p> <ol> <li>\u5224\u65b7\u6578\u5b57 <code>target - nums[i]</code> \u662f\u5426\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u82e5\u662f\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u9019\u5169\u500b\u5143\u7d20\u7684\u7d22\u5f15\u3002</li> <li>\u5c07\u9375\u503c\u5c0d <code>nums[i]</code> \u548c\u7d22\u5f15 <code>i</code> \u65b0\u589e\u9032\u96dc\u6e4a\u8868\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> \u5716 10-10 \u00a0 \u8f14\u52a9\u96dc\u6e4a\u8868\u6c42\u89e3\u5169\u6578\u4e4b\u548c </p> <p>\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff0c\u50c5\u9700\u55ae\u5c64\u8ff4\u5708\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868\"\"\"\n    # \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    dic = {}\n    # \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    unordered_map&lt;int, int&gt; dic;\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    Dictionary&lt;int, int&gt; dic = [];\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    hashTable := map[int]int{}\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    var dic: [Int: Int] = [:]\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfunction twoSumHashTable(nums, target) {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    let m = {};\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    let m: Map&lt;number, number&gt; = new Map();\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a \u8f14\u52a9\u96dc\u6e4a\u8868 */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n  Map&lt;int, int&gt; dic = HashMap();\n  // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    let mut dic = HashMap::new();\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32),\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u96dc\u6e4a\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u65bc uthash.h \u5be6\u73fe\n} HashTable;\n\n/* \u96dc\u6e4a\u8868\u67e5\u8a62 */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* \u96dc\u6e4a\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(HashTable **h, int key, int val) {\n    HashTable *t = find(*h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(*h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(&amp;hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfun twoSumHashTable(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    val dic = HashMap&lt;Int, Int&gt;()\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (i in 0..&lt;size) {\n        if (dic.containsKey(target - nums[i])) {\n            return intArrayOf(dic[target - nums[i]]!!, i)\n        }\n        dic[nums[i]] = i\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 ###\ndef two_sum_hash_table(nums, target)\n  # \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n  dic = {}\n  # \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n  for i in 0...nums.length\n    return [dic[target - nums[i]], i] if dic.has_key?(target - nums[i])\n\n    dic[nums[i]] = i\n  end\n\n  []\nend\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\n    var size: usize = nums.len;\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    var dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\n    defer dic.deinit();\n    var i: usize = 0;\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    while (i &lt; size) : (i += 1) {\n        if (dic.contains(target - nums[i])) {\n            return [_]i32{dic.get(target - nums[i]).?, @intCast(i)};\n        }\n        try dic.put(nums[i], @intCast(i));\n    }\n    return null;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6b64\u65b9\u6cd5\u900f\u904e\u96dc\u6e4a\u67e5\u8a62\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u57f7\u884c\u6548\u7387\u3002</p> <p>\u7531\u65bc\u9700\u8981\u7dad\u8b77\u4e00\u500b\u984d\u5916\u7684\u96dc\u6e4a\u8868\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002\u5118\u7ba1\u5982\u6b64\uff0c\u8a72\u65b9\u6cd5\u7684\u6574\u9ad4\u6642\u7a7a\u6548\u7387\u66f4\u70ba\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u984c\u7684\u6700\u512a\u89e3\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.5 \u00a0 \u91cd\u8b58\u641c\u5c0b\u6f14\u7b97\u6cd5","text":"<p>\u641c\u5c0b\u6f14\u7b97\u6cd5\uff08searching algorithm\uff09\u7528\u65bc\u5728\u8cc7\u6599\u7d50\u69cb\uff08\u4f8b\u5982\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u6a39\u6216\u5716\uff09\u4e2d\u641c\u7d22\u4e00\u500b\u6216\u4e00\u7d44\u6eff\u8db3\u7279\u5b9a\u689d\u4ef6\u7684\u5143\u7d20\u3002</p> <p>\u641c\u5c0b\u6f14\u7b97\u6cd5\u53ef\u6839\u64da\u5be6\u73fe\u601d\u8def\u5206\u70ba\u4ee5\u4e0b\u5169\u985e\u3002</p> <ul> <li>\u900f\u904e\u8d70\u8a2a\u8cc7\u6599\u7d50\u69cb\u4f86\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\uff0c\u4f8b\u5982\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u6a39\u548c\u5716\u7684\u8d70\u8a2a\u7b49\u3002</li> <li>\u5229\u7528\u8cc7\u6599\u7d44\u7e54\u7d50\u69cb\u6216\u8cc7\u6599\u5305\u542b\u7684\u5148\u9a57\u8cc7\u8a0a\uff0c\u5be6\u73fe\u9ad8\u6548\u5143\u7d20\u67e5\u8a62\uff0c\u4f8b\u5982\u4e8c\u5206\u641c\u5c0b\u3001\u96dc\u6e4a\u67e5\u8a62\u548c\u4e8c\u5143\u641c\u5c0b\u6a39\u67e5\u8a62\u7b49\u3002</li> </ul> <p>\u4e0d\u96e3\u767c\u73fe\uff0c\u9019\u4e9b\u77e5\u8b58\u9ede\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u7bc0\u4e2d\u4ecb\u7d39\u904e\uff0c\u56e0\u6b64\u641c\u5c0b\u6f14\u7b97\u6cd5\u5c0d\u65bc\u6211\u5011\u4f86\u8aaa\u4e26\u4e0d\u964c\u751f\u3002\u5728\u672c\u7bc0\u4e2d\uff0c\u6211\u5011\u5c07\u5f9e\u66f4\u52a0\u7cfb\u7d71\u7684\u8996\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5be9\u8996\u641c\u5c0b\u6f14\u7b97\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1051","title":"10.5.1 \u00a0 \u66b4\u529b\u641c\u5c0b","text":"<p>\u66b4\u529b\u641c\u5c0b\u900f\u904e\u8d70\u8a2a\u8cc7\u6599\u7d50\u69cb\u7684\u6bcf\u500b\u5143\u7d20\u4f86\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\u3002</p> <ul> <li>\u201c\u7dda\u6027\u641c\u5c0b\u201d\u9069\u7528\u65bc\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u7b49\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u3002\u5b83\u5f9e\u8cc7\u6599\u7d50\u69cb\u7684\u4e00\u7aef\u958b\u59cb\uff0c\u9010\u500b\u8a2a\u554f\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6a19\u5143\u7d20\u6216\u5230\u9054\u53e6\u4e00\u7aef\u4ecd\u6c92\u6709\u627e\u5230\u76ee\u6a19\u5143\u7d20\u70ba\u6b62\u3002</li> <li>\u201c\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\u201d\u548c\u201c\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u201d\u662f\u5716\u548c\u6a39\u7684\u5169\u7a2e\u8d70\u8a2a\u7b56\u7565\u3002\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\u5f9e\u521d\u59cb\u7bc0\u9ede\u958b\u59cb\u9010\u5c64\u641c\u5c0b\uff0c\u7531\u8fd1\u53ca\u9060\u5730\u8a2a\u554f\u5404\u500b\u7bc0\u9ede\u3002\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u5f9e\u521d\u59cb\u7bc0\u9ede\u958b\u59cb\uff0c\u6cbf\u8457\u4e00\u689d\u8def\u5f91\u8d70\u5230\u982d\uff0c\u518d\u56de\u6eaf\u4e26\u5617\u8a66\u5176\u4ed6\u8def\u5f91\uff0c\u76f4\u5230\u8d70\u8a2a\u5b8c\u6574\u500b\u8cc7\u6599\u7d50\u69cb\u3002</li> </ul> <p>\u66b4\u529b\u641c\u5c0b\u7684\u512a\u9ede\u662f\u7c21\u55ae\u4e14\u901a\u7528\u6027\u597d\uff0c\u7121\u9808\u5c0d\u8cc7\u6599\u505a\u9810\u8655\u7406\u548c\u85c9\u52a9\u984d\u5916\u7684\u8cc7\u6599\u7d50\u69cb\u3002</p> <p>\u7136\u800c\uff0c\u6b64\u985e\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u70ba\u5143\u7d20\u6578\u91cf\uff0c\u56e0\u6b64\u5728\u8cc7\u6599\u91cf\u8f03\u5927\u7684\u60c5\u6cc1\u4e0b\u6548\u80fd\u8f03\u5dee\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1052","title":"10.5.2 \u00a0 \u81ea\u9069\u61c9\u641c\u5c0b","text":"<p>\u81ea\u9069\u61c9\u641c\u5c0b\u5229\u7528\u8cc7\u6599\u7684\u7279\u6709\u5c6c\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u4f86\u6700\u4f73\u5316\u641c\u5c0b\u904e\u7a0b\uff0c\u5f9e\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\u3002</p> <ul> <li>\u201c\u4e8c\u5206\u641c\u5c0b\u201d\u5229\u7528\u8cc7\u6599\u7684\u6709\u5e8f\u6027\u5be6\u73fe\u9ad8\u6548\u67e5\u8a62\uff0c\u50c5\u9069\u7528\u65bc\u9663\u5217\u3002</li> <li>\u201c\u96dc\u6e4a\u67e5\u8a62\u201d\u5229\u7528\u96dc\u6e4a\u8868\u5c07\u641c\u5c0b\u8cc7\u6599\u548c\u76ee\u6a19\u8cc7\u6599\u5efa\u7acb\u70ba\u9375\u503c\u5c0d\u5c0d\u6620\uff0c\u5f9e\u800c\u5be6\u73fe\u67e5\u8a62\u64cd\u4f5c\u3002</li> <li>\u201c\u6a39\u67e5\u8a62\u201d\u5728\u7279\u5b9a\u7684\u6a39\u7d50\u69cb\uff08\u4f8b\u5982\u4e8c\u5143\u641c\u5c0b\u6a39\uff09\u4e2d\uff0c\u57fa\u65bc\u6bd4\u8f03\u7bc0\u9ede\u503c\u4f86\u5feb\u901f\u6392\u9664\u7bc0\u9ede\uff0c\u5f9e\u800c\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\u3002</li> </ul> <p>\u6b64\u985e\u6f14\u7b97\u6cd5\u7684\u512a\u9ede\u662f\u6548\u7387\u9ad8\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</p> <p>\u7136\u800c\uff0c\u4f7f\u7528\u9019\u4e9b\u6f14\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5c0d\u8cc7\u6599\u9032\u884c\u9810\u8655\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u641c\u5c0b\u9700\u8981\u9810\u5148\u5c0d\u9663\u5217\u9032\u884c\u6392\u5e8f\uff0c\u96dc\u6e4a\u67e5\u8a62\u548c\u6a39\u67e5\u8a62\u90fd\u9700\u8981\u85c9\u52a9\u984d\u5916\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u7dad\u8b77\u9019\u4e9b\u8cc7\u6599\u7d50\u69cb\u4e5f\u9700\u8981\u984d\u5916\u7684\u6642\u9593\u548c\u7a7a\u9593\u958b\u92b7\u3002</p> <p>Tip</p> <p>\u81ea\u9069\u61c9\u641c\u5c0b\u6f14\u7b97\u6cd5\u5e38\u88ab\u7a31\u70ba\u67e5\u8a62\u6f14\u7b97\u6cd5\uff0c\u4e3b\u8981\u7528\u65bc\u5728\u7279\u5b9a\u8cc7\u6599\u7d50\u69cb\u4e2d\u5feb\u901f\u6aa2\u7d22\u76ee\u6a19\u5143\u7d20\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1053","title":"10.5.3 \u00a0 \u641c\u5c0b\u65b9\u6cd5\u9078\u53d6","text":"<p>\u7d66\u5b9a\u5927\u5c0f\u70ba \\(n\\) \u7684\u4e00\u7d44\u8cc7\u6599\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u7dda\u6027\u641c\u5c0b\u3001\u4e8c\u5206\u641c\u5c0b\u3001\u6a39\u67e5\u8a62\u3001\u96dc\u6e4a\u67e5\u8a62\u7b49\u591a\u7a2e\u65b9\u6cd5\u5f9e\u4e2d\u641c\u7d22\u76ee\u6a19\u5143\u7d20\u3002\u5404\u500b\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u5716 10-11 \u6240\u793a\u3002</p> <p></p> <p> \u5716 10-11 \u00a0 \u591a\u7a2e\u641c\u5c0b\u7b56\u7565 </p> <p>\u4e0a\u8ff0\u5e7e\u7a2e\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u8207\u7279\u6027\u5982\u8868 10-1 \u6240\u793a\u3002</p> <p> \u8868 10-1 \u00a0 \u67e5\u8a62\u6f14\u7b97\u6cd5\u6548\u7387\u5c0d\u6bd4 </p> \u7dda\u6027\u641c\u5c0b \u4e8c\u5206\u641c\u5c0b \u6a39\u67e5\u8a62 \u96dc\u6e4a\u67e5\u8a62 \u67e5\u8a62\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u522a\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u984d\u5916\u7a7a\u9593 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u8cc7\u6599\u9810\u8655\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6a39 \\(O(n \\log n)\\) \u5efa\u96dc\u6e4a\u8868 \\(O(n)\\) \u8cc7\u6599\u662f\u5426\u6709\u5e8f \u7121\u5e8f \u6709\u5e8f \u6709\u5e8f \u7121\u5e8f <p>\u641c\u5c0b\u6f14\u7b97\u6cd5\u7684\u9078\u64c7\u9084\u53d6\u6c7a\u65bc\u8cc7\u6599\u9ad4\u91cf\u3001\u641c\u5c0b\u6548\u80fd\u8981\u6c42\u3001\u8cc7\u6599\u67e5\u8a62\u8207\u66f4\u65b0\u983b\u7387\u7b49\u3002</p> <p>\u7dda\u6027\u641c\u5c0b</p> <ul> <li>\u901a\u7528\u6027\u8f03\u597d\uff0c\u7121\u9808\u4efb\u4f55\u8cc7\u6599\u9810\u8655\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u5011\u50c5\u9700\u67e5\u8a62\u4e00\u6b21\u8cc7\u6599\uff0c\u90a3\u9ebc\u5176\u4ed6\u4e09\u7a2e\u65b9\u6cd5\u7684\u8cc7\u6599\u9810\u8655\u7406\u7684\u6642\u9593\u6bd4\u7dda\u6027\u641c\u5c0b\u7684\u6642\u9593\u9084\u8981\u66f4\u9577\u3002</li> <li>\u9069\u7528\u65bc\u9ad4\u91cf\u8f03\u5c0f\u7684\u8cc7\u6599\uff0c\u6b64\u60c5\u6cc1\u4e0b\u6642\u9593\u8907\u96dc\u5ea6\u5c0d\u6548\u7387\u5f71\u97ff\u8f03\u5c0f\u3002</li> <li>\u9069\u7528\u65bc\u8cc7\u6599\u66f4\u65b0\u983b\u7387\u8f03\u9ad8\u7684\u5834\u666f\uff0c\u56e0\u70ba\u8a72\u65b9\u6cd5\u4e0d\u9700\u8981\u5c0d\u8cc7\u6599\u9032\u884c\u4efb\u4f55\u984d\u5916\u7dad\u8b77\u3002</li> </ul> <p>\u4e8c\u5206\u641c\u5c0b</p> <ul> <li>\u9069\u7528\u65bc\u5927\u8cc7\u6599\u91cf\u7684\u60c5\u6cc1\uff0c\u6548\u7387\u8868\u73fe\u7a69\u5b9a\uff0c\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \u3002</li> <li>\u8cc7\u6599\u91cf\u4e0d\u80fd\u904e\u5927\uff0c\u56e0\u70ba\u5132\u5b58\u9663\u5217\u9700\u8981\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> <li>\u4e0d\u9069\u7528\u65bc\u9ad8\u983b\u589e\u522a\u8cc7\u6599\u7684\u5834\u666f\uff0c\u56e0\u70ba\u7dad\u8b77\u6709\u5e8f\u9663\u5217\u7684\u958b\u92b7\u8f03\u5927\u3002</li> </ul> <p>\u96dc\u6e4a\u67e5\u8a62</p> <ul> <li>\u9069\u5408\u5c0d\u67e5\u8a62\u6548\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u5834\u666f\uff0c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</li> <li>\u4e0d\u9069\u5408\u9700\u8981\u6709\u5e8f\u8cc7\u6599\u6216\u7bc4\u570d\u67e5\u8a62\u7684\u5834\u666f\uff0c\u56e0\u70ba\u96dc\u6e4a\u8868\u7121\u6cd5\u7dad\u8b77\u8cc7\u6599\u7684\u6709\u5e8f\u6027\u3002</li> <li>\u5c0d\u96dc\u6e4a\u51fd\u5f0f\u548c\u96dc\u6e4a\u885d\u7a81\u8655\u7406\u7b56\u7565\u7684\u4f9d\u8cf4\u6027\u8f03\u9ad8\uff0c\u5177\u6709\u8f03\u5927\u7684\u6548\u80fd\u52a3\u5316\u98a8\u96aa\u3002</li> <li>\u4e0d\u9069\u5408\u8cc7\u6599\u91cf\u904e\u5927\u7684\u60c5\u6cc1\uff0c\u56e0\u70ba\u96dc\u6e4a\u8868\u9700\u8981\u984d\u5916\u7a7a\u9593\u4f86\u6700\u5927\u7a0b\u5ea6\u5730\u6e1b\u5c11\u885d\u7a81\uff0c\u5f9e\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8a62\u6548\u80fd\u3002</li> </ul> <p>\u6a39\u67e5\u8a62</p> <ul> <li>\u9069\u7528\u65bc\u6d77\u91cf\u8cc7\u6599\uff0c\u56e0\u70ba\u6a39\u7bc0\u9ede\u5728\u8a18\u61b6\u9ad4\u4e2d\u662f\u5206\u6563\u5132\u5b58\u7684\u3002</li> <li>\u9069\u5408\u9700\u8981\u7dad\u8b77\u6709\u5e8f\u8cc7\u6599\u6216\u7bc4\u570d\u67e5\u8a62\u7684\u5834\u666f\u3002</li> <li>\u5728\u6301\u7e8c\u589e\u522a\u7bc0\u9ede\u7684\u904e\u7a0b\u4e2d\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u53ef\u80fd\u7522\u751f\u50be\u659c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u82e5\u4f7f\u7528 AVL \u6a39\u6216\u7d05\u9ed1\u6a39\uff0c\u5247\u5404\u9805\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a69\u5b9a\u57f7\u884c\uff0c\u4f46\u7dad\u8b77\u6a39\u5e73\u8861\u7684\u64cd\u4f5c\u6703\u589e\u52a0\u984d\u5916\u7684\u958b\u92b7\u3002</li> </ul>"},{"location":"chapter_searching/summary/","title":"10.6 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u4e8c\u5206\u641c\u5c0b\u4f9d\u8cf4\u8cc7\u6599\u7684\u6709\u5e8f\u6027\uff0c\u900f\u904e\u8ff4\u5708\u9010\u6b65\u7e2e\u6e1b\u4e00\u534a\u641c\u5c0b\u5340\u9593\u4f86\u9032\u884c\u67e5\u8a62\u3002\u5b83\u8981\u6c42\u8f38\u5165\u8cc7\u6599\u6709\u5e8f\uff0c\u4e14\u50c5\u9069\u7528\u65bc\u9663\u5217\u6216\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u66b4\u529b\u641c\u5c0b\u900f\u904e\u8d70\u8a2a\u8cc7\u6599\u7d50\u69cb\u4f86\u5b9a\u4f4d\u8cc7\u6599\u3002\u7dda\u6027\u641c\u5c0b\u9069\u7528\u65bc\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\uff0c\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\u548c\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u9069\u7528\u65bc\u5716\u548c\u6a39\u3002\u6b64\u985e\u6f14\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u7121\u9808\u5c0d\u8cc7\u6599\u9032\u884c\u9810\u8655\u7406\uff0c\u4f46\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\) \u8f03\u9ad8\u3002</li> <li>\u96dc\u6e4a\u67e5\u8a62\u3001\u6a39\u67e5\u8a62\u548c\u4e8c\u5206\u641c\u5c0b\u5c6c\u65bc\u9ad8\u6548\u641c\u5c0b\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u8cc7\u6599\u7d50\u69cb\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\u3002\u6b64\u985e\u6f14\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u85c9\u52a9\u984d\u5916\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u5be6\u969b\u4e2d\uff0c\u6211\u5011\u9700\u8981\u5c0d\u8cc7\u6599\u9ad4\u91cf\u3001\u641c\u5c0b\u6548\u80fd\u8981\u6c42\u3001\u8cc7\u6599\u67e5\u8a62\u548c\u66f4\u65b0\u983b\u7387\u7b49\u56e0\u7d20\u9032\u884c\u5177\u9ad4\u5206\u6790\uff0c\u5f9e\u800c\u9078\u64c7\u5408\u9069\u7684\u641c\u5c0b\u65b9\u6cd5\u3002</li> <li>\u7dda\u6027\u641c\u5c0b\u9069\u7528\u65bc\u5c0f\u578b\u6216\u983b\u7e41\u66f4\u65b0\u7684\u8cc7\u6599\uff1b\u4e8c\u5206\u641c\u5c0b\u9069\u7528\u65bc\u5927\u578b\u3001\u6392\u5e8f\u7684\u8cc7\u6599\uff1b\u96dc\u6e4a\u67e5\u8a62\u9069\u7528\u65bc\u5c0d\u67e5\u8a62\u6548\u7387\u8981\u6c42\u8f03\u9ad8\u4e14\u7121\u9808\u7bc4\u570d\u67e5\u8a62\u7684\u8cc7\u6599\uff1b\u6a39\u67e5\u8a62\u9069\u7528\u65bc\u9700\u8981\u7dad\u8b77\u9806\u5e8f\u548c\u652f\u6301\u7bc4\u570d\u67e5\u8a62\u7684\u5927\u578b\u52d5\u614b\u8cc7\u6599\u3002</li> <li>\u7528\u96dc\u6e4a\u67e5\u8a62\u66ff\u63db\u7dda\u6027\u67e5\u8a62\u662f\u4e00\u7a2e\u5e38\u7528\u7684\u6700\u4f73\u5316\u57f7\u884c\u6642\u9593\u7684\u7b56\u7565\uff0c\u53ef\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</li> </ul>"},{"location":"chapter_sorting/","title":"\u7b2c 11 \u7ae0 \u00a0 \u6392\u5e8f","text":"<p>Abstract</p> <p>\u6392\u5e8f\u7336\u5982\u4e00\u628a\u5c07\u6df7\u4e82\u8b8a\u70ba\u79e9\u5e8f\u7684\u9b54\u6cd5\u9470\u5319\uff0c\u4f7f\u6211\u5011\u80fd\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u7406\u89e3\u8207\u8655\u7406\u8cc7\u6599\u3002</p> <p>\u7121\u8ad6\u662f\u7c21\u55ae\u7684\u5347\u5e8f\uff0c\u9084\u662f\u8907\u96dc\u7684\u5206\u985e\u6392\u5217\uff0c\u6392\u5e8f\u90fd\u5411\u6211\u5011\u5c55\u793a\u4e86\u8cc7\u6599\u7684\u548c\u8ae7\u7f8e\u611f\u3002</p>"},{"location":"chapter_sorting/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>11.1 \u00a0 \u6392\u5e8f\u6f14\u7b97\u6cd5</li> <li>11.2 \u00a0 \u9078\u64c7\u6392\u5e8f</li> <li>11.3 \u00a0 \u6ce1\u6cab\u6392\u5e8f</li> <li>11.4 \u00a0 \u63d2\u5165\u6392\u5e8f</li> <li>11.5 \u00a0 \u5feb\u901f\u6392\u5e8f</li> <li>11.6 \u00a0 \u5408\u4f75\u6392\u5e8f</li> <li>11.7 \u00a0 \u5806\u7a4d\u6392\u5e8f</li> <li>11.8 \u00a0 \u6876\u6392\u5e8f</li> <li>11.9 \u00a0 \u8a08\u6578\u6392\u5e8f</li> <li>11.10 \u00a0 \u57fa\u6578\u6392\u5e8f</li> <li>11.11 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_sorting/bubble_sort/","title":"11.3 \u00a0 \u6ce1\u6cab\u6392\u5e8f","text":"<p>\u6ce1\u6cab\u6392\u5e8f\uff08bubble sort\uff09\u900f\u904e\u9023\u7e8c\u5730\u6bd4\u8f03\u8207\u4ea4\u63db\u76f8\u9130\u5143\u7d20\u5be6\u73fe\u6392\u5e8f\u3002\u9019\u500b\u904e\u7a0b\u5c31\u50cf\u6c23\u6ce1\u5f9e\u5e95\u90e8\u5347\u5230\u9802\u90e8\u4e00\u6a23\uff0c\u56e0\u6b64\u5f97\u540d\u6ce1\u6cab\u6392\u5e8f\u3002</p> <p>\u5982\u5716 11-4 \u6240\u793a\uff0c\u5192\u6ce1\u904e\u7a0b\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u63db\u64cd\u4f5c\u4f86\u6a21\u64ec\uff1a\u5f9e\u9663\u5217\u6700\u5de6\u7aef\u958b\u59cb\u5411\u53f3\u8d70\u8a2a\uff0c\u4f9d\u6b21\u6bd4\u8f03\u76f8\u9130\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 &gt; \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u63db\u4e8c\u8005\u3002\u8d70\u8a2a\u5b8c\u6210\u5f8c\uff0c\u6700\u5927\u7684\u5143\u7d20\u6703\u88ab\u79fb\u52d5\u5230\u9663\u5217\u7684\u6700\u53f3\u7aef\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-4 \u00a0 \u5229\u7528\u5143\u7d20\u4ea4\u63db\u64cd\u4f5c\u6a21\u64ec\u5192\u6ce1 </p>"},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8a2d\u9663\u5217\u7684\u9577\u5ea6\u70ba \\(n\\) \uff0c\u6ce1\u6cab\u6392\u5e8f\u7684\u6b65\u9a5f\u5982\u5716 11-5 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5c0d \\(n\\) \u500b\u5143\u7d20\u57f7\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c07\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u6b63\u78ba\u4f4d\u7f6e\u3002</li> <li>\u63a5\u4e0b\u4f86\uff0c\u5c0d\u5269\u9918 \\(n - 1\\) \u500b\u5143\u7d20\u57f7\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c07\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u6b63\u78ba\u4f4d\u7f6e\u3002</li> <li>\u4ee5\u6b64\u985e\u63a8\uff0c\u7d93\u904e \\(n - 1\\) \u8f2a\u201c\u5192\u6ce1\u201d\u5f8c\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u63db\u81f3\u6b63\u78ba\u4f4d\u7f6e\u3002</li> <li>\u50c5\u5269\u7684\u4e00\u500b\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u7121\u9808\u6392\u5e8f\uff0c\u56e0\u6b64\u9663\u5217\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> <p></p> <p> \u5716 11-5 \u00a0 \u6ce1\u6cab\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"\u6ce1\u6cab\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in range(n - 1, 0, -1):\n        # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                // \u9019\u88e1\u4f7f\u7528\u4e86 std::swap() \u51fd\u5f0f\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid BubbleSort(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums.swapAt(j, j + 1)\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfunction bubbleSort(nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums.swap(j, j + 1);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfun bubbleSort(nums: IntArray) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### \u6ce1\u6cab\u6392\u5e8f ###\ndef bubble_sort(nums)\n  n = nums.length\n  # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for i in (n - 1).downto(1)\n    # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n      end\n    end\n  end\nend\n</code></pre> bubble_sort.zig<pre><code>// \u6ce1\u6cab\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2 \u00a0 \u6548\u7387\u6700\u4f73\u5316","text":"<p>\u6211\u5011\u767c\u73fe\uff0c\u5982\u679c\u67d0\u8f2a\u201c\u5192\u6ce1\u201d\u4e2d\u6c92\u6709\u57f7\u884c\u4efb\u4f55\u4ea4\u63db\u64cd\u4f5c\uff0c\u8aaa\u660e\u9663\u5217\u5df2\u7d93\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7d50\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u500b\u6a19\u8a8c\u4f4d <code>flag</code> \u4f86\u76e3\u6e2c\u9019\u7a2e\u60c5\u6cc1\uff0c\u4e00\u65e6\u51fa\u73fe\u5c31\u7acb\u5373\u8fd4\u56de\u3002</p> <p>\u7d93\u904e\u6700\u4f73\u5316\uff0c\u6ce1\u6cab\u6392\u5e8f\u7684\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u548c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u4ecd\u70ba \\(O(n^2)\\) \uff1b\u4f46\u7576\u8f38\u5165\u9663\u5217\u5b8c\u5168\u6709\u5e8f\u6642\uff0c\u53ef\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"\u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09\"\"\"\n    n = len(nums)\n    # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in range(n - 1, 0, -1):\n        flag = False  # \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n        if not flag:\n            break  # \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n</code></pre> bubble_sort.cpp<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                // \u9019\u88e1\u4f7f\u7528\u4e86 std::swap() \u51fd\u5f0f\n                swap(nums[j], nums[j + 1]);\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nvoid BubbleSortWithFlag(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag) break;     // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if flag == false { // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        var flag = false // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums.swapAt(j, j + 1)\n                flag = true // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if !flag { // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n    // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n      }\n    }\n    if (!flag) break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09 */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums.swap(j, j + 1);\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if !flag {\n            break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n        };\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        bool flag = false;\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09 */\nfun bubbleSortWithFlag(nums: IntArray) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        var flag = false // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                flag = true // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag) break // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09###\ndef bubble_sort_with_flag(nums)\n  n = nums.length\n  # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for i in (n - 1).downto(1)\n    flag = false # \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n\n    # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n        flag = true # \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n      end\n    end\n\n    break unless flag # \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n  end\nend\n</code></pre> bubble_sort.zig<pre><code>// \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var flag = false;   // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        var j: usize = 0;\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true;\n            }\n        }\n        if (!flag) break;   // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\)\u3001\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5404\u8f2a\u201c\u5192\u6ce1\u201d\u8d70\u8a2a\u7684\u9663\u5217\u9577\u5ea6\u4f9d\u6b21\u70ba \\(n - 1\\)\u3001\\(n - 2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u7e3d\u548c\u70ba \\((n - 1) n / 2\\) \u3002\u5728\u5f15\u5165 <code>flag</code> \u6700\u4f73\u5316\u5f8c\uff0c\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054\u5230 \\(O(n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u7531\u65bc\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u63db\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/","title":"11.8 \u00a0 \u6876\u6392\u5e8f","text":"<p>\u524d\u8ff0\u5e7e\u7a2e\u6392\u5e8f\u6f14\u7b97\u6cd5\u90fd\u5c6c\u65bc\u201c\u57fa\u65bc\u6bd4\u8f03\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\u201d\uff0c\u5b83\u5011\u900f\u904e\u6bd4\u8f03\u5143\u7d20\u9593\u7684\u5927\u5c0f\u4f86\u5be6\u73fe\u6392\u5e8f\u3002\u6b64\u985e\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u7121\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5c07\u63a2\u8a0e\u5e7e\u7a2e\u201c\u975e\u6bd4\u8f03\u6392\u5e8f\u6f14\u7b97\u6cd5\u201d\uff0c\u5b83\u5011\u7684\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u9054\u5230\u7dda\u6027\u968e\u3002</p> <p>\u6876\u6392\u5e8f\uff08bucket sort\uff09\u662f\u5206\u6cbb\u7b56\u7565\u7684\u4e00\u500b\u5178\u578b\u61c9\u7528\u3002\u5b83\u900f\u904e\u8a2d\u5b9a\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u9806\u5e8f\u7684\u6876\uff0c\u6bcf\u500b\u6876\u5c0d\u61c9\u4e00\u500b\u6578\u64da\u7bc4\u570d\uff0c\u5c07\u8cc7\u6599\u5e73\u5747\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\uff1b\u7136\u5f8c\uff0c\u5728\u6bcf\u500b\u6876\u5167\u90e8\u5206\u5225\u57f7\u884c\u6392\u5e8f\uff1b\u6700\u7d42\u6309\u7167\u6876\u7684\u9806\u5e8f\u5c07\u6240\u6709\u8cc7\u6599\u5408\u4f75\u3002</p>"},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8003\u616e\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217\uff0c\u5176\u5143\u7d20\u662f\u7bc4\u570d \\([0, 1)\\) \u5167\u7684\u6d6e\u9ede\u6578\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u5716 11-13 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316 \\(k\\) \u500b\u6876\uff0c\u5c07 \\(n\\) \u500b\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u500b\u6876\u4e2d\u3002</li> <li>\u5c0d\u6bcf\u500b\u6876\u5206\u5225\u57f7\u884c\u6392\u5e8f\uff08\u9019\u88e1\u63a1\u7528\u7a0b\u5f0f\u8a9e\u8a00\u7684\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff09\u3002</li> <li>\u6309\u7167\u6876\u5f9e\u5c0f\u5230\u5927\u7684\u9806\u5e8f\u5408\u4f75\u7d50\u679c\u3002</li> </ol> <p></p> <p> \u5716 11-13 \u00a0 \u6876\u6392\u5e8f\u6f14\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"\u6876\u6392\u5e8f\"\"\"\n    # \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for num in nums:\n        # \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        i = int(num * k)\n        # \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].append(num)\n    # 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for bucket in buckets:\n        # \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort()\n    # 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        int i = num * k;\n        // \u5c07 num \u65b0\u589e\u9032\u6876 bucket_idx\n        buckets[i].push_back(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        int i = (int) (num * k);\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets.get(i).add(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (List&lt;Float&gt; bucket : buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        Collections.sort(bucket);\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* \u6876\u6392\u5e8f */\nvoid BucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    foreach (float num in nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        int i = (int)(num * k);\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].Add(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.Sort();\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for _, num := range nums {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        i := int(num * float64(k))\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for i := 0; i &lt; k; i++ {\n        // \u4f7f\u7528\u5167\u5efa\u5207\u7247\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        sort.Float64s(buckets[i])\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for num in nums {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        let i = Int(num * Double(k))\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].append(num)\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for i in buckets.indices {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        buckets[i].sort()\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n  for (double _num in nums) {\n    // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 _num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n    int i = (_num * k).toInt();\n    // \u5c07 _num \u65b0\u589e\u9032\u6876 bucket_idx\n    buckets[i].add(_num);\n  }\n  // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* \u6876\u6392\u5e8f */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for &amp;num in nums.iter() {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        let i = (num * k as f64) as usize;\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for bucket in &amp;mut buckets {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    let mut i = 0;\n    for bucket in buckets.iter() {\n        for &amp;num in bucket.iter() {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int n) {\n    int k = n / 2;                                 // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\n    int *sizes = malloc(k * sizeof(int));          // \u8a18\u9304\u6bcf\u500b\u6876\u7684\u5927\u5c0f\n    float **buckets = malloc(k * sizeof(float *)); // \u52d5\u614b\u9663\u5217\u7684\u9663\u5217\uff08\u6876\uff09\n    // \u70ba\u6bcf\u500b\u6876\u9810\u5206\u914d\u8db3\u5920\u7684\u7a7a\u9593\n    for (int i = 0; i &lt; k; ++i) {\n        buckets[i] = (float *)malloc(n * sizeof(float));\n        sizes[i] = 0;\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (int i = 0; i &lt; n; ++i) {\n        int idx = (int)(nums[i] * k);\n        buckets[idx][sizes[idx]++] = nums[i];\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (int i = 0; i &lt; k; ++i) {\n        qsort(buckets[i], sizes[i], sizeof(float), compare);\n    }\n    // 3. \u5408\u4f75\u6392\u5e8f\u5f8c\u7684\u6876\n    int idx = 0;\n    for (int i = 0; i &lt; k; ++i) {\n        for (int j = 0; j &lt; sizes[i]; ++j) {\n            nums[idx++] = buckets[i][j];\n        }\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        free(buckets[i]);\n    }\n}\n</code></pre> bucket_sort.kt<pre><code>/* \u6876\u6392\u5e8f */\nfun bucketSort(nums: FloatArray) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    val k = nums.size / 2\n    val buckets = mutableListOf&lt;MutableList&lt;Float&gt;&gt;()\n    for (i in 0..&lt;k) {\n        buckets.add(mutableListOf())\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (num in nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        val i = (num * k).toInt()\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].add(num)\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (bucket in buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort()\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    var i = 0\n    for (bucket in buckets) {\n        for (num in bucket) {\n            nums[i++] = num\n        }\n    }\n}\n</code></pre> bucket_sort.rb<pre><code>### \u6876\u6392\u5e8f ###\ndef bucket_sort(nums)\n  # \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n  k = nums.length / 2\n  buckets = Array.new(k) { [] }\n\n  # 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n  nums.each do |num|\n    # \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n    i = (num * k).to_i\n    # \u5c07 num \u65b0\u589e\u9032\u6876 i\n    buckets[i] &lt;&lt; num\n  end\n\n  # 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n  buckets.each do |bucket|\n    # \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n    bucket.sort!\n  end\n\n  # 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n  i = 0\n  buckets.each do |bucket|\n    bucket.each do |num|\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> bucket_sort.zig<pre><code>[class]{}-[func]{bucketSort}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<p>\u6876\u6392\u5e8f\u9069\u7528\u65bc\u8655\u7406\u9ad4\u91cf\u5f88\u5927\u7684\u8cc7\u6599\u3002\u4f8b\u5982\uff0c\u8f38\u5165\u8cc7\u6599\u5305\u542b 100 \u842c\u500b\u5143\u7d20\uff0c\u7531\u65bc\u7a7a\u9593\u9650\u5236\uff0c\u7cfb\u7d71\u8a18\u61b6\u9ad4\u7121\u6cd5\u4e00\u6b21\u6027\u8f09\u5165\u6240\u6709\u8cc7\u6599\u3002\u6b64\u6642\uff0c\u53ef\u4ee5\u5c07\u8cc7\u6599\u5206\u6210 1000 \u500b\u6876\uff0c\u7136\u5f8c\u5206\u5225\u5c0d\u6bcf\u500b\u6876\u9032\u884c\u6392\u5e8f\uff0c\u6700\u5f8c\u5c07\u7d50\u679c\u5408\u4f75\u3002</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + k)\\) \uff1a\u5047\u8a2d\u5143\u7d20\u5728\u5404\u500b\u6876\u5167\u5e73\u5747\u5206\u4f48\uff0c\u90a3\u9ebc\u6bcf\u500b\u6876\u5167\u7684\u5143\u7d20\u6578\u91cf\u70ba \\(\\frac{n}{k}\\) \u3002\u5047\u8a2d\u6392\u5e8f\u55ae\u500b\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u6642\u9593\uff0c\u5247\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u6642\u9593\u3002\u7576\u6876\u6578\u91cf \\(k\\) \u6bd4\u8f03\u5927\u6642\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u5247\u8da8\u5411\u65bc \\(O(n)\\) \u3002\u5408\u4f75\u7d50\u679c\u6642\u9700\u8981\u8d70\u8a2a\u6240\u6709\u6876\u548c\u5143\u7d20\uff0c\u82b1\u8cbb \\(O(n + k)\\) \u6642\u9593\u3002\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u6240\u6709\u8cc7\u6599\u88ab\u5206\u914d\u5230\u4e00\u500b\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8a72\u6876\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + k)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9700\u8981\u85c9\u52a9 \\(k\\) \u500b\u6876\u548c\u7e3d\u5171 \\(n\\) \u500b\u5143\u7d20\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u6876\u6392\u5e8f\u662f\u5426\u7a69\u5b9a\u53d6\u6c7a\u65bc\u6392\u5e8f\u6876\u5167\u5143\u7d20\u7684\u6f14\u7b97\u6cd5\u662f\u5426\u7a69\u5b9a\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/#1183","title":"11.8.3 \u00a0 \u5982\u4f55\u5be6\u73fe\u5e73\u5747\u5206\u914d","text":"<p>\u6876\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u7406\u8ad6\u4e0a\u53ef\u4ee5\u9054\u5230 \\(O(n)\\) \uff0c\u95dc\u9375\u5728\u65bc\u5c07\u5143\u7d20\u5747\u52fb\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\uff0c\u56e0\u70ba\u5be6\u969b\u8cc7\u6599\u5f80\u5f80\u4e0d\u662f\u5747\u52fb\u5206\u4f48\u7684\u3002\u4f8b\u5982\uff0c\u6211\u5011\u60f3\u8981\u5c07\u6dd8\u5bf6\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u50f9\u683c\u7bc4\u570d\u5e73\u5747\u5206\u914d\u5230 10 \u500b\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u50f9\u683c\u5206\u4f48\u4e0d\u5747\uff0c\u4f4e\u65bc 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u65bc 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c07\u50f9\u683c\u5340\u9593\u5e73\u5747\u5283\u5206\u70ba 10 \u500b\uff0c\u5404\u500b\u6876\u4e2d\u7684\u5546\u54c1\u6578\u91cf\u5dee\u8ddd\u6703\u975e\u5e38\u5927\u3002</p> <p>\u70ba\u5be6\u73fe\u5e73\u5747\u5206\u914d\uff0c\u6211\u5011\u53ef\u4ee5\u5148\u8a2d\u5b9a\u4e00\u689d\u5927\u81f4\u7684\u5206\u754c\u7dda\uff0c\u5c07\u8cc7\u6599\u7c97\u7565\u5730\u5206\u5230 3 \u500b\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u7562\u5f8c\uff0c\u518d\u5c07\u5546\u54c1\u8f03\u591a\u7684\u6876\u7e7c\u7e8c\u5283\u5206\u70ba 3 \u500b\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6578\u91cf\u5927\u81f4\u76f8\u7b49\u3002</p> <p>\u5982\u5716 11-14 \u6240\u793a\uff0c\u9019\u7a2e\u65b9\u6cd5\u672c\u8cea\u4e0a\u662f\u5efa\u7acb\u4e00\u68f5\u905e\u8ff4\u6a39\uff0c\u76ee\u6a19\u662f\u8b93\u8449\u7bc0\u9ede\u7684\u503c\u5118\u53ef\u80fd\u5e73\u5747\u3002\u7576\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f2a\u5c07\u8cc7\u6599\u5283\u5206\u70ba 3 \u500b\u6876\uff0c\u5177\u9ad4\u5283\u5206\u65b9\u5f0f\u53ef\u6839\u64da\u8cc7\u6599\u7279\u9ede\u9748\u6d3b\u9078\u64c7\u3002</p> <p></p> <p> \u5716 11-14 \u00a0 \u905e\u8ff4\u5283\u5206\u6876 </p> <p>\u5982\u679c\u6211\u5011\u63d0\u524d\u77e5\u9053\u5546\u54c1\u50f9\u683c\u7684\u6a5f\u7387\u5206\u4f48\uff0c\u5247\u53ef\u4ee5\u6839\u64da\u8cc7\u6599\u6a5f\u7387\u5206\u4f48\u8a2d\u5b9a\u6bcf\u500b\u6876\u7684\u50f9\u683c\u5206\u754c\u7dda\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8cc7\u6599\u5206\u4f48\u4e26\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7d71\u8a08\uff0c\u4e5f\u53ef\u4ee5\u6839\u64da\u8cc7\u6599\u7279\u9ede\u63a1\u7528\u67d0\u7a2e\u6a5f\u7387\u6a21\u578b\u9032\u884c\u8fd1\u4f3c\u3002</p> <p>\u5982\u5716 11-15 \u6240\u793a\uff0c\u6211\u5011\u5047\u8a2d\u5546\u54c1\u50f9\u683c\u670d\u5f9e\u6b63\u614b\u5206\u4f48\uff0c\u9019\u6a23\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8a2d\u5b9a\u50f9\u683c\u5340\u9593\uff0c\u5f9e\u800c\u5c07\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\u3002</p> <p></p> <p> \u5716 11-15 \u00a0 \u6839\u64da\u6a5f\u7387\u5206\u4f48\u5283\u5206\u6876 </p>"},{"location":"chapter_sorting/counting_sort/","title":"11.9 \u00a0 \u8a08\u6578\u6392\u5e8f","text":"<p>\u8a08\u6578\u6392\u5e8f\uff08counting sort\uff09\u900f\u904e\u7d71\u8a08\u5143\u7d20\u6578\u91cf\u4f86\u5be6\u73fe\u6392\u5e8f\uff0c\u901a\u5e38\u61c9\u7528\u65bc\u6574\u6578\u9663\u5217\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1 \u00a0 \u7c21\u55ae\u5be6\u73fe","text":"<p>\u5148\u4f86\u770b\u4e00\u500b\u7c21\u55ae\u7684\u4f8b\u5b50\u3002\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217 <code>nums</code> \uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8ca0\u6574\u6578\u201d\uff0c\u8a08\u6578\u6392\u5e8f\u7684\u6574\u9ad4\u6d41\u7a0b\u5982\u5716 11-16 \u6240\u793a\u3002</p> <ol> <li>\u8d70\u8a2a\u9663\u5217\uff0c\u627e\u51fa\u5176\u4e2d\u7684\u6700\u5927\u6578\u5b57\uff0c\u8a18\u70ba \\(m\\) \uff0c\u7136\u5f8c\u5efa\u7acb\u4e00\u500b\u9577\u5ea6\u70ba \\(m + 1\\) \u7684\u8f14\u52a9\u9663\u5217 <code>counter</code> \u3002</li> <li>\u85c9\u52a9 <code>counter</code> \u7d71\u8a08 <code>nums</code> \u4e2d\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\uff0c\u5176\u4e2d <code>counter[num]</code> \u5c0d\u61c9\u6578\u5b57 <code>num</code> \u7684\u51fa\u73fe\u6b21\u6578\u3002\u7d71\u8a08\u65b9\u6cd5\u5f88\u7c21\u55ae\uff0c\u53ea\u9700\u8d70\u8a2a <code>nums</code>\uff08\u8a2d\u7576\u524d\u6578\u5b57\u70ba <code>num</code>\uff09\uff0c\u6bcf\u8f2a\u5c07 <code>counter[num]</code> \u589e\u52a0 \\(1\\) \u5373\u53ef\u3002</li> <li>\u7531\u65bc <code>counter</code> \u7684\u5404\u500b\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u7576\u65bc\u6240\u6709\u6578\u5b57\u5df2\u7d93\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u8d70\u8a2a <code>counter</code> \uff0c\u6839\u64da\u5404\u6578\u5b57\u51fa\u73fe\u6b21\u6578\u5f9e\u5c0f\u5230\u5927\u7684\u9806\u5e8f\u586b\u5165 <code>nums</code> \u5373\u53ef\u3002</li> </ol> <p></p> <p> \u5716 11-16 \u00a0 \u8a08\u6578\u6392\u5e8f\u6d41\u7a0b </p> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"\u8a08\u6578\u6392\u5e8f\"\"\"\n    # \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\n    # 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    m = 0\n    for num in nums:\n        m = max(m, num)\n    # 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid countingSortNaive(int[] nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid CountingSortNaive(int[] nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfunc countingSortNaive(nums []int) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    var i = 0\n    for num in 0 ..&lt; m + 1 {\n        for _ in 0 ..&lt; counter[num] {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfunction countingSortNaive(nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfunction countingSortNaive(nums: number[]): void {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73fe\u6b21\u6578\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = *nums.iter().max().unwrap();\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid countingSortNaive(int nums[], int size) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int *counter = calloc(m + 1, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfun countingSortNaive(nums: IntArray) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    var i = 0\n    for (num in 0..&lt;m + 1) {\n        var j = 0\n        while (j &lt; counter[num]) {\n            nums[i] = num\n            j++\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### \u8a08\u6578\u6392\u5e8f ###\ndef counting_sort_naive(nums)\n  # \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\n  # 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n  m = 0\n  nums.each { |num| m = [m, num].max }\n  # 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n  i = 0\n  for num in 0...(m + 1)\n    (0...counter[num]).each do\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSortNaive}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u8a08\u6578\u6392\u5e8f\u8207\u6876\u6392\u5e8f\u7684\u95dc\u806f</p> <p>\u5f9e\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u8a08\u6578\u6392\u5e8f\u4e2d\u7684\u8a08\u6578\u9663\u5217 <code>counter</code> \u7684\u6bcf\u500b\u7d22\u5f15\u8996\u70ba\u4e00\u500b\u6876\uff0c\u5c07\u7d71\u8a08\u6578\u91cf\u7684\u904e\u7a0b\u770b\u4f5c\u5c07\u5404\u500b\u5143\u7d20\u5206\u914d\u5230\u5c0d\u61c9\u7684\u6876\u4e2d\u3002\u672c\u8cea\u4e0a\uff0c\u8a08\u6578\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u8cc7\u6599\u4e0b\u7684\u4e00\u500b\u7279\u4f8b\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2 \u00a0 \u5b8c\u6574\u5be6\u73fe","text":"<p>\u7d30\u5fc3\u7684\u8b80\u8005\u53ef\u80fd\u767c\u73fe\u4e86\uff0c\u5982\u679c\u8f38\u5165\u8cc7\u6599\u662f\u7269\u4ef6\uff0c\u4e0a\u8ff0\u6b65\u9a5f <code>3.</code> \u5c31\u5931\u6548\u4e86\u3002\u5047\u8a2d\u8f38\u5165\u8cc7\u6599\u662f\u5546\u54c1\u7269\u4ef6\uff0c\u6211\u5011\u60f3\u6309\u7167\u5546\u54c1\u50f9\u683c\uff08\u985e\u5225\u7684\u6210\u54e1\u8b8a\u6578\uff09\u5c0d\u5546\u54c1\u9032\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u6f14\u7b97\u6cd5\u53ea\u80fd\u7d66\u51fa\u50f9\u683c\u7684\u6392\u5e8f\u7d50\u679c\u3002</p> <p>\u90a3\u9ebc\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u8cc7\u6599\u7684\u6392\u5e8f\u7d50\u679c\u5462\uff1f\u6211\u5011\u9996\u5148\u8a08\u7b97 <code>counter</code> \u7684\u201c\u524d\u7db4\u548c\u201d\u3002\u9867\u540d\u601d\u7fa9\uff0c\u7d22\u5f15 <code>i</code> \u8655\u7684\u524d\u7db4\u548c <code>prefix[i]</code> \u7b49\u65bc\u9663\u5217\u524d <code>i</code> \u500b\u5143\u7d20\u4e4b\u548c\uff1a</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>\u524d\u7db4\u548c\u5177\u6709\u660e\u78ba\u7684\u610f\u7fa9\uff0c<code>prefix[num] - 1</code> \u4ee3\u8868\u5143\u7d20 <code>num</code> \u5728\u7d50\u679c\u9663\u5217 <code>res</code> \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\u3002\u9019\u500b\u8cc7\u8a0a\u975e\u5e38\u95dc\u9375\uff0c\u56e0\u70ba\u5b83\u544a\u8a34\u6211\u5011\u5404\u500b\u5143\u7d20\u61c9\u8a72\u51fa\u73fe\u5728\u7d50\u679c\u9663\u5217\u7684\u54ea\u500b\u4f4d\u7f6e\u3002\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5012\u5e8f\u8d70\u8a2a\u539f\u9663\u5217 <code>nums</code> \u7684\u6bcf\u500b\u5143\u7d20 <code>num</code> \uff0c\u5728\u6bcf\u8f2a\u8fed\u4ee3\u4e2d\u57f7\u884c\u4ee5\u4e0b\u5169\u6b65\u3002</p> <ol> <li>\u5c07 <code>num</code> \u586b\u5165\u9663\u5217 <code>res</code> \u7684\u7d22\u5f15 <code>prefix[num] - 1</code> \u8655\u3002</li> <li>\u4ee4\u524d\u7db4\u548c <code>prefix[num]</code> \u6e1b\u5c0f \\(1\\) \uff0c\u5f9e\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e <code>num</code> \u7684\u7d22\u5f15\u3002</li> </ol> <p>\u8d70\u8a2a\u5b8c\u6210\u5f8c\uff0c\u9663\u5217 <code>res</code> \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7d50\u679c\uff0c\u6700\u5f8c\u4f7f\u7528 <code>res</code> \u8986\u84cb\u539f\u9663\u5217 <code>nums</code> \u5373\u53ef\u3002\u5716 11-17 \u5c55\u793a\u4e86\u5b8c\u6574\u7684\u8a08\u6578\u6392\u5e8f\u6d41\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-17 \u00a0 \u8a08\u6578\u6392\u5e8f\u6b65\u9a5f </p> <p>\u8a08\u6578\u6392\u5e8f\u7684\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"\u8a08\u6578\u6392\u5e8f\"\"\"\n    # \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\n    # 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    m = max(nums)\n    # 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    # \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num] -= 1  # \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    # \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--;              // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid CountingSort(int[] nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        res[counter[num]-1] = num\n        // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n        counter[num]--\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for i in 0 ..&lt; m {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let num = nums[i]\n        res[counter[num] - 1] = num // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num] -= 1 // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73fe\u6b21\u6578\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n  // \u5373 counter[_num]-1 \u662f _num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n  // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // \u5c07 _num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n    counter[_num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e _num \u7684\u7d22\u5f15\n  }\n  // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = *nums.iter().max().unwrap() as usize;\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    let mut counter = vec![0; m + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for i in 0..m {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num as usize] -= 1; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    nums.copy_from_slice(&amp;res)\n}\n</code></pre> counting_sort.c<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--;              // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    memcpy(nums, res, size * sizeof(int));\n    // 5. \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(res);\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfun countingSort(nums: IntArray) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (i in 0..&lt;m) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    val n = nums.size\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val num = nums[i]\n        res[counter[num] - 1] = num // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]-- // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (i in 0..&lt;n) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### \u8a08\u6578\u6392\u5e8f ###\ndef counting_sort(nums)\n  # \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\n  # 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n  m = nums.max\n  # 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n  # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n  (0...m).each { |i| counter[i + 1] += counter[i] }\n  # 4. \u5012\u5e8f\u8d70\u8a2a nums, \u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n  # \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n  n = nums.length\n  res = Array.new(n, 0)\n  (n - 1).downto(0).each do |i|\n    num = nums[i]\n    res[counter[num] - 1] = num # \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n    counter[num] -= 1 # \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n  end\n  # \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n  (0...n).each { |i| nums[i] = res[i] }\nend\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSort}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + m)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f \uff1a\u6d89\u53ca\u8d70\u8a2a <code>nums</code> \u548c\u8d70\u8a2a <code>counter</code> \uff0c\u90fd\u4f7f\u7528\u7dda\u6027\u6642\u9593\u3002\u4e00\u822c\u60c5\u6cc1\u4e0b \\(n \\gg m\\) \uff0c\u6642\u9593\u8907\u96dc\u5ea6\u8da8\u65bc \\(O(n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + m)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u85c9\u52a9\u4e86\u9577\u5ea6\u5206\u5225\u70ba \\(n\\) \u548c \\(m\\) \u7684\u9663\u5217 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u7531\u65bc\u5411 <code>res</code> \u4e2d\u586b\u5145\u5143\u7d20\u7684\u9806\u5e8f\u662f\u201c\u5f9e\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u8d70\u8a2a <code>nums</code> \u53ef\u4ee5\u907f\u514d\u6539\u8b8a\u76f8\u7b49\u5143\u7d20\u4e4b\u9593\u7684\u76f8\u5c0d\u4f4d\u7f6e\uff0c\u5f9e\u800c\u5be6\u73fe\u7a69\u5b9a\u6392\u5e8f\u3002\u5be6\u969b\u4e0a\uff0c\u6b63\u5e8f\u8d70\u8a2a <code>nums</code> \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u78ba\u7684\u6392\u5e8f\u7d50\u679c\uff0c\u4f46\u7d50\u679c\u662f\u975e\u7a69\u5b9a\u7684\u3002</li> </ul>"},{"location":"chapter_sorting/counting_sort/#1194","title":"11.9.4 \u00a0 \u4fb7\u9650\u6027","text":"<p>\u770b\u5230\u9019\u88e1\uff0c\u4f60\u4e5f\u8a31\u6703\u89ba\u5f97\u8a08\u6578\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u50c5\u900f\u904e\u7d71\u8a08\u6578\u91cf\u5c31\u53ef\u4ee5\u5be6\u73fe\u9ad8\u6548\u7684\u6392\u5e8f\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8a08\u6578\u6392\u5e8f\u7684\u524d\u7f6e\u689d\u4ef6\u76f8\u5c0d\u8f03\u70ba\u56b4\u683c\u3002</p> <p>\u8a08\u6578\u6392\u5e8f\u53ea\u9069\u7528\u65bc\u975e\u8ca0\u6574\u6578\u3002\u82e5\u60f3\u5c07\u5176\u7528\u65bc\u5176\u4ed6\u578b\u5225\u7684\u8cc7\u6599\uff0c\u9700\u8981\u78ba\u4fdd\u9019\u4e9b\u8cc7\u6599\u53ef\u4ee5\u8f49\u63db\u70ba\u975e\u8ca0\u6574\u6578\uff0c\u4e26\u4e14\u5728\u8f49\u63db\u904e\u7a0b\u4e2d\u4e0d\u80fd\u6539\u8b8a\u5404\u500b\u5143\u7d20\u4e4b\u9593\u7684\u76f8\u5c0d\u5927\u5c0f\u95dc\u4fc2\u3002\u4f8b\u5982\uff0c\u5c0d\u65bc\u5305\u542b\u8ca0\u6578\u7684\u6574\u6578\u9663\u5217\uff0c\u53ef\u4ee5\u5148\u7d66\u6240\u6709\u6578\u5b57\u52a0\u4e0a\u4e00\u500b\u5e38\u6578\uff0c\u5c07\u5168\u90e8\u6578\u5b57\u8f49\u5316\u70ba\u6b63\u6578\uff0c\u6392\u5e8f\u5b8c\u6210\u5f8c\u518d\u8f49\u63db\u56de\u53bb\u3002</p> <p>\u8a08\u6578\u6392\u5e8f\u9069\u7528\u65bc\u8cc7\u6599\u91cf\u5927\u4f46\u8cc7\u6599\u7bc4\u570d\u8f03\u5c0f\u7684\u60c5\u6cc1\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5247\u6703\u4f54\u7528\u904e\u591a\u7a7a\u9593\u3002\u800c\u7576 \\(n \\ll m\\) \u6642\uff0c\u8a08\u6578\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u6642\u9593\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\u9084\u8981\u6162\u3002</p>"},{"location":"chapter_sorting/heap_sort/","title":"11.7 \u00a0 \u5806\u7a4d\u6392\u5e8f","text":"<p>Tip</p> <p>\u95b1\u8b80\u672c\u7bc0\u524d\uff0c\u8acb\u78ba\u4fdd\u5df2\u5b78\u5b8c\u201c\u5806\u7a4d\u201c\u7ae0\u7bc0\u3002</p> <p>\u5806\u7a4d\u6392\u5e8f\uff08heap sort\uff09\u662f\u4e00\u7a2e\u57fa\u65bc\u5806\u7a4d\u8cc7\u6599\u7d50\u69cb\u5be6\u73fe\u7684\u9ad8\u6548\u6392\u5e8f\u6f14\u7b97\u6cd5\u3002\u6211\u5011\u53ef\u4ee5\u5229\u7528\u5df2\u7d93\u5b78\u904e\u7684\u201c\u5efa\u5806\u7a4d\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u7a4d\u64cd\u4f5c\u201d\u5be6\u73fe\u5806\u7a4d\u6392\u5e8f\u3002</p> <ol> <li>\u8f38\u5165\u9663\u5217\u4e26\u5efa\u7acb\u5c0f\u9802\u5806\u7a4d\uff0c\u6b64\u6642\u6700\u5c0f\u5143\u7d20\u4f4d\u65bc\u5806\u7a4d\u9802\u3002</li> <li>\u4e0d\u65b7\u57f7\u884c\u51fa\u5806\u7a4d\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8a18\u9304\u51fa\u5806\u7a4d\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u5f9e\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002</li> </ol> <p>\u4ee5\u4e0a\u65b9\u6cd5\u96d6\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u85c9\u52a9\u4e00\u500b\u984d\u5916\u9663\u5217\u4f86\u5132\u5b58\u5f48\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f03\u6d6a\u8cbb\u7a7a\u9593\u3002\u5728\u5be6\u969b\u4e2d\uff0c\u6211\u5011\u901a\u5e38\u4f7f\u7528\u4e00\u7a2e\u66f4\u52a0\u512a\u96c5\u7684\u5be6\u73fe\u65b9\u5f0f\u3002</p>"},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8a2d\u9663\u5217\u7684\u9577\u5ea6\u70ba \\(n\\) \uff0c\u5806\u7a4d\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u5716 11-12 \u6240\u793a\u3002</p> <ol> <li>\u8f38\u5165\u9663\u5217\u4e26\u5efa\u7acb\u5927\u9802\u5806\u7a4d\u3002\u5b8c\u6210\u5f8c\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u65bc\u5806\u7a4d\u9802\u3002</li> <li>\u5c07\u5806\u7a4d\u9802\u5143\u7d20\uff08\u7b2c\u4e00\u500b\u5143\u7d20\uff09\u8207\u5806\u7a4d\u5e95\u5143\u7d20\uff08\u6700\u5f8c\u4e00\u500b\u5143\u7d20\uff09\u4ea4\u63db\u3002\u5b8c\u6210\u4ea4\u63db\u5f8c\uff0c\u5806\u7a4d\u7684\u9577\u5ea6\u6e1b \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6578\u91cf\u52a0 \\(1\\) \u3002</li> <li>\u5f9e\u5806\u7a4d\u9802\u5143\u7d20\u958b\u59cb\uff0c\u5f9e\u9802\u5230\u5e95\u57f7\u884c\u5806\u7a4d\u5316\u64cd\u4f5c\uff08sift down\uff09\u3002\u5b8c\u6210\u5806\u7a4d\u5316\u5f8c\uff0c\u5806\u7a4d\u7684\u6027\u8cea\u5f97\u5230\u4fee\u5fa9\u3002</li> <li>\u8ff4\u5708\u57f7\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\u3002\u8ff4\u5708 \\(n - 1\\) \u8f2a\u5f8c\uff0c\u5373\u53ef\u5b8c\u6210\u9663\u5217\u6392\u5e8f\u3002</li> </ol> <p>Tip</p> <p>\u5be6\u969b\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u7a4d\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u500b\u5f48\u51fa\u5143\u7d20\u7684\u6b65\u9a5f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-12 \u00a0 \u5806\u7a4d\u6392\u5e8f\u6b65\u9a5f </p> <p>\u5728\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e2d\uff0c\u6211\u5011\u4f7f\u7528\u4e86\u8207\u201c\u5806\u7a4d\u201d\u7ae0\u7bc0\u76f8\u540c\u7684\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 <code>sift_down()</code> \u51fd\u5f0f\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u65bc\u5806\u7a4d\u7684\u9577\u5ea6\u6703\u96a8\u8457\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u6e1b\u5c0f\uff0c\u56e0\u6b64\u6211\u5011\u9700\u8981\u7d66 <code>sift_down()</code> \u51fd\u5f0f\u65b0\u589e\u4e00\u500b\u9577\u5ea6\u53c3\u6578 \\(n\\) \uff0c\u7528\u65bc\u6307\u5b9a\u5806\u7a4d\u7684\u7576\u524d\u6709\u6548\u9577\u5ea6\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"\u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\"\"\"\n    while True:\n        # \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u63db\u5169\u7bc0\u9ede\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"\u5806\u7a4d\u6392\u5e8f\"\"\"\n    # \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for i in range(len(nums) - 1, 0, -1):\n        # \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        nums[0], nums[i] = nums[i], nums[0]\n        # \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(nums[i], nums[ma]);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        swap(nums[0], nums[i]);\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid HeapSort(int[] nums) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        nums.swapAt(i, ma)\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for i in nums.indices.dropFirst().reversed() {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        nums.swapAt(0, i)\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfunction heapSort(nums) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u63db\u5169\u7bc0\u9ede\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n    i = ma;\n  }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid heapSort(List&lt;int&gt; nums) {\n  // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        nums.swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in (0..nums.len() / 2).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for i in (1..nums.len()).rev() {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        nums.swap(0, i);\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (int i = n - 1; i &gt; 0; --i) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.kt<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfun siftDown(nums: IntArray, n: Int, li: Int) {\n    var i = li\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        val l = 2 * i + 1\n        val r = 2 * i + 2\n        var ma = i\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) \n            ma = l\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) \n            ma = r\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) \n            break\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        val temp = nums[i]\n        nums[i] = nums[ma]\n        nums[ma] = temp\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfun heapSort(nums: IntArray) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (i in nums.size / 2 - 1 downTo 0) {\n        siftDown(nums, nums.size, i)\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (i in nums.size - 1 downTo 1) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        val temp = nums[0]\n        nums[0] = nums[i]\n        nums[i] = temp\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.rb<pre><code>### \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 ###\ndef sift_down(nums, n, i)\n  while true\n    # \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n    l = 2 * i + 1\n    r = 2 * i + 2\n    ma = i\n    ma = l if l &lt; n &amp;&amp; nums[l] &gt; nums[ma]\n    ma = r if r &lt; n &amp;&amp; nums[r] &gt; nums[ma]\n    # \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n    break if ma == i\n    # \u4ea4\u63db\u5169\u7bc0\u9ede\n    nums[i], nums[ma] = nums[ma], nums[i]\n    # \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n    i = ma\n  end\nend\n\n### \u5806\u7a4d\u6392\u5e8f ###\ndef heap_sort(nums)\n  # \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n  (nums.length / 2 - 1).downto(0) do |i|\n    sift_down(nums, nums.length, i)\n  end\n  # \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n  (nums.length - 1).downto(1) do |i|\n    # \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    nums[0], nums[i] = nums[i], nums[0]\n    # \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n    sift_down(nums, i, 0)\n  end\nend\n</code></pre> heap_sort.zig<pre><code>[class]{}-[func]{siftDown}\n\n[class]{}-[func]{heapSort}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5efa\u5806\u7a4d\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002\u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \uff0c\u5171\u8ff4\u5708 \\(n - 1\\) \u8f2a\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u5e7e\u500b\u6307\u6a19\u8b8a\u6578\u4f7f\u7528 \\(O(1)\\) \u7a7a\u9593\u3002\u5143\u7d20\u4ea4\u63db\u548c\u5806\u7a4d\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u9663\u5217\u4e0a\u9032\u884c\u7684\u3002</li> <li>\u975e\u7a69\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u63db\u5806\u7a4d\u9802\u5143\u7d20\u548c\u5806\u7a4d\u5e95\u5143\u7d20\u6642\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5c0d\u4f4d\u7f6e\u53ef\u80fd\u767c\u751f\u8b8a\u5316\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/","title":"11.4 \u00a0 \u63d2\u5165\u6392\u5e8f","text":"<p>\u63d2\u5165\u6392\u5e8f\uff08insertion sort\uff09\u662f\u4e00\u7a2e\u7c21\u55ae\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u8207\u624b\u52d5\u6574\u7406\u4e00\u526f\u724c\u7684\u904e\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002</p> <p>\u5177\u9ad4\u4f86\u8aaa\uff0c\u6211\u5011\u5728\u672a\u6392\u5e8f\u5340\u9593\u9078\u64c7\u4e00\u500b\u57fa\u6e96\u5143\u7d20\uff0c\u5c07\u8a72\u5143\u7d20\u8207\u5176\u5de6\u5074\u5df2\u6392\u5e8f\u5340\u9593\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f03\u5927\u5c0f\uff0c\u4e26\u5c07\u8a72\u5143\u7d20\u63d2\u5165\u5230\u6b63\u78ba\u7684\u4f4d\u7f6e\u3002</p> <p>\u5716 11-6 \u5c55\u793a\u4e86\u9663\u5217\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u6d41\u7a0b\u3002\u8a2d\u57fa\u6e96\u5143\u7d20\u70ba <code>base</code> \uff0c\u6211\u5011\u9700\u8981\u5c07\u5f9e\u76ee\u6a19\u7d22\u5f15\u5230 <code>base</code> \u4e4b\u9593\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\uff0c\u7136\u5f8c\u5c07 <code>base</code> \u8ce6\u503c\u7d66\u76ee\u6a19\u7d22\u5f15\u3002</p> <p></p> <p> \u5716 11-6 \u00a0 \u55ae\u6b21\u63d2\u5165\u64cd\u4f5c </p>"},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u6574\u9ad4\u6d41\u7a0b\u5982\u5716 11-7 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72c0\u614b\u4e0b\uff0c\u9663\u5217\u7684\u7b2c 1 \u500b\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u9078\u53d6\u9663\u5217\u7684\u7b2c 2 \u500b\u5143\u7d20\u4f5c\u70ba <code>base</code> \uff0c\u5c07\u5176\u63d2\u5165\u5230\u6b63\u78ba\u4f4d\u7f6e\u5f8c\uff0c\u9663\u5217\u7684\u524d 2 \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9078\u53d6\u7b2c 3 \u500b\u5143\u7d20\u4f5c\u70ba <code>base</code> \uff0c\u5c07\u5176\u63d2\u5165\u5230\u6b63\u78ba\u4f4d\u7f6e\u5f8c\uff0c\u9663\u5217\u7684\u524d 3 \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u985e\u63a8\uff0c\u5728\u6700\u5f8c\u4e00\u8f2a\u4e2d\uff0c\u9078\u53d6\u6700\u5f8c\u4e00\u500b\u5143\u7d20\u4f5c\u70ba <code>base</code> \uff0c\u5c07\u5176\u63d2\u5165\u5230\u6b63\u78ba\u4f4d\u7f6e\u5f8c\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u5716 11-7 \u00a0 \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n    # \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j -= 1\n        nums[j + 1] = base  # \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n</code></pre> insertion_sort.cpp<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base;        // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid InsertionSort(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = bas;         // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--\n        }\n        nums[j+1] = base // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for i in nums.indices.dropFirst() {\n        let base = nums[i]\n        var j = i - 1\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j -= 1\n        }\n        nums[j + 1] = base // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n      j--;\n    }\n    nums[j + 1] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i], (i - 1) as i32);\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.kt<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfun insertionSort(nums: IntArray) {\n    //\u5916\u8ff4\u5708: \u5df2\u6392\u5e8f\u5143\u7d20\u70ba 1, 2, ..., n\n    for (i in nums.indices) {\n        val base = nums[i]\n        var j = i - 1\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j] // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--\n        }\n        nums[j + 1] = base        // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.rb<pre><code>### \u63d2\u5165\u6392\u5e8f ###\ndef insertion_sort(nums)\n  n = nums.length\n  # \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n  for i in 1...n\n    base = nums[i]\n    j = i - 1\n    # \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n    while j &gt;= 0 &amp;&amp; nums[j] &gt; base\n      nums[j + 1] = nums[j] # \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n      j -= 1\n    end\n    nums[j + 1] = base # \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n  end\nend\n</code></pre> insertion_sort.zig<pre><code>// \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    var i: usize = 1;\n    while (i &lt; nums.len) : (i += 1) {\n        var base = nums[i];\n        var j: usize = i;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 1 and nums[j - 1] &gt; base) : (j -= 1) {\n            nums[j] = nums[j - 1];  // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n        }\n        nums[j] = base;             // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\)\u3001\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u5225\u9700\u8981\u8ff4\u5708 \\(n - 1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\((n - 1) n / 2\\) \uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u8cc7\u6599\u6642\uff0c\u63d2\u5165\u64cd\u4f5c\u6703\u63d0\u524d\u7d42\u6b62\u3002\u7576\u8f38\u5165\u9663\u5217\u5b8c\u5168\u6709\u5e8f\u6642\uff0c\u63d2\u5165\u6392\u5e8f\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u904e\u7a0b\u4e2d\uff0c\u6211\u5011\u6703\u5c07\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u5074\uff0c\u4e0d\u6703\u6539\u8b8a\u5b83\u5011\u7684\u9806\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/#1143","title":"11.4.3 \u00a0 \u63d2\u5165\u6392\u5e8f\u7684\u512a\u52e2","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u800c\u6211\u5011\u5373\u5c07\u5b78\u7fd2\u7684\u5feb\u901f\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002\u5118\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u66f4\u9ad8\uff0c\u4f46\u5728\u8cc7\u6599\u91cf\u8f03\u5c0f\u7684\u60c5\u6cc1\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002</p> <p>\u9019\u500b\u7d50\u8ad6\u8207\u7dda\u6027\u67e5\u8a62\u548c\u4e8c\u5206\u641c\u5c0b\u7684\u9069\u7528\u60c5\u6cc1\u7684\u7d50\u8ad6\u985e\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u9019\u985e \\(O(n \\log n)\\) \u7684\u6f14\u7b97\u6cd5\u5c6c\u65bc\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u55ae\u5143\u8a08\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u8cc7\u6599\u91cf\u8f03\u5c0f\u6642\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6578\u503c\u6bd4\u8f03\u63a5\u8fd1\uff0c\u8907\u96dc\u5ea6\u4e0d\u4f54\u4e3b\u5c0e\u5730\u4f4d\uff0c\u6bcf\u8f2a\u4e2d\u7684\u55ae\u5143\u64cd\u4f5c\u6578\u91cf\u8d77\u5230\u6c7a\u5b9a\u6027\u4f5c\u7528\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\u63a1\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u70ba\uff1a\u5c0d\u65bc\u9577\u9663\u5217\uff0c\u63a1\u7528\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5c0d\u65bc\u77ed\u9663\u5217\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002</p> <p>\u96d6\u7136\u6ce1\u6cab\u6392\u5e8f\u3001\u9078\u64c7\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n^2)\\) \uff0c\u4f46\u5728\u5be6\u969b\u60c5\u6cc1\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u983b\u7387\u986f\u8457\u9ad8\u65bc\u6ce1\u6cab\u6392\u5e8f\u548c\u9078\u64c7\u6392\u5e8f\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u6ce1\u6cab\u6392\u5e8f\u57fa\u65bc\u5143\u7d20\u4ea4\u63db\u5be6\u73fe\uff0c\u9700\u8981\u85c9\u52a9\u4e00\u500b\u81e8\u6642\u8b8a\u6578\uff0c\u5171\u6d89\u53ca 3 \u500b\u55ae\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u65bc\u5143\u7d20\u8ce6\u503c\u5be6\u73fe\uff0c\u50c5\u9700 1 \u500b\u55ae\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u6ce1\u6cab\u6392\u5e8f\u7684\u8a08\u7b97\u958b\u92b7\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002</li> <li>\u9078\u64c7\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u6cc1\u4e0b\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n^2)\\) \u3002\u5982\u679c\u7d66\u5b9a\u4e00\u7d44\u90e8\u5206\u6709\u5e8f\u7684\u8cc7\u6599\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9078\u64c7\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u9078\u64c7\u6392\u5e8f\u4e0d\u7a69\u5b9a\uff0c\u7121\u6cd5\u61c9\u7528\u65bc\u591a\u7d1a\u6392\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/","title":"11.6 \u00a0 \u5408\u4f75\u6392\u5e8f","text":"<p>\u5408\u4f75\u6392\u5e8f\uff08merge sort\uff09\u662f\u4e00\u7a2e\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u5305\u542b\u5716 11-10 \u6240\u793a\u7684\u201c\u5283\u5206\u201d\u548c\u201c\u5408\u4f75\u201d\u968e\u6bb5\u3002</p> <ol> <li>\u5283\u5206\u968e\u6bb5\uff1a\u900f\u904e\u905e\u8ff4\u4e0d\u65b7\u5730\u5c07\u9663\u5217\u5f9e\u4e2d\u9ede\u8655\u5206\u958b\uff0c\u5c07\u9577\u9663\u5217\u7684\u6392\u5e8f\u554f\u984c\u8f49\u63db\u70ba\u77ed\u9663\u5217\u7684\u6392\u5e8f\u554f\u984c\u3002</li> <li>\u5408\u4f75\u968e\u6bb5\uff1a\u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u5283\u5206\uff0c\u958b\u59cb\u5408\u4f75\uff0c\u6301\u7e8c\u5730\u5c07\u5de6\u53f3\u5169\u500b\u8f03\u77ed\u7684\u6709\u5e8f\u9663\u5217\u5408\u4f75\u70ba\u4e00\u500b\u8f03\u9577\u7684\u6709\u5e8f\u9663\u5217\uff0c\u76f4\u81f3\u7d50\u675f\u3002</li> </ol> <p></p> <p> \u5716 11-10 \u00a0 \u5408\u4f75\u6392\u5e8f\u7684\u5283\u5206\u8207\u5408\u4f75\u968e\u6bb5 </p>"},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5982\u5716 11-11 \u6240\u793a\uff0c\u201c\u5283\u5206\u968e\u6bb5\u201d\u5f9e\u9802\u81f3\u5e95\u905e\u8ff4\u5730\u5c07\u9663\u5217\u5f9e\u4e2d\u9ede\u5207\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\u3002</p> <ol> <li>\u8a08\u7b97\u9663\u5217\u4e2d\u9ede <code>mid</code> \uff0c\u905e\u8ff4\u5283\u5206\u5de6\u5b50\u9663\u5217\uff08\u5340\u9593 <code>[left, mid]</code> \uff09\u548c\u53f3\u5b50\u9663\u5217\uff08\u5340\u9593 <code>[mid + 1, right]</code> \uff09\u3002</li> <li>\u905e\u8ff4\u57f7\u884c\u6b65\u9a5f <code>1.</code> \uff0c\u76f4\u81f3\u5b50\u9663\u5217\u5340\u9593\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u3002</li> </ol> <p>\u201c\u5408\u4f75\u968e\u6bb5\u201d\u5f9e\u5e95\u81f3\u9802\u5730\u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u5408\u4f75\u70ba\u4e00\u500b\u6709\u5e8f\u9663\u5217\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5f9e\u9577\u5ea6\u70ba 1 \u7684\u5b50\u9663\u5217\u958b\u59cb\u5408\u4f75\uff0c\u5408\u4f75\u968e\u6bb5\u4e2d\u7684\u6bcf\u500b\u5b50\u9663\u5217\u90fd\u662f\u6709\u5e8f\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-11 \u00a0 \u5408\u4f75\u6392\u5e8f\u6b65\u9a5f </p> <p>\u89c0\u5bdf\u767c\u73fe\uff0c\u5408\u4f75\u6392\u5e8f\u8207\u4e8c\u5143\u6a39\u5f8c\u5e8f\u8d70\u8a2a\u7684\u905e\u8ff4\u9806\u5e8f\u662f\u4e00\u81f4\u7684\u3002</p> <ul> <li>\u5f8c\u5e8f\u8d70\u8a2a\uff1a\u5148\u905e\u8ff4\u5de6\u5b50\u6a39\uff0c\u518d\u905e\u8ff4\u53f3\u5b50\u6a39\uff0c\u6700\u5f8c\u8655\u7406\u6839\u7bc0\u9ede\u3002</li> <li>\u5408\u4f75\u6392\u5e8f\uff1a\u5148\u905e\u8ff4\u5de6\u5b50\u9663\u5217\uff0c\u518d\u905e\u8ff4\u53f3\u5b50\u9663\u5217\uff0c\u6700\u5f8c\u8655\u7406\u5408\u4f75\u3002</li> </ul> <p>\u5408\u4f75\u6392\u5e8f\u7684\u5be6\u73fe\u5982\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\u3002\u8acb\u6ce8\u610f\uff0c<code>nums</code> \u7684\u5f85\u5408\u4f75\u5340\u9593\u70ba <code>[left, right]</code> \uff0c\u800c <code>tmp</code> \u7684\u5c0d\u61c9\u5340\u9593\u70ba <code>[0, right - left]</code> \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"\u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\"\"\"\n    # \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    # \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    tmp = [0] * (right - left + 1)\n    # \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k = left, mid + 1, 0\n    # \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"\u5408\u4f75\u6392\u5e8f\"\"\"\n    # \u7d42\u6b62\u689d\u4ef6\n    if left &gt;= right:\n        return  # \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    # \u5283\u5206\u968e\u6bb5\n    mid = (left + right) // 2 # \u8a08\u7b97\u4e2d\u9ede\n    merge_sort(nums, left, mid)  # \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    merge_sort(nums, mid + 1, right)  # \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    # \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    vector&lt;int&gt; tmp(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right)\n        return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    int mid = left + (right - left) / 2;    // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid);      // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right)\n        return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    int mid = left + (right - left) / 2; // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid MergeSort(int[] nums, int left, int right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return;       // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    int mid = left + (right - left) / 2;    // \u8a08\u7b97\u4e2d\u9ede\n    MergeSort(nums, left, mid);      // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    MergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfunc merge(nums []int, left, mid, right int) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    tmp := make([]int, right-left+1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k := left, mid+1, 0\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if left &gt;= right {\n        return\n    }\n    // \u5283\u5206\u968e\u6bb5\n    mid := left + (right - left) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left, j = mid + 1, k = 0\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n        }\n        k += 1\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if left &gt;= right { // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n        return\n    }\n    // \u5283\u5206\u968e\u6bb5\n    let mid = left + (right - left) / 2 // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums: &amp;nums, left: left, right: mid) // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfunction merge(nums, left, mid, right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    let mid = Math.floor(left + (right - left) / 2); // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    let mid = Math.floor(left + (right - left) / 2); // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n  // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n  int i = left, j = mid + 1, k = 0;\n  // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // \u7d42\u6b62\u689d\u4ef6\n  if (left &gt;= right) return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n  // \u5283\u5206\u968e\u6bb5\n  int mid = left + (right - left) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\n  mergeSort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n  mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n  // \u5408\u4f75\u968e\u6bb5\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if left &gt;= right {\n        return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    }\n\n    // \u5283\u5206\u968e\u6bb5\n    let mid = left + (right - left) / 2; // \u8a08\u7b97\u4e2d\u9ede\n    merge_sort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    merge_sort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid merge(int *nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(tmp);\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right)\n        return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    int mid = left + (right - left) / 2;    // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid);      // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.kt<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfun merge(nums: IntArray, left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    val tmp = IntArray(right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left\n    var j = mid + 1\n    var k = 0\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++]\n        else\n            tmp[k++] = nums[j++]\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++]\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++]\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (l in tmp.indices) {\n        nums[left + l] = tmp[l]\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfun mergeSort(nums: IntArray, left: Int, right: Int) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return  // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    val mid = left + (right - left) / 2 // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid) // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right) // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.rb<pre><code>### \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 ###\ndef merge(nums, left, mid, right)\n  # \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n  # \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp\uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n  tmp = Array.new(right - left + 1, 0)\n  # \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n  i, j, k = left, mid + 1, 0\n  # \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n  while i &lt;= mid &amp;&amp; j &lt;= right\n    if nums[i] &lt;= nums[j]\n      tmp[k] = nums[i]\n      i += 1\n    else\n      tmp[k] = nums[j]\n      j += 1\n    end\n    k += 1\n  end\n  # \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n  while i &lt;= mid\n    tmp[k] = nums[i]\n    i += 1\n    k += 1\n  end\n  while j &lt;= right\n    tmp[k] = nums[j]\n    j += 1\n    k += 1\n  end\n  # \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n  (0...tmp.length).each do |k|\n    nums[left + k] = tmp[k]\n  end\nend\n\n### \u5408\u4f75\u6392\u5e8f ###\ndef merge_sort(nums, left, right)\n  # \u7d42\u6b62\u689d\u4ef6\n  # \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n  return if left &gt;= right\n  # \u5283\u5206\u968e\u6bb5\n  mid = left + (right - left) / 2 # \u8a08\u7b97\u4e2d\u9ede\n  merge_sort(nums, left, mid) # \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n  merge_sort(nums, mid + 1, right) # \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n  # \u5408\u4f75\u968e\u6bb5\n  merge(nums, left, mid, right)\nend\n</code></pre> merge_sort.zig<pre><code>// \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\n// \u5de6\u5b50\u9663\u5217\u5340\u9593 [left, mid]\n// \u53f3\u5b50\u9663\u5217\u5340\u9593 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n    // \u521d\u59cb\u5316\u8f14\u52a9\u9663\u5217\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var tmp = try mem_allocator.alloc(i32, right + 1 - left);\n    std.mem.copy(i32, tmp, nums[left..right+1]);\n    // \u5de6\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7d50\u675f\u7d22\u5f15\n    var leftStart = left - left;\n    var leftEnd = mid - left;\n    // \u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7d50\u675f\u7d22\u5f15\n    var rightStart = mid + 1 - left;\n    var rightEnd = right - left;\n    // i, j \u5206\u5225\u6307\u5411\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\u7684\u9996\u5143\u7d20\n    var i = leftStart;\n    var j = rightStart;\n    // \u900f\u904e\u8986\u84cb\u539f\u9663\u5217 nums \u4f86\u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\n    var k = left;\n    while (k &lt;= right) : (k += 1) {\n        // \u82e5\u201c\u5de6\u5b50\u9663\u5217\u5df2\u5168\u90e8\u5408\u4f75\u5b8c\u201d\uff0c\u5247\u9078\u53d6\u53f3\u5b50\u9663\u5217\u5143\u7d20\uff0c\u4e26\u4e14 j++\n        if (i &gt; leftEnd) {\n            nums[k] = tmp[j];\n            j += 1;\n        // \u5426\u5247\uff0c\u82e5\u201c\u53f3\u5b50\u9663\u5217\u5df2\u5168\u90e8\u5408\u4f75\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u9663\u5217\u5143\u7d20 &lt;= \u53f3\u5b50\u9663\u5217\u5143\u7d20\u201d\uff0c\u5247\u9078\u53d6\u5de6\u5b50\u9663\u5217\u5143\u7d20\uff0c\u4e26\u4e14 i++\n        } else if  (j &gt; rightEnd or tmp[i] &lt;= tmp[j]) {\n            nums[k] = tmp[i];\n            i += 1;\n        // \u5426\u5247\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u672a\u5168\u90e8\u5408\u4f75\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u9663\u5217\u5143\u7d20 &gt; \u53f3\u5b50\u9663\u5217\u5143\u7d20\u201d\uff0c\u5247\u9078\u53d6\u53f3\u5b50\u9663\u5217\u5143\u7d20\uff0c\u4e26\u4e14 j++\n        } else {\n            nums[k] = tmp[j];\n            j += 1;\n        }\n    }\n}\n\n// \u5408\u4f75\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return;              // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    var mid = left + (right - left) / 2;    // \u8a08\u7b97\u4e2d\u9ede\n    try mergeSort(nums, left, mid);         // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    try mergeSort(nums, mid + 1, right);    // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    try merge(nums, left, mid, right);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5283\u5206\u7522\u751f\u9ad8\u5ea6\u70ba \\(\\log n\\) \u7684\u905e\u8ff4\u6a39\uff0c\u6bcf\u5c64\u5408\u4f75\u7684\u7e3d\u64cd\u4f5c\u6578\u91cf\u70ba \\(n\\) \uff0c\u56e0\u6b64\u7e3d\u9ad4\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u905e\u8ff4\u6df1\u5ea6\u70ba \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002\u5408\u4f75\u64cd\u4f5c\u9700\u8981\u85c9\u52a9\u8f14\u52a9\u9663\u5217\u5be6\u73fe\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u4f75\u904e\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u8b8a\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/#1163","title":"11.6.3 \u00a0 \u93c8\u7d50\u4e32\u5217\u6392\u5e8f","text":"<p>\u5c0d\u65bc\u93c8\u7d50\u4e32\u5217\uff0c\u5408\u4f75\u6392\u5e8f\u76f8\u8f03\u65bc\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\u5177\u6709\u986f\u8457\u512a\u52e2\uff0c\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u6392\u5e8f\u4efb\u52d9\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(1)\\) \u3002</p> <ul> <li>\u5283\u5206\u968e\u6bb5\uff1a\u53ef\u4ee5\u4f7f\u7528\u201c\u8fed\u4ee3\u201d\u66ff\u4ee3\u201c\u905e\u8ff4\u201d\u4f86\u5be6\u73fe\u93c8\u7d50\u4e32\u5217\u5283\u5206\u5de5\u4f5c\uff0c\u5f9e\u800c\u7701\u53bb\u905e\u8ff4\u4f7f\u7528\u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002</li> <li>\u5408\u4f75\u968e\u6bb5\uff1a\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\uff0c\u7bc0\u9ede\u589e\u522a\u64cd\u4f5c\u50c5\u9700\u6539\u8b8a\u5f15\u7528\uff08\u6307\u6a19\uff09\u5373\u53ef\u5be6\u73fe\uff0c\u56e0\u6b64\u5408\u4f75\u968e\u6bb5\uff08\u5c07\u5169\u500b\u77ed\u6709\u5e8f\u93c8\u7d50\u4e32\u5217\u5408\u4f75\u70ba\u4e00\u500b\u9577\u6709\u5e8f\u93c8\u7d50\u4e32\u5217\uff09\u7121\u9808\u5efa\u7acb\u984d\u5916\u93c8\u7d50\u4e32\u5217\u3002</li> </ul> <p>\u5177\u9ad4\u5be6\u73fe\u7d30\u7bc0\u6bd4\u8f03\u8907\u96dc\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u67e5\u95b1\u76f8\u95dc\u8cc7\u6599\u9032\u884c\u5b78\u7fd2\u3002</p>"},{"location":"chapter_sorting/quick_sort/","title":"11.5 \u00a0 \u5feb\u901f\u6392\u5e8f","text":"<p>\u5feb\u901f\u6392\u5e8f\uff08quick sort\uff09\u662f\u4e00\u7a2e\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u57f7\u884c\u9ad8\u6548\uff0c\u61c9\u7528\u5ee3\u6cdb\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u201c\u54e8\u5175\u5283\u5206\u201d\uff0c\u5176\u76ee\u6a19\u662f\uff1a\u9078\u64c7\u9663\u5217\u4e2d\u7684\u67d0\u500b\u5143\u7d20\u4f5c\u70ba\u201c\u57fa\u6e96\u6578\u201d\uff0c\u5c07\u6240\u6709\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u5074\uff0c\u800c\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u5074\u3002\u5177\u9ad4\u4f86\u8aaa\uff0c\u54e8\u5175\u5283\u5206\u7684\u6d41\u7a0b\u5982\u5716 11-8 \u6240\u793a\u3002</p> <ol> <li>\u9078\u53d6\u9663\u5217\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u70ba\u57fa\u6e96\u6578\uff0c\u521d\u59cb\u5316\u5169\u500b\u6307\u6a19 <code>i</code> \u548c <code>j</code> \u5206\u5225\u6307\u5411\u9663\u5217\u7684\u5169\u7aef\u3002</li> <li>\u8a2d\u5b9a\u4e00\u500b\u8ff4\u5708\uff0c\u5728\u6bcf\u8f2a\u4e2d\u4f7f\u7528 <code>i</code>\uff08<code>j</code>\uff09\u5206\u5225\u5c0b\u627e\u7b2c\u4e00\u500b\u6bd4\u57fa\u6e96\u6578\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u5f8c\u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\u3002</li> <li>\u8ff4\u5708\u57f7\u884c\u6b65\u9a5f <code>2.</code> \uff0c\u76f4\u5230 <code>i</code> \u548c <code>j</code> \u76f8\u9047\u6642\u505c\u6b62\uff0c\u6700\u5f8c\u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u500b\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-8 \u00a0 \u54e8\u5175\u5283\u5206\u6b65\u9a5f </p> <p>\u54e8\u5175\u5283\u5206\u5b8c\u6210\u5f8c\uff0c\u539f\u9663\u5217\u88ab\u5283\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u9663\u5217\u3001\u57fa\u6e96\u6578\u3001\u53f3\u5b50\u9663\u5217\uff0c\u4e14\u6eff\u8db3\u201c\u5de6\u5b50\u9663\u5217\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u6e96\u6578 \\(\\leq\\) \u53f3\u5b50\u9663\u5217\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u5011\u63a5\u4e0b\u4f86\u53ea\u9700\u5c0d\u9019\u5169\u500b\u5b50\u9663\u5217\u9032\u884c\u6392\u5e8f\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565</p> <p>\u54e8\u5175\u5283\u5206\u7684\u5be6\u8cea\u662f\u5c07\u4e00\u500b\u8f03\u9577\u9663\u5217\u7684\u6392\u5e8f\u554f\u984c\u7c21\u5316\u70ba\u5169\u500b\u8f03\u77ed\u9663\u5217\u7684\u6392\u5e8f\u554f\u984c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5283\u5206\"\"\"\n    # \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u63db\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u54e8\u5175\u5283\u5206 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums[i], nums[j]); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums[i], nums[left]);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;                   // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nint partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;             // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nint Partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;             // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u54e8\u5175\u5283\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u63db\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u54e8\u5175\u5283\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        nums.swapAt(i, j) // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    nums.swapAt(i, left) // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.js<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\npartition(nums, left, right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u63db\n        this.swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\npartition(nums: number[], left: number, right: number): number {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u63db\n        this.swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n  return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u54e8\u5175\u5283\u5206 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    nums.swap(i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    i // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nint partition(int nums[], int left, int right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n        swap(nums, i, j);\n    }\n    // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    swap(nums, i, left);\n    // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n    return i;\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nfun swap(nums: IntArray, i: Int, j: Int) {\n    val temp = nums[i]\n    nums[i] = nums[j]\n    nums[j] = temp\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j)  // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left)   // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i              // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rb<pre><code>### \u54e8\u5175\u5283\u5206 ###\ndef partition(nums, left, right)\n  # \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    end\n    # \u5143\u7d20\u4ea4\u63db\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n  nums[i], nums[left] = nums[left], nums[i]\n  i # \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\nend\n</code></pre> quick_sort.zig<pre><code>// \u5143\u7d20\u4ea4\u63db\nfn swap(nums: []i32, i: usize, j: usize) void {\n    var tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n// \u54e8\u5175\u5283\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;               // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5feb\u901f\u6392\u5e8f\u7684\u6574\u9ad4\u6d41\u7a0b\u5982\u5716 11-9 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5c0d\u539f\u9663\u5217\u57f7\u884c\u4e00\u6b21\u201c\u54e8\u5175\u5283\u5206\u201d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u3002</li> <li>\u7136\u5f8c\uff0c\u5c0d\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u5206\u5225\u905e\u8ff4\u57f7\u884c\u201c\u54e8\u5175\u5283\u5206\u201d\u3002</li> <li>\u6301\u7e8c\u905e\u8ff4\uff0c\u76f4\u81f3\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\uff0c\u5f9e\u800c\u5b8c\u6210\u6574\u500b\u9663\u5217\u7684\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u5716 11-9 \u00a0 \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n    # \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if left &gt;= right:\n        return\n    # \u54e8\u5175\u5283\u5206\n    pivot = self.partition(nums, left, right)\n    # \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5283\u5206\n    int pivot = partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5283\u5206\n    int pivot = partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5283\u5206\n    int pivot = Partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5283\u5206\n    pivot := q.partition(nums, left, right)\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5283\u5206\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5283\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5283\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n  if (left &gt;= right) return;\n  // \u54e8\u5175\u5283\u5206\n  int pivot = _partition(nums, left, right);\n  // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if left &gt;= right {\n        return;\n    }\n    // \u54e8\u5175\u5283\u5206\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int nums[], int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5283\u5206\n    int pivot = partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) return\n    // \u54e8\u5175\u5283\u5206\n    val pivot = partition(nums, left, right)\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>### \u5feb\u901f\u6392\u5e8f\u985e\u5225 ###\ndef quick_sort(nums, left, right)\n  # \u5b50\u9663\u5217\u9577\u5ea6\u4e0d\u70ba 1 \u6642\u905e\u8ff4\n  if left &lt; right\n    # \u54e8\u5175\u5283\u5206\n    pivot = partition(nums, left, right)\n    # \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quick_sort(nums, left, pivot - 1)\n    quick_sort(nums, pivot + 1, right)\n  end\n  nums\nend\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5283\u5206\n    var pivot = partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5728\u5e73\u5747\u60c5\u6cc1\u4e0b\uff0c\u54e8\u5175\u5283\u5206\u7684\u905e\u8ff4\u5c64\u6578\u70ba \\(\\log n\\) \uff0c\u6bcf\u5c64\u4e2d\u7684\u7e3d\u8ff4\u5708\u6578\u70ba \\(n\\) \uff0c\u7e3d\u9ad4\u4f7f\u7528 \\(O(n \\log n)\\) \u6642\u9593\u3002\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u6bcf\u8f2a\u54e8\u5175\u5283\u5206\u64cd\u4f5c\u90fd\u5c07\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217\u5283\u5206\u70ba\u9577\u5ea6\u70ba \\(0\\) \u548c \\(n - 1\\) \u7684\u5169\u500b\u5b50\u9663\u5217\uff0c\u6b64\u6642\u905e\u8ff4\u5c64\u6578\u9054\u5230 \\(n\\) \uff0c\u6bcf\u5c64\u4e2d\u7684\u8ff4\u5708\u6578\u70ba \\(n\\) \uff0c\u7e3d\u9ad4\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u5728\u8f38\u5165\u9663\u5217\u5b8c\u5168\u5012\u5e8f\u7684\u60c5\u6cc1\u4e0b\uff0c\u9054\u5230\u6700\u5dee\u905e\u8ff4\u6df1\u5ea6 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002\u6392\u5e8f\u64cd\u4f5c\u662f\u5728\u539f\u9663\u5217\u4e0a\u9032\u884c\u7684\uff0c\u672a\u85c9\u52a9\u984d\u5916\u9663\u5217\u3002</li> <li>\u975e\u7a69\u5b9a\u6392\u5e8f\uff1a\u5728\u54e8\u5175\u5283\u5206\u7684\u6700\u5f8c\u4e00\u6b65\uff0c\u57fa\u6e96\u6578\u53ef\u80fd\u6703\u88ab\u4ea4\u63db\u81f3\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u5074\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1153","title":"11.5.3 \u00a0 \u5feb\u901f\u6392\u5e8f\u70ba\u4ec0\u9ebc\u5feb","text":"<p>\u5f9e\u540d\u7a31\u4e0a\u5c31\u80fd\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u61c9\u8a72\u5177\u6709\u4e00\u5b9a\u7684\u512a\u52e2\u3002\u5118\u7ba1\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u8207\u201c\u5408\u4f75\u6392\u5e8f\u201d\u548c\u201c\u5806\u7a4d\u6392\u5e8f\u201d\u76f8\u540c\uff0c\u4f46\u901a\u5e38\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u51fa\u73fe\u6700\u5dee\u60c5\u6cc1\u7684\u6a5f\u7387\u5f88\u4f4e\uff1a\u96d6\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u6c92\u6709\u5408\u4f75\u6392\u5e8f\u7a69\u5b9a\uff0c\u4f46\u5728\u7d55\u5927\u591a\u6578\u60c5\u6cc1\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u80fd\u5728 \\(O(n \\log n)\\) \u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e0b\u57f7\u884c\u3002</li> <li>\u5feb\u53d6\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u5728\u57f7\u884c\u54e8\u5175\u5283\u5206\u64cd\u4f5c\u6642\uff0c\u7cfb\u7d71\u53ef\u5c07\u6574\u500b\u5b50\u9663\u5217\u8f09\u5165\u5230\u5feb\u53d6\uff0c\u56e0\u6b64\u8a2a\u554f\u5143\u7d20\u7684\u6548\u7387\u8f03\u9ad8\u3002\u800c\u50cf\u201c\u5806\u7a4d\u6392\u5e8f\u201d\u9019\u985e\u6f14\u7b97\u6cd5\u9700\u8981\u8df3\u8e8d\u5f0f\u8a2a\u554f\u5143\u7d20\uff0c\u5f9e\u800c\u7f3a\u4e4f\u9019\u4e00\u7279\u6027\u3002</li> <li>\u8907\u96dc\u5ea6\u7684\u5e38\u6578\u4fc2\u6578\u5c0f\uff1a\u5728\u4e0a\u8ff0\u4e09\u7a2e\u6f14\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6bd4\u8f03\u3001\u8ce6\u503c\u3001\u4ea4\u63db\u7b49\u64cd\u4f5c\u7684\u7e3d\u6578\u91cf\u6700\u5c11\u3002\u9019\u8207\u201c\u63d2\u5165\u6392\u5e8f\u201d\u6bd4\u201c\u6ce1\u6cab\u6392\u5e8f\u201d\u66f4\u5feb\u7684\u539f\u56e0\u985e\u4f3c\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1154","title":"11.5.4 \u00a0 \u57fa\u6e96\u6578\u6700\u4f73\u5316","text":"<p>\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f38\u5165\u4e0b\u7684\u6642\u9593\u6548\u7387\u53ef\u80fd\u964d\u4f4e\u3002\u8209\u4e00\u500b\u6975\u7aef\u4f8b\u5b50\uff0c\u5047\u8a2d\u8f38\u5165\u9663\u5217\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u65bc\u6211\u5011\u9078\u64c7\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u70ba\u57fa\u6e96\u6578\uff0c\u90a3\u9ebc\u5728\u54e8\u5175\u5283\u5206\u5b8c\u6210\u5f8c\uff0c\u57fa\u6e96\u6578\u88ab\u4ea4\u63db\u81f3\u9663\u5217\u6700\u53f3\u7aef\uff0c\u5c0e\u81f4\u5de6\u5b50\u9663\u5217\u9577\u5ea6\u70ba \\(n - 1\\)\u3001\u53f3\u5b50\u9663\u5217\u9577\u5ea6\u70ba \\(0\\) \u3002\u5982\u6b64\u905e\u8ff4\u4e0b\u53bb\uff0c\u6bcf\u8f2a\u54e8\u5175\u5283\u5206\u5f8c\u90fd\u6709\u4e00\u500b\u5b50\u9663\u5217\u7684\u9577\u5ea6\u70ba \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u70ba\u201c\u6ce1\u6cab\u6392\u5e8f\u201d\u7684\u8fd1\u4f3c\u5f62\u5f0f\u3002</p> <p>\u70ba\u4e86\u5118\u91cf\u907f\u514d\u9019\u7a2e\u60c5\u6cc1\u767c\u751f\uff0c\u6211\u5011\u53ef\u4ee5\u6700\u4f73\u5316\u54e8\u5175\u5283\u5206\u4e2d\u7684\u57fa\u6e96\u6578\u7684\u9078\u53d6\u7b56\u7565\u3002\u4f8b\u5982\uff0c\u6211\u5011\u53ef\u4ee5\u96a8\u6a5f\u9078\u53d6\u4e00\u500b\u5143\u7d20\u4f5c\u70ba\u57fa\u6e96\u6578\u3002\u7136\u800c\uff0c\u5982\u679c\u904b\u6c23\u4e0d\u4f73\uff0c\u6bcf\u6b21\u90fd\u9078\u5230\u4e0d\u7406\u60f3\u7684\u57fa\u6e96\u6578\uff0c\u6548\u7387\u4ecd\u7136\u4e0d\u76e1\u5982\u4eba\u610f\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7a0b\u5f0f\u8a9e\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u507d\u96a8\u6a5f\u6578\u201d\u3002\u5982\u679c\u6211\u5011\u91dd\u5c0d\u507d\u96a8\u6a5f\u6578\u5e8f\u5217\u69cb\u5efa\u4e00\u500b\u7279\u5b9a\u7684\u6e2c\u8a66\u6a23\u4f8b\uff0c\u90a3\u9ebc\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002</p> <p>\u70ba\u4e86\u9032\u4e00\u6b65\u6539\u9032\uff0c\u6211\u5011\u53ef\u4ee5\u5728\u9663\u5217\u4e2d\u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\uff08\u901a\u5e38\u70ba\u9663\u5217\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u9ede\u5143\u7d20\uff09\uff0c\u4e26\u5c07\u9019\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\u4f5c\u70ba\u57fa\u6e96\u6578\u3002\u9019\u6a23\u4e00\u4f86\uff0c\u57fa\u6e96\u6578\u201c\u65e2\u4e0d\u592a\u5c0f\u4e5f\u4e0d\u592a\u5927\u201d\u7684\u6a5f\u7387\u5c07\u5927\u5e45\u63d0\u5347\u3002\u7576\u7136\uff0c\u6211\u5011\u9084\u53ef\u4ee5\u9078\u53d6\u66f4\u591a\u5019\u9078\u5143\u7d20\uff0c\u4ee5\u9032\u4e00\u6b65\u63d0\u9ad8\u6f14\u7b97\u6cd5\u7684\u7a69\u5065\u6027\u3002\u63a1\u7528\u9019\u7a2e\u65b9\u6cd5\u5f8c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6a5f\u7387\u5927\u5927\u964d\u4f4e\u3002</p> <p>\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"\u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\"\"\"\n    l, m, r = nums[left], nums[mid], nums[right]\n    if (l &lt;= m &lt;= r) or (r &lt;= m &lt;= l):\n        return mid  # m \u5728 l \u548c r \u4e4b\u9593\n    if (m &lt;= l &lt;= r) or (r &lt;= l &lt;= m):\n        return left  # l \u5728 m \u548c r \u4e4b\u9593\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09\"\"\"\n    # \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    # \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u63db\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums[left], nums[med]);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums[i], nums[j]); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums[i], nums[left]);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;                   // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint medianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;             // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint Partition(int[] nums, int left, int right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    Swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;             // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    l, m, r := nums[left], nums[mid], nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m \u5728 l \u548c r \u4e4b\u9593\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l \u5728 m \u548c r \u4e4b\u9593\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- //\u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ //\u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        //\u5143\u7d20\u4ea4\u63db\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    //\u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    nums[i], nums[left] = nums[left], nums[i]\n    return i //\u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    let l = nums[left]\n    let m = nums[mid]\n    let r = nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m \u5728 l \u548c r \u4e4b\u9593\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l \u5728 m \u548c r \u4e4b\u9593\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    let med = medianThree(nums: nums, left: left, mid: left + (right - left) / 2, right: right)\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    nums.swapAt(left, med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nmedianThree(nums, left, mid, right) {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m \u5728 l \u548c r \u4e4b\u9593\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l \u5728 m \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        this.swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m \u5728 l \u548c r \u4e4b\u9593\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l \u5728 m \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        this.swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  int l = nums[left], m = nums[mid], r = nums[right];\n  if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n    return mid; // m \u5728 l \u548c r \u4e4b\u9593\n  if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n    return left; // l \u5728 m \u548c r \u4e4b\u9593\n  return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n  _swap(nums, left, med);\n  // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n  return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    let (l, m, r) = (nums[left], nums[mid], nums[right]);\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    }\n    right\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    nums.swap(left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    nums.swap(i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    i // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint medianThree(int nums[], int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint partitionMedian(int nums[], int left, int right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;            // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.kt<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nfun medianThree(nums: IntArray, left: Int, mid: Int, right: Int): Int {\n    val l = nums[left]\n    val m = nums[mid]\n    val r = nums[right]\n    if ((m in l..r) || (m in r..l))\n        return mid  // m \u5728 l \u548c r \u4e4b\u9593\n    if ((l in m..r) || (l in r..m))\n        return left // l \u5728 m \u548c r \u4e4b\u9593\n    return right\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nfun partitionMedian(nums: IntArray, left: Int, right: Int): Int {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    val med = medianThree(nums, left, (left + right) / 2, right)\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums, left, med)\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--                      // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++                      // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j)             // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left)              // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i                         // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rb<pre><code>### \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 ###\ndef median_three(nums, left, mid, right)\n  # \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n  _l, _m, _r = nums[left], nums[mid], nums[right]\n  # m \u5728 l \u548c r \u4e4b\u9593\n  return mid if (_l &lt;= _m &amp;&amp; _m &lt;= _r) || (_r &lt;= _m &amp;&amp; _m &lt;= _l)\n  # l \u5728 m \u548c r \u4e4b\u9593\n  return left if (_m &lt;= _l &amp;&amp; _l &lt;= _r) || (_r &lt;= _l &amp;&amp; _l &lt;= _m)\n  return right\nend\n\n### \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09###\ndef partition(nums, left, right)\n  ### \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n  med = median_three(nums, left, (left + right) / 2, right)\n  # \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u65b7\n  nums[left], nums[med] = nums[med], nums[left]\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    end\n    # \u5143\u7d20\u4ea4\u63db\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n  nums[i], nums[left] = nums[left], nums[i]\n  i # \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\nend\n</code></pre> quick_sort.zig<pre><code>// \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n    var l = nums[left];\n    var m = nums[mid];\n    var r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n// \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    var med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;               // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5 \u00a0 \u5c3e\u905e\u8ff4\u6700\u4f73\u5316","text":"<p>\u5728\u67d0\u4e9b\u8f38\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u4f54\u7528\u7a7a\u9593\u8f03\u591a\u3002\u4ee5\u5b8c\u5168\u6709\u5e8f\u7684\u8f38\u5165\u9663\u5217\u70ba\u4f8b\uff0c\u8a2d\u905e\u8ff4\u4e2d\u7684\u5b50\u9663\u5217\u9577\u5ea6\u70ba \\(m\\) \uff0c\u6bcf\u8f2a\u54e8\u5175\u5283\u5206\u64cd\u4f5c\u90fd\u5c07\u7522\u751f\u9577\u5ea6\u70ba \\(0\\) \u7684\u5de6\u5b50\u9663\u5217\u548c\u9577\u5ea6\u70ba \\(m - 1\\) \u7684\u53f3\u5b50\u9663\u5217\uff0c\u9019\u610f\u5473\u8457\u6bcf\u4e00\u5c64\u905e\u8ff4\u547c\u53eb\u6e1b\u5c11\u7684\u554f\u984c\u898f\u6a21\u975e\u5e38\u5c0f\uff08\u53ea\u6e1b\u5c11\u4e00\u500b\u5143\u7d20\uff09\uff0c\u905e\u8ff4\u6a39\u7684\u9ad8\u5ea6\u6703\u9054\u5230 \\(n - 1\\) \uff0c\u6b64\u6642\u9700\u8981\u4f54\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002</p> <p>\u70ba\u4e86\u9632\u6b62\u5806\u758a\u5e40\u7a7a\u9593\u7684\u7d2f\u7a4d\uff0c\u6211\u5011\u53ef\u4ee5\u5728\u6bcf\u8f2a\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u5f8c\uff0c\u6bd4\u8f03\u5169\u500b\u5b50\u9663\u5217\u7684\u9577\u5ea6\uff0c\u50c5\u5c0d\u8f03\u77ed\u7684\u5b50\u9663\u5217\u9032\u884c\u905e\u8ff4\u3002\u7531\u65bc\u8f03\u77ed\u5b50\u9663\u5217\u7684\u9577\u5ea6\u4e0d\u6703\u8d85\u904e \\(n / 2\\) \uff0c\u56e0\u6b64\u9019\u7a2e\u65b9\u6cd5\u80fd\u78ba\u4fdd\u905e\u8ff4\u6df1\u5ea6\u4e0d\u8d85\u904e \\(\\log n\\) \uff0c\u5f9e\u800c\u5c07\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(\\log n)\\) \u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09\"\"\"\n    # \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while left &lt; right:\n        # \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        pivot = self.partition(nums, left, right)\n        # \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1  # \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1  # \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1;                 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1;                 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        int pivot = Partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1;  // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            QuickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    for left &lt; right {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        pivot := q.partition(nums, left, right)\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1                 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            q.quickSort(nums, pivot+1, right) // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1                 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nquickSort(nums, left, right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n  while (left &lt; right) {\n    // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n    int pivot = _partition(nums, left, right);\n    // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n      left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n    } else {\n      quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n      right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            quickSortTailCall(nums, left, pivot - 1);\n            // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n            left = pivot + 1;\n        } else {\n            // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            quickSortTailCall(nums, pivot + 1, right);\n            // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n            right = pivot - 1;\n        }\n    }\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nfun quickSortTailCall(nums: IntArray, left: Int, right: Int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    var l = left\n    var r = right\n    while (l &lt; r) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        val pivot = partition(nums, l, r)\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - l &lt; r - pivot) {\n            quickSort(nums, l, pivot - 1) // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            l = pivot + 1 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, r) // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            r = pivot - 1 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.rb<pre><code>### \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09###\ndef quick_sort(nums, left, right)\n  # \u5b50\u9663\u5217\u9577\u5ea6\u4e0d\u70ba 1 \u6642\u905e\u8ff4\n  while left &lt; right\n    # \u54e8\u5175\u5283\u5206\n    pivot = partition(nums, left, right)\n    # \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n    if pivot - left &lt; right - pivot\n      quick_sort(nums, left, pivot - 1)\n      left = pivot + 1 # \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n    else\n      quick_sort(nums, pivot + 1, right)\n      right = pivot - 1 # \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n    end\n  end\nend\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\n    var left = left_;\n    var right = right_;\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        var pivot = partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1);   // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1;                   // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right);  // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1;                  // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/radix_sort/","title":"11.10 \u00a0 \u57fa\u6578\u6392\u5e8f","text":"<p>\u4e0a\u4e00\u7bc0\u4ecb\u7d39\u4e86\u8a08\u6578\u6392\u5e8f\uff0c\u5b83\u9069\u7528\u65bc\u8cc7\u6599\u91cf \\(n\\) \u8f03\u5927\u4f46\u8cc7\u6599\u7bc4\u570d \\(m\\) \u8f03\u5c0f\u7684\u60c5\u6cc1\u3002\u5047\u8a2d\u6211\u5011\u9700\u8981\u5c0d \\(n = 10^6\\) \u500b\u5b78\u865f\u9032\u884c\u6392\u5e8f\uff0c\u800c\u5b78\u865f\u662f\u4e00\u500b \\(8\\) \u4f4d\u6578\u5b57\uff0c\u9019\u610f\u5473\u8457\u8cc7\u6599\u7bc4\u570d \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8a08\u6578\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u8a18\u61b6\u9ad4\u7a7a\u9593\uff0c\u800c\u57fa\u6578\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u9019\u7a2e\u60c5\u6cc1\u3002</p> <p>\u57fa\u6578\u6392\u5e8f\uff08radix sort\uff09\u7684\u6838\u5fc3\u601d\u60f3\u8207\u8a08\u6578\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u900f\u904e\u7d71\u8a08\u500b\u6578\u4f86\u5be6\u73fe\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u790e\u4e0a\uff0c\u57fa\u6578\u6392\u5e8f\u5229\u7528\u6578\u5b57\u5404\u4f4d\u4e4b\u9593\u7684\u905e\u9032\u95dc\u4fc2\uff0c\u4f9d\u6b21\u5c0d\u6bcf\u4e00\u4f4d\u9032\u884c\u6392\u5e8f\uff0c\u5f9e\u800c\u5f97\u5230\u6700\u7d42\u7684\u6392\u5e8f\u7d50\u679c\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u4ee5\u5b78\u865f\u8cc7\u6599\u70ba\u4f8b\uff0c\u5047\u8a2d\u6578\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6578\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u5716 11-18 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4f4d\u6578 \\(k = 1\\) \u3002</li> <li>\u5c0d\u5b78\u865f\u7684\u7b2c \\(k\\) \u4f4d\u57f7\u884c\u201c\u8a08\u6578\u6392\u5e8f\u201d\u3002\u5b8c\u6210\u5f8c\uff0c\u8cc7\u6599\u6703\u6839\u64da\u7b2c \\(k\\) \u4f4d\u5f9e\u5c0f\u5230\u5927\u6392\u5e8f\u3002</li> <li>\u5c07 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u5f8c\u8fd4\u56de\u6b65\u9a5f <code>2.</code> \u7e7c\u7e8c\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u5f8c\u7d50\u675f\u3002</li> </ol> <p></p> <p> \u5716 11-18 \u00a0 \u57fa\u6578\u6392\u5e8f\u6f14\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4e0b\u9762\u5256\u6790\u7a0b\u5f0f\u78bc\u5be6\u73fe\u3002\u5c0d\u65bc\u4e00\u500b \\(d\\) \u9032\u4f4d\u5236\u7684\u6578\u5b57 \\(x\\) \uff0c\u8981\u7372\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8a08\u7b97\u516c\u5f0f\uff1a</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>\u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5c0d\u6d6e\u9ede\u6578 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\: d\\) \u8868\u793a\u5c0d \\(d\\) \u53d6\u6a21\uff08\u53d6\u9918\uff09\u3002\u5c0d\u65bc\u5b78\u865f\u8cc7\u6599\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002</p> <p>\u6b64\u5916\uff0c\u6211\u5011\u9700\u8981\u5c0f\u5e45\u6539\u52d5\u8a08\u6578\u6392\u5e8f\u7a0b\u5f0f\u78bc\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u64da\u6578\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u9032\u884c\u6392\u5e8f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"\u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n    # \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"\u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n    # \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    counter = [0] * 10\n    n = len(nums)\n    # \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for i in range(n):\n        d = digit(nums[i], exp)  # \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d] += 1  # \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    # \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]  # \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1  # \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    # \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"\u57fa\u6578\u6392\u5e8f\"\"\"\n    # \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    m = max(nums)\n    # \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    exp = 1\n    while exp &lt;= m:\n        # \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # \u5373 exp = 10^(k-1)\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++;                // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    int m = *max_element(nums.begin(), nums.end());\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    int[] counter = new int[10];\n    int n = nums.length;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++;                // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.cs<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint Digit(int num, int exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++;                // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid RadixSort(int[] nums) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    counter := make([]int, 10)\n    n := len(nums)\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++             // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]    // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--        // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfunc radixSort(nums []int) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    (num / exp) % 10\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    var counter = Array(repeating: 0, count: 10)\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d] += 1 // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i] // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1 // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfunction radixSort(nums) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* \u7372\u53d6\u5143\u7d20 _num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int _num, int exp) {\n  // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n  return (_num ~/ exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n    counter[d]++; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n  }\n  // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n    res[j] = nums[i]; // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n    counter[d]--; // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n  }\n  // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid radixSort(List&lt;int&gt; nums) {\n  // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n  // dart \u4e2d int \u7684\u9577\u5ea6\u662f 64 \u4f4d\u7684\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // \u5373 exp = 10^(k-1)\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return ((num / exp) % 10) as usize;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for i in 0..n {\n        let d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d] += 1; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1; // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    nums.copy_from_slice(&amp;res);\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    let m = *nums.into_iter().max().unwrap();\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    memset(counter, 0, sizeof(int) * 10); // \u521d\u59cb\u5316\u70ba 0 \u4ee5\u652f\u6301\u5f8c\u7e8c\u8a18\u61b6\u9ad4\u91cb\u653e\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (int i = 0; i &lt; size; i++) {\n        // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        int d = digit(nums[i], exp);\n        // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n        counter[d]++;\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(res);\n    free(counter);\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    int max = INT32_MIN;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.kt<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfun digit(num: Int, exp: Int): Int {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfun countingSortDigit(nums: IntArray, exp: Int) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    val counter = IntArray(10)\n    val n = nums.size\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (i in 0..&lt;n) {\n        val d = digit(nums[i], exp) // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++                // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (i in 1..9) {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val d = digit(nums[i], exp)\n        val j = counter[d] - 1 // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (i in 0..&lt;n)\n        nums[i] = res[i]\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfun radixSort(nums: IntArray) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    var m = Int.MIN_VALUE\n    for (num in nums) if (num &gt; m) m = num\n    var exp = 1\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    while (exp &lt;= m) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n        exp *= 10\n    }\n}\n</code></pre> radix_sort.rb<pre><code>### \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) ###\ndef digit(num, exp)\n  # \u8f49\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n  (num / exp) % 10\nend\n\n### \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09###\ndef counting_sort_digit(nums, exp)\n  # \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n  counter = Array.new(10, 0)\n  n = nums.length\n  # \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  for i in 0...n\n    d = digit(nums[i], exp) # \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n    counter[d] += 1 # \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n  end\n  # \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n  (1...10).each { |i| counter[i] += counter[i - 1] }\n  # \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n  res = Array.new(n, 0)\n  for i in (n - 1).downto(0)\n    d = digit(nums[i], exp)\n    j = counter[d] - 1 # \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n    res[j] = nums[i] # \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n    counter[d] -= 1 # \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n  end\n  # \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n  (0...n).each { |i| nums[i] = res[i] }\nend\n\n### \u57fa\u6578\u6392\u5e8f ###\ndef radix_sort(nums)\n  # \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n  m = nums.max\n  # \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n  exp = 1\n  while exp &lt;= m\n    # \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n    # k = 1 -&gt; exp = 1\n    # k = 2 -&gt; exp = 10\n    # \u5373 exp = 10^(k-1)\n    counting_sort_digit(nums, exp)\n    exp *= 10\n  end\nend\n</code></pre> radix_sort.zig<pre><code>// \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return @mod(@divFloor(num, exp), 10);\n}\n\n// \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    // defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var counter = try mem_allocator.alloc(usize, 10);\n    @memset(counter, 0);\n    var n = nums.len;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (nums) |num| {\n        var d: u32 = @bitCast(digit(num, exp)); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d] += 1; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    var i: usize = 1;\n    while (i &lt; 10) : (i += 1) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    var res = try mem_allocator.alloc(i32, n);\n    i = n - 1;\n    while (i &gt;= 0) : (i -= 1) {\n        var d: u32 = @bitCast(digit(nums[i], exp));\n        var j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1;        // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n        if (i == 0) break;\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        nums[i] = res[i];\n    }\n}\n\n// \u57fa\u6578\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    var m: i32 = std.math.minInt(i32);\n    for (nums) |num| {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    var exp: i32 = 1;\n    while (exp &lt;= m) : (exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        try countingSortDigit(nums, exp);    \n    }\n} \n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u70ba\u4ec0\u9ebc\u5f9e\u6700\u4f4e\u4f4d\u958b\u59cb\u6392\u5e8f\uff1f</p> <p>\u5728\u9023\u7e8c\u7684\u6392\u5e8f\u8f2a\u6b21\u4e2d\uff0c\u5f8c\u4e00\u8f2a\u6392\u5e8f\u6703\u8986\u84cb\u524d\u4e00\u8f2a\u6392\u5e8f\u7684\u7d50\u679c\u3002\u8209\u4f8b\u4f86\u8aaa\uff0c\u5982\u679c\u7b2c\u4e00\u8f2a\u6392\u5e8f\u7d50\u679c \\(a &lt; b\\) \uff0c\u800c\u7b2c\u4e8c\u8f2a\u6392\u5e8f\u7d50\u679c \\(a &gt; b\\) \uff0c\u90a3\u9ebc\u7b2c\u4e8c\u8f2a\u7684\u7d50\u679c\u5c07\u53d6\u4ee3\u7b2c\u4e00\u8f2a\u7684\u7d50\u679c\u3002\u7531\u65bc\u6578\u5b57\u7684\u9ad8\u4f4d\u512a\u5148\u9806\u5e8f\u9ad8\u65bc\u4f4e\u4f4d\uff0c\u56e0\u6b64\u61c9\u8a72\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<p>\u76f8\u8f03\u65bc\u8a08\u6578\u6392\u5e8f\uff0c\u57fa\u6578\u6392\u5e8f\u9069\u7528\u65bc\u6578\u503c\u7bc4\u570d\u8f03\u5927\u7684\u60c5\u6cc1\uff0c\u4f46\u524d\u63d0\u662f\u8cc7\u6599\u5fc5\u9808\u53ef\u4ee5\u8868\u793a\u70ba\u56fa\u5b9a\u4f4d\u6578\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6578\u4e0d\u80fd\u904e\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u9ede\u6578\u4e0d\u9069\u5408\u4f7f\u7528\u57fa\u6578\u6392\u5e8f\uff0c\u56e0\u70ba\u5176\u4f4d\u6578 \\(k\\) \u904e\u5927\uff0c\u53ef\u80fd\u5c0e\u81f4\u6642\u9593\u8907\u96dc\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(nk)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u8a2d\u8cc7\u6599\u91cf\u70ba \\(n\\)\u3001\u8cc7\u6599\u70ba \\(d\\) \u9032\u4f4d\u5236\u3001\u6700\u5927\u4f4d\u6578\u70ba \\(k\\) \uff0c\u5247\u5c0d\u67d0\u4e00\u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u6642\u9593\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u6642\u9593\u3002\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5c0d\u8f03\u5c0f\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u8da8\u5411 \\(O(n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + d)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u8207\u8a08\u6578\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6578\u6392\u5e8f\u9700\u8981\u85c9\u52a9\u9577\u5ea6\u70ba \\(n\\) \u548c \\(d\\) \u7684\u9663\u5217 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u7576\u8a08\u6578\u6392\u5e8f\u7a69\u5b9a\u6642\uff0c\u57fa\u6578\u6392\u5e8f\u4e5f\u7a69\u5b9a\uff1b\u7576\u8a08\u6578\u6392\u5e8f\u4e0d\u7a69\u5b9a\u6642\uff0c\u57fa\u6578\u6392\u5e8f\u7121\u6cd5\u4fdd\u8b49\u5f97\u5230\u6b63\u78ba\u7684\u6392\u5e8f\u7d50\u679c\u3002</li> </ul>"},{"location":"chapter_sorting/selection_sort/","title":"11.2 \u00a0 \u9078\u64c7\u6392\u5e8f","text":"<p>\u9078\u64c7\u6392\u5e8f\uff08selection sort\uff09\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u7c21\u55ae\uff1a\u958b\u555f\u4e00\u500b\u8ff4\u5708\uff0c\u6bcf\u8f2a\u5f9e\u672a\u6392\u5e8f\u5340\u9593\u9078\u64c7\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c07\u5176\u653e\u5230\u5df2\u6392\u5e8f\u5340\u9593\u7684\u672b\u5c3e\u3002</p> <p>\u8a2d\u9663\u5217\u7684\u9577\u5ea6\u70ba \\(n\\) \uff0c\u9078\u64c7\u6392\u5e8f\u7684\u6f14\u7b97\u6cd5\u6d41\u7a0b\u5982\u5716 11-2 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72c0\u614b\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u5340\u9593\u70ba \\([0, n-1]\\) \u3002</li> <li>\u9078\u53d6\u5340\u9593 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c07\u5176\u8207\u7d22\u5f15 \\(0\\) \u8655\u7684\u5143\u7d20\u4ea4\u63db\u3002\u5b8c\u6210\u5f8c\uff0c\u9663\u5217\u524d 1 \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9078\u53d6\u5340\u9593 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c07\u5176\u8207\u7d22\u5f15 \\(1\\) \u8655\u7684\u5143\u7d20\u4ea4\u63db\u3002\u5b8c\u6210\u5f8c\uff0c\u9663\u5217\u524d 2 \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u985e\u63a8\u3002\u7d93\u904e \\(n - 1\\) \u8f2a\u9078\u64c7\u8207\u4ea4\u63db\u5f8c\uff0c\u9663\u5217\u524d \\(n - 1\\) \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u50c5\u5269\u7684\u4e00\u500b\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u7121\u9808\u6392\u5e8f\uff0c\u56e0\u6b64\u9663\u5217\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-2 \u00a0 \u9078\u64c7\u6392\u5e8f\u6b65\u9a5f </p> <p>\u5728\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u6211\u5011\u7528 \\(k\\) \u4f86\u8a18\u9304\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"\u9078\u64c7\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for i in range(n - 1):\n        # \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        # \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for i := 0; i &lt; n-1; i++ {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n                k = j\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for i in nums.indices.dropLast() {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n  for (int i = 0; i &lt; n - 1; i++) {\n    // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n    }\n    // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfn selection_sort(nums: &amp;mut [i32]) {\n    if nums.is_empty() {\n        return;\n    }\n    let n = nums.len();\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for i in 0..n - 1 {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        let mut k = i;\n        for j in i + 1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.kt<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfun selectionSort(nums: IntArray) {\n    val n = nums.size\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (i in 0..&lt;n - 1) {\n        var k = i\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        for (j in i + 1..&lt;n) {\n            if (nums[j] &lt; nums[k])\n                k = j // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        val temp = nums[i]\n        nums[i] = nums[k]\n        nums[k] = temp\n    }\n}\n</code></pre> selection_sort.rb<pre><code>### \u9078\u64c7\u6392\u5e8f ###\ndef selection_sort(nums)\n  n = nums.length\n  # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n  for i in 0...(n - 1)\n    # \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n    k = i\n    for j in (i + 1)...n\n      if nums[j] &lt; nums[k]\n        k = j # \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n      end\n    end\n    # \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n    nums[i], nums[k] = nums[k], nums[i]\n  end\nend\n</code></pre> selection_sort.zig<pre><code>[class]{}-[func]{selectionSort}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5916\u8ff4\u5708\u5171 \\(n - 1\\) \u8f2a\uff0c\u7b2c\u4e00\u8f2a\u7684\u672a\u6392\u5e8f\u5340\u9593\u9577\u5ea6\u70ba \\(n\\) \uff0c\u6700\u5f8c\u4e00\u8f2a\u7684\u672a\u6392\u5e8f\u5340\u9593\u9577\u5ea6\u70ba \\(2\\) \uff0c\u5373\u5404\u8f2a\u5916\u8ff4\u5708\u5206\u5225\u5305\u542b \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(3\\)\u3001\\(2\\) \u8f2a\u5167\u8ff4\u5708\uff0c\u6c42\u548c\u70ba \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u975e\u7a69\u5b9a\u6392\u5e8f\uff1a\u5982\u5716 11-3 \u6240\u793a\uff0c\u5143\u7d20 <code>nums[i]</code> \u6709\u53ef\u80fd\u88ab\u4ea4\u63db\u81f3\u8207\u5176\u76f8\u7b49\u7684\u5143\u7d20\u7684\u53f3\u908a\uff0c\u5c0e\u81f4\u5169\u8005\u7684\u76f8\u5c0d\u9806\u5e8f\u767c\u751f\u6539\u8b8a\u3002</li> </ul> <p> \u5716 11-3 \u00a0 \u9078\u64c7\u6392\u5e8f\u975e\u7a69\u5b9a\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1 \u00a0 \u6392\u5e8f\u6f14\u7b97\u6cd5","text":"<p>\u6392\u5e8f\u6f14\u7b97\u6cd5\uff08sorting algorithm\uff09\u7528\u65bc\u5c0d\u4e00\u7d44\u8cc7\u6599\u6309\u7167\u7279\u5b9a\u9806\u5e8f\u9032\u884c\u6392\u5217\u3002\u6392\u5e8f\u6f14\u7b97\u6cd5\u6709\u8457\u5ee3\u6cdb\u7684\u61c9\u7528\uff0c\u56e0\u70ba\u6709\u5e8f\u8cc7\u6599\u901a\u5e38\u80fd\u5920\u88ab\u66f4\u9ad8\u6548\u5730\u67e5\u8a62\u3001\u5206\u6790\u548c\u8655\u7406\u3002</p> <p>\u5982\u5716 11-1 \u6240\u793a\uff0c\u6392\u5e8f\u6f14\u7b97\u6cd5\u4e2d\u7684\u8cc7\u6599\u578b\u5225\u53ef\u4ee5\u662f\u6574\u6578\u3001\u6d6e\u9ede\u6578\u3001\u5b57\u5143\u6216\u5b57\u4e32\u7b49\u3002\u6392\u5e8f\u7684\u5224\u65b7\u898f\u5247\u53ef\u6839\u64da\u9700\u6c42\u8a2d\u5b9a\uff0c\u5982\u6578\u5b57\u5927\u5c0f\u3001\u5b57\u5143 ASCII \u78bc\u9806\u5e8f\u6216\u81ea\u5b9a\u7fa9\u898f\u5247\u3002</p> <p></p> <p> \u5716 11-1 \u00a0 \u8cc7\u6599\u578b\u5225\u548c\u5224\u65b7\u898f\u5247\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1 \u00a0 \u8a55\u50f9\u7dad\u5ea6","text":"<p>\u57f7\u884c\u6548\u7387\uff1a\u6211\u5011\u671f\u671b\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5118\u91cf\u4f4e\uff0c\u4e14\u7e3d\u9ad4\u64cd\u4f5c\u6578\u91cf\u8f03\u5c11\uff08\u6642\u9593\u8907\u96dc\u5ea6\u4e2d\u7684\u5e38\u6578\u9805\u8b8a\u5c0f\uff09\u3002\u5c0d\u65bc\u5927\u8cc7\u6599\u91cf\u7684\u60c5\u6cc1\uff0c\u57f7\u884c\u6548\u7387\u986f\u5f97\u5c24\u70ba\u91cd\u8981\u3002</p> <p>\u5c31\u5730\u6027\uff1a\u9867\u540d\u601d\u7fa9\uff0c\u539f\u5730\u6392\u5e8f\u900f\u904e\u5728\u539f\u9663\u5217\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5be6\u73fe\u6392\u5e8f\uff0c\u7121\u9808\u85c9\u52a9\u984d\u5916\u7684\u8f14\u52a9\u9663\u5217\uff0c\u5f9e\u800c\u7bc0\u7701\u8a18\u61b6\u9ad4\u3002\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u8cc7\u6599\u642c\u904b\u64cd\u4f5c\u8f03\u5c11\uff0c\u57f7\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002</p> <p>\u7a69\u5b9a\u6027\uff1a\u7a69\u5b9a\u6392\u5e8f\u5728\u5b8c\u6210\u6392\u5e8f\u5f8c\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u9663\u5217\u4e2d\u7684\u76f8\u5c0d\u9806\u5e8f\u4e0d\u767c\u751f\u6539\u8b8a\u3002</p> <p>\u7a69\u5b9a\u6392\u5e8f\u662f\u591a\u7d1a\u6392\u5e8f\u5834\u666f\u7684\u5fc5\u8981\u689d\u4ef6\u3002\u5047\u8a2d\u6211\u5011\u6709\u4e00\u500b\u5132\u5b58\u5b78\u751f\u8cc7\u8a0a\u7684\u8868\u683c\uff0c\u7b2c 1 \u5217\u548c\u7b2c 2 \u5217\u5206\u5225\u662f\u59d3\u540d\u548c\u5e74\u9f61\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u975e\u7a69\u5b9a\u6392\u5e8f\u53ef\u80fd\u5c0e\u81f4\u8f38\u5165\u8cc7\u6599\u7684\u6709\u5e8f\u6027\u55aa\u5931\uff1a</p> <pre><code># \u8f38\u5165\u8cc7\u6599\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# \u5047\u8a2d\u4f7f\u7528\u975e\u7a69\u5b9a\u6392\u5e8f\u6f14\u7b97\u6cd5\u6309\u5e74\u9f61\u6392\u5e8f\u4e32\u5217\uff0c\n# \u7d50\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5c0d\u4f4d\u7f6e\u6539\u8b8a\uff0c\n# \u8f38\u5165\u8cc7\u6599\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8cea\u4e1f\u5931\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>\u81ea\u9069\u61c9\u6027\uff1a\u81ea\u9069\u61c9\u6392\u5e8f\u80fd\u5920\u5229\u7528\u8f38\u5165\u8cc7\u6599\u5df2\u6709\u7684\u9806\u5e8f\u8cc7\u8a0a\u4f86\u6e1b\u5c11\u8a08\u7b97\u91cf\uff0c\u9054\u5230\u66f4\u512a\u7684\u6642\u9593\u6548\u7387\u3002\u81ea\u9069\u61c9\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u512a\u65bc\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <p>\u662f\u5426\u57fa\u65bc\u6bd4\u8f03\uff1a\u57fa\u65bc\u6bd4\u8f03\u7684\u6392\u5e8f\u4f9d\u8cf4\u6bd4\u8f03\u904b\u7b97\u5b50\uff08\\(&lt;\\)\u3001\\(=\\)\u3001\\(&gt;\\)\uff09\u4f86\u5224\u65b7\u5143\u7d20\u7684\u76f8\u5c0d\u9806\u5e8f\uff0c\u5f9e\u800c\u6392\u5e8f\u6574\u500b\u9663\u5217\uff0c\u7406\u8ad6\u6700\u512a\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002\u800c\u975e\u6bd4\u8f03\u6392\u5e8f\u4e0d\u4f7f\u7528\u6bd4\u8f03\u904b\u7b97\u5b50\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054 \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5c0d\u8f03\u5dee\u3002</p>"},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2 \u00a0 \u7406\u60f3\u6392\u5e8f\u6f14\u7b97\u6cd5","text":"<p>\u57f7\u884c\u5feb\u3001\u539f\u5730\u3001\u7a69\u5b9a\u3001\u81ea\u9069\u61c9\u3001\u901a\u7528\u6027\u597d\u3002\u986f\u7136\uff0c\u8fc4\u4eca\u70ba\u6b62\u5c1a\u672a\u767c\u73fe\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9078\u64c7\u6392\u5e8f\u6f14\u7b97\u6cd5\u6642\uff0c\u9700\u8981\u6839\u64da\u5177\u9ad4\u7684\u8cc7\u6599\u7279\u9ede\u548c\u554f\u984c\u9700\u6c42\u4f86\u6c7a\u5b9a\u3002</p> <p>\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5c07\u5171\u540c\u5b78\u7fd2\u5404\u7a2e\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u4e26\u57fa\u65bc\u4e0a\u8ff0\u8a55\u50f9\u7dad\u5ea6\u5c0d\u5404\u500b\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u512a\u7f3a\u9ede\u9032\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_sorting/summary/","title":"11.11 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_sorting/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u6ce1\u6cab\u6392\u5e8f\u900f\u904e\u4ea4\u63db\u76f8\u9130\u5143\u7d20\u4f86\u5be6\u73fe\u6392\u5e8f\u3002\u900f\u904e\u65b0\u589e\u4e00\u500b\u6a19\u8a8c\u4f4d\u4f86\u5be6\u73fe\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u6ce1\u6cab\u6392\u5e8f\u7684\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u5230 \\(O(n)\\) \u3002</li> <li>\u63d2\u5165\u6392\u5e8f\u6bcf\u8f2a\u5c07\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593\u7684\u6b63\u78ba\u4f4d\u7f6e\uff0c\u5f9e\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u96d6\u7136\u63d2\u5165\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u4f46\u7531\u65bc\u55ae\u5143\u64cd\u4f5c\u76f8\u5c0d\u8f03\u5c11\uff0c\u56e0\u6b64\u5728\u5c0f\u8cc7\u6599\u91cf\u7684\u6392\u5e8f\u4efb\u52d9\u4e2d\u975e\u5e38\u53d7\u6b61\u8fce\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\u57fa\u65bc\u54e8\u5175\u5283\u5206\u64cd\u4f5c\u5be6\u73fe\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5283\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9078\u53d6\u5230\u6700\u5dee\u7684\u57fa\u6e96\u6578\uff0c\u5c0e\u81f4\u6642\u9593\u8907\u96dc\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6578\u57fa\u6e96\u6578\u6216\u96a8\u6a5f\u57fa\u6e96\u6578\u53ef\u4ee5\u964d\u4f4e\u9019\u7a2e\u52a3\u5316\u7684\u6a5f\u7387\u3002\u5c3e\u905e\u8ff4\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u6e1b\u5c11\u905e\u8ff4\u6df1\u5ea6\uff0c\u5c07\u7a7a\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u5230 \\(O(\\log n)\\) \u3002</li> <li>\u5408\u4f75\u6392\u5e8f\u5305\u62ec\u5283\u5206\u548c\u5408\u4e26\u5169\u500b\u968e\u6bb5\uff0c\u5178\u578b\u5730\u9ad4\u73fe\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5408\u4f75\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u9663\u5217\u9700\u8981\u5efa\u7acb\u8f14\u52a9\u9663\u5217\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u93c8\u7d50\u4e32\u5217\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u6700\u4f73\u5316\u81f3 \\(O(1)\\) \u3002</li> <li>\u6876\u6392\u5e8f\u5305\u542b\u4e09\u500b\u6b65\u9a5f\uff1a\u8cc7\u6599\u5206\u6876\u3001\u6876\u5167\u6392\u5e8f\u548c\u5408\u4e26\u7d50\u679c\u3002\u5b83\u540c\u6a23\u9ad4\u73fe\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9069\u7528\u65bc\u8cc7\u6599\u9ad4\u91cf\u5f88\u5927\u7684\u60c5\u6cc1\u3002\u6876\u6392\u5e8f\u7684\u95dc\u9375\u5728\u65bc\u5c0d\u8cc7\u6599\u9032\u884c\u5e73\u5747\u5206\u914d\u3002</li> <li>\u8a08\u6578\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u500b\u7279\u4f8b\uff0c\u5b83\u900f\u904e\u7d71\u8a08\u8cc7\u6599\u51fa\u73fe\u7684\u6b21\u6578\u4f86\u5be6\u73fe\u6392\u5e8f\u3002\u8a08\u6578\u6392\u5e8f\u9069\u7528\u65bc\u8cc7\u6599\u91cf\u5927\u4f46\u8cc7\u6599\u7bc4\u570d\u6709\u9650\u7684\u60c5\u6cc1\uff0c\u4e26\u4e14\u8981\u6c42\u8cc7\u6599\u80fd\u5920\u8f49\u63db\u70ba\u6b63\u6574\u6578\u3002</li> <li>\u57fa\u6578\u6392\u5e8f\u900f\u904e\u9010\u4f4d\u6392\u5e8f\u4f86\u5be6\u73fe\u8cc7\u6599\u6392\u5e8f\uff0c\u8981\u6c42\u8cc7\u6599\u80fd\u5920\u8868\u793a\u70ba\u56fa\u5b9a\u4f4d\u6578\u7684\u6578\u5b57\u3002</li> <li>\u7e3d\u7684\u4f86\u8aaa\uff0c\u6211\u5011\u5e0c\u671b\u627e\u5230\u4e00\u7a2e\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a69\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u81ea\u9069\u61c9\u6027\u7b49\u512a\u9ede\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u8cc7\u6599\u7d50\u69cb\u548c\u6f14\u7b97\u6cd5\u4e00\u6a23\uff0c\u6c92\u6709\u4e00\u7a2e\u6392\u5e8f\u6f14\u7b97\u6cd5\u80fd\u5920\u540c\u6642\u6eff\u8db3\u6240\u6709\u9019\u4e9b\u689d\u4ef6\u3002\u5728\u5be6\u969b\u61c9\u7528\u4e2d\uff0c\u6211\u5011\u9700\u8981\u6839\u64da\u8cc7\u6599\u7684\u7279\u6027\u4f86\u9078\u64c7\u5408\u9069\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\u3002</li> <li>\u5716 11-19 \u5c0d\u6bd4\u4e86\u4e3b\u6d41\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6548\u7387\u3001\u7a69\u5b9a\u6027\u3001\u5c31\u5730\u6027\u548c\u81ea\u9069\u61c9\u6027\u7b49\u3002</li> </ul> <p> \u5716 11-19 \u00a0 \u6392\u5e8f\u6f14\u7b97\u6cd5\u5c0d\u6bd4 </p>"},{"location":"chapter_sorting/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6392\u5e8f\u6f14\u7b97\u6cd5\u7a69\u5b9a\u6027\u5728\u4ec0\u9ebc\u60c5\u6cc1\u4e0b\u662f\u5fc5\u9700\u7684\uff1f</p> <p>\u5728\u73fe\u5be6\u4e2d\uff0c\u6211\u5011\u6709\u53ef\u80fd\u57fa\u65bc\u7269\u4ef6\u7684\u67d0\u500b\u5c6c\u6027\u9032\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b78\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u5169\u500b\u5c6c\u6027\uff0c\u6211\u5011\u5e0c\u671b\u5be6\u73fe\u4e00\u500b\u591a\u7d1a\u6392\u5e8f\uff1a\u5148\u6309\u7167\u59d3\u540d\u9032\u884c\u6392\u5e8f\uff0c\u5f97\u5230 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> \uff1b\u518d\u5c0d\u8eab\u9ad8\u9032\u884c\u6392\u5e8f\u3002\u7531\u65bc\u6392\u5e8f\u6f14\u7b97\u6cd5\u4e0d\u7a69\u5b9a\uff0c\u56e0\u6b64\u53ef\u80fd\u5f97\u5230 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> \u3002</p> <p>\u53ef\u4ee5\u767c\u73fe\uff0c\u5b78\u751f D \u548c C \u7684\u4f4d\u7f6e\u767c\u751f\u4e86\u4ea4\u63db\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u58de\u4e86\uff0c\u800c\u9019\u662f\u6211\u5011\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002</p> <p>Q\uff1a\u54e8\u5175\u5283\u5206\u4e2d\u201c\u5f9e\u53f3\u5f80\u5de6\u67e5\u8a62\u201d\u8207\u201c\u5f9e\u5de6\u5f80\u53f3\u67e5\u8a62\u201d\u7684\u9806\u5e8f\u53ef\u4ee5\u4ea4\u63db\u55ce\uff1f</p> <p>\u4e0d\u884c\uff0c\u7576\u6211\u5011\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u70ba\u57fa\u6e96\u6578\u6642\uff0c\u5fc5\u9808\u5148\u201c\u5f9e\u53f3\u5f80\u5de6\u67e5\u8a62\u201d\u518d\u201c\u5f9e\u5de6\u5f80\u53f3\u67e5\u8a62\u201d\u3002\u9019\u500b\u7d50\u8ad6\u6709\u4e9b\u53cd\u76f4\u89ba\uff0c\u6211\u5011\u4f86\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002</p> <p>\u54e8\u5175\u5283\u5206 <code>partition()</code> \u7684\u6700\u5f8c\u4e00\u6b65\u662f\u4ea4\u63db <code>nums[left]</code> \u548c <code>nums[i]</code> \u3002\u5b8c\u6210\u4ea4\u63db\u5f8c\uff0c\u57fa\u6e96\u6578\u5de6\u908a\u7684\u5143\u7d20\u90fd <code>&lt;=</code> \u57fa\u6e96\u6578\uff0c\u9019\u5c31\u8981\u6c42\u6700\u5f8c\u4e00\u6b65\u4ea4\u63db\u524d <code>nums[left] &gt;= nums[i]</code> \u5fc5\u9808\u6210\u7acb\u3002\u5047\u8a2d\u6211\u5011\u5148\u201c\u5f9e\u5de6\u5f80\u53f3\u67e5\u8a62\u201d\uff0c\u90a3\u9ebc\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u6e96\u6578\u66f4\u5927\u7684\u5143\u7d20\uff0c\u5247\u6703\u5728 <code>i == j</code> \u6642\u8df3\u51fa\u8ff4\u5708\uff0c\u6b64\u6642\u53ef\u80fd <code>nums[j] == nums[i] &gt; nums[left]</code>\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u6b64\u6642\u6700\u5f8c\u4e00\u6b65\u4ea4\u63db\u64cd\u4f5c\u6703\u628a\u4e00\u500b\u6bd4\u57fa\u6e96\u6578\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\uff0c\u5c0e\u81f4\u54e8\u5175\u5283\u5206\u5931\u6557\u3002</p> <p>\u8209\u500b\u4f8b\u5b50\uff0c\u7d66\u5b9a\u9663\u5217 <code>[0, 0, 0, 0, 1]</code> \uff0c\u5982\u679c\u5148\u201c\u5f9e\u5de6\u5411\u53f3\u67e5\u8a62\u201d\uff0c\u54e8\u5175\u5283\u5206\u5f8c\u9663\u5217\u70ba <code>[1, 0, 0, 0, 0]</code> \uff0c\u9019\u500b\u7d50\u679c\u662f\u4e0d\u6b63\u78ba\u7684\u3002</p> <p>\u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u5011\u9078\u64c7 <code>nums[right]</code> \u70ba\u57fa\u6e96\u6578\uff0c\u90a3\u9ebc\u6b63\u597d\u53cd\u904e\u4f86\uff0c\u5fc5\u9808\u5148\u201c\u5f9e\u5de6\u5f80\u53f3\u67e5\u8a62\u201d\u3002</p> <p>Q\uff1a\u95dc\u65bc\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff0c\u70ba\u4ec0\u9ebc\u9078\u77ed\u7684\u9663\u5217\u80fd\u4fdd\u8b49\u905e\u8ff4\u6df1\u5ea6\u4e0d\u8d85\u904e \\(\\log n\\) \uff1f</p> <p>\u905e\u8ff4\u6df1\u5ea6\u5c31\u662f\u7576\u524d\u672a\u8fd4\u56de\u7684\u905e\u8ff4\u65b9\u6cd5\u7684\u6578\u91cf\u3002\u6bcf\u8f2a\u54e8\u5175\u5283\u5206\u6211\u5011\u5c07\u539f\u9663\u5217\u5283\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\u3002\u5728\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\u5f8c\uff0c\u5411\u4e0b\u905e\u8ff4\u7684\u5b50\u9663\u5217\u9577\u5ea6\u6700\u5927\u70ba\u539f\u9663\u5217\u9577\u5ea6\u7684\u4e00\u534a\u3002\u5047\u8a2d\u6700\u5dee\u60c5\u6cc1\uff0c\u4e00\u76f4\u70ba\u4e00\u534a\u9577\u5ea6\uff0c\u90a3\u9ebc\u6700\u7d42\u7684\u905e\u8ff4\u6df1\u5ea6\u5c31\u662f \\(\\log n\\) \u3002</p> <p>\u56de\u9867\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u5011\u6709\u53ef\u80fd\u6703\u9023\u7e8c\u5730\u905e\u8ff4\u9577\u5ea6\u8f03\u5927\u7684\u9663\u5217\uff0c\u6700\u5dee\u60c5\u6cc1\u4e0b\u70ba \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \u3002\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\u53ef\u4ee5\u907f\u514d\u9019\u7a2e\u60c5\u6cc1\u51fa\u73fe\u3002</p> <p>Q\uff1a\u7576\u9663\u5217\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u6642\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(n^2)\\) \u55ce\uff1f\u8a72\u5982\u4f55\u8655\u7406\u9019\u7a2e\u9000\u5316\u60c5\u6cc1\uff1f</p> <p>\u662f\u7684\u3002\u5c0d\u65bc\u9019\u7a2e\u60c5\u6cc1\uff0c\u53ef\u4ee5\u8003\u616e\u900f\u904e\u54e8\u5175\u5283\u5206\u5c07\u9663\u5217\u5283\u5206\u70ba\u4e09\u500b\u90e8\u5206\uff1a\u5c0f\u65bc\u3001\u7b49\u65bc\u3001\u5927\u65bc\u57fa\u6e96\u6578\u3002\u50c5\u5411\u4e0b\u905e\u8ff4\u5c0f\u65bc\u548c\u5927\u65bc\u7684\u5169\u90e8\u5206\u3002\u5728\u8a72\u65b9\u6cd5\u4e0b\uff0c\u8f38\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u9663\u5217\uff0c\u50c5\u4e00\u8f2a\u54e8\u5175\u5283\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002</p> <p>Q\uff1a\u6876\u6392\u5e8f\u7684\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u70ba\u4ec0\u9ebc\u662f \\(O(n^2)\\) \uff1f</p> <p>\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u500b\u6876\u4e2d\u3002\u5982\u679c\u6211\u5011\u63a1\u7528\u4e00\u500b \\(O(n^2)\\) \u6f14\u7b97\u6cd5\u4f86\u6392\u5e8f\u9019\u4e9b\u5143\u7d20\uff0c\u5247\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_stack_and_queue/","title":"\u7b2c 5 \u7ae0 \u00a0 \u5806\u758a\u8207\u4f47\u5217","text":"<p>Abstract</p> <p>\u5806\u758a\u5982\u540c\u758a\u8c93\u8c93\uff0c\u800c\u4f47\u5217\u5c31\u50cf\u8c93\u8c93\u6392\u968a\u3002</p> <p>\u5169\u8005\u5206\u5225\u4ee3\u8868\u5148\u5165\u5f8c\u51fa\u548c\u5148\u5165\u5148\u51fa\u7684\u908f\u8f2f\u95dc\u4fc2\u3002</p>"},{"location":"chapter_stack_and_queue/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>5.1 \u00a0 \u5806\u758a</li> <li>5.2 \u00a0 \u4f47\u5217</li> <li>5.3 \u00a0 \u96d9\u5411\u4f47\u5217</li> <li>5.4 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_stack_and_queue/deque/","title":"5.3 \u00a0 \u96d9\u5411\u4f47\u5217","text":"<p>\u5728\u4f47\u5217\u4e2d\uff0c\u6211\u5011\u50c5\u80fd\u522a\u9664\u982d\u90e8\u5143\u7d20\u6216\u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\u3002\u5982\u5716 5-7 \u6240\u793a\uff0c\u96d9\u5411\u4f47\u5217\uff08double-ended queue\uff09\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u9748\u6d3b\u6027\uff0c\u5141\u8a31\u5728\u982d\u90e8\u548c\u5c3e\u90e8\u57f7\u884c\u5143\u7d20\u7684\u65b0\u589e\u6216\u522a\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u5716 5-7 \u00a0 \u96d9\u5411\u4f47\u5217\u7684\u64cd\u4f5c </p>"},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1 \u00a0 \u96d9\u5411\u4f47\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u96d9\u5411\u4f47\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-3 \u6240\u793a\uff0c\u5177\u9ad4\u7684\u65b9\u6cd5\u540d\u7a31\u9700\u8981\u6839\u64da\u6240\u4f7f\u7528\u7684\u7a0b\u5f0f\u8a9e\u8a00\u4f86\u78ba\u5b9a\u3002</p> <p> \u8868 5-3 \u00a0 \u96d9\u5411\u4f47\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u6642\u9593\u8907\u96dc\u5ea6 <code>push_first()</code> \u5c07\u5143\u7d20\u65b0\u589e\u81f3\u4f47\u5217\u9996 \\(O(1)\\) <code>push_last()</code> \u5c07\u5143\u7d20\u65b0\u589e\u81f3\u4f47\u5217\u5c3e \\(O(1)\\) <code>pop_first()</code> \u522a\u9664\u4f47\u5217\u9996\u5143\u7d20 \\(O(1)\\) <code>pop_last()</code> \u522a\u9664\u4f47\u5217\u5c3e\u5143\u7d20 \\(O(1)\\) <code>peek_first()</code> \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 \\(O(1)\\) <code>peek_last()</code> \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 \\(O(1)\\) <p>\u540c\u6a23\u5730\uff0c\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u5df2\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217\u985e\u5225\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig deque.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217\ndeq: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u5217\ndeq.append(2)      # \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeq.append(5)\ndeq.append(4)\ndeq.appendleft(3)  # \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeq.appendleft(1)\n\n# \u8a2a\u554f\u5143\u7d20\nfront: int = deq[0]  # \u4f47\u5217\u9996\u5143\u7d20\nrear: int = deq[-1]  # \u4f47\u5217\u5c3e\u5143\u7d20\n\n# \u5143\u7d20\u51fa\u5217\npop_front: int = deq.popleft()  # \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\npop_rear: int = deq.pop()       # \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n# \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\nsize: int = len(deq)\n\n# \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\nis_empty: bool = len(deq) == 0\n</code></pre> deque.cpp<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\ndeque&lt;int&gt; deque;\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push_back(2);   // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.push_front(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint front = deque.front(); // \u4f47\u5217\u9996\u5143\u7d20\nint back = deque.back();   // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\ndeque.pop_front();  // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\ndeque.pop_back();   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.offerLast(2);   // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.offerFirst(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u4f47\u5217\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\nint popFirst = deque.pollFirst();  // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\nint popLast = deque.pollLast();    // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c07\u93c8\u7d50\u4e32\u5217 LinkedList \u770b\u4f5c\u96d9\u5411\u4f47\u5217\u4f86\u4f7f\u7528\nLinkedList&lt;int&gt; deque = new();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.AddLast(2);   // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.AddFirst(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u4f47\u5217\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\ndeque.RemoveFirst();  // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\ndeque.RemoveLast();   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = deque.Count;\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c07 list \u4f5c\u70ba\u96d9\u5411\u4f47\u5217\u4f7f\u7528\ndeque := list.New()\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.PushBack(2)      // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.PushFront(1)\n\n/* \u8a2a\u554f\u5143\u7d20 */\nfront := deque.Front() // \u4f47\u5217\u9996\u5143\u7d20\nrear := deque.Back()   // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\ndeque.Remove(front)    // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\ndeque.Remove(rear)     // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nsize := deque.Len()\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// Swift \u6c92\u6709\u5167\u5efa\u7684\u96d9\u5411\u4f47\u5217\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u96d9\u5411\u4f47\u5217\u4f86\u4f7f\u7528\nvar deque: [Int] = []\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.append(2) // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.insert(1, at: 0)\n\n/* \u8a2a\u554f\u5143\u7d20 */\nlet peekFirst = deque.first! // \u4f47\u5217\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u4f7f\u7528 Array \u6a21\u64ec\u6642 popFirst \u7684\u8907\u96dc\u5ea6\u70ba O(n)\nlet popFirst = deque.removeFirst() // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\nlet popLast = deque.removeLast() // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nlet size = deque.count\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// JavaScript \u6c92\u6709\u5167\u5efa\u7684\u96d9\u7aef\u4f47\u5217\uff0c\u53ea\u80fd\u628a Array \u7576\u4f5c\u96d9\u7aef\u4f47\u5217\u4f86\u4f7f\u7528\nconst deque = [];\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cunshift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nconst peekFirst = deque[0];\nconst peekLast = deque[deque.length - 1];\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cshift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\nconst popFront = deque.shift();\nconst popBack = deque.pop();\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nconst size = deque.length;\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// TypeScript \u6c92\u6709\u5167\u5efa\u7684\u96d9\u7aef\u4f47\u5217\uff0c\u53ea\u80fd\u628a Array \u7576\u4f5c\u96d9\u7aef\u4f47\u5217\u4f86\u4f7f\u7528\nconst deque: number[] = [];\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cunshift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconst peekLast: number = deque[deque.length - 1];\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cshift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\nconst popFront: number = deque.shift() as number;\nconst popBack: number = deque.pop() as number;\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nconst size: number = deque.length;\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u7fa9\u70ba\u96d9\u5411\u4f47\u5217\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.addLast(2);  // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.addFirst(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint peekFirst = deque.first; // \u4f47\u5217\u9996\u5143\u7d20\nint peekLast = deque.last;   // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\nint popFirst = deque.removeFirst(); // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\nint popLast = deque.removeLast();   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = deque.length;\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push_back(2);  // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.push_front(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nif let Some(front) = deque.front() { // \u4f47\u5217\u9996\u5143\u7d20\n}\nif let Some(rear) = deque.back() {   // \u4f47\u5217\u5c3e\u5143\u7d20\n}\n\n/* \u5143\u7d20\u51fa\u5217 */\nif let Some(pop_front) = deque.pop_front() { // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\n}\nif let Some(pop_rear) = deque.pop_back() {   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n}\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u96d9\u5411\u4f47\u5217\n</code></pre> deque.kt<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\nval deque = LinkedList&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.offerLast(2)  // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.offerLast(5)\ndeque.offerLast(4)\ndeque.offerFirst(3) // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.offerFirst(1)\n\n/* \u8a2a\u554f\u5143\u7d20 */\nval peekFirst = deque.peekFirst() // \u4f47\u5217\u9996\u5143\u7d20\nval peekLast = deque.peekLast()   // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\nval popFirst = deque.pollFirst() // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\nval popLast = deque.pollLast()   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nval size = deque.size\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nval isEmpty = deque.isEmpty()\n</code></pre> deque.rb<pre><code># \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217\n# Ruby \u6c92\u6709\u5167\u76f4\u7684\u96d9\u7aef\u4f47\u5217\uff0c\u53ea\u80fd\u628a Array \u7576\u4f5c\u96d9\u7aef\u4f47\u5217\u4f86\u4f7f\u7528\ndeque = []\n\n# \u5143\u7d20\u5982\u968a\ndeque &lt;&lt; 2\ndeque &lt;&lt; 5\ndeque &lt;&lt; 4\n# \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cArray#unshift \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\ndeque.unshift(3)\ndeque.unshift(1)\n\n# \u8a2a\u554f\u5143\u7d20\npeek_first = deque.first\npeek_last = deque.last\n\n# \u5143\u7d20\u51fa\u5217\n# \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0c Array#shift \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\npop_front = deque.shift\npop_back = deque.pop\n\n# \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\nsize = deque.length\n\n# \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\nis_empty = size.zero?\n</code></pre> deque.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2 \u00a0 \u96d9\u5411\u4f47\u5217\u5be6\u73fe *","text":"<p>\u96d9\u5411\u4f47\u5217\u7684\u5be6\u73fe\u8207\u4f47\u5217\u985e\u4f3c\uff0c\u53ef\u4ee5\u9078\u64c7\u93c8\u7d50\u4e32\u5217\u6216\u9663\u5217\u4f5c\u70ba\u5e95\u5c64\u8cc7\u6599\u7d50\u69cb\u3002</p>"},{"location":"chapter_stack_and_queue/deque/#1","title":"1. \u00a0 \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7684\u5be6\u73fe","text":"<p>\u56de\u9867\u4e0a\u4e00\u7bc0\u5167\u5bb9\uff0c\u6211\u5011\u4f7f\u7528\u666e\u901a\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u4f86\u5be6\u73fe\u4f47\u5217\uff0c\u56e0\u70ba\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u522a\u9664\u982d\u7bc0\u9ede\uff08\u5c0d\u61c9\u51fa\u5217\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e\u65b0\u7bc0\u9ede\uff08\u5c0d\u61c9\u5165\u5217\u64cd\u4f5c\uff09\u3002</p> <p>\u5c0d\u65bc\u96d9\u5411\u4f47\u5217\u800c\u8a00\uff0c\u982d\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u57f7\u884c\u5165\u5217\u548c\u51fa\u5217\u64cd\u4f5c\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u96d9\u5411\u4f47\u5217\u9700\u8981\u5be6\u73fe\u53e6\u4e00\u500b\u5c0d\u7a31\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u70ba\u6b64\uff0c\u6211\u5011\u63a1\u7528\u201c\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u201d\u4f5c\u70ba\u96d9\u5411\u4f47\u5217\u7684\u5e95\u5c64\u8cc7\u6599\u7d50\u69cb\u3002</p> <p>\u5982\u5716 5-8 \u6240\u793a\uff0c\u6211\u5011\u5c07\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7684\u982d\u7bc0\u9ede\u548c\u5c3e\u7bc0\u9ede\u8996\u70ba\u96d9\u5411\u4f47\u5217\u7684\u4f47\u5217\u9996\u548c\u4f47\u5217\u5c3e\uff0c\u540c\u6642\u5be6\u73fe\u5728\u5169\u7aef\u65b0\u589e\u548c\u522a\u9664\u7bc0\u9ede\u7684\u529f\u80fd\u3002</p> LinkedListDequepush_last()push_first()pop_last()pop_first() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 5-8 \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u96d9\u5411\u4f47\u5217\u7684\u5165\u5217\u51fa\u5217\u64cd\u4f5c </p> <p>\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n        self.prev: ListNode | None = None  # \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n\nclass LinkedListDeque:\n    \"\"\"\u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._front: ListNode | None = None  # \u982d\u7bc0\u9ede front\n        self._rear: ListNode | None = None  # \u5c3e\u7bc0\u9ede rear\n        self._size: int = 0  # \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"\u5165\u5217\u64cd\u4f5c\"\"\"\n        node = ListNode(num)\n        # \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if self.is_empty():\n            self._front = self._rear = node\n        # \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        elif is_front:\n            # \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # \u66f4\u65b0\u982d\u7bc0\u9ede\n        # \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        else:\n            # \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        self._size += 1  # \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n\n    def push_first(self, num: int):\n        \"\"\"\u4f47\u5217\u9996\u5165\u5217\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"\u4f47\u5217\u5c3e\u5165\u5217\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"\u51fa\u5217\u64cd\u4f5c\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        # \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if is_front:\n            val: int = self._front.val  # \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            # \u522a\u9664\u982d\u7bc0\u9ede\n            fnext: ListNode | None = self._front.next\n            if fnext != None:\n                fnext.prev = None\n                self._front.next = None\n            self._front = fnext  # \u66f4\u65b0\u982d\u7bc0\u9ede\n        # \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        else:\n            val: int = self._rear.val  # \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            # \u522a\u9664\u5c3e\u7bc0\u9ede\n            rprev: ListNode | None = self._rear.prev\n            if rprev != None:\n                rprev.next = None\n                self._rear.prev = None\n            self._rear = rprev  # \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        self._size -= 1  # \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u4f47\u5217\u9996\u51fa\u5217\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u4f47\u5217\u5c3e\u51fa\u5217\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nstruct DoublyListNode {\n    int val;              // \u7bc0\u9ede\u503c\n    DoublyListNode *next; // \u5f8c\u7e7c\u7bc0\u9ede\u6307\u6a19\n    DoublyListNode *prev; // \u524d\u9a45\u7bc0\u9ede\u6307\u6a19\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    int queSize = 0;              // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* \u6790\u69cb\u65b9\u6cd5 */\n    ~LinkedListDeque() {\n        // \u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u522a\u9664\u7bc0\u9ede\uff0c\u91cb\u653e\u8a18\u61b6\u9ad4\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"\u4f47\u5217\u70ba\u7a7a\");\n        int val;\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if (isFront) {\n            val = front-&gt;val; // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            DoublyListNode *fNext = front-&gt;next;\n            if (fNext != nullptr) {\n                fNext-&gt;prev = nullptr;\n                front-&gt;next = nullptr;\n            }\n            delete front;\n            front = fNext; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        } else {\n            val = rear-&gt;val; // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            DoublyListNode *rPrev = rear-&gt;prev;\n            if (rPrev != nullptr) {\n                rPrev-&gt;next = nullptr;\n                rear-&gt;prev = nullptr;\n            }\n            delete rear;\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n        return rear-&gt;val;\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n    int val; // \u7bc0\u9ede\u503c\n    ListNode next; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n    ListNode prev; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    private ListNode front, rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    private int queSize = 0; // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if (isFront) {\n            val = front.val; // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            ListNode fNext = front.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front.next = null;\n            }\n            front = fNext; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        } else {\n            val = rear.val; // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            ListNode rPrev = rear.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear.prev = null;\n            }\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode(int val) {\n    public int val = val;       // \u7bc0\u9ede\u503c\n    public ListNode? next = null; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n    public ListNode? prev = null; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    ListNode? front, rear; // \u982d\u7bc0\u9ede front, \u5c3e\u7bc0\u9ede rear\n    int queSize = 0;      // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front!.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede                           \n        }\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n\n        queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if (isFront) {\n            val = front?.val; // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            ListNode? fNext = front?.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front!.next = null;\n            }\n            front = fNext;   // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        else {\n            val = rear?.val;  // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            ListNode? rPrev = rear?.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear!.prev = null;\n            }\n            rear = rPrev;    // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n\n        queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\ntype linkedListDeque struct {\n    // \u4f7f\u7528\u5167\u5efa\u5305 list\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u96d9\u7aef\u4f47\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* \u4f47\u5217\u9996\u5143\u7d20\u5165\u5217 */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* \u4f47\u5217\u5c3e\u5143\u7d20\u5165\u5217 */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217 */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217 */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u7372\u53d6 List \u7528\u65bc\u5217\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var next: ListNode? // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n    weak var prev: ListNode? // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? // \u982d\u7bc0\u9ede front\n    private var rear: ListNode? // \u5c3e\u7bc0\u9ede rear\n    private var _size: Int // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        else if isFront {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        _size += 1 // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        let val: Int\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if isFront {\n            val = front!.val // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            let fNext = front?.next\n            if fNext != nil {\n                fNext?.prev = nil\n                front?.next = nil\n            }\n            front = fNext // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        else {\n            val = rear!.val // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            let rPrev = rear?.prev\n            if rPrev != nil {\n                rPrev?.next = nil\n                rear?.prev = nil\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        _size -= 1 // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return rear!.val\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n    prev; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528 (\u6307\u6a19)\n    next; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528 (\u6307\u6a19)\n    val; // \u7bc0\u9ede\u503c\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    #front; // \u982d\u7bc0\u9ede front\n    #rear; // \u5c3e\u7bc0\u9ede rear\n    #queSize; // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        this.#queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        this.#queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // \u5132\u5b58\u5c3e\u7bc0\u9ede\u503c\n        // \u522a\u9664\u5c3e\u7bc0\u9ede\n        let temp = this.#rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.#rear.prev = null;\n        }\n        this.#rear = temp; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // \u5132\u5b58\u5c3e\u7bc0\u9ede\u503c\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        let temp = this.#front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.#front.next = null;\n        }\n        this.#front = temp; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5217\u5370\u96d9\u5411\u4f47\u5217 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n    prev: ListNode; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528 (\u6307\u6a19)\n    next: ListNode; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528 (\u6307\u6a19)\n    val: number; // \u7bc0\u9ede\u503c\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    private front: ListNode; // \u982d\u7bc0\u9ede front\n    private rear: ListNode; // \u5c3e\u7bc0\u9ede rear\n    private queSize: number; // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        this.queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        this.queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear.val; // \u5132\u5b58\u5c3e\u7bc0\u9ede\u503c\n        // \u522a\u9664\u5c3e\u7bc0\u9ede\n        let temp: ListNode = this.rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.rear.prev = null;\n        }\n        this.rear = temp; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        this.queSize--;\n        return value;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // \u5132\u5b58\u5c3e\u7bc0\u9ede\u503c\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        let temp: ListNode = this.front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.front.next = null;\n        }\n        this.front = temp; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        this.queSize--;\n        return value;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5217\u5370\u96d9\u5411\u4f47\u5217 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n  int val; // \u7bc0\u9ede\u503c\n  ListNode? next; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n  ListNode? prev; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u5c0d\u5217 */\nclass LinkedListDeque {\n  late ListNode? _front; // \u982d\u7bc0\u9ede _front\n  late ListNode? _rear; // \u5c3e\u7bc0\u9ede _rear\n  int _queSize = 0; // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u9577\u5ea6 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* \u5165\u5217\u64cd\u4f5c */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n      _front = _rear = node;\n    } else if (isFront) {\n      // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n      // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n    } else {\n      // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n      // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    }\n    _queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n  }\n\n  /* \u4f47\u5217\u9996\u5165\u5217 */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* \u4f47\u5217\u5c3e\u5165\u5217 */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* \u51fa\u5217\u64cd\u4f5c */\n  int? pop(bool isFront) {\n    // \u82e5\u4f47\u5217\u70ba\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    if (isFront) {\n      // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n      val = _front!.val; // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n      // \u522a\u9664\u982d\u7bc0\u9ede\n      ListNode? fNext = _front!.next;\n      if (fNext != null) {\n        fNext.prev = null;\n        _front!.next = null;\n      }\n      _front = fNext; // \u66f4\u65b0\u982d\u7bc0\u9ede\n    } else {\n      // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n      val = _rear!.val; // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n      // \u522a\u9664\u5c3e\u7bc0\u9ede\n      ListNode? rPrev = _rear!.prev;\n      if (rPrev != null) {\n        rPrev.next = null;\n        _rear!.prev = null;\n      }\n      _rear = rPrev; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    }\n    _queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    return val;\n  }\n\n  /* \u4f47\u5217\u9996\u51fa\u5217 */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // \u7bc0\u9ede\u503c\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5f8c\u7e7c\u7bc0\u9ede\u6307\u6a19\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u524d\u9a45\u7bc0\u9ede\u6307\u6a19\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u982d\u7bc0\u9ede front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u7bc0\u9ede rear\n    que_size: usize,                         // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    pub fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        if is_front {\n            match self.front.take() {\n                // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // \u66f4\u65b0\u982d\u7bc0\u9ede\n                }\n            }\n        }\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        else {\n            match self.rear.take() {\n                // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n                }\n            }\n        }\n        self.que_size += 1; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    pub fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // \u82e5\u4f47\u5217\u70ba\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n        if self.is_empty() {\n            return None;\n        };\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // \u66f4\u65b0\u982d\u7bc0\u9ede\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n                Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n            })\n        }\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n                Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n            })\n        }\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\ntypedef struct DoublyListNode {\n    int val;                     // \u7bc0\u9ede\u503c\n    struct DoublyListNode *next; // \u5f8c\u7e7c\u7bc0\u9ede\n    struct DoublyListNode *prev; // \u524d\u9a45\u7bc0\u9ede\n} DoublyListNode;\n\n/* \u5efa\u69cb\u5b50 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\ntypedef struct {\n    DoublyListNode *front, *rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    int queSize;                  // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n} LinkedListDeque;\n\n/* \u5efa\u69cb\u5b50 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // \u91cb\u653e\u6240\u6709\u7bc0\u9ede\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91cb\u653e deque \u7d50\u69cb\u9ad4\n    free(deque);\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* \u5165\u5217 */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n    else if (isFront) {\n        // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n    }\n    // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n    else {\n        // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n}\n\n/* \u4f47\u5217\u9996\u5165\u5217 */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* \u4f47\u5217\u5c3e\u5165\u5217 */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* \u51fa\u5217 */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n    if (isFront) {\n        val = peekFirst(deque); // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        if (fNext) {\n            fNext-&gt;prev = NULL;\n            deque-&gt;front-&gt;next = NULL;\n        }\n        delDoublyListNode(deque-&gt;front);\n        deque-&gt;front = fNext; // \u66f4\u65b0\u982d\u7bc0\u9ede\n    }\n    // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n    else {\n        val = peekLast(deque); // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        if (rPrev) {\n            rPrev-&gt;next = NULL;\n            deque-&gt;rear-&gt;prev = NULL;\n        }\n        delDoublyListNode(deque-&gt;rear);\n        deque-&gt;rear = rPrev; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    }\n    deque-&gt;queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    return val;\n}\n\n/* \u4f47\u5217\u9996\u51fa\u5217 */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* \u4f47\u5217\u5c3e\u51fa\u5217 */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* \u5217\u5370\u4f47\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // \u8907\u88fd\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u8cc7\u6599\u5230\u9663\u5217\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode(var _val: Int) {\n    // \u7bc0\u9ede\u503c\n    var next: ListNode? = null // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n    var prev: ListNode? = null // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? = null // \u982d\u7bc0\u9ede front\n    private var rear: ListNode? = null // \u5c3e\u7bc0\u9ede rear\n    private var queSize: Int = 0 // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        } else if (isFront) {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u982d\u7bc0\u9ede\n            // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize++ // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) \n            throw IndexOutOfBoundsException()\n        val _val: Int\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if (isFront) {\n            _val = front!!._val // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            val fNext = front!!.next\n            if (fNext != null) {\n                fNext.prev = null\n                front!!.next = null\n            }\n            front = fNext // \u66f4\u65b0\u982d\u7bc0\u9ede\n            // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        } else {\n            _val = rear!!._val // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            val rPrev = rear!!.prev\n            if (rPrev != null) {\n                rPrev.next = null\n                rear!!.prev = null\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize-- // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return _val\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!._val\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>=begin\nFile: linkedlist_deque.rb\nCreated Time: 2024-04-06\nAuthor: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)\n=end\n\n### \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\nclass ListNode\n  attr_accessor :val\n  attr_accessor :next # \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n  attr_accessor :prev # \u524d\u8ec0\u7bc0\u9ede\u5f15\u7528\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(val)\n    @val = val\n  end\nend\n\n### \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 ###\nclass LinkedListDeque\n  ### \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 ###\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @front = nil  # \u982d\u7bc0\u9ede front\n    @rear = nil   # \u5c3e\u7bc0\u9ede rear\n    @size = 0     # \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n  end\n\n  ### \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### \u5165\u5217\u64cd\u4f5c ###\n  def push(num, is_front)\n    node = ListNode.new(num)\n    # \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c \u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n    if is_empty?\n      @front = @rear = node\n    # \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n    elsif is_front\n      # \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n      @front.prev = node\n      node.next = @front\n      @front = node # \u66f4\u65b0\u982d\u7bc0\u9ede\n    # \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n    else\n      # \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n      @rear.next = node\n      node.prev = @rear\n      @rear = node # \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    end\n    @size += 1 # \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n  end\n\n  ### \u4f47\u5217\u9996\u5165\u5217 ###\n  def push_first(num)\n    push(num, true)\n  end\n\n  ### \u4f47\u5217\u5c3e\u5165\u5217 ###\n  def push_last(num)\n    push(num, false)\n  end\n\n  ### \u51fa\u5217\u64cd\u4f5c ###\n  def pop(is_front)\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    # \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n    if is_front\n      val = @front.val # \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n      # \u522a\u9664\u982d\u7bc0\u9ede\n      fnext = @front.next\n      unless fnext.nil?\n        fnext.prev = nil\n        @front.next = nil\n      end\n      @front = fnext # \u66f4\u65b0\u982d\u7bc0\u9ede\n    # \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n    else\n      val = @rear.val # \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n      # \u522a\u9664\u5c3e\u7bc0\u9ede\n      rprev = @rear.prev\n      unless rprev.nil?\n        rprev.next = nil\n        @rear.prev = nil\n      end\n      @rear = rprev # \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    end\n    @size -= 1 # \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n\n    val\n  end\n\n  ### \u4f47\u5217\u9996\u51fa\u5217 ###\n  def pop_first\n    pop(true)\n  end\n\n  ### \u4f47\u5217\u9996\u51fa\u5217 ###\n  def pop_last\n    pop(false)\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 ###\n  def peek_first\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @front.val\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 ###\n  def peek_last\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @rear.val\n  end\n\n  ### \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 ###\n  def to_array\n    node = @front\n    res = Array.new(size, 0)\n    for i in 0...size\n      res[i] = node.val\n      node = node.next\n    end\n    res\n  end\nend\n</code></pre> linkedlist_deque.zig<pre><code>// \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\nfn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = undefined,     // \u7bc0\u9ede\u503c\n        next: ?*Self = null,    // \u5f8c\u7e7c\u7bc0\u9ede\u6307\u6a19\n        prev: ?*Self = null,    // \u524d\u9a45\u7bc0\u9ede\u6307\u6a19\n\n        // Initialize a list node with specific value\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n\n// \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217\nfn LinkedListDeque(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*ListNode(T) = null,                    // \u982d\u7bc0\u9ede front\n        rear: ?*ListNode(T) = null,                     // \u5c3e\u7bc0\u9ede rear\n        que_size: usize = 0,                             // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u4f47\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u5165\u5217\u64cd\u4f5c\n        pub fn push(self: *Self, num: T, is_front: bool) !void {\n            var node = try self.mem_allocator.create(ListNode(T));\n            node.init(num);\n            // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n            if (self.isEmpty()) {\n                self.front = node;\n                self.rear = node;\n            // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n            } else if (is_front) {\n                // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n                self.front.?.prev = node;\n                node.next = self.front;\n                self.front = node;  // \u66f4\u65b0\u982d\u7bc0\u9ede\n            // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n            } else {\n                // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n                self.rear.?.next = node;\n                node.prev = self.rear;\n                self.rear = node;   // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n            }\n            self.que_size += 1;      // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        } \n\n        // \u4f47\u5217\u9996\u5165\u5217\n        pub fn pushFirst(self: *Self, num: T) !void {\n            try self.push(num, true);\n        } \n\n        // \u4f47\u5217\u5c3e\u5165\u5217\n        pub fn pushLast(self: *Self, num: T) !void {\n            try self.push(num, false);\n        } \n\n        // \u51fa\u5217\u64cd\u4f5c\n        pub fn pop(self: *Self, is_front: bool) T {\n            if (self.isEmpty()) @panic(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n            var val: T = undefined;\n            // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n            if (is_front) {\n                val = self.front.?.val;     // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n                // \u522a\u9664\u982d\u7bc0\u9ede\n                var fNext = self.front.?.next;\n                if (fNext != null) {\n                    fNext.?.prev = null;\n                    self.front.?.next = null;\n                }\n                self.front = fNext;         // \u66f4\u65b0\u982d\u7bc0\u9ede\n            // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n            } else {\n                val = self.rear.?.val;      // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n                // \u522a\u9664\u5c3e\u7bc0\u9ede\n                var rPrev = self.rear.?.prev;\n                if (rPrev != null) {\n                    rPrev.?.next = null;\n                    self.rear.?.prev = null;\n                }\n                self.rear = rPrev;          // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n            }\n            self.que_size -= 1;              // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n            return val;\n        } \n\n        // \u4f47\u5217\u9996\u51fa\u5217\n        pub fn popFirst(self: *Self) T {\n            return self.pop(true);\n        } \n\n        // \u4f47\u5217\u5c3e\u51fa\u5217\n        pub fn popLast(self: *Self) T {\n            return self.pop(false);\n        } \n\n        // \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\n        pub fn peekFirst(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20\n        pub fn peekLast(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n            return self.rear.?.val;\n        }\n\n        // \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#2","title":"2. \u00a0 \u57fa\u65bc\u9663\u5217\u7684\u5be6\u73fe","text":"<p>\u5982\u5716 5-9 \u6240\u793a\uff0c\u8207\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u4f47\u5217\u985e\u4f3c\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u4f7f\u7528\u74b0\u5f62\u9663\u5217\u4f86\u5be6\u73fe\u96d9\u5411\u4f47\u5217\u3002</p> ArrayDequepush_last()push_first()pop_last()pop_first() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 5-9 \u00a0 \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u96d9\u5411\u4f47\u5217\u7684\u5165\u5217\u51fa\u5217\u64cd\u4f5c </p> <p>\u5728\u4f47\u5217\u7684\u5be6\u73fe\u57fa\u790e\u4e0a\uff0c\u50c5\u9700\u589e\u52a0\u201c\u4f47\u5217\u9996\u5165\u5217\u201d\u548c\u201c\u4f47\u5217\u5c3e\u51fa\u5217\u201d\u7684\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"\u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"\u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15\"\"\"\n        # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        # \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        # \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"\u4f47\u5217\u9996\u5165\u5217\"\"\"\n        if self._size == self.capacity():\n            print(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        self._front = self.index(self._front - 1)\n        # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"\u4f47\u5217\u5c3e\u5165\u5217\"\"\"\n        if self._size == self.capacity():\n            print(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        # \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        rear = self.index(self._front + self._size)\n        # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u4f47\u5217\u9996\u51fa\u5217\"\"\"\n        num = self.peek_first()\n        # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u4f47\u5217\u5c3e\u51fa\u5217\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        # \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        # \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;        // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize;      // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    int index(int i) {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\" &lt;&lt; endl;\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\" &lt;&lt; endl;\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    int popFirst() {\n        int num = peekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    private int[] nums; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private int front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private int queSize; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    private int index(int i) {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    public int popFirst() {\n        int num = peekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    public int[] toArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    int[] nums;  // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;   // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    int Index(int i) {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = Index(front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        int rear = Index(front + queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    public int[] ToArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\ntype arrayDeque struct {\n    nums        []int // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    front       int   // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    queSize     int   // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n    queCapacity int   // \u4f47\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7d0d\u5143\u7d20\u6578\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u4f47\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u4f47\u5217\u9996\u5165\u5217 */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n        return\n    }\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n    q.front = q.index(q.front - 1)\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* \u4f47\u5217\u5c3e\u5165\u5217 */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n        return\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    rear := q.index(q.front + q.queSize)\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u4f47\u5217\u9996\u51fa\u5217 */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    if num == nil {\n        return nil\n    }\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* \u4f47\u5217\u5c3e\u51fa\u5217 */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    if num == nil {\n        return nil\n    }\n    q.queSize--\n    return num\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* \u7372\u53d6 Slice \u7528\u65bc\u5217\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n    // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    private var nums: [Int] // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private var front: Int // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private var _size: Int // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    private func index(i: Int) -&gt; Int {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        (i + capacity()) % capacity()\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = index(i: front - 1)\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num\n        _size += 1\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        let rear = index(i: front + size())\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    func toArray() -&gt; [Int] {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    #nums; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    #front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    #queSize; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    index(i) {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u96d9\u5411\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        this.#front = this.index(this.#front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u96d9\u5411\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        const rear = this.index(this.#front + this.#queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    popFirst() {\n        const num = this.peekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    toArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    private nums: number[]; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private front: number; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private queSize: number; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    index(i: number): number {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u96d9\u5411\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        this.front = this.index(this.front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u96d9\u5411\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        const rear: number = this.index(this.front + this.queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.nums[this.front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    toArray(): number[] {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n  late int _front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n  late int _queSize; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n  /* \u5efa\u69cb\u5b50 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n  int index(int i) {\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + capacity()) % capacity();\n  }\n\n  /* \u4f47\u5217\u9996\u5165\u5217 */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n    }\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe _front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n    _front = index(_front - 1);\n    // \u5c07 _num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* \u4f47\u5217\u5c3e\u5165\u5217 */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    int rear = index(_front + _queSize);\n    // \u5c07 _num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u4f47\u5217\u9996\u51fa\u5217 */\n  int popFirst() {\n    int _num = peekFirst();\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n    }\n    // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n  List&lt;int&gt; toArray() {\n    // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nstruct ArrayDeque {\n    nums: Vec&lt;i32&gt;,  // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    front: usize,    // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    que_size: usize, // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n}\n\nimpl ArrayDeque {\n    /* \u5efa\u69cb\u5b50 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![0; capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    pub fn push_first(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        self.front = self.index(self.front as i32 - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    pub fn push_last(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    fn pop_first(&amp;mut self) -&gt; i32 {\n        let num = self.peek_first();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    fn pop_last(&amp;mut self) -&gt; i32 {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fn peek_first(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        };\n        self.nums[self.front]\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    fn peek_last(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        };\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        let mut res = vec![0; self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;       // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e + 1\n    int queCapacity; // \u4f47\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u5efa\u69cb\u5b50 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // \u521d\u59cb\u5316\u9663\u5217\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u56de\u5230\u982d\u90e8\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u4f47\u5217\u9996\u5165\u5217 */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\\r\\n\");\n        return;\n    }\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u56de\u5230\u5c3e\u90e8\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // \u5c07 num \u65b0\u589e\u5230\u4f47\u5217\u9996\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u4f47\u5217\u5c3e\u5165\u5217 */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\\r\\n\");\n        return;\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n    // \u8a2a\u554f\u7570\u5e38\uff1a\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n    // \u8a2a\u554f\u7570\u5e38\uff1a\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* \u4f47\u5217\u9996\u51fa\u5217 */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u4f47\u5217\u5c3e\u51fa\u5217 */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\nint *toArray(ArrayDeque *deque, int *queSize) {\n    *queSize = deque-&gt;queSize;\n    int *res = (int *)calloc(deque-&gt;queSize, sizeof(int));\n    int j = deque-&gt;front;\n    for (int i = 0; i &lt; deque-&gt;queSize; i++) {\n        res[i] = deque-&gt;nums[j % deque-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_deque.kt<pre><code>/* \u5efa\u69cb\u5b50 */\nclass ArrayDeque(capacity: Int) {\n    private var nums: IntArray = IntArray(capacity) // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private var front: Int = 0 // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private var queSize: Int = 0 // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    private fun index(i: Int): Int {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity()\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1)\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num\n        queSize++\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        val rear = index(front + queSize)\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    fun toArray(): IntArray {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>### \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 ###\nclass ArrayDeque\n  ### \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 ###\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(capacity)\n    @nums = Array.new(capacity, 0)\n    @front = 0\n    @size = 0\n  end\n\n  ### \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf ###\n  def capacity\n    @nums.length\n  end\n\n  ### \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### \u4f47\u5217\u9996\u5165\u5217 ###\n  def push_first(num)\n    if size == capacity\n      puts '\u96d9\u5411\u4f47\u5217\u5df2\u6eff'\n      return\n    end\n\n    # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n    # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n    @front = index(@front - 1)\n    # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n    @nums[@front] = num\n    @size += 1\n  end\n\n  ### \u4f47\u5217\u5c3e\u5165\u5217 ###\n  def push_last(num)\n    if size == capacity\n      puts '\u96d9\u5411\u4f47\u5217\u5df2\u6eff'\n      return\n    end\n\n    # \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    rear = index(@front + size)\n    # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### \u4f47\u5217\u9996\u51fa\u5217 ###\n  def pop_first\n    num = peek_first\n    # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    @front = index(@front + 1)\n    @size -= 1\n    num\n  end\n\n  ### \u4f47\u5217\u5c3e\u51fa\u5217 ###\n  def pop_last\n    num = peek_last\n    @size -= 1\n    num\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 ###\n  def peek_first\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 ###\n  def peek_last\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    # \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last = index(@front + size - 1)\n    @nums[last]\n  end\n\n  ### \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 ###\n  def to_array\n    # \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    res = []\n    for i in 0...size\n      res &lt;&lt; @nums[index(@front + i)]\n    end\n    res\n  end\n\n  private\n\n  ### \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 ###\n  def index(i)\n    # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n    # \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n    # \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n    (i + capacity) % capacity\n  end\nend\n</code></pre> array_deque.zig<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3 \u00a0 \u96d9\u5411\u4f47\u5217\u61c9\u7528","text":"<p>\u96d9\u5411\u4f47\u5217\u517c\u5177\u5806\u758a\u8207\u4f47\u5217\u7684\u908f\u8f2f\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5be6\u73fe\u9019\u5169\u8005\u7684\u6240\u6709\u61c9\u7528\u5834\u666f\uff0c\u540c\u6642\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002</p> <p>\u6211\u5011\u77e5\u9053\uff0c\u8edf\u9ad4\u7684\u201c\u64a4\u92b7\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u5806\u758a\u4f86\u5be6\u73fe\uff1a\u7cfb\u7d71\u5c07\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c <code>push</code> \u5230\u5806\u758a\u4e2d\uff0c\u7136\u5f8c\u900f\u904e <code>pop</code> \u5be6\u73fe\u64a4\u92b7\u3002\u7136\u800c\uff0c\u8003\u616e\u5230\u7cfb\u7d71\u8cc7\u6e90\u7684\u9650\u5236\uff0c\u8edf\u9ad4\u901a\u5e38\u6703\u9650\u5236\u64a4\u92b7\u7684\u6b65\u6578\uff08\u4f8b\u5982\u50c5\u5141\u8a31\u5132\u5b58 \\(50\\) \u6b65\uff09\u3002\u7576\u5806\u758a\u7684\u9577\u5ea6\u8d85\u904e \\(50\\) \u6642\uff0c\u8edf\u9ad4\u9700\u8981\u5728\u5806\u758a\u5e95\uff08\u4f47\u5217\u9996\uff09\u57f7\u884c\u522a\u9664\u64cd\u4f5c\u3002\u4f46\u5806\u758a\u7121\u6cd5\u5be6\u73fe\u8a72\u529f\u80fd\uff0c\u6b64\u6642\u5c31\u9700\u8981\u4f7f\u7528\u96d9\u5411\u4f47\u5217\u4f86\u66ff\u4ee3\u5806\u758a\u3002\u8acb\u6ce8\u610f\uff0c\u201c\u64a4\u92b7\u201d\u7684\u6838\u5fc3\u908f\u8f2f\u4ecd\u7136\u9075\u5faa\u5806\u758a\u7684\u5148\u5165\u5f8c\u51fa\u539f\u5247\uff0c\u53ea\u662f\u96d9\u5411\u4f47\u5217\u80fd\u5920\u66f4\u52a0\u9748\u6d3b\u5730\u5be6\u73fe\u4e00\u4e9b\u984d\u5916\u908f\u8f2f\u3002</p>"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 \u4f47\u5217","text":"<p>\u4f47\u5217\uff08queue\uff09\u662f\u4e00\u7a2e\u9075\u5faa\u5148\u5165\u5148\u51fa\u898f\u5247\u7684\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u3002\u9867\u540d\u601d\u7fa9\uff0c\u4f47\u5217\u6a21\u64ec\u4e86\u6392\u968a\u73fe\u8c61\uff0c\u5373\u65b0\u4f86\u7684\u4eba\u4e0d\u65b7\u52a0\u5165\u4f47\u5217\u5c3e\u90e8\uff0c\u800c\u4f4d\u65bc\u4f47\u5217\u982d\u90e8\u7684\u4eba\u9010\u500b\u96e2\u958b\u3002</p> <p>\u5982\u5716 5-4 \u6240\u793a\uff0c\u6211\u5011\u5c07\u4f47\u5217\u982d\u90e8\u7a31\u70ba\u201c\u4f47\u5217\u9996\u201d\uff0c\u5c3e\u90e8\u7a31\u70ba\u201c\u4f47\u5217\u5c3e\u201d\uff0c\u5c07\u628a\u5143\u7d20\u52a0\u5165\u5217\u5c3e\u7684\u64cd\u4f5c\u7a31\u70ba\u201c\u5165\u5217\u201d\uff0c\u522a\u9664\u4f47\u5217\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u7a31\u70ba\u201c\u51fa\u5217\u201d\u3002</p> <p></p> <p> \u5716 5-4 \u00a0 \u4f47\u5217\u7684\u5148\u5165\u5148\u51fa\u898f\u5247 </p>"},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1 \u00a0 \u4f47\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u4f47\u5217\u7684\u5e38\u898b\u64cd\u4f5c\u5982\u8868 5-2 \u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7a0b\u5f0f\u8a9e\u8a00\u7684\u65b9\u6cd5\u540d\u7a31\u53ef\u80fd\u6703\u6709\u6240\u4e0d\u540c\u3002\u6211\u5011\u5728\u6b64\u63a1\u7528\u8207\u5806\u758a\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002</p> <p> \u8868 5-2 \u00a0 \u4f47\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u6642\u9593\u8907\u96dc\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u5217\uff0c\u5373\u5c07\u5143\u7d20\u65b0\u589e\u81f3\u4f47\u5217\u5c3e \\(O(1)\\) <code>pop()</code> \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217 \\(O(1)\\) <code>peek()</code> \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 \\(O(1)\\) <p>\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u73fe\u6210\u7684\u4f47\u5217\u985e\u5225\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig queue.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u4f47\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u5011\u4e00\u822c\u5c07\u96d9\u5411\u4f47\u5217\u985e\u5225 deque \u7576\u4f5c\u4f47\u5217\u4f7f\u7528\n# \u96d6\u7136 queue.Queue() \u662f\u7d14\u6b63\u7684\u4f47\u5217\u985e\u5225\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u63a8\u85a6\nque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u5217\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\nfront: int = que[0]\n\n# \u5143\u7d20\u51fa\u5217\npop: int = que.popleft()\n\n# \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\nsize: int = len(que)\n\n# \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\nqueue&lt;int&gt; queue;\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint front = queue.front();\n\n/* \u5143\u7d20\u51fa\u5217 */\nqueue.pop();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek = queue.peek();\n\n/* \u5143\u7d20\u51fa\u5217 */\nint pop = queue.poll();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\nQueue&lt;int&gt; queue = new();\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n\n/* \u5143\u7d20\u51fa\u5217 */\nint pop = queue.Dequeue();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = queue.Count;\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c07 list \u4f5c\u70ba\u4f47\u5217\u4f86\u4f7f\u7528\nqueue := list.New()\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\npeek := queue.Front()\n\n/* \u5143\u7d20\u51fa\u5217 */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nsize := queue.Len()\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// Swift \u6c92\u6709\u5167\u5efa\u7684\u4f47\u5217\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u4f47\u5217\u4f86\u4f7f\u7528\nvar queue: [Int] = []\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nlet peek = queue.first!\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u7531\u65bc\u662f\u9663\u5217\uff0c\u56e0\u6b64 removeFirst \u7684\u8907\u96dc\u5ea6\u70ba O(n)\nlet pool = queue.removeFirst()\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nlet size = queue.count\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// JavaScript \u6c92\u6709\u5167\u5efa\u7684\u4f47\u5217\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u4f47\u5217\u4f86\u4f7f\u7528\nconst queue = [];\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u5e95\u5c64\u662f\u9663\u5217\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\nconst pop = queue.shift();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// TypeScript \u6c92\u6709\u5167\u5efa\u7684\u4f47\u5217\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u4f47\u5217\u4f86\u4f7f\u7528\nconst queue: number[] = [];\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u5e95\u5c64\u662f\u9663\u5217\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\nconst pop = queue.shift();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u4f47\u5217\u985e\u5225 Qeque \u662f\u96d9\u5411\u4f47\u5217\uff0c\u4e5f\u53ef\u4f5c\u70ba\u4f47\u5217\u4f7f\u7528\nQueue&lt;int&gt; queue = Queue();\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek = queue.first;\n\n/* \u5143\u7d20\u51fa\u5217 */\nint pop = queue.removeFirst();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = queue.length;\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// \u5728 Rust \u4e2d\u4f7f\u7528\u96d9\u5411\u4f47\u5217\u4f5c\u70ba\u666e\u901a\u4f47\u5217\u4f86\u4f7f\u7528\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nif let Some(front) = deque.front() {\n}\n\n/* \u5143\u7d20\u51fa\u5217 */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u4f47\u5217\n</code></pre> queue.kt<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\nval queue = LinkedList&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.offer(1)\nqueue.offer(3)\nqueue.offer(2)\nqueue.offer(5)\nqueue.offer(4)\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nval peek = queue.peek()\n\n/* \u5143\u7d20\u51fa\u5217 */\nval pop = queue.poll()\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nval size = queue.size\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nval isEmpty = queue.isEmpty()\n</code></pre> queue.rb<pre><code># \u521d\u59cb\u5316\u4f47\u5217\n# Ruby \u5167\u5efa\u7684\u4f47\u5217\uff08Thread::Queue) \u6c92\u6709 peek \u548c\u8d70\u8a2a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u4f47\u5217\u4f86\u4f7f\u7528\nqueue = []\n\n# \u5143\u7d20\u5165\u5217\nqueue.push(1)\nqueue.push(3)\nqueue.push(2)\nqueue.push(5)\nqueue.push(4)\n\n# \u8a2a\u554f\u4f47\u5217\u5143\u7d20\npeek = queue.first\n\n# \u5143\u7d20\u51fa\u5217\n# \u6e05\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cArray#shift \u65b9\u6cd5\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\npop = queue.shift\n\n# \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\nsize = queue.length\n\n# \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\nis_empty = queue.empty?\n</code></pre> queue.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2 \u00a0 \u4f47\u5217\u5be6\u73fe","text":"<p>\u70ba\u4e86\u5be6\u73fe\u4f47\u5217\uff0c\u6211\u5011\u9700\u8981\u4e00\u7a2e\u8cc7\u6599\u7d50\u69cb\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u65b0\u589e\u5143\u7d20\uff0c\u4e26\u5728\u53e6\u4e00\u7aef\u522a\u9664\u5143\u7d20\uff0c\u93c8\u7d50\u4e32\u5217\u548c\u9663\u5217\u90fd\u7b26\u5408\u8981\u6c42\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#1","title":"1. \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u7684\u5be6\u73fe","text":"<p>\u5982\u5716 5-5 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u7684\u201c\u982d\u7bc0\u9ede\u201d\u548c\u201c\u5c3e\u7bc0\u9ede\u201d\u5206\u5225\u8996\u70ba\u201c\u4f47\u5217\u9996\u201d\u548c\u201c\u4f47\u5217\u5c3e\u201d\uff0c\u898f\u5b9a\u4f47\u5217\u5c3e\u50c5\u53ef\u65b0\u589e\u7bc0\u9ede\uff0c\u4f47\u5217\u9996\u50c5\u53ef\u522a\u9664\u7bc0\u9ede\u3002</p> LinkedListQueuepush()pop() <p></p> <p></p> <p></p> <p> \u5716 5-5 \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u4f47\u5217\u7684\u5165\u5217\u51fa\u5217\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u7528\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u4f47\u5217\u7684\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._front: ListNode | None = None  # \u982d\u7bc0\u9ede front\n        self._rear: ListNode | None = None  # \u5c3e\u7bc0\u9ede rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u5217\"\"\"\n        # \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        node = ListNode(num)\n        # \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u5217\"\"\"\n        num = self.peek()\n        # \u522a\u9664\u982d\u7bc0\u9ede\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u4f47\u5217\u70ba\u7a7a\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f49\u5316\u70ba\u4e32\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // \u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u522a\u9664\u7bc0\u9ede\uff0c\u91cb\u653e\u8a18\u61b6\u9ad4\n        freeMemoryLinkedList(front);\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u5217 */\n    void push(int num) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        ListNode *node = new ListNode(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    int pop() {\n        int num = peek();\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"\u4f47\u5217\u70ba\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Vector \u4e26\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    private ListNode front, rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5217 */\n    public void push(int num) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        ListNode node = new ListNode(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (front == null) {\n            front = node;\n            rear = node;\n        // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    public int pop() {\n        int num = peek();\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u5217 */\n    public void Push(int num) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        ListNode node = new(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (front == null) {\n            front = node;\n            rear = node;\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    public int Pop() {\n        int num = Peek();\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\ntype linkedListQueue struct {\n    // \u4f7f\u7528\u5167\u5efa\u5305 list \u4f86\u5be6\u73fe\u4f47\u5217\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u4f47\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u5217 */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u5217 */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u7372\u53d6 List \u7528\u65bc\u5217\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    private var front: ListNode? // \u982d\u7bc0\u9ede\n    private var rear: ListNode? // \u5c3e\u7bc0\u9ede\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u5217 */\n    func push(num: Int) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        let node = ListNode(x: num)\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* \u51fa\u5217 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    #front; // \u982d\u7bc0\u9ede #front\n    #rear; // \u5c3e\u7bc0\u9ede #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u5217 */\n    push(num) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        const node = new ListNode(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    pop() {\n        const num = this.peek();\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peek() {\n        if (this.size === 0) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        return this.#front.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    private front: ListNode | null; // \u982d\u7bc0\u9ede front\n    private rear: ListNode | null; // \u5c3e\u7bc0\u9ede rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u5217 */\n    push(num: number): void {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        const node = new ListNode(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.size === 0) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        return this.front!.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n  ListNode? _front; // \u982d\u7bc0\u9ede _front\n  ListNode? _rear; // \u5c3e\u7bc0\u9ede _rear\n  int _queSize = 0; // \u4f47\u5217\u9577\u5ea6\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u5217 */\n  void push(int _num) {\n    // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e _num\n    final node = ListNode(_num);\n    // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* \u51fa\u5217 */\n  int pop() {\n    final int _num = peek();\n    // \u522a\u9664\u982d\u7bc0\u9ede\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('\u4f47\u5217\u70ba\u7a7a');\n    }\n    return _front!.val;\n  }\n\n  /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u982d\u7bc0\u9ede front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u7bc0\u9ede rear\n    que_size: usize,                         // \u4f47\u5217\u7684\u9577\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u5217 */\n    pub fn push(&amp;mut self, num: T) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u5217 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* \u5efa\u69cb\u5b50 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // \u91cb\u653e\u6240\u6709\u7bc0\u9ede\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91cb\u653e queue \u7d50\u69cb\u9ad4\n    free(queue);\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* \u5165\u5217 */\nvoid push(LinkedListQueue *queue, int num) {\n    // \u5c3e\u7bc0\u9ede\u8655\u65b0\u589e node\n    ListNode *node = newListNode(num);\n    // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* \u51fa\u5217 */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* \u5217\u5370\u4f47\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // \u8907\u88fd\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u8cc7\u6599\u5230\u9663\u5217\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue(\n    // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u5217 */\n    fun push(num: Int) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        val node = ListNode(num)\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (front == null) {\n            front = node\n            rear = node\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* \u51fa\u5217 */\n    fun pop(): Int {\n        val num = peek()\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>### \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u982d\u73fe\u7684\u4f47\u5217 ###\nclass LinkedListQueue\n  ### \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 ###\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @front = nil  # \u982d\u7bc0\u9ede front\n    @rear = nil   # \u5c3e\u7bc0\u9ede rear\n    @size = 0\n  end\n\n  ### \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    @front.nil?\n  end\n\n  ### \u5165\u5217 ###\n  def push(num)\n    # \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n    node = ListNode.new(num)\n\n    # \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\uff0c\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n    if @front.nil?\n      @front = node\n      @rear = node\n    # \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u4ee4\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n    else\n      @rear.next = node\n      @rear = node\n    end\n\n    @size += 1\n  end\n\n  ### \u51fa\u5217 ###\n  def pop\n    num = peek\n    # \u522a\u9664\u982d\u7bc0\u9ede\n    @front = @front.next\n    @size -= 1\n    num\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 ###\n  def peek\n    raise IndexError, '\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @front.val\n  end\n\n  ### \u5c07\u93c8\u7d50\u4e32\u5217\u70ba Array \u4e26\u8fd4\u56de ###\n  def to_array\n    queue = []\n    temp = @front\n    while temp\n      queue &lt;&lt; temp.val\n      temp = temp.next\n    end\n    queue\n  end\nend\n</code></pre> linkedlist_queue.zig<pre><code>// \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217\nfn LinkedListQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*inc.ListNode(T) = null,                // \u982d\u7bc0\u9ede front\n        rear: ?*inc.ListNode(T) = null,                 // \u5c3e\u7bc0\u9ede rear\n        que_size: usize = 0,                            // \u4f47\u5217\u7684\u9577\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u4f47\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u4f47\u5217\u70ba\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u5165\u5217\n        pub fn push(self: *Self, num: T) !void {\n            // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n            if (self.front == null) {\n                self.front = node;\n                self.rear = node;\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n            } else {\n                self.rear.?.next = node;\n                self.rear = node;\n            }\n            self.que_size += 1;\n        } \n\n        // \u51fa\u5217\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            self.front = self.front.?.next;\n            self.que_size -= 1;\n            return num;\n        } \n\n        // \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#2","title":"2. \u00a0 \u57fa\u65bc\u9663\u5217\u7684\u5be6\u73fe","text":"<p>\u5728\u9663\u5217\u4e2d\u522a\u9664\u9996\u5143\u7d20\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u9019\u6703\u5c0e\u81f4\u51fa\u5217\u64cd\u4f5c\u6548\u7387\u8f03\u4f4e\u3002\u7136\u800c\uff0c\u6211\u5011\u53ef\u4ee5\u63a1\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u4f86\u907f\u514d\u9019\u500b\u554f\u984c\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u4e00\u500b\u8b8a\u6578 <code>front</code> \u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u4e26\u7dad\u8b77\u4e00\u500b\u8b8a\u6578 <code>size</code> \u7528\u65bc\u8a18\u9304\u4f47\u5217\u9577\u5ea6\u3002\u5b9a\u7fa9 <code>rear = front + size</code> \uff0c\u9019\u500b\u516c\u5f0f\u8a08\u7b97\u51fa\u7684 <code>rear</code> \u6307\u5411\u4f47\u5217\u5c3e\u5143\u7d20\u4e4b\u5f8c\u7684\u4e0b\u4e00\u500b\u4f4d\u7f6e\u3002</p> <p>\u57fa\u65bc\u6b64\u8a2d\u8a08\uff0c\u9663\u5217\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u5340\u9593\u70ba <code>[front, rear - 1]</code>\uff0c\u5404\u7a2e\u64cd\u4f5c\u7684\u5be6\u73fe\u65b9\u6cd5\u5982\u5716 5-6 \u6240\u793a\u3002</p> <ul> <li>\u5165\u5217\u64cd\u4f5c\uff1a\u5c07\u8f38\u5165\u5143\u7d20\u8ce6\u503c\u7d66 <code>rear</code> \u7d22\u5f15\u8655\uff0c\u4e26\u5c07 <code>size</code> \u589e\u52a0 1 \u3002</li> <li>\u51fa\u5217\u64cd\u4f5c\uff1a\u53ea\u9700\u5c07 <code>front</code> \u589e\u52a0 1 \uff0c\u4e26\u5c07 <code>size</code> \u6e1b\u5c11 1 \u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u5230\uff0c\u5165\u5217\u548c\u51fa\u5217\u64cd\u4f5c\u90fd\u53ea\u9700\u9032\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(1)\\) \u3002</p> ArrayQueuepush()pop() <p></p> <p></p> <p></p> <p> \u5716 5-6 \u00a0 \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u4f47\u5217\u7684\u5165\u5217\u51fa\u5217\u64cd\u4f5c </p> <p>\u4f60\u53ef\u80fd\u6703\u767c\u73fe\u4e00\u500b\u554f\u984c\uff1a\u5728\u4e0d\u65b7\u9032\u884c\u5165\u5217\u548c\u51fa\u5217\u7684\u904e\u7a0b\u4e2d\uff0c<code>front</code> \u548c <code>rear</code> \u90fd\u5728\u5411\u53f3\u79fb\u52d5\uff0c\u7576\u5b83\u5011\u5230\u9054\u9663\u5217\u5c3e\u90e8\u6642\u5c31\u7121\u6cd5\u7e7c\u7e8c\u79fb\u52d5\u4e86\u3002\u70ba\u4e86\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u9663\u5217\u8996\u70ba\u9996\u5c3e\u76f8\u63a5\u7684\u201c\u74b0\u5f62\u9663\u5217\u201d\u3002</p> <p>\u5c0d\u65bc\u74b0\u5f62\u9663\u5217\uff0c\u6211\u5011\u9700\u8981\u8b93 <code>front</code> \u6216 <code>rear</code> \u5728\u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u76f4\u63a5\u56de\u5230\u9663\u5217\u982d\u90e8\u7e7c\u7e8c\u8d70\u8a2a\u3002\u9019\u7a2e\u9031\u671f\u6027\u898f\u5f8b\u53ef\u4ee5\u900f\u904e\u201c\u53d6\u9918\u64cd\u4f5c\u201d\u4f86\u5be6\u73fe\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"\u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._nums: list[int] = [0] * size  # \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n        self._front: int = 0  # \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n        self._size: int = 0  # \u4f47\u5217\u9577\u5ea6\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u5217\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"\u4f47\u5217\u5df2\u6eff\")\n        # \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        rear: int = (self._front + self._size) % self.capacity()\n        # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u5217\"\"\"\n        num: int = self.peek()\n        # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u4f47\u5217\u70ba\u7a7a\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u4e32\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n  private:\n    int *nums;       // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;       // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize;     // \u4f47\u5217\u9577\u5ea6\n    int queCapacity; // \u4f47\u5217\u5bb9\u91cf\n\n  public:\n    ArrayQueue(int capacity) {\n        // \u521d\u59cb\u5316\u9663\u5217\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5217 */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"\u4f47\u5217\u5df2\u6eff\" &lt;&lt; endl;\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        int rear = (front + queSize) % queCapacity;\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    int pop() {\n        int num = peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"\u4f47\u5217\u70ba\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u5c07\u9663\u5217\u8f49\u5316\u70ba Vector \u4e26\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    private int[] nums; // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private int front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private int queSize; // \u4f47\u5217\u9577\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u5217 */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        int rear = (front + queSize) % capacity();\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    public int pop() {\n        int num = peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    public int[] toArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    int[] nums;  // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;   // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize; // \u4f47\u5217\u9577\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u5217 */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        int rear = (front + queSize) % Capacity();\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    public int Pop() {\n        int num = Peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    public int[] ToArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\ntype arrayQueue struct {\n    nums        []int // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    front       int   // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    queSize     int   // \u4f47\u5217\u9577\u5ea6\n    queCapacity int   // \u4f47\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7d0d\u5143\u7d20\u6578\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u4f47\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u5165\u5217 */\nfunc (q *arrayQueue) push(num int) {\n    // \u7576 rear == queCapacity \u8868\u793a\u4f47\u5217\u5df2\u6eff\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n    rear := (q.front + q.queSize) % q.queCapacity\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u51fa\u5217 */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    if num == nil {\n        return nil\n    }\n\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u7372\u53d6 Slice \u7528\u65bc\u5217\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    private var nums: [Int] // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private var front: Int // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private var _size: Int // \u4f47\u5217\u9577\u5ea6\n\n    init(capacity: Int) {\n        // \u521d\u59cb\u5316\u9663\u5217\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u5217 */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        let rear = (front + size()) % capacity()\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u51fa\u5217 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    func toArray() -&gt; [Int] {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    #nums; // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    #front = 0; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    #queSize = 0; // \u4f47\u5217\u9577\u5ea6\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5165\u5217 */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        const rear = (this.#front + this.size) % this.capacity;\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    pop() {\n        const num = this.peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peek() {\n        if (this.isEmpty()) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    private nums: number[]; // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private front: number; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private queSize: number; // \u4f47\u5217\u9577\u5ea6\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5165\u5217 */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        const rear = (this.front + this.queSize) % this.capacity;\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    pop(): number {\n        const num = this.peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        return this.nums[this.front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray(): number[] {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n  late int _front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n  late int _queSize; // \u4f47\u5217\u9577\u5ea6\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u5217 */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"\u4f47\u5217\u5df2\u6eff\");\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n    int rear = (_front + _queSize) % capaCity();\n    // \u5c07 _num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u51fa\u5217 */\n  int pop() {\n    int _num = peek();\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u4f47\u5217\u70ba\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8fd4\u56de Array */\n  List&lt;int&gt; toArray() {\n    // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nstruct ArrayQueue {\n    nums: Vec&lt;i32&gt;,    // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    front: i32,        // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    que_size: i32,     // \u4f47\u5217\u9577\u5ea6\n    que_capacity: i32, // \u4f47\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n    /* \u5efa\u69cb\u5b50 */\n    fn new(capacity: i32) -&gt; ArrayQueue {\n        ArrayQueue {\n            nums: vec![0; capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u5165\u5217 */\n    fn push(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u5217 */\n    fn pop(&amp;mut self) -&gt; i32 {\n        let num = self.peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![0; self.que_size as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;       // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e + 1\n    int queCapacity; // \u4f47\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u5efa\u69cb\u5b50 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // \u521d\u59cb\u5316\u9663\u5217\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* \u5165\u5217 */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"\u4f47\u5217\u5df2\u6eff\\r\\n\");\n        return;\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* \u51fa\u5217 */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\nint *toArray(ArrayQueue *queue, int *queSize) {\n    *queSize = queue-&gt;queSize;\n    int *res = (int *)calloc(queue-&gt;queSize, sizeof(int));\n    int j = queue-&gt;front;\n    for (int i = 0; i &lt; queue-&gt;queSize; i++) {\n        res[i] = queue-&gt;nums[j % queue-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_queue.kt<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue(capacity: Int) {\n    private val nums: IntArray = IntArray(capacity) // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private var front: Int = 0 // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private var queSize: Int = 0 // \u4f47\u5217\u9577\u5ea6\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u5165\u5217 */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        val rear = (front + queSize) % capacity()\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u51fa\u5217 */\n    fun pop(): Int {\n        val num = peek()\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    fun toArray(): IntArray {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>### \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 ###\nclass ArrayQueue\n  ### \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 ###\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(size)\n    @nums = Array.new(size, 0) # \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    @front = 0 # \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    @size = 0 # \u4f47\u5217\u9577\u5ea6\n  end\n\n  ### \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf ###\n  def capacity\n    @nums.length\n  end\n\n  ### \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### \u5165\u5217 ###\n  def push(num)\n    raise IndexError, '\u4f47\u5217\u5df2\u6eff' if size == capacity\n\n    # \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n    rear = (@front + size) % capacity\n    # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### \u51fa\u5217 ###\n  def pop\n    num = peek\n    # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n    @front = (@front + 1) % capacity\n    @size -= 1\n    num\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 ###\n  def peek\n    raise IndexError, '\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### \u8fd4\u56de\u4e32\u5217\u7528\u65bc\u5217\u5370 ###\n  def to_array\n    res = Array.new(size, 0)\n    j = @front\n\n    for i in 0...size\n      res[i] = @nums[j % capacity]\n      j += 1\n    end\n\n    res\n  end\nend\n</code></pre> array_queue.zig<pre><code>// \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217\nfn ArrayQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        nums: []T = undefined,                          // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217     \n        cap: usize = 0,                                 // \u4f47\u5217\u5bb9\u91cf\n        front: usize = 0,                               // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n        queSize: usize = 0,                             // \u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e + 1\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u9663\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.cap = cap;\n            self.nums = try self.mem_allocator.alloc(T, self.cap);\n            @memset(self.nums, @as(T, 0));\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.cap;\n        }\n\n        // \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.queSize;\n        }\n\n        // \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.queSize == 0;\n        }\n\n        // \u5165\u5217\n        pub fn push(self: *Self, num: T) !void {\n            if (self.size() == self.capacity()) {\n                std.debug.print(\"\u4f47\u5217\u5df2\u6eff\\n\", .{});\n                return;\n            }\n            // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n            // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n            var rear = (self.front + self.queSize) % self.capacity();\n            // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n            self.nums[rear] = num;\n            self.queSize += 1;\n        } \n\n        // \u51fa\u5217\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n            self.front = (self.front + 1) % self.capacity();\n            self.queSize -= 1;\n            return num;\n        } \n\n        // \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u4f47\u5217\u70ba\u7a7a\");\n            return self.nums[self.front];\n        } \n\n        // \u8fd4\u56de\u9663\u5217\n        pub fn toArray(self: *Self) ![]T {\n            // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            var j: usize = self.front;\n            while (i &lt; self.size()) : ({ i += 1; j += 1; }) {\n                res[i] = self.nums[j % self.capacity()];\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u4ee5\u4e0a\u5be6\u73fe\u7684\u4f47\u5217\u4ecd\u7136\u5177\u6709\u4fb7\u9650\u6027\uff1a\u5176\u9577\u5ea6\u4e0d\u53ef\u8b8a\u3002\u7136\u800c\uff0c\u9019\u500b\u554f\u984c\u4e0d\u96e3\u89e3\u6c7a\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u9663\u5217\u66ff\u63db\u70ba\u52d5\u614b\u9663\u5217\uff0c\u5f9e\u800c\u5f15\u5165\u64f4\u5bb9\u6a5f\u5236\u3002\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u5617\u8a66\u81ea\u884c\u5be6\u73fe\u3002</p> <p>\u5169\u7a2e\u5be6\u73fe\u7684\u5c0d\u6bd4\u7d50\u8ad6\u8207\u5806\u758a\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d05\u8ff0\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3 \u00a0 \u4f47\u5217\u5178\u578b\u61c9\u7528","text":"<ul> <li>\u6dd8\u5bf6\u8a02\u55ae\u3002\u8cfc\u7269\u8005\u4e0b\u55ae\u5f8c\uff0c\u8a02\u55ae\u5c07\u52a0\u5165\u5217\u5217\u4e2d\uff0c\u7cfb\u7d71\u96a8\u5f8c\u6703\u6839\u64da\u9806\u5e8f\u8655\u7406\u4f47\u5217\u4e2d\u7684\u8a02\u55ae\u3002\u5728\u96d9\u5341\u4e00\u671f\u9593\uff0c\u77ed\u6642\u9593\u5167\u6703\u7522\u751f\u6d77\u91cf\u8a02\u55ae\uff0c\u9ad8\u4f75\u767c\u6210\u70ba\u5de5\u7a0b\u5e2b\u5011\u9700\u8981\u91cd\u9ede\u653b\u514b\u7684\u554f\u984c\u3002</li> <li>\u5404\u985e\u5f85\u8fa6\u4e8b\u9805\u3002\u4efb\u4f55\u9700\u8981\u5be6\u73fe\u201c\u5148\u4f86\u5f8c\u5230\u201d\u529f\u80fd\u7684\u5834\u666f\uff0c\u4f8b\u5982\u5370\u8868\u6a5f\u7684\u4efb\u52d9\u4f47\u5217\u3001\u9910\u5ef3\u7684\u51fa\u9910\u4f47\u5217\u7b49\uff0c\u4f47\u5217\u5728\u9019\u4e9b\u5834\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7dad\u8b77\u8655\u7406\u9806\u5e8f\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/stack/","title":"5.1 \u00a0 \u5806\u758a","text":"<p>\u5806\u758a\uff08stack\uff09\u662f\u4e00\u7a2e\u9075\u5faa\u5148\u5165\u5f8c\u51fa\u908f\u8f2f\u7684\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u5c07\u5806\u758a\u985e\u6bd4\u70ba\u684c\u9762\u4e0a\u7684\u4e00\u758a\u76e4\u5b50\uff0c\u5982\u679c\u60f3\u53d6\u51fa\u5e95\u90e8\u7684\u76e4\u5b50\uff0c\u5247\u9700\u8981\u5148\u5c07\u4e0a\u9762\u7684\u76e4\u5b50\u4f9d\u6b21\u79fb\u8d70\u3002\u6211\u5011\u5c07\u76e4\u5b50\u66ff\u63db\u70ba\u5404\u7a2e\u578b\u5225\u7684\u5143\u7d20\uff08\u5982\u6574\u6578\u3001\u5b57\u5143\u3001\u7269\u4ef6\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u5806\u758a\u9019\u7a2e\u8cc7\u6599\u7d50\u69cb\u3002</p> <p>\u5982\u5716 5-1 \u6240\u793a\uff0c\u6211\u5011\u628a\u5806\u7a4d\u758a\u5143\u7d20\u7684\u9802\u90e8\u7a31\u70ba\u201c\u5806\u758a\u9802\u201d\uff0c\u5e95\u90e8\u7a31\u70ba\u201c\u5806\u758a\u5e95\u201d\u3002\u5c07\u628a\u5143\u7d20\u65b0\u589e\u5230\u5806\u758a\u9802\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u5165\u5806\u758a\u201d\uff0c\u522a\u9664\u5806\u758a\u9802\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u51fa\u5806\u758a\u201d\u3002</p> <p></p> <p> \u5716 5-1 \u00a0 \u5806\u758a\u7684\u5148\u5165\u5f8c\u51fa\u898f\u5247 </p>"},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1 \u00a0 \u5806\u758a\u7684\u5e38\u7528\u64cd\u4f5c","text":"<p>\u5806\u758a\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-1 \u6240\u793a\uff0c\u5177\u9ad4\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u64da\u6240\u4f7f\u7528\u7684\u7a0b\u5f0f\u8a9e\u8a00\u4f86\u78ba\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u5011\u4ee5\u5e38\u898b\u7684 <code>push()</code>\u3001<code>pop()</code>\u3001<code>peek()</code> \u547d\u540d\u70ba\u4f8b\u3002</p> <p> \u8868 5-1 \u00a0 \u5806\u758a\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5 \u63cf\u8ff0 \u6642\u9593\u8907\u96dc\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u5806\u758a\uff08\u65b0\u589e\u81f3\u5806\u758a\u9802\uff09 \\(O(1)\\) <code>pop()</code> \u5806\u758a\u9802\u5143\u7d20\u51fa\u5806\u758a \\(O(1)\\) <code>peek()</code> \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 \\(O(1)\\) <p>\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8a9e\u8a00\u53ef\u80fd\u6c92\u6709\u5c08\u9580\u63d0\u4f9b\u5806\u758a\u985e\u5225\uff0c\u9019\u6642\u6211\u5011\u53ef\u4ee5\u5c07\u8a72\u8a9e\u8a00\u7684\u201c\u9663\u5217\u201d\u6216\u201c\u93c8\u7d50\u4e32\u5217\u201d\u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\uff0c\u4e26\u5728\u7a0b\u5f0f\u908f\u8f2f\u4e0a\u5ffd\u7565\u8207\u5806\u758a\u7121\u95dc\u7684\u64cd\u4f5c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig stack.py<pre><code># \u521d\u59cb\u5316\u5806\u758a\n# Python \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a list \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nstack: list[int] = []\n\n# \u5143\u7d20\u5165\u5806\u758a\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\npeek: int = stack[-1]\n\n# \u5143\u7d20\u51fa\u5806\u758a\npop: int = stack.pop()\n\n# \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\nsize: int = len(stack)\n\n# \u5224\u65b7\u662f\u5426\u70ba\u7a7a\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\nstack&lt;int&gt; stack;\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint top = stack.top();\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nstack.pop(); // \u7121\u8fd4\u56de\u503c\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek = stack.peek();\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nint pop = stack.pop();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\nStack&lt;int&gt; stack = new();\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek = stack.Peek();\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nint pop = stack.Pop();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size = stack.Count;\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// \u5728 Go \u4e2d\uff0c\u63a8\u85a6\u5c07 Slice \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nvar stack []int\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\npeek := stack[len(stack)-1]\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nsize := len(stack)\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// Swift \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nvar stack: [Int] = []\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nlet peek = stack.last!\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nlet pop = stack.removeLast()\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nlet size = stack.count\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// JavaScript \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nconst stack = [];\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nconst pop = stack.pop();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// TypeScript \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nconst stack: number[] = [];\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nconst pop = stack.pop();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// Dart \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a List \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nList&lt;int&gt; stack = [];\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek = stack.last;\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nint pop = stack.removeLast();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size = stack.length;\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// \u628a Vec \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nlet top = stack.last().unwrap();\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nlet pop = stack.pop().unwrap();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nlet size = stack.len();\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u5806\u758a\n</code></pre> stack.kt<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\nval stack = Stack&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1)\nstack.push(3)\nstack.push(2)\nstack.push(5)\nstack.push(4)\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nval peek = stack.peek()\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nval pop = stack.pop()\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nval size = stack.size\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nval isEmpty = stack.isEmpty()\n</code></pre> stack.rb<pre><code># \u521d\u59cb\u5316\u5806\u758a\n# Ruby \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nstack = []\n\n# \u5143\u7d20\u5165\u5806\u758a\nstack &lt;&lt; 1\nstack &lt;&lt; 3\nstack &lt;&lt; 2\nstack &lt;&lt; 5\nstack &lt;&lt; 4\n\n# \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\npeek = stack.last\n\n# \u5143\u7d20\u51fa\u5806\u758a\npop = stack.pop\n\n# \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\nsize = stack.length\n\n# \u5224\u65b7\u662f\u5426\u70ba\u7a7a\nis_empty = stack.empty?\n</code></pre> stack.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2 \u00a0 \u5806\u758a\u7684\u5be6\u73fe","text":"<p>\u70ba\u4e86\u6df1\u5165\u77ad\u89e3\u5806\u758a\u7684\u57f7\u884c\u6a5f\u5236\uff0c\u6211\u5011\u4f86\u5617\u8a66\u81ea\u5df1\u5be6\u73fe\u4e00\u500b\u5806\u758a\u985e\u5225\u3002</p> <p>\u5806\u758a\u9075\u5faa\u5148\u5165\u5f8c\u51fa\u7684\u539f\u5247\uff0c\u56e0\u6b64\u6211\u5011\u53ea\u80fd\u5728\u5806\u758a\u9802\u65b0\u589e\u6216\u522a\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u65b0\u589e\u548c\u522a\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u5806\u758a\u53ef\u4ee5\u8996\u70ba\u4e00\u7a2e\u53d7\u9650\u5236\u7684\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u6211\u5011\u53ef\u4ee5\u201c\u906e\u853d\u201d\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u7684\u90e8\u5206\u7121\u95dc\u64cd\u4f5c\uff0c\u4f7f\u5176\u5c0d\u5916\u8868\u73fe\u7684\u908f\u8f2f\u7b26\u5408\u5806\u758a\u7684\u7279\u6027\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#1","title":"1. \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u7684\u5be6\u73fe","text":"<p>\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u5806\u758a\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u7684\u982d\u7bc0\u9ede\u8996\u70ba\u5806\u758a\u9802\uff0c\u5c3e\u7bc0\u9ede\u8996\u70ba\u5806\u758a\u5e95\u3002</p> <p>\u5982\u5716 5-2 \u6240\u793a\uff0c\u5c0d\u65bc\u5165\u5806\u758a\u64cd\u4f5c\uff0c\u6211\u5011\u53ea\u9700\u5c07\u5143\u7d20\u63d2\u5165\u93c8\u7d50\u4e32\u5217\u982d\u90e8\uff0c\u9019\u7a2e\u7bc0\u9ede\u63d2\u5165\u65b9\u6cd5\u88ab\u7a31\u70ba\u201c\u982d\u63d2\u6cd5\u201d\u3002\u800c\u5c0d\u65bc\u51fa\u5806\u758a\u64cd\u4f5c\uff0c\u53ea\u9700\u5c07\u982d\u7bc0\u9ede\u5f9e\u93c8\u7d50\u4e32\u5217\u4e2d\u522a\u9664\u5373\u53ef\u3002</p> LinkedListStackpush()pop() <p></p> <p></p> <p></p> <p> \u5716 5-2 \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u5806\u758a\u7684\u5165\u5806\u758a\u51fa\u5806\u758a\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u5806\u758a\u7684\u793a\u4f8b\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, val: int):\n        \"\"\"\u5165\u5806\u758a\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u5806\u758a\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u5806\u758a\u70ba\u7a7a\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f49\u5316\u70ba\u4e32\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    int stkSize;        // \u5806\u758a\u7684\u9577\u5ea6\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // \u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u522a\u9664\u7bc0\u9ede\uff0c\u91cb\u653e\u8a18\u61b6\u9ad4\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u5806\u758a\u70ba\u7a7a\");\n        return stackTop-&gt;val;\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    private ListNode stackPeek; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    private int stkSize = 0; // \u5806\u758a\u7684\u9577\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    ListNode? stackPeek;  // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    int stkSize = 0;   // \u5806\u758a\u7684\u9577\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\ntype linkedListStack struct {\n    // \u4f7f\u7528\u5167\u5efa\u5305 list \u4f86\u5be6\u73fe\u5806\u758a\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u5806\u758a */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u5806\u758a */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u5806\u758a */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u7372\u53d6 List \u7528\u65bc\u5217\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    private var _peek: ListNode? // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    private var _size: Int // \u5806\u758a\u7684\u9577\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u5806\u758a */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* \u51fa\u5806\u758a */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u5806\u758a\u70ba\u7a7a\")\n        }\n        return _peek!.val\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    #stackPeek; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    #stkSize = 0; // \u5806\u758a\u7684\u9577\u5ea6\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.#stackPeek.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    private stkSize: number = 0; // \u5806\u758a\u7684\u9577\u5ea6\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.stackPeek.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u985e\u5225\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n  ListNode? _stackPeek; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n  int _stkSize = 0; // \u5806\u758a\u7684\u9577\u5ea6\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* \u5165\u5806\u758a */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* \u51fa\u5806\u758a */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"\u5806\u758a\u70ba\u7a7a\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba List \u4e26\u8fd4\u56de */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    stk_size: usize,                              // \u5806\u758a\u7684\u9577\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* \u51fa\u5806\u758a */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            match old_head.borrow_mut().next.take() {\n                Some(new_head) =&gt; {\n                    self.stack_peek = Some(new_head);\n                }\n                None =&gt; {\n                    self.stack_peek = None;\n                }\n            }\n            self.stk_size -= 1;\n            Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.push(node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\ntypedef struct {\n    ListNode *top; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    int size;      // \u5806\u758a\u7684\u9577\u5ea6\n} LinkedListStack;\n\n/* \u5efa\u69cb\u5b50 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* \u5165\u5806\u758a */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // \u66f4\u65b0\u65b0\u52a0\u7bc0\u9ede\u6307\u6a19\u57df\n    node-&gt;val = num;     // \u66f4\u65b0\u65b0\u52a0\u7bc0\u9ede\u8cc7\u6599\u57df\n    s-&gt;top = node;       // \u66f4\u65b0\u5806\u758a\u9802\n    s-&gt;size++;           // \u66f4\u65b0\u5806\u758a\u5927\u5c0f\n}\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"\u5806\u758a\u70ba\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* \u51fa\u5806\u758a */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    private var stkSize: Int = 0 // \u5806\u758a\u7684\u9577\u5ea6\n) {\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u5806\u758a */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* \u51fa\u5806\u758a */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--\n        return num\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?._val\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?._val!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>### \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a ###\nclass LinkedListStack\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @size = 0\n  end\n\n  ### \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    @peek.nil?\n  end\n\n  ### \u5165\u5806\u758a ###\n  def push(val)\n    node = ListNode.new(val)\n    node.next = @peek\n    @peek = node\n    @size += 1\n  end\n\n  ### \u51fa\u5806\u758a ###\n  def pop\n    num = peek\n    @peek = @peek.next\n    @size -= 1\n    num\n  end\n\n  ### \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 ###\n  def peek\n    raise IndexError, '\u5806\u758a\u70ba\u7a7a' if is_empty?\n\n    @peek.val\n  end\n\n  ### \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u53cd\u56de ###\n  def to_array\n    arr = []\n    node = @peek\n    while node\n      arr &lt;&lt; node.val\n      node = node.next\n    end\n    arr.reverse\n  end\nend\n</code></pre> linkedlist_stack.zig<pre><code>// \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a\nfn LinkedListStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack_top: ?*inc.ListNode(T) = null,             // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n        stk_size: usize = 0,                             // \u5806\u758a\u7684\u9577\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,    // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u5806\u758a\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.stack_top = null;\n            self.stk_size = 0;\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stk_size;\n        }\n\n        // \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u5806\u758a\u70ba\u7a7a\");\n            return self.stack_top.?.val;\n        }  \n\n        // \u5165\u5806\u758a\n        pub fn push(self: *Self, num: T) !void {\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            node.next = self.stack_top;\n            self.stack_top = node;\n            self.stk_size += 1;\n        } \n\n        // \u51fa\u5806\u758a\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            self.stack_top = self.stack_top.?.next;\n            self.stk_size -= 1;\n            return num;\n        } \n\n        // \u5c07\u5806\u758a\u8f49\u63db\u70ba\u9663\u5217\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.stack_top;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[res.len - i - 1] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#2","title":"2. \u00a0 \u57fa\u65bc\u9663\u5217\u7684\u5be6\u73fe","text":"<p>\u4f7f\u7528\u9663\u5217\u5be6\u73fe\u5806\u758a\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u9663\u5217\u7684\u5c3e\u90e8\u4f5c\u70ba\u5806\u758a\u9802\u3002\u5982\u5716 5-3 \u6240\u793a\uff0c\u5165\u5806\u758a\u8207\u51fa\u5806\u758a\u64cd\u4f5c\u5206\u5225\u5c0d\u61c9\u5728\u9663\u5217\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\u8207\u522a\u9664\u5143\u7d20\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(1)\\) \u3002</p> ArrayStackpush()pop() <p></p> <p></p> <p></p> <p> \u5716 5-3 \u00a0 \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u5806\u758a\u7684\u5165\u5806\u758a\u51fa\u5806\u758a\u64cd\u4f5c </p> <p>\u7531\u65bc\u5165\u5806\u758a\u7684\u5143\u7d20\u53ef\u80fd\u6703\u6e90\u6e90\u4e0d\u65b7\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u52d5\u614b\u9663\u5217\uff0c\u9019\u6a23\u5c31\u7121\u9808\u81ea\u884c\u8655\u7406\u9663\u5217\u64f4\u5bb9\u554f\u984c\u3002\u4ee5\u4e0b\u70ba\u793a\u4f8b\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_stack.py<pre><code>class ArrayStack:\n    \"\"\"\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self.size() == 0\n\n    def push(self, item: int):\n        \"\"\"\u5165\u5806\u758a\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u5806\u758a\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u5806\u758a\u70ba\u7a7a\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u5806\u758a\u70ba\u7a7a\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u4e32\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u5806\u758a\u70ba\u7a7a\");\n        return stack.back();\n    }\n\n    /* \u8fd4\u56de Vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\n        stack = [];\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\ntype arrayStack struct {\n    data []int // \u8cc7\u6599\n}\n\n/* \u521d\u59cb\u5316\u5806\u758a */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // \u8a2d\u5b9a\u5806\u758a\u7684\u9577\u5ea6\u70ba 0\uff0c\u5bb9\u91cf\u70ba 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* \u5806\u758a\u7684\u9577\u5ea6 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* \u5806\u758a\u662f\u5426\u70ba\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* \u5165\u5806\u758a */\nfunc (s *arrayStack) push(v int) {\n    // \u5207\u7247\u6703\u81ea\u52d5\u64f4\u5bb9\n    s.data = append(s.data, v)\n}\n\n/* \u51fa\u5806\u758a */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* \u7372\u53d6\u5806\u758a\u9802\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* \u7372\u53d6 Slice \u7528\u65bc\u5217\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // \u521d\u59cb\u5316\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\n        stack = []\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* \u5165\u5806\u758a */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* \u51fa\u5806\u758a */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u5806\u758a\u70ba\u7a7a\")\n        }\n        return stack.removeLast()\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u5806\u758a\u70ba\u7a7a\")\n        }\n        return stack.last!\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    pop() {\n        if (this.isEmpty()) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.#stack.pop();\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    top() {\n        if (this.isEmpty()) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.stack.pop();\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* \u5165\u5806\u758a */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* \u51fa\u5806\u758a */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"\u5806\u758a\u70ba\u7a7a\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u5806\u758a\u70ba\u7a7a\");\n    }\n    return _stack.last;\n  }\n\n  /* \u5c07\u5806\u758a\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* \u521d\u59cb\u5316\u5806\u758a */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* \u5165\u5806\u758a */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"\u5806\u758a\u70ba\u7a7a\")\n        };\n        self.stack.last()\n    }\n\n    /* \u8fd4\u56de &amp;Vec */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* \u5efa\u69cb\u5b50 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // \u521d\u59cb\u5316\u4e00\u500b\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u64f4\u5bb9\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* \u5165\u5806\u758a */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"\u5806\u758a\u5df2\u6eff\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"\u5806\u758a\u70ba\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* \u51fa\u5806\u758a */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    // \u521d\u59cb\u5316\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\n    private val stack = mutableListOf&lt;Int&gt;()\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u5806\u758a */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* \u51fa\u5806\u758a */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toTypedArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>### \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a ###\nclass ArrayStack\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @stack = []\n  end\n\n  ### \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 ###\n  def size\n    @stack.length\n  end\n\n  ### \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    @stack.empty?\n  end\n\n  ### \u5165\u5806\u758a ###\n  def push(item)\n    @stack &lt;&lt; item\n  end\n\n  ### \u51fa\u5806\u758a ###\n  def pop\n    raise IndexError, '\u5806\u758a\u70ba\u7a7a' if is_empty?\n\n    @stack.pop\n  end\n\n  ### \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 ###\n  def peek\n    raise IndexError, '\u5806\u758a\u70ba\u7a7a' if is_empty?\n\n    @stack.last\n  end\n\n  ### \u8fd4\u56de\u4e32\u5217\u7528\u65bc\u5217\u5370 ###\n  def to_array\n    @stack\n  end\nend\n</code></pre> array_stack.zig<pre><code>// \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\nfn ArrayStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack: ?std.ArrayList(T) = null,     \n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u5806\u758a\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n            if (self.stack == null) {\n                self.stack = std.ArrayList(T).init(allocator);\n            }\n        }\n\n        // \u6790\u69cb\u65b9\u6cd5\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.stack == null) return;\n            self.stack.?.deinit();\n        }\n\n        // \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stack.?.items.len;\n        }\n\n        // \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u5806\u758a\u70ba\u7a7a\");\n            return self.stack.?.items[self.size() - 1];\n        }  \n\n        // \u5165\u5806\u758a\n        pub fn push(self: *Self, num: T) !void {\n            try self.stack.?.append(num);\n        } \n\n        // \u51fa\u5806\u758a\n        pub fn pop(self: *Self) T {\n            var num = self.stack.?.pop();\n            return num;\n        } \n\n        // \u8fd4\u56de ArrayList\n        pub fn toList(self: *Self) std.ArrayList(T) {\n            return self.stack.?;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3 \u00a0 \u5169\u7a2e\u5be6\u73fe\u5c0d\u6bd4","text":"<p>\u652f\u6301\u64cd\u4f5c</p> <p>\u5169\u7a2e\u5be6\u73fe\u90fd\u652f\u6301\u5806\u758a\u5b9a\u7fa9\u4e2d\u7684\u5404\u9805\u64cd\u4f5c\u3002\u9663\u5217\u5be6\u73fe\u984d\u5916\u652f\u6301\u96a8\u6a5f\u8a2a\u554f\uff0c\u4f46\u9019\u5df2\u8d85\u51fa\u4e86\u5806\u758a\u7684\u5b9a\u7fa9\u7bc4\u7587\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u6703\u7528\u5230\u3002</p> <p>\u6642\u9593\u6548\u7387</p> <p>\u5728\u57fa\u65bc\u9663\u5217\u7684\u5be6\u73fe\u4e2d\uff0c\u5165\u5806\u758a\u548c\u51fa\u5806\u758a\u64cd\u4f5c\u90fd\u5728\u9810\u5148\u5206\u914d\u597d\u7684\u9023\u7e8c\u8a18\u61b6\u9ad4\u4e2d\u9032\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u5feb\u53d6\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f03\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u5806\u758a\u6642\u8d85\u51fa\u9663\u5217\u5bb9\u91cf\uff0c\u6703\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\uff0c\u5c0e\u81f4\u8a72\u6b21\u5165\u5806\u758a\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u8b8a\u70ba \\(O(n)\\) \u3002</p> <p>\u5728\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u7684\u5be6\u73fe\u4e2d\uff0c\u93c8\u7d50\u4e32\u5217\u7684\u64f4\u5bb9\u975e\u5e38\u9748\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u9663\u5217\u64f4\u5bb9\u6642\u6548\u7387\u964d\u4f4e\u7684\u554f\u984c\u3002\u4f46\u662f\uff0c\u5165\u5806\u758a\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u7bc0\u9ede\u7269\u4ef6\u4e26\u4fee\u6539\u6307\u6a19\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5c0d\u8f03\u4f4e\u3002\u4e0d\u904e\uff0c\u5982\u679c\u5165\u5806\u758a\u5143\u7d20\u672c\u8eab\u5c31\u662f\u7bc0\u9ede\u7269\u4ef6\uff0c\u90a3\u9ebc\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9a5f\uff0c\u5f9e\u800c\u63d0\u9ad8\u6548\u7387\u3002</p> <p>\u7d9c\u4e0a\u6240\u8ff0\uff0c\u7576\u5165\u5806\u758a\u8207\u51fa\u5806\u758a\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u8cc7\u6599\u578b\u5225\u6642\uff0c\u4f8b\u5982 <code>int</code> \u6216 <code>double</code> \uff0c\u6211\u5011\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7d50\u8ad6\u3002</p> <ul> <li>\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\u5728\u89f8\u767c\u64f4\u5bb9\u6642\u6548\u7387\u6703\u964d\u4f4e\uff0c\u4f46\u7531\u65bc\u64f4\u5bb9\u662f\u4f4e\u983b\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a69\u5b9a\u7684\u6548\u7387\u8868\u73fe\u3002</li> </ul> <p>\u7a7a\u9593\u6548\u7387</p> <p>\u5728\u521d\u59cb\u5316\u4e32\u5217\u6642\uff0c\u7cfb\u7d71\u6703\u70ba\u4e32\u5217\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8a72\u5bb9\u91cf\u53ef\u80fd\u8d85\u51fa\u5be6\u969b\u9700\u6c42\uff1b\u4e26\u4e14\uff0c\u64f4\u5bb9\u6a5f\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u9032\u884c\u64f4\u5bb9\u7684\uff0c\u64f4\u5bb9\u5f8c\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5be6\u969b\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u9593\u6d6a\u8cbb\u3002</p> <p>\u7136\u800c\uff0c\u7531\u65bc\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u9700\u8981\u984d\u5916\u5132\u5b58\u6307\u6a19\uff0c\u56e0\u6b64\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u4f54\u7528\u7684\u7a7a\u9593\u76f8\u5c0d\u8f03\u5927\u3002</p> <p>\u7d9c\u4e0a\uff0c\u6211\u5011\u4e0d\u80fd\u7c21\u55ae\u5730\u78ba\u5b9a\u54ea\u7a2e\u5be6\u73fe\u66f4\u52a0\u7bc0\u7701\u8a18\u61b6\u9ad4\uff0c\u9700\u8981\u91dd\u5c0d\u5177\u9ad4\u60c5\u6cc1\u9032\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4 \u00a0 \u5806\u758a\u7684\u5178\u578b\u61c9\u7528","text":"<ul> <li>\u700f\u89bd\u5668\u4e2d\u7684\u5f8c\u9000\u8207\u524d\u9032\u3001\u8edf\u9ad4\u4e2d\u7684\u64a4\u92b7\u8207\u53cd\u64a4\u92b7\u3002\u6bcf\u7576\u6211\u5011\u958b\u555f\u65b0\u7684\u7db2\u9801\uff0c\u700f\u89bd\u5668\u5c31\u6703\u5c0d\u4e0a\u4e00\u500b\u7db2\u9801\u57f7\u884c\u5165\u5806\u758a\uff0c\u9019\u6a23\u6211\u5011\u5c31\u53ef\u4ee5\u901a\u904e\u5f8c\u9000\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u500b\u7db2\u9801\u3002\u5f8c\u9000\u64cd\u4f5c\u5be6\u969b\u4e0a\u662f\u5728\u57f7\u884c\u51fa\u5806\u758a\u3002\u5982\u679c\u8981\u540c\u6642\u652f\u6301\u5f8c\u9000\u548c\u524d\u9032\uff0c\u90a3\u9ebc\u9700\u8981\u5169\u500b\u5806\u758a\u4f86\u914d\u5408\u5be6\u73fe\u3002</li> <li>\u7a0b\u5f0f\u8a18\u61b6\u9ad4\u7ba1\u7406\u3002\u6bcf\u6b21\u547c\u53eb\u51fd\u5f0f\u6642\uff0c\u7cfb\u7d71\u90fd\u6703\u5728\u5806\u758a\u9802\u65b0\u589e\u4e00\u500b\u5806\u758a\u5e40\uff0c\u7528\u65bc\u8a18\u9304\u51fd\u5f0f\u7684\u4e0a\u4e0b\u6587\u8cc7\u8a0a\u3002\u5728\u905e\u8ff4\u51fd\u5f0f\u4e2d\uff0c\u5411\u4e0b\u905e\u63a8\u968e\u6bb5\u6703\u4e0d\u65b7\u57f7\u884c\u5165\u5806\u758a\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u968e\u6bb5\u5247\u6703\u4e0d\u65b7\u57f7\u884c\u51fa\u5806\u758a\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/","title":"5.4 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_stack_and_queue/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u5806\u758a\u662f\u4e00\u7a2e\u9075\u5faa\u5148\u5165\u5f8c\u51fa\u539f\u5247\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u53ef\u900f\u904e\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u4f86\u5be6\u73fe\u3002</li> <li>\u5728\u6642\u9593\u6548\u7387\u65b9\u9762\uff0c\u5806\u758a\u7684\u9663\u5217\u5be6\u73fe\u5177\u6709\u8f03\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u64f4\u5bb9\u904e\u7a0b\u4e2d\uff0c\u55ae\u6b21\u5165\u5806\u758a\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u6703\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5806\u758a\u7684\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u5177\u6709\u66f4\u70ba\u7a69\u5b9a\u7684\u6548\u7387\u8868\u73fe\u3002</li> <li>\u5728\u7a7a\u9593\u6548\u7387\u65b9\u9762\uff0c\u5806\u758a\u7684\u9663\u5217\u5be6\u73fe\u53ef\u80fd\u5c0e\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u9593\u6d6a\u8cbb\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u6240\u4f54\u7528\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u6bd4\u9663\u5217\u5143\u7d20\u66f4\u5927\u3002</li> <li>\u4f47\u5217\u662f\u4e00\u7a2e\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5247\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u540c\u6a23\u53ef\u4ee5\u900f\u904e\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u4f86\u5be6\u73fe\u3002\u5728\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u7684\u5c0d\u6bd4\u4e0a\uff0c\u4f47\u5217\u7684\u7d50\u8ad6\u8207\u524d\u8ff0\u5806\u758a\u7684\u7d50\u8ad6\u76f8\u4f3c\u3002</li> <li>\u96d9\u5411\u4f47\u5217\u662f\u4e00\u7a2e\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u4f47\u5217\uff0c\u5b83\u5141\u8a31\u5728\u5169\u7aef\u9032\u884c\u5143\u7d20\u7684\u65b0\u589e\u548c\u522a\u9664\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u700f\u89bd\u5668\u7684\u524d\u9032\u5f8c\u9000\u662f\u5426\u662f\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\uff1f</p> <p>\u700f\u89bd\u5668\u7684\u524d\u9032\u5f8c\u9000\u529f\u80fd\u672c\u8cea\u4e0a\u662f\u201c\u5806\u758a\u201d\u7684\u9ad4\u73fe\u3002\u7576\u7528\u6236\u8a2a\u554f\u4e00\u500b\u65b0\u9801\u9762\u6642\uff0c\u8a72\u9801\u9762\u6703\u88ab\u65b0\u589e\u5230\u5806\u758a\u9802\uff1b\u7576\u7528\u6236\u9ede\u9078\u5f8c\u9000\u6309\u9215\u6642\uff0c\u8a72\u9801\u9762\u6703\u5f9e\u5806\u758a\u9802\u5f48\u51fa\u3002\u4f7f\u7528\u96d9\u5411\u4f47\u5217\u53ef\u4ee5\u65b9\u4fbf\u5730\u5be6\u73fe\u4e00\u4e9b\u984d\u5916\u64cd\u4f5c\uff0c\u9019\u500b\u5728\u201c\u96d9\u5411\u4f47\u5217\u201d\u7ae0\u7bc0\u6709\u63d0\u5230\u3002</p> <p>Q\uff1a\u5728\u51fa\u5806\u758a\u5f8c\uff0c\u662f\u5426\u9700\u8981\u91cb\u653e\u51fa\u5806\u758a\u7bc0\u9ede\u7684\u8a18\u61b6\u9ad4\uff1f</p> <p>\u5982\u679c\u5f8c\u7e8c\u4ecd\u9700\u8981\u4f7f\u7528\u5f48\u51fa\u7bc0\u9ede\uff0c\u5247\u4e0d\u9700\u8981\u91cb\u653e\u8a18\u61b6\u9ad4\u3002\u82e5\u4e4b\u5f8c\u4e0d\u9700\u8981\u7528\u5230\uff0c<code>Java</code> \u548c <code>Python</code> \u7b49\u8a9e\u8a00\u64c1\u6709\u81ea\u52d5\u5783\u573e\u56de\u6536\u6a5f\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52d5\u91cb\u653e\u8a18\u61b6\u9ad4\uff1b\u5728 <code>C</code> \u548c <code>C++</code> \u4e2d\u9700\u8981\u624b\u52d5\u91cb\u653e\u8a18\u61b6\u9ad4\u3002</p> <p>Q\uff1a\u96d9\u5411\u4f47\u5217\u50cf\u662f\u5169\u500b\u5806\u758a\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u9ebc\uff1f</p> <p>\u96d9\u5411\u4f47\u5217\u5c31\u50cf\u662f\u5806\u758a\u548c\u4f47\u5217\u7684\u7d44\u5408\u6216\u5169\u500b\u5806\u758a\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73fe\u7684\u662f\u5806\u758a + \u4f47\u5217\u7684\u908f\u8f2f\uff0c\u56e0\u6b64\u53ef\u4ee5\u5be6\u73fe\u5806\u758a\u8207\u4f47\u5217\u7684\u6240\u6709\u61c9\u7528\uff0c\u4e26\u4e14\u66f4\u52a0\u9748\u6d3b\u3002</p> <p>Q\uff1a\u64a4\u92b7\uff08undo\uff09\u548c\u53cd\u64a4\u92b7\uff08redo\uff09\u5177\u9ad4\u662f\u5982\u4f55\u5be6\u73fe\u7684\uff1f</p> <p>\u4f7f\u7528\u5169\u500b\u5806\u758a\uff0c\u5806\u758a <code>A</code> \u7528\u65bc\u64a4\u92b7\uff0c\u5806\u758a <code>B</code> \u7528\u65bc\u53cd\u64a4\u92b7\u3002</p> <ol> <li>\u6bcf\u7576\u4f7f\u7528\u8005\u57f7\u884c\u4e00\u500b\u64cd\u4f5c\uff0c\u5c07\u9019\u500b\u64cd\u4f5c\u58d3\u5165\u5806\u758a <code>A</code> \uff0c\u4e26\u6e05\u7a7a\u5806\u758a <code>B</code> \u3002</li> <li>\u7576\u7528\u6236\u57f7\u884c\u201c\u64a4\u92b7\u201d\u6642\uff0c\u5f9e\u5806\u758a <code>A</code> \u4e2d\u5f48\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u4e26\u5c07\u5176\u58d3\u5165\u5806\u758a <code>B</code> \u3002</li> <li>\u7576\u7528\u6236\u57f7\u884c\u201c\u53cd\u64a4\u92b7\u201d\u6642\uff0c\u5f9e\u5806\u758a <code>B</code> \u4e2d\u5f48\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u4e26\u5c07\u5176\u58d3\u5165\u5806\u758a <code>A</code> \u3002</li> </ol>"},{"location":"chapter_tree/","title":"\u7b2c 7 \u7ae0 \u00a0 \u6a39","text":"<p>Abstract</p> <p>\u53c3\u5929\u5927\u6a39\u5145\u6eff\u751f\u547d\u529b\uff0c\u6839\u6df1\u8449\u8302\uff0c\u5206\u679d\u6276\u758f\u3002</p> <p>\u5b83\u70ba\u6211\u5011\u5c55\u73fe\u4e86\u8cc7\u6599\u5206\u6cbb\u7684\u751f\u52d5\u5f62\u614b\u3002</p>"},{"location":"chapter_tree/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>7.1 \u00a0 \u4e8c\u5143\u6a39</li> <li>7.2 \u00a0 \u4e8c\u5143\u6a39\u8d70\u8a2a</li> <li>7.3 \u00a0 \u4e8c\u5143\u6a39\u9663\u5217\u8868\u793a</li> <li>7.4 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39</li> <li>7.5 \u00a0 AVL *</li> <li>7.6 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3 \u00a0 \u4e8c\u5143\u6a39\u9663\u5217\u8868\u793a","text":"<p>\u5728\u93c8\u7d50\u4e32\u5217\u8868\u793a\u4e0b\uff0c\u4e8c\u5143\u6a39\u7684\u5132\u5b58\u55ae\u5143\u70ba\u7bc0\u9ede <code>TreeNode</code> \uff0c\u7bc0\u9ede\u4e4b\u9593\u900f\u904e\u6307\u6a19\u76f8\u9023\u7dda\u3002\u4e0a\u4e00\u7bc0\u4ecb\u7d39\u4e86\u93c8\u7d50\u4e32\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u7684\u5404\u9805\u57fa\u672c\u64cd\u4f5c\u3002</p> <p>\u90a3\u9ebc\uff0c\u6211\u5011\u80fd\u5426\u7528\u9663\u5217\u4f86\u8868\u793a\u4e8c\u5143\u6a39\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1 \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u5143\u6a39","text":"<p>\u5148\u5206\u6790\u4e00\u500b\u7c21\u55ae\u6848\u4f8b\u3002\u7d66\u5b9a\u4e00\u68f5\u5b8c\u7f8e\u4e8c\u5143\u6a39\uff0c\u6211\u5011\u5c07\u6240\u6709\u7bc0\u9ede\u6309\u7167\u5c64\u5e8f\u8d70\u8a2a\u7684\u9806\u5e8f\u5132\u5b58\u5728\u4e00\u500b\u9663\u5217\u4e2d\uff0c\u5247\u6bcf\u500b\u7bc0\u9ede\u90fd\u5c0d\u61c9\u552f\u4e00\u7684\u9663\u5217\u7d22\u5f15\u3002</p> <p>\u6839\u64da\u5c64\u5e8f\u8d70\u8a2a\u7684\u7279\u6027\uff0c\u6211\u5011\u53ef\u4ee5\u63a8\u5c0e\u51fa\u7236\u7bc0\u9ede\u7d22\u5f15\u8207\u5b50\u7bc0\u9ede\u7d22\u5f15\u4e4b\u9593\u7684\u201c\u5c0d\u6620\u516c\u5f0f\u201d\uff1a\u82e5\u67d0\u7bc0\u9ede\u7684\u7d22\u5f15\u70ba \\(i\\) \uff0c\u5247\u8a72\u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7d22\u5f15\u70ba \\(2i + 1\\) \uff0c\u53f3\u5b50\u7bc0\u9ede\u7d22\u5f15\u70ba \\(2i + 2\\) \u3002\u5716 7-12 \u5c55\u793a\u4e86\u5404\u500b\u7bc0\u9ede\u7d22\u5f15\u4e4b\u9593\u7684\u5c0d\u6620\u95dc\u4fc2\u3002</p> <p></p> <p> \u5716 7-12 \u00a0 \u5b8c\u7f8e\u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a </p> <p>\u5c0d\u6620\u516c\u5f0f\u7684\u89d2\u8272\u76f8\u7576\u65bc\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u7bc0\u9ede\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002\u7d66\u5b9a\u9663\u5217\u4e2d\u7684\u4efb\u610f\u4e00\u500b\u7bc0\u9ede\uff0c\u6211\u5011\u90fd\u53ef\u4ee5\u900f\u904e\u5c0d\u6620\u516c\u5f0f\u4f86\u8a2a\u554f\u5b83\u7684\u5de6\uff08\u53f3\uff09\u5b50\u7bc0\u9ede\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2 \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u5143\u6a39","text":"<p>\u5b8c\u7f8e\u4e8c\u5143\u6a39\u662f\u4e00\u500b\u7279\u4f8b\uff0c\u5728\u4e8c\u5143\u6a39\u7684\u4e2d\u9593\u5c64\u901a\u5e38\u5b58\u5728\u8a31\u591a <code>None</code> \u3002\u7531\u65bc\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u4e26\u4e0d\u5305\u542b\u9019\u4e9b <code>None</code> \uff0c\u56e0\u6b64\u6211\u5011\u7121\u6cd5\u50c5\u6191\u8a72\u5e8f\u5217\u4f86\u63a8\u6e2c <code>None</code> \u7684\u6578\u91cf\u548c\u5206\u4f48\u4f4d\u7f6e\u3002\u9019\u610f\u5473\u8457\u5b58\u5728\u591a\u7a2e\u4e8c\u5143\u6a39\u7d50\u69cb\u90fd\u7b26\u5408\u8a72\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u3002</p> <p>\u5982\u5716 7-13 \u6240\u793a\uff0c\u7d66\u5b9a\u4e00\u68f5\u975e\u5b8c\u7f8e\u4e8c\u5143\u6a39\uff0c\u4e0a\u8ff0\u9663\u5217\u8868\u793a\u65b9\u6cd5\u5df2\u7d93\u5931\u6548\u3002</p> <p></p> <p> \u5716 7-13 \u00a0 \u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u5c0d\u61c9\u591a\u7a2e\u4e8c\u5143\u6a39\u53ef\u80fd\u6027 </p> <p>\u70ba\u4e86\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u8003\u616e\u5728\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u4e2d\u986f\u5f0f\u5730\u5beb\u51fa\u6240\u6709 <code>None</code> \u3002\u5982\u5716 7-14 \u6240\u793a\uff0c\u9019\u6a23\u8655\u7406\u5f8c\uff0c\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u5143\u6a39\u4e86\u3002\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a\n# \u4f7f\u7528 None \u4f86\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c INT_MAX \u6a19\u8a18\u7a7a\u4f4d\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88dd\u985e\u5225 Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u4f86\u6a19\u8a18\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u578b\u5225 \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u4f86\u6a19\u8a18\u7a7a\u4f4d\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 any \u578b\u5225\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u4f86\u6a19\u8a18\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u578b\u5225 \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u4f86\u6a19\u8a18\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 null \u4f86\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 null \u4f86\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u578b\u5225 \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u4f86\u6a19\u8a18\u7a7a\u4f4d\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 None \u4f86\u6a19\u8a18\u7a7a\u4f4d\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6a19\u8a18\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u7bc0\u9ede\u503c\u4e0d\u80fd\u70ba INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 null \u4f86\u8868\u793a\u7a7a\u4f4d\nval tree = arrayOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>### \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a ###\n# \u4f7f\u7528 nil \u4f86\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>\n</code></pre> <p></p> <p> \u5716 7-14 \u00a0 \u4efb\u610f\u578b\u5225\u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a </p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u5b8c\u5168\u4e8c\u5143\u6a39\u975e\u5e38\u9069\u5408\u4f7f\u7528\u9663\u5217\u4f86\u8868\u793a\u3002\u56de\u9867\u5b8c\u5168\u4e8c\u5143\u6a39\u7684\u5b9a\u7fa9\uff0c<code>None</code> \u53ea\u51fa\u73fe\u5728\u6700\u5e95\u5c64\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u56e0\u6b64\u6240\u6709 <code>None</code> \u4e00\u5b9a\u51fa\u73fe\u5728\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u7684\u672b\u5c3e\u3002</p> <p>\u9019\u610f\u5473\u8457\u4f7f\u7528\u9663\u5217\u8868\u793a\u5b8c\u5168\u4e8c\u5143\u6a39\u6642\uff0c\u53ef\u4ee5\u7701\u7565\u5132\u5b58\u6240\u6709 <code>None</code> \uff0c\u975e\u5e38\u65b9\u4fbf\u3002\u5716 7-15 \u7d66\u51fa\u4e86\u4e00\u500b\u4f8b\u5b50\u3002</p> <p></p> <p> \u5716 7-15 \u00a0 \u5b8c\u5168\u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a </p> <p>\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e86\u4e00\u68f5\u57fa\u65bc\u9663\u5217\u8868\u793a\u7684\u4e8c\u5143\u6a39\uff0c\u5305\u62ec\u4ee5\u4e0b\u5e7e\u7a2e\u64cd\u4f5c\u3002</p> <ul> <li>\u7d66\u5b9a\u67d0\u7bc0\u9ede\uff0c\u7372\u53d6\u5b83\u7684\u503c\u3001\u5de6\uff08\u53f3\uff09\u5b50\u7bc0\u9ede\u3001\u7236\u7bc0\u9ede\u3002</li> <li>\u7372\u53d6\u524d\u5e8f\u8d70\u8a2a\u3001\u4e2d\u5e8f\u8d70\u8a2a\u3001\u5f8c\u5e8f\u8d70\u8a2a\u3001\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"\u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"\u4e32\u5217\u5bb9\u91cf\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int | None:\n        \"\"\"\u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c\"\"\"\n        # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"\u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"\u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"\u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"\u5c64\u5e8f\u8d70\u8a2a\"\"\"\n        self.res = []\n        # \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\"\"\"\n        if self.val(i) is None:\n            return\n        # \u524d\u5e8f\u8d70\u8a2a\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # \u4e2d\u5e8f\u8d70\u8a2a\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # \u5f8c\u5e8f\u8d70\u8a2a\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"\u524d\u5e8f\u8d70\u8a2a\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"\u4e2d\u5e8f\u8d70\u8a2a\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"\u5f8c\u5e8f\u8d70\u8a2a\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    int size() {\n        return tree.size();\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    int val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (val(i) == INT_MAX)\n            return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* \u5efa\u69cb\u5b50 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    public int size() {\n        return tree.size();\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    public Integer val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (val(i) == null)\n            return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    public int? Val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (!Val(i).HasValue)\n            return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* \u4e32\u5217\u5bb9\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* \u5c64\u5e8f\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n    if abt.val(i) == nil {\n        return\n    }\n    // \u524d\u5e8f\u8d70\u8a2a\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // \u4e2d\u5e8f\u8d70\u8a2a\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // \u5f8c\u5e8f\u8d70\u8a2a\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* \u524d\u5e8f\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* \u5efa\u69cb\u5b50 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    func val(i: Int) -&gt; Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        guard let val = val(i: i) else {\n            return\n        }\n        // \u524d\u5e8f\u8d70\u8a2a\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    val(i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    levelOrder() {\n        let res = [];\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    #dfs(i, order, res) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    val(i: number): number | null {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    levelOrder(): number[] {\n        let res = [];\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* \u5efa\u69cb\u5b50 */\n  ArrayBinaryTree(this._tree);\n\n  /* \u4e32\u5217\u5bb9\u91cf */\n  int size() {\n    return _tree.length;\n  }\n\n  /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n  int? val(int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* \u5c64\u5e8f\u8d70\u8a2a */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n    if (val(i) == null) {\n      return;\n    }\n    // \u524d\u5e8f\u8d70\u8a2a\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // \u4e2d\u5e8f\u8d70\u8a2a\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // \u5f8c\u5e8f\u8d70\u8a2a\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* \u524d\u5e8f\u8d70\u8a2a */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* \u4e2d\u5e8f\u8d70\u8a2a */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* \u5f8c\u5e8f\u8d70\u8a2a */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* \u5efa\u69cb\u5b50 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        self.tree.iter().filter_map(|&amp;x| x).collect()\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    fn dfs(&amp;self, i: i32, order: &amp;'static str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // \u524d\u5e8f\u8d70\u8a2a\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u7d50\u69cb\u9ad4 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* \u5efa\u69cb\u5b50 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* \u4e32\u5217\u5bb9\u91cf */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* \u5c64\u5e8f\u8d70\u8a2a */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n    if (val(abt, i) == INT_MAX)\n        return;\n    // \u524d\u5e8f\u8d70\u8a2a\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // \u4e2d\u5e8f\u8d70\u8a2a\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // \u5f8c\u5e8f\u8d70\u8a2a\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u8d70\u8a2a */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree(val tree: MutableList&lt;Int?&gt;) {\n    /* \u4e32\u5217\u5bb9\u91cf */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    fun _val(i: Int): Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    fun levelOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (i in 0..&lt;size()) {\n            if (_val(i) != null)\n                res.add(_val(i))\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (_val(i) == null)\n            return\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (\"pre\" == order)\n            res.add(_val(i))\n        dfs(left(i), order, res)\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (\"in\" == order)\n            res.add(_val(i))\n        dfs(right(i), order, res)\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (\"post\" == order)\n            res.add(_val(i))\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    fun preOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    fun inOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    fun postOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>### \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 ###\nclass ArrayBinaryTree\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(arr)\n    @tree = arr.to_a\n  end\n\n  ### \u4e32\u5217\u5bb9\u91cf ###\n  def size\n    @tree.length\n  end\n\n  ### \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c ###\n  def val(i)\n    # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de nil \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    return if i &lt; 0 || i &gt;= size\n\n    @tree[i]\n  end\n\n  ### \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 ###\n  def left(i)\n    2 * i + 1\n  end\n\n  ### \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 ###\n  def right(i)\n    2 * i + 2\n  end\n\n  ### \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 ###\n  def parent(i)\n    (i - 1) / 2\n  end\n\n  ### \u5c64\u5e8f\u8d70\u8a2a ###\n  def level_order\n    @res = []\n\n    # \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n    for i in 0...size\n      @res &lt;&lt; val(i) unless val(i).nil?\n    end\n\n    @res\n  end\n\n  ### \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a ###\n  def dfs(i, order)\n    return if val(i).nil?\n    # \u524d\u5e8f\u8d70\u8a2a\n    @res &lt;&lt; val(i) if order == :pre\n    dfs(left(i), order)\n    # \u4e2d\u5e8f\u8d70\u8a2a\n    @res &lt;&lt; val(i) if order == :in\n    dfs(right(i), order)\n    # \u5f8c\u5e8f\u8d70\u8a2a\n    @res &lt;&lt; val(i) if order == :post\n  end\n\n  ### \u524d\u5e8f\u8d70\u8a2a ###\n  def pre_order\n    @res = []\n    dfs(0, :pre)\n    @res\n  end\n\n  ### \u4e2d\u5e8f\u8d70\u8a2a ###\n  def in_order\n    @res = []\n    dfs(0, :in)\n    @res\n  end\n\n  ### \u5f8c\u5e8f\u8d70\u8a2a ###\n  def post_order\n    @res = []\n    dfs(0, :post)\n    @res\n  end\nend\n</code></pre> array_binary_tree.zig<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3 \u00a0 \u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a\u4e3b\u8981\u6709\u4ee5\u4e0b\u512a\u9ede\u3002</p> <ul> <li>\u9663\u5217\u5132\u5b58\u5728\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e2d\uff0c\u5c0d\u5feb\u53d6\u53cb\u597d\uff0c\u8a2a\u554f\u8207\u8d70\u8a2a\u901f\u5ea6\u8f03\u5feb\u3002</li> <li>\u4e0d\u9700\u8981\u5132\u5b58\u6307\u6a19\uff0c\u6bd4\u8f03\u7bc0\u7701\u7a7a\u9593\u3002</li> <li>\u5141\u8a31\u96a8\u6a5f\u8a2a\u554f\u7bc0\u9ede\u3002</li> </ul> <p>\u7136\u800c\uff0c\u9663\u5217\u8868\u793a\u4e5f\u5b58\u5728\u4e00\u4e9b\u4fb7\u9650\u6027\u3002</p> <ul> <li>\u9663\u5217\u5132\u5b58\u9700\u8981\u9023\u7e8c\u8a18\u61b6\u9ad4\u7a7a\u9593\uff0c\u56e0\u6b64\u4e0d\u9069\u5408\u5132\u5b58\u8cc7\u6599\u91cf\u904e\u5927\u7684\u6a39\u3002</li> <li>\u589e\u522a\u7bc0\u9ede\u9700\u8981\u900f\u904e\u9663\u5217\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u5be6\u73fe\uff0c\u6548\u7387\u8f03\u4f4e\u3002</li> <li>\u7576\u4e8c\u5143\u6a39\u4e2d\u5b58\u5728\u5927\u91cf <code>None</code> \u6642\uff0c\u9663\u5217\u4e2d\u5305\u542b\u7684\u7bc0\u9ede\u8cc7\u6599\u6bd4\u91cd\u8f03\u4f4e\uff0c\u7a7a\u9593\u5229\u7528\u7387\u8f03\u4f4e\u3002</li> </ul>"},{"location":"chapter_tree/avl_tree/","title":"7.5 \u00a0 AVL \u6a39 *","text":"<p>\u5728\u201c\u4e8c\u5143\u641c\u5c0b\u6a39\u201d\u7ae0\u7bc0\u4e2d\u6211\u5011\u63d0\u5230\uff0c\u5728\u591a\u6b21\u63d2\u5165\u548c\u522a\u9664\u64cd\u4f5c\u5f8c\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u53ef\u80fd\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5c07\u5f9e \\(O(\\log n)\\) \u52a3\u5316\u70ba \\(O(n)\\) \u3002</p> <p>\u5982\u5716 7-24 \u6240\u793a\uff0c\u7d93\u904e\u5169\u6b21\u522a\u9664\u7bc0\u9ede\u64cd\u4f5c\uff0c\u9019\u68f5\u4e8c\u5143\u641c\u5c0b\u6a39\u4fbf\u6703\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u3002</p> <p></p> <p> \u5716 7-24 \u00a0 AVL \u6a39\u5728\u522a\u9664\u7bc0\u9ede\u5f8c\u767c\u751f\u9000\u5316 </p> <p>\u518d\u4f8b\u5982\uff0c\u5728\u5716 7-25 \u6240\u793a\u7684\u5b8c\u7f8e\u4e8c\u5143\u6a39\u4e2d\u63d2\u5165\u5169\u500b\u7bc0\u9ede\u5f8c\uff0c\u6a39\u5c07\u56b4\u91cd\u5411\u5de6\u50be\u659c\uff0c\u67e5\u8a62\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e5f\u96a8\u4e4b\u52a3\u5316\u3002</p> <p></p> <p> \u5716 7-25 \u00a0 AVL \u6a39\u5728\u63d2\u5165\u7bc0\u9ede\u5f8c\u767c\u751f\u9000\u5316 </p> <p>1962 \u5e74 G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u8ad6\u6587\u201cAn algorithm for the organization of information\u201d\u4e2d\u63d0\u51fa\u4e86 AVL \u6a39\u3002\u8ad6\u6587\u4e2d\u8a73\u7d30\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u78ba\u4fdd\u5728\u6301\u7e8c\u65b0\u589e\u548c\u522a\u9664\u7bc0\u9ede\u5f8c\uff0cAVL \u6a39\u4e0d\u6703\u9000\u5316\uff0c\u5f9e\u800c\u4f7f\u5f97\u5404\u7a2e\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7d1a\u5225\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u5728\u9700\u8981\u983b\u7e41\u9032\u884c\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u7684\u5834\u666f\u4e2d\uff0cAVL \u6a39\u80fd\u59cb\u7d42\u4fdd\u6301\u9ad8\u6548\u7684\u8cc7\u6599\u64cd\u4f5c\u6548\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u61c9\u7528\u50f9\u503c\u3002</p>"},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1 \u00a0 AVL \u6a39\u5e38\u898b\u8853\u8a9e","text":"<p>AVL \u6a39\u65e2\u662f\u4e8c\u5143\u641c\u5c0b\u6a39\uff0c\u4e5f\u662f\u5e73\u8861\u4e8c\u5143\u6a39\uff0c\u540c\u6642\u6eff\u8db3\u9019\u5169\u985e\u4e8c\u5143\u6a39\u7684\u6240\u6709\u6027\u8cea\uff0c\u56e0\u6b64\u662f\u4e00\u7a2e\u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39\uff08balanced binary search tree\uff09\u3002</p>"},{"location":"chapter_tree/avl_tree/#1","title":"1. \u00a0 \u7bc0\u9ede\u9ad8\u5ea6","text":"<p>\u7531\u65bc AVL \u6a39\u7684\u76f8\u95dc\u64cd\u4f5c\u9700\u8981\u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6\uff0c\u56e0\u6b64\u6211\u5011\u9700\u8981\u70ba\u7bc0\u9ede\u985e\u5225\u65b0\u589e <code>height</code> \u8b8a\u6578\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"AVL \u6a39\u7bc0\u9ede\u985e\u5225\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # \u7bc0\u9ede\u503c\n        self.height: int = 0                # \u7bc0\u9ede\u9ad8\u5ea6\n        self.left: TreeNode | None = None   # \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n        self.right: TreeNode | None = None  # \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nstruct TreeNode {\n    int val{};          // \u7bc0\u9ede\u503c\n    int height = 0;     // \u7bc0\u9ede\u9ad8\u5ea6\n    TreeNode *left{};   // \u5de6\u5b50\u7bc0\u9ede\n    TreeNode *right{};  // \u53f3\u5b50\u7bc0\u9ede\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    public int val;        // \u7bc0\u9ede\u503c\n    public int height;     // \u7bc0\u9ede\u9ad8\u5ea6\n    public TreeNode left;  // \u5de6\u5b50\u7bc0\u9ede\n    public TreeNode right; // \u53f3\u5b50\u7bc0\u9ede\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u7bc0\u9ede\u503c\n    public int height;      // \u7bc0\u9ede\u9ad8\u5ea6\n    public TreeNode? left;  // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntype TreeNode struct {\n    Val    int       // \u7bc0\u9ede\u503c\n    Height int       // \u7bc0\u9ede\u9ad8\u5ea6\n    Left   *TreeNode // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    Right  *TreeNode // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var height: Int // \u7bc0\u9ede\u9ad8\u5ea6\n    var left: TreeNode? // \u5de6\u5b50\u7bc0\u9ede\n    var right: TreeNode? // \u53f3\u5b50\u7bc0\u9ede\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    val; // \u7bc0\u9ede\u503c\n    height; //\u7bc0\u9ede\u9ad8\u5ea6\n    left; // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    right; // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    val: number;            // \u7bc0\u9ede\u503c\n    height: number;         // \u7bc0\u9ede\u9ad8\u5ea6\n    left: TreeNode | null;  // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    right: TreeNode | null; // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n  int val;         // \u7bc0\u9ede\u503c\n  int height;      // \u7bc0\u9ede\u9ad8\u5ea6\n  TreeNode? left;  // \u5de6\u5b50\u7bc0\u9ede\n  TreeNode? right; // \u53f3\u5b50\u7bc0\u9ede\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL \u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct TreeNode {\n    val: i32,                               // \u7bc0\u9ede\u503c\n    height: i32,                            // \u7bc0\u9ede\u9ad8\u5ea6\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u7bc0\u9ede\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u7bc0\u9ede\n}\n\nimpl TreeNode {\n    /* \u5efa\u69cb\u5b50 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* \u5efa\u69cb\u5b50 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode(val _val: Int) {  // \u7bc0\u9ede\u503c\n    val height: Int = 0          // \u7bc0\u9ede\u9ad8\u5ea6\n    val left: TreeNode? = null   // \u5de6\u5b50\u7bc0\u9ede\n    val right: TreeNode? = null  // \u53f3\u5b50\u7bc0\u9ede\n}\n</code></pre> <pre><code>### AVL \u6a39\u7bc0\u9ede\u985e\u5225 ###\nclass TreeNode\n  attr_accessor :val    # \u7bc0\u9ede\u503c\n  attr_accessor :height # \u7bc0\u9ede\u9ad8\u5ea6\n  attr_accessor :left   # \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n  attr_accessor :right  # \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n\n  def initialize(val)\n    @val = val\n    @height = 0\n  end\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u201c\u7bc0\u9ede\u9ad8\u5ea6\u201d\u662f\u6307\u5f9e\u8a72\u7bc0\u9ede\u5230\u5b83\u7684\u6700\u9060\u8449\u7bc0\u9ede\u7684\u8ddd\u96e2\uff0c\u5373\u6240\u7d93\u904e\u7684\u201c\u908a\u201d\u7684\u6578\u91cf\u3002\u9700\u8981\u7279\u5225\u6ce8\u610f\u7684\u662f\uff0c\u8449\u7bc0\u9ede\u7684\u9ad8\u5ea6\u70ba \\(0\\) \uff0c\u800c\u7a7a\u7bc0\u9ede\u7684\u9ad8\u5ea6\u70ba \\(-1\\) \u3002\u6211\u5011\u5c07\u5efa\u7acb\u5169\u500b\u5de5\u5177\u51fd\u5f0f\uff0c\u5206\u5225\u7528\u65bc\u7372\u53d6\u548c\u66f4\u65b0\u7bc0\u9ede\u7684\u9ad8\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6\"\"\"\n    # \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"\u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\"\"\"\n    # \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint height(TreeNode node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    node?.height ?? -1\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nheight(node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\n#updateHeight(node) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n  // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n  return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n  // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nfun height(node: TreeNode?): Int {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node?.height ?: -1\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nfun updateHeight(node: TreeNode?) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node?.height = max(height(node?.left), height(node?.right)) + 1\n}\n</code></pre> avl_tree.rb<pre><code>### \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 ###\ndef height(node)\n  # \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n  return node.height unless node.nil?\n\n  -1\nend\n\n### \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 ###\ndef update_height(node)\n  # \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n  node.height = [height(node.left), height(node.right)].max + 1\nend\n</code></pre> avl_tree.zig<pre><code>// \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    _ = self;\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2","title":"2. \u00a0 \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50","text":"<p>\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\uff08balance factor\uff09\u5b9a\u7fa9\u70ba\u7bc0\u9ede\u5de6\u5b50\u6a39\u7684\u9ad8\u5ea6\u6e1b\u53bb\u53f3\u5b50\u6a39\u7684\u9ad8\u5ea6\uff0c\u540c\u6642\u898f\u5b9a\u7a7a\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\u70ba \\(0\\) \u3002\u6211\u5011\u540c\u6a23\u5c07\u7372\u53d6\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88dd\u6210\u51fd\u5f0f\uff0c\u65b9\u4fbf\u5f8c\u7e8c\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u7372\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n    # \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if node is None:\n        return 0\n    # \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == nullptr)\n        return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == null)\n        return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == null) return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if node == nil {\n        return 0\n    }\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    guard let node = node else { return 0 }\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node === null) return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node === null) return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n  // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n  if (node == null) return 0;\n  // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n        None =&gt; 0,\n        // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == NULL) {\n        return 0;\n    }\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nfun balanceFactor(node: TreeNode?): Int {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == null) return 0\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>### \u7372\u53d6\u5e73\u8861\u56e0\u5b50 ###\ndef balance_factor(node)\n  # \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n  return 0 if node.nil?\n\n  # \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n  height(node.left) - height(node.right)\nend\n</code></pre> avl_tree.zig<pre><code>// \u7372\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == null) return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return self.height(node.?.left) - self.height(node.?.right);\n}\n</code></pre> <p>Tip</p> <p>\u8a2d\u5e73\u8861\u56e0\u5b50\u70ba \\(f\\) \uff0c\u5247\u4e00\u68f5 AVL \u6a39\u7684\u4efb\u610f\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6eff\u8db3 \\(-1 \\le f \\le 1\\) \u3002</p>"},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2 \u00a0 AVL \u6a39\u65cb\u8f49","text":"<p>AVL \u6a39\u7684\u7279\u9ede\u5728\u65bc\u201c\u65cb\u8f49\u201d\u64cd\u4f5c\uff0c\u5b83\u80fd\u5920\u5728\u4e0d\u5f71\u97ff\u4e8c\u5143\u6a39\u7684\u4e2d\u5e8f\u8d70\u8a2a\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u7bc0\u9ede\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u65cb\u8f49\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u201c\u4e8c\u5143\u641c\u5c0b\u6a39\u201d\u7684\u6027\u8cea\uff0c\u4e5f\u80fd\u4f7f\u6a39\u91cd\u65b0\u8b8a\u70ba\u201c\u5e73\u8861\u4e8c\u5143\u6a39\u201d\u3002</p> <p>\u6211\u5011\u5c07\u5e73\u8861\u56e0\u5b50\u7d55\u5c0d\u503c \\(&gt; 1\\) \u7684\u7bc0\u9ede\u7a31\u70ba\u201c\u5931\u8861\u7bc0\u9ede\u201d\u3002\u6839\u64da\u7bc0\u9ede\u5931\u8861\u60c5\u6cc1\u7684\u4e0d\u540c\uff0c\u65cb\u8f49\u64cd\u4f5c\u5206\u70ba\u56db\u7a2e\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\u3002\u4e0b\u9762\u8a73\u7d30\u4ecb\u7d39\u9019\u4e9b\u65cb\u8f49\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/avl_tree/#1_1","title":"1. \u00a0 \u53f3\u65cb","text":"<p>\u5982\u5716 7-26 \u6240\u793a\uff0c\u7bc0\u9ede\u4e0b\u65b9\u70ba\u5e73\u8861\u56e0\u5b50\u3002\u5f9e\u5e95\u81f3\u9802\u770b\uff0c\u4e8c\u5143\u6a39\u4e2d\u9996\u500b\u5931\u8861\u7bc0\u9ede\u662f\u201c\u7bc0\u9ede 3\u201d\u3002\u6211\u5011\u95dc\u6ce8\u4ee5\u8a72\u5931\u8861\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\u7684\u5b50\u6a39\uff0c\u5c07\u8a72\u7bc0\u9ede\u8a18\u70ba <code>node</code> \uff0c\u5176\u5de6\u5b50\u7bc0\u9ede\u8a18\u70ba <code>child</code> \uff0c\u57f7\u884c\u201c\u53f3\u65cb\u201d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u5f8c\uff0c\u5b50\u6a39\u6062\u5fa9\u5e73\u8861\uff0c\u4e26\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u6027\u8cea\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 7-26 \u00a0 \u53f3\u65cb\u64cd\u4f5c\u6b65\u9a5f </p> <p>\u5982\u5716 7-27 \u6240\u793a\uff0c\u7576\u7bc0\u9ede <code>child</code> \u6709\u53f3\u5b50\u7bc0\u9ede\uff08\u8a18\u70ba <code>grand_child</code> \uff09\u6642\uff0c\u9700\u8981\u5728\u53f3\u65cb\u4e2d\u65b0\u589e\u4e00\u6b65\uff1a\u5c07 <code>grand_child</code> \u4f5c\u70ba <code>node</code> \u7684\u5de6\u5b50\u7bc0\u9ede\u3002</p> <p></p> <p> \u5716 7-27 \u00a0 \u6709 grand_child \u7684\u53f3\u65cb\u64cd\u4f5c </p> <p>\u201c\u5411\u53f3\u65cb\u8f49\u201d\u662f\u4e00\u7a2e\u5f62\u8c61\u5316\u7684\u8aaa\u6cd5\uff0c\u5be6\u969b\u4e0a\u9700\u8981\u900f\u904e\u4fee\u6539\u7bc0\u9ede\u6307\u6a19\u4f86\u5be6\u73fe\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n    child = node.left\n    grand_child = child.right\n    # \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node\n    node.left = grand_child\n    # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.Right = node\n    node.Left = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child?.right = node\n    node?.left = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n  child.right = node;\n  node.left = grandChild;\n  // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node\n    node.left = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### \u53f3\u65cb\u64cd\u4f5c ###\ndef right_rotate(node)\n  child = node.left\n  grand_child = child.right\n  # \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n  child.right = node\n  node.left = grand_child\n  # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  update_height(node)\n  update_height(child)\n  # \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  child\nend\n</code></pre> avl_tree.zig<pre><code>// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.left;\n    var grandChild = child.?.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.?.right = node;\n    node.?.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_1","title":"2. \u00a0 \u5de6\u65cb","text":"<p>\u76f8\u61c9\u5730\uff0c\u5982\u679c\u8003\u616e\u4e0a\u8ff0\u5931\u8861\u4e8c\u5143\u6a39\u7684\u201c\u6620\u8c61\u201d\uff0c\u5247\u9700\u8981\u57f7\u884c\u5716 7-28 \u6240\u793a\u7684\u201c\u5de6\u65cb\u201d\u64cd\u4f5c\u3002</p> <p></p> <p> \u5716 7-28 \u00a0 \u5de6\u65cb\u64cd\u4f5c </p> <p>\u540c\u7406\uff0c\u5982\u5716 7-29 \u6240\u793a\uff0c\u7576\u7bc0\u9ede <code>child</code> \u6709\u5de6\u5b50\u7bc0\u9ede\uff08\u8a18\u70ba <code>grand_child</code> \uff09\u6642\uff0c\u9700\u8981\u5728\u5de6\u65cb\u4e2d\u65b0\u589e\u4e00\u6b65\uff1a\u5c07 <code>grand_child</code> \u4f5c\u70ba <code>node</code> \u7684\u53f3\u5b50\u7bc0\u9ede\u3002</p> <p></p> <p> \u5716 7-29 \u00a0 \u6709 grand_child \u7684\u5de6\u65cb\u64cd\u4f5c </p> <p>\u53ef\u4ee5\u89c0\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u908f\u8f2f\u4e0a\u662f\u6620\u8c61\u5c0d\u7a31\u7684\uff0c\u5b83\u5011\u5206\u5225\u89e3\u6c7a\u7684\u5169\u7a2e\u5931\u8861\u60c5\u6cc1\u4e5f\u662f\u5c0d\u7a31\u7684\u3002\u57fa\u65bc\u5c0d\u7a31\u6027\uff0c\u6211\u5011\u53ea\u9700\u5c07\u53f3\u65cb\u7684\u5be6\u73fe\u7a0b\u5f0f\u78bc\u4e2d\u7684\u6240\u6709\u7684 <code>left</code> \u66ff\u63db\u70ba <code>right</code> \uff0c\u5c07\u6240\u6709\u7684 <code>right</code> \u66ff\u63db\u70ba <code>left</code> \uff0c\u5373\u53ef\u5f97\u5230\u5de6\u65cb\u7684\u5be6\u73fe\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n    child = node.right\n    grand_child = child.left\n    # \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node\n    node.right = grand_child\n    # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.Left = node\n    node.Right = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child?.left = node\n    node?.right = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n  child.left = node;\n  node.right = grandChild;\n  // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node\n    node.right = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### \u5de6\u65cb\u64cd\u4f5c ###\ndef left_rotate(node)\n  child = node.right\n  grand_child = child.left\n  # \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n  child.left = node\n  node.right = grand_child\n  # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  update_height(node)\n  update_height(child)\n  # \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  child\nend\n</code></pre> avl_tree.zig<pre><code>// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.right;\n    var grandChild = child.?.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.?.left = node;\n    node.?.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3","title":"3. \u00a0 \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb","text":"<p>\u5c0d\u65bc\u5716 7-30 \u4e2d\u7684\u5931\u8861\u7bc0\u9ede 3 \uff0c\u50c5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u7121\u6cd5\u4f7f\u5b50\u6a39\u6062\u5fa9\u5e73\u8861\u3002\u6b64\u6642\u9700\u8981\u5148\u5c0d <code>child</code> \u57f7\u884c\u201c\u5de6\u65cb\u201d\uff0c\u518d\u5c0d <code>node</code> \u57f7\u884c\u201c\u53f3\u65cb\u201d\u3002</p> <p></p> <p> \u5716 7-30 \u00a0 \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb </p>"},{"location":"chapter_tree/avl_tree/#4","title":"4. \u00a0 \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb","text":"<p>\u5982\u5716 7-31 \u6240\u793a\uff0c\u5c0d\u65bc\u4e0a\u8ff0\u5931\u8861\u4e8c\u5143\u6a39\u7684\u6620\u8c61\u60c5\u6cc1\uff0c\u9700\u8981\u5148\u5c0d <code>child</code> \u57f7\u884c\u201c\u53f3\u65cb\u201d\uff0c\u518d\u5c0d <code>node</code> \u57f7\u884c\u201c\u5de6\u65cb\u201d\u3002</p> <p></p> <p> \u5716 7-31 \u00a0 \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb </p>"},{"location":"chapter_tree/avl_tree/#5","title":"5. \u00a0 \u65cb\u8f49\u7684\u9078\u64c7","text":"<p>\u5716 7-32 \u5c55\u793a\u7684\u56db\u7a2e\u5931\u8861\u60c5\u6cc1\u8207\u4e0a\u8ff0\u6848\u4f8b\u9010\u500b\u5c0d\u61c9\uff0c\u5206\u5225\u9700\u8981\u63a1\u7528\u53f3\u65cb\u3001\u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\u3001\u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\u3001\u5de6\u65cb\u7684\u64cd\u4f5c\u3002</p> <p></p> <p> \u5716 7-32 \u00a0 AVL \u6a39\u7684\u56db\u7a2e\u65cb\u8f49\u60c5\u6cc1 </p> <p>\u5982\u4e0b\u8868\u6240\u793a\uff0c\u6211\u5011\u900f\u904e\u5224\u65b7\u5931\u8861\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f03\u9ad8\u4e00\u5074\u5b50\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8ca0\u865f\uff0c\u4f86\u78ba\u5b9a\u5931\u8861\u7bc0\u9ede\u5c6c\u65bc\u5716 7-32 \u4e2d\u7684\u54ea\u7a2e\u60c5\u6cc1\u3002</p> <p> \u8868 7-3 \u00a0 \u56db\u7a2e\u65cb\u8f49\u60c5\u6cc1\u7684\u9078\u64c7\u689d\u4ef6 </p> \u5931\u8861\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50 \u61c9\u63a1\u7528\u7684\u65cb\u8f49\u65b9\u6cd5 \\(&gt; 1\\) \uff08\u5de6\u504f\u6a39\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(&gt; 1\\) \uff08\u5de6\u504f\u6a39\uff09 \\(&lt;0\\) \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6a39\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6a39\uff09 \\(&gt;0\\) \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb <p>\u70ba\u4e86\u4fbf\u65bc\u4f7f\u7528\uff0c\u6211\u5011\u5c07\u65cb\u8f49\u64cd\u4f5c\u5c01\u88dd\u6210\u4e00\u500b\u51fd\u5f0f\u3002\u6709\u4e86\u9019\u500b\u51fd\u5f0f\uff0c\u6211\u5011\u5c31\u80fd\u5c0d\u5404\u7a2e\u5931\u8861\u60c5\u6cc1\u9032\u884c\u65cb\u8f49\uff0c\u4f7f\u5931\u8861\u7bc0\u9ede\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\"\"\"\n    # \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    balance_factor = self.balance_factor(node)\n    # \u5de6\u504f\u6a39\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # \u53f3\u65cb\n            return self.right_rotate(node)\n        else:\n            # \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # \u53f3\u504f\u6a39\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # \u5de6\u65cb\n            return self.left_rotate(node)\n        else:\n            # \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    int _balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactorInt = BalanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return RightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return LeftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    // Go \u63a8\u85a6\u77ed\u8b8a\u6578\uff0c\u9019\u88e1 bf \u6307\u4ee3 t.balanceFactor\n    bf := t.balanceFactor(node)\n    // \u5de6\u504f\u6a39\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // \u53f3\u65cb\n            return t.rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // \u5de6\u65cb\n            return t.leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    let balanceFactor = balanceFactor(node: node)\n    // \u5de6\u504f\u6a39\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // \u53f3\u65cb\n            return rightRotate(node: node)\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // \u5de6\u65cb\n            return leftRotate(node: node)\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n#rotate(node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.#rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.#leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n  // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n  int factor = balanceFactor(node);\n  // \u5de6\u504f\u6a39\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // \u53f3\u65cb\n      return rightRotate(node);\n    } else {\n      // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // \u53f3\u504f\u6a39\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // \u5de6\u65cb\n      return leftRotate(node);\n    } else {\n      // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    let balance_factor = Self::balance_factor(node.clone());\n    // \u5de6\u504f\u6a39\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // \u53f3\u65cb\n            Self::right_rotate(Some(node))\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // \u53f3\u504f\u6a39\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // \u5de6\u65cb\n            Self::left_rotate(Some(node))\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    int bf = balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nfun rotate(node: TreeNode): TreeNode {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    val balanceFactor = balanceFactor(node)\n    // \u5de6\u504f\u6a39\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 ###\ndef rotate(node)\n  # \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n  balance_factor = balance_factor(node)\n  # \u5de6\u904d\u6a39\n  if balance_factor &gt; 1\n    if balance_factor(node.left) &gt;= 0\n      # \u53f3\u65cb\n      return right_rotate(node)\n    else\n      # \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n      node.left = left_rotate(node.left)\n      return right_rotate(node)\n    end\n  # \u53f3\u904d\u6a39\n  elsif balance_factor &lt; -1\n    if balance_factor(node.right) &lt;= 0\n      # \u5de6\u65cb\n      return left_rotate(node)\n    else\n      # \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n      node.right = right_rotate(node.right)\n      return left_rotate(node)\n    end\n  end\n  # \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n  node\nend\n</code></pre> avl_tree.zig<pre><code>// \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    var balance_factor = self.balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balance_factor &gt; 1) {\n        if (self.balanceFactor(node.?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return self.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.?.left = self.leftRotate(node.?.left);\n            return self.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balance_factor &lt; -1) {\n        if (self.balanceFactor(node.?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return self.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.?.right = self.rightRotate(node.?.right);\n            return self.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3 \u00a0 AVL \u6a39\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#1_2","title":"1. \u00a0 \u63d2\u5165\u7bc0\u9ede","text":"<p>AVL \u6a39\u7684\u7bc0\u9ede\u63d2\u5165\u64cd\u4f5c\u8207\u4e8c\u5143\u641c\u5c0b\u6a39\u5728\u4e3b\u9ad4\u4e0a\u985e\u4f3c\u3002\u552f\u4e00\u7684\u5340\u5225\u5728\u65bc\uff0c\u5728 AVL \u6a39\u4e2d\u63d2\u5165\u7bc0\u9ede\u5f8c\uff0c\u5f9e\u8a72\u7bc0\u9ede\u5230\u6839\u7bc0\u9ede\u7684\u8def\u5f91\u4e0a\u53ef\u80fd\u6703\u51fa\u73fe\u4e00\u7cfb\u5217\u5931\u8861\u7bc0\u9ede\u3002\u56e0\u6b64\uff0c\u6211\u5011\u9700\u8981\u5f9e\u9019\u500b\u7bc0\u9ede\u958b\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u7bc0\u9ede\u6062\u5fa9\u5e73\u8861\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"\u63d2\u5165\u7bc0\u9ede\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"\u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    UpdateHeight(node);  // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u51fd\u5f0f\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    }\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    this.#updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n  updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n                }\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n\n            /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u51fd\u5f0f\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node;\n    }\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfun insert(_val: Int) {\n    root = insertHelper(root, _val)\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfun insertHelper(n: TreeNode?, _val: Int): TreeNode {\n    if (n == null)\n        return TreeNode(_val)\n    var node = n\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (_val &lt; node._val)\n        node.left = insertHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = insertHelper(node.right, _val)\n    else\n        return node // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node) // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### \u63d2\u5165\u7bc0\u9ede ###\ndef insert(val)\n  @root = insert_helper(@root, val)\nend\n\n### \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09###\ndef insert_helper(node, val)\n  return TreeNode.new(val) if node.nil?\n  # 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede\n  if val &lt; node.val\n    node.left = insert_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = insert_helper(node.right, val)\n  else\n    # \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n  end\n  # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  update_height(node)\n  # 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n  rotate(node)\nend\n</code></pre> avl_tree.zig<pre><code>// \u63d2\u5165\u7bc0\u9ede\nfn insert(self: *Self, val: T) !void {\n    self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) {\n        var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n        tmp_node.init(val);\n        return tmp_node;\n    }\n    // 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede\n    if (val &lt; node.?.val) {\n        node.?.left = try self.insertHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = try self.insertHelper(node.?.right, val);\n    } else {\n        return node;            // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    self.updateHeight(node);    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    // 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_2","title":"2. \u00a0 \u522a\u9664\u7bc0\u9ede","text":"<p>\u985e\u4f3c\u5730\uff0c\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u522a\u9664\u7bc0\u9ede\u65b9\u6cd5\u7684\u57fa\u790e\u4e0a\uff0c\u9700\u8981\u5f9e\u5e95\u81f3\u9802\u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u7bc0\u9ede\u6062\u5fa9\u5e73\u8861\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"\u522a\u9664\u7bc0\u9ede\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"\u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return None\n    # 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if child is None:\n                return None\n            # \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else:\n                node = child\n        else:\n            # \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u51fd\u5f0f\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n                return nil\n            } else {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n                node = child\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if child == nil {\n                return nil\n            }\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else {\n                node = child\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child === null) return null;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else node = child;\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child === null) {\n                return null;\n            } else {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n      if (child == null)\n        return null;\n      // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n      else\n        node = child;\n    } else {\n      // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n                    None =&gt; {\n                        return None;\n                    }\n                    // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n\n            /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u522a\u9664\u7bc0\u9ede */\n// \u7531\u65bc\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u8655\u7121\u6cd5\u4f7f\u7528 remove \u95dc\u9375\u8a5e\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u51fd\u5f0f\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfun remove(_val: Int) {\n    root = removeHelper(root, _val)\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfun removeHelper(n: TreeNode?, _val: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (_val &lt; node._val)\n        node.left = removeHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = removeHelper(node.right, _val)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null)\n                node.left\n            else\n                node.right\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == null)\n                return null\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else\n                node = child\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp._val)\n            node._val = temp._val\n        }\n    }\n    updateHeight(node) // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### \u522a\u9664\u7bc0\u9ede ###\ndef remove(val)\n  @root = remove_helper(@root, val)\nend\n\n### \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09###\ndef remove_helper(node, val)\n  return if node.nil?\n  # 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664\n  if val &lt; node.val\n    node.left = remove_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = remove_helper(node.right, val)\n  else\n    if node.left.nil? || node.right.nil?\n      child = node.left || node.right\n      # \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n      return if child.nil?\n      # \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n      node = child\n    else\n      # \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n      temp = node.right\n      while !temp.left.nil?\n        temp = temp.left\n      end\n      node.right = remove_helper(node.right, temp.val)\n      node.val = temp.val\n    end\n  end\n  # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  update_height(node)\n  # 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n  rotate(node)\nend\n</code></pre> avl_tree.zig<pre><code>// \u522a\u9664\u7bc0\u9ede\nfn remove(self: *Self, val: T) void {\n   self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) return null;\n    // 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664\n    if (val &lt; node.?.val) {\n        node.?.left = self.removeHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = self.removeHelper(node.?.right, val);\n    } else {\n        if (node.?.left == null or node.?.right == null) {\n            var child = if (node.?.left != null) node.?.left else node.?.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == null) {\n                return null;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            } else {\n                node = child;\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            var temp = node.?.right;\n            while (temp.?.left != null) {\n                temp = temp.?.left;\n            }\n            node.?.right = self.removeHelper(node.?.right, temp.?.val);\n            node.?.val = temp.?.val;\n        }\n    }\n    self.updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    // 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3_1","title":"3. \u00a0 \u67e5\u8a62\u7bc0\u9ede","text":"<p>AVL \u6a39\u7684\u7bc0\u9ede\u67e5\u8a62\u64cd\u4f5c\u8207\u4e8c\u5143\u641c\u5c0b\u6a39\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d05\u8ff0\u3002</p>"},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4 \u00a0 AVL \u6a39\u5178\u578b\u61c9\u7528","text":"<ul> <li>\u7d44\u7e54\u548c\u5132\u5b58\u5927\u578b\u8cc7\u6599\uff0c\u9069\u7528\u65bc\u9ad8\u983b\u67e5\u8a62\u3001\u4f4e\u983b\u589e\u522a\u7684\u5834\u666f\u3002</li> <li>\u7528\u65bc\u69cb\u5efa\u8cc7\u6599\u5eab\u4e2d\u7684\u7d22\u5f15\u7cfb\u7d71\u3002</li> <li>\u7d05\u9ed1\u6a39\u4e5f\u662f\u4e00\u7a2e\u5e38\u898b\u7684\u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39\u3002\u76f8\u8f03\u65bc AVL \u6a39\uff0c\u7d05\u9ed1\u6a39\u7684\u5e73\u8861\u689d\u4ef6\u66f4\u5bec\u9b06\uff0c\u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede\u6240\u9700\u7684\u65cb\u8f49\u64cd\u4f5c\u66f4\u5c11\uff0c\u7bc0\u9ede\u589e\u522a\u64cd\u4f5c\u7684\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> </ul>"},{"location":"chapter_tree/binary_search_tree/","title":"7.4 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39","text":"<p>\u5982\u5716 7-16 \u6240\u793a\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\uff08binary search tree\uff09\u6eff\u8db3\u4ee5\u4e0b\u689d\u4ef6\u3002</p> <ol> <li>\u5c0d\u65bc\u6839\u7bc0\u9ede\uff0c\u5de6\u5b50\u6a39\u4e2d\u6240\u6709\u7bc0\u9ede\u7684\u503c \\(&lt;\\) \u6839\u7bc0\u9ede\u7684\u503c \\(&lt;\\) \u53f3\u5b50\u6a39\u4e2d\u6240\u6709\u7bc0\u9ede\u7684\u503c\u3002</li> <li>\u4efb\u610f\u7bc0\u9ede\u7684\u5de6\u3001\u53f3\u5b50\u6a39\u4e5f\u662f\u4e8c\u5143\u641c\u5c0b\u6a39\uff0c\u5373\u540c\u6a23\u6eff\u8db3\u689d\u4ef6 <code>1.</code> \u3002</li> </ol> <p></p> <p> \u5716 7-16 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39 </p>"},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u64cd\u4f5c","text":"<p>\u6211\u5011\u5c07\u4e8c\u5143\u641c\u5c0b\u6a39\u5c01\u88dd\u70ba\u4e00\u500b\u985e\u5225 <code>BinarySearchTree</code> \uff0c\u4e26\u5ba3\u544a\u4e00\u500b\u6210\u54e1\u8b8a\u6578 <code>root</code> \uff0c\u6307\u5411\u6a39\u7684\u6839\u7bc0\u9ede\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#1","title":"1. \u00a0 \u67e5\u8a62\u7bc0\u9ede","text":"<p>\u7d66\u5b9a\u76ee\u6a19\u7bc0\u9ede\u503c <code>num</code> \uff0c\u53ef\u4ee5\u6839\u64da\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u6027\u8cea\u4f86\u67e5\u8a62\u3002\u5982\u5716 7-17 \u6240\u793a\uff0c\u6211\u5011\u5ba3\u544a\u4e00\u500b\u7bc0\u9ede <code>cur</code> \uff0c\u5f9e\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede <code>root</code> \u51fa\u767c\uff0c\u8ff4\u5708\u6bd4\u8f03\u7bc0\u9ede\u503c <code>cur.val</code> \u548c <code>num</code> \u4e4b\u9593\u7684\u5927\u5c0f\u95dc\u4fc2\u3002</p> <ul> <li>\u82e5 <code>cur.val &lt; num</code> \uff0c\u8aaa\u660e\u76ee\u6a19\u7bc0\u9ede\u5728 <code>cur</code> \u7684\u53f3\u5b50\u6a39\u4e2d\uff0c\u56e0\u6b64\u57f7\u884c <code>cur = cur.right</code> \u3002</li> <li>\u82e5 <code>cur.val &gt; num</code> \uff0c\u8aaa\u660e\u76ee\u6a19\u7bc0\u9ede\u5728 <code>cur</code> \u7684\u5de6\u5b50\u6a39\u4e2d\uff0c\u56e0\u6b64\u57f7\u884c <code>cur = cur.left</code> \u3002</li> <li>\u82e5 <code>cur.val = num</code> \uff0c\u8aaa\u660e\u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\u4e26\u8fd4\u56de\u8a72\u7bc0\u9ede\u3002</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 7-17 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u67e5\u8a62\u7bc0\u9ede\u793a\u4f8b </p> <p>\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u67e5\u8a62\u64cd\u4f5c\u8207\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f2a\u6392\u9664\u4e00\u534a\u60c5\u6cc1\u3002\u8ff4\u5708\u6b21\u6578\u6700\u591a\u70ba\u4e8c\u5143\u6a39\u7684\u9ad8\u5ea6\uff0c\u7576\u4e8c\u5143\u6a39\u5e73\u8861\u6642\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6642\u9593\u3002\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"\u67e5\u8a62\u7bc0\u9ede\"\"\"\n    cur = self._root\n    # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while cur is not None:\n        # \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        elif cur.val &gt; num:\n            cur = cur.left\n        # \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            node = node.Left\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while cur != nil {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else {\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nsearch(num) {\n    let cur = this.root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  while (cur != null) {\n    // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n    else\n      break;\n  }\n  // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            cur = cur-&gt;left;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        cur = if (cur._val &lt; num)\n            cur.right\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur._val &gt; num)\n            cur.left\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else\n            break\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>### \u67e5\u8a62\u7bc0\u9ede ###\ndef search(num)\n  cur = @root\n\n  # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  while !cur.nil?\n    # \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if cur.val &lt; num\n      cur = cur.right\n    # \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    elsif cur.val &gt; num\n      cur = cur.left\n    # \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n    else\n      break\n    end\n  end\n\n  cur\nend\n</code></pre> binary_search_tree.zig<pre><code>// \u67e5\u8a62\u7bc0\u9ede\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n    var cur = self.root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        } else if (cur.?.val &gt; num) {\n            cur = cur.?.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        } else {\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#2","title":"2. \u00a0 \u63d2\u5165\u7bc0\u9ede","text":"<p>\u7d66\u5b9a\u4e00\u500b\u5f85\u63d2\u5165\u5143\u7d20 <code>num</code> \uff0c\u70ba\u4e86\u4fdd\u6301\u4e8c\u5143\u641c\u5c0b\u6a39\u201c\u5de6\u5b50\u6a39 &lt; \u6839\u7bc0\u9ede &lt; \u53f3\u5b50\u6a39\u201d\u7684\u6027\u8cea\uff0c\u63d2\u5165\u64cd\u4f5c\u6d41\u7a0b\u5982\u5716 7-18 \u6240\u793a\u3002</p> <ol> <li>\u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\uff1a\u8207\u67e5\u8a62\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5f9e\u6839\u7bc0\u9ede\u51fa\u767c\uff0c\u6839\u64da\u7576\u524d\u7bc0\u9ede\u503c\u548c <code>num</code> \u7684\u5927\u5c0f\u95dc\u4fc2\u8ff4\u5708\u5411\u4e0b\u641c\u5c0b\uff0c\u76f4\u5230\u8d8a\u904e\u8449\u7bc0\u9ede\uff08\u8d70\u8a2a\u81f3 <code>None</code> \uff09\u6642\u8df3\u51fa\u8ff4\u5708\u3002</li> <li>\u5728\u8a72\u4f4d\u7f6e\u63d2\u5165\u7bc0\u9ede\uff1a\u521d\u59cb\u5316\u7bc0\u9ede <code>num</code> \uff0c\u5c07\u8a72\u7bc0\u9ede\u7f6e\u65bc <code>None</code> \u7684\u4f4d\u7f6e\u3002</li> </ol> <p></p> <p> \u5716 7-18 \u00a0 \u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u63d2\u5165\u7bc0\u9ede </p> <p>\u5728\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u5169\u9ede\u3002</p> <ul> <li>\u4e8c\u5143\u641c\u5c0b\u6a39\u4e0d\u5141\u8a31\u5b58\u5728\u91cd\u8907\u7bc0\u9ede\uff0c\u5426\u5247\u5c07\u9055\u53cd\u5176\u5b9a\u7fa9\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u7bc0\u9ede\u5728\u6a39\u4e2d\u5df2\u5b58\u5728\uff0c\u5247\u4e0d\u57f7\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002</li> <li>\u70ba\u4e86\u5be6\u73fe\u63d2\u5165\u7bc0\u9ede\uff0c\u6211\u5011\u9700\u8981\u85c9\u52a9\u7bc0\u9ede <code>pre</code> \u5132\u5b58\u4e0a\u4e00\u8f2a\u8ff4\u5708\u7684\u7bc0\u9ede\u3002\u9019\u6a23\u5728\u8d70\u8a2a\u81f3 <code>None</code> \u6642\uff0c\u6211\u5011\u53ef\u4ee5\u7372\u53d6\u5230\u5176\u7236\u7bc0\u9ede\uff0c\u5f9e\u800c\u5b8c\u6210\u7bc0\u9ede\u63d2\u5165\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"\u63d2\u5165\u7bc0\u9ede\"\"\"\n    # \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur.val == num:\n            return\n        pre = cur\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else:\n            cur = cur.left\n    # \u63d2\u5165\u7bc0\u9ede\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid Insert(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur.left;\n    }\n\n    // \u63d2\u5165\u7bc0\u9ede\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // \u5f85\u63d2\u5165\u7bc0\u9ede\u4e4b\u524d\u7684\u7bc0\u9ede\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfunc insert(num: Int) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\ninsert(num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\ninsert(num: number): void {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int _num) {\n  // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n    if (cur.val == _num) return;\n    pre = cur;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u63d2\u5165\u7bc0\u9ede\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\npub fn insert(&amp;mut self, num: i32) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n            Ordering::Equal =&gt; return,\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(BinarySearchTree *bst, int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfun insert(num: Int) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur._val == num)\n            return\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        cur = if (cur._val &lt; num)\n            cur.right\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur.left\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    val node = TreeNode(num)\n    if (pre?._val!! &lt; num)\n        pre.right = node\n    else\n        pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>### \u63d2\u5165\u7bc0\u9ede ###\ndef insert(num)\n  # \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n  if @root.nil?\n    @root = TreeNode.new(num)\n    return\n  end\n\n  # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  cur, pre = @root, nil\n  while !cur.nil?\n    # \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n    return if cur.val == num\n\n    pre = cur\n    # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if cur.val &lt; num\n      cur = cur.right\n    # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else\n      cur = cur.left\n    end\n  end\n\n  # \u63d2\u5165\u7bc0\u9ede\n  node = TreeNode.new(num)\n  if pre.val &lt; num\n    pre.right = node\n  else\n    pre.left = node\n  end\nend\n</code></pre> binary_search_tree.zig<pre><code>// \u63d2\u5165\u7bc0\u9ede\nfn insert(self: *Self, num: T) !void {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (self.root == null) {\n        self.root = try self.mem_allocator.create(inc.TreeNode(T));\n        return;\n    }\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.?.val == num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    var node = try self.mem_allocator.create(inc.TreeNode(T));\n    node.init(num);\n    if (pre.?.val &lt; num) {\n        pre.?.right = node;\n    } else {\n        pre.?.left = node;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u8207\u67e5\u8a62\u7bc0\u9ede\u76f8\u540c\uff0c\u63d2\u5165\u7bc0\u9ede\u4f7f\u7528 \\(O(\\log n)\\) \u6642\u9593\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#3","title":"3. \u00a0 \u522a\u9664\u7bc0\u9ede","text":"<p>\u5148\u5728\u4e8c\u5143\u6a39\u4e2d\u67e5\u8a62\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u518d\u5c07\u5176\u522a\u9664\u3002\u8207\u63d2\u5165\u7bc0\u9ede\u985e\u4f3c\uff0c\u6211\u5011\u9700\u8981\u4fdd\u8b49\u5728\u522a\u9664\u64cd\u4f5c\u5b8c\u6210\u5f8c\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u201c\u5de6\u5b50\u6a39 &lt; \u6839\u7bc0\u9ede &lt; \u53f3\u5b50\u6a39\u201d\u7684\u6027\u8cea\u4ecd\u7136\u6eff\u8db3\u3002\u56e0\u6b64\uff0c\u6211\u5011\u6839\u64da\u76ee\u6a19\u7bc0\u9ede\u7684\u5b50\u7bc0\u9ede\u6578\u91cf\uff0c\u5206 0\u30011 \u548c 2 \u4e09\u7a2e\u60c5\u6cc1\uff0c\u57f7\u884c\u5c0d\u61c9\u7684\u522a\u9664\u7bc0\u9ede\u64cd\u4f5c\u3002</p> <p>\u5982\u5716 7-19 \u6240\u793a\uff0c\u7576\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u5ea6\u70ba \\(0\\) \u6642\uff0c\u8868\u793a\u8a72\u7bc0\u9ede\u662f\u8449\u7bc0\u9ede\uff0c\u53ef\u4ee5\u76f4\u63a5\u522a\u9664\u3002</p> <p></p> <p> \u5716 7-19 \u00a0 \u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u522a\u9664\u7bc0\u9ede\uff08\u5ea6\u70ba 0 \uff09 </p> <p>\u5982\u5716 7-20 \u6240\u793a\uff0c\u7576\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u5ea6\u70ba \\(1\\) \u6642\uff0c\u5c07\u5f85\u522a\u9664\u7bc0\u9ede\u66ff\u63db\u70ba\u5176\u5b50\u7bc0\u9ede\u5373\u53ef\u3002</p> <p></p> <p> \u5716 7-20 \u00a0 \u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u522a\u9664\u7bc0\u9ede\uff08\u5ea6\u70ba 1 \uff09 </p> <p>\u7576\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u5ea6\u70ba \\(2\\) \u6642\uff0c\u6211\u5011\u7121\u6cd5\u76f4\u63a5\u522a\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u500b\u7bc0\u9ede\u66ff\u63db\u8a72\u7bc0\u9ede\u3002\u7531\u65bc\u8981\u4fdd\u6301\u4e8c\u5143\u641c\u5c0b\u6a39\u201c\u5de6\u5b50\u6a39 \\(&lt;\\) \u6839\u7bc0\u9ede \\(&lt;\\) \u53f3\u5b50\u6a39\u201d\u7684\u6027\u8cea\uff0c\u56e0\u6b64\u9019\u500b\u7bc0\u9ede\u53ef\u4ee5\u662f\u53f3\u5b50\u6a39\u7684\u6700\u5c0f\u7bc0\u9ede\u6216\u5de6\u5b50\u6a39\u7684\u6700\u5927\u7bc0\u9ede\u3002</p> <p>\u5047\u8a2d\u6211\u5011\u9078\u64c7\u53f3\u5b50\u6a39\u7684\u6700\u5c0f\u7bc0\u9ede\uff08\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\uff09\uff0c\u5247\u522a\u9664\u64cd\u4f5c\u6d41\u7a0b\u5982\u5716 7-21 \u6240\u793a\u3002</p> <ol> <li>\u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\u5728\u201c\u4e2d\u5e8f\u8d70\u8a2a\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\uff0c\u8a18\u70ba <code>tmp</code> \u3002</li> <li>\u7528 <code>tmp</code> \u7684\u503c\u8986\u84cb\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u503c\uff0c\u4e26\u5728\u6a39\u4e2d\u905e\u8ff4\u522a\u9664\u7bc0\u9ede <code>tmp</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 7-21 \u00a0 \u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u522a\u9664\u7bc0\u9ede\uff08\u5ea6\u70ba 2 \uff09 </p> <p>\u522a\u9664\u7bc0\u9ede\u64cd\u4f5c\u540c\u6a23\u4f7f\u7528 \\(O(\\log n)\\) \u6642\u9593\uff0c\u5176\u4e2d\u67e5\u8a62\u5f85\u522a\u9664\u7bc0\u9ede\u9700\u8981 \\(O(\\log n)\\) \u6642\u9593\uff0c\u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u5f8c\u7e7c\u7bc0\u9ede\u9700\u8981 \\(O(\\log n)\\) \u6642\u9593\u3002\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"\u522a\u9664\u7bc0\u9ede\"\"\"\n    # \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self._root is None:\n        return\n    # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if cur.val == num:\n            break\n        pre = cur\n        # \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else:\n            cur = cur.left\n    # \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if cur is None:\n        return\n\n    # \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if cur.left is None or cur.right is None:\n        # \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        child = cur.left or cur.right\n        # \u522a\u9664\u7bc0\u9ede cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            self._root = child\n    # \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else:\n        # \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        self.remove(tmp.val)\n        # \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == nullptr)\n        return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = nullptr / \u8a72\u5b50\u7bc0\u9ede\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child;\n        }\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        delete cur;\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        remove(tmp-&gt;val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child;\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        remove(tmp.val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid Remove(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        TreeNode? child = cur.left ?? cur.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child;\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        Remove(tmp.val!.Value);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5f85\u522a\u9664\u7bc0\u9ede\u4e4b\u524d\u7684\u7bc0\u9ede\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728\u53f3\u5b50\u6a39\u4e2d\n            cur = cur.Right\n        } else {\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728\u5de6\u5b50\u6a39\u4e2d\n            cur = cur.Left\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u70ba 0 \u6216 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // \u53d6\u51fa\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u5b50\u7bc0\u9ede\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // \u522a\u9664\u7bc0\u9ede cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            bst.root = child\n        }\n        // \u5b50\u7bc0\u9ede\u6578\u70ba 2\n    } else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d\u5f85\u522a\u9664\u7bc0\u9ede cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        bst.remove(tmp.Val.(int))\n        // \u7528 tmp \u8986\u84cb cur\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfunc remove(num: Int) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        let child = cur?.left ?? cur?.right\n        // \u522a\u9664\u7bc0\u9ede cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        remove(num: tmp!.val)\n        // \u7528 tmp \u8986\u84cb cur\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nremove(num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        const child = cur.left !== null ? cur.left : cur.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            this.root = child;\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        this.remove(tmp.val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nremove(num: number): void {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            this.root = child;\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        this.remove(tmp!.val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int _num) {\n  // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n    if (cur.val == _num) break;\n    pre = cur;\n    // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n  if (cur == null) return;\n  // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n    TreeNode? child = cur.left ?? cur.right;\n    // \u522a\u9664\u7bc0\u9ede cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n      _root = child;\n    }\n  } else {\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n    remove(tmp.val);\n    // \u7528 tmp \u8986\u84cb cur\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u522a\u9664\u7bc0\u9ede */\npub fn remove(&amp;mut self, num: i32) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n            Ordering::Equal =&gt; break,\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = nullptr / \u8a72\u5b50\u7bc0\u9ede\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // \u522a\u9664\u7bc0\u9ede cur\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n                self.root = child;\n            }\n        }\n        // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n        (Some(_), Some(_)) =&gt; {\n            // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmp_val = tmp.unwrap().borrow().val;\n            // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n            self.remove(tmp_val);\n            // \u7528 tmp \u8986\u84cb cur\n            cur.borrow_mut().val = tmp_val;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u522a\u9664\u7bc0\u9ede */\n// \u7531\u65bc\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u8655\u7121\u6cd5\u4f7f\u7528 remove \u95dc\u9375\u8a5e\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 root \u7684\u53f3\u5b50\u6a39\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 root \u7684\u5de6\u5b50\u6a39\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == NULL)\n        return;\n    // \u5224\u65b7\u5f85\u522a\u9664\u7bc0\u9ede\u662f\u5426\u5b58\u5728\u5b50\u7bc0\u9ede\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1 */\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = nullptr / \u8a72\u5b50\u7bc0\u9ede\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        free(cur);\n    } else {\n        /* \u5b50\u7bc0\u9ede\u6578\u91cf = 2 */\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        removeItem(bst, tmp-&gt;val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfun remove(num: Int) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return\n    var cur = root\n    var pre: TreeNode? = null\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur._val == num)\n            break\n        pre = cur\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        cur = if (cur._val &lt; num)\n            cur.right\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur.left\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        val child = if (cur.left != null)\n            cur.left\n        else\n            cur.right\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur != root) {\n            if (pre!!.left == cur)\n                pre.left = child\n            else\n                pre.right = child\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child\n        }\n        // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    } else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        remove(tmp._val)\n        // \u7528 tmp \u8986\u84cb cur\n        cur._val = tmp._val\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>### \u522a\u9664\u7bc0\u9ede ###\ndef remove(num)\n  # \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  return if @root.nil?\n\n  # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  cur, pre = @root, nil\n  while !cur.nil?\n    # \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n    break if cur.val == num\n\n    pre = cur\n    # \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if cur.val &lt; num\n      cur = cur.right\n    # \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else\n      cur = cur.left\n    end\n  end\n  # \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  return if cur.nil?\n\n  # \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n  if cur.left.nil? || cur.right.nil?\n    # \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n    child = cur.left || cur.right\n    # \u522a\u9664\u7bc0\u9ede cur\n    if cur != @root\n      if pre.left == cur\n        pre.left = child\n      else\n        pre.right = child\n      end\n    else\n      # \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n      @root = child\n    end\n  # \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n  else\n    # \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n    tmp = cur.right\n    while !tmp.left.nil?\n      tmp = tmp.left\n    end\n    # \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n    remove(tmp.val)\n    # \u7528 tmp \u8986\u84cb cur\n    cur.val = tmp.val\n  end\nend\n</code></pre> binary_search_tree.zig<pre><code>// \u522a\u9664\u7bc0\u9ede\nfn remove(self: *Self, num: T) void {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (self.root == null) return;\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.?.val == num) break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.?.left == null or cur.?.right == null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (pre.?.left == cur) {\n            pre.?.left = child;\n        } else {\n            pre.?.right = child;\n        }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    } else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        var tmp = cur.?.right;\n        while (tmp.?.left != null) {\n            tmp = tmp.?.left;\n        }\n        var tmp_val = tmp.?.val;\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        self.remove(tmp.?.val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.?.val = tmp_val;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#4","title":"4. \u00a0 \u4e2d\u5e8f\u8d70\u8a2a\u6709\u5e8f","text":"<p>\u5982\u5716 7-22 \u6240\u793a\uff0c\u4e8c\u5143\u6a39\u7684\u4e2d\u5e8f\u8d70\u8a2a\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u8d70\u8a2a\u9806\u5e8f\uff0c\u800c\u4e8c\u5143\u641c\u5c0b\u6a39\u6eff\u8db3\u201c\u5de6\u5b50\u7bc0\u9ede \\(&lt;\\) \u6839\u7bc0\u9ede \\(&lt;\\) \u53f3\u5b50\u7bc0\u9ede\u201d\u7684\u5927\u5c0f\u95dc\u4fc2\u3002</p> <p>\u9019\u610f\u5473\u8457\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u9032\u884c\u4e2d\u5e8f\u8d70\u8a2a\u6642\uff0c\u7e3d\u662f\u6703\u512a\u5148\u8d70\u8a2a\u4e0b\u4e00\u500b\u6700\u5c0f\u7bc0\u9ede\uff0c\u5f9e\u800c\u5f97\u51fa\u4e00\u500b\u91cd\u8981\u6027\u8cea\uff1a\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u4e2d\u5e8f\u8d70\u8a2a\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002</p> <p>\u5229\u7528\u4e2d\u5e8f\u8d70\u8a2a\u5347\u5e8f\u7684\u6027\u8cea\uff0c\u6211\u5011\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u7372\u53d6\u6709\u5e8f\u8cc7\u6599\u50c5\u9700 \\(O(n)\\) \u6642\u9593\uff0c\u7121\u9808\u9032\u884c\u984d\u5916\u7684\u6392\u5e8f\u64cd\u4f5c\uff0c\u975e\u5e38\u9ad8\u6548\u3002</p> <p></p> <p> \u5716 7-22 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u4e2d\u5e8f\u8d70\u8a2a\u5e8f\u5217 </p>"},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u6548\u7387","text":"<p>\u7d66\u5b9a\u4e00\u7d44\u8cc7\u6599\uff0c\u6211\u5011\u8003\u616e\u4f7f\u7528\u9663\u5217\u6216\u4e8c\u5143\u641c\u5c0b\u6a39\u5132\u5b58\u3002\u89c0\u5bdf\u8868 7-2 \uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u5404\u9805\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u662f\u5c0d\u6578\u968e\uff0c\u5177\u6709\u7a69\u5b9a\u4e14\u9ad8\u6548\u7684\u6548\u80fd\u3002\u53ea\u6709\u5728\u9ad8\u983b\u65b0\u589e\u3001\u4f4e\u983b\u67e5\u8a62\u522a\u9664\u8cc7\u6599\u7684\u5834\u666f\u4e0b\uff0c\u9663\u5217\u6bd4\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u6548\u7387\u66f4\u9ad8\u3002</p> <p> \u8868 7-2 \u00a0 \u9663\u5217\u8207\u641c\u5c0b\u6a39\u7684\u6548\u7387\u5c0d\u6bd4 </p> \u7121\u5e8f\u9663\u5217 \u4e8c\u5143\u641c\u5c0b\u6a39 \u67e5\u8a62\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u522a\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) <p>\u5728\u7406\u60f3\u60c5\u6cc1\u4e0b\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u9019\u6a23\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f2a\u8ff4\u5708\u5167\u67e5\u8a62\u4efb\u610f\u7bc0\u9ede\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u5011\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u4e0d\u65b7\u5730\u63d2\u5165\u548c\u522a\u9664\u7bc0\u9ede\uff0c\u53ef\u80fd\u5c0e\u81f4\u4e8c\u5143\u6a39\u9000\u5316\u70ba\u5716 7-23 \u6240\u793a\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u9019\u6642\u5404\u7a2e\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e5f\u6703\u9000\u5316\u70ba \\(O(n)\\) \u3002</p> <p></p> <p> \u5716 7-23 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u9000\u5316 </p>"},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u5e38\u898b\u61c9\u7528","text":"<ul> <li>\u7528\u4f5c\u7cfb\u7d71\u4e2d\u7684\u591a\u7d1a\u7d22\u5f15\uff0c\u5be6\u73fe\u9ad8\u6548\u7684\u67e5\u8a62\u3001\u63d2\u5165\u3001\u522a\u9664\u64cd\u4f5c\u3002</li> <li>\u4f5c\u70ba\u67d0\u4e9b\u641c\u5c0b\u6f14\u7b97\u6cd5\u7684\u5e95\u5c64\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u7528\u65bc\u5132\u5b58\u8cc7\u6599\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72c0\u614b\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree/","title":"7.1 \u00a0 \u4e8c\u5143\u6a39","text":"<p>\u4e8c\u5143\u6a39\uff08binary tree\uff09\u662f\u4e00\u7a2e\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u4ee3\u8868\u201c\u7956\u5148\u201d\u8207\u201c\u5f8c\u4ee3\u201d\u4e4b\u9593\u7684\u6d3e\u751f\u95dc\u4fc2\uff0c\u9ad4\u73fe\u4e86\u201c\u4e00\u5206\u70ba\u4e8c\u201d\u7684\u5206\u6cbb\u908f\u8f2f\u3002\u8207\u93c8\u7d50\u4e32\u5217\u985e\u4f3c\uff0c\u4e8c\u5143\u6a39\u7684\u57fa\u672c\u55ae\u5143\u662f\u7bc0\u9ede\uff0c\u6bcf\u500b\u7bc0\u9ede\u5305\u542b\u503c\u3001\u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\u548c\u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"\u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u7bc0\u9ede\u503c\n        self.left: TreeNode | None = None  # \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n        self.right: TreeNode | None = None # \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct TreeNode {\n    int val;          // \u7bc0\u9ede\u503c\n    TreeNode *left;   // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    TreeNode *right;  // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    int val;         // \u7bc0\u9ede\u503c\n    TreeNode left;   // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    TreeNode right;  // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u7bc0\u9ede\u503c\n    public TreeNode? left;  // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* \u5efa\u69cb\u5b50 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n        Right: nil, // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n        Val:   v,   // \u7bc0\u9ede\u503c\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var left: TreeNode? // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    var right: TreeNode? // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    val; // \u7bc0\u9ede\u503c\n    left; // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    right; // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // \u7bc0\u9ede\u503c\n        this.left = left === undefined ? null : left; // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n        this.right = right === undefined ? null : right; // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n  int val;         // \u7bc0\u9ede\u503c\n  TreeNode? left;  // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n  TreeNode? right; // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct TreeNode {\n    val: i32,                               // \u7bc0\u9ede\u503c\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n\nimpl TreeNode {\n    /* \u5efa\u69cb\u5b50 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct TreeNode {\n    int val;                // \u7bc0\u9ede\u503c\n    int height;             // \u7bc0\u9ede\u9ad8\u5ea6\n    struct TreeNode *left;  // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    struct TreeNode *right; // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n} TreeNode;\n\n/* \u5efa\u69cb\u5b50 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode(val _val: Int) {  // \u7bc0\u9ede\u503c\n    val left: TreeNode? = null   // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    val right: TreeNode? = null  // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n</code></pre> <pre><code>### \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 ###\nclass TreeNode\n  attr_accessor :val    # \u7bc0\u9ede\u503c\n  attr_accessor :left   # \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n  attr_accessor :right  # \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n\n  def initialize(val)\n    @val = val\n  end\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u6bcf\u500b\u7bc0\u9ede\u90fd\u6709\u5169\u500b\u5f15\u7528\uff08\u6307\u6a19\uff09\uff0c\u5206\u5225\u6307\u5411\u5de6\u5b50\u7bc0\u9ede\uff08left-child node\uff09\u548c\u53f3\u5b50\u7bc0\u9ede\uff08right-child node\uff09\uff0c\u8a72\u7bc0\u9ede\u88ab\u7a31\u70ba\u9019\u5169\u500b\u5b50\u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\uff08parent node\uff09\u3002\u7576\u7d66\u5b9a\u4e00\u500b\u4e8c\u5143\u6a39\u7684\u7bc0\u9ede\u6642\uff0c\u6211\u5011\u5c07\u8a72\u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u53ca\u5176\u4ee5\u4e0b\u7bc0\u9ede\u5f62\u6210\u7684\u6a39\u7a31\u70ba\u8a72\u7bc0\u9ede\u7684\u5de6\u5b50\u6a39\uff08left subtree\uff09\uff0c\u540c\u7406\u53ef\u5f97\u53f3\u5b50\u6a39\uff08right subtree\uff09\u3002</p> <p>\u5728\u4e8c\u5143\u6a39\u4e2d\uff0c\u9664\u8449\u7bc0\u9ede\u5916\uff0c\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\u90fd\u5305\u542b\u5b50\u7bc0\u9ede\u548c\u975e\u7a7a\u5b50\u6a39\u3002\u5982\u5716 7-1 \u6240\u793a\uff0c\u5982\u679c\u5c07\u201c\u7bc0\u9ede 2\u201d\u8996\u70ba\u7236\u7bc0\u9ede\uff0c\u5247\u5176\u5de6\u5b50\u7bc0\u9ede\u548c\u53f3\u5b50\u7bc0\u9ede\u5206\u5225\u662f\u201c\u7bc0\u9ede 4\u201d\u548c\u201c\u7bc0\u9ede 5\u201d\uff0c\u5de6\u5b50\u6a39\u662f\u201c\u7bc0\u9ede 4 \u53ca\u5176\u4ee5\u4e0b\u7bc0\u9ede\u5f62\u6210\u7684\u6a39\u201d\uff0c\u53f3\u5b50\u6a39\u662f\u201c\u7bc0\u9ede 5 \u53ca\u5176\u4ee5\u4e0b\u7bc0\u9ede\u5f62\u6210\u7684\u6a39\u201d\u3002</p> <p></p> <p> \u5716 7-1 \u00a0 \u7236\u7bc0\u9ede\u3001\u5b50\u7bc0\u9ede\u3001\u5b50\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1 \u00a0 \u4e8c\u5143\u6a39\u5e38\u898b\u8853\u8a9e","text":"<p>\u4e8c\u5143\u6a39\u7684\u5e38\u7528\u8853\u8a9e\u5982\u5716 7-2 \u6240\u793a\u3002</p> <ul> <li>\u6839\u7bc0\u9ede\uff08root node\uff09\uff1a\u4f4d\u65bc\u4e8c\u5143\u6a39\u9802\u5c64\u7684\u7bc0\u9ede\uff0c\u6c92\u6709\u7236\u7bc0\u9ede\u3002</li> <li>\u8449\u7bc0\u9ede\uff08leaf node\uff09\uff1a\u6c92\u6709\u5b50\u7bc0\u9ede\u7684\u7bc0\u9ede\uff0c\u5176\u5169\u500b\u6307\u6a19\u5747\u6307\u5411 <code>None</code> \u3002</li> <li>\u908a\uff08edge\uff09\uff1a\u9023\u7dda\u5169\u500b\u7bc0\u9ede\u7684\u7dda\u6bb5\uff0c\u5373\u7bc0\u9ede\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002</li> <li>\u7bc0\u9ede\u6240\u5728\u7684\u5c64\uff08level\uff09\uff1a\u5f9e\u9802\u81f3\u5e95\u905e\u589e\uff0c\u6839\u7bc0\u9ede\u6240\u5728\u5c64\u70ba 1 \u3002</li> <li>\u7bc0\u9ede\u7684\u5ea6\uff08degree\uff09\uff1a\u7bc0\u9ede\u7684\u5b50\u7bc0\u9ede\u7684\u6578\u91cf\u3002\u5728\u4e8c\u5143\u6a39\u4e2d\uff0c\u5ea6\u7684\u53d6\u503c\u7bc4\u570d\u662f 0\u30011\u30012 \u3002</li> <li>\u4e8c\u5143\u6a39\u7684\u9ad8\u5ea6\uff08height\uff09\uff1a\u5f9e\u6839\u7bc0\u9ede\u5230\u6700\u9060\u8449\u7bc0\u9ede\u6240\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u3002</li> <li>\u7bc0\u9ede\u7684\u6df1\u5ea6\uff08depth\uff09\uff1a\u5f9e\u6839\u7bc0\u9ede\u5230\u8a72\u7bc0\u9ede\u6240\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u3002</li> <li>\u7bc0\u9ede\u7684\u9ad8\u5ea6\uff08height\uff09\uff1a\u5f9e\u8ddd\u96e2\u8a72\u7bc0\u9ede\u6700\u9060\u7684\u8449\u7bc0\u9ede\u5230\u8a72\u7bc0\u9ede\u6240\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u3002</li> </ul> <p></p> <p> \u5716 7-2 \u00a0 \u4e8c\u5143\u6a39\u7684\u5e38\u7528\u8853\u8a9e </p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u6211\u5011\u901a\u5e38\u5c07\u201c\u9ad8\u5ea6\u201d\u548c\u201c\u6df1\u5ea6\u201d\u5b9a\u7fa9\u70ba\u201c\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u984c\u76ee\u6216\u6559\u6750\u53ef\u80fd\u6703\u5c07\u5176\u5b9a\u7fa9\u70ba\u201c\u7d93\u904e\u7684\u7bc0\u9ede\u7684\u6578\u91cf\u201d\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002</p>"},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2 \u00a0 \u4e8c\u5143\u6a39\u57fa\u672c\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_tree/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e8c\u5143\u6a39","text":"<p>\u8207\u93c8\u7d50\u4e32\u5217\u985e\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u7bc0\u9ede\uff0c\u7136\u5f8c\u69cb\u5efa\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># \u521d\u59cb\u5316\u4e8c\u5143\u6a39\n# \u521d\u59cb\u5316\u7bc0\u9ede\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// \u521d\u59cb\u5316\u7bc0\u9ede\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// \u521d\u59cb\u5316\u7bc0\u9ede\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// \u521d\u59cb\u5316\u7bc0\u9ede\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code># \u521d\u59cb\u5316\u4e8c\u5143\u6a39\n# \u521d\u59cb\u5316\u7bc0\u9ede\nn1 = TreeNode.new(1)\nn2 = TreeNode.new(2)\nn3 = TreeNode.new(3)\nn4 = TreeNode.new(4)\nn5 = TreeNode.new(5)\n# \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/binary_tree/#2","title":"2. \u00a0 \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede","text":"<p>\u8207\u93c8\u7d50\u4e32\u5217\u985e\u4f3c\uff0c\u5728\u4e8c\u5143\u6a39\u4e2d\u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede\u53ef\u4ee5\u900f\u904e\u4fee\u6539\u6307\u6a19\u4f86\u5be6\u73fe\u3002\u5716 7-3 \u7d66\u51fa\u4e86\u4e00\u500b\u793a\u4f8b\u3002</p> <p></p> <p> \u5716 7-3 \u00a0 \u5728\u4e8c\u5143\u6a39\u4e2d\u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede\np = TreeNode(0)\n# \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = p\np.left = n2\n# \u522a\u9664\u7bc0\u9ede P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nTreeNode P = new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u522a\u9664\u7bc0\u9ede P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P\nP.left = n2\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nlet P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nconst P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// \u522a\u9664\u7bc0\u9ede p\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1-&gt;left = n2;\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P\nP.left = n2\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code># \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede\n_p = TreeNode.new(0)\n# \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede _p\nn1.left = _p\n_p.left = n2\n# \u522a\u9664\u7bc0\u9ede\nn1.left = n2\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>Tip</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u7bc0\u9ede\u53ef\u80fd\u6703\u6539\u8b8a\u4e8c\u5143\u6a39\u7684\u539f\u6709\u908f\u8f2f\u7d50\u69cb\uff0c\u800c\u522a\u9664\u7bc0\u9ede\u901a\u5e38\u610f\u5473\u8457\u522a\u9664\u8a72\u7bc0\u9ede\u53ca\u5176\u6240\u6709\u5b50\u6a39\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u5143\u6a39\u4e2d\uff0c\u63d2\u5165\u8207\u522a\u9664\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5be6\u73fe\u6709\u5be6\u969b\u610f\u7fa9\u7684\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3 \u00a0 \u5e38\u898b\u4e8c\u5143\u6a39\u578b\u5225","text":""},{"location":"chapter_tree/binary_tree/#1_1","title":"1. \u00a0 \u5b8c\u7f8e\u4e8c\u5143\u6a39","text":"<p>\u5982\u5716 7-4 \u6240\u793a\uff0c\u5b8c\u7f8e\u4e8c\u5143\u6a39\uff08perfect binary tree\uff09\u6240\u6709\u5c64\u7684\u7bc0\u9ede\u90fd\u88ab\u5b8c\u5168\u586b\u6eff\u3002\u5728\u5b8c\u7f8e\u4e8c\u5143\u6a39\u4e2d\uff0c\u8449\u7bc0\u9ede\u7684\u5ea6\u70ba \\(0\\) \uff0c\u5176\u9918\u6240\u6709\u7bc0\u9ede\u7684\u5ea6\u90fd\u70ba \\(2\\) \uff1b\u82e5\u6a39\u7684\u9ad8\u5ea6\u70ba \\(h\\) \uff0c\u5247\u7bc0\u9ede\u7e3d\u6578\u70ba \\(2^{h+1} - 1\\) \uff0c\u5448\u73fe\u6a19\u6e96\u7684\u6307\u6578\u7d1a\u95dc\u4fc2\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u898b\u7684\u7d30\u80de\u5206\u88c2\u73fe\u8c61\u3002</p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u5728\u4e2d\u6587\u793e\u7fa4\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u5143\u6a39\u5e38\u88ab\u7a31\u70ba\u6eff\u4e8c\u5143\u6a39\u3002</p> <p></p> <p> \u5716 7-4 \u00a0 \u5b8c\u7f8e\u4e8c\u5143\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#2_1","title":"2. \u00a0 \u5b8c\u5168\u4e8c\u5143\u6a39","text":"<p>\u5982\u5716 7-5 \u6240\u793a\uff0c\u5b8c\u5168\u4e8c\u5143\u6a39\uff08complete binary tree\uff09\u53ea\u6709\u6700\u5e95\u5c64\u7684\u7bc0\u9ede\u672a\u88ab\u586b\u6eff\uff0c\u4e14\u6700\u5e95\u5c64\u7bc0\u9ede\u5118\u91cf\u9760\u5de6\u586b\u5145\u3002\u8acb\u6ce8\u610f\uff0c\u5b8c\u7f8e\u4e8c\u5143\u6a39\u4e5f\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u5143\u6a39\u3002</p> <p></p> <p> \u5716 7-5 \u00a0 \u5b8c\u5168\u4e8c\u5143\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#3","title":"3. \u00a0 \u5b8c\u6eff\u4e8c\u5143\u6a39","text":"<p>\u5982\u5716 7-6 \u6240\u793a\uff0c\u5b8c\u6eff\u4e8c\u5143\u6a39\uff08full binary tree\uff09\u9664\u4e86\u8449\u7bc0\u9ede\u4e4b\u5916\uff0c\u5176\u9918\u6240\u6709\u7bc0\u9ede\u90fd\u6709\u5169\u500b\u5b50\u7bc0\u9ede\u3002</p> <p></p> <p> \u5716 7-6 \u00a0 \u5b8c\u6eff\u4e8c\u5143\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#4","title":"4. \u00a0 \u5e73\u8861\u4e8c\u5143\u6a39","text":"<p>\u5982\u5716 7-7 \u6240\u793a\uff0c\u5e73\u8861\u4e8c\u5143\u6a39\uff08balanced binary tree\uff09\u4e2d\u4efb\u610f\u7bc0\u9ede\u7684\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7d55\u5c0d\u503c\u4e0d\u8d85\u904e 1 \u3002</p> <p></p> <p> \u5716 7-7 \u00a0 \u5e73\u8861\u4e8c\u5143\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4 \u00a0 \u4e8c\u5143\u6a39\u7684\u9000\u5316","text":"<p>\u5716 7-8 \u5c55\u793a\u4e86\u4e8c\u5143\u6a39\u7684\u7406\u60f3\u7d50\u69cb\u8207\u9000\u5316\u7d50\u69cb\u3002\u7576\u4e8c\u5143\u6a39\u7684\u6bcf\u5c64\u7bc0\u9ede\u90fd\u88ab\u586b\u6eff\u6642\uff0c\u9054\u5230\u201c\u5b8c\u7f8e\u4e8c\u5143\u6a39\u201d\uff1b\u800c\u7576\u6240\u6709\u7bc0\u9ede\u90fd\u504f\u5411\u4e00\u5074\u6642\uff0c\u4e8c\u5143\u6a39\u9000\u5316\u70ba\u201c\u93c8\u7d50\u4e32\u5217\u201d\u3002</p> <ul> <li>\u5b8c\u7f8e\u4e8c\u5143\u6a39\u662f\u7406\u60f3\u60c5\u6cc1\uff0c\u53ef\u4ee5\u5145\u5206\u767c\u63ee\u4e8c\u5143\u6a39\u201c\u5206\u6cbb\u201d\u7684\u512a\u52e2\u3002</li> <li>\u93c8\u7d50\u4e32\u5217\u5247\u662f\u53e6\u4e00\u500b\u6975\u7aef\uff0c\u5404\u9805\u64cd\u4f5c\u90fd\u8b8a\u70ba\u7dda\u6027\u64cd\u4f5c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</li> </ul> <p></p> <p> \u5716 7-8 \u00a0 \u4e8c\u5143\u6a39\u7684\u6700\u4f73\u7d50\u69cb\u8207\u6700\u5dee\u7d50\u69cb </p> <p>\u5982\u8868 7-1 \u6240\u793a\uff0c\u5728\u6700\u4f73\u7d50\u69cb\u548c\u6700\u5dee\u7d50\u69cb\u4e0b\uff0c\u4e8c\u5143\u6a39\u7684\u8449\u7bc0\u9ede\u6578\u91cf\u3001\u7bc0\u9ede\u7e3d\u6578\u3001\u9ad8\u5ea6\u7b49\u9054\u5230\u6975\u5927\u503c\u6216\u6975\u5c0f\u503c\u3002</p> <p> \u8868 7-1 \u00a0 \u4e8c\u5143\u6a39\u7684\u6700\u4f73\u7d50\u69cb\u8207\u6700\u5dee\u7d50\u69cb </p> \u5b8c\u7f8e\u4e8c\u5143\u6a39 \u93c8\u7d50\u4e32\u5217 \u7b2c \\(i\\) \u5c64\u7684\u7bc0\u9ede\u6578\u91cf \\(2^{i-1}\\) \\(1\\) \u9ad8\u5ea6\u70ba \\(h\\) \u7684\u6a39\u7684\u8449\u7bc0\u9ede\u6578\u91cf \\(2^h\\) \\(1\\) \u9ad8\u5ea6\u70ba \\(h\\) \u7684\u6a39\u7684\u7bc0\u9ede\u7e3d\u6578 \\(2^{h+1} - 1\\) \\(h + 1\\) \u7bc0\u9ede\u7e3d\u6578\u70ba \\(n\\) \u7684\u6a39\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 \u4e8c\u5143\u6a39\u8d70\u8a2a","text":"<p>\u5f9e\u7269\u7406\u7d50\u69cb\u7684\u89d2\u5ea6\u4f86\u770b\uff0c\u6a39\u662f\u4e00\u7a2e\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u56e0\u6b64\u5176\u8d70\u8a2a\u65b9\u5f0f\u662f\u900f\u904e\u6307\u6a19\u9010\u500b\u8a2a\u554f\u7bc0\u9ede\u3002\u7136\u800c\uff0c\u6a39\u662f\u4e00\u7a2e\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u9019\u4f7f\u5f97\u8d70\u8a2a\u6a39\u6bd4\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u66f4\u52a0\u8907\u96dc\uff0c\u9700\u8981\u85c9\u52a9\u641c\u5c0b\u6f14\u7b97\u6cd5\u4f86\u5be6\u73fe\u3002</p> <p>\u4e8c\u5143\u6a39\u5e38\u898b\u7684\u8d70\u8a2a\u65b9\u5f0f\u5305\u62ec\u5c64\u5e8f\u8d70\u8a2a\u3001\u524d\u5e8f\u8d70\u8a2a\u3001\u4e2d\u5e8f\u8d70\u8a2a\u548c\u5f8c\u5e8f\u8d70\u8a2a\u7b49\u3002</p>"},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1 \u00a0 \u5c64\u5e8f\u8d70\u8a2a","text":"<p>\u5982\u5716 7-9 \u6240\u793a\uff0c\u5c64\u5e8f\u8d70\u8a2a\uff08level-order traversal\uff09\u5f9e\u9802\u90e8\u5230\u5e95\u90e8\u9010\u5c64\u8d70\u8a2a\u4e8c\u5143\u6a39\uff0c\u4e26\u5728\u6bcf\u4e00\u5c64\u6309\u7167\u5f9e\u5de6\u5230\u53f3\u7684\u9806\u5e8f\u8a2a\u554f\u7bc0\u9ede\u3002</p> <p>\u5c64\u5e8f\u8d70\u8a2a\u672c\u8cea\u4e0a\u5c6c\u65bc\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\uff08breadth-first traversal\uff09\uff0c\u4e5f\u7a31\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\uff08breadth-first search, BFS\uff09\uff0c\u5b83\u9ad4\u73fe\u4e86\u4e00\u7a2e\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u64f4\u5c55\u201d\u7684\u9010\u5c64\u8d70\u8a2a\u65b9\u5f0f\u3002</p> <p></p> <p> \u5716 7-9 \u00a0 \u4e8c\u5143\u6a39\u7684\u5c64\u5e8f\u8d70\u8a2a </p>"},{"location":"chapter_tree/binary_tree_traversal/#1","title":"1. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u901a\u5e38\u85c9\u52a9\u201c\u4f47\u5217\u201d\u4f86\u5be6\u73fe\u3002\u4f47\u5217\u9075\u5faa\u201c\u5148\u9032\u5148\u51fa\u201d\u7684\u898f\u5247\uff0c\u800c\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u5247\u9075\u5faa\u201c\u9010\u5c64\u63a8\u9032\u201d\u7684\u898f\u5247\uff0c\u5169\u8005\u80cc\u5f8c\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"\u5c64\u5e8f\u8d70\u8a2a\"\"\"\n    # \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # \u968a\u5217\u51fa\u968a\n        res.append(node.val)  # \u5132\u5b58\u7bc0\u9ede\u503c\n        if node.left is not None:\n            queue.append(node.left)  # \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if node.right is not None:\n            queue.append(node.right)  # \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // \u968a\u5217\u51fa\u968a\n        vec.push_back(node-&gt;val); // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // \u968a\u5217\u51fa\u968a\n        list.add(node.val);           // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left != null)\n            queue.offer(node.left);   // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node.right != null)\n            queue.offer(node.right);  // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // \u968a\u5217\u51fa\u968a\n        list.Add(node.val!.Value);       // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left != null)\n            queue.Enqueue(node.left);    // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node.right != null)\n            queue.Enqueue(node.right);   // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfunc levelOrder(root *TreeNode) []any {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    queue := list.New()\n    queue.PushBack(root)\n    // \u521d\u59cb\u5316\u4e00\u500b\u5207\u7247\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // \u968a\u5217\u51fa\u968a\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // \u5132\u5b58\u7bc0\u9ede\u503c\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    var queue: [TreeNode] = [root]\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // \u968a\u5217\u51fa\u968a\n        list.append(node.val) // \u5132\u5b58\u7bc0\u9ede\u503c\n        if let left = node.left {\n            queue.append(left) // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n        if let right = node.right {\n            queue.append(right) // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfunction levelOrder(root) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // \u968a\u5217\u51fa\u968a\n        list.push(node.val); // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left) queue.push(node.left); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node.right) queue.push(node.right); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // \u968a\u5217\u51fa\u968a\n        list.push(node.val); // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left) {\n            queue.push(node.left); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n        if (node.right) {\n            queue.push(node.right); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // \u968a\u5217\u51fa\u968a\n    res.add(node!.val); // \u5132\u5b58\u7bc0\u9ede\u503c\n    if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n    if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    let mut que = VecDeque::new();\n    que.push_back(root.clone());\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // \u968a\u5217\u51fa\u968a\n        vec.push(node.borrow().val); // \u5132\u5b58\u7bc0\u9ede\u503c\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(left.clone()); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(right.clone()); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nint *levelOrder(TreeNode *root, int *size) {\n    /* \u8f14\u52a9\u4f47\u5217 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* \u8f14\u52a9\u4f47\u5217 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // \u4f47\u5217\u6307\u6a19\n    front = 0, rear = 0;\n    // \u52a0\u5165\u6839\u7bc0\u9ede\n    queue[rear++] = root;\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    /* \u8f14\u52a9\u9663\u5217 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // \u9663\u5217\u6307\u6a19\n    index = 0;\n    while (front &lt; rear) {\n        // \u968a\u5217\u51fa\u968a\n        node = queue[front++];\n        // \u5132\u5b58\u7bc0\u9ede\u503c\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // \u66f4\u65b0\u9663\u5217\u9577\u5ea6\u7684\u503c\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // \u91cb\u653e\u8f14\u52a9\u9663\u5217\u7a7a\u9593\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    val list = mutableListOf&lt;Int&gt;()\n    while (queue.isNotEmpty()) {\n        val node = queue.poll()      // \u968a\u5217\u51fa\u968a\n        list.add(node?._val!!)       // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left != null)\n            queue.offer(node.left)   // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node.right != null)\n            queue.offer(node.right)  // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>### \u5c64\u5e8f\u8d70\u8a2a ###\ndef level_order(root)\n  # \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n  queue = [root]\n  # \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n  res = []\n  while !queue.empty?\n    node = queue.shift # \u968a\u5217\u51fa\u968a\n    res &lt;&lt; node.val # \u5132\u5b58\u7bc0\u9ede\u503c\n    queue &lt;&lt; node.left unless node.left.nil? # \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n    queue &lt;&lt; node.right unless node.right.nil? # \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n  end\n  res\nend\n</code></pre> binary_tree_bfs.zig<pre><code>// \u5c64\u5e8f\u8d70\u8a2a\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    const L = std.TailQueue(*inc.TreeNode(T));\n    var queue = L{};\n    var root_node = try mem_allocator.create(L.Node);\n    root_node.data = root;\n    queue.append(root_node); \n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    var list = std.ArrayList(T).init(std.heap.page_allocator);\n    while (queue.len &gt; 0) {\n        var queue_node = queue.popFirst().?;    // \u968a\u5217\u51fa\u968a\n        var node = queue_node.data;\n        try list.append(node.val);              // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.left.?;\n            queue.append(tmp_node);             // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n        if (node.right != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.right.?;\n            queue.append(tmp_node);             // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n        }        \n    }\n    return list;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/binary_tree_traversal/#2","title":"2. \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1a\u6240\u6709\u7bc0\u9ede\u88ab\u8a2a\u554f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\uff0c\u5176\u4e2d \\(n\\) \u70ba\u7bc0\u9ede\u6578\u91cf\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u5373\u6eff\u4e8c\u5143\u6a39\u6642\uff0c\u8d70\u8a2a\u5230\u6700\u5e95\u5c64\u4e4b\u524d\uff0c\u4f47\u5217\u4e2d\u6700\u591a\u540c\u6642\u5b58\u5728 \\((n + 1) / 2\\) \u500b\u7bc0\u9ede\uff0c\u4f54\u7528 \\(O(n)\\) \u7a7a\u9593\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2 \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a","text":"<p>\u76f8\u61c9\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u5f8c\u5e8f\u8d70\u8a2a\u90fd\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\uff08depth-first traversal\uff09\uff0c\u4e5f\u7a31\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\uff08depth-first search, DFS\uff09\uff0c\u5b83\u9ad4\u73fe\u4e86\u4e00\u7a2e\u201c\u5148\u8d70\u5230\u76e1\u982d\uff0c\u518d\u56de\u6eaf\u7e7c\u7e8c\u201d\u7684\u8d70\u8a2a\u65b9\u5f0f\u3002</p> <p>\u5716 7-10 \u5c55\u793a\u4e86\u5c0d\u4e8c\u5143\u6a39\u9032\u884c\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u7684\u5de5\u4f5c\u539f\u7406\u3002\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u5c31\u50cf\u662f\u7e5e\u8457\u6574\u68f5\u4e8c\u5143\u6a39\u7684\u5916\u570d\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u6bcf\u500b\u7bc0\u9ede\u90fd\u6703\u9047\u5230\u4e09\u500b\u4f4d\u7f6e\uff0c\u5206\u5225\u5c0d\u61c9\u524d\u5e8f\u8d70\u8a2a\u3001\u4e2d\u5e8f\u8d70\u8a2a\u548c\u5f8c\u5e8f\u8d70\u8a2a\u3002</p> <p></p> <p> \u5716 7-10 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a </p>"},{"location":"chapter_tree/binary_tree_traversal/#1_1","title":"1. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u5be6\u73fe\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"\u524d\u5e8f\u8d70\u8a2a\"\"\"\n    if root is None:\n        return\n    # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"\u4e2d\u5e8f\u8d70\u8a2a\"\"\"\n    if root is None:\n        return\n    # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"\u5f8c\u5e8f\u8d70\u8a2a\"\"\"\n    if root is None:\n        return\n    # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfunction preOrder(root) {\n    if (root === null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunction inOrder(root) {\n    if (root === null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunction postOrder(root) {\n    if (root === null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n            let node = node.borrow();\n            res.push(node.val);\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            res.push(node.val);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n            res.push(node.val);\n        }\n    }\n\n    dfs(root, &amp;mut result);\n\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.add(root._val)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root.left)\n    list.add(root._val)\n    inOrder(root.right)\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root._val)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>### \u524d\u5e8f\u8d70\u8a2a ###\ndef pre_order(root)\n  return if root.nil?\n\n  # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n  $res &lt;&lt; root.val\n  pre_order(root.left)\n  pre_order(root.right)\nend\n\n### \u4e2d\u5e8f\u8d70\u8a2a ###\ndef in_order(root)\n  return if root.nil?\n\n  # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n  in_order(root.left)\n  $res &lt;&lt; root.val\n  in_order(root.right)\nend\n\n### \u5f8c\u5e8f\u8d70\u8a2a ###\ndef post_order(root)\n  return if root.nil?\n\n  # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n  post_order(root.left)\n  post_order(root.right)\n  $res &lt;&lt; root.val\nend\n</code></pre> binary_tree_dfs.zig<pre><code>// \u524d\u5e8f\u8d70\u8a2a\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    try list.append(root.?.val);\n    try preOrder(T, root.?.left);\n    try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u8d70\u8a2a\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    try inOrder(T, root.?.left);\n    try list.append(root.?.val);\n    try inOrder(T, root.?.right);\n}\n\n// \u5f8c\u5e8f\u8d70\u8a2a\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    try postOrder(T, root.?.left);\n    try postOrder(T, root.?.right);\n    try list.append(root.?.val);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>Tip</p> <p>\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u4e5f\u53ef\u4ee5\u57fa\u65bc\u8fed\u4ee3\u5be6\u73fe\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u81ea\u884c\u7814\u7a76\u3002</p> <p>\u5716 7-11 \u5c55\u793a\u4e86\u524d\u5e8f\u8d70\u8a2a\u4e8c\u5143\u6a39\u7684\u905e\u8ff4\u904e\u7a0b\uff0c\u5176\u53ef\u5206\u70ba\u201c\u905e\u201d\u548c\u201c\u8ff4\u201d\u5169\u500b\u9006\u5411\u7684\u90e8\u5206\u3002</p> <ol> <li>\u201c\u905e\u201d\u8868\u793a\u958b\u555f\u65b0\u65b9\u6cd5\uff0c\u7a0b\u5f0f\u5728\u6b64\u904e\u7a0b\u4e2d\u8a2a\u554f\u4e0b\u4e00\u500b\u7bc0\u9ede\u3002</li> <li>\u201c\u8ff4\u201d\u8868\u793a\u51fd\u5f0f\u8fd4\u56de\uff0c\u4ee3\u8868\u7576\u524d\u7bc0\u9ede\u5df2\u7d93\u8a2a\u554f\u5b8c\u7562\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 7-11 \u00a0 \u524d\u5e8f\u8d70\u8a2a\u7684\u905e\u8ff4\u904e\u7a0b </p>"},{"location":"chapter_tree/binary_tree_traversal/#2_1","title":"2. \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1a\u6240\u6709\u7bc0\u9ede\u88ab\u8a2a\u554f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u5373\u6a39\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u6642\uff0c\u905e\u8ff4\u6df1\u5ea6\u9054\u5230 \\(n\\) \uff0c\u7cfb\u7d71\u4f54\u7528 \\(O(n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002</li> </ul>"},{"location":"chapter_tree/summary/","title":"7.6 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_tree/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u4e8c\u5143\u6a39\u662f\u4e00\u7a2e\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u9ad4\u73fe\u201c\u4e00\u5206\u70ba\u4e8c\u201d\u7684\u5206\u6cbb\u908f\u8f2f\u3002\u6bcf\u500b\u4e8c\u5143\u6a39\u7bc0\u9ede\u5305\u542b\u4e00\u500b\u503c\u4ee5\u53ca\u5169\u500b\u6307\u6a19\uff0c\u5206\u5225\u6307\u5411\u5176\u5de6\u5b50\u7bc0\u9ede\u548c\u53f3\u5b50\u7bc0\u9ede\u3002</li> <li>\u5c0d\u65bc\u4e8c\u5143\u6a39\u4e2d\u7684\u67d0\u500b\u7bc0\u9ede\uff0c\u5176\u5de6\uff08\u53f3\uff09\u5b50\u7bc0\u9ede\u53ca\u5176\u4ee5\u4e0b\u5f62\u6210\u7684\u6a39\u88ab\u7a31\u70ba\u8a72\u7bc0\u9ede\u7684\u5de6\uff08\u53f3\uff09\u5b50\u6a39\u3002</li> <li>\u4e8c\u5143\u6a39\u7684\u76f8\u95dc\u8853\u8a9e\u5305\u62ec\u6839\u7bc0\u9ede\u3001\u8449\u7bc0\u9ede\u3001\u5c64\u3001\u5ea6\u3001\u908a\u3001\u9ad8\u5ea6\u548c\u6df1\u5ea6\u7b49\u3002</li> <li>\u4e8c\u5143\u6a39\u7684\u521d\u59cb\u5316\u3001\u7bc0\u9ede\u63d2\u5165\u548c\u7bc0\u9ede\u522a\u9664\u64cd\u4f5c\u8207\u93c8\u7d50\u4e32\u5217\u64cd\u4f5c\u65b9\u6cd5\u985e\u4f3c\u3002</li> <li>\u5e38\u898b\u7684\u4e8c\u5143\u6a39\u578b\u5225\u6709\u5b8c\u7f8e\u4e8c\u5143\u6a39\u3001\u5b8c\u5168\u4e8c\u5143\u6a39\u3001\u5b8c\u6eff\u4e8c\u5143\u6a39\u548c\u5e73\u8861\u4e8c\u5143\u6a39\u3002\u5b8c\u7f8e\u4e8c\u5143\u6a39\u662f\u6700\u7406\u60f3\u7684\u72c0\u614b\uff0c\u800c\u93c8\u7d50\u4e32\u5217\u662f\u9000\u5316\u5f8c\u7684\u6700\u5dee\u72c0\u614b\u3002</li> <li>\u4e8c\u5143\u6a39\u53ef\u4ee5\u7528\u9663\u5217\u8868\u793a\uff0c\u65b9\u6cd5\u662f\u5c07\u7bc0\u9ede\u503c\u548c\u7a7a\u4f4d\u6309\u5c64\u5e8f\u8d70\u8a2a\u9806\u5e8f\u6392\u5217\uff0c\u4e26\u6839\u64da\u7236\u7bc0\u9ede\u8207\u5b50\u7bc0\u9ede\u4e4b\u9593\u7684\u7d22\u5f15\u5c0d\u6620\u95dc\u4fc2\u4f86\u5be6\u73fe\u6307\u6a19\u3002</li> <li>\u4e8c\u5143\u6a39\u7684\u5c64\u5e8f\u8d70\u8a2a\u662f\u4e00\u7a2e\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\u65b9\u6cd5\uff0c\u5b83\u9ad4\u73fe\u4e86\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u64f4\u5c55\u201d\u7684\u9010\u5c64\u8d70\u8a2a\u65b9\u5f0f\uff0c\u901a\u5e38\u900f\u904e\u4f47\u5217\u4f86\u5be6\u73fe\u3002</li> <li>\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a\u7686\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\uff0c\u5b83\u5011\u9ad4\u73fe\u4e86\u201c\u5148\u8d70\u5230\u76e1\u982d\uff0c\u518d\u56de\u6eaf\u7e7c\u7e8c\u201d\u7684\u8d70\u8a2a\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u905e\u8ff4\u4f86\u5be6\u73fe\u3002</li> <li>\u4e8c\u5143\u641c\u5c0b\u6a39\u662f\u4e00\u7a2e\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8a62\u8cc7\u6599\u7d50\u69cb\uff0c\u5176\u67e5\u8a62\u3001\u63d2\u5165\u548c\u522a\u9664\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(\\log n)\\) \u3002\u7576\u4e8c\u5143\u641c\u5c0b\u6a39\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u6642\uff0c\u5404\u9805\u6642\u9593\u8907\u96dc\u5ea6\u6703\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>AVL \u6a39\uff0c\u4e5f\u7a31\u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39\uff0c\u5b83\u900f\u904e\u65cb\u8f49\u64cd\u4f5c\u78ba\u4fdd\u5728\u4e0d\u65b7\u63d2\u5165\u548c\u522a\u9664\u7bc0\u9ede\u5f8c\u6a39\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002</li> <li>AVL \u6a39\u7684\u65cb\u8f49\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u522a\u9664\u7bc0\u9ede\u5f8c\uff0cAVL \u6a39\u6703\u5f9e\u5e95\u5411\u9802\u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\u3002</li> </ul>"},{"location":"chapter_tree/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u5c0d\u65bc\u53ea\u6709\u4e00\u500b\u7bc0\u9ede\u7684\u4e8c\u5143\u6a39\uff0c\u6a39\u7684\u9ad8\u5ea6\u548c\u6839\u7bc0\u9ede\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u55ce\uff1f</p> <p>\u662f\u7684\uff0c\u56e0\u70ba\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u7fa9\u70ba\u201c\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u201d\u3002</p> <p>Q\uff1a\u4e8c\u5143\u6a39\u4e2d\u7684\u63d2\u5165\u8207\u522a\u9664\u4e00\u822c\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\uff0c\u9019\u88e1\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u9ebc\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u70ba\u8cc7\u6e90\u7684\u5b50\u7bc0\u9ede\u7684\u8cc7\u6e90\u91cb\u653e\u55ce\uff1f</p> <p>\u62ff\u4e8c\u5143\u641c\u5c0b\u6a39\u4f86\u8209\u4f8b\uff0c\u522a\u9664\u7bc0\u9ede\u64cd\u4f5c\u8981\u5206\u4e09\u7a2e\u60c5\u6cc1\u8655\u7406\uff0c\u5176\u4e2d\u6bcf\u7a2e\u60c5\u6cc1\u90fd\u9700\u8981\u9032\u884c\u591a\u500b\u6b65\u9a5f\u7684\u7bc0\u9ede\u64cd\u4f5c\u3002</p> <p>Q\uff1a\u70ba\u4ec0\u9ebc DFS \u8d70\u8a2a\u4e8c\u5143\u6a39\u6709\u524d\u3001\u4e2d\u3001\u5f8c\u4e09\u7a2e\u9806\u5e8f\uff0c\u5206\u5225\u6709\u4ec0\u9ebc\u7528\u5462\uff1f</p> <p>\u8207\u9806\u5e8f\u548c\u9006\u5e8f\u8d70\u8a2a\u9663\u5217\u985e\u4f3c\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a\u662f\u4e09\u7a2e\u4e8c\u5143\u6a39\u8d70\u8a2a\u65b9\u6cd5\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u5b83\u5011\u5f97\u5230\u4e00\u500b\u7279\u5b9a\u9806\u5e8f\u7684\u8d70\u8a2a\u7d50\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\uff0c\u7531\u65bc\u7bc0\u9ede\u5927\u5c0f\u6eff\u8db3 <code>\u5de6\u5b50\u7bc0\u9ede\u503c &lt; \u6839\u7bc0\u9ede\u503c &lt; \u53f3\u5b50\u7bc0\u9ede\u503c</code> \uff0c\u56e0\u6b64\u6211\u5011\u53ea\u8981\u6309\u7167\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u512a\u5148\u9806\u5e8f\u8d70\u8a2a\u6a39\uff0c\u5c31\u53ef\u4ee5\u7372\u5f97\u6709\u5e8f\u7684\u7bc0\u9ede\u5e8f\u5217\u3002</p> <p>Q\uff1a\u53f3\u65cb\u64cd\u4f5c\u662f\u8655\u7406\u5931\u8861\u7bc0\u9ede <code>node</code>\u3001<code>child</code>\u3001<code>grand_child</code> \u4e4b\u9593\u7684\u95dc\u4fc2\uff0c\u90a3 <code>node</code> \u7684\u7236\u7bc0\u9ede\u548c <code>node</code> \u539f\u4f86\u7684\u9023\u7dda\u4e0d\u9700\u8981\u7dad\u8b77\u55ce\uff1f\u53f3\u65cb\u64cd\u4f5c\u5f8c\u8c48\u4e0d\u662f\u65b7\u6389\u4e86\uff1f</p> <p>\u6211\u5011\u9700\u8981\u5f9e\u905e\u8ff4\u7684\u8996\u89d2\u4f86\u770b\u9019\u500b\u554f\u984c\u3002\u53f3\u65cb\u64cd\u4f5c <code>right_rotate(root)</code> \u50b3\u5165\u7684\u662f\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\uff0c\u6700\u7d42 <code>return child</code> \u8fd4\u56de\u65cb\u8f49\u4e4b\u5f8c\u7684\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\u3002\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\u548c\u5176\u7236\u7bc0\u9ede\u7684\u9023\u7dda\u662f\u5728\u8a72\u51fd\u5f0f\u8fd4\u56de\u5f8c\u5b8c\u6210\u7684\uff0c\u4e0d\u5c6c\u65bc\u53f3\u65cb\u64cd\u4f5c\u7684\u7dad\u8b77\u7bc4\u570d\u3002</p> <p>Q\uff1a\u5728 C++ \u4e2d\uff0c\u51fd\u5f0f\u88ab\u5283\u5206\u5230 <code>private</code> \u548c <code>public</code> \u4e2d\uff0c\u9019\u65b9\u9762\u6709\u4ec0\u9ebc\u8003\u91cf\u55ce\uff1f\u70ba\u4ec0\u9ebc\u8981\u5c07 <code>height()</code> \u51fd\u5f0f\u548c <code>updateHeight()</code> \u51fd\u5f0f\u5206\u5225\u653e\u5728 <code>public</code> \u548c <code>private</code> \u4e2d\u5462\uff1f</p> <p>\u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u7bc4\u570d\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u985e\u5225\u5167\u90e8\u4f7f\u7528\uff0c\u90a3\u9ebc\u5c31\u8a2d\u8a08\u70ba <code>private</code> \u3002\u4f8b\u5982\uff0c\u4f7f\u7528\u8005\u55ae\u7368\u547c\u53eb <code>updateHeight()</code> \u662f\u6c92\u6709\u610f\u7fa9\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u522a\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800c <code>height()</code> \u662f\u8a2a\u554f\u7bc0\u9ede\u9ad8\u5ea6\uff0c\u985e\u4f3c\u65bc <code>vector.size()</code> \uff0c\u56e0\u6b64\u8a2d\u5b9a\u6210 <code>public</code> \u4ee5\u4fbf\u4f7f\u7528\u3002</p> <p>Q\uff1a\u5982\u4f55\u5f9e\u4e00\u7d44\u8f38\u5165\u8cc7\u6599\u69cb\u5efa\u4e00\u68f5\u4e8c\u5143\u641c\u5c0b\u6a39\uff1f\u6839\u7bc0\u9ede\u7684\u9078\u64c7\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f</p> <p>\u662f\u7684\uff0c\u69cb\u5efa\u6a39\u7684\u65b9\u6cd5\u5df2\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u7a0b\u5f0f\u78bc\u4e2d\u7684 <code>build_tree()</code> \u65b9\u6cd5\u4e2d\u7d66\u51fa\u3002\u81f3\u65bc\u6839\u7bc0\u9ede\u7684\u9078\u64c7\uff0c\u6211\u5011\u901a\u5e38\u6703\u5c07\u8f38\u5165\u8cc7\u6599\u6392\u5e8f\uff0c\u7136\u5f8c\u5c07\u4e2d\u9ede\u5143\u7d20\u4f5c\u70ba\u6839\u7bc0\u9ede\uff0c\u518d\u905e\u8ff4\u5730\u69cb\u5efa\u5de6\u53f3\u5b50\u6a39\u3002\u9019\u6a23\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8b49\u6a39\u7684\u5e73\u8861\u6027\u3002</p> <p>Q\uff1a\u5728 Java \u4e2d\uff0c\u5b57\u4e32\u5c0d\u6bd4\u662f\u5426\u4e00\u5b9a\u8981\u7528 <code>equals()</code> \u65b9\u6cd5\uff1f</p> <p>\u5728 Java \u4e2d\uff0c\u5c0d\u65bc\u57fa\u672c\u8cc7\u6599\u578b\u5225\uff0c<code>==</code> \u7528\u65bc\u5c0d\u6bd4\u5169\u500b\u8b8a\u6578\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002\u5c0d\u65bc\u5f15\u7528\u578b\u5225\uff0c\u5169\u7a2e\u7b26\u865f\u7684\u5de5\u4f5c\u539f\u7406\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li><code>==</code> \uff1a\u7528\u4f86\u6bd4\u8f03\u5169\u500b\u8b8a\u6578\u662f\u5426\u6307\u5411\u540c\u4e00\u500b\u7269\u4ef6\uff0c\u5373\u5b83\u5011\u5728\u8a18\u61b6\u9ad4\u4e2d\u7684\u4f4d\u7f6e\u662f\u5426\u76f8\u540c\u3002</li> <li><code>equals()</code>\uff1a\u7528\u4f86\u5c0d\u6bd4\u5169\u500b\u7269\u4ef6\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002</li> </ul> <p>\u56e0\u6b64\uff0c\u5982\u679c\u8981\u5c0d\u6bd4\u503c\uff0c\u6211\u5011\u61c9\u8a72\u4f7f\u7528 <code>equals()</code> \u3002\u7136\u800c\uff0c\u900f\u904e <code>String a = \"hi\"; String b = \"hi\";</code> \u521d\u59cb\u5316\u7684\u5b57\u4e32\u90fd\u5132\u5b58\u5728\u5b57\u4e32\u5e38\u6578\u6c60\u4e2d\uff0c\u5b83\u5011\u6307\u5411\u540c\u4e00\u500b\u7269\u4ef6\uff0c\u56e0\u6b64\u4e5f\u53ef\u4ee5\u7528 <code>a == b</code> \u4f86\u6bd4\u8f03\u5169\u500b\u5b57\u4e32\u7684\u5167\u5bb9\u3002</p> <p>Q\uff1a\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u5230\u6700\u5e95\u5c64\u4e4b\u524d\uff0c\u4f47\u5217\u4e2d\u7684\u7bc0\u9ede\u6578\u91cf\u662f \\(2^h\\) \u55ce\uff1f</p> <p>\u662f\u7684\uff0c\u4f8b\u5982\u9ad8\u5ea6 \\(h = 2\\) \u7684\u6eff\u4e8c\u5143\u6a39\uff0c\u5176\u7bc0\u9ede\u7e3d\u6578 \\(n = 7\\) \uff0c\u5247\u5e95\u5c64\u7bc0\u9ede\u6578\u91cf \\(4 = 2^h = (n + 1) / 2\\) \u3002</p>"}]}
\ No newline at end of file
+{"config":{"lang":["en"],"separator":"[\\s\\u200b\\u3000\\-\u3001\u3002\uff0c\uff0e\uff1f\uff01\uff1b]+","pipeline":["stemmer"]},"docs":[{"location":"chapter_appendix/","title":"\u7b2c 16 \u7ae0 \u00a0 \u9644\u9304","text":""},{"location":"chapter_appendix/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>16.1 \u00a0 \u7a0b\u5f0f\u8a2d\u8a08\u74b0\u5883\u5b89\u88dd</li> <li>16.2 \u00a0 \u4e00\u8d77\u53c3\u8207\u5275\u4f5c</li> <li>16.3 \u00a0 \u8853\u8a9e\u8868</li> </ul>"},{"location":"chapter_appendix/contribution/","title":"16.2 \u00a0 \u4e00\u8d77\u53c3\u8207\u5275\u4f5c","text":"<p>\u7531\u65bc\u7b46\u8005\u80fd\u529b\u6709\u9650\uff0c\u66f8\u4e2d\u96e3\u514d\u5b58\u5728\u4e00\u4e9b\u907a\u6f0f\u548c\u932f\u8aa4\uff0c\u8acb\u60a8\u8ad2\u89e3\u3002\u5982\u679c\u60a8\u767c\u73fe\u4e86\u7b46\u8aa4\u3001\u9023\u7d50\u5931\u6548\u3001\u5167\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u7fa9\u3001\u89e3\u91cb\u4e0d\u6e05\u6670\u6216\u884c\u6587\u7d50\u69cb\u4e0d\u5408\u7406\u7b49\u554f\u984c\uff0c\u8acb\u5354\u52a9\u6211\u5011\u9032\u884c\u4fee\u6b63\uff0c\u4ee5\u7d66\u8b80\u8005\u63d0\u4f9b\u66f4\u512a\u8cea\u7684\u5b78\u7fd2\u8cc7\u6e90\u3002</p> <p>\u6240\u6709\u64b0\u7a3f\u4eba\u7684 GitHub ID \u5c07\u5728\u672c\u66f8\u5009\u5eab\u3001\u7db2\u9801\u7248\u548c PDF \u7248\u7684\u4e3b\u9801\u4e0a\u9032\u884c\u5c55\u793a\uff0c\u4ee5\u611f\u8b1d\u4ed6\u5011\u5c0d\u958b\u6e90\u793e\u7fa4\u7684\u7121\u79c1\u5949\u737b\u3002</p> <p>\u958b\u6e90\u7684\u9b45\u529b</p> <p>\u7d19\u8cea\u5716\u66f8\u7684\u5169\u6b21\u5370\u5237\u7684\u9593\u9694\u6642\u9593\u5f80\u5f80\u8f03\u4e45\uff0c\u5167\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002</p> <p>\u800c\u5728\u672c\u958b\u6e90\u66f8\u4e2d\uff0c\u5167\u5bb9\u66f4\u8fed\u7684\u6642\u9593\u88ab\u7e2e\u77ed\u81f3\u6578\u65e5\u751a\u81f3\u5e7e\u500b\u5c0f\u6642\u3002</p>"},{"location":"chapter_appendix/contribution/#1","title":"1. \u00a0 \u5167\u5bb9\u5fae\u8abf","text":"<p>\u5982\u5716 16-3 \u6240\u793a\uff0c\u6bcf\u500b\u9801\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u201c\u7de8\u8f2f\u5716\u793a\u201d\u3002\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9a5f\u4fee\u6539\u6587\u5b57\u6216\u7a0b\u5f0f\u78bc\u3002</p> <ol> <li>\u9ede\u9078\u201c\u7de8\u8f2f\u5716\u793a\u201d\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u5009\u5eab\u201d\u7684\u63d0\u793a\uff0c\u8acb\u540c\u610f\u8a72\u64cd\u4f5c\u3002</li> <li>\u4fee\u6539 Markdown \u6e90\u6a94\u6848\u5167\u5bb9\uff0c\u6aa2\u67e5\u5167\u5bb9\u7684\u6b63\u78ba\u6027\uff0c\u4e26\u5118\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7d71\u4e00\u3002</li> <li>\u5728\u9801\u9762\u5e95\u90e8\u586b\u5beb\u4fee\u6539\u8aaa\u660e\uff0c\u7136\u5f8c\u9ede\u9078\u201cPropose file change\u201d\u6309\u9215\u3002\u9801\u9762\u8df3\u8f49\u5f8c\uff0c\u9ede\u9078\u201cCreate pull request\u201d\u6309\u9215\u5373\u53ef\u767c\u8d77\u62c9\u53d6\u8acb\u6c42\u3002</li> </ol> <p></p> <p> \u5716 16-3 \u00a0 \u9801\u9762\u7de8\u8f2f\u6309\u9375 </p> <p>\u5716\u7247\u7121\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u900f\u904e\u65b0\u5efa Issue \u6216\u8a55\u8ad6\u7559\u8a00\u4f86\u63cf\u8ff0\u554f\u984c\uff0c\u6211\u5011\u6703\u76e1\u5feb\u91cd\u65b0\u7e6a\u88fd\u4e26\u66ff\u63db\u5716\u7247\u3002</p>"},{"location":"chapter_appendix/contribution/#2","title":"2. \u00a0 \u5167\u5bb9\u5275\u4f5c","text":"<p>\u5982\u679c\u60a8\u6709\u8208\u8da3\u53c3\u8207\u6b64\u958b\u6e90\u5c08\u6848\uff0c\u5305\u62ec\u5c07\u7a0b\u5f0f\u78bc\u7ffb\u8b6f\u6210\u5176\u4ed6\u7a0b\u5f0f\u8a9e\u8a00\u3001\u64f4\u5c55\u6587\u7ae0\u5167\u5bb9\u7b49\uff0c\u90a3\u9ebc\u9700\u8981\u5be6\u65bd\u4ee5\u4e0b Pull Request \u5de5\u4f5c\u6d41\u7a0b\u3002</p> <ol> <li>\u767b\u5165 GitHub \uff0c\u5c07\u672c\u66f8\u7684\u7a0b\u5f0f\u78bc\u5009\u5eab Fork \u5230\u500b\u4eba\u5e33\u865f\u4e0b\u3002</li> <li>\u9032\u5165\u60a8\u7684 Fork \u5009\u5eab\u7db2\u9801\uff0c\u4f7f\u7528 <code>git clone</code> \u547d\u4ee4\u5c07\u5009\u5eab\u514b\u9686\u81f3\u672c\u5730\u3002</li> <li>\u5728\u672c\u5730\u9032\u884c\u5167\u5bb9\u5275\u4f5c\uff0c\u4e26\u9032\u884c\u5b8c\u6574\u6e2c\u8a66\uff0c\u9a57\u8b49\u7a0b\u5f0f\u78bc\u7684\u6b63\u78ba\u6027\u3002</li> <li>\u5c07\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u5f8c Push \u81f3\u9060\u7aef\u5009\u5eab\u3002</li> <li>\u91cd\u65b0\u6574\u7406\u5009\u5eab\u7db2\u9801\uff0c\u9ede\u9078\u201cCreate pull request\u201d\u6309\u9215\u5373\u53ef\u767c\u8d77\u62c9\u53d6\u8acb\u6c42\u3002</li> </ol>"},{"location":"chapter_appendix/contribution/#3-docker","title":"3. \u00a0 Docker \u90e8\u7f72","text":"<p>\u5728 <code>hello-algo</code> \u6839\u76ee\u9304\u4e0b\uff0c\u57f7\u884c\u4ee5\u4e0b Docker \u6307\u4ee4\u78bc\uff0c\u5373\u53ef\u5728 <code>http://localhost:8000</code> \u8a2a\u554f\u672c\u5c08\u6848\uff1a</p> <pre><code>docker-compose up -d\n</code></pre> <p>\u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u522a\u9664\u90e8\u7f72\uff1a</p> <pre><code>docker-compose down\n</code></pre>"},{"location":"chapter_appendix/installation/","title":"16.1 \u00a0 \u7a0b\u5f0f\u8a2d\u8a08\u74b0\u5883\u5b89\u88dd","text":""},{"location":"chapter_appendix/installation/#1611-ide","title":"16.1.1 \u00a0 \u5b89\u88dd IDE","text":"<p>\u63a8\u85a6\u4f7f\u7528\u958b\u6e90\u3001\u8f15\u91cf\u7684 VS Code \u4f5c\u70ba\u672c\u5730\u6574\u5408\u958b\u767c\u74b0\u5883\uff08IDE\uff09\u3002\u8a2a\u554f VS Code \u5b98\u7db2\uff0c\u6839\u64da\u4f5c\u696d\u7cfb\u7d71\u9078\u64c7\u76f8\u61c9\u7248\u672c\u7684 VS Code \u9032\u884c\u4e0b\u8f09\u548c\u5b89\u88dd\u3002</p> <p></p> <p> \u5716 16-1 \u00a0 \u5f9e\u5b98\u7db2\u4e0b\u8f09 VS Code </p> <p>VS Code \u64c1\u6709\u5f37\u5927\u7684\u64f4\u5c55\u5305\u751f\u614b\u7cfb\u7d71\uff0c\u652f\u6301\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u7684\u57f7\u884c\u548c\u9664\u932f\u3002\u4ee5 Python \u70ba\u4f8b\uff0c\u5b89\u88dd\u201cPython Extension Pack\u201d\u64f4\u5c55\u5305\u4e4b\u5f8c\uff0c\u5373\u53ef\u9032\u884c Python \u7a0b\u5f0f\u78bc\u9664\u932f\u3002\u5b89\u88dd\u6b65\u9a5f\u5982\u5716 16-2 \u6240\u793a\u3002</p> <p></p> <p> \u5716 16-2 \u00a0 \u5b89\u88dd VS Code \u64f4\u5c55\u5305 </p>"},{"location":"chapter_appendix/installation/#1612","title":"16.1.2 \u00a0 \u5b89\u88dd\u8a9e\u8a00\u74b0\u5883","text":""},{"location":"chapter_appendix/installation/#1-python","title":"1. \u00a0 Python \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Miniconda3 \uff0c\u9700\u8981 Python 3.10 \u6216\u66f4\u65b0\u7248\u672c\u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>python</code> \uff0c\u5b89\u88dd Python Extension Pack \u3002</li> <li>\uff08\u53ef\u9078\uff09\u5728\u547d\u4ee4\u5217\u8f38\u5165 <code>pip install black</code> \uff0c\u5b89\u88dd\u7a0b\u5f0f\u78bc\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#2-cc","title":"2. \u00a0 C/C++ \u74b0\u5883","text":"<ol> <li>Windows \u7cfb\u7d71\u9700\u8981\u5b89\u88dd MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff1bMacOS \u81ea\u5e36 Clang \uff0c\u7121\u9808\u5b89\u88dd\u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>c++</code> \uff0c\u5b89\u88dd C/C++ Extension Pack \u3002</li> <li>\uff08\u53ef\u9078\uff09\u958b\u555f Settings \u9801\u9762\uff0c\u641c\u5c0b <code>Clang_format_fallback Style</code> \u7a0b\u5f0f\u78bc\u683c\u5f0f\u5316\u9078\u9805\uff0c\u8a2d\u5b9a\u70ba <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code> \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#3-java","title":"3. \u00a0 Java \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd OpenJDK\uff08\u7248\u672c\u9700\u6eff\u8db3 &gt; JDK 9\uff09\u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>java</code> \uff0c\u5b89\u88dd Extension Pack for Java \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#4-c","title":"4. \u00a0 C# \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd .Net 8.0 \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>C# Dev Kit</code> \uff0c\u5b89\u88dd C# Dev Kit \uff08\u914d\u7f6e\u6559\u7a0b\uff09\u3002</li> <li>\u4e5f\u53ef\u4f7f\u7528 Visual Studio\uff08\u5b89\u88dd\u6559\u7a0b\uff09\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#5-go","title":"5. \u00a0 Go \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd go \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>go</code> \uff0c\u5b89\u88dd Go \u3002</li> <li>\u6309\u5feb\u6377\u9375 <code>Ctrl + Shift + P</code> \u64a5\u51fa\u547d\u4ee4\u6b04\uff0c\u8f38\u5165 go \uff0c\u9078\u64c7 <code>Go: Install/Update Tools</code> \uff0c\u5168\u90e8\u52fe\u9078\u4e26\u5b89\u88dd\u5373\u53ef\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#6-swift","title":"6. \u00a0 Swift \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Swift \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>swift</code> \uff0c\u5b89\u88dd Swift for Visual Studio Code \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#7-javascript","title":"7. \u00a0 JavaScript \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Node.js \u3002</li> <li>\uff08\u53ef\u9078\uff09\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>Prettier</code> \uff0c\u5b89\u88dd\u7a0b\u5f0f\u78bc\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#8-typescript","title":"8. \u00a0 TypeScript \u74b0\u5883","text":"<ol> <li>\u540c JavaScript \u74b0\u5883\u5b89\u88dd\u6b65\u9a5f\u3002</li> <li>\u5b89\u88dd TypeScript Execute (tsx) \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>typescript</code> \uff0c\u5b89\u88dd Pretty TypeScript Errors \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#9-dart","title":"9. \u00a0 Dart \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Dart \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>dart</code> \uff0c\u5b89\u88dd Dart \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#10-rust","title":"10. \u00a0 Rust \u74b0\u5883","text":"<ol> <li>\u4e0b\u8f09\u4e26\u5b89\u88dd Rust \u3002</li> <li>\u5728 VS Code \u7684\u64f4\u5145\u529f\u80fd\u5e02\u5834\u4e2d\u641c\u7d22 <code>rust</code> \uff0c\u5b89\u88dd rust-analyzer \u3002</li> </ol>"},{"location":"chapter_appendix/terminology/","title":"16.3 \u00a0 \u8853\u8a9e\u8868","text":"<p>\u8868 16-1 \u5217\u51fa\u4e86\u66f8\u4e2d\u51fa\u73fe\u7684\u91cd\u8981\u8853\u8a9e\uff0c\u503c\u5f97\u6ce8\u610f\u4ee5\u4e0b\u5e7e\u9ede\u3002</p> <ul> <li>\u5efa\u8b70\u8a18\u4f4f\u540d\u8a5e\u7684\u82f1\u6587\u53eb\u6cd5\uff0c\u4ee5\u4fbf\u95b1\u8b80\u82f1\u6587\u6587\u737b\u3002</li> <li>\u90e8\u5206\u540d\u8a5e\u5728\u7c21\u9ad4\u4e2d\u6587\u548c\u7e41\u9ad4\u4e2d\u6587\u4e0b\u7684\u53eb\u6cd5\u4e0d\u540c\u3002</li> </ul> <p> \u8868 16-1 \u00a0 \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u91cd\u8981\u540d\u8a5e </p> English \u7c21\u9ad4\u4e2d\u6587 \u7e41\u9ad4\u4e2d\u6587 algorithm \u7b97\u6cd5 \u6f14\u7b97\u6cd5 data structure \u6570\u636e\u7ed3\u6784 \u8cc7\u6599\u7d50\u69cb code \u4ee3\u7801 \u7a0b\u5f0f\u78bc file \u6587\u4ef6 \u6a94\u6848 function \u51fd\u6570 \u51fd\u5f0f method \u65b9\u6cd5 \u65b9\u6cd5 variable \u53d8\u91cf \u8b8a\u6578 asymptotic complexity analysis \u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 \u6f38\u8fd1\u8907\u96dc\u5ea6\u5206\u6790 time complexity \u65f6\u95f4\u590d\u6742\u5ea6 \u6642\u9593\u8907\u96dc\u5ea6 space complexity \u7a7a\u95f4\u590d\u6742\u5ea6 \u7a7a\u9593\u8907\u96dc\u5ea6 loop \u5faa\u73af \u8ff4\u5708 iteration \u8fed\u4ee3 \u8fed\u4ee3 recursion \u9012\u5f52 \u905e\u8ff4 tail recursion \u5c3e\u9012\u5f52 \u5c3e\u905e\u8ff4 recursion tree \u9012\u5f52\u6811 \u905e\u8ff4\u6a39 big-\\(O\\) notation \u5927 \\(O\\) \u8bb0\u53f7 \u5927 \\(O\\) \u8a18\u865f asymptotic upper bound \u6e10\u8fd1\u4e0a\u754c \u6f38\u8fd1\u4e0a\u754c sign-magnitude \u539f\u7801 \u539f\u78bc 1\u2019s complement \u53cd\u7801 \u4e00\u88dc\u6578 2\u2019s complement \u8865\u7801 \u4e8c\u88dc\u6578 array \u6570\u7ec4 \u9663\u5217 index \u7d22\u5f15 \u7d22\u5f15 linked list \u94fe\u8868 \u93c8\u7d50\u4e32\u5217 linked list node, list node \u94fe\u8868\u8282\u70b9 \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede head node \u5934\u8282\u70b9 \u982d\u7bc0\u9ede tail node \u5c3e\u8282\u70b9 \u5c3e\u7bc0\u9ede list \u5217\u8868 \u4e32\u5217 dynamic array \u52a8\u6001\u6570\u7ec4 \u52d5\u614b\u9663\u5217 hard disk \u786c\u76d8 \u786c\u789f random-access memory (RAM) \u5185\u5b58 \u8a18\u61b6\u9ad4 cache memory \u7f13\u5b58 \u5feb\u53d6 cache miss \u7f13\u5b58\u672a\u547d\u4e2d \u5feb\u53d6\u672a\u547d\u4e2d cache hit rate \u7f13\u5b58\u547d\u4e2d\u7387 \u5feb\u53d6\u547d\u4e2d\u7387 stack \u6808 \u5806\u758a top of the stack \u6808\u9876 \u5806\u758a\u9802 bottom of the stack \u6808\u5e95 \u5806\u758a\u5e95 queue \u961f\u5217 \u4f47\u5217 double-ended queue \u53cc\u5411\u961f\u5217 \u96d9\u5411\u4f47\u5217 front of the queue \u961f\u9996 \u4f47\u5217\u9996 rear of the queue \u961f\u5c3e \u4f47\u5217\u5c3e hash table \u54c8\u5e0c\u8868 \u96dc\u6e4a\u8868 hash set \u54c8\u5e0c\u96c6\u5408 \u96dc\u6e4a\u96c6\u5408 bucket \u6876 \u6876 hash function \u54c8\u5e0c\u51fd\u6570 \u96dc\u6e4a\u51fd\u5f0f hash collision \u54c8\u5e0c\u51b2\u7a81 \u96dc\u6e4a\u885d\u7a81 load factor \u8d1f\u8f7d\u56e0\u5b50 \u8ca0\u8f09\u56e0\u5b50 separate chaining \u94fe\u5f0f\u5730\u5740 \u93c8\u7d50\u4f4d\u5740 open addressing \u5f00\u653e\u5bfb\u5740 \u958b\u653e\u5b9a\u5740 linear probing \u7ebf\u6027\u63a2\u6d4b \u7dda\u6027\u63a2\u67e5 lazy deletion \u61d2\u5220\u9664 \u61f6\u522a\u9664 binary tree \u4e8c\u53c9\u6811 \u4e8c\u5143\u6a39 tree node \u6811\u8282\u70b9 \u6a39\u7bc0\u9ede left-child node \u5de6\u5b50\u8282\u70b9 \u5de6\u5b50\u7bc0\u9ede right-child node \u53f3\u5b50\u8282\u70b9 \u53f3\u5b50\u7bc0\u9ede parent node \u7236\u8282\u70b9 \u7236\u7bc0\u9ede left subtree \u5de6\u5b50\u6811 \u5de6\u5b50\u6a39 right subtree \u53f3\u5b50\u6811 \u53f3\u5b50\u6a39 root node \u6839\u8282\u70b9 \u6839\u7bc0\u9ede leaf node \u53f6\u8282\u70b9 \u8449\u7bc0\u9ede edge \u8fb9 \u908a level \u5c42 \u5c64 degree \u5ea6 \u5ea6 height \u9ad8\u5ea6 \u9ad8\u5ea6 depth \u6df1\u5ea6 \u6df1\u5ea6 perfect binary tree \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u5b8c\u7f8e\u4e8c\u5143\u6a39 complete binary tree \u5b8c\u5168\u4e8c\u53c9\u6811 \u5b8c\u5168\u4e8c\u5143\u6a39 full binary tree \u5b8c\u6ee1\u4e8c\u53c9\u6811 \u5b8c\u6eff\u4e8c\u5143\u6a39 balanced binary tree \u5e73\u8861\u4e8c\u53c9\u6811 \u5e73\u8861\u4e8c\u5143\u6a39 binary search tree \u4e8c\u53c9\u641c\u7d22\u6811 \u4e8c\u5143\u641c\u5c0b\u6a39 AVL tree AVL \u6811 AVL \u6a39 red-black tree \u7ea2\u9ed1\u6811 \u7d05\u9ed1\u6a39 level-order traversal \u5c42\u5e8f\u904d\u5386 \u5c64\u5e8f\u8d70\u8a2a breadth-first traversal \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a depth-first traversal \u6df1\u5ea6\u4f18\u5148\u904d\u5386 \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a binary search tree \u4e8c\u53c9\u641c\u7d22\u6811 \u4e8c\u5143\u641c\u5c0b\u6a39 balanced binary search tree \u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 \u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39 balance factor \u5e73\u8861\u56e0\u5b50 \u5e73\u8861\u56e0\u5b50 heap \u5806 \u5806\u7a4d max heap \u5927\u9876\u5806 \u5927\u9802\u5806\u7a4d min heap \u5c0f\u9876\u5806 \u5c0f\u9802\u5806\u7a4d priority queue \u4f18\u5148\u961f\u5217 \u512a\u5148\u4f47\u5217 heapify \u5806\u5316 \u5806\u7a4d\u5316 top-\\(k\\) problem Top-\\(k\\) \u95ee\u9898 Top-\\(k\\) \u554f\u984c graph \u56fe \u5716 vertex \u9876\u70b9 \u9802\u9ede undirected graph \u65e0\u5411\u56fe \u7121\u5411\u5716 directed graph \u6709\u5411\u56fe \u6709\u5411\u5716 connected graph \u8fde\u901a\u56fe \u9023\u901a\u5716 disconnected graph \u975e\u8fde\u901a\u56fe \u975e\u9023\u901a\u5716 weighted graph \u6709\u6743\u56fe \u6709\u6b0a\u5716 adjacency \u90bb\u63a5 \u9130\u63a5 path \u8def\u5f84 \u8def\u5f91 in-degree \u5165\u5ea6 \u5165\u5ea6 out-degree \u51fa\u5ea6 \u51fa\u5ea6 adjacency matrix \u90bb\u63a5\u77e9\u9635 \u9130\u63a5\u77e9\u9663 adjacency list \u90bb\u63a5\u8868 \u9130\u63a5\u8868 breadth-first search \u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 \u5ee3\u5ea6\u512a\u5148\u641c\u5c0b depth-first search \u6df1\u5ea6\u4f18\u5148\u641c\u7d22 \u6df1\u5ea6\u512a\u5148\u641c\u5c0b binary search \u4e8c\u5206\u67e5\u627e \u4e8c\u5206\u641c\u5c0b searching algorithm \u641c\u7d22\u7b97\u6cd5 \u641c\u5c0b\u6f14\u7b97\u6cd5 sorting algorithm \u6392\u5e8f\u7b97\u6cd5 \u6392\u5e8f\u6f14\u7b97\u6cd5 selection sort \u9009\u62e9\u6392\u5e8f \u9078\u64c7\u6392\u5e8f bubble sort \u5192\u6ce1\u6392\u5e8f \u6ce1\u6cab\u6392\u5e8f insertion sort \u63d2\u5165\u6392\u5e8f \u63d2\u5165\u6392\u5e8f quick sort \u5feb\u901f\u6392\u5e8f \u5feb\u901f\u6392\u5e8f merge sort \u5f52\u5e76\u6392\u5e8f \u5408\u4f75\u6392\u5e8f heap sort \u5806\u6392\u5e8f \u5806\u7a4d\u6392\u5e8f bucket sort \u6876\u6392\u5e8f \u6876\u6392\u5e8f counting sort \u8ba1\u6570\u6392\u5e8f \u8a08\u6578\u6392\u5e8f radix sort \u57fa\u6570\u6392\u5e8f \u57fa\u6578\u6392\u5e8f divide and conquer \u5206\u6cbb \u5206\u6cbb hanota problem \u6c49\u8bfa\u5854\u95ee\u9898 \u6cb3\u5167\u5854\u554f\u984c backtracking algorithm \u56de\u6eaf\u7b97\u6cd5 \u56de\u6eaf\u6f14\u7b97\u6cd5 constraint \u7ea6\u675f \u7d04\u675f solution \u89e3 \u89e3 state \u72b6\u6001 \u72c0\u614b pruning \u526a\u679d \u526a\u679d permutations problem \u5168\u6392\u5217\u95ee\u9898 \u5168\u6392\u5217\u554f\u984c subset-sum problem \u5b50\u96c6\u548c\u95ee\u9898 \u5b50\u96c6\u5408\u554f\u984c \\(n\\)-queens problem \\(n\\) \u7687\u540e\u95ee\u9898 \\(n\\) \u7687\u540e\u554f\u984c dynamic programming \u52a8\u6001\u89c4\u5212 \u52d5\u614b\u898f\u5283 initial state \u521d\u59cb\u72b6\u6001 \u521d\u59cb\u72c0\u614b state-transition equation \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \u72c0\u614b\u8f49\u79fb\u65b9\u7a0b knapsack problem \u80cc\u5305\u95ee\u9898 \u80cc\u5305\u554f\u984c edit distance problem \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 \u7de8\u8f2f\u8ddd\u96e2\u554f\u984c greedy algorithm \u8d2a\u5fc3\u7b97\u6cd5 \u8caa\u5a6a\u6f14\u7b97\u6cd5"},{"location":"chapter_array_and_linkedlist/","title":"\u7b2c 4 \u7ae0 \u00a0 \u9663\u5217\u8207\u93c8\u7d50\u4e32\u5217","text":"<p>Abstract</p> <p>\u8cc7\u6599\u7d50\u69cb\u7684\u4e16\u754c\u5982\u540c\u4e00\u5835\u539a\u5be6\u7684\u78da\u7246\u3002</p> <p>\u9663\u5217\u7684\u78da\u584a\u6574\u9f4a\u6392\u5217\uff0c\u9010\u500b\u7dca\u8cbc\u3002\u93c8\u7d50\u4e32\u5217\u7684\u78da\u584a\u5206\u6563\u5404\u8655\uff0c\u9023\u7dda\u7684\u85e4\u8513\u81ea\u7531\u5730\u7a7f\u68ad\u65bc\u78da\u7e2b\u4e4b\u9593\u3002</p>"},{"location":"chapter_array_and_linkedlist/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>4.1 \u00a0 \u9663\u5217</li> <li>4.2 \u00a0 \u93c8\u7d50\u4e32\u5217</li> <li>4.3 \u00a0 \u4e32\u5217</li> <li>4.4 \u00a0 \u8a18\u61b6\u9ad4\u8207\u5feb\u53d6 *</li> <li>4.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/","title":"4.1 \u00a0 \u9663\u5217","text":"<p>\u9663\u5217\uff08array\uff09\u662f\u4e00\u7a2e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u5176\u5c07\u76f8\u540c\u578b\u5225\u7684\u5143\u7d20\u5132\u5b58\u5728\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e2d\u3002\u6211\u5011\u5c07\u5143\u7d20\u5728\u9663\u5217\u4e2d\u7684\u4f4d\u7f6e\u7a31\u70ba\u8a72\u5143\u7d20\u7684\u7d22\u5f15\uff08index\uff09\u3002\u5716 4-1 \u5c55\u793a\u4e86\u9663\u5217\u7684\u4e3b\u8981\u6982\u5ff5\u548c\u5132\u5b58\u65b9\u5f0f\u3002</p> <p></p> <p> \u5716 4-1 \u00a0 \u9663\u5217\u5b9a\u7fa9\u8207\u5132\u5b58\u65b9\u5f0f </p>"},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1 \u00a0 \u9663\u5217\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/array/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u9663\u5217","text":"<p>\u6211\u5011\u53ef\u4ee5\u6839\u64da\u9700\u6c42\u9078\u7528\u9663\u5217\u7684\u5169\u7a2e\u521d\u59cb\u5316\u65b9\u5f0f\uff1a\u7121\u521d\u59cb\u503c\u3001\u7d66\u5b9a\u521d\u59cb\u503c\u3002\u5728\u672a\u6307\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u6cc1\u4e0b\uff0c\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u6703\u5c07\u9663\u5217\u5143\u7d20\u521d\u59cb\u5316\u70ba \\(0\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code># \u521d\u59cb\u5316\u9663\u5217\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]  \n</code></pre> array.cpp<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\n// \u5132\u5b58\u5728\u5806\u758a\u4e0a\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// \u5132\u5b58\u5728\u5806\u7a4d\u4e0a\uff08\u9700\u8981\u624b\u52d5\u91cb\u653e\u7a7a\u9593\uff09\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u9577\u5ea6\u6642\uff08[5]int\uff09\u70ba\u9663\u5217\uff0c\u4e0d\u6307\u5b9a\u9577\u5ea6\u6642\uff08[]int\uff09\u70ba\u5207\u7247\n// \u7531\u65bc Go \u7684\u9663\u5217\u88ab\u8a2d\u8a08\u70ba\u5728\u7de8\u8b6f\u671f\u78ba\u5b9a\u9577\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u6578\u4f86\u6307\u5b9a\u9577\u5ea6\n// \u70ba\u4e86\u65b9\u4fbf\u5be6\u73fe\u64f4\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c07\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u9663\u5217\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nlet arr: [i32; 5] = [0; 5]; // [0, 0, 0, 0, 0]\nlet slice: &amp;[i32] = &amp;[0; 5];\n// \u5728 Rust \u4e2d\uff0c\u6307\u5b9a\u9577\u5ea6\u6642\uff08[i32; 5]\uff09\u70ba\u9663\u5217\uff0c\u4e0d\u6307\u5b9a\u9577\u5ea6\u6642\uff08&amp;[i32]\uff09\u70ba\u5207\u7247\n// \u7531\u65bc Rust \u7684\u9663\u5217\u88ab\u8a2d\u8a08\u70ba\u5728\u7de8\u8b6f\u671f\u78ba\u5b9a\u9577\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u6578\u4f86\u6307\u5b9a\u9577\u5ea6\n// Vector \u662f Rust \u4e00\u822c\u60c5\u6cc1\u4e0b\u7528\u4f5c\u52d5\u614b\u9663\u5217\u7684\u578b\u5225\n// \u70ba\u4e86\u65b9\u4fbf\u5be6\u73fe\u64f4\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c07 vector \u770b\u4f5c\u9663\u5217\uff08array\uff09\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>/* \u521d\u59cb\u5316\u9663\u5217 */\nvar arr = IntArray(5) // { 0, 0, 0, 0, 0 }\nvar nums = intArrayOf(1, 3, 2, 5, 4)\n</code></pre> array.rb<pre><code># \u521d\u59cb\u5316\u9663\u5217\narr = Array.new(5, 0)\nnums = [1, 3, 2, 5, 4]\n</code></pre> array.zig<pre><code>// \u521d\u59cb\u5316\u9663\u5217\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#2","title":"2. \u00a0 \u8a2a\u554f\u5143\u7d20","text":"<p>\u9663\u5217\u5143\u7d20\u88ab\u5132\u5b58\u5728\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e2d\uff0c\u9019\u610f\u5473\u8457\u8a08\u7b97\u9663\u5217\u5143\u7d20\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7d66\u5b9a\u9663\u5217\u8a18\u61b6\u9ad4\u4f4d\u5740\uff08\u9996\u5143\u7d20\u8a18\u61b6\u9ad4\u4f4d\u5740\uff09\u548c\u67d0\u500b\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u5716 4-2 \u6240\u793a\u7684\u516c\u5f0f\u8a08\u7b97\u5f97\u5230\u8a72\u5143\u7d20\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\uff0c\u5f9e\u800c\u76f4\u63a5\u8a2a\u554f\u8a72\u5143\u7d20\u3002</p> <p></p> <p> \u5716 4-2 \u00a0 \u9663\u5217\u5143\u7d20\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u8a08\u7b97 </p> <p>\u89c0\u5bdf\u5716 4-2 \uff0c\u6211\u5011\u767c\u73fe\u9663\u5217\u9996\u500b\u5143\u7d20\u7684\u7d22\u5f15\u70ba \\(0\\) \uff0c\u9019\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89ba\uff0c\u56e0\u70ba\u5f9e \\(1\\) \u958b\u59cb\u8a08\u6578\u6703\u66f4\u81ea\u7136\u3002\u4f46\u5f9e\u4f4d\u5740\u8a08\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8cea\u4e0a\u662f\u8a18\u61b6\u9ad4\u4f4d\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u500b\u5143\u7d20\u7684\u4f4d\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u5b83\u7684\u7d22\u5f15\u70ba \\(0\\) \u662f\u5408\u7406\u7684\u3002</p> <p>\u5728\u9663\u5217\u4e2d\u8a2a\u554f\u5143\u7d20\u975e\u5e38\u9ad8\u6548\uff0c\u6211\u5011\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u5167\u96a8\u6a5f\u8a2a\u554f\u9663\u5217\u4e2d\u7684\u4efb\u610f\u4e00\u500b\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"\u96a8\u6a5f\u8a2a\u554f\u5143\u7d20\"\"\"\n    # \u5728\u5340\u9593 [0, len(nums)-1] \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    random_index = random.randint(0, len(nums) - 1)\n    # \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u5340\u9593 [0, size) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    int randomIndex = rand() % size;\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint randomAccess(int[] nums) {\n    // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // \u5728\u5340\u9593 [0, nums.Length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    int randomIndex = random.Next(nums.Length);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    randomIndex := rand.Intn(len(nums))\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // \u5728\u5340\u9593 [0, nums.count) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    let randomIndex = nums.indices.randomElement()!\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfunction randomAccess(nums) {\n    // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n    // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint randomAccess(List&lt;int&gt; nums) {\n  // \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n  int randomIndex = Random().nextInt(nums.length);\n  // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // \u5728\u5340\u9593 [0, nums.len()) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u5340\u9593 [0, size) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    int randomIndex = rand() % size;\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 */\nfun randomAccess(nums: IntArray): Int {\n    // \u5728\u5340\u9593 [0, nums.size) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20 ###\ndef random_access(nums)\n  # \u5728\u5340\u9593 [0, nums.length) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6578\u5b57\n  random_index = Random.rand(0...nums.length)\n\n  # \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n  nums[random_index]\nend\n</code></pre> array.zig<pre><code>// \u96a8\u6a5f\u8a2a\u554f\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n    // \u5728\u5340\u9593 [0, nums.len) \u4e2d\u96a8\u6a5f\u62bd\u53d6\u4e00\u500b\u6574\u6578\n    var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n    // \u7372\u53d6\u4e26\u8fd4\u56de\u96a8\u6a5f\u5143\u7d20\n    var randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#3","title":"3. \u00a0 \u63d2\u5165\u5143\u7d20","text":"<p>\u9663\u5217\u5143\u7d20\u5728\u8a18\u61b6\u9ad4\u4e2d\u662f\u201c\u7dca\u6328\u8457\u7684\u201d\uff0c\u5b83\u5011\u4e4b\u9593\u6c92\u6709\u7a7a\u9593\u518d\u5b58\u653e\u4efb\u4f55\u8cc7\u6599\u3002\u5982\u5716 4-3 \u6240\u793a\uff0c\u5982\u679c\u60f3\u5728\u9663\u5217\u4e2d\u9593\u63d2\u5165\u4e00\u500b\u5143\u7d20\uff0c\u5247\u9700\u8981\u5c07\u8a72\u5143\u7d20\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u4e4b\u5f8c\u518d\u628a\u5143\u7d20\u8ce6\u503c\u7d66\u8a72\u7d22\u5f15\u3002</p> <p></p> <p> \u5716 4-3 \u00a0 \u9663\u5217\u63d2\u5165\u5143\u7d20\u793a\u4f8b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u65bc\u9663\u5217\u7684\u9577\u5ea6\u662f\u56fa\u5b9a\u7684\uff0c\u56e0\u6b64\u63d2\u5165\u4e00\u500b\u5143\u7d20\u5fc5\u5b9a\u6703\u5c0e\u81f4\u9663\u5217\u5c3e\u90e8\u5143\u7d20\u201c\u4e1f\u5931\u201d\u3002\u6211\u5011\u5c07\u9019\u500b\u554f\u984c\u7684\u89e3\u6c7a\u65b9\u6848\u7559\u5728\u201c\u4e32\u5217\u201d\u7ae0\u7bc0\u4e2d\u8a0e\u8ad6\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"\u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num\"\"\"\n    # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).reversed() {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // \u5c07 _num \u8ce6\u7d66 index \u8655\u5143\u7d20\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &amp;mut [i32], num: i32, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num ###\ndef insert(nums, num, index)\n  # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n  nums[index] = num\nend\n</code></pre> array.zig<pre><code>// \u5728\u9663\u5217\u7684\u7d22\u5f15 index \u8655\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    var i = nums.len - 1;\n    while (i &gt; index) : (i -= 1) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c07 num \u8ce6\u7d66 index \u8655\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#4","title":"4. \u00a0 \u522a\u9664\u5143\u7d20","text":"<p>\u540c\u7406\uff0c\u5982\u5716 4-4 \u6240\u793a\uff0c\u82e5\u60f3\u522a\u9664\u7d22\u5f15 \\(i\\) \u8655\u7684\u5143\u7d20\uff0c\u5247\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\u3002</p> <p></p> <p> \u5716 4-4 \u00a0 \u9663\u5217\u522a\u9664\u5143\u7d20\u793a\u4f8b </p> <p>\u8acb\u6ce8\u610f\uff0c\u522a\u9664\u5143\u7d20\u5b8c\u6210\u5f8c\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u8b8a\u5f97\u201c\u7121\u610f\u7fa9\u201d\u4e86\uff0c\u6240\u4ee5\u6211\u5011\u7121\u9808\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"\u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20\"\"\"\n    # \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfn remove(nums: &amp;mut [i32], index: usize) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u4f54\u7528\u4e86 remove \u95dc\u9375\u8a5e\nvoid removeItem(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 */\nfun remove(nums: IntArray, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20 ###\ndef remove(nums, index)\n  # \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n  for i in index...(nums.length - 1)\n    nums[i] = nums[i + 1]\n  end\nend\n</code></pre> array.zig<pre><code>// \u522a\u9664\u7d22\u5f15 index \u8655\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    var i = index;\n    while (i &lt; nums.len - 1) : (i += 1) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u7e3d\u7684\u4f86\u770b\uff0c\u9663\u5217\u7684\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u9ede\u3002</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u9ad8\uff1a\u9663\u5217\u7684\u63d2\u5165\u548c\u522a\u9664\u7684\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u70ba\u9663\u5217\u9577\u5ea6\u3002</li> <li>\u4e1f\u5931\u5143\u7d20\uff1a\u7531\u65bc\u9663\u5217\u7684\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u5f8c\uff0c\u8d85\u51fa\u9663\u5217\u9577\u5ea6\u7bc4\u570d\u7684\u5143\u7d20\u6703\u4e1f\u5931\u3002</li> <li>\u8a18\u61b6\u9ad4\u6d6a\u8cbb\uff1a\u6211\u5011\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u500b\u6bd4\u8f03\u9577\u7684\u9663\u5217\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u9019\u6a23\u5728\u63d2\u5165\u8cc7\u6599\u6642\uff0c\u4e1f\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u201c\u7121\u610f\u7fa9\u201d\u7684\uff0c\u4f46\u9019\u6a23\u505a\u6703\u9020\u6210\u90e8\u5206\u8a18\u61b6\u9ad4\u7a7a\u9593\u6d6a\u8cbb\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#5","title":"5. \u00a0 \u8d70\u8a2a\u9663\u5217","text":"<p>\u5728\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\uff0c\u6211\u5011\u65e2\u53ef\u4ee5\u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u8d70\u8a2a\u7372\u53d6\u9663\u5217\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"\u8d70\u8a2a\u9663\u5217\"\"\"\n    count = 0\n    # \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for i in range(len(nums)):\n        count += nums[i]\n    # \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for num in nums:\n        count += num\n    # \u540c\u6642\u8d70\u8a2a\u8cc7\u6599\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfunc traverse(nums []int) {\n    count := 0\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for _, num := range nums {\n        count += num\n    }\n    // \u540c\u6642\u8d70\u8a2a\u8cc7\u6599\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for num in nums {\n        count += num\n    }\n    // \u540c\u6642\u8d70\u8a2a\u8cc7\u6599\u7d22\u5f15\u548c\u5143\u7d20\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfunction traverse(nums) {\n    let count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* \u8d70\u8a2a\u9663\u5217\u5143\u7d20 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n  for (int _num in nums) {\n    count += _num;\n  }\n  // \u900f\u904e forEach \u65b9\u6cd5\u8d70\u8a2a\u9663\u5217\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u8d70\u8a2a\u9663\u5217 */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (j in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### \u8d70\u8a2a\u9663\u5217 ###\ndef traverse(nums)\n  count = 0\n\n  # \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n  for num in nums\n    count += num\n  end\nend\n</code></pre> array.zig<pre><code>// \u8d70\u8a2a\u9663\u5217\nfn traverse(nums: []i32) void {\n    var count: i32 = 0;\n    // \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u9663\u5217\n    var i: i32 = 0;\n    while (i &lt; nums.len) : (i += 1) {\n        count += nums[i];\n    }\n    count = 0;\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\u5143\u7d20\n    for (nums) |num| {\n        count += num;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#6","title":"6. \u00a0 \u67e5\u8a62\u5143\u7d20","text":"<p>\u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20\u9700\u8981\u8d70\u8a2a\u9663\u5217\uff0c\u6bcf\u8f2a\u5224\u65b7\u5143\u7d20\u503c\u662f\u5426\u5339\u914d\uff0c\u82e5\u5339\u914d\u5247\u8f38\u51fa\u5c0d\u61c9\u7d22\u5f15\u3002</p> <p>\u56e0\u70ba\u9663\u5217\u662f\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u6240\u4ee5\u4e0a\u8ff0\u67e5\u8a62\u64cd\u4f5c\u88ab\u7a31\u70ba\u201c\u7dda\u6027\u67e5\u8a62\u201d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target)\n            return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20 ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> array.zig<pre><code>// \u5728\u9663\u5217\u4e2d\u67e5\u8a62\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n    for (nums, 0..) |num, i| {\n        if (num == target) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#7","title":"7. \u00a0 \u64f4\u5bb9\u9663\u5217","text":"<p>\u5728\u8907\u96dc\u7684\u7cfb\u7d71\u74b0\u5883\u4e2d\uff0c\u7a0b\u5f0f\u96e3\u4ee5\u4fdd\u8b49\u9663\u5217\u4e4b\u5f8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u662f\u53ef\u7528\u7684\uff0c\u5f9e\u800c\u7121\u6cd5\u5b89\u5168\u5730\u64f4\u5c55\u9663\u5217\u5bb9\u91cf\u3002\u56e0\u6b64\u5728\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\uff0c\u9663\u5217\u7684\u9577\u5ea6\u662f\u4e0d\u53ef\u8b8a\u7684\u3002</p> <p>\u5982\u679c\u6211\u5011\u5e0c\u671b\u64f4\u5bb9\u9663\u5217\uff0c\u5247\u9700\u91cd\u65b0\u5efa\u7acb\u4e00\u500b\u66f4\u5927\u7684\u9663\u5217\uff0c\u7136\u5f8c\u628a\u539f\u9663\u5217\u5143\u7d20\u4f9d\u6b21\u8907\u88fd\u5230\u65b0\u9663\u5217\u3002\u9019\u662f\u4e00\u500b \\(O(n)\\) \u7684\u64cd\u4f5c\uff0c\u5728\u9663\u5217\u5f88\u5927\u7684\u60c5\u6cc1\u4e0b\u975e\u5e38\u8017\u6642\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"\u64f4\u5c55\u9663\u5217\u9577\u5ea6\"\"\"\n    # \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    res = [0] * (len(nums) + enlarge)\n    # \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res\n</code></pre> array.cpp<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    int *res = new int[size + enlarge];\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    delete[] nums;\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.java<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    int[] res = new int[nums.length + enlarge];\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.cs<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    int[] res = new int[nums.Length + enlarge];\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.go<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    res := make([]int, len(nums)+enlarge)\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for i, num := range nums {\n        res[i] = num\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res\n}\n</code></pre> array.swift<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res\n}\n</code></pre> array.js<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\n// \u8acb\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52d5\u614b\u9663\u5217\uff0c\u53ef\u4ee5\u76f4\u63a5\u64f4\u5c55\n// \u70ba\u4e86\u65b9\u4fbf\u5b78\u7fd2\uff0c\u672c\u51fd\u5f0f\u5c07 Array \u770b\u4f5c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u7684\u9663\u5217\nfunction extend(nums, enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.ts<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\n// \u8acb\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52d5\u614b\u9663\u5217\uff0c\u53ef\u4ee5\u76f4\u63a5\u64f4\u5c55\n// \u70ba\u4e86\u65b9\u4fbf\u5b78\u7fd2\uff0c\u672c\u51fd\u5f0f\u5c07 Array \u770b\u4f5c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u7684\u9663\u5217\nfunction extend(nums: number[], enlarge: number): number[] {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.dart<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n  return res;\n}\n</code></pre> array.rs<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nfn extend(nums: &amp;[i32], enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\n    for i in 0..nums.len() {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    res\n}\n</code></pre> array.c<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u521d\u59cb\u5316\u64f4\u5c55\u5f8c\u7684\u7a7a\u9593\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> array.kt<pre><code>/* \u64f4\u5c55\u9663\u5217\u9577\u5ea6 */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    val res = IntArray(nums.size + enlarge)\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res\n}\n</code></pre> array.rb<pre><code>### \u64f4\u5c55\u9663\u5217\u9577\u5ea6 ###\n# \u8acb\u6ce8\u610f\uff0cRuby \u7684 Array \u662f\u52d5\u614b\u9663\u5217\uff0c\u53ef\u4ee5\u76f4\u63a5\u64f4\u5c55\n# \u70ba\u4e86\u65b9\u4fbf\u5b78\u7fd2\uff0c\u672c\u51fd\u5f0f\u5c07 Array \u770b\u4f5c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u7684\u9663\u5217\ndef extend(nums, enlarge)\n  # \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n  res = Array.new(nums.length + enlarge, 0)\n\n  # \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n  res\nend\n</code></pre> array.zig<pre><code>// \u64f4\u5c55\u9663\u5217\u9577\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n    // \u521d\u59cb\u5316\u4e00\u500b\u64f4\u5c55\u9577\u5ea6\u5f8c\u7684\u9663\u5217\n    var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n    @memset(res, 0);\n    // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n    std.mem.copy(i32, res, nums);\n    // \u8fd4\u56de\u64f4\u5c55\u5f8c\u7684\u65b0\u9663\u5217\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2 \u00a0 \u9663\u5217\u7684\u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u9663\u5217\u5132\u5b58\u5728\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u5167\uff0c\u4e14\u5143\u7d20\u578b\u5225\u76f8\u540c\u3002\u9019\u7a2e\u505a\u6cd5\u5305\u542b\u8c50\u5bcc\u7684\u5148\u9a57\u8cc7\u8a0a\uff0c\u7cfb\u7d71\u53ef\u4ee5\u5229\u7528\u9019\u4e9b\u8cc7\u8a0a\u4f86\u6700\u4f73\u5316\u8cc7\u6599\u7d50\u69cb\u7684\u64cd\u4f5c\u6548\u7387\u3002</p> <ul> <li>\u7a7a\u9593\u6548\u7387\u9ad8\uff1a\u9663\u5217\u70ba\u8cc7\u6599\u5206\u914d\u4e86\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u584a\uff0c\u7121\u9808\u984d\u5916\u7684\u7d50\u69cb\u958b\u92b7\u3002</li> <li>\u652f\u6301\u96a8\u6a5f\u8a2a\u554f\uff1a\u9663\u5217\u5141\u8a31\u5728 \\(O(1)\\) \u6642\u9593\u5167\u8a2a\u554f\u4efb\u4f55\u5143\u7d20\u3002</li> <li>\u5feb\u53d6\u5340\u57df\u6027\uff1a\u7576\u8a2a\u554f\u9663\u5217\u5143\u7d20\u6642\uff0c\u8a08\u7b97\u6a5f\u4e0d\u50c5\u6703\u8f09\u5165\u5b83\uff0c\u9084\u6703\u5feb\u53d6\u5176\u5468\u570d\u7684\u5176\u4ed6\u8cc7\u6599\uff0c\u5f9e\u800c\u85c9\u52a9\u9ad8\u901f\u5feb\u53d6\u4f86\u63d0\u5347\u5f8c\u7e8c\u64cd\u4f5c\u7684\u57f7\u884c\u901f\u5ea6\u3002</li> </ul> <p>\u9023\u7e8c\u7a7a\u9593\u5132\u5b58\u662f\u4e00\u628a\u96d9\u5203\u528d\uff0c\u5176\u5b58\u5728\u4ee5\u4e0b\u4fb7\u9650\u6027\u3002</p> <ul> <li>\u63d2\u5165\u8207\u522a\u9664\u6548\u7387\u4f4e\uff1a\u7576\u9663\u5217\u4e2d\u5143\u7d20\u8f03\u591a\u6642\uff0c\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u9700\u8981\u79fb\u52d5\u5927\u91cf\u7684\u5143\u7d20\u3002</li> <li>\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff1a\u9663\u5217\u5728\u521d\u59cb\u5316\u5f8c\u9577\u5ea6\u5c31\u56fa\u5b9a\u4e86\uff0c\u64f4\u5bb9\u9663\u5217\u9700\u8981\u5c07\u6240\u6709\u8cc7\u6599\u8907\u88fd\u5230\u65b0\u9663\u5217\uff0c\u958b\u92b7\u5f88\u5927\u3002</li> <li>\u7a7a\u9593\u6d6a\u8cbb\uff1a\u5982\u679c\u9663\u5217\u5206\u914d\u7684\u5927\u5c0f\u8d85\u904e\u5be6\u969b\u6240\u9700\uff0c\u90a3\u9ebc\u591a\u9918\u7684\u7a7a\u9593\u5c31\u88ab\u6d6a\u8cbb\u4e86\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3 \u00a0 \u9663\u5217\u5178\u578b\u61c9\u7528","text":"<p>\u9663\u5217\u662f\u4e00\u7a2e\u57fa\u790e\u4e14\u5e38\u898b\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u65e2\u983b\u7e41\u61c9\u7528\u5728\u5404\u985e\u6f14\u7b97\u6cd5\u4e4b\u4e2d\uff0c\u4e5f\u53ef\u7528\u65bc\u5be6\u73fe\u5404\u7a2e\u8907\u96dc\u8cc7\u6599\u7d50\u69cb\u3002</p> <ul> <li>\u96a8\u6a5f\u8a2a\u554f\uff1a\u5982\u679c\u6211\u5011\u60f3\u96a8\u6a5f\u62bd\u53d6\u4e00\u4e9b\u6a23\u672c\uff0c\u90a3\u9ebc\u53ef\u4ee5\u7528\u9663\u5217\u5132\u5b58\uff0c\u4e26\u751f\u6210\u4e00\u500b\u96a8\u6a5f\u5e8f\u5217\uff0c\u6839\u64da\u7d22\u5f15\u5be6\u73fe\u96a8\u6a5f\u62bd\u6a23\u3002</li> <li>\u6392\u5e8f\u548c\u641c\u5c0b\uff1a\u9663\u5217\u662f\u6392\u5e8f\u548c\u641c\u5c0b\u6f14\u7b97\u6cd5\u6700\u5e38\u7528\u7684\u8cc7\u6599\u7d50\u69cb\u3002\u5feb\u901f\u6392\u5e8f\u3001\u5408\u4f75\u6392\u5e8f\u3001\u4e8c\u5206\u641c\u5c0b\u7b49\u90fd\u4e3b\u8981\u5728\u9663\u5217\u4e0a\u9032\u884c\u3002</li> <li>\u67e5\u8a62\u8868\uff1a\u7576\u9700\u8981\u5feb\u901f\u67e5\u8a62\u4e00\u500b\u5143\u7d20\u6216\u5176\u5c0d\u61c9\u95dc\u4fc2\u6642\uff0c\u53ef\u4ee5\u4f7f\u7528\u9663\u5217\u4f5c\u70ba\u67e5\u8a62\u8868\u3002\u5047\u5982\u6211\u5011\u60f3\u5be6\u73fe\u5b57\u5143\u5230 ASCII \u78bc\u7684\u5c0d\u6620\uff0c\u5247\u53ef\u4ee5\u5c07\u5b57\u5143\u7684 ASCII \u78bc\u503c\u4f5c\u70ba\u7d22\u5f15\uff0c\u5c0d\u61c9\u7684\u5143\u7d20\u5b58\u653e\u5728\u9663\u5217\u4e2d\u7684\u5c0d\u61c9\u4f4d\u7f6e\u3002</li> <li>\u6a5f\u5668\u5b78\u7fd2\uff1a\u795e\u7d93\u7db2\u8def\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9663\u3001\u5f35\u91cf\u4e4b\u9593\u7684\u7dda\u6027\u4ee3\u6578\u904b\u7b97\uff0c\u9019\u4e9b\u8cc7\u6599\u90fd\u662f\u4ee5\u9663\u5217\u7684\u5f62\u5f0f\u69cb\u5efa\u7684\u3002\u9663\u5217\u662f\u795e\u7d93\u7db2\u8def\u7a0b\u5f0f\u8a2d\u8a08\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u8cc7\u6599\u7d50\u69cb\u5be6\u73fe\uff1a\u9663\u5217\u53ef\u4ee5\u7528\u65bc\u5be6\u73fe\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u5806\u7a4d\u3001\u5716\u7b49\u8cc7\u6599\u7d50\u69cb\u3002\u4f8b\u5982\uff0c\u5716\u7684\u9130\u63a5\u77e9\u9663\u8868\u793a\u5be6\u969b\u4e0a\u662f\u4e00\u500b\u4e8c\u7dad\u9663\u5217\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2 \u00a0 \u93c8\u7d50\u4e32\u5217","text":"<p>\u8a18\u61b6\u9ad4\u7a7a\u9593\u662f\u6240\u6709\u7a0b\u5f0f\u7684\u516c\u5171\u8cc7\u6e90\uff0c\u5728\u4e00\u500b\u8907\u96dc\u7684\u7cfb\u7d71\u57f7\u884c\u74b0\u5883\u4e0b\uff0c\u7a7a\u9592\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u53ef\u80fd\u6563\u843d\u5728\u8a18\u61b6\u9ad4\u5404\u8655\u3002\u6211\u5011\u77e5\u9053\uff0c\u5132\u5b58\u9663\u5217\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u5fc5\u9808\u662f\u9023\u7e8c\u7684\uff0c\u800c\u7576\u9663\u5217\u975e\u5e38\u5927\u6642\uff0c\u8a18\u61b6\u9ad4\u53ef\u80fd\u7121\u6cd5\u63d0\u4f9b\u5982\u6b64\u5927\u7684\u9023\u7e8c\u7a7a\u9593\u3002\u6b64\u6642\u93c8\u7d50\u4e32\u5217\u7684\u9748\u6d3b\u6027\u512a\u52e2\u5c31\u9ad4\u73fe\u51fa\u4f86\u4e86\u3002</p> <p>\u93c8\u7d50\u4e32\u5217\uff08linked list\uff09\u662f\u4e00\u7a2e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u5176\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\u90fd\u662f\u4e00\u500b\u7bc0\u9ede\u7269\u4ef6\uff0c\u5404\u500b\u7bc0\u9ede\u900f\u904e\u201c\u5f15\u7528\u201d\u76f8\u9023\u7dda\u3002\u5f15\u7528\u8a18\u9304\u4e86\u4e0b\u4e00\u500b\u7bc0\u9ede\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\uff0c\u900f\u904e\u5b83\u53ef\u4ee5\u5f9e\u7576\u524d\u7bc0\u9ede\u8a2a\u554f\u5230\u4e0b\u4e00\u500b\u7bc0\u9ede\u3002</p> <p>\u93c8\u7d50\u4e32\u5217\u7684\u8a2d\u8a08\u4f7f\u5f97\u5404\u500b\u7bc0\u9ede\u53ef\u4ee5\u5206\u6563\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u5404\u8655\uff0c\u5b83\u5011\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u7121\u9808\u9023\u7e8c\u3002</p> <p></p> <p> \u5716 4-5 \u00a0 \u93c8\u7d50\u4e32\u5217\u5b9a\u7fa9\u8207\u5132\u5b58\u65b9\u5f0f </p> <p>\u89c0\u5bdf\u5716 4-5 \uff0c\u93c8\u7d50\u4e32\u5217\u7684\u7d44\u6210\u55ae\u4f4d\u662f\u7bc0\u9ede\uff08node\uff09\u7269\u4ef6\u3002\u6bcf\u500b\u7bc0\u9ede\u90fd\u5305\u542b\u5169\u9805\u8cc7\u6599\uff1a\u7bc0\u9ede\u7684\u201c\u503c\u201d\u548c\u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u201c\u5f15\u7528\u201d\u3002</p> <ul> <li>\u93c8\u7d50\u4e32\u5217\u7684\u9996\u500b\u7bc0\u9ede\u88ab\u7a31\u70ba\u201c\u982d\u7bc0\u9ede\u201d\uff0c\u6700\u5f8c\u4e00\u500b\u7bc0\u9ede\u88ab\u7a31\u70ba\u201c\u5c3e\u7bc0\u9ede\u201d\u3002</li> <li>\u5c3e\u7bc0\u9ede\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5b83\u5728 Java\u3001C++ \u548c Python \u4e2d\u5206\u5225\u88ab\u8a18\u70ba <code>null</code>\u3001<code>nullptr</code> \u548c <code>None</code> \u3002</li> <li>\u5728 C\u3001C++\u3001Go \u548c Rust \u7b49\u652f\u6301\u6307\u6a19\u7684\u8a9e\u8a00\u4e2d\uff0c\u4e0a\u8ff0\u201c\u5f15\u7528\u201d\u61c9\u88ab\u66ff\u63db\u70ba\u201c\u6307\u6a19\u201d\u3002</li> </ul> <p>\u5982\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\uff0c\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede <code>ListNode</code> \u9664\u4e86\u5305\u542b\u503c\uff0c\u9084\u9700\u984d\u5916\u5132\u5b58\u4e00\u500b\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002\u56e0\u6b64\u5728\u76f8\u540c\u8cc7\u6599\u91cf\u4e0b\uff0c\u93c8\u7d50\u4e32\u5217\u6bd4\u9663\u5217\u4f54\u7528\u66f4\u591a\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class ListNode:\n    \"\"\"\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # \u7bc0\u9ede\u503c\n        self.next: ListNode | None = None # \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct ListNode {\n    int val;         // \u7bc0\u9ede\u503c\n    ListNode *next;  // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n    ListNode(int x) : val(x), next(nullptr) {}  // \u5efa\u69cb\u5b50\n};\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    int val;        // \u7bc0\u9ede\u503c\n    ListNode next;  // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u5efa\u69cb\u5b50\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode(int x) {  //\u5efa\u69cb\u5b50\n    int val = x;         // \u7bc0\u9ede\u503c\n    ListNode? next;      // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntype ListNode struct {\n    Val  int       // \u7bc0\u9ede\u503c\n    Next *ListNode // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n}\n\n// NewListNode \u5efa\u69cb\u5b50\uff0c\u5efa\u7acb\u4e00\u500b\u65b0\u7684\u93c8\u7d50\u4e32\u5217\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var next: ListNode? // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n\n    init(x: Int) { // \u5efa\u69cb\u5b50\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // \u7bc0\u9ede\u503c\n        this.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // \u7bc0\u9ede\u503c\n        this.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n  int val; // \u7bc0\u9ede\u503c\n  ListNode? next; // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n  ListNode(this.val, [this.next]); // \u5efa\u69cb\u5b50\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u7bc0\u9ede\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct ListNode {\n    int val;               // \u7bc0\u9ede\u503c\n    struct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n} ListNode;\n\n/* \u5efa\u69cb\u5b50 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\n// \u5efa\u69cb\u5b50\nclass ListNode(x: Int) {\n    val _val: Int = x          // \u7bc0\u9ede\u503c\n    val next: ListNode? = null // \u6307\u5411\u4e0b\u4e00\u500b\u7bc0\u9ede\u7684\u5f15\u7528\n}\n</code></pre> <pre><code># \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\nclass ListNode\n  attr_accessor :val  # \u7bc0\u9ede\u503c\n  attr_accessor :next # \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n\n  def initialize(val=0, next_node=nil)\n    @val = val\n    @next = next_node\n  end\nend\n</code></pre> <pre><code>// \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u7bc0\u9ede\u503c\n        next: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u6307\u6a19\n\n        // \u5efa\u69cb\u5b50\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1 \u00a0 \u93c8\u7d50\u4e32\u5217\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217","text":"<p>\u5efa\u7acb\u93c8\u7d50\u4e32\u5217\u5206\u70ba\u5169\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\u7269\u4ef6\uff0c\u7b2c\u4e8c\u6b65\u662f\u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\u95dc\u4fc2\u3002\u521d\u59cb\u5316\u5b8c\u6210\u5f8c\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5f9e\u93c8\u7d50\u4e32\u5217\u7684\u982d\u7bc0\u9ede\u51fa\u767c\uff0c\u900f\u904e\u5f15\u7528\u6307\u5411 <code>next</code> \u4f9d\u6b21\u8a2a\u554f\u6240\u6709\u7bc0\u9ede\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code># \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\\\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>/* \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nval n0 = ListNode(1)\nval n1 = ListNode(3)\nval n2 = ListNode(2)\nval n3 = ListNode(5)\nval n4 = ListNode(4)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rb<pre><code># \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nn0 = ListNode.new(1)\nn1 = ListNode.new(3)\nn2 = ListNode.new(2)\nn3 = ListNode.new(5)\nn4 = ListNode.new(4)\n# \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.zig<pre><code>// \u521d\u59cb\u5316\u93c8\u7d50\u4e32\u5217\n// \u521d\u59cb\u5316\u5404\u500b\u7bc0\u9ede\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\nn0.next = &amp;n1;\nn1.next = &amp;n2;\nn2.next = &amp;n3;\nn3.next = &amp;n4;\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u9663\u5217\u6574\u9ad4\u662f\u4e00\u500b\u8b8a\u6578\uff0c\u6bd4\u5982\u9663\u5217 <code>nums</code> \u5305\u542b\u5143\u7d20 <code>nums[0]</code> \u548c <code>nums[1]</code> \u7b49\uff0c\u800c\u93c8\u7d50\u4e32\u5217\u662f\u7531\u591a\u500b\u7368\u7acb\u7684\u7bc0\u9ede\u7269\u4ef6\u7d44\u6210\u7684\u3002\u6211\u5011\u901a\u5e38\u5c07\u982d\u7bc0\u9ede\u7576\u4f5c\u93c8\u7d50\u4e32\u5217\u7684\u4ee3\u7a31\uff0c\u6bd4\u5982\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u4e2d\u7684\u93c8\u7d50\u4e32\u5217\u53ef\u8a18\u4f5c\u93c8\u7d50\u4e32\u5217 <code>n0</code> \u3002</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#2","title":"2. \u00a0 \u63d2\u5165\u7bc0\u9ede","text":"<p>\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u63d2\u5165\u7bc0\u9ede\u975e\u5e38\u5bb9\u6613\u3002\u5982\u5716 4-6 \u6240\u793a\uff0c\u5047\u8a2d\u6211\u5011\u60f3\u5728\u76f8\u9130\u7684\u5169\u500b\u7bc0\u9ede <code>n0</code> \u548c <code>n1</code> \u4e4b\u9593\u63d2\u5165\u4e00\u500b\u65b0\u7bc0\u9ede <code>P</code> \uff0c\u5247\u53ea\u9700\u6539\u8b8a\u5169\u500b\u7bc0\u9ede\u5f15\u7528\uff08\u6307\u6a19\uff09\u5373\u53ef\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</p> <p>\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5728\u9663\u5217\u4e2d\u63d2\u5165\u5143\u7d20\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u5728\u5927\u8cc7\u6599\u91cf\u4e0b\u7684\u6548\u7387\u8f03\u4f4e\u3002</p> <p></p> <p> \u5716 4-6 \u00a0 \u93c8\u7d50\u4e32\u5217\u63d2\u5165\u7bc0\u9ede\u793a\u4f8b </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"\u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede _p ###\n# Ruby \u7684 `p` \u662f\u4e00\u500b\u5167\u5efa\u51fd\u5f0f\uff0c `P` \u662f\u4e00\u500b\u5e38\u6578\uff0c\u6240\u4ee5\u53ef\u4ee5\u4f7f\u7528 `_p` \u4ee3\u66ff\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u63d2\u5165\u7bc0\u9ede P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n    var n1 = n0.?.next;\n    P.?.next = n1;\n    n0.?.next = P;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#3","title":"3. \u00a0 \u522a\u9664\u7bc0\u9ede","text":"<p>\u5982\u5716 4-7 \u6240\u793a\uff0c\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u522a\u9664\u7bc0\u9ede\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u8b8a\u4e00\u500b\u7bc0\u9ede\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\u5373\u53ef\u3002</p> <p>\u8acb\u6ce8\u610f\uff0c\u5118\u7ba1\u5728\u522a\u9664\u64cd\u4f5c\u5b8c\u6210\u5f8c\u7bc0\u9ede <code>P</code> \u4ecd\u7136\u6307\u5411 <code>n1</code> \uff0c\u4f46\u5be6\u969b\u4e0a\u8d70\u8a2a\u6b64\u93c8\u7d50\u4e32\u5217\u5df2\u7d93\u7121\u6cd5\u8a2a\u554f\u5230 <code>P</code> \uff0c\u9019\u610f\u5473\u8457 <code>P</code> \u5df2\u7d93\u4e0d\u518d\u5c6c\u65bc\u8a72\u93c8\u7d50\u4e32\u5217\u4e86\u3002</p> <p></p> <p> \u5716 4-7 \u00a0 \u93c8\u7d50\u4e32\u5217\u522a\u9664\u7bc0\u9ede </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"\u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    if n0.borrow().next.is_none() {\n        return;\n    };\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\n// \u6ce8\u610f\uff1astdio.h \u4f54\u7528\u4e86 remove \u95dc\u9375\u8a5e\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede */\nfun remove(n0: ListNode?) {\n    if (n0?.next == null)\n        return\n    // n0 -&gt; P -&gt; n1\n    val p = n0.next\n    val n1 = p?.next\n    n0.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> linked_list.zig<pre><code>// \u522a\u9664\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede n0 \u4e4b\u5f8c\u7684\u9996\u500b\u7bc0\u9ede\nfn remove(n0: ?*inc.ListNode(i32)) void {\n    if (n0.?.next == null) return;\n    // n0 -&gt; P -&gt; n1\n    var P = n0.?.next;\n    var n1 = P.?.next;\n    n0.?.next = n1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#4","title":"4. \u00a0 \u8a2a\u554f\u7bc0\u9ede","text":"<p>\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u8a2a\u554f\u7bc0\u9ede\u7684\u6548\u7387\u8f03\u4f4e\u3002\u5982\u4e0a\u4e00\u7bc0\u6240\u8ff0\uff0c\u6211\u5011\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u4e0b\u8a2a\u554f\u9663\u5217\u4e2d\u7684\u4efb\u610f\u5143\u7d20\u3002\u93c8\u7d50\u4e32\u5217\u5247\u4e0d\u7136\uff0c\u7a0b\u5f0f\u9700\u8981\u5f9e\u982d\u7bc0\u9ede\u51fa\u767c\uff0c\u9010\u500b\u5411\u5f8c\u8d70\u8a2a\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6a19\u7bc0\u9ede\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u7684\u7b2c \\(i\\) \u500b\u7bc0\u9ede\u9700\u8981\u8ff4\u5708 \\(i - 1\\) \u8f2a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n    if index &lt;= 0 {\n        return head;\n    };\n    if let Some(node) = &amp;head.borrow().next {\n        return access(node.clone(), index - 1);\n    }\n\n    return head;\n}\n</code></pre> linked_list.c<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        if (h == null)\n            return null\n        h = h.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> linked_list.zig<pre><code>// \u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u4e2d\u7d22\u5f15\u70ba index \u7684\u7bc0\u9ede\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n    var head = node;\n    var i: i32 = 0;\n    while (i &lt; index) : (i += 1) {\n        head = head.?.next;\n        if (head == null) return null;\n    }\n    return head;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#5","title":"5. \u00a0 \u67e5\u8a62\u7bc0\u9ede","text":"<p>\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\uff0c\u67e5\u8a62\u5176\u4e2d\u503c\u70ba <code>target</code> \u7684\u7bc0\u9ede\uff0c\u8f38\u51fa\u8a72\u7bc0\u9ede\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u7d22\u5f15\u3002\u6b64\u904e\u7a0b\u4e5f\u5c6c\u65bc\u7dda\u6027\u67e5\u8a62\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T, index: i32) -&gt; i32 {\n    if head.borrow().val == target {\n        return index;\n    };\n    if let Some(node) = &amp;head.borrow_mut().next {\n        return find(node.clone(), target, index + 1);\n    }\n    return -1;\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h._val == target)\n            return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u503c\u70ba target \u7684\u9996\u500b\u7bc0\u9ede\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n    var head = node;\n    var index: i32 = 0;\n    while (head != null) {\n        if (head.?.val == target) return index;\n        head = head.?.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","title":"4.2.2 \u00a0 \u9663\u5217 vs. \u93c8\u7d50\u4e32\u5217","text":"<p>\u8868 4-1 \u7e3d\u7d50\u4e86\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u7684\u5404\u9805\u7279\u9ede\u4e26\u5c0d\u6bd4\u4e86\u64cd\u4f5c\u6548\u7387\u3002\u7531\u65bc\u5b83\u5011\u63a1\u7528\u5169\u7a2e\u76f8\u53cd\u7684\u5132\u5b58\u7b56\u7565\uff0c\u56e0\u6b64\u5404\u7a2e\u6027\u8cea\u548c\u64cd\u4f5c\u6548\u7387\u4e5f\u5448\u73fe\u5c0d\u7acb\u7684\u7279\u9ede\u3002</p> <p> \u8868 4-1 \u00a0 \u9663\u5217\u8207\u93c8\u7d50\u4e32\u5217\u7684\u6548\u7387\u5c0d\u6bd4 </p> \u9663\u5217 \u93c8\u7d50\u4e32\u5217 \u5132\u5b58\u65b9\u5f0f \u9023\u7e8c\u8a18\u61b6\u9ad4\u7a7a\u9593 \u5206\u6563\u8a18\u61b6\u9ad4\u7a7a\u9593 \u5bb9\u91cf\u64f4\u5c55 \u9577\u5ea6\u4e0d\u53ef\u8b8a \u53ef\u9748\u6d3b\u64f4\u5c55 \u8a18\u61b6\u9ad4\u6548\u7387 \u5143\u7d20\u4f54\u7528\u8a18\u61b6\u9ad4\u5c11\u3001\u4f46\u53ef\u80fd\u6d6a\u8cbb\u7a7a\u9593 \u5143\u7d20\u4f54\u7528\u8a18\u61b6\u9ad4\u591a \u8a2a\u554f\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \u65b0\u589e\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \u522a\u9664\u5143\u7d20 \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3 \u00a0 \u5e38\u898b\u93c8\u7d50\u4e32\u5217\u578b\u5225","text":"<p>\u5982\u5716 4-8 \u6240\u793a\uff0c\u5e38\u898b\u7684\u93c8\u7d50\u4e32\u5217\u578b\u5225\u5305\u62ec\u4e09\u7a2e\u3002</p> <ul> <li>\u55ae\u5411\u93c8\u7d50\u4e32\u5217\uff1a\u5373\u524d\u9762\u4ecb\u7d39\u7684\u666e\u901a\u93c8\u7d50\u4e32\u5217\u3002\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\u5169\u9805\u8cc7\u6599\u3002\u6211\u5011\u5c07\u9996\u500b\u7bc0\u9ede\u7a31\u70ba\u982d\u7bc0\u9ede\uff0c\u5c07\u6700\u5f8c\u4e00\u500b\u7bc0\u9ede\u7a31\u70ba\u5c3e\u7bc0\u9ede\uff0c\u5c3e\u7bc0\u9ede\u6307\u5411\u7a7a <code>None</code> \u3002</li> <li>\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\uff1a\u5982\u679c\u6211\u5011\u4ee4\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u7684\u5c3e\u7bc0\u9ede\u6307\u5411\u982d\u7bc0\u9ede\uff08\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5247\u5f97\u5230\u4e00\u500b\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u3002\u5728\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u4e2d\uff0c\u4efb\u610f\u7bc0\u9ede\u90fd\u53ef\u4ee5\u8996\u4f5c\u982d\u7bc0\u9ede\u3002</li> <li>\u96d9\u5411\u93c8\u7d50\u4e32\u5217\uff1a\u8207\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u76f8\u6bd4\uff0c\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u8a18\u9304\u4e86\u5169\u500b\u65b9\u5411\u7684\u5f15\u7528\u3002\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7684\u7bc0\u9ede\u5b9a\u7fa9\u540c\u6642\u5305\u542b\u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\uff08\u4e0b\u4e00\u500b\u7bc0\u9ede\uff09\u548c\u524d\u9a45\u7bc0\u9ede\uff08\u4e0a\u4e00\u500b\u7bc0\u9ede\uff09\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002\u76f8\u8f03\u65bc\u55ae\u5411\u93c8\u7d50\u4e32\u5217\uff0c\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u66f4\u5177\u9748\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u5169\u500b\u65b9\u5411\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\uff0c\u4f46\u76f8\u61c9\u5730\u4e5f\u9700\u8981\u4f54\u7528\u66f4\u591a\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class ListNode:\n    \"\"\"\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u7bc0\u9ede\u503c\n        self.next: ListNode | None = None  # \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n        self.prev: ListNode | None = None  # \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct ListNode {\n    int val;         // \u7bc0\u9ede\u503c\n    ListNode *next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n    ListNode *prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u5efa\u69cb\u5b50\n};\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    int val;        // \u7bc0\u9ede\u503c\n    ListNode next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u5efa\u69cb\u5b50\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode(int x) {  // \u5efa\u69cb\u5b50\n    int val = x;    // \u7bc0\u9ede\u503c\n    ListNode next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntype DoublyListNode struct {\n    Val  int             // \u7bc0\u9ede\u503c\n    Next *DoublyListNode // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n    Prev *DoublyListNode // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n}\n\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var next: ListNode? // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    var prev: ListNode? // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n\n    init(x: Int) { // \u5efa\u69cb\u5b50\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // \u7bc0\u9ede\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // \u7bc0\u9ede\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\nclass ListNode {\n    int val;        // \u7bc0\u9ede\u503c\n    ListNode? next;  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode? prev;  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n    ListNode(this.val, [this.next, this.prev]);  // \u5efa\u69cb\u5b50\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u578b\u5225 */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u7bc0\u9ede\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n}\n\n/* \u5efa\u69cb\u5b50 */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct ListNode {\n    int val;               // \u7bc0\u9ede\u503c\n    struct ListNode *next; // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n    struct ListNode *prev; // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n} ListNode;\n\n/* \u5efa\u69cb\u5b50 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225 */\n// \u5efa\u69cb\u5b50\nclass ListNode(x: Int) {\n    val _val: Int = x           // \u7bc0\u9ede\u503c\n    val next: ListNode? = null  // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n    val prev: ListNode? = null  // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n}\n</code></pre> <pre><code># \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\nclass ListNode\n  attr_accessor :val    # \u7bc0\u9ede\u503c\n  attr_accessor :next   # \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u5f15\u7528\n  attr_accessor :prev   # \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u5f15\u7528\n\n  def initialize(val=0, next_node=nil, prev_node=nil)\n    @val = val\n    @next = next_node\n    @prev = prev_node\n  end\nend\n</code></pre> <pre><code>// \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u985e\u5225\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u7bc0\u9ede\u503c\n        next: ?*Self = null, // \u6307\u5411\u5f8c\u7e7c\u7bc0\u9ede\u7684\u6307\u6a19\n        prev: ?*Self = null, // \u6307\u5411\u524d\u9a45\u7bc0\u9ede\u7684\u6307\u6a19\n\n        // \u5efa\u69cb\u5b50\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n</code></pre> <p></p> <p> \u5716 4-8 \u00a0 \u5e38\u898b\u93c8\u7d50\u4e32\u5217\u7a2e\u985e </p>"},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4 \u00a0 \u93c8\u7d50\u4e32\u5217\u5178\u578b\u61c9\u7528","text":"<p>\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u901a\u5e38\u7528\u65bc\u5be6\u73fe\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u548c\u5716\u7b49\u8cc7\u6599\u7d50\u69cb\u3002</p> <ul> <li>\u5806\u758a\u8207\u4f47\u5217\uff1a\u7576\u63d2\u5165\u548c\u522a\u9664\u64cd\u4f5c\u90fd\u5728\u93c8\u7d50\u4e32\u5217\u7684\u4e00\u7aef\u9032\u884c\u6642\uff0c\u5b83\u8868\u73fe\u7684\u7279\u6027\u70ba\u5148\u9032\u5f8c\u51fa\uff0c\u5c0d\u61c9\u5806\u758a\uff1b\u7576\u63d2\u5165\u64cd\u4f5c\u5728\u93c8\u7d50\u4e32\u5217\u7684\u4e00\u7aef\u9032\u884c\uff0c\u522a\u9664\u64cd\u4f5c\u5728\u93c8\u7d50\u4e32\u5217\u7684\u53e6\u4e00\u7aef\u9032\u884c\uff0c\u5b83\u8868\u73fe\u7684\u7279\u6027\u70ba\u5148\u9032\u5148\u51fa\uff0c\u5c0d\u61c9\u4f47\u5217\u3002</li> <li>\u96dc\u6e4a\u8868\uff1a\u93c8\u5f0f\u4f4d\u5740\u662f\u89e3\u6c7a\u96dc\u6e4a\u885d\u7a81\u7684\u4e3b\u6d41\u65b9\u6848\u4e4b\u4e00\uff0c\u5728\u8a72\u65b9\u6848\u4e2d\uff0c\u6240\u6709\u885d\u7a81\u7684\u5143\u7d20\u90fd\u6703\u88ab\u653e\u5230\u4e00\u500b\u93c8\u7d50\u4e32\u5217\u4e2d\u3002</li> <li>\u5716\uff1a\u9130\u63a5\u8868\u662f\u8868\u793a\u5716\u7684\u4e00\u7a2e\u5e38\u7528\u65b9\u5f0f\uff0c\u5176\u4e2d\u5716\u7684\u6bcf\u500b\u9802\u9ede\u90fd\u8207\u4e00\u500b\u93c8\u7d50\u4e32\u5217\u76f8\u95dc\u806f\uff0c\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\u90fd\u4ee3\u8868\u8207\u8a72\u9802\u9ede\u76f8\u9023\u7684\u5176\u4ed6\u9802\u9ede\u3002</li> </ul> <p>\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5e38\u7528\u65bc\u9700\u8981\u5feb\u901f\u67e5\u8a62\u524d\u4e00\u500b\u548c\u5f8c\u4e00\u500b\u5143\u7d20\u7684\u5834\u666f\u3002</p> <ul> <li>\u9ad8\u968e\u8cc7\u6599\u7d50\u69cb\uff1a\u6bd4\u5982\u5728\u7d05\u9ed1\u6a39\u3001B \u6a39\u4e2d\uff0c\u6211\u5011\u9700\u8981\u8a2a\u554f\u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\uff0c\u9019\u53ef\u4ee5\u900f\u904e\u5728\u7bc0\u9ede\u4e2d\u5132\u5b58\u4e00\u500b\u6307\u5411\u7236\u7bc0\u9ede\u7684\u5f15\u7528\u4f86\u5be6\u73fe\uff0c\u985e\u4f3c\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u3002</li> <li>\u700f\u89bd\u5668\u6b77\u53f2\uff1a\u5728\u7db2\u9801\u700f\u89bd\u5668\u4e2d\uff0c\u7576\u7528\u6236\u9ede\u9078\u524d\u9032\u6216\u5f8c\u9000\u6309\u9215\u6642\uff0c\u700f\u89bd\u5668\u9700\u8981\u77e5\u9053\u4f7f\u7528\u8005\u8a2a\u554f\u904e\u7684\u524d\u4e00\u500b\u548c\u5f8c\u4e00\u500b\u7db2\u9801\u3002\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7684\u7279\u6027\u4f7f\u5f97\u9019\u7a2e\u64cd\u4f5c\u8b8a\u5f97\u7c21\u55ae\u3002</li> <li>LRU \u6f14\u7b97\u6cd5\uff1a\u5728\u5feb\u53d6\u6dd8\u6c70\uff08LRU\uff09\u6f14\u7b97\u6cd5\u4e2d\uff0c\u6211\u5011\u9700\u8981\u5feb\u901f\u627e\u5230\u6700\u8fd1\u6700\u5c11\u4f7f\u7528\u7684\u8cc7\u6599\uff0c\u4ee5\u53ca\u652f\u6301\u5feb\u901f\u65b0\u589e\u548c\u522a\u9664\u7bc0\u9ede\u3002\u9019\u6642\u5019\u4f7f\u7528\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5c31\u975e\u5e38\u5408\u9069\u3002</li> </ul> <p>\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u5e38\u7528\u65bc\u9700\u8981\u9031\u671f\u6027\u64cd\u4f5c\u7684\u5834\u666f\uff0c\u6bd4\u5982\u4f5c\u696d\u7cfb\u7d71\u7684\u8cc7\u6e90\u6392\u7a0b\u3002</p> <ul> <li>\u6642\u9593\u7247\u8f2a\u8f49\u6392\u7a0b\u6f14\u7b97\u6cd5\uff1a\u5728\u4f5c\u696d\u7cfb\u7d71\u4e2d\uff0c\u6642\u9593\u7247\u8f2a\u8f49\u6392\u7a0b\u6f14\u7b97\u6cd5\u662f\u4e00\u7a2e\u5e38\u898b\u7684 CPU \u6392\u7a0b\u6f14\u7b97\u6cd5\uff0c\u5b83\u9700\u8981\u5c0d\u4e00\u7d44\u7a0b\u5e8f\u9032\u884c\u8ff4\u5708\u3002\u6bcf\u500b\u7a0b\u5e8f\u88ab\u8ce6\u4e88\u4e00\u500b\u6642\u9593\u7247\uff0c\u7576\u6642\u9593\u7247\u7528\u5b8c\u6642\uff0cCPU \u5c07\u5207\u63db\u5230\u4e0b\u4e00\u500b\u7a0b\u5e8f\u3002\u9019\u7a2e\u8ff4\u5708\u64cd\u4f5c\u53ef\u4ee5\u900f\u904e\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u4f86\u5be6\u73fe\u3002</li> <li>\u8cc7\u6599\u7de9\u885d\u5340\uff1a\u5728\u67d0\u4e9b\u8cc7\u6599\u7de9\u885d\u5340\u7684\u5be6\u73fe\u4e2d\uff0c\u4e5f\u53ef\u80fd\u6703\u4f7f\u7528\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u3002\u6bd4\u5982\u5728\u97f3\u8a0a\u3001\u5f71\u7247\u64ad\u653e\u5668\u4e2d\uff0c\u8cc7\u6599\u6d41\u53ef\u80fd\u6703\u88ab\u5206\u6210\u591a\u500b\u7de9\u885d\u584a\u4e26\u653e\u5165\u4e00\u500b\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\uff0c\u4ee5\u4fbf\u5be6\u73fe\u7121\u7e2b\u64ad\u653e\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/list/","title":"4.3 \u00a0 \u4e32\u5217","text":"<p>\u4e32\u5217\uff08list\uff09\u662f\u4e00\u500b\u62bd\u8c61\u7684\u8cc7\u6599\u7d50\u69cb\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u5143\u7d20\u7684\u6709\u5e8f\u96c6\u5408\uff0c\u652f\u6301\u5143\u7d20\u8a2a\u554f\u3001\u4fee\u6539\u3001\u65b0\u589e\u3001\u522a\u9664\u548c\u8d70\u8a2a\u7b49\u64cd\u4f5c\uff0c\u7121\u9808\u4f7f\u7528\u8005\u8003\u616e\u5bb9\u91cf\u9650\u5236\u7684\u554f\u984c\u3002\u4e32\u5217\u53ef\u4ee5\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u6216\u9663\u5217\u5be6\u73fe\u3002</p> <ul> <li>\u93c8\u7d50\u4e32\u5217\u5929\u7136\u53ef\u4ee5\u770b\u4f5c\u4e00\u500b\u4e32\u5217\uff0c\u5176\u652f\u6301\u5143\u7d20\u589e\u522a\u67e5\u6539\u64cd\u4f5c\uff0c\u4e26\u4e14\u53ef\u4ee5\u9748\u6d3b\u52d5\u614b\u64f4\u5bb9\u3002</li> <li>\u9663\u5217\u4e5f\u652f\u6301\u5143\u7d20\u589e\u522a\u67e5\u6539\uff0c\u4f46\u7531\u65bc\u5176\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u53ea\u80fd\u770b\u4f5c\u4e00\u500b\u5177\u6709\u9577\u5ea6\u9650\u5236\u7684\u4e32\u5217\u3002</li> </ul> <p>\u7576\u4f7f\u7528\u9663\u5217\u5be6\u73fe\u4e32\u5217\u6642\uff0c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u7684\u6027\u8cea\u6703\u5c0e\u81f4\u4e32\u5217\u7684\u5be6\u7528\u6027\u964d\u4f4e\u3002\u9019\u662f\u56e0\u70ba\u6211\u5011\u901a\u5e38\u7121\u6cd5\u4e8b\u5148\u78ba\u5b9a\u9700\u8981\u5132\u5b58\u591a\u5c11\u8cc7\u6599\uff0c\u5f9e\u800c\u96e3\u4ee5\u9078\u64c7\u5408\u9069\u7684\u4e32\u5217\u9577\u5ea6\u3002\u82e5\u9577\u5ea6\u904e\u5c0f\uff0c\u5247\u5f88\u53ef\u80fd\u7121\u6cd5\u6eff\u8db3\u4f7f\u7528\u9700\u6c42\uff1b\u82e5\u9577\u5ea6\u904e\u5927\uff0c\u5247\u6703\u9020\u6210\u8a18\u61b6\u9ad4\u7a7a\u9593\u6d6a\u8cbb\u3002</p> <p>\u70ba\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u52d5\u614b\u9663\u5217\uff08dynamic array\uff09\u4f86\u5be6\u73fe\u4e32\u5217\u3002\u5b83\u7e7c\u627f\u4e86\u9663\u5217\u7684\u5404\u9805\u512a\u9ede\uff0c\u4e26\u4e14\u53ef\u4ee5\u5728\u7a0b\u5f0f\u57f7\u884c\u904e\u7a0b\u4e2d\u9032\u884c\u52d5\u614b\u64f4\u5bb9\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u7684\u6a19\u6e96\u5eab\u63d0\u4f9b\u7684\u4e32\u5217\u662f\u57fa\u65bc\u52d5\u614b\u9663\u5217\u5be6\u73fe\u7684\uff0c\u4f8b\u5982 Python \u4e2d\u7684 <code>list</code> \u3001Java \u4e2d\u7684 <code>ArrayList</code> \u3001C++ \u4e2d\u7684 <code>vector</code> \u548c C# \u4e2d\u7684 <code>List</code> \u7b49\u3002\u5728\u63a5\u4e0b\u4f86\u7684\u8a0e\u8ad6\u4e2d\uff0c\u6211\u5011\u5c07\u628a\u201c\u4e32\u5217\u201d\u548c\u201c\u52d5\u614b\u9663\u5217\u201d\u8996\u70ba\u7b49\u540c\u7684\u6982\u5ff5\u3002</p>"},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1 \u00a0 \u4e32\u5217\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e32\u5217","text":"<p>\u6211\u5011\u901a\u5e38\u4f7f\u7528\u201c\u7121\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u9019\u5169\u7a2e\u521d\u59cb\u5316\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u521d\u59cb\u5316\u4e32\u5217\n# \u7121\u521d\u59cb\u503c\nnums1: list[int] = []\n# \u6709\u521d\u59cb\u503c\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 nums\n// \u7121\u521d\u59cb\u503c\nvector&lt;int&gt; nums1;\n// \u6709\u521d\u59cb\u503c\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u9663\u5217\u7684\u5143\u7d20\u578b\u5225\u9700\u70ba int[] \u7684\u5305\u88dd\u985e\u5225 Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nnums1 := []int{}\n// \u6709\u521d\u59cb\u503c\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nlet nums1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nconst nums1 = [];\n// \u6709\u521d\u59cb\u503c\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nconst nums1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// \u6709\u521d\u59cb\u503c\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u521d\u59cb\u5316\u4e32\u5217 */\n// \u7121\u521d\u59cb\u503c\nvar nums1 = listOf&lt;Int&gt;()\n// \u6709\u521d\u59cb\u503c\nvar numbers = arrayOf(1, 3, 2, 5, 4)\nvar nums = numbers.toMutableList()\n</code></pre> list.rb<pre><code># \u521d\u59cb\u5316\u4e32\u5217\n# \u7121\u521d\u59cb\u503c\nnums1 = []\n# \u6709\u521d\u59cb\u503c\nnums = [1, 3, 2, 5, 4]\n</code></pre> list.zig<pre><code>// \u521d\u59cb\u5316\u4e32\u5217\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&amp;[_]i32{ 1, 3, 2, 5, 4 });\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#2","title":"2. \u00a0 \u8a2a\u554f\u5143\u7d20","text":"<p>\u4e32\u5217\u672c\u8cea\u4e0a\u662f\u9663\u5217\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u5167\u8a2a\u554f\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u8a2a\u554f\u5143\u7d20\nnum: int = nums[1]  # \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0    # \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.cpp<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nint num = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.java<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nint num = nums.get(1);  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums.set(1, 0);  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.cs<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nint num = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list_test.go<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nnum := nums[1]  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0     // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.swift<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nlet num = nums[1] // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.js<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nconst num = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.ts<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nconst num: number = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.dart<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nint num = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.rs<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nlet num: i32 = nums[1];  // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;             // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u8a2a\u554f\u5143\u7d20 */\nval num = nums[1]       // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0             // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.rb<pre><code># \u8a2a\u554f\u5143\u7d20\nnum = nums[1] # \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0 # \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> list.zig<pre><code>// \u8a2a\u554f\u5143\u7d20\nvar num = nums.items[1]; // \u8a2a\u554f\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\n\n// \u66f4\u65b0\u5143\u7d20\nnums.items[1] = 0; // \u5c07\u7d22\u5f15 1 \u8655\u7684\u5143\u7d20\u66f4\u65b0\u70ba 0\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#3","title":"3. \u00a0 \u63d2\u5165\u8207\u522a\u9664\u5143\u7d20","text":"<p>\u76f8\u8f03\u65bc\u9663\u5217\uff0c\u4e32\u5217\u53ef\u4ee5\u81ea\u7531\u5730\u65b0\u589e\u8207\u522a\u9664\u5143\u7d20\u3002\u5728\u4e32\u5217\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u522a\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u8207\u9663\u5217\u76f8\u540c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u6e05\u7a7a\u4e32\u5217\nnums.clear()\n\n# \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n# \u522a\u9664\u5143\u7d20\nnums.pop(3)        # \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.cpp<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.insert(nums.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.erase(nums.begin() + 3);      // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.java<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.remove(3);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.cs<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.Clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.Insert(3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.RemoveAt(3);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list_test.go<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums = nil\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums = append(nums[:3], nums[4:]...) // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.swift<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.removeAll()\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.remove(at: 3) // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.js<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6); // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.splice(3, 1);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.ts<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6); // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.splice(3, 1);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.dart<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.removeAt(3); // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.rs<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.remove(3);    // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u6e05\u7a7a\u4e32\u5217 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n/* \u522a\u9664\u5143\u7d20 */\nnums.remove(3);  // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.rb<pre><code># \u6e05\u7a7a\u4e32\u5217\nnums.clear\n\n# \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\nnums &lt;&lt; 1\nnums &lt;&lt; 3\nnums &lt;&lt; 2\nnums &lt;&lt; 5\nnums &lt;&lt; 4\n\n# \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6) # \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n# \u522a\u9664\u5143\u7d20\nnums.delete_at(3) # \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> list.zig<pre><code>// \u6e05\u7a7a\u4e32\u5217\nnums.clearRetainingCapacity();\n\n// \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\ntry nums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u8655\u63d2\u5165\u6578\u5b57 6\n\n// \u522a\u9664\u5143\u7d20\n_ = nums.orderedRemove(3); // \u522a\u9664\u7d22\u5f15 3 \u8655\u7684\u5143\u7d20\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#4","title":"4. \u00a0 \u8d70\u8a2a\u4e32\u5217","text":"<p>\u8207\u9663\u5217\u4e00\u6a23\uff0c\u4e32\u5217\u53ef\u4ee5\u6839\u64da\u7d22\u5f15\u8d70\u8a2a\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u8d70\u8a2a\u5404\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217 */\nvar count = 0\nfor (i in nums.indices) {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20 */\nfor (num in nums) {\n    count += num\n}\n</code></pre> list.rb<pre><code># \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217\ncount = 0\nfor i in 0...nums.length\n    count += nums[i]\nend\n\n# \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20\ncount = 0\nfor num in nums\n    count += num\nend\n</code></pre> list.zig<pre><code>// \u900f\u904e\u7d22\u5f15\u8d70\u8a2a\u4e32\u5217\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; nums.items.len) : (i += 1) {\n    count += nums[i];\n}\n\n// \u76f4\u63a5\u8d70\u8a2a\u4e32\u5217\u5143\u7d20\ncount = 0;\nfor (nums.items) |num| {\n    count += num;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#5","title":"5. \u00a0 \u62fc\u63a5\u4e32\u5217","text":"<p>\u7d66\u5b9a\u4e00\u500b\u65b0\u4e32\u5217 <code>nums1</code> \uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5176\u62fc\u63a5\u5230\u539f\u4e32\u5217\u7684\u5c3e\u90e8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u62fc\u63a5\u5169\u500b\u4e32\u5217\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.cpp<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.cs<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list_test.go<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.swift<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.js<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.ts<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.dart<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.rs<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u62fc\u63a5\u5169\u500b\u4e32\u5217 */\nval nums1 = intArrayOf(6, 8, 7, 10, 9).toMutableList()\nnums.addAll(nums1)  // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> list.rb<pre><code># \u62fc\u63a5\u5169\u500b\u4e32\u5217\nnums1 = [6, 8, 7, 10, 9]\nnums += nums1\n</code></pre> list.zig<pre><code>// \u62fc\u63a5\u5169\u500b\u4e32\u5217\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&amp;[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // \u5c07\u4e32\u5217 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u5f8c\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#6","title":"6. \u00a0 \u6392\u5e8f\u4e32\u5217","text":"<p>\u5b8c\u6210\u4e32\u5217\u6392\u5e8f\u5f8c\uff0c\u6211\u5011\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u9663\u5217\u985e\u5225\u6f14\u7b97\u6cd5\u984c\u4e2d\u7d93\u5e38\u8003\u67e5\u7684\u201c\u4e8c\u5206\u641c\u5c0b\u201d\u548c\u201c\u96d9\u6307\u6a19\u201d\u6f14\u7b97\u6cd5\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u6392\u5e8f\u4e32\u5217\nnums.sort()  # \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cpp<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nsort(nums.begin(), nums.end());  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.java<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nCollections.sort(nums);  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cs<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.Sort(); // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list_test.go<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nsort.Ints(nums)  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.swift<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort() // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.js<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.ts<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.dart<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort(); // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rs<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort(); // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u52d5\u614b\u9663\u5217\n</code></pre> list.kt<pre><code>/* \u6392\u5e8f\u4e32\u5217 */\nnums.sort() // \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rb<pre><code># \u6392\u5e8f\u4e32\u5217\nnums = nums.sort { |a, b| a &lt;=&gt; b } # \u6392\u5e8f\u5f8c\uff0c\u4e32\u5217\u5143\u7d20\u5f9e\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.zig<pre><code>// \u6392\u5e8f\u4e32\u5217\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2 \u00a0 \u4e32\u5217\u5be6\u73fe","text":"<p>\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u5167\u5efa\u4e86\u4e32\u5217\uff0c\u4f8b\u5982 Java\u3001C++\u3001Python \u7b49\u3002\u5b83\u5011\u7684\u5be6\u73fe\u6bd4\u8f03\u8907\u96dc\uff0c\u5404\u500b\u53c3\u6578\u7684\u8a2d\u5b9a\u4e5f\u975e\u5e38\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u64f4\u5bb9\u500d\u6578\u7b49\u3002\u611f\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u67e5\u95b1\u539f\u59cb\u78bc\u9032\u884c\u5b78\u7fd2\u3002</p> <p>\u70ba\u4e86\u52a0\u6df1\u5c0d\u4e32\u5217\u5de5\u4f5c\u539f\u7406\u7684\u7406\u89e3\uff0c\u6211\u5011\u5617\u8a66\u5be6\u73fe\u4e00\u500b\u7c21\u6613\u7248\u4e32\u5217\uff0c\u5305\u62ec\u4ee5\u4e0b\u4e09\u500b\u91cd\u9ede\u8a2d\u8a08\u3002</p> <ul> <li>\u521d\u59cb\u5bb9\u91cf\uff1a\u9078\u53d6\u4e00\u500b\u5408\u7406\u7684\u9663\u5217\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u5011\u9078\u64c7 10 \u4f5c\u70ba\u521d\u59cb\u5bb9\u91cf\u3002</li> <li>\u6578\u91cf\u8a18\u9304\uff1a\u5ba3\u544a\u4e00\u500b\u8b8a\u6578 <code>size</code> \uff0c\u7528\u65bc\u8a18\u9304\u4e32\u5217\u7576\u524d\u5143\u7d20\u6578\u91cf\uff0c\u4e26\u96a8\u8457\u5143\u7d20\u63d2\u5165\u548c\u522a\u9664\u5be6\u6642\u66f4\u65b0\u3002\u6839\u64da\u6b64\u8b8a\u6578\uff0c\u6211\u5011\u53ef\u4ee5\u5b9a\u4f4d\u4e32\u5217\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65b7\u662f\u5426\u9700\u8981\u64f4\u5bb9\u3002</li> <li>\u64f4\u5bb9\u6a5f\u5236\uff1a\u82e5\u63d2\u5165\u5143\u7d20\u6642\u4e32\u5217\u5bb9\u91cf\u5df2\u6eff\uff0c\u5247\u9700\u8981\u9032\u884c\u64f4\u5bb9\u3002\u5148\u6839\u64da\u64f4\u5bb9\u500d\u6578\u5efa\u7acb\u4e00\u500b\u66f4\u5927\u7684\u9663\u5217\uff0c\u518d\u5c07\u7576\u524d\u9663\u5217\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52d5\u81f3\u65b0\u9663\u5217\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u5011\u898f\u5b9a\u6bcf\u6b21\u5c07\u9663\u5217\u64f4\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_list.py<pre><code>class MyList:\n    \"\"\"\u4e32\u5217\u985e\u5225\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._capacity: int = 10  # \u4e32\u5217\u5bb9\u91cf\n        self._arr: list[int] = [0] * self._capacity  # \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n        self._size: int = 0  # \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n        self._extend_ratio: int = 2  # \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4e32\u5217\u5bb9\u91cf\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"\u8a2a\u554f\u5143\u7d20\"\"\"\n        # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"\u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\"\"\"\n        # \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"\u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        # \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"\u522a\u9664\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        num = self._arr[index]\n        # \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self._size -= 1\n        # \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num\n\n    def extend_capacity(self):\n        \"\"\"\u4e32\u5217\u64f4\u5bb9\"\"\"\n        # \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 _extend_ratio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6709\u6548\u9577\u5ea6\u7684\u4e32\u5217\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n  private:\n    int *arr;             // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    int arrCapacity = 10; // \u4e32\u5217\u5bb9\u91cf\n    int arrSize = 0;      // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    int extendRatio = 2;   // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u6790\u69cb\u65b9\u6cd5 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    int size() {\n        return arrSize;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    void add(int num) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size() == capacity())\n            extendCapacity();\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba Vector \u7528\u65bc\u5217\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private int[] arr; // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private int capacity = 10; // \u4e32\u5217\u5bb9\u91cf\n    private int size = 0; // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private int extendRatio = 2; // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09 */\n    public int size() {\n        return size;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    public int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    public void add(int num) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size == capacity())\n            extendCapacity();\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    public void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        capacity = arr.length;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    public int[] toArray() {\n        int size = size();\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private int[] arr;           // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private int arrCapacity = 10;    // \u4e32\u5217\u5bb9\u91cf\n    private int arrSize = 0;         // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private readonly int extendRatio = 2;  // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    public int Get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    public void Add(int num) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    public void ExtendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba arrCapacity * extendRatio \u7684\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        arrCapacity = arr.Length;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    public int[] ToArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* \u4e32\u5217\u985e\u5225 */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // \u4e32\u5217\u5bb9\u91cf\n        arr:         make([]int, 10), // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n        arrSize:     0,               // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n        extendRatio: 2,               // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n    }\n}\n\n/* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* \u8a2a\u554f\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n    // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    l.arrSize++\n}\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    l.arrSize++\n}\n\n/* \u522a\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    num := l.arr[index]\n    // \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    l.arrSize--\n    // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n    return num\n}\n\n/* \u4e32\u5217\u64f4\u5bb9 */\nfunc (l *myList) extendCapacity() {\n    // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n    l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u9577\u5ea6\u7684\u4e32\u5217 */\nfunc (l *myList) toArray() []int {\n    // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private var arr: [Int] // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private var _capacity: Int // \u4e32\u5217\u5bb9\u91cf\n    private var _size: Int // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private let extendRatio: Int // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    func get(index: Int) -&gt; Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5247\u4e1f\u64f2\u932f\u8aa4\uff0c\u4e0b\u540c\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    func add(num: Int) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        _size += 1\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        _size += 1\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        let num = arr[index]\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        _size -= 1\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    func extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        _capacity = arr.count\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    #arr = new Array(); // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    #capacity = 10; // \u4e32\u5217\u5bb9\u91cf\n    #size = 0; // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    #extendRatio = 2; // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    size() {\n        return this.#size;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    get(index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.#arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.#arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    add(num) {\n        // \u5982\u679c\u9577\u5ea6\u7b49\u65bc\u5bb9\u91cf\uff0c\u5247\u9700\u8981\u64f4\u5bb9\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c07\u65b0\u5143\u7d20\u65b0\u589e\u5230\u4e32\u5217\u5c3e\u90e8\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.#arr[index];\n        // \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        this.#size--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        this.#capacity = this.#arr.length;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    toArray() {\n        let size = this.size();\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private arr: Array&lt;number&gt;; // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private _capacity: number = 10; // \u4e32\u5217\u5bb9\u91cf\n    private _size: number = 0; // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private extendRatio: number = 2; // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    public get(index: number): number {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    public add(num: number): void {\n        // \u5982\u679c\u9577\u5ea6\u7b49\u65bc\u5bb9\u91cf\uff0c\u5247\u9700\u8981\u64f4\u5bb9\n        if (this._size === this._capacity) this.extendCapacity();\n        // \u5c07\u65b0\u5143\u7d20\u65b0\u589e\u5230\u4e32\u5217\u5c3e\u90e8\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.arr[index];\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        this._size--;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    public extendCapacity(): void {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba size \u7684\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        this._capacity = this.arr.length;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    public toArray(): number[] {\n        let size = this.size();\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n  late List&lt;int&gt; _arr; // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n  int _capacity = 10; // \u4e32\u5217\u5bb9\u91cf\n  int _size = 0; // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n  int _extendRatio = 2; // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n  /* \u5efa\u69cb\u5b50 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n  int size() =&gt; _size;\n\n  /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n  int capacity() =&gt; _capacity;\n\n  /* \u8a2a\u554f\u5143\u7d20 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    return _arr[index];\n  }\n\n  /* \u66f4\u65b0\u5143\u7d20 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    _arr[index] = _num;\n  }\n\n  /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n  void add(int _num) {\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    _size++;\n  }\n\n  /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if (_size == _capacity) extendCapacity();\n    // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    _size++;\n  }\n\n  /* \u522a\u9664\u5143\u7d20 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    int _num = _arr[index];\n    // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    _size--;\n    // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n    return _num;\n  }\n\n  /* \u4e32\u5217\u64f4\u5bb9 */\n  void extendCapacity() {\n    // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 _extendRatio \u500d\u7684\u65b0\u9663\u5217\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // \u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n    List.copyRange(_newNums, 0, _arr);\n    // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n    _arr = _newNums;\n    // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n    _capacity = _arr.length;\n  }\n\n  /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* \u4e32\u5217\u985e\u5225 */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    capacity: usize,     // \u4e32\u5217\u5bb9\u91cf\n    size: usize,         // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    extend_ratio: usize, // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* \u5efa\u69cb\u5b50 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = Vec::new();\n        vec.resize(capacity, 0);\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        return self.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    pub fn add(&amp;mut self, num: i32) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self.size += 1;\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self.size += 1;\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        let num = self.arr[index];\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for j in (index..self.size - 1) {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        self.size -= 1;\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    pub fn extend_capacity(&amp;mut self) {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extend_ratio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        self.capacity = new_capacity;\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    pub fn to_array(&amp;mut self) -&gt; Vec&lt;i32&gt; {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* \u4e32\u5217\u985e\u5225 */\ntypedef struct {\n    int *arr;        // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    int capacity;    // \u4e32\u5217\u5bb9\u91cf\n    int size;        // \u4e32\u5217\u5927\u5c0f\n    int extendRatio; // \u4e32\u5217\u6bcf\u6b21\u64f4\u5bb9\u7684\u500d\u6578\n} MyList;\n\n/* \u5efa\u69cb\u5b50 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* \u7372\u53d6\u4e32\u5217\u9577\u5ea6 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u64f4\u5bb9\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u64f4\u5bb9\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* \u522a\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u4f54\u7528\u4e86 remove \u95dc\u9375\u8a5e\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* \u4e32\u5217\u64f4\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n    // \u5148\u5206\u914d\u7a7a\u9593\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // \u8907\u88fd\u820a\u8cc7\u6599\u5230\u65b0\u8cc7\u6599\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // \u91cb\u653e\u820a\u8cc7\u6599\n    free(temp);\n\n    // \u66f4\u65b0\u65b0\u8cc7\u6599\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* \u5c07\u4e32\u5217\u8f49\u63db\u70ba Array \u7528\u65bc\u5217\u5370 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* \u4e32\u5217\u985e\u5225 */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n    private var capacity: Int = 10 // \u4e32\u5217\u5bb9\u91cf\n    private var size: Int = 0 // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n    private var extendRatio: Int = 2 // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09 */\n    fun size(): Int {\n        return size\n    }\n\n    /* \u7372\u53d6\u4e32\u5217\u5bb9\u91cf */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* \u8a2a\u554f\u5143\u7d20 */\n    fun get(index: Int): Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 */\n    fun add(num: Int) {\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size++\n    }\n\n    /* \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n        if (size == capacity())\n            extendCapacity()\n        // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size++\n    }\n\n    /* \u522a\u9664\u5143\u7d20 */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        val num = arr[index]\n        // \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n        size--\n        // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u4e32\u5217\u64f4\u5bb9 */\n    fun extendCapacity() {\n        // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extendRatio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n        arr = arr.copyOf(capacity() * extendRatio)\n        // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n        capacity = arr.size\n    }\n\n    /* \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 */\n    fun toArray(): IntArray {\n        val size = size()\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### \u4e32\u5217\u985e\u5225 ###\nclass MyList\n  attr_reader :size       # \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n  attr_reader :capacity   # \u7372\u53d6\u4e32\u5217\u5bb9\u91cf\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new(capacity)\n  end\n\n  ### \u8a2a\u554f\u5143\u7d20 ###\n  def get(index)\n    # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### \u8a2a\u554f\u5143\u7d20 ###\n  def set(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20 ###\n  def add(num)\n    # \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    @size += 1\n  end\n\n  ### \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20 ###\n  def insert(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n\n    # \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n    extend_capacity if size == capacity\n\n    # \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j]\n    end\n    @arr[index] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    @size += 1\n  end\n\n  ### \u522a\u9664\u5143\u7d20 ###\n  def remove(index)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # \u5c07\u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n    @size -= 1\n\n    # \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n    num\n  end\n\n  ### \u4e32\u5217\u64f4\u5bb9 ###\n  def extend_capacity\n    # \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba\u539f\u9663\u5217 extend_ratio \u500d\u7684\u65b0\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n    @capacity = arr.length\n  end\n\n  ### \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217 ###\n  def to_array\n    sz = size\n    # \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    arr = Array.new(sz)\n    for i in 0...sz\n      arr[i] = get(i)\n    end\n    arr\n  end\nend\n</code></pre> my_list.zig<pre><code>// \u4e32\u5217\u985e\u5225\nfn MyList(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        arr: []T = undefined,                        // \u9663\u5217\uff08\u5132\u5b58\u4e32\u5217\u5143\u7d20\uff09\n        arrCapacity: usize = 10,                     // \u4e32\u5217\u5bb9\u91cf\n        numSize: usize = 0,                           // \u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n        extendRatio: usize = 2,                       // \u6bcf\u6b21\u4e32\u5217\u64f4\u5bb9\u7684\u500d\u6578\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined, // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u4e32\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n            @memset(self.arr, @as(T, 0));\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u4e32\u5217\u9577\u5ea6\uff08\u7576\u524d\u5143\u7d20\u6578\u91cf\uff09\n        pub fn size(self: *Self) usize {\n            return self.numSize;\n        }\n\n        // \u7372\u53d6\u4e32\u5217\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.arrCapacity;\n        }\n\n        // \u8a2a\u554f\u5143\u7d20\n        pub fn get(self: *Self, index: usize) T {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            return self.arr[index];\n        }  \n\n        // \u66f4\u65b0\u5143\u7d20\n        pub fn set(self: *Self, index: usize, num: T) void {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5247\u4e1f\u64f2\u7570\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            self.arr[index] = num;\n        }  \n\n        // \u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\n        pub fn add(self: *Self, num: T) !void {\n            // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            self.arr[self.size()] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n            self.numSize += 1;\n        }  \n\n        // \u5728\u4e2d\u9593\u63d2\u5165\u5143\u7d20\n        pub fn insert(self: *Self, index: usize, num: T) !void {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            // \u5143\u7d20\u6578\u91cf\u8d85\u51fa\u5bb9\u91cf\u6642\uff0c\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            // \u5c07\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n            var j = self.size() - 1;\n            while (j &gt;= index) : (j -= 1) {\n                self.arr[j + 1] = self.arr[j];\n            }\n            self.arr[index] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n            self.numSize += 1;\n        }\n\n        // \u522a\u9664\u5143\u7d20\n        pub fn remove(self: *Self, index: usize) T {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            var num = self.arr[index];\n            // \u5c07\u7d22\u5f15 index \u4e4b\u5f8c\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52d5\u4e00\u4f4d\n            var j = index;\n            while (j &lt; self.size() - 1) : (j += 1) {\n                self.arr[j] = self.arr[j + 1];\n            }\n            // \u66f4\u65b0\u5143\u7d20\u6578\u91cf\n            self.numSize -= 1;\n            // \u8fd4\u56de\u88ab\u522a\u9664\u7684\u5143\u7d20\n            return num;\n        }\n\n        // \u4e32\u5217\u64f4\u5bb9\n        pub fn extendCapacity(self: *Self) !void {\n            // \u65b0\u5efa\u4e00\u500b\u9577\u5ea6\u70ba size * extendRatio \u7684\u9663\u5217\uff0c\u4e26\u5c07\u539f\u9663\u5217\u8907\u88fd\u5230\u65b0\u9663\u5217\n            var newCapacity = self.capacity() * self.extendRatio;\n            var extend = try self.mem_allocator.alloc(T, newCapacity);\n            @memset(extend, @as(T, 0));\n            // \u5c07\u539f\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\u8907\u88fd\u5230\u65b0\u9663\u5217\n            std.mem.copy(T, extend, self.arr);\n            self.arr = extend;\n            // \u66f4\u65b0\u4e32\u5217\u5bb9\u91cf\n            self.arrCapacity = newCapacity;\n        }\n\n        // \u5c07\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217\n        pub fn toArray(self: *Self) ![]T {\n            // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n            var arr = try self.mem_allocator.alloc(T, self.size());\n           @memset(arr, @as(T, 0));\n            for (arr, 0..) |*num, i| {\n                num.* = self.get(i);\n            }\n            return arr;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 \u8a18\u61b6\u9ad4\u8207\u5feb\u53d6 *","text":"<p>\u5728\u672c\u7ae0\u7684\u524d\u5169\u7bc0\u4e2d\uff0c\u6211\u5011\u63a2\u8a0e\u4e86\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u9019\u5169\u7a2e\u57fa\u790e\u4e14\u91cd\u8981\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u5b83\u5011\u5206\u5225\u4ee3\u8868\u4e86\u201c\u9023\u7e8c\u5132\u5b58\u201d\u548c\u201c\u5206\u6563\u5132\u5b58\u201d\u5169\u7a2e\u7269\u7406\u7d50\u69cb\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u7269\u7406\u7d50\u69cb\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u6c7a\u5b9a\u4e86\u7a0b\u5f0f\u5c0d\u8a18\u61b6\u9ad4\u548c\u5feb\u53d6\u7684\u4f7f\u7528\u6548\u7387\uff0c\u9032\u800c\u5f71\u97ff\u6f14\u7b97\u6cd5\u7a0b\u5f0f\u7684\u6574\u9ad4\u6548\u80fd\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","title":"4.4.1 \u00a0 \u8a08\u7b97\u6a5f\u5132\u5b58\u88dd\u7f6e","text":"<p>\u8a08\u7b97\u6a5f\u4e2d\u5305\u62ec\u4e09\u7a2e\u985e\u578b\u7684\u5132\u5b58\u88dd\u7f6e\uff1a\u786c\u789f\uff08hard disk\uff09\u3001\u8a18\u61b6\u9ad4\uff08random-access memory, RAM\uff09\u3001\u5feb\u53d6\uff08cache memory\uff09\u3002\u8868 4-2 \u5c55\u793a\u4e86\u5b83\u5011\u5728\u8a08\u7b97\u6a5f\u7cfb\u7d71\u4e2d\u7684\u4e0d\u540c\u89d2\u8272\u548c\u6548\u80fd\u7279\u9ede\u3002</p> <p> \u8868 4-2 \u00a0 \u8a08\u7b97\u6a5f\u7684\u5132\u5b58\u88dd\u7f6e </p> \u786c\u789f \u8a18\u61b6\u9ad4 \u5feb\u53d6 \u7528\u9014 \u9577\u671f\u5132\u5b58\u8cc7\u6599\uff0c\u5305\u62ec\u4f5c\u696d\u7cfb\u7d71\u3001\u7a0b\u5f0f\u3001\u6a94\u6848\u7b49 \u81e8\u6642\u5132\u5b58\u7576\u524d\u57f7\u884c\u7684\u7a0b\u5f0f\u548c\u6b63\u5728\u8655\u7406\u7684\u8cc7\u6599 \u5132\u5b58\u7d93\u5e38\u8a2a\u554f\u7684\u8cc7\u6599\u548c\u6307\u4ee4\uff0c\u6e1b\u5c11 CPU \u8a2a\u554f\u8a18\u61b6\u9ad4\u7684\u6b21\u6578 \u6613\u5931\u6027 \u65b7\u96fb\u5f8c\u8cc7\u6599\u4e0d\u6703\u4e1f\u5931 \u65b7\u96fb\u5f8c\u8cc7\u6599\u6703\u4e1f\u5931 \u65b7\u96fb\u5f8c\u8cc7\u6599\u6703\u4e1f\u5931 \u5bb9\u91cf \u8f03\u5927\uff0cTB \u7d1a\u5225 \u8f03\u5c0f\uff0cGB \u7d1a\u5225 \u975e\u5e38\u5c0f\uff0cMB \u7d1a\u5225 \u901f\u5ea6 \u8f03\u6162\uff0c\u5e7e\u767e\u5230\u5e7e\u5343 MB/s \u8f03\u5feb\uff0c\u5e7e\u5341 GB/s \u975e\u5e38\u5feb\uff0c\u5e7e\u5341\u5230\u5e7e\u767e GB/s \u50f9\u683c \u8f03\u4fbf\u5b9c\uff0c\u5e7e\u6bdb\u5230\u5e7e\u5143 / GB \u8f03\u8cb4\uff0c\u5e7e\u5341\u5230\u5e7e\u767e\u5143 / GB \u975e\u5e38\u8cb4\uff0c\u96a8 CPU \u6253\u5305\u8a08\u50f9 <p>\u6211\u5011\u53ef\u4ee5\u5c07\u8a08\u7b97\u6a5f\u5132\u5b58\u7cfb\u7d71\u60f3\u8c61\u70ba\u5716 4-9 \u6240\u793a\u7684\u91d1\u5b57\u5854\u7d50\u69cb\u3002\u8d8a\u9760\u8fd1\u91d1\u5b57\u5854\u9802\u7aef\u7684\u5132\u5b58\u88dd\u7f6e\u7684\u901f\u5ea6\u8d8a\u5feb\u3001\u5bb9\u91cf\u8d8a\u5c0f\u3001\u6210\u672c\u8d8a\u9ad8\u3002\u9019\u7a2e\u591a\u5c64\u7d1a\u7684\u8a2d\u8a08\u4e26\u975e\u5076\u7136\uff0c\u800c\u662f\u8a08\u7b97\u6a5f\u79d1\u5b78\u5bb6\u548c\u5de5\u7a0b\u5e2b\u5011\u7d93\u904e\u6df1\u601d\u719f\u616e\u7684\u7d50\u679c\u3002</p> <ul> <li>\u786c\u789f\u96e3\u4ee5\u88ab\u8a18\u61b6\u9ad4\u53d6\u4ee3\u3002\u9996\u5148\uff0c\u8a18\u61b6\u9ad4\u4e2d\u7684\u8cc7\u6599\u5728\u65b7\u96fb\u5f8c\u6703\u4e1f\u5931\uff0c\u56e0\u6b64\u5b83\u4e0d\u9069\u5408\u9577\u671f\u5132\u5b58\u8cc7\u6599\uff1b\u5176\u6b21\uff0c\u8a18\u61b6\u9ad4\u7684\u6210\u672c\u662f\u786c\u789f\u7684\u5e7e\u5341\u500d\uff0c\u9019\u4f7f\u5f97\u5b83\u96e3\u4ee5\u5728\u6d88\u8cbb\u8005\u5e02\u5834\u666e\u53ca\u3002</li> <li>\u5feb\u53d6\u7684\u5927\u5bb9\u91cf\u548c\u9ad8\u901f\u5ea6\u96e3\u4ee5\u517c\u5f97\u3002\u96a8\u8457 L1\u3001L2\u3001L3 \u5feb\u53d6\u7684\u5bb9\u91cf\u9010\u6b65\u589e\u5927\uff0c\u5176\u7269\u7406\u5c3a\u5bf8\u6703\u8b8a\u5927\uff0c\u8207 CPU \u6838\u5fc3\u4e4b\u9593\u7684\u7269\u7406\u8ddd\u96e2\u6703\u8b8a\u9060\uff0c\u5f9e\u800c\u5c0e\u81f4\u8cc7\u6599\u50b3\u8f38\u6642\u9593\u589e\u52a0\uff0c\u5143\u7d20\u8a2a\u554f\u5ef6\u9072\u8b8a\u9ad8\u3002\u5728\u7576\u524d\u6280\u8853\u4e0b\uff0c\u591a\u5c64\u7d1a\u7684\u5feb\u53d6\u7d50\u69cb\u662f\u5bb9\u91cf\u3001\u901f\u5ea6\u548c\u6210\u672c\u4e4b\u9593\u7684\u6700\u4f73\u5e73\u8861\u9ede\u3002</li> </ul> <p></p> <p> \u5716 4-9 \u00a0 \u8a08\u7b97\u6a5f\u5132\u5b58\u7cfb\u7d71 </p> <p>Tip</p> <p>\u8a08\u7b97\u6a5f\u7684\u5132\u5b58\u5c64\u6b21\u7d50\u69cb\u9ad4\u73fe\u4e86\u901f\u5ea6\u3001\u5bb9\u91cf\u548c\u6210\u672c\u4e09\u8005\u4e4b\u9593\u7684\u7cbe\u5999\u5e73\u8861\u3002\u5be6\u969b\u4e0a\uff0c\u9019\u7a2e\u6b0a\u8861\u666e\u904d\u5b58\u5728\u65bc\u6240\u6709\u5de5\u696d\u9818\u57df\uff0c\u5b83\u8981\u6c42\u6211\u5011\u5728\u4e0d\u540c\u7684\u512a\u52e2\u548c\u9650\u5236\u4e4b\u9593\u627e\u5230\u6700\u4f73\u5e73\u8861\u9ede\u3002</p> <p>\u7e3d\u7684\u4f86\u8aaa\uff0c\u786c\u789f\u7528\u65bc\u9577\u671f\u5132\u5b58\u5927\u91cf\u8cc7\u6599\uff0c\u8a18\u61b6\u9ad4\u7528\u65bc\u81e8\u6642\u5132\u5b58\u7a0b\u5f0f\u57f7\u884c\u4e2d\u6b63\u5728\u8655\u7406\u7684\u8cc7\u6599\uff0c\u800c\u5feb\u53d6\u5247\u7528\u65bc\u5132\u5b58\u7d93\u5e38\u8a2a\u554f\u7684\u8cc7\u6599\u548c\u6307\u4ee4\uff0c\u4ee5\u63d0\u9ad8\u7a0b\u5f0f\u57f7\u884c\u6548\u7387\u3002\u4e09\u8005\u5171\u540c\u5354\u4f5c\uff0c\u78ba\u4fdd\u8a08\u7b97\u6a5f\u7cfb\u7d71\u9ad8\u6548\u57f7\u884c\u3002</p> <p>\u5982\u5716 4-10 \u6240\u793a\uff0c\u5728\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u8cc7\u6599\u6703\u5f9e\u786c\u789f\u4e2d\u88ab\u8b80\u53d6\u5230\u8a18\u61b6\u9ad4\u4e2d\uff0c\u4f9b CPU \u8a08\u7b97\u4f7f\u7528\u3002\u5feb\u53d6\u53ef\u4ee5\u770b\u4f5c CPU \u7684\u4e00\u90e8\u5206\uff0c\u5b83\u900f\u904e\u667a\u6167\u5730\u5f9e\u8a18\u61b6\u9ad4\u8f09\u5165\u8cc7\u6599\uff0c\u7d66 CPU \u63d0\u4f9b\u9ad8\u901f\u7684\u8cc7\u6599\u8b80\u53d6\uff0c\u5f9e\u800c\u986f\u8457\u63d0\u5347\u7a0b\u5f0f\u7684\u57f7\u884c\u6548\u7387\uff0c\u6e1b\u5c11\u5c0d\u8f03\u6162\u7684\u8a18\u61b6\u9ad4\u7684\u4f9d\u8cf4\u3002</p> <p></p> <p> \u5716 4-10 \u00a0 \u786c\u789f\u3001\u8a18\u61b6\u9ad4\u548c\u5feb\u53d6\u4e4b\u9593\u7684\u8cc7\u6599\u6d41\u901a </p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","title":"4.4.2 \u00a0 \u8cc7\u6599\u7d50\u69cb\u7684\u8a18\u61b6\u9ad4\u6548\u7387","text":"<p>\u5728\u8a18\u61b6\u9ad4\u7a7a\u9593\u5229\u7528\u65b9\u9762\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u5404\u81ea\u5177\u6709\u512a\u52e2\u548c\u4fb7\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u8a18\u61b6\u9ad4\u662f\u6709\u9650\u7684\uff0c\u4e14\u540c\u4e00\u584a\u8a18\u61b6\u9ad4\u4e0d\u80fd\u88ab\u591a\u500b\u7a0b\u5f0f\u5171\u4eab\uff0c\u56e0\u6b64\u6211\u5011\u5e0c\u671b\u8cc7\u6599\u7d50\u69cb\u80fd\u5920\u5118\u53ef\u80fd\u9ad8\u6548\u5730\u5229\u7528\u7a7a\u9593\u3002\u9663\u5217\u7684\u5143\u7d20\u7dca\u5bc6\u6392\u5217\uff0c\u4e0d\u9700\u8981\u984d\u5916\u7684\u7a7a\u9593\u4f86\u5132\u5b58\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\uff0c\u56e0\u6b64\u7a7a\u9593\u6548\u7387\u66f4\u9ad8\u3002\u7136\u800c\uff0c\u9663\u5217\u9700\u8981\u4e00\u6b21\u6027\u5206\u914d\u8db3\u5920\u7684\u9023\u7e8c\u8a18\u61b6\u9ad4\u7a7a\u9593\uff0c\u9019\u53ef\u80fd\u5c0e\u81f4\u8a18\u61b6\u9ad4\u6d6a\u8cbb\uff0c\u9663\u5217\u64f4\u5bb9\u4e5f\u9700\u8981\u984d\u5916\u7684\u6642\u9593\u548c\u7a7a\u9593\u6210\u672c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u93c8\u7d50\u4e32\u5217\u4ee5\u201c\u7bc0\u9ede\u201d\u70ba\u55ae\u4f4d\u9032\u884c\u52d5\u614b\u8a18\u61b6\u9ad4\u5206\u914d\u548c\u56de\u6536\uff0c\u63d0\u4f9b\u4e86\u66f4\u5927\u7684\u9748\u6d3b\u6027\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5728\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u96a8\u8457\u53cd\u8986\u7533\u8acb\u8207\u91cb\u653e\u8a18\u61b6\u9ad4\uff0c\u7a7a\u9592\u8a18\u61b6\u9ad4\u7684\u788e\u7247\u5316\u7a0b\u5ea6\u6703\u8d8a\u4f86\u8d8a\u9ad8\uff0c\u5f9e\u800c\u5c0e\u81f4\u8a18\u61b6\u9ad4\u7684\u5229\u7528\u6548\u7387\u964d\u4f4e\u3002\u9663\u5217\u7531\u65bc\u5176\u9023\u7e8c\u7684\u5132\u5b58\u65b9\u5f0f\uff0c\u76f8\u5c0d\u4e0d\u5bb9\u6613\u5c0e\u81f4\u8a18\u61b6\u9ad4\u788e\u7247\u5316\u3002\u76f8\u53cd\uff0c\u93c8\u7d50\u4e32\u5217\u7684\u5143\u7d20\u662f\u5206\u6563\u5132\u5b58\u7684\uff0c\u5728\u983b\u7e41\u7684\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u4e2d\uff0c\u66f4\u5bb9\u6613\u5c0e\u81f4\u8a18\u61b6\u9ad4\u788e\u7247\u5316\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","title":"4.4.3 \u00a0 \u8cc7\u6599\u7d50\u69cb\u7684\u5feb\u53d6\u6548\u7387","text":"<p>\u5feb\u53d6\u96d6\u7136\u5728\u7a7a\u9593\u5bb9\u91cf\u4e0a\u9060\u5c0f\u65bc\u8a18\u61b6\u9ad4\uff0c\u4f46\u5b83\u6bd4\u8a18\u61b6\u9ad4\u5feb\u5f97\u591a\uff0c\u5728\u7a0b\u5f0f\u57f7\u884c\u901f\u5ea6\u4e0a\u8d77\u8457\u81f3\u95dc\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7531\u65bc\u5feb\u53d6\u7684\u5bb9\u91cf\u6709\u9650\uff0c\u53ea\u80fd\u5132\u5b58\u4e00\u5c0f\u90e8\u5206\u983b\u7e41\u8a2a\u554f\u7684\u8cc7\u6599\uff0c\u56e0\u6b64\u7576 CPU \u5617\u8a66\u8a2a\u554f\u7684\u8cc7\u6599\u4e0d\u5728\u5feb\u53d6\u4e2d\u6642\uff0c\u5c31\u6703\u767c\u751f\u5feb\u53d6\u672a\u547d\u4e2d\uff08cache miss\uff09\uff0c\u6b64\u6642 CPU \u4e0d\u5f97\u4e0d\u5f9e\u901f\u5ea6\u8f03\u6162\u7684\u8a18\u61b6\u9ad4\u4e2d\u8f09\u5165\u6240\u9700\u8cc7\u6599\u3002</p> <p>\u986f\u7136\uff0c\u201c\u5feb\u53d6\u672a\u547d\u4e2d\u201d\u8d8a\u5c11\uff0cCPU \u8b80\u5beb\u8cc7\u6599\u7684\u6548\u7387\u5c31\u8d8a\u9ad8\uff0c\u7a0b\u5f0f\u6548\u80fd\u4e5f\u5c31\u8d8a\u597d\u3002\u6211\u5011\u5c07 CPU \u5f9e\u5feb\u53d6\u4e2d\u6210\u529f\u7372\u53d6\u8cc7\u6599\u7684\u6bd4\u4f8b\u7a31\u70ba\u5feb\u53d6\u547d\u4e2d\u7387\uff08cache hit rate\uff09\uff0c\u9019\u500b\u6307\u6a19\u901a\u5e38\u7528\u4f86\u8861\u91cf\u5feb\u53d6\u6548\u7387\u3002</p> <p>\u70ba\u4e86\u5118\u53ef\u80fd\u9054\u5230\u66f4\u9ad8\u7684\u6548\u7387\uff0c\u5feb\u53d6\u6703\u63a1\u53d6\u4ee5\u4e0b\u8cc7\u6599\u8f09\u5165\u6a5f\u5236\u3002</p> <ul> <li>\u5feb\u53d6\u884c\uff1a\u5feb\u53d6\u4e0d\u662f\u55ae\u500b\u4f4d\u5143\u7d44\u5730\u5132\u5b58\u8207\u8f09\u5165\u8cc7\u6599\uff0c\u800c\u662f\u4ee5\u5feb\u53d6\u884c\u70ba\u55ae\u4f4d\u3002\u76f8\u6bd4\u65bc\u55ae\u500b\u4f4d\u5143\u7d44\u7684\u50b3\u8f38\uff0c\u5feb\u53d6\u884c\u7684\u50b3\u8f38\u5f62\u5f0f\u66f4\u52a0\u9ad8\u6548\u3002</li> <li>\u9810\u53d6\u6a5f\u5236\uff1a\u8655\u7406\u5668\u6703\u5617\u8a66\u9810\u6e2c\u8cc7\u6599\u8a2a\u554f\u6a21\u5f0f\uff08\u4f8b\u5982\u9806\u5e8f\u8a2a\u554f\u3001\u56fa\u5b9a\u6b65\u9577\u8df3\u8e8d\u8a2a\u554f\u7b49\uff09\uff0c\u4e26\u6839\u64da\u7279\u5b9a\u6a21\u5f0f\u5c07\u8cc7\u6599\u8f09\u5165\u81f3\u5feb\u53d6\u4e4b\u4e2d\uff0c\u5f9e\u800c\u63d0\u5347\u547d\u4e2d\u7387\u3002</li> <li>\u7a7a\u9593\u5340\u57df\u6027\uff1a\u5982\u679c\u4e00\u500b\u6578\u64da\u88ab\u8a2a\u554f\uff0c\u90a3\u9ebc\u5b83\u9644\u8fd1\u7684\u8cc7\u6599\u53ef\u80fd\u8fd1\u671f\u4e5f\u6703\u88ab\u8a2a\u554f\u3002\u56e0\u6b64\uff0c\u5feb\u53d6\u5728\u8f09\u5165\u67d0\u4e00\u8cc7\u6599\u6642\uff0c\u4e5f\u6703\u8f09\u5165\u5176\u9644\u8fd1\u7684\u8cc7\u6599\uff0c\u4ee5\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> <li>\u6642\u9593\u5340\u57df\u6027\uff1a\u5982\u679c\u4e00\u500b\u6578\u64da\u88ab\u8a2a\u554f\uff0c\u90a3\u9ebc\u5b83\u5728\u4e0d\u4e45\u7684\u5c07\u4f86\u5f88\u53ef\u80fd\u518d\u6b21\u88ab\u8a2a\u554f\u3002\u5feb\u53d6\u5229\u7528\u9019\u4e00\u539f\u7406\uff0c\u900f\u904e\u4fdd\u7559\u6700\u8fd1\u8a2a\u554f\u904e\u7684\u8cc7\u6599\u4f86\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u5c0d\u5feb\u53d6\u7684\u5229\u7528\u6548\u7387\u662f\u4e0d\u540c\u7684\uff0c\u4e3b\u8981\u9ad4\u73fe\u5728\u4ee5\u4e0b\u5e7e\u500b\u65b9\u9762\u3002</p> <ul> <li>\u4f54\u7528\u7a7a\u9593\uff1a\u93c8\u7d50\u4e32\u5217\u5143\u7d20\u6bd4\u9663\u5217\u5143\u7d20\u4f54\u7528\u7a7a\u9593\u66f4\u591a\uff0c\u5c0e\u81f4\u5feb\u53d6\u4e2d\u5bb9\u7d0d\u7684\u6709\u6548\u8cc7\u6599\u91cf\u66f4\u5c11\u3002</li> <li>\u5feb\u53d6\u884c\uff1a\u93c8\u7d50\u4e32\u5217\u8cc7\u6599\u5206\u6563\u5728\u8a18\u61b6\u9ad4\u5404\u8655\uff0c\u800c\u5feb\u53d6\u662f\u201c\u6309\u884c\u8f09\u5165\u201d\u7684\uff0c\u56e0\u6b64\u8f09\u5165\u5230\u7121\u6548\u8cc7\u6599\u7684\u6bd4\u4f8b\u66f4\u9ad8\u3002</li> <li>\u9810\u53d6\u6a5f\u5236\uff1a\u9663\u5217\u6bd4\u93c8\u7d50\u4e32\u5217\u7684\u8cc7\u6599\u8a2a\u554f\u6a21\u5f0f\u66f4\u5177\u201c\u53ef\u9810\u6e2c\u6027\u201d\uff0c\u5373\u7cfb\u7d71\u66f4\u5bb9\u6613\u731c\u51fa\u5373\u5c07\u88ab\u8f09\u5165\u7684\u8cc7\u6599\u3002</li> <li>\u7a7a\u9593\u5340\u57df\u6027\uff1a\u9663\u5217\u88ab\u5132\u5b58\u5728\u96c6\u4e2d\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e2d\uff0c\u56e0\u6b64\u88ab\u8f09\u5165\u8cc7\u6599\u9644\u8fd1\u7684\u8cc7\u6599\u66f4\u6709\u53ef\u80fd\u5373\u5c07\u88ab\u8a2a\u554f\u3002</li> </ul> <p>\u7e3d\u9ad4\u800c\u8a00\uff0c\u9663\u5217\u5177\u6709\u66f4\u9ad8\u7684\u5feb\u53d6\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u5728\u64cd\u4f5c\u6548\u7387\u4e0a\u901a\u5e38\u512a\u65bc\u93c8\u7d50\u4e32\u5217\u3002\u9019\u4f7f\u5f97\u5728\u89e3\u6c7a\u6f14\u7b97\u6cd5\u554f\u984c\u6642\uff0c\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u8cc7\u6599\u7d50\u69cb\u5f80\u5f80\u66f4\u53d7\u6b61\u8fce\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9ad8\u5feb\u53d6\u6548\u7387\u4e26\u4e0d\u610f\u5473\u8457\u9663\u5217\u5728\u6240\u6709\u60c5\u6cc1\u4e0b\u90fd\u512a\u65bc\u93c8\u7d50\u4e32\u5217\u3002\u5be6\u969b\u61c9\u7528\u4e2d\u9078\u64c7\u54ea\u7a2e\u8cc7\u6599\u7d50\u69cb\uff0c\u61c9\u6839\u64da\u5177\u9ad4\u9700\u6c42\u4f86\u6c7a\u5b9a\u3002\u4f8b\u5982\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u90fd\u53ef\u4ee5\u5be6\u73fe\u201c\u5806\u758a\u201d\u8cc7\u6599\u7d50\u69cb\uff08\u4e0b\u4e00\u7ae0\u6703\u8a73\u7d30\u4ecb\u7d39\uff09\uff0c\u4f46\u5b83\u5011\u9069\u7528\u65bc\u4e0d\u540c\u5834\u666f\u3002</p> <ul> <li>\u5728\u505a\u6f14\u7b97\u6cd5\u984c\u6642\uff0c\u6211\u5011\u6703\u50be\u5411\u65bc\u9078\u64c7\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\uff0c\u56e0\u70ba\u5b83\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u64cd\u4f5c\u6548\u7387\u548c\u96a8\u6a5f\u8a2a\u554f\u7684\u80fd\u529b\uff0c\u4ee3\u50f9\u50c5\u662f\u9700\u8981\u9810\u5148\u70ba\u9663\u5217\u5206\u914d\u4e00\u5b9a\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> <li>\u5982\u679c\u8cc7\u6599\u91cf\u975e\u5e38\u5927\u3001\u52d5\u614b\u6027\u5f88\u9ad8\u3001\u5806\u758a\u7684\u9810\u671f\u5927\u5c0f\u96e3\u4ee5\u4f30\u8a08\uff0c\u90a3\u9ebc\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a\u66f4\u52a0\u5408\u9069\u3002\u93c8\u7d50\u4e32\u5217\u80fd\u5920\u5c07\u5927\u91cf\u8cc7\u6599\u5206\u6563\u5132\u5b58\u65bc\u8a18\u61b6\u9ad4\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u4e26\u4e14\u907f\u514d\u4e86\u9663\u5217\u64f4\u5bb9\u7522\u751f\u7684\u984d\u5916\u958b\u92b7\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/","title":"4.5 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_array_and_linkedlist/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u662f\u5169\u7a2e\u57fa\u672c\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u5206\u5225\u4ee3\u8868\u8cc7\u6599\u5728\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u4e2d\u7684\u5169\u7a2e\u5132\u5b58\u65b9\u5f0f\uff1a\u9023\u7e8c\u7a7a\u9593\u5132\u5b58\u548c\u5206\u6563\u7a7a\u9593\u5132\u5b58\u3002\u5169\u8005\u7684\u7279\u9ede\u5448\u73fe\u51fa\u4e92\u88dc\u7684\u7279\u6027\u3002</li> <li>\u9663\u5217\u652f\u6301\u96a8\u6a5f\u8a2a\u554f\u3001\u4f54\u7528\u8a18\u61b6\u9ad4\u8f03\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u522a\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u5f8c\u9577\u5ea6\u4e0d\u53ef\u8b8a\u3002</li> <li>\u93c8\u7d50\u4e32\u5217\u900f\u904e\u66f4\u6539\u5f15\u7528\uff08\u6307\u6a19\uff09\u5be6\u73fe\u9ad8\u6548\u7684\u7bc0\u9ede\u63d2\u5165\u8207\u522a\u9664\uff0c\u4e14\u53ef\u4ee5\u9748\u6d3b\u8abf\u6574\u9577\u5ea6\uff1b\u4f46\u7bc0\u9ede\u8a2a\u554f\u6548\u7387\u4f4e\u3001\u4f54\u7528\u8a18\u61b6\u9ad4\u8f03\u591a\u3002\u5e38\u898b\u7684\u93c8\u7d50\u4e32\u5217\u578b\u5225\u5305\u62ec\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u3001\u74b0\u5f62\u93c8\u7d50\u4e32\u5217\u3001\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u3002</li> <li>\u4e32\u5217\u662f\u4e00\u7a2e\u652f\u6301\u589e\u522a\u67e5\u6539\u7684\u5143\u7d20\u6709\u5e8f\u96c6\u5408\uff0c\u901a\u5e38\u57fa\u65bc\u52d5\u614b\u9663\u5217\u5be6\u73fe\u3002\u5b83\u4fdd\u7559\u4e86\u9663\u5217\u7684\u512a\u52e2\uff0c\u540c\u6642\u53ef\u4ee5\u9748\u6d3b\u8abf\u6574\u9577\u5ea6\u3002</li> <li>\u4e32\u5217\u7684\u51fa\u73fe\u5927\u5e45\u63d0\u9ad8\u4e86\u9663\u5217\u7684\u5be6\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5c0e\u81f4\u90e8\u5206\u8a18\u61b6\u9ad4\u7a7a\u9593\u6d6a\u8cbb\u3002</li> <li>\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u8cc7\u6599\u4e3b\u8981\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u4e2d\u3002\u9663\u5217\u53ef\u63d0\u4f9b\u66f4\u9ad8\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u6548\u7387\uff0c\u800c\u93c8\u7d50\u4e32\u5217\u5247\u5728\u8a18\u61b6\u9ad4\u4f7f\u7528\u4e0a\u66f4\u52a0\u9748\u6d3b\u3002</li> <li>\u5feb\u53d6\u900f\u904e\u5feb\u53d6\u884c\u3001\u9810\u53d6\u6a5f\u5236\u4ee5\u53ca\u7a7a\u9593\u5340\u57df\u6027\u548c\u6642\u9593\u5340\u57df\u6027\u7b49\u8cc7\u6599\u8f09\u5165\u6a5f\u5236\uff0c\u70ba CPU \u63d0\u4f9b\u5feb\u901f\u8cc7\u6599\u8a2a\u554f\uff0c\u986f\u8457\u63d0\u5347\u7a0b\u5f0f\u7684\u57f7\u884c\u6548\u7387\u3002</li> <li>\u7531\u65bc\u9663\u5217\u5177\u6709\u66f4\u9ad8\u7684\u5feb\u53d6\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u901a\u5e38\u6bd4\u93c8\u7d50\u4e32\u5217\u66f4\u9ad8\u6548\u3002\u5728\u9078\u64c7\u8cc7\u6599\u7d50\u69cb\u6642\uff0c\u61c9\u6839\u64da\u5177\u9ad4\u9700\u6c42\u548c\u5834\u666f\u505a\u51fa\u6070\u7576\u9078\u64c7\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u9663\u5217\u5132\u5b58\u5728\u5806\u758a\u4e0a\u548c\u5132\u5b58\u5728\u5806\u7a4d\u4e0a\uff0c\u5c0d\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u662f\u5426\u6709\u5f71\u97ff\uff1f</p> <p>\u5132\u5b58\u5728\u5806\u758a\u4e0a\u548c\u5806\u7a4d\u4e0a\u7684\u9663\u5217\u90fd\u88ab\u5132\u5b58\u5728\u9023\u7e8c\u8a18\u61b6\u9ad4\u7a7a\u9593\u5167\uff0c\u8cc7\u6599\u64cd\u4f5c\u6548\u7387\u57fa\u672c\u4e00\u81f4\u3002\u7136\u800c\uff0c\u5806\u758a\u548c\u5806\u7a4d\u5177\u6709\u5404\u81ea\u7684\u7279\u9ede\uff0c\u5f9e\u800c\u5c0e\u81f4\u4ee5\u4e0b\u4e0d\u540c\u9ede\u3002</p> <ol> <li>\u5206\u914d\u548c\u91cb\u653e\u6548\u7387\uff1a\u5806\u758a\u662f\u4e00\u584a\u8f03\u5c0f\u7684\u8a18\u61b6\u9ad4\uff0c\u5206\u914d\u7531\u7de8\u8b6f\u5668\u81ea\u52d5\u5b8c\u6210\uff1b\u800c\u5806\u7a4d\u8a18\u61b6\u9ad4\u76f8\u5c0d\u66f4\u5927\uff0c\u53ef\u4ee5\u5728\u7a0b\u5f0f\u78bc\u4e2d\u52d5\u614b\u5206\u914d\uff0c\u66f4\u5bb9\u6613\u788e\u7247\u5316\u3002\u56e0\u6b64\uff0c\u5806\u7a4d\u4e0a\u7684\u5206\u914d\u548c\u91cb\u653e\u64cd\u4f5c\u901a\u5e38\u6bd4\u5806\u758a\u4e0a\u7684\u6162\u3002</li> <li>\u5927\u5c0f\u9650\u5236\uff1a\u5806\u758a\u8a18\u61b6\u9ad4\u76f8\u5c0d\u8f03\u5c0f\uff0c\u5806\u7a4d\u7684\u5927\u5c0f\u4e00\u822c\u53d7\u9650\u65bc\u53ef\u7528\u8a18\u61b6\u9ad4\u3002\u56e0\u6b64\u5806\u7a4d\u66f4\u52a0\u9069\u5408\u5132\u5b58\u5927\u578b\u9663\u5217\u3002</li> <li>\u9748\u6d3b\u6027\uff1a\u5806\u758a\u4e0a\u7684\u9663\u5217\u7684\u5927\u5c0f\u9700\u8981\u5728\u7de8\u8b6f\u6642\u78ba\u5b9a\uff0c\u800c\u5806\u7a4d\u4e0a\u7684\u9663\u5217\u7684\u5927\u5c0f\u53ef\u4ee5\u5728\u57f7\u884c\u6642\u52d5\u614b\u78ba\u5b9a\u3002</li> </ol> <p>Q\uff1a\u70ba\u4ec0\u9ebc\u9663\u5217\u8981\u6c42\u76f8\u540c\u578b\u5225\u7684\u5143\u7d20\uff0c\u800c\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u537b\u6c92\u6709\u5f37\u8abf\u76f8\u540c\u578b\u5225\u5462\uff1f</p> <p>\u93c8\u7d50\u4e32\u5217\u7531\u7bc0\u9ede\u7d44\u6210\uff0c\u7bc0\u9ede\u4e4b\u9593\u900f\u904e\u5f15\u7528\uff08\u6307\u6a19\uff09\u9023\u7dda\uff0c\u5404\u500b\u7bc0\u9ede\u53ef\u4ee5\u5132\u5b58\u4e0d\u540c\u578b\u5225\u7684\u8cc7\u6599\uff0c\u4f8b\u5982 <code>int</code>\u3001<code>double</code>\u3001<code>string</code>\u3001<code>object</code> \u7b49\u3002</p> <p>\u76f8\u5c0d\u5730\uff0c\u9663\u5217\u5143\u7d20\u5247\u5fc5\u9808\u662f\u76f8\u540c\u578b\u5225\u7684\uff0c\u9019\u6a23\u624d\u80fd\u900f\u904e\u8a08\u7b97\u504f\u79fb\u91cf\u4f86\u7372\u53d6\u5c0d\u61c9\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u9663\u5217\u540c\u6642\u5305\u542b <code>int</code> \u548c <code>long</code> \u5169\u7a2e\u578b\u5225\uff0c\u55ae\u500b\u5143\u7d20\u5206\u5225\u4f54\u7528 4 \u4f4d\u5143\u7d44\u548c 8 \u4f4d\u5143\u7d44 \uff0c\u6b64\u6642\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8a08\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u70ba\u9663\u5217\u4e2d\u5305\u542b\u4e86\u5169\u7a2e\u201c\u5143\u7d20\u9577\u5ea6\u201d\u3002</p> <pre><code># \u5143\u7d20\u8a18\u61b6\u9ad4\u4f4d\u5740 = \u9663\u5217\u8a18\u61b6\u9ad4\u4f4d\u5740\uff08\u9996\u5143\u7d20\u8a18\u61b6\u9ad4\u4f4d\u5740\uff09 + \u5143\u7d20\u9577\u5ea6 * \u5143\u7d20\u7d22\u5f15\n</code></pre> <p>Q\uff1a\u522a\u9664\u7bc0\u9ede <code>P</code> \u5f8c\uff0c\u662f\u5426\u9700\u8981\u628a <code>P.next</code> \u8a2d\u70ba <code>None</code> \u5462\uff1f</p> <p>\u4e0d\u4fee\u6539 <code>P.next</code> \u4e5f\u53ef\u4ee5\u3002\u5f9e\u8a72\u93c8\u7d50\u4e32\u5217\u7684\u89d2\u5ea6\u770b\uff0c\u5f9e\u982d\u7bc0\u9ede\u8d70\u8a2a\u5230\u5c3e\u7bc0\u9ede\u5df2\u7d93\u4e0d\u6703\u9047\u5230 <code>P</code> \u4e86\u3002\u9019\u610f\u5473\u8457\u7bc0\u9ede <code>P</code> \u5df2\u7d93\u5f9e\u93c8\u7d50\u4e32\u5217\u4e2d\u522a\u9664\u4e86\uff0c\u6b64\u6642\u7bc0\u9ede <code>P</code> \u6307\u5411\u54ea\u88e1\u90fd\u4e0d\u6703\u5c0d\u8a72\u93c8\u7d50\u4e32\u5217\u7522\u751f\u5f71\u97ff\u3002</p> <p>\u5f9e\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\uff08\u505a\u984c\uff09\u7684\u89d2\u5ea6\u770b\uff0c\u4e0d\u65b7\u958b\u6c92\u6709\u95dc\u4fc2\uff0c\u53ea\u8981\u4fdd\u8b49\u7a0b\u5f0f\u7684\u908f\u8f2f\u662f\u6b63\u78ba\u7684\u5c31\u884c\u3002\u5f9e\u6a19\u6e96\u5eab\u7684\u89d2\u5ea6\u770b\uff0c\u65b7\u958b\u66f4\u52a0\u5b89\u5168\u3001\u908f\u8f2f\u66f4\u52a0\u6e05\u6670\u3002\u5982\u679c\u4e0d\u65b7\u958b\uff0c\u5047\u8a2d\u88ab\u522a\u9664\u7bc0\u9ede\u672a\u88ab\u6b63\u5e38\u56de\u6536\uff0c\u90a3\u9ebc\u5b83\u6703\u5f71\u97ff\u5f8c\u7e7c\u7bc0\u9ede\u7684\u8a18\u61b6\u9ad4\u56de\u6536\u3002</p> <p>Q\uff1a\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u63d2\u5165\u548c\u522a\u9664\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u522a\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u7684\u6642\u9593\u67e5\u8a62\u5143\u7d20\uff0c\u90a3\u70ba\u4ec0\u9ebc\u6642\u9593\u8907\u96dc\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f</p> <p>\u5982\u679c\u662f\u5148\u67e5\u8a62\u5143\u7d20\u3001\u518d\u522a\u9664\u5143\u7d20\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u78ba\u5be6\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u93c8\u7d50\u4e32\u5217\u7684 \\(O(1)\\) \u589e\u522a\u7684\u512a\u52e2\u53ef\u4ee5\u5728\u5176\u4ed6\u61c9\u7528\u4e0a\u5f97\u5230\u9ad4\u73fe\u3002\u4f8b\u5982\uff0c\u96d9\u5411\u4f47\u5217\u9069\u5408\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\uff0c\u6211\u5011\u7dad\u8b77\u4e00\u500b\u6307\u6a19\u8b8a\u6578\u59cb\u7d42\u6307\u5411\u982d\u7bc0\u9ede\u3001\u5c3e\u7bc0\u9ede\uff0c\u6bcf\u6b21\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002</p> <p>Q\uff1a\u5716\u201c\u93c8\u7d50\u4e32\u5217\u5b9a\u7fa9\u8207\u5132\u5b58\u65b9\u5f0f\u201d\u4e2d\uff0c\u6dfa\u85cd\u8272\u7684\u5132\u5b58\u7bc0\u9ede\u6307\u6a19\u662f\u4f54\u7528\u4e00\u584a\u8a18\u61b6\u9ad4\u4f4d\u5740\u55ce\uff1f\u9084\u662f\u548c\u7bc0\u9ede\u503c\u5404\u4f54\u4e00\u534a\u5462\uff1f</p> <p>\u8a72\u793a\u610f\u5716\u53ea\u662f\u5b9a\u6027\u8868\u793a\uff0c\u5b9a\u91cf\u8868\u793a\u9700\u8981\u6839\u64da\u5177\u9ad4\u60c5\u6cc1\u9032\u884c\u5206\u6790\u3002</p> <ul> <li>\u4e0d\u540c\u578b\u5225\u7684\u7bc0\u9ede\u503c\u4f54\u7528\u7684\u7a7a\u9593\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 <code>int</code>\u3001<code>long</code>\u3001<code>double</code> \u548c\u4f8b\u9805\u7269\u4ef6\u7b49\u3002</li> <li>\u6307\u6a19\u8b8a\u6578\u4f54\u7528\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u5927\u5c0f\u6839\u64da\u6240\u4f7f\u7528\u7684\u4f5c\u696d\u7cfb\u7d71\u53ca\u7de8\u8b6f\u74b0\u5883\u800c\u5b9a\uff0c\u5927\u591a\u70ba 8 \u4f4d\u5143\u7d44\u6216 4 \u4f4d\u5143\u7d44\u3002</li> </ul> <p>Q\uff1a\u5728\u4e32\u5217\u672b\u5c3e\u65b0\u589e\u5143\u7d20\u662f\u5426\u6642\u6642\u523b\u523b\u90fd\u70ba \\(O(1)\\) \uff1f</p> <p>\u5982\u679c\u65b0\u589e\u5143\u7d20\u6642\u8d85\u51fa\u4e32\u5217\u9577\u5ea6\uff0c\u5247\u9700\u8981\u5148\u64f4\u5bb9\u4e32\u5217\u518d\u65b0\u589e\u3002\u7cfb\u7d71\u6703\u7533\u8acb\u4e00\u584a\u65b0\u7684\u8a18\u61b6\u9ad4\uff0c\u4e26\u5c07\u539f\u4e32\u5217\u7684\u6240\u6709\u5143\u7d20\u642c\u904b\u904e\u53bb\uff0c\u9019\u6642\u5019\u6642\u9593\u8907\u96dc\u5ea6\u5c31\u6703\u662f \\(O(n)\\) \u3002</p> <p>Q\uff1a\u201c\u4e32\u5217\u7684\u51fa\u73fe\u6975\u5927\u5730\u63d0\u9ad8\u4e86\u9663\u5217\u7684\u5be6\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5c0e\u81f4\u90e8\u5206\u8a18\u61b6\u9ad4\u7a7a\u9593\u6d6a\u8cbb\u201d\uff0c\u9019\u88e1\u7684\u7a7a\u9593\u6d6a\u8cbb\u662f\u6307\u984d\u5916\u589e\u52a0\u7684\u8b8a\u6578\u5982\u5bb9\u91cf\u3001\u9577\u5ea6\u3001\u64f4\u5bb9\u500d\u6578\u6240\u4f54\u7684\u8a18\u61b6\u9ad4\u55ce\uff1f</p> <p>\u9019\u88e1\u7684\u7a7a\u9593\u6d6a\u8cbb\u4e3b\u8981\u6709\u5169\u65b9\u9762\u542b\u7fa9\uff1a\u4e00\u65b9\u9762\uff0c\u4e32\u5217\u90fd\u6703\u8a2d\u5b9a\u4e00\u500b\u521d\u59cb\u9577\u5ea6\uff0c\u6211\u5011\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u9019\u9ebc\u591a\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u70ba\u4e86\u9632\u6b62\u983b\u7e41\u64f4\u5bb9\uff0c\u64f4\u5bb9\u4e00\u822c\u6703\u4e58\u4ee5\u4e00\u500b\u4fc2\u6578\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u9019\u6a23\u4e00\u4f86\uff0c\u4e5f\u6703\u51fa\u73fe\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u5011\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6eff\u5b83\u5011\u3002</p> <p>Q\uff1a\u5728 Python \u4e2d\u521d\u59cb\u5316 <code>n = [1, 2, 3]</code> \u5f8c\uff0c\u9019 3 \u500b\u5143\u7d20\u7684\u4f4d\u5740\u662f\u76f8\u9023\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316 <code>m = [2, 1, 3]</code> \u6703\u767c\u73fe\u5b83\u5011\u6bcf\u500b\u5143\u7d20\u7684 id \u4e26\u4e0d\u662f\u9023\u7e8c\u7684\uff0c\u800c\u662f\u5206\u5225\u8ddf <code>n</code> \u4e2d\u7684\u76f8\u540c\u3002\u9019\u4e9b\u5143\u7d20\u7684\u4f4d\u5740\u4e0d\u9023\u7e8c\uff0c\u90a3\u9ebc <code>m</code> \u9084\u662f\u9663\u5217\u55ce\uff1f</p> <p>\u5047\u5982\u628a\u4e32\u5217\u5143\u7d20\u63db\u6210\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede <code>n = [n1, n2, n3, n4, n5]</code> \uff0c\u901a\u5e38\u60c5\u6cc1\u4e0b\u9019 5 \u500b\u7bc0\u9ede\u7269\u4ef6\u4e5f\u5206\u6563\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u5404\u8655\u3002\u7136\u800c\uff0c\u7d66\u5b9a\u4e00\u500b\u4e32\u5217\u7d22\u5f15\uff0c\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u5167\u7372\u53d6\u7bc0\u9ede\u8a18\u61b6\u9ad4\u4f4d\u5740\uff0c\u5f9e\u800c\u8a2a\u554f\u5230\u5c0d\u61c9\u7684\u7bc0\u9ede\u3002\u9019\u662f\u56e0\u70ba\u9663\u5217\u4e2d\u5132\u5b58\u7684\u662f\u7bc0\u9ede\u7684\u5f15\u7528\uff0c\u800c\u975e\u7bc0\u9ede\u672c\u8eab\u3002</p> <p>\u8207\u8a31\u591a\u8a9e\u8a00\u4e0d\u540c\uff0cPython \u4e2d\u7684\u6578\u5b57\u4e5f\u88ab\u5305\u88dd\u70ba\u7269\u4ef6\uff0c\u4e32\u5217\u4e2d\u5132\u5b58\u7684\u4e0d\u662f\u6578\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5c0d\u6578\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u5011\u6703\u767c\u73fe\u5169\u500b\u9663\u5217\u4e2d\u7684\u76f8\u540c\u6578\u5b57\u64c1\u6709\u540c\u4e00\u500b id \uff0c\u4e26\u4e14\u9019\u4e9b\u6578\u5b57\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u7121\u9808\u9023\u7e8c\u3002</p> <p>Q\uff1aC++ STL \u88e1\u9762\u7684 <code>std::list</code> \u5df2\u7d93\u5be6\u73fe\u4e86\u96d9\u5411\u93c8\u7d50\u4e32\u5217\uff0c\u4f46\u597d\u50cf\u4e00\u4e9b\u6f14\u7b97\u6cd5\u66f8\u4e0a\u4e0d\u600e\u9ebc\u76f4\u63a5\u4f7f\u7528\u5b83\uff0c\u662f\u4e0d\u662f\u56e0\u70ba\u6709\u4ec0\u9ebc\u4fb7\u9650\u6027\u5462\uff1f</p> <p>\u4e00\u65b9\u9762\uff0c\u6211\u5011\u5f80\u5f80\u66f4\u9752\u775e\u4f7f\u7528\u9663\u5217\u5be6\u73fe\u6f14\u7b97\u6cd5\uff0c\u800c\u53ea\u5728\u5fc5\u8981\u6642\u624d\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\uff0c\u4e3b\u8981\u6709\u5169\u500b\u539f\u56e0\u3002</p> <ul> <li>\u7a7a\u9593\u958b\u92b7\uff1a\u7531\u65bc\u6bcf\u500b\u5143\u7d20\u9700\u8981\u5169\u500b\u984d\u5916\u7684\u6307\u6a19\uff08\u4e00\u500b\u7528\u65bc\u524d\u4e00\u500b\u5143\u7d20\uff0c\u4e00\u500b\u7528\u65bc\u5f8c\u4e00\u500b\u5143\u7d20\uff09\uff0c\u6240\u4ee5 <code>std::list</code> \u901a\u5e38\u6bd4 <code>std::vector</code> \u66f4\u4f54\u7528\u7a7a\u9593\u3002</li> <li>\u5feb\u53d6\u4e0d\u53cb\u597d\uff1a\u7531\u65bc\u8cc7\u6599\u4e0d\u662f\u9023\u7e8c\u5b58\u653e\u7684\uff0c\u56e0\u6b64 <code>std::list</code> \u5c0d\u5feb\u53d6\u7684\u5229\u7528\u7387\u8f03\u4f4e\u3002\u4e00\u822c\u60c5\u6cc1\u4e0b\uff0c<code>std::vector</code> \u7684\u6548\u80fd\u6703\u66f4\u597d\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5fc5\u8981\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\u7684\u60c5\u6cc1\u4e3b\u8981\u662f\u4e8c\u5143\u6a39\u548c\u5716\u3002\u5806\u758a\u548c\u4f47\u5217\u5f80\u5f80\u6703\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u63d0\u4f9b\u7684 <code>stack</code> \u548c <code>queue</code> \uff0c\u800c\u975e\u93c8\u7d50\u4e32\u5217\u3002</p> <p>Q\uff1a\u64cd\u4f5c <code>res = [[0]] * n</code> \u751f\u6210\u4e86\u4e00\u500b\u4e8c\u7dad\u4e32\u5217\uff0c\u5176\u4e2d\u6bcf\u4e00\u500b <code>[0]</code> \u90fd\u662f\u7368\u7acb\u7684\u55ce\uff1f</p> <p>\u4e0d\u662f\u7368\u7acb\u7684\u3002\u6b64\u4e8c\u7dad\u4e32\u5217\u4e2d\uff0c\u6240\u6709\u7684 <code>[0]</code> \u5be6\u969b\u4e0a\u662f\u540c\u4e00\u500b\u7269\u4ef6\u7684\u5f15\u7528\u3002\u5982\u679c\u6211\u5011\u4fee\u6539\u5176\u4e2d\u4e00\u500b\u5143\u7d20\uff0c\u6703\u767c\u73fe\u6240\u6709\u7684\u5c0d\u61c9\u5143\u7d20\u90fd\u6703\u96a8\u4e4b\u6539\u8b8a\u3002</p> <p>\u5982\u679c\u5e0c\u671b\u4e8c\u7dad\u4e32\u5217\u4e2d\u7684\u6bcf\u500b <code>[0]</code> \u90fd\u662f\u7368\u7acb\u7684\uff0c\u53ef\u4ee5\u4f7f\u7528 <code>res = [[0] for _ in range(n)]</code> \u4f86\u5be6\u73fe\u3002\u9019\u7a2e\u65b9\u5f0f\u7684\u539f\u7406\u662f\u521d\u59cb\u5316\u4e86 \\(n\\) \u500b\u7368\u7acb\u7684 <code>[0]</code> \u4e32\u5217\u7269\u4ef6\u3002</p> <p>Q\uff1a\u64cd\u4f5c <code>res = [0] * n</code> \u751f\u6210\u4e86\u4e00\u500b\u4e32\u5217\uff0c\u5176\u4e2d\u6bcf\u4e00\u500b\u6574\u6578 0 \u90fd\u662f\u7368\u7acb\u7684\u55ce\uff1f</p> <p>\u5728\u8a72\u4e32\u5217\u4e2d\uff0c\u6240\u6709\u6574\u6578 0 \u90fd\u662f\u540c\u4e00\u500b\u7269\u4ef6\u7684\u5f15\u7528\u3002\u9019\u662f\u56e0\u70ba Python \u5c0d\u5c0f\u6574\u6578\uff08\u901a\u5e38\u662f -5 \u5230 256\uff09\u63a1\u7528\u4e86\u5feb\u53d6\u6c60\u6a5f\u5236\uff0c\u4ee5\u4fbf\u6700\u5927\u5316\u7269\u4ef6\u8907\u7528\uff0c\u5f9e\u800c\u63d0\u5347\u6548\u80fd\u3002</p> <p>\u96d6\u7136\u5b83\u5011\u6307\u5411\u540c\u4e00\u500b\u7269\u4ef6\uff0c\u4f46\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u7368\u7acb\u4fee\u6539\u4e32\u5217\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\uff0c\u9019\u662f\u56e0\u70ba Python \u7684\u6574\u6578\u662f\u201c\u4e0d\u53ef\u8b8a\u7269\u4ef6\u201d\u3002\u7576\u6211\u5011\u4fee\u6539\u67d0\u500b\u5143\u7d20\u6642\uff0c\u5be6\u969b\u4e0a\u662f\u5207\u63db\u70ba\u53e6\u4e00\u500b\u7269\u4ef6\u7684\u5f15\u7528\uff0c\u800c\u4e0d\u662f\u6539\u8b8a\u539f\u6709\u7269\u4ef6\u672c\u8eab\u3002</p> <p>\u7136\u800c\uff0c\u7576\u4e32\u5217\u5143\u7d20\u662f\u201c\u53ef\u8b8a\u7269\u4ef6\u201d\u6642\uff08\u4f8b\u5982\u4e32\u5217\u3001\u5b57\u5178\u6216\u985e\u5225\u4f8b\u9805\u7b49\uff09\uff0c\u4fee\u6539\u67d0\u500b\u5143\u7d20\u6703\u76f4\u63a5\u6539\u8b8a\u8a72\u7269\u4ef6\u672c\u8eab\uff0c\u6240\u6709\u5f15\u7528\u8a72\u7269\u4ef6\u7684\u5143\u7d20\u90fd\u6703\u7522\u751f\u76f8\u540c\u8b8a\u5316\u3002</p>"},{"location":"chapter_backtracking/","title":"\u7b2c 13 \u7ae0 \u00a0 \u56de\u6eaf","text":"<p>Abstract</p> <p>\u6211\u5011\u5982\u540c\u8ff7\u5bae\u4e2d\u7684\u63a2\u7d22\u8005\uff0c\u5728\u524d\u9032\u7684\u9053\u8def\u4e0a\u53ef\u80fd\u6703\u9047\u5230\u56f0\u96e3\u3002</p> <p>\u56de\u6eaf\u7684\u529b\u91cf\u8b93\u6211\u5011\u80fd\u5920\u91cd\u65b0\u958b\u59cb\uff0c\u4e0d\u65b7\u5617\u8a66\uff0c\u6700\u7d42\u627e\u5230\u901a\u5f80\u5149\u660e\u7684\u51fa\u53e3\u3002</p>"},{"location":"chapter_backtracking/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>13.1 \u00a0 \u56de\u6eaf\u6f14\u7b97\u6cd5</li> <li>13.2 \u00a0 \u5168\u6392\u5217\u554f\u984c</li> <li>13.3 \u00a0 \u5b50\u96c6\u548c\u554f\u984c</li> <li>13.4 \u00a0 N \u7687\u540e\u554f\u984c</li> <li>13.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/","title":"13.1 \u00a0 \u56de\u6eaf\u6f14\u7b97\u6cd5","text":"<p>\u56de\u6eaf\u6f14\u7b97\u6cd5\uff08backtracking algorithm\uff09\u662f\u4e00\u7a2e\u900f\u904e\u7aae\u8209\u4f86\u89e3\u6c7a\u554f\u984c\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5f9e\u4e00\u500b\u521d\u59cb\u72c0\u614b\u51fa\u767c\uff0c\u66b4\u529b\u641c\u5c0b\u6240\u6709\u53ef\u80fd\u7684\u89e3\u6c7a\u65b9\u6848\uff0c\u7576\u9047\u5230\u6b63\u78ba\u7684\u89e3\u5247\u5c07\u5176\u8a18\u9304\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5617\u8a66\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9078\u64c7\u90fd\u7121\u6cd5\u627e\u5230\u89e3\u70ba\u6b62\u3002</p> <p>\u56de\u6eaf\u6f14\u7b97\u6cd5\u901a\u5e38\u63a1\u7528\u201c\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u201d\u4f86\u8d70\u8a2a\u89e3\u7a7a\u9593\u3002\u5728\u201c\u4e8c\u5143\u6a39\u201d\u7ae0\u7bc0\u4e2d\uff0c\u6211\u5011\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u5f8c\u5e8f\u8d70\u8a2a\u90fd\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u3002\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5229\u7528\u524d\u5e8f\u8d70\u8a2a\u69cb\u9020\u4e00\u500b\u56de\u6eaf\u554f\u984c\uff0c\u9010\u6b65\u77ad\u89e3\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p>\u4f8b\u984c\u4e00</p> <p>\u7d66\u5b9a\u4e00\u68f5\u4e8c\u5143\u6a39\uff0c\u641c\u5c0b\u4e26\u8a18\u9304\u6240\u6709\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\uff0c\u8acb\u8fd4\u56de\u7bc0\u9ede\u4e32\u5217\u3002</p> <p>\u5c0d\u65bc\u6b64\u984c\uff0c\u6211\u5011\u524d\u5e8f\u8d70\u8a2a\u9019\u68f5\u6a39\uff0c\u4e26\u5224\u65b7\u7576\u524d\u7bc0\u9ede\u7684\u503c\u662f\u5426\u70ba \\(7\\) \uff0c\u82e5\u662f\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u7684\u503c\u52a0\u5165\u7d50\u679c\u4e32\u5217 <code>res</code> \u4e4b\u4e2d\u3002\u76f8\u95dc\u904e\u7a0b\u5be6\u73fe\u5982\u5716 13-1 \u548c\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # \u8a18\u9304\u89e3\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // \u8a18\u9304\u89e3\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // \u8a18\u9304\u89e3\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // \u8a18\u9304\u89e3\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // \u8a18\u9304\u89e3\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.as_ref());\n        pre_order(res, node.borrow().right.as_ref());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.kt<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    if (root._val == 7) {\n        // \u8a18\u9304\u89e3\n        res!!.add(root)\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n}\n</code></pre> preorder_traversal_i_compact.rb<pre><code>### \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e00 ###\ndef pre_order(root)\n  return unless root\n\n  # \u8a18\u9304\u89e3\n  $res &lt;&lt; root if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\nend\n</code></pre> preorder_traversal_i_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 13-1 \u00a0 \u5728\u524d\u5e8f\u8d70\u8a2a\u4e2d\u641c\u7d22\u7bc0\u9ede </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1311","title":"13.1.1 \u00a0 \u5617\u8a66\u8207\u56de\u9000","text":"<p>\u4e4b\u6240\u4ee5\u7a31\u4e4b\u70ba\u56de\u6eaf\u6f14\u7b97\u6cd5\uff0c\u662f\u56e0\u70ba\u8a72\u6f14\u7b97\u6cd5\u5728\u641c\u5c0b\u89e3\u7a7a\u9593\u6642\u6703\u63a1\u7528\u201c\u5617\u8a66\u201d\u8207\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u7576\u6f14\u7b97\u6cd5\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\u9047\u5230\u67d0\u500b\u72c0\u614b\u7121\u6cd5\u7e7c\u7e8c\u524d\u9032\u6216\u7121\u6cd5\u5f97\u5230\u6eff\u8db3\u689d\u4ef6\u7684\u89e3\u6642\uff0c\u5b83\u6703\u64a4\u92b7\u4e0a\u4e00\u6b65\u7684\u9078\u64c7\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72c0\u614b\uff0c\u4e26\u5617\u8a66\u5176\u4ed6\u53ef\u80fd\u7684\u9078\u64c7\u3002</p> <p>\u5c0d\u65bc\u4f8b\u984c\u4e00\uff0c\u8a2a\u554f\u6bcf\u500b\u7bc0\u9ede\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5617\u8a66\u201d\uff0c\u800c\u8d8a\u904e\u8449\u7bc0\u9ede\u6216\u8fd4\u56de\u7236\u7bc0\u9ede\u7684 <code>return</code> \u5247\u8868\u793a\u201c\u56de\u9000\u201d\u3002</p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u56de\u9000\u4e26\u4e0d\u50c5\u50c5\u5305\u62ec\u51fd\u5f0f\u8fd4\u56de\u3002\u70ba\u89e3\u91cb\u9019\u4e00\u9ede\uff0c\u6211\u5011\u5c0d\u4f8b\u984c\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002</p> <p>\u4f8b\u984c\u4e8c</p> <p>\u5728\u4e8c\u5143\u6a39\u4e2d\u641c\u7d22\u6240\u6709\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\uff0c\u8acb\u8fd4\u56de\u6839\u7bc0\u9ede\u5230\u9019\u4e9b\u7bc0\u9ede\u7684\u8def\u5f91\u3002</p> <p>\u5728\u4f8b\u984c\u4e00\u7a0b\u5f0f\u78bc\u7684\u57fa\u790e\u4e0a\uff0c\u6211\u5011\u9700\u8981\u85c9\u52a9\u4e00\u500b\u4e32\u5217 <code>path</code> \u8a18\u9304\u8a2a\u554f\u904e\u7684\u7bc0\u9ede\u8def\u5f91\u3002\u7576\u8a2a\u554f\u5230\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\u6642\uff0c\u5247\u8907\u88fd <code>path</code> \u4e26\u65b0\u589e\u9032\u7d50\u679c\u4e32\u5217 <code>res</code> \u3002\u8d70\u8a2a\u5b8c\u6210\u5f8c\uff0c<code>res</code> \u4e2d\u5132\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c\"\"\"\n    if root is None:\n        return\n    # \u5617\u8a66\n    path.append(root)\n    if root.val == 7:\n        # \u8a18\u9304\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // \u5617\u8a66\n    path.add(root);\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // \u5617\u8a66\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // \u5617\u8a66\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8a18\u9304\u89e3\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u5617\u8a66\n    path.append(root)\n    if root.val == 7 {\n        // \u8a18\u9304\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push(root);\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push(root);\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // \u5617\u8a66\n  path.add(root);\n  if (root.val == 7) {\n    // \u8a18\u9304\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5617\u8a66\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8a18\u9304\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // \u56de\u9000\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // \u5617\u8a66\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.kt<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    // \u5617\u8a66\n    path!!.add(root)\n    if (root._val == 7) {\n        // \u8a18\u9304\u89e3\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // \u56de\u9000\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_ii_compact.rb<pre><code>### \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e8c ###\ndef pre_order(root)\n  return unless root\n\n  # \u5617\u8a66\n  $path &lt;&lt; root\n\n  # \u8a18\u9304\u89e3\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # \u56de\u9000\n  $path.pop\nend\n</code></pre> preorder_traversal_ii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5728\u6bcf\u6b21\u201c\u5617\u8a66\u201d\u4e2d\uff0c\u6211\u5011\u900f\u904e\u5c07\u7576\u524d\u7bc0\u9ede\u65b0\u589e\u9032 <code>path</code> \u4f86\u8a18\u9304\u8def\u5f91\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u5011\u9700\u8981\u5c07\u8a72\u7bc0\u9ede\u5f9e <code>path</code> \u4e2d\u5f48\u51fa\uff0c\u4ee5\u6062\u5fa9\u672c\u6b21\u5617\u8a66\u4e4b\u524d\u7684\u72c0\u614b\u3002</p> <p>\u89c0\u5bdf\u5716 13-2 \u6240\u793a\u7684\u904e\u7a0b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5617\u8a66\u548c\u56de\u9000\u7406\u89e3\u70ba\u201c\u524d\u9032\u201d\u8207\u201c\u64a4\u92b7\u201d\uff0c\u5169\u500b\u64cd\u4f5c\u4e92\u70ba\u9006\u5411\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 13-2 \u00a0 \u5617\u8a66\u8207\u56de\u9000 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1312","title":"13.1.2 \u00a0 \u526a\u679d","text":"<p>\u8907\u96dc\u7684\u56de\u6eaf\u554f\u984c\u901a\u5e38\u5305\u542b\u4e00\u500b\u6216\u591a\u500b\u7d04\u675f\u689d\u4ef6\uff0c\u7d04\u675f\u689d\u4ef6\u901a\u5e38\u53ef\u7528\u65bc\u201c\u526a\u679d\u201d\u3002</p> <p>\u4f8b\u984c\u4e09</p> <p>\u5728\u4e8c\u5143\u6a39\u4e2d\u641c\u7d22\u6240\u6709\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\uff0c\u8acb\u8fd4\u56de\u6839\u7bc0\u9ede\u5230\u9019\u4e9b\u7bc0\u9ede\u7684\u8def\u5f91\uff0c\u4e26\u8981\u6c42\u8def\u5f91\u4e2d\u4e0d\u5305\u542b\u503c\u70ba \\(3\\) \u7684\u7bc0\u9ede\u3002</p> <p>\u70ba\u4e86\u6eff\u8db3\u4ee5\u4e0a\u7d04\u675f\u689d\u4ef6\uff0c\u6211\u5011\u9700\u8981\u65b0\u589e\u526a\u679d\u64cd\u4f5c\uff1a\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\uff0c\u82e5\u9047\u5230\u503c\u70ba \\(3\\) \u7684\u7bc0\u9ede\uff0c\u5247\u63d0\u524d\u8fd4\u56de\uff0c\u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09\"\"\"\n    # \u526a\u679d\n    if root is None or root.val == 3:\n        return\n    # \u5617\u8a66\n    path.append(root)\n    if root.val == 7:\n        # \u8a18\u9304\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid preOrder(TreeNode root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.add(root);\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid PreOrder(TreeNode? root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8a18\u9304\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // \u526a\u679d\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // \u5617\u8a66\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8a18\u9304\u89e3\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfunc preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // \u5617\u8a66\n    path.append(root)\n    if root.val == 7 {\n        // \u8a18\u9304\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfunction preOrder(root, path, res) {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push(root);\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path.push(root);\n    if (root.val === 7) {\n        // \u8a18\u9304\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // \u5617\u8a66\n  path.add(root);\n  if (root.val == 7) {\n    // \u8a18\u9304\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    // \u526a\u679d\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5617\u8a66\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8a18\u9304\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // \u56de\u9000\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // \u5617\u8a66\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8a18\u9304\u89e3\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.kt<pre><code>/* \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 */\nfun preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    if (root == null || root._val == 3) {\n        return\n    }\n    // \u5617\u8a66\n    path!!.add(root)\n    if (root._val == 7) {\n        // \u8a18\u9304\u89e3\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // \u56de\u9000\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_iii_compact.rb<pre><code>### \u524d\u5e8f\u8d70\u8a2a\uff1a\u4f8b\u984c\u4e09 ###\ndef pre_order(root)\n  # \u526a\u679d\n  return if !root || root.val == 3\n\n  # \u5617\u8a66\n  $path.append(root)\n\n  # \u8a18\u9304\u89e3\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # \u56de\u9000\n  $path.pop\nend\n</code></pre> preorder_traversal_iii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u201c\u526a\u679d\u201d\u662f\u4e00\u500b\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8a5e\u3002\u5982\u5716 13-3 \u6240\u793a\uff0c\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\uff0c\u6211\u5011\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\u7684\u641c\u5c0b\u5206\u652f\uff0c\u907f\u514d\u8a31\u591a\u7121\u610f\u7fa9\u7684\u5617\u8a66\uff0c\u5f9e\u800c\u63d0\u9ad8\u4e86\u641c\u5c0b\u6548\u7387\u3002</p> <p></p> <p> \u5716 13-3 \u00a0 \u6839\u64da\u7d04\u675f\u689d\u4ef6\u526a\u679d </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1313","title":"13.1.3 \u00a0 \u6846\u67b6\u7a0b\u5f0f\u78bc","text":"<p>\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5617\u8a66\u5c07\u56de\u6eaf\u7684\u201c\u5617\u8a66\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u9ad4\u6846\u67b6\u63d0\u7149\u51fa\u4f86\uff0c\u63d0\u5347\u7a0b\u5f0f\u78bc\u7684\u901a\u7528\u6027\u3002</p> <p>\u5728\u4ee5\u4e0b\u6846\u67b6\u7a0b\u5f0f\u78bc\u4e2d\uff0c<code>state</code> \u8868\u793a\u554f\u984c\u7684\u7576\u524d\u72c0\u614b\uff0c<code>choices</code> \u8868\u793a\u7576\u524d\u72c0\u614b\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9078\u64c7\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6\"\"\"\n    # \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if is_solution(state):\n        # \u8a18\u9304\u89e3\n        record_solution(state, res)\n        # \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices:\n        # \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (IsSolution(state)) {\n        // \u8a18\u9304\u89e3\n        RecordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    foreach (Choice choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if isSolution(state) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if isSolution(state: state) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state: state, res: &amp;res)\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n  if (isSolution(state)) {\n    // \u8a18\u9304\u89e3\n    recordSolution(state, res);\n    // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (Choice choice in choices) {\n    // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if is_solution(state) {\n        // \u8a18\u9304\u89e3\n        record_solution(state, res);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res, numRes);\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; numChoices; i++) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, &amp;choices[i])) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 */\nfun backtrack(state: State?, choices: List&lt;Choice?&gt;, res: List&lt;State?&gt;?) {\n    // \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6 ###\ndef backtrack(state, choices, res)\n    # \u5224\u65b7\u662f\u5426\u70ba\u89e3\n    if is_solution?(state)\n        # \u8a18\u9304\u89e3\n        record_solution(state, res)\n        return\n    end\n\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices\n        # \u526a\u679d\uff1a\u5224\u65b7\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid?(state, choice)\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice)\n        end\n    end\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u57fa\u65bc\u6846\u67b6\u7a0b\u5f0f\u78bc\u4f86\u89e3\u6c7a\u4f8b\u984c\u4e09\u3002\u72c0\u614b <code>state</code> \u70ba\u7bc0\u9ede\u8d70\u8a2a\u8def\u5f91\uff0c\u9078\u64c7 <code>choices</code> \u70ba\u7576\u524d\u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u548c\u53f3\u5b50\u7bc0\u9ede\uff0c\u7d50\u679c <code>res</code> \u662f\u8def\u5f91\u4e32\u5217\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"\u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"\u8a18\u9304\u89e3\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"\u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u66f4\u65b0\u72c0\u614b\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u6062\u5fa9\u72c0\u614b\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09\"\"\"\n    # \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if is_solution(state):\n        # \u8a18\u9304\u89e3\n        record_solution(state, res)\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices:\n        # \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice)\n            # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, [choice.left, choice.right], res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (TreeNode *choice : choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (TreeNode choice : choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (IsSolution(state)) {\n        // \u8a18\u9304\u89e3\n        RecordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    foreach (TreeNode choice in choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            MakeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* \u8a18\u9304\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, append([]*TreeNode{}, *state...))\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if isSolution(state) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for _, choice := range *choices {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* \u8a18\u9304\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state: &amp;state, choice: choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfunction backtrack(state, choices, res) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n  if (isSolution(state)) {\n    // \u8a18\u9304\u89e3\n    recordSolution(state, res);\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (TreeNode? choice in choices) {\n    // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      makeChoice(state, choice);\n      // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, [choice!.left, choice.right], res);\n      // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.last().unwrap().borrow().val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nfn record_solution(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    res.push(state.clone());\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return choice.is_some() &amp;&amp; choice.unwrap().borrow().val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfn backtrack(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    choices: &amp;Vec&lt;Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if is_solution(state) {\n        // \u8a18\u9304\u89e3\n        record_solution(state, res);\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for &amp;choice in choices.iter() {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            make_choice(state, choice.unwrap().clone());\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(\n                state,\n                &amp;vec![\n                    choice.unwrap().borrow().left.as_ref(),\n                    choice.unwrap().borrow().right.as_ref(),\n                ],\n                res,\n            );\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undo_choice(state, choice.unwrap().clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* \u8a18\u9304\u89e3 */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nvoid backtrack(TreeNode *choices[2]) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution()) {\n        // \u8a18\u9304\u89e3\n        recordSolution();\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.kt<pre><code>/* \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 */\nfun isSolution(state: MutableList&lt;TreeNode?&gt;): Boolean {\n    return state.isNotEmpty() &amp;&amp; state[state.size - 1]?._val == 7\n}\n\n/* \u8a18\u9304\u89e3 */\nfun recordSolution(state: MutableList&lt;TreeNode?&gt;?, res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;) {\n    res.add(state!!.toMutableList())\n}\n\n/* \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 */\nfun isValid(state: MutableList&lt;TreeNode?&gt;?, choice: TreeNode?): Boolean {\n    return choice != null &amp;&amp; choice._val != 3\n}\n\n/* \u66f4\u65b0\u72c0\u614b */\nfun makeChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.add(choice)\n}\n\n/* \u6062\u5fa9\u72c0\u614b */\nfun undoChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 */\nfun backtrack(\n    state: MutableList&lt;TreeNode?&gt;,\n    choices: MutableList&lt;TreeNode?&gt;,\n    res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;\n) {\n    // \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n    if (isSolution(state)) {\n        // \u8a18\u9304\u89e3\n        recordSolution(state, res)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            makeChoice(state, choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, mutableListOf(choice!!.left, choice.right), res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.rb<pre><code>### \u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5426\u70ba\u89e3 ###\ndef is_solution?(state)\n  !state.empty? &amp;&amp; state.last.val == 7\nend\n\n### \u8a18\u9304\u89e3 ###\ndef record_solution(state, res)\n  res &lt;&lt; state.dup\nend\n\n### \u5224\u65b7\u5728\u7576\u524d\u72c0\u614b\u4e0b\uff0c\u8a72\u9078\u64c7\u662f\u5426\u5408\u6cd5 ###\ndef is_valid?(state, choice)\n  choice &amp;&amp; choice.val != 3\nend\n\n### \u66f4\u65b0\u72c0\u614b ###\ndef make_choice(state, choice)\n  state &lt;&lt; choice\nend\n\n### \u6062\u5fa9\u72c0\u614b ###\ndef undo_choice(state, choice)\n  state.pop\nend\n\n### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u4f8b\u984c\u4e09 ###\ndef backtrack(state, choices, res)\n  # \u6aa2\u67e5\u662f\u5426\u70ba\u89e3\n  record_solution(state, res) if is_solution?(state)\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for choice in choices\n    # \u526a\u679d\uff1a\u6aa2\u67e5\u9078\u64c7\u662f\u5426\u5408\u6cd5\n    if is_valid?(state, choice)\n      # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      make_choice(state, choice)\n      # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, [choice.left, choice.right], res)\n      # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      undo_choice(state, choice)\n    end\n  end\nend\n</code></pre> preorder_traversal_iii_template.zig<pre><code>[class]{}-[func]{isSolution}\n\n[class]{}-[func]{recordSolution}\n\n[class]{}-[func]{isValid}\n\n[class]{}-[func]{makeChoice}\n\n[class]{}-[func]{undoChoice}\n\n[class]{}-[func]{backtrack}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6839\u64da\u984c\u610f\uff0c\u6211\u5011\u5728\u627e\u5230\u503c\u70ba \\(7\\) \u7684\u7bc0\u9ede\u5f8c\u61c9\u8a72\u7e7c\u7e8c\u641c\u5c0b\uff0c\u56e0\u6b64\u9700\u8981\u5c07\u8a18\u9304\u89e3\u4e4b\u5f8c\u7684 <code>return</code> \u8a9e\u53e5\u522a\u9664\u3002\u5716 13-4 \u5c0d\u6bd4\u4e86\u4fdd\u7559\u6216\u522a\u9664 <code>return</code> \u8a9e\u53e5\u7684\u641c\u5c0b\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 13-4 \u00a0 \u4fdd\u7559\u8207\u522a\u9664 return \u7684\u641c\u5c0b\u904e\u7a0b\u5c0d\u6bd4 </p> <p>\u76f8\u6bd4\u57fa\u65bc\u524d\u5e8f\u8d70\u8a2a\u7684\u7a0b\u5f0f\u78bc\u5be6\u73fe\uff0c\u57fa\u65bc\u56de\u6eaf\u6f14\u7b97\u6cd5\u6846\u67b6\u7684\u7a0b\u5f0f\u78bc\u5be6\u73fe\u96d6\u7136\u986f\u5f97\u56c9\u5506\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u56de\u6eaf\u554f\u984c\u53ef\u4ee5\u5728\u8a72\u6846\u67b6\u4e0b\u89e3\u6c7a\u3002\u6211\u5011\u53ea\u9700\u6839\u64da\u5177\u9ad4\u554f\u984c\u4f86\u5b9a\u7fa9 <code>state</code> \u548c <code>choices</code> \uff0c\u4e26\u5be6\u73fe\u6846\u67b6\u4e2d\u7684\u5404\u500b\u65b9\u6cd5\u5373\u53ef\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1314","title":"13.1.4 \u00a0 \u5e38\u7528\u8853\u8a9e","text":"<p>\u70ba\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u6211\u5011\u7e3d\u7d50\u4e00\u4e0b\u56de\u6eaf\u6f14\u7b97\u6cd5\u4e2d\u5e38\u7528\u8853\u8a9e\u7684\u542b\u7fa9\uff0c\u4e26\u5c0d\u7167\u4f8b\u984c\u4e09\u7d66\u51fa\u5c0d\u61c9\u793a\u4f8b\uff0c\u5982\u8868 13-1 \u6240\u793a\u3002</p> <p> \u8868 13-1 \u00a0 \u5e38\u898b\u7684\u56de\u6eaf\u6f14\u7b97\u6cd5\u8853\u8a9e </p> \u540d\u8a5e \u5b9a\u7fa9 \u4f8b\u984c\u4e09 \u89e3\uff08solution\uff09 \u89e3\u662f\u6eff\u8db3\u554f\u984c\u7279\u5b9a\u689d\u4ef6\u7684\u7b54\u6848\uff0c\u53ef\u80fd\u6709\u4e00\u500b\u6216\u591a\u500b \u6839\u7bc0\u9ede\u5230\u7bc0\u9ede \\(7\\) \u7684\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\u7684\u6240\u6709\u8def\u5f91 \u7d04\u675f\u689d\u4ef6\uff08constraint\uff09 \u7d04\u675f\u689d\u4ef6\u662f\u554f\u984c\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u689d\u4ef6\uff0c\u901a\u5e38\u7528\u65bc\u526a\u679d \u8def\u5f91\u4e2d\u4e0d\u5305\u542b\u7bc0\u9ede \\(3\\) \u72c0\u614b\uff08state\uff09 \u72c0\u614b\u8868\u793a\u554f\u984c\u5728\u67d0\u4e00\u6642\u523b\u7684\u60c5\u6cc1\uff0c\u5305\u62ec\u5df2\u7d93\u505a\u51fa\u7684\u9078\u64c7 \u7576\u524d\u5df2\u8a2a\u554f\u7684\u7bc0\u9ede\u8def\u5f91\uff0c\u5373 <code>path</code> \u7bc0\u9ede\u4e32\u5217 \u5617\u8a66\uff08attempt\uff09 \u5617\u8a66\u662f\u6839\u64da\u53ef\u7528\u9078\u64c7\u4f86\u63a2\u7d22\u89e3\u7a7a\u9593\u7684\u904e\u7a0b\uff0c\u5305\u62ec\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\uff0c\u6aa2\u67e5\u662f\u5426\u70ba\u89e3 \u905e\u8ff4\u8a2a\u554f\u5de6\uff08\u53f3\uff09\u5b50\u7bc0\u9ede\uff0c\u5c07\u7bc0\u9ede\u65b0\u589e\u9032 <code>path</code> \uff0c\u5224\u65b7\u7bc0\u9ede\u7684\u503c\u662f\u5426\u70ba \\(7\\) \u56de\u9000\uff08backtracking\uff09 \u56de\u9000\u6307\u9047\u5230\u4e0d\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\u7684\u72c0\u614b\u6642\uff0c\u64a4\u92b7\u524d\u9762\u505a\u51fa\u7684\u9078\u64c7\uff0c\u56de\u5230\u4e0a\u4e00\u500b\u72c0\u614b \u7576\u8d8a\u904e\u8449\u7bc0\u9ede\u3001\u7d50\u675f\u7bc0\u9ede\u8a2a\u554f\u3001\u9047\u5230\u503c\u70ba \\(3\\) \u7684\u7bc0\u9ede\u6642\u7d42\u6b62\u641c\u5c0b\uff0c\u51fd\u5f0f\u8fd4\u56de \u526a\u679d\uff08pruning\uff09 \u526a\u679d\u662f\u6839\u64da\u554f\u984c\u7279\u6027\u548c\u7d04\u675f\u689d\u4ef6\u907f\u514d\u7121\u610f\u7fa9\u7684\u641c\u5c0b\u8def\u5f91\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u5c0b\u6548\u7387 \u7576\u9047\u5230\u503c\u70ba \\(3\\) \u7684\u7bc0\u9ede\u6642\uff0c\u5247\u4e0d\u518d\u7e7c\u7e8c\u641c\u5c0b <p>Tip</p> <p>\u554f\u984c\u3001\u89e3\u3001\u72c0\u614b\u7b49\u6982\u5ff5\u662f\u901a\u7528\u7684\uff0c\u5728\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52d5\u614b\u898f\u5283\u3001\u8caa\u5a6a\u7b49\u6f14\u7b97\u6cd5\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1315","title":"13.1.5 \u00a0 \u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u56de\u6eaf\u6f14\u7b97\u6cd5\u672c\u8cea\u4e0a\u662f\u4e00\u7a2e\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u6f14\u7b97\u6cd5\uff0c\u5b83\u5617\u8a66\u6240\u6709\u53ef\u80fd\u7684\u89e3\u6c7a\u65b9\u6848\u76f4\u5230\u627e\u5230\u6eff\u8db3\u689d\u4ef6\u7684\u89e3\u3002\u9019\u7a2e\u65b9\u6cd5\u7684\u512a\u9ede\u5728\u65bc\u80fd\u5920\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u6c7a\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002</p> <p>\u7136\u800c\uff0c\u5728\u8655\u7406\u5927\u898f\u6a21\u6216\u8005\u8907\u96dc\u554f\u984c\u6642\uff0c\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u57f7\u884c\u6548\u7387\u53ef\u80fd\u96e3\u4ee5\u63a5\u53d7\u3002</p> <ul> <li>\u6642\u9593\uff1a\u56de\u6eaf\u6f14\u7b97\u6cd5\u901a\u5e38\u9700\u8981\u8d70\u8a2a\u72c0\u614b\u7a7a\u9593\u7684\u6240\u6709\u53ef\u80fd\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u9054\u5230\u6307\u6578\u968e\u6216\u968e\u4e58\u968e\u3002</li> <li>\u7a7a\u9593\uff1a\u5728\u905e\u8ff4\u547c\u53eb\u4e2d\u9700\u8981\u5132\u5b58\u7576\u524d\u7684\u72c0\u614b\uff08\u4f8b\u5982\u8def\u5f91\u3001\u7528\u65bc\u526a\u679d\u7684\u8f14\u52a9\u8b8a\u6578\u7b49\uff09\uff0c\u7576\u6df1\u5ea6\u5f88\u5927\u6642\uff0c\u7a7a\u9593\u9700\u6c42\u53ef\u80fd\u6703\u8b8a\u5f97\u5f88\u5927\u3002</li> </ul> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u6f14\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u5c0b\u554f\u984c\u548c\u7d04\u675f\u6eff\u8db3\u554f\u984c\u7684\u6700\u4f73\u89e3\u6c7a\u65b9\u6848\u3002\u5c0d\u65bc\u9019\u4e9b\u554f\u984c\uff0c\u7531\u65bc\u7121\u6cd5\u9810\u6e2c\u54ea\u4e9b\u9078\u64c7\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u5011\u5fc5\u9808\u5c0d\u6240\u6709\u53ef\u80fd\u7684\u9078\u64c7\u9032\u884c\u8d70\u8a2a\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u95dc\u9375\u662f\u5982\u4f55\u6700\u4f73\u5316\u6548\u7387\uff0c\u5e38\u898b\u7684\u6548\u7387\u6700\u4f73\u5316\u65b9\u6cd5\u6709\u5169\u7a2e\u3002</p> <ul> <li>\u526a\u679d\uff1a\u907f\u514d\u641c\u5c0b\u90a3\u4e9b\u80af\u5b9a\u4e0d\u6703\u7522\u751f\u89e3\u7684\u8def\u5f91\uff0c\u5f9e\u800c\u7bc0\u7701\u6642\u9593\u548c\u7a7a\u9593\u3002</li> <li>\u555f\u767c\u5f0f\u641c\u5c0b\uff1a\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8a08\u503c\uff0c\u5f9e\u800c\u512a\u5148\u641c\u5c0b\u6700\u6709\u53ef\u80fd\u7522\u751f\u6709\u6548\u89e3\u7684\u8def\u5f91\u3002</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/#1316","title":"13.1.6 \u00a0 \u56de\u6eaf\u5178\u578b\u4f8b\u984c","text":"<p>\u56de\u6eaf\u6f14\u7b97\u6cd5\u53ef\u7528\u65bc\u89e3\u6c7a\u8a31\u591a\u641c\u5c0b\u554f\u984c\u3001\u7d04\u675f\u6eff\u8db3\u554f\u984c\u548c\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\u3002</p> <p>\u641c\u5c0b\u554f\u984c\uff1a\u9019\u985e\u554f\u984c\u7684\u76ee\u6a19\u662f\u627e\u5230\u6eff\u8db3\u7279\u5b9a\u689d\u4ef6\u7684\u89e3\u6c7a\u65b9\u6848\u3002</p> <ul> <li>\u5168\u6392\u5217\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u500b\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7d44\u5408\u3002</li> <li>\u5b50\u96c6\u548c\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u500b\u96c6\u5408\u548c\u4e00\u500b\u76ee\u6a19\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u70ba\u76ee\u6a19\u548c\u7684\u5b50\u96c6\u3002</li> <li>\u6cb3\u5167\u5854\u554f\u984c\uff1a\u7d66\u5b9a\u4e09\u6839\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5713\u76e4\uff0c\u8981\u6c42\u5c07\u6240\u6709\u5713\u76e4\u5f9e\u4e00\u6839\u67f1\u5b50\u79fb\u52d5\u5230\u53e6\u4e00\u6839\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52d5\u4e00\u500b\u5713\u76e4\uff0c\u4e14\u4e0d\u80fd\u5c07\u5927\u5713\u76e4\u653e\u5728\u5c0f\u5713\u76e4\u4e0a\u3002</li> </ul> <p>\u7d04\u675f\u6eff\u8db3\u554f\u984c\uff1a\u9019\u985e\u554f\u984c\u7684\u76ee\u6a19\u662f\u627e\u5230\u6eff\u8db3\u6240\u6709\u7d04\u675f\u689d\u4ef6\u7684\u89e3\u3002</p> <ul> <li>\\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76e4\u4e0a\u653e\u7f6e \\(n\\) \u500b\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u5011\u4e92\u4e0d\u653b\u64ca\u3002</li> <li>\u6578\u7368\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7db2\u683c\u4e2d\u586b\u5165\u6578\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u500b \\(3 \\times 3\\) \u5b50\u7db2\u683c\u4e2d\u7684\u6578\u5b57\u4e0d\u91cd\u8907\u3002</li> <li>\u5716\u8457\u8272\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u500b\u7121\u5411\u5716\uff0c\u7528\u6700\u5c11\u7684\u984f\u8272\u7d66\u5716\u7684\u6bcf\u500b\u9802\u9ede\u8457\u8272\uff0c\u4f7f\u5f97\u76f8\u9130\u9802\u9ede\u984f\u8272\u4e0d\u540c\u3002</li> </ul> <p>\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\uff1a\u9019\u985e\u554f\u984c\u7684\u76ee\u6a19\u662f\u5728\u4e00\u500b\u7d44\u5408\u7a7a\u9593\u4e2d\u627e\u5230\u6eff\u8db3\u67d0\u4e9b\u689d\u4ef6\u7684\u6700\u512a\u89e3\u3002</p> <ul> <li>0-1 \u80cc\u5305\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u7d44\u7269\u54c1\u548c\u4e00\u500b\u80cc\u5305\uff0c\u6bcf\u500b\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u50f9\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5167\uff0c\u9078\u64c7\u7269\u54c1\u4f7f\u5f97\u7e3d\u50f9\u503c\u6700\u5927\u3002</li> <li>\u65c5\u884c\u5546\u554f\u984c\uff1a\u5728\u4e00\u500b\u5716\u4e2d\uff0c\u5f9e\u4e00\u500b\u9ede\u51fa\u767c\uff0c\u8a2a\u554f\u6240\u6709\u5176\u4ed6\u9ede\u6070\u597d\u4e00\u6b21\u5f8c\u8fd4\u56de\u8d77\u9ede\uff0c\u6c42\u6700\u77ed\u8def\u5f91\u3002</li> <li>\u6700\u5927\u5718\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u500b\u7121\u5411\u5716\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u5716\uff0c\u5373\u5b50\u5716\u4e2d\u7684\u4efb\u610f\u5169\u500b\u9802\u9ede\u4e4b\u9593\u90fd\u6709\u908a\u76f8\u9023\u3002</li> </ul> <p>\u8acb\u6ce8\u610f\uff0c\u5c0d\u65bc\u8a31\u591a\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\uff0c\u56de\u6eaf\u4e0d\u662f\u6700\u512a\u89e3\u6c7a\u65b9\u6848\u3002</p> <ul> <li>0-1 \u80cc\u5305\u554f\u984c\u901a\u5e38\u4f7f\u7528\u52d5\u614b\u898f\u5283\u89e3\u6c7a\uff0c\u4ee5\u9054\u5230\u66f4\u9ad8\u7684\u6642\u9593\u6548\u7387\u3002</li> <li>\u65c5\u884c\u5546\u662f\u4e00\u500b\u8457\u540d\u7684 NP-Hard \u554f\u984c\uff0c\u5e38\u7528\u89e3\u6cd5\u6709\u907a\u50b3\u6f14\u7b97\u6cd5\u548c\u87fb\u7fa4\u6f14\u7b97\u6cd5\u7b49\u3002</li> <li>\u6700\u5927\u5718\u554f\u984c\u662f\u5716\u8ad6\u4e2d\u7684\u4e00\u500b\u7d93\u5178\u554f\u984c\uff0c\u53ef\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7b49\u555f\u767c\u5f0f\u6f14\u7b97\u6cd5\u4f86\u89e3\u6c7a\u3002</li> </ul>"},{"location":"chapter_backtracking/n_queens_problem/","title":"13.4 \u00a0 n \u7687\u540e\u554f\u984c","text":"<p>Question</p> <p>\u6839\u64da\u570b\u969b\u8c61\u68cb\u7684\u898f\u5247\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u64ca\u8207\u540c\u8655\u4e00\u884c\u3001\u4e00\u5217\u6216\u4e00\u689d\u659c\u7dda\u4e0a\u7684\u68cb\u5b50\u3002\u7d66\u5b9a \\(n\\) \u500b\u7687\u540e\u548c\u4e00\u500b \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5c0b\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u9593\u7121\u6cd5\u76f8\u4e92\u653b\u64ca\u7684\u64fa\u653e\u65b9\u6848\u3002</p> <p>\u5982\u5716 13-15 \u6240\u793a\uff0c\u7576 \\(n = 4\\) \u6642\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u5169\u500b\u89e3\u3002\u5f9e\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76e4\u5171\u6709 \\(n^2\\) \u500b\u683c\u5b50\uff0c\u7d66\u51fa\u4e86\u6240\u6709\u7684\u9078\u64c7 <code>choices</code> \u3002\u5728\u9010\u500b\u653e\u7f6e\u7687\u540e\u7684\u904e\u7a0b\u4e2d\uff0c\u68cb\u76e4\u72c0\u614b\u5728\u4e0d\u65b7\u5730\u8b8a\u5316\uff0c\u6bcf\u500b\u6642\u523b\u7684\u68cb\u76e4\u5c31\u662f\u72c0\u614b <code>state</code> \u3002</p> <p></p> <p> \u5716 13-15 \u00a0 4 \u7687\u540e\u554f\u984c\u7684\u89e3 </p> <p>\u5716 13-16 \u5c55\u793a\u4e86\u672c\u984c\u7684\u4e09\u500b\u7d04\u675f\u689d\u4ef6\uff1a\u591a\u500b\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u3001\u540c\u4e00\u689d\u5c0d\u89d2\u7dda\u4e0a\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5c0d\u89d2\u7dda\u5206\u70ba\u4e3b\u5c0d\u89d2\u7dda <code>\\</code> \u548c\u6b21\u5c0d\u89d2\u7dda <code>/</code> \u5169\u7a2e\u3002</p> <p></p> <p> \u5716 13-16 \u00a0 n \u7687\u540e\u554f\u984c\u7684\u7d04\u675f\u689d\u4ef6 </p>"},{"location":"chapter_backtracking/n_queens_problem/#1","title":"1. \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565","text":"<p>\u7687\u540e\u7684\u6578\u91cf\u548c\u68cb\u76e4\u7684\u884c\u6578\u90fd\u70ba \\(n\\) \uff0c\u56e0\u6b64\u6211\u5011\u5bb9\u6613\u5f97\u5230\u4e00\u500b\u63a8\u8ad6\uff1a\u68cb\u76e4\u6bcf\u884c\u90fd\u5141\u8a31\u4e14\u53ea\u5141\u8a31\u653e\u7f6e\u4e00\u500b\u7687\u540e\u3002</p> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u6211\u5011\u53ef\u4ee5\u63a1\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u5f9e\u7b2c\u4e00\u884c\u958b\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u500b\u7687\u540e\uff0c\u76f4\u81f3\u6700\u5f8c\u4e00\u884c\u7d50\u675f\u3002</p> <p>\u5716 13-17 \u6240\u793a\u70ba 4 \u7687\u540e\u554f\u984c\u7684\u9010\u884c\u653e\u7f6e\u904e\u7a0b\u3002\u53d7\u756b\u5e45\u9650\u5236\uff0c\u5716 13-17 \u50c5\u5c55\u958b\u4e86\u7b2c\u4e00\u884c\u7684\u5176\u4e2d\u4e00\u500b\u641c\u5c0b\u5206\u652f\uff0c\u4e26\u4e14\u5c07\u4e0d\u6eff\u8db3\u5217\u7d04\u675f\u548c\u5c0d\u89d2\u7dda\u7d04\u675f\u7684\u65b9\u6848\u90fd\u9032\u884c\u4e86\u526a\u679d\u3002</p> <p></p> <p> \u5716 13-17 \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565 </p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u9010\u884c\u653e\u7f6e\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73fe\u591a\u500b\u7687\u540e\u7684\u6240\u6709\u641c\u5c0b\u5206\u652f\u3002</p>"},{"location":"chapter_backtracking/n_queens_problem/#2","title":"2. \u00a0 \u5217\u8207\u5c0d\u89d2\u7dda\u526a\u679d","text":"<p>\u70ba\u4e86\u6eff\u8db3\u5217\u7d04\u675f\uff0c\u6211\u5011\u53ef\u4ee5\u5229\u7528\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u5e03\u6797\u578b\u9663\u5217 <code>cols</code> \u8a18\u9304\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u6c7a\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u5011\u900f\u904e <code>cols</code> \u5c07\u5df2\u6709\u7687\u540e\u7684\u5217\u9032\u884c\u526a\u679d\uff0c\u4e26\u5728\u56de\u6eaf\u4e2d\u52d5\u614b\u66f4\u65b0 <code>cols</code> \u7684\u72c0\u614b\u3002</p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u77e9\u9663\u7684\u8d77\u9ede\u4f4d\u65bc\u5de6\u4e0a\u89d2\uff0c\u5176\u4e2d\u884c\u7d22\u5f15\u5f9e\u4e0a\u5230\u4e0b\u589e\u52a0\uff0c\u5217\u7d22\u5f15\u5f9e\u5de6\u5230\u53f3\u589e\u52a0\u3002</p> <p>\u90a3\u9ebc\uff0c\u5982\u4f55\u8655\u7406\u5c0d\u89d2\u7dda\u7d04\u675f\u5462\uff1f\u8a2d\u68cb\u76e4\u4e2d\u67d0\u500b\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u70ba \\((row, col)\\) \uff0c\u9078\u5b9a\u77e9\u9663\u4e2d\u7684\u67d0\u689d\u4e3b\u5c0d\u89d2\u7dda\uff0c\u6211\u5011\u767c\u73fe\u8a72\u5c0d\u89d2\u7dda\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u6e1b\u5217\u7d22\u5f15\u90fd\u76f8\u7b49\uff0c\u5373\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u6240\u6709\u683c\u5b50\u7684 \\(row - col\\) \u70ba\u6046\u5b9a\u503c\u3002</p> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u5982\u679c\u5169\u500b\u683c\u5b50\u6eff\u8db3 \\(row_1 - col_1 = row_2 - col_2\\) \uff0c\u5247\u5b83\u5011\u4e00\u5b9a\u8655\u5728\u540c\u4e00\u689d\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u3002\u5229\u7528\u8a72\u898f\u5f8b\uff0c\u6211\u5011\u53ef\u4ee5\u85c9\u52a9\u5716 13-18 \u6240\u793a\u7684\u9663\u5217 <code>diags1</code> \u8a18\u9304\u6bcf\u689d\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002</p> <p>\u540c\u7406\uff0c\u6b21\u5c0d\u89d2\u7dda\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 \\(row + col\\) \u662f\u6046\u5b9a\u503c\u3002\u6211\u5011\u540c\u6a23\u4e5f\u53ef\u4ee5\u85c9\u52a9\u9663\u5217 <code>diags2</code> \u4f86\u8655\u7406\u6b21\u5c0d\u89d2\u7dda\u7d04\u675f\u3002</p> <p></p> <p> \u5716 13-18 \u00a0 \u8655\u7406\u5217\u7d04\u675f\u548c\u5c0d\u89d2\u7dda\u7d04\u675f </p>"},{"location":"chapter_backtracking/n_queens_problem/#3","title":"3. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u8acb\u6ce8\u610f\uff0c\\(n\\) \u7dad\u65b9\u9663\u4e2d \\(row - col\\) \u7684\u7bc4\u570d\u662f \\([-n + 1, n - 1]\\) \uff0c\\(row + col\\) \u7684\u7bc4\u570d\u662f \\([0, 2n - 2]\\) \uff0c\u6240\u4ee5\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\u7684\u6578\u91cf\u90fd\u70ba \\(2n - 1\\) \uff0c\u5373\u9663\u5217 <code>diags1</code> \u548c <code>diags2</code> \u7684\u9577\u5ea6\u90fd\u70ba \\(2n - 1\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e\"\"\"\n    # \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # \u8d70\u8a2a\u6240\u6709\u5217\n    for col in range(n):\n        # \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"\u6c42\u89e3 n \u7687\u540e\"\"\"\n    # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    diags1 = [False] * (2 * n - 1)  # \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    diags2 = [False] * (2 * n - 1)  # \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags1 = new boolean[2 * n - 1]; // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags2 = new boolean[2 * n - 1]; // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    bool[] diags1 = new bool[2 * n - 1]; // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool[] diags2 = new bool[2 * n - 1]; // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for col := 0; col &lt; n; col++ {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunc nQueens(n int) [][][]string {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if row == n {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for col in 0 ..&lt; n {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunction nQueens(n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u5217\n  for (int col = 0; col &lt; n; col++) {\n    // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // \u653e\u7f6e\u4e0b\u4e00\u884c\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfn backtrack(\n    row: usize,\n    n: usize,\n    state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool],\n    diags1: &amp;mut [bool],\n    diags2: &amp;mut [bool],\n) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if row == n {\n        res.push(state.clone());\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for col in 0..n {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = vec![vec![\"#\".to_string(); n]; n];\n    let mut cols = vec![false; n]; // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    let mut diags1 = vec![false; 2 * n - 1]; // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut diags2 = vec![false; 2 * n - 1]; // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(\n        0,\n        n,\n        &amp;mut state,\n        &amp;mut res,\n        &amp;mut cols,\n        &amp;mut diags1,\n        &amp;mut diags2,\n    );\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e */\nfun backtrack(\n    row: Int,\n    n: Int,\n    state: MutableList&lt;MutableList&lt;String&gt;&gt;,\n    res: MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;,\n    cols: BooleanArray,\n    diags1: BooleanArray,\n    diags2: BooleanArray\n) {\n    // \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n    if (row == n) {\n        val copyState = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n        for (sRow in state) {\n            copyState.add(sRow.toMutableList())\n        }\n        res.add(copyState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u5217\n    for (col in 0..&lt;n) {\n        // \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n        val diag1 = row - col + n - 1\n        val diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n            state[row][col] = \"Q\"\n            diags2[diag2] = true\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            diags2[diag2] = false\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfun nQueens(n: Int): MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    val state = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val row = mutableListOf&lt;String&gt;()\n        for (j in 0..&lt;n) {\n            row.add(\"#\")\n        }\n        state.add(row)\n    }\n    val cols = BooleanArray(n) // \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n    val diags1 = BooleanArray(2 * n - 1) // \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    val diags2 = BooleanArray(2 * n - 1) // \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n    val res = mutableListOf&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;()\n\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n}\n</code></pre> n_queens.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1an \u7687\u540e ###\ndef backtrack(row, n, state, res, cols, diags1, diags2)\n  # \u7576\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u6642\uff0c\u8a18\u9304\u89e3\n  if row == n\n    res &lt;&lt; state.map { |row| row.dup }\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u5217\n  for col in 0...n\n    # \u8a08\u7b97\u8a72\u683c\u5b50\u5c0d\u61c9\u7684\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\n    diag1 = row - col + n - 1\n    diag2 = row + col\n    # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8a72\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5c0d\u89d2\u7dda\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u5b58\u5728\u7687\u540e\n    if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]\n      # \u5617\u8a66\uff1a\u5c07\u7687\u540e\u653e\u7f6e\u5728\u8a72\u683c\u5b50\n      state[row][col] = \"Q\"\n      cols[col] = diags1[diag1] = diags2[diag2] = true\n      # \u653e\u7f6e\u4e0b\u4e00\u884c\n      backtrack(row + 1, n, state, res, cols, diags1, diags2)\n      # \u56de\u9000\uff1a\u5c07\u8a72\u683c\u5b50\u6062\u5fa9\u70ba\u7a7a\u4f4d\n      state[row][col] = \"#\"\n      cols[col] = diags1[diag1] = diags2[diag2] = false\n    end\n  end\nend\n\n### \u6c42\u89e3 n \u7687\u540e ###\ndef n_queens(n)\n  # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76e4\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n  state = Array.new(n) { Array.new(n, \"#\") }\n  cols = Array.new(n, false) # \u8a18\u9304\u5217\u662f\u5426\u6709\u7687\u540e\n  diags1 = Array.new(2 * n - 1, false) # \u8a18\u9304\u4e3b\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n  diags2 = Array.new(2 * n - 1, false) # \u8a18\u9304\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u6709\u7687\u540e\n  res = []\n  backtrack(0, n, state, res, cols, diags1, diags2)\n\n  res\nend\n</code></pre> n_queens.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{nQueens}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u616e\u5217\u7d04\u675f\uff0c\u5247\u5f9e\u7b2c\u4e00\u884c\u5230\u6700\u5f8c\u4e00\u884c\u5206\u5225\u6709 \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u500b\u9078\u64c7\uff0c\u4f7f\u7528 \\(O(n!)\\) \u6642\u9593\u3002\u7576\u8a18\u9304\u89e3\u6642\uff0c\u9700\u8981\u8907\u88fd\u77e9\u9663 <code>state</code> \u4e26\u65b0\u589e\u9032 <code>res</code> \uff0c\u8907\u88fd\u64cd\u4f5c\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002\u56e0\u6b64\uff0c\u7e3d\u9ad4\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n! \\cdot n^2)\\) \u3002\u5be6\u969b\u4e0a\uff0c\u6839\u64da\u5c0d\u89d2\u7dda\u7d04\u675f\u7684\u526a\u679d\u4e5f\u80fd\u5920\u5927\u5e45\u7e2e\u5c0f\u641c\u5c0b\u7a7a\u9593\uff0c\u56e0\u800c\u641c\u5c0b\u6548\u7387\u5f80\u5f80\u512a\u65bc\u4ee5\u4e0a\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <p>\u9663\u5217 <code>state</code> \u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u9593\uff0c\u9663\u5217 <code>cols</code>\u3001<code>diags1</code> \u548c <code>diags2</code> \u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u9593\u3002\u6700\u5927\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002\u56e0\u6b64\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/","title":"13.2 \u00a0 \u5168\u6392\u5217\u554f\u984c","text":"<p>\u5168\u6392\u5217\u554f\u984c\u662f\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u4e00\u500b\u5178\u578b\u61c9\u7528\u3002\u5b83\u7684\u5b9a\u7fa9\u662f\u5728\u7d66\u5b9a\u4e00\u500b\u96c6\u5408\uff08\u5982\u4e00\u500b\u9663\u5217\u6216\u5b57\u4e32\uff09\u7684\u60c5\u6cc1\u4e0b\uff0c\u627e\u51fa\u5176\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u8868 13-2 \u5217\u8209\u4e86\u5e7e\u500b\u793a\u4f8b\u8cc7\u6599\uff0c\u5305\u62ec\u8f38\u5165\u9663\u5217\u548c\u5c0d\u61c9\u7684\u6240\u6709\u6392\u5217\u3002</p> <p> \u8868 13-2 \u00a0 \u5168\u6392\u5217\u793a\u4f8b </p> \u8f38\u5165\u9663\u5217 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1321","title":"13.2.1 \u00a0 \u7121\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u8f38\u5165\u4e00\u500b\u6574\u6578\u9663\u5217\uff0c\u5176\u4e2d\u4e0d\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u5f9e\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u904e\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9078\u64c7\u7684\u7d50\u679c\u3002\u5047\u8a2d\u8f38\u5165\u9663\u5217\u70ba \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u5011\u5148\u9078\u64c7 \\(1\\) \uff0c\u518d\u9078\u64c7 \\(3\\) \uff0c\u6700\u5f8c\u9078\u64c7 \\(2\\) \uff0c\u5247\u7372\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u92b7\u4e00\u500b\u9078\u64c7\uff0c\u4e4b\u5f8c\u7e7c\u7e8c\u5617\u8a66\u5176\u4ed6\u9078\u64c7\u3002</p> <p>\u5f9e\u56de\u6eaf\u7a0b\u5f0f\u78bc\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9078\u96c6\u5408 <code>choices</code> \u662f\u8f38\u5165\u9663\u5217\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72c0\u614b <code>state</code> \u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9078\u64c7\u7684\u5143\u7d20\u3002\u8acb\u6ce8\u610f\uff0c\u6bcf\u500b\u5143\u7d20\u53ea\u5141\u8a31\u88ab\u9078\u64c7\u4e00\u6b21\uff0c\u56e0\u6b64 <code>state</code> \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u61c9\u8a72\u662f\u552f\u4e00\u7684\u3002</p> <p>\u5982\u5716 13-5 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u641c\u5c0b\u904e\u7a0b\u5c55\u958b\u6210\u4e00\u68f5\u905e\u8ff4\u6a39\uff0c\u6a39\u4e2d\u7684\u6bcf\u500b\u7bc0\u9ede\u4ee3\u8868\u7576\u524d\u72c0\u614b <code>state</code> \u3002\u5f9e\u6839\u7bc0\u9ede\u958b\u59cb\uff0c\u7d93\u904e\u4e09\u8f2a\u9078\u64c7\u5f8c\u5230\u9054\u8449\u7bc0\u9ede\uff0c\u6bcf\u500b\u8449\u7bc0\u9ede\u90fd\u5c0d\u61c9\u4e00\u500b\u6392\u5217\u3002</p> <p></p> <p> \u5716 13-5 \u00a0 \u5168\u6392\u5217\u7684\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_backtracking/permutations_problem/#1","title":"1. \u00a0 \u91cd\u8907\u9078\u64c7\u526a\u679d","text":"<p>\u70ba\u4e86\u5be6\u73fe\u6bcf\u500b\u5143\u7d20\u53ea\u88ab\u9078\u64c7\u4e00\u6b21\uff0c\u6211\u5011\u8003\u616e\u5f15\u5165\u4e00\u500b\u5e03\u6797\u578b\u9663\u5217 <code>selected</code> \uff0c\u5176\u4e2d <code>selected[i]</code> \u8868\u793a <code>choices[i]</code> \u662f\u5426\u5df2\u88ab\u9078\u64c7\uff0c\u4e26\u57fa\u65bc\u5b83\u5be6\u73fe\u4ee5\u4e0b\u526a\u679d\u64cd\u4f5c\u3002</p> <ul> <li>\u5728\u505a\u51fa\u9078\u64c7 <code>choice[i]</code> \u5f8c\uff0c\u6211\u5011\u5c31\u5c07 <code>selected[i]</code> \u8ce6\u503c\u70ba \\(\\text{True}\\) \uff0c\u4ee3\u8868\u5b83\u5df2\u88ab\u9078\u64c7\u3002</li> <li>\u8d70\u8a2a\u9078\u64c7\u4e32\u5217 <code>choices</code> \u6642\uff0c\u8df3\u904e\u6240\u6709\u5df2\u88ab\u9078\u64c7\u7684\u7bc0\u9ede\uff0c\u5373\u526a\u679d\u3002</li> </ul> <p>\u5982\u5716 13-6 \u6240\u793a\uff0c\u5047\u8a2d\u6211\u5011\u7b2c\u4e00\u8f2a\u9078\u64c7 1 \uff0c\u7b2c\u4e8c\u8f2a\u9078\u64c7 3 \uff0c\u7b2c\u4e09\u8f2a\u9078\u64c7 2 \uff0c\u5247\u9700\u8981\u5728\u7b2c\u4e8c\u8f2a\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f2a\u526a\u6389\u5143\u7d20 1 \u548c\u5143\u7d20 3 \u7684\u5206\u652f\u3002</p> <p></p> <p> \u5716 13-6 \u00a0 \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b </p> <p>\u89c0\u5bdf\u5716 13-6 \u767c\u73fe\uff0c\u8a72\u526a\u679d\u64cd\u4f5c\u5c07\u641c\u5c0b\u7a7a\u9593\u5927\u5c0f\u5f9e \\(O(n^n)\\) \u6e1b\u5c0f\u81f3 \\(O(n!)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u60f3\u6e05\u695a\u4ee5\u4e0a\u8cc7\u8a0a\u4e4b\u5f8c\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u7a0b\u5f0f\u78bc\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u70ba\u4e86\u7e2e\u77ed\u6574\u9ad4\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u4e0d\u55ae\u7368\u5be6\u73fe\u6846\u67b6\u7a0b\u5f0f\u78bc\u4e2d\u7684\u5404\u500b\u51fd\u5f0f\uff0c\u800c\u662f\u5c07\u5b83\u5011\u5c55\u958b\u5728 <code>backtrack()</code> \u51fd\u5f0f\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n    # \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if not selected[i]:\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = True\n            state.append(choice)\n            # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.push_back(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.add(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.Add(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if !(*selected)[i] {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrackI(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if !selected[i] {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true\n            state.append(choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n    if (!selected[i]) {\n      // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      selected[i] = true;\n      state.add(choice);\n      // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if !selected[i] {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n        if (!selected[i]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            selected[i] = true\n            state.add(choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfun permutationsI(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_i.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I ###\ndef backtrack(state, choices, selected, res)\n  # \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  choices.each_with_index do |choice, i|\n    # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20\n    unless selected[i]\n      # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      selected[i] = true\n      state &lt;&lt; choice\n      # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, choices, selected, res)\n      # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### \u5168\u6392\u5217 I ###\ndef permutations_i(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> permutations_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsI}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_backtracking/permutations_problem/#1322","title":"13.2.2 \u00a0 \u8003\u616e\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u8f38\u5165\u4e00\u500b\u6574\u6578\u9663\u5217\uff0c\u9663\u5217\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u8907\u7684\u6392\u5217\u3002</p> <p>\u5047\u8a2d\u8f38\u5165\u9663\u5217\u70ba \\([1, 1, 2]\\) \u3002\u70ba\u4e86\u65b9\u4fbf\u5340\u5206\u5169\u500b\u91cd\u8907\u5143\u7d20 \\(1\\) \uff0c\u6211\u5011\u5c07\u7b2c\u4e8c\u500b \\(1\\) \u8a18\u70ba \\(\\hat{1}\\) \u3002</p> <p>\u5982\u5716 13-7 \u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u662f\u91cd\u8907\u7684\u3002</p> <p></p> <p> \u5716 13-7 \u00a0 \u91cd\u8907\u6392\u5217 </p> <p>\u90a3\u9ebc\u5982\u4f55\u53bb\u9664\u91cd\u8907\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u8003\u616e\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u96c6\u5408\uff0c\u76f4\u63a5\u5c0d\u6392\u5217\u7d50\u679c\u9032\u884c\u53bb\u91cd\u3002\u7136\u800c\u9019\u6a23\u505a\u4e0d\u5920\u512a\u96c5\uff0c\u56e0\u70ba\u751f\u6210\u91cd\u8907\u6392\u5217\u7684\u641c\u5c0b\u5206\u652f\u6c92\u6709\u5fc5\u8981\uff0c\u61c9\u7576\u63d0\u524d\u8b58\u5225\u4e26\u526a\u679d\uff0c\u9019\u6a23\u53ef\u4ee5\u9032\u4e00\u6b65\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u89c0\u5bdf\u5716 13-8 \uff0c\u5728\u7b2c\u4e00\u8f2a\u4e2d\uff0c\u9078\u64c7 \\(1\\) \u6216\u9078\u64c7 \\(\\hat{1}\\) \u662f\u7b49\u50f9\u7684\uff0c\u5728\u9019\u5169\u500b\u9078\u64c7\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u8907\u7684\u3002\u56e0\u6b64\u61c9\u8a72\u628a \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f2a\u9078\u64c7 \\(2\\) \u4e4b\u5f8c\uff0c\u7b2c\u4e8c\u8f2a\u9078\u64c7\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u6703\u7522\u751f\u91cd\u8907\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u61c9\u5c07\u7b2c\u4e8c\u8f2a\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u6211\u5011\u7684\u76ee\u6a19\u662f\u5728\u67d0\u4e00\u8f2a\u9078\u64c7\u4e2d\uff0c\u4fdd\u8b49\u591a\u500b\u76f8\u7b49\u7684\u5143\u7d20\u50c5\u88ab\u9078\u64c7\u4e00\u6b21\u3002</p> <p></p> <p> \u5716 13-8 \u00a0 \u91cd\u8907\u6392\u5217\u526a\u679d </p>"},{"location":"chapter_backtracking/permutations_problem/#2_1","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5728\u4e0a\u4e00\u984c\u7684\u7a0b\u5f0f\u78bc\u7684\u57fa\u790e\u4e0a\uff0c\u6211\u5011\u8003\u616e\u5728\u6bcf\u4e00\u8f2a\u9078\u64c7\u4e2d\u958b\u555f\u4e00\u500b\u96dc\u6e4a\u96c6\u5408 <code>duplicated</code> \uff0c\u7528\u65bc\u8a18\u9304\u8a72\u8f2a\u4e2d\u5df2\u7d93\u5617\u8a66\u904e\u7684\u5143\u7d20\uff0c\u4e26\u5c07\u91cd\u8907\u5143\u7d20\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n    # \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if not selected[i] and choice not in duplicated:\n            # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice)  # \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = True\n            state.append(choice)\n            # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.emplace(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push_back(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.add(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.Add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.Add(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrackII(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if !selected[i], !duplicated.contains(choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.insert(choice) // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.append(choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      duplicated.add(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n      selected[i] = true;\n      state.add(choice);\n      // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.insert(choice); // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated[choice] = true; // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    val duplicated = HashSet&lt;Int&gt;()\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n            duplicated.add(choice) // \u8a18\u9304\u9078\u64c7\u904e\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.add(choice)\n            // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n            backtrack(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfun permutationsII(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_ii.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II ###\ndef backtrack(state, choices, selected, res)\n  # \u7576\u72c0\u614b\u9577\u5ea6\u7b49\u65bc\u5143\u7d20\u6578\u91cf\u6642\uff0c\u8a18\u9304\u89e3\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  duplicated = Set.new\n  choices.each_with_index do |choice, i|\n    # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u5143\u7d20 \u4e14 \u4e0d\u5141\u8a31\u91cd\u8907\u9078\u64c7\u76f8\u7b49\u5143\u7d20\n    if !selected[i] &amp;&amp; !duplicated.include?(choice)\n      # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n      duplicated.add(choice)\n      selected[i] = true\n      state &lt;&lt; choice\n      # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n      backtrack(state, choices, selected, res)\n      # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### \u5168\u6392\u5217 II ###\ndef permutations_ii(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> permutations_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsII}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5047\u8a2d\u5143\u7d20\u5169\u5169\u4e4b\u9593\u4e92\u4e0d\u76f8\u540c\uff0c\u5247 \\(n\\) \u500b\u5143\u7d20\u5171\u6709 \\(n!\\)  \u7a2e\u6392\u5217\uff08\u968e\u4e58\uff09\uff1b\u5728\u8a18\u9304\u7d50\u679c\u6642\uff0c\u9700\u8981\u8907\u88fd\u9577\u5ea6\u70ba \\(n\\) \u7684\u4e32\u5217\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n!n)\\) \u3002</p> <p>\u6700\u5927\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002<code>selected</code> \u4f7f\u7528 \\(O(n)\\) \u7a7a\u9593\u3002\u540c\u4e00\u6642\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u500b <code>duplicated</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u9593\u3002\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#3","title":"3. \u00a0 \u5169\u7a2e\u526a\u679d\u5c0d\u6bd4","text":"<p>\u8acb\u6ce8\u610f\uff0c\u96d6\u7136 <code>selected</code> \u548c <code>duplicated</code> \u90fd\u7528\u65bc\u526a\u679d\uff0c\u4f46\u5169\u8005\u7684\u76ee\u6a19\u4e0d\u540c\u3002</p> <ul> <li>\u91cd\u8907\u9078\u64c7\u526a\u679d\uff1a\u6574\u500b\u641c\u5c0b\u904e\u7a0b\u4e2d\u53ea\u6709\u4e00\u500b <code>selected</code> \u3002\u5b83\u8a18\u9304\u7684\u662f\u7576\u524d\u72c0\u614b\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u5176\u4f5c\u7528\u662f\u907f\u514d\u67d0\u500b\u5143\u7d20\u5728 <code>state</code> \u4e2d\u91cd\u8907\u51fa\u73fe\u3002</li> <li>\u76f8\u7b49\u5143\u7d20\u526a\u679d\uff1a\u6bcf\u8f2a\u9078\u64c7\uff08\u6bcf\u500b\u547c\u53eb\u7684 <code>backtrack</code> \u51fd\u5f0f\uff09\u90fd\u5305\u542b\u4e00\u500b <code>duplicated</code> \u3002\u5b83\u8a18\u9304\u7684\u662f\u5728\u672c\u8f2a\u8d70\u8a2a\uff08<code>for</code> \u8ff4\u5708\uff09\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9078\u64c7\u904e\uff0c\u5176\u4f5c\u7528\u662f\u4fdd\u8b49\u76f8\u7b49\u5143\u7d20\u53ea\u88ab\u9078\u64c7\u4e00\u6b21\u3002</li> </ul> <p>\u5716 13-9 \u5c55\u793a\u4e86\u5169\u500b\u526a\u679d\u689d\u4ef6\u7684\u751f\u6548\u7bc4\u570d\u3002\u6ce8\u610f\uff0c\u6a39\u4e2d\u7684\u6bcf\u500b\u7bc0\u9ede\u4ee3\u8868\u4e00\u500b\u9078\u64c7\uff0c\u5f9e\u6839\u7bc0\u9ede\u5230\u8449\u7bc0\u9ede\u7684\u8def\u5f91\u4e0a\u7684\u5404\u500b\u7bc0\u9ede\u69cb\u6210\u4e00\u500b\u6392\u5217\u3002</p> <p></p> <p> \u5716 13-9 \u00a0 \u5169\u7a2e\u526a\u679d\u689d\u4ef6\u7684\u4f5c\u7528\u7bc4\u570d </p>"},{"location":"chapter_backtracking/subset_sum_problem/","title":"13.3 \u00a0 \u5b50\u96c6\u548c\u554f\u984c","text":""},{"location":"chapter_backtracking/subset_sum_problem/#1331","title":"13.3.1 \u00a0 \u7121\u91cd\u8907\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u6b63\u6574\u6578\u9663\u5217 <code>nums</code> \u548c\u4e00\u500b\u76ee\u6a19\u6b63\u6574\u6578 <code>target</code> \uff0c\u8acb\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7d44\u5408\uff0c\u4f7f\u5f97\u7d44\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u65bc <code>target</code> \u3002\u7d66\u5b9a\u9663\u5217\u7121\u91cd\u8907\u5143\u7d20\uff0c\u6bcf\u500b\u5143\u7d20\u53ef\u4ee5\u88ab\u9078\u53d6\u591a\u6b21\u3002\u8acb\u4ee5\u4e32\u5217\u5f62\u5f0f\u8fd4\u56de\u9019\u4e9b\u7d44\u5408\uff0c\u4e32\u5217\u4e2d\u4e0d\u61c9\u5305\u542b\u91cd\u8907\u7d44\u5408\u3002</p> <p>\u4f8b\u5982\uff0c\u8f38\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6a19\u6574\u6578 \\(9\\) \uff0c\u89e3\u70ba \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u5169\u9ede\u3002</p> <ul> <li>\u8f38\u5165\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u7121\u9650\u6b21\u91cd\u8907\u9078\u53d6\u3002</li> <li>\u5b50\u96c6\u4e0d\u5340\u5206\u5143\u7d20\u9806\u5e8f\uff0c\u6bd4\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u500b\u5b50\u96c6\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#1","title":"1. \u00a0 \u53c3\u8003\u5168\u6392\u5217\u89e3\u6cd5","text":"<p>\u985e\u4f3c\u65bc\u5168\u6392\u5217\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u904e\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9078\u64c7\u7684\u7d50\u679c\uff0c\u4e26\u5728\u9078\u64c7\u904e\u7a0b\u4e2d\u5be6\u6642\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u7576\u5143\u7d20\u548c\u7b49\u65bc <code>target</code> \u6642\uff0c\u5c31\u5c07\u5b50\u96c6\u8a18\u9304\u81f3\u7d50\u679c\u4e32\u5217\u3002</p> <p>\u800c\u8207\u5168\u6392\u5217\u554f\u984c\u4e0d\u540c\u7684\u662f\uff0c\u672c\u984c\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u7121\u9650\u6b21\u9078\u53d6\uff0c\u56e0\u6b64\u7121\u9808\u85c9\u52a9 <code>selected</code> \u5e03\u6797\u4e32\u5217\u4f86\u8a18\u9304\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9078\u64c7\u3002\u6211\u5011\u53ef\u4ee5\u5c0d\u5168\u6392\u5217\u7a0b\u5f0f\u78bc\u9032\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u984c\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if total == target:\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i in range(len(choices)):\n        # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if total + choices[i] &gt; target:\n            continue\n        # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res)\n        # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09\"\"\"\n    state = []  # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    total = 0  # \u5b50\u96c6\u548c\n    res = []  # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push_back(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    int total = 0;           // \u5b50\u96c6\u548c\n    vector&lt;vector&lt;int&gt;&gt; res; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.Add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        Backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i := 0; i &lt; len(*choices); i++ {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        *state = append(*state, (*choices)[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    total := 0              // \u5b50\u96c6\u548c\n    res := make([][]int, 0) // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if total == target {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i in choices.indices {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    let total = 0 // \u5b50\u96c6\u548c\n    var res: [[Int]] = [] // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, total, choices, res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (int i = 0; i &lt; choices.length; i++) {\n    // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n    state.add(choices[i]);\n    // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target, total + choices[i], choices, res);\n    // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  int total = 0; // \u5143\u7d20\u548c\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    total: i32,\n    choices: &amp;[i32],\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if total == target {\n        res.push(state.clone());\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for i in 0..choices.len() {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    let total = 0; // \u5b50\u96c6\u548c\n    let mut res = Vec::new(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(&amp;mut state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state[stateSize++] = choices[i];\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // \u521d\u59cb\u5316\u89e3\u7684\u6578\u91cf\u70ba0\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    total: Int,\n    choices: IntArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (total == target) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (i in choices.indices) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n        if (total + choices[i] &gt; target) {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target, total + choices[i], choices, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09 */\nfun subsetSumINaive(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    val total = 0 // \u5b50\u96c6\u548c\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n}\n</code></pre> subset_sum_i_naive.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I ###\ndef backtrack(state, target, total, choices, res)\n  # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if total == target\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for i in 0...choices.length\n    # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u8df3\u904e\u8a72\u9078\u64c7\n    next if total + choices[i] &gt; target\n    # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n    state &lt;&lt; choices[i]\n    # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target, total + choices[i], choices, res)\n    # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.pop\n  end\nend\n\n### \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u8907\u5b50\u96c6\uff09###\ndef subset_sum_i_naive(nums, target)\n  state = [] # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  total = 0 # \u5b50\u96c6\u548c\n  res = [] # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, total, nums, res)\n  res\nend\n</code></pre> subset_sum_i_naive.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumINaive}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5411\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u8f38\u5165\u9663\u5217 \\([3, 4, 5]\\) \u548c\u76ee\u6a19\u5143\u7d20 \\(9\\) \uff0c\u8f38\u51fa\u7d50\u679c\u70ba \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u96d6\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u70ba \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u8907\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002</p> <p>\u9019\u662f\u56e0\u70ba\u641c\u5c0b\u904e\u7a0b\u662f\u5340\u5206\u9078\u64c7\u9806\u5e8f\u7684\uff0c\u7136\u800c\u5b50\u96c6\u4e0d\u5340\u5206\u9078\u64c7\u9806\u5e8f\u3002\u5982\u5716 13-10 \u6240\u793a\uff0c\u5148\u9078 \\(4\\) \u5f8c\u9078 \\(5\\) \u8207\u5148\u9078 \\(5\\) \u5f8c\u9078 \\(4\\) \u662f\u4e0d\u540c\u7684\u5206\u652f\uff0c\u4f46\u5c0d\u61c9\u540c\u4e00\u500b\u5b50\u96c6\u3002</p> <p></p> <p> \u5716 13-10 \u00a0 \u5b50\u96c6\u641c\u5c0b\u8207\u8d8a\u754c\u526a\u679d </p> <p>\u70ba\u4e86\u53bb\u9664\u91cd\u8907\u5b50\u96c6\uff0c\u4e00\u7a2e\u76f4\u63a5\u7684\u601d\u8def\u662f\u5c0d\u7d50\u679c\u4e32\u5217\u9032\u884c\u53bb\u91cd\u3002\u4f46\u9019\u500b\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u6709\u5169\u65b9\u9762\u539f\u56e0\u3002</p> <ul> <li>\u7576\u9663\u5217\u5143\u7d20\u8f03\u591a\uff0c\u5c24\u5176\u662f\u7576 <code>target</code> \u8f03\u5927\u6642\uff0c\u641c\u5c0b\u904e\u7a0b\u6703\u7522\u751f\u5927\u91cf\u7684\u91cd\u8907\u5b50\u96c6\u3002</li> <li>\u6bd4\u8f03\u5b50\u96c6\uff08\u9663\u5217\uff09\u7684\u7570\u540c\u975e\u5e38\u8017\u6642\uff0c\u9700\u8981\u5148\u6392\u5e8f\u9663\u5217\uff0c\u518d\u6bd4\u8f03\u9663\u5217\u4e2d\u6bcf\u500b\u5143\u7d20\u7684\u7570\u540c\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#2","title":"2. \u00a0 \u91cd\u8907\u5b50\u96c6\u526a\u679d","text":"<p>\u6211\u5011\u8003\u616e\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\u900f\u904e\u526a\u679d\u9032\u884c\u53bb\u91cd\u3002\u89c0\u5bdf\u5716 13-11 \uff0c\u91cd\u8907\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u9806\u5e8f\u9078\u64c7\u9663\u5217\u5143\u7d20\u6642\u7522\u751f\u7684\uff0c\u4f8b\u5982\u4ee5\u4e0b\u60c5\u6cc1\u3002</p> <ol> <li>\u7576\u7b2c\u4e00\u8f2a\u548c\u7b2c\u4e8c\u8f2a\u5206\u5225\u9078\u64c7 \\(3\\) \u548c \\(4\\) \u6642\uff0c\u6703\u751f\u6210\u5305\u542b\u9019\u5169\u500b\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8a18\u70ba \\([3, 4, \\dots]\\) \u3002</li> <li>\u4e4b\u5f8c\uff0c\u7576\u7b2c\u4e00\u8f2a\u9078\u64c7 \\(4\\) \u6642\uff0c\u5247\u7b2c\u4e8c\u8f2a\u61c9\u8a72\u8df3\u904e \\(3\\) \uff0c\u56e0\u70ba\u8a72\u9078\u64c7\u7522\u751f\u7684\u5b50\u96c6 \\([4, 3, \\dots]\\) \u548c\u7b2c <code>1.</code> \u6b65\u4e2d\u751f\u6210\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u8907\u3002</li> </ol> <p>\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c64\u7684\u9078\u64c7\u90fd\u662f\u5f9e\u5de6\u5230\u53f3\u88ab\u9010\u500b\u5617\u8a66\u7684\uff0c\u56e0\u6b64\u8d8a\u9760\u53f3\u7684\u5206\u652f\u88ab\u526a\u6389\u7684\u8d8a\u591a\u3002</p> <ol> <li>\u524d\u5169\u8f2a\u9078\u64c7 \\(3\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([3, 5, \\dots]\\) \u3002</li> <li>\u524d\u5169\u8f2a\u9078\u64c7 \\(4\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([4, 5, \\dots]\\) \u3002</li> <li>\u82e5\u7b2c\u4e00\u8f2a\u9078\u64c7 \\(5\\) \uff0c\u5247\u7b2c\u4e8c\u8f2a\u61c9\u8a72\u8df3\u904e \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u70ba\u5b50\u96c6 \\([5, 3, \\dots]\\) \u548c \\([5, 4, \\dots]\\) \u8207\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\u4e2d\u63cf\u8ff0\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u8907\u3002</li> </ol> <p></p> <p> \u5716 13-11 \u00a0 \u4e0d\u540c\u9078\u64c7\u9806\u5e8f\u5c0e\u81f4\u7684\u91cd\u8907\u5b50\u96c6 </p> <p>\u7e3d\u7d50\u4f86\u770b\uff0c\u7d66\u5b9a\u8f38\u5165\u9663\u5217 \\([x_1, x_2, \\dots, x_n]\\) \uff0c\u8a2d\u641c\u5c0b\u904e\u7a0b\u4e2d\u7684\u9078\u64c7\u5e8f\u5217\u70ba \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) \uff0c\u5247\u8a72\u9078\u64c7\u5e8f\u5217\u9700\u8981\u6eff\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4e0d\u6eff\u8db3\u8a72\u689d\u4ef6\u7684\u9078\u64c7\u5e8f\u5217\u90fd\u6703\u9020\u6210\u91cd\u8907\uff0c\u61c9\u7576\u526a\u679d\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#3","title":"3. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u70ba\u5be6\u73fe\u8a72\u526a\u679d\uff0c\u6211\u5011\u521d\u59cb\u5316\u8b8a\u6578 <code>start</code> \uff0c\u7528\u65bc\u6307\u793a\u8d70\u8a2a\u8d77\u59cb\u9ede\u3002\u7576\u505a\u51fa\u9078\u64c7 \\(x_{i}\\) \u5f8c\uff0c\u8a2d\u5b9a\u4e0b\u4e00\u8f2a\u5f9e\u7d22\u5f15 \\(i\\) \u958b\u59cb\u8d70\u8a2a\u3002\u9019\u6a23\u505a\u5c31\u53ef\u4ee5\u8b93\u9078\u64c7\u5e8f\u5217\u6eff\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u5f9e\u800c\u4fdd\u8b49\u5b50\u96c6\u552f\u4e00\u3002</p> <p>\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u5011\u9084\u5c0d\u7a0b\u5f0f\u78bc\u9032\u884c\u4e86\u4ee5\u4e0b\u5169\u9805\u6700\u4f73\u5316\u3002</p> <ul> <li>\u5728\u958b\u555f\u641c\u5c0b\u524d\uff0c\u5148\u5c07\u9663\u5217 <code>nums</code> \u6392\u5e8f\u3002\u5728\u8d70\u8a2a\u6240\u6709\u9078\u64c7\u6642\uff0c\u7576\u5b50\u96c6\u548c\u8d85\u904e <code>target</code> \u6642\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\uff0c\u56e0\u70ba\u5f8c\u908a\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e <code>target</code> \u3002</li> <li>\u7701\u53bb\u5143\u7d20\u548c\u8b8a\u6578 <code>total</code> \uff0c\u900f\u904e\u5728 <code>target</code> \u4e0a\u57f7\u884c\u6e1b\u6cd5\u4f86\u7d71\u8a08\u5143\u7d20\u548c\uff0c\u7576 <code>target</code> \u7b49\u65bc \\(0\\) \u6642\u8a18\u9304\u89e3\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    # \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        # \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0:\n            break\n        # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res)\n        # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\n    state = []  # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5c0d nums \u9032\u884c\u6392\u5e8f\n    start = 0  # \u8d70\u8a2a\u8d77\u59cb\u9ede\n    res = []  # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0;                  // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        Backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    start := 0              // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    res := make([][]int, 0) // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for i in choices.indices.dropFirst(start) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    let start = 0 // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    var res: [[Int]] = [] // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums, target) {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    const start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    const start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n    // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target - choices[i], choices, i, res);\n    // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n  int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    let start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    let mut res = Vec::new(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0;                           // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    for (i in start..&lt;choices.size) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfun subsetSumI(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort() // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    val start = 0 // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_i.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I ###\ndef backtrack(state, target, choices, start, res)\n  # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  # \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n  for i in start...choices.length\n    # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n    # \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n    break if target - choices[i] &lt; 0\n    # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n    state &lt;&lt; choices[i]\n    # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target - choices[i], choices, i, res)\n    # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.pop\n  end\nend\n\n### \u6c42\u89e3\u5b50\u96c6\u548c I ###\ndef subset_sum_i(nums, target)\n  state = [] # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  nums.sort! # \u5c0d nums \u9032\u884c\u6392\u5e8f\n  start = 0 # \u8d70\u8a2a\u8d77\u59cb\u9ede\n  res = [] # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> subset_sum_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumI}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 13-12 \u6240\u793a\u70ba\u5c07\u9663\u5217 \\([3, 4, 5]\\) \u548c\u76ee\u6a19\u5143\u7d20 \\(9\\) \u8f38\u5165\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u5f8c\u7684\u6574\u9ad4\u56de\u6eaf\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 13-12 \u00a0 \u5b50\u96c6\u548c I \u56de\u6eaf\u904e\u7a0b </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1332","title":"13.3.2 \u00a0 \u8003\u616e\u91cd\u8907\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u6b63\u6574\u6578\u9663\u5217 <code>nums</code> \u548c\u4e00\u500b\u76ee\u6a19\u6b63\u6574\u6578 <code>target</code> \uff0c\u8acb\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7d44\u5408\uff0c\u4f7f\u5f97\u7d44\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u65bc <code>target</code> \u3002\u7d66\u5b9a\u9663\u5217\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u6bcf\u500b\u5143\u7d20\u53ea\u53ef\u88ab\u9078\u64c7\u4e00\u6b21\u3002\u8acb\u4ee5\u4e32\u5217\u5f62\u5f0f\u8fd4\u56de\u9019\u4e9b\u7d44\u5408\uff0c\u4e32\u5217\u4e2d\u4e0d\u61c9\u5305\u542b\u91cd\u8907\u7d44\u5408\u3002</p> <p>\u76f8\u6bd4\u65bc\u4e0a\u984c\uff0c\u672c\u984c\u7684\u8f38\u5165\u9663\u5217\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u9019\u5f15\u5165\u4e86\u65b0\u7684\u554f\u984c\u3002\u4f8b\u5982\uff0c\u7d66\u5b9a\u9663\u5217 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6a19\u5143\u7d20 \\(9\\) \uff0c\u5247\u73fe\u6709\u7a0b\u5f0f\u78bc\u7684\u8f38\u51fa\u7d50\u679c\u70ba \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u51fa\u73fe\u4e86\u91cd\u8907\u5b50\u96c6\u3002</p> <p>\u9020\u6210\u9019\u7a2e\u91cd\u8907\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f2a\u4e2d\u88ab\u591a\u6b21\u9078\u64c7\u3002\u5728\u5716 13-13 \u4e2d\uff0c\u7b2c\u4e00\u8f2a\u5171\u6709\u4e09\u500b\u9078\u64c7\uff0c\u5176\u4e2d\u5169\u500b\u90fd\u70ba \\(4\\) \uff0c\u6703\u7522\u751f\u5169\u500b\u91cd\u8907\u7684\u641c\u5c0b\u5206\u652f\uff0c\u5f9e\u800c\u8f38\u51fa\u91cd\u8907\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f2a\u7684\u5169\u500b \\(4\\) \u4e5f\u6703\u7522\u751f\u91cd\u8907\u5b50\u96c6\u3002</p> <p></p> <p> \u5716 13-13 \u00a0 \u76f8\u7b49\u5143\u7d20\u5c0e\u81f4\u7684\u91cd\u8907\u5b50\u96c6 </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u70ba\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u6211\u5011\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f2a\u4e2d\u53ea\u80fd\u88ab\u9078\u64c7\u4e00\u6b21\u3002\u5be6\u73fe\u65b9\u5f0f\u6bd4\u8f03\u5de7\u5999\uff1a\u7531\u65bc\u9663\u5217\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u9130\u7684\u3002\u9019\u610f\u5473\u8457\u5728\u67d0\u8f2a\u9078\u64c7\u4e2d\uff0c\u82e5\u7576\u524d\u5143\u7d20\u8207\u5176\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u5247\u8aaa\u660e\u5b83\u5df2\u7d93\u88ab\u9078\u64c7\u904e\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u904e\u7576\u524d\u5143\u7d20\u3002</p> <p>\u8207\u6b64\u540c\u6642\uff0c\u672c\u984c\u898f\u5b9a\u6bcf\u500b\u9663\u5217\u5143\u7d20\u53ea\u80fd\u88ab\u9078\u64c7\u4e00\u6b21\u3002\u5e78\u904b\u7684\u662f\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u5229\u7528\u8b8a\u6578 <code>start</code> \u4f86\u6eff\u8db3\u8a72\u7d04\u675f\uff1a\u7576\u505a\u51fa\u9078\u64c7 \\(x_{i}\\) \u5f8c\uff0c\u8a2d\u5b9a\u4e0b\u4e00\u8f2a\u5f9e\u7d22\u5f15 \\(i + 1\\) \u958b\u59cb\u5411\u5f8c\u8d70\u8a2a\u3002\u9019\u6a23\u65e2\u80fd\u53bb\u9664\u91cd\u8907\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u8907\u9078\u64c7\u5143\u7d20\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#2_1","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    # \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    # \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        # \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0:\n            break\n        # \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\n    state = []  # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5c0d nums \u9032\u884c\u6392\u5e8f\n    start = 0  # \u8d70\u8a2a\u8d77\u59cb\u9ede\n    res = []  # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0;                  // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    start := 0              // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    res := make([][]int, 0) // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for i in choices.indices.dropFirst(start) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    let start = 0 // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    var res: [[Int]] = [] // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums, target) {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    const start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    const start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    const res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n  // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n    // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n  int start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    let start = 0; // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    let mut res = Vec::new(); // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u8df3\u904e\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // \u958b\u59cb\u56de\u6eaf\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.kt<pre><code>/* \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    // \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n    for (i in start..&lt;choices.size) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n        // \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i])\n        // \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        // \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfun subsetSumII(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // \u72c0\u614b\uff08\u5b50\u96c6\uff09\n    nums.sort() // \u5c0d nums \u9032\u884c\u6392\u5e8f\n    val start = 0 // \u8d70\u8a2a\u8d77\u59cb\u9ede\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_ii.rb<pre><code>### \u56de\u6eaf\u6f14\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II ###\ndef backtrack(state, target, choices, start, res)\n  # \u5b50\u96c6\u548c\u7b49\u65bc target \u6642\uff0c\u8a18\u9304\u89e3\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  # \u526a\u679d\u4e8c\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u751f\u6210\u91cd\u8907\u5b50\u96c6\n  # \u526a\u679d\u4e09\uff1a\u5f9e start \u958b\u59cb\u8d70\u8a2a\uff0c\u907f\u514d\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\n  for i in start...choices.length\n    # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u904e target \uff0c\u5247\u76f4\u63a5\u7d50\u675f\u8ff4\u5708\n    # \u9019\u662f\u56e0\u70ba\u9663\u5217\u5df2\u6392\u5e8f\uff0c\u5f8c\u908a\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u904e target\n    break if target - choices[i] &lt; 0\n    # \u526a\u679d\u56db\uff1a\u5982\u679c\u8a72\u5143\u7d20\u8207\u5de6\u908a\u5143\u7d20\u76f8\u7b49\uff0c\u8aaa\u660e\u8a72\u641c\u5c0b\u5206\u652f\u91cd\u8907\uff0c\u76f4\u63a5\u8df3\u904e\n    next if i &gt; start &amp;&amp; choices[i] == choices[i - 1]\n    # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0 target, start\n    state &lt;&lt; choices[i]\n    # \u9032\u884c\u4e0b\u4e00\u8f2a\u9078\u64c7\n    backtrack(state, target - choices[i], choices, i + 1, res)\n    # \u56de\u9000\uff1a\u64a4\u92b7\u9078\u64c7\uff0c\u6062\u5fa9\u5230\u4e4b\u524d\u7684\u72c0\u614b\n    state.pop\n  end\nend\n\n### \u6c42\u89e3\u5b50\u96c6\u548c II ###\ndef subset_sum_ii(nums, target)\n  state = [] # \u72c0\u614b\uff08\u5b50\u96c6\uff09\n  nums.sort! # \u5c0d nums \u9032\u884c\u6392\u5e8f\n  start = 0 # \u8d70\u8a2a\u8d77\u59cb\u9ede\n  res = [] # \u7d50\u679c\u4e32\u5217\uff08\u5b50\u96c6\u4e32\u5217\uff09\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> subset_sum_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumII}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 13-14 \u5c55\u793a\u4e86\u9663\u5217 \\([4, 4, 5]\\) \u548c\u76ee\u6a19\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u904e\u7a0b\uff0c\u5171\u5305\u542b\u56db\u7a2e\u526a\u679d\u64cd\u4f5c\u3002\u8acb\u4f60\u5c07\u5716\u793a\u8207\u7a0b\u5f0f\u78bc\u8a3b\u91cb\u76f8\u7d50\u5408\uff0c\u7406\u89e3\u6574\u500b\u641c\u5c0b\u904e\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u7a2e\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002</p> <p></p> <p> \u5716 13-14 \u00a0 \u5b50\u96c6\u548c II \u56de\u6eaf\u904e\u7a0b </p>"},{"location":"chapter_backtracking/summary/","title":"13.5 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_backtracking/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u672c\u8cea\u662f\u7aae\u8209\u6cd5\uff0c\u900f\u904e\u5c0d\u89e3\u7a7a\u9593\u9032\u884c\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u4f86\u5c0b\u627e\u7b26\u5408\u689d\u4ef6\u7684\u89e3\u3002\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\uff0c\u9047\u5230\u6eff\u8db3\u689d\u4ef6\u7684\u89e3\u5247\u8a18\u9304\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u8d70\u8a2a\u5b8c\u6210\u5f8c\u7d50\u675f\u3002</li> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u7684\u641c\u5c0b\u904e\u7a0b\u5305\u62ec\u5617\u8a66\u8207\u56de\u9000\u5169\u500b\u90e8\u5206\u3002\u5b83\u900f\u904e\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u4f86\u5617\u8a66\u5404\u7a2e\u9078\u64c7\uff0c\u7576\u9047\u5230\u4e0d\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\u7684\u60c5\u6cc1\u6642\uff0c\u5247\u64a4\u92b7\u4e0a\u4e00\u6b65\u7684\u9078\u64c7\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72c0\u614b\uff0c\u4e26\u7e7c\u7e8c\u5617\u8a66\u5176\u4ed6\u9078\u64c7\u3002\u5617\u8a66\u8207\u56de\u9000\u662f\u5169\u500b\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002</li> <li>\u56de\u6eaf\u554f\u984c\u901a\u5e38\u5305\u542b\u591a\u500b\u7d04\u675f\u689d\u4ef6\uff0c\u5b83\u5011\u53ef\u7528\u65bc\u5be6\u73fe\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7d50\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u5c0b\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u5c0b\u6548\u7387\u3002</li> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u65bc\u89e3\u6c7a\u641c\u5c0b\u554f\u984c\u548c\u7d04\u675f\u6eff\u8db3\u554f\u984c\u3002\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\u96d6\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u6f14\u7b97\u6cd5\u89e3\u6c7a\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u6548\u7387\u66f4\u9ad8\u6216\u6548\u679c\u66f4\u597d\u7684\u89e3\u6cd5\u3002</li> <li>\u5168\u6392\u5217\u554f\u984c\u65e8\u5728\u641c\u5c0b\u7d66\u5b9a\u96c6\u5408\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u5011\u85c9\u52a9\u4e00\u500b\u9663\u5217\u4f86\u8a18\u9304\u6bcf\u500b\u5143\u7d20\u662f\u5426\u88ab\u9078\u64c7\uff0c\u526a\u6389\u91cd\u8907\u9078\u64c7\u540c\u4e00\u5143\u7d20\u7684\u641c\u5c0b\u5206\u652f\uff0c\u78ba\u4fdd\u6bcf\u500b\u5143\u7d20\u53ea\u88ab\u9078\u64c7\u4e00\u6b21\u3002</li> <li>\u5728\u5168\u6392\u5217\u554f\u984c\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff0c\u5247\u6700\u7d42\u7d50\u679c\u6703\u51fa\u73fe\u91cd\u8907\u6392\u5217\u3002\u6211\u5011\u9700\u8981\u7d04\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f2a\u4e2d\u53ea\u80fd\u88ab\u9078\u64c7\u4e00\u6b21\uff0c\u9019\u901a\u5e38\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u96c6\u5408\u4f86\u5be6\u73fe\u3002</li> <li>\u5b50\u96c6\u548c\u554f\u984c\u7684\u76ee\u6a19\u662f\u5728\u7d66\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u70ba\u76ee\u6a19\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u5340\u5206\u5143\u7d20\u9806\u5e8f\uff0c\u800c\u641c\u5c0b\u904e\u7a0b\u6703\u8f38\u51fa\u6240\u6709\u9806\u5e8f\u7684\u7d50\u679c\uff0c\u7522\u751f\u91cd\u8907\u5b50\u96c6\u3002\u6211\u5011\u5728\u56de\u6eaf\u524d\u5c07\u8cc7\u6599\u9032\u884c\u6392\u5e8f\uff0c\u4e26\u8a2d\u5b9a\u4e00\u500b\u8b8a\u6578\u4f86\u6307\u793a\u6bcf\u4e00\u8f2a\u7684\u8d70\u8a2a\u8d77\u59cb\u9ede\uff0c\u5f9e\u800c\u5c07\u751f\u6210\u91cd\u8907\u5b50\u96c6\u7684\u641c\u5c0b\u5206\u652f\u9032\u884c\u526a\u679d\u3002</li> <li>\u5c0d\u65bc\u5b50\u96c6\u548c\u554f\u984c\uff0c\u9663\u5217\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u6703\u7522\u751f\u91cd\u8907\u96c6\u5408\u3002\u6211\u5011\u5229\u7528\u9663\u5217\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u689d\u4ef6\uff0c\u900f\u904e\u5224\u65b7\u76f8\u9130\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5be6\u73fe\u526a\u679d\uff0c\u5f9e\u800c\u78ba\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f2a\u4e2d\u53ea\u80fd\u88ab\u9078\u4e2d\u4e00\u6b21\u3002</li> <li>\\(n\\) \u7687\u540e\u554f\u984c\u65e8\u5728\u5c0b\u627e\u5c07 \\(n\\) \u500b\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76e4\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u5169\u5169\u4e4b\u9593\u7121\u6cd5\u653b\u64ca\u5c0d\u65b9\u3002\u8a72\u554f\u984c\u7684\u7d04\u675f\u689d\u4ef6\u6709\u884c\u7d04\u675f\u3001\u5217\u7d04\u675f\u3001\u4e3b\u5c0d\u89d2\u7dda\u548c\u6b21\u5c0d\u89d2\u7dda\u7d04\u675f\u3002\u70ba\u6eff\u8db3\u884c\u7d04\u675f\uff0c\u6211\u5011\u63a1\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8b49\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u500b\u7687\u540e\u3002</li> <li>\u5217\u7d04\u675f\u548c\u5c0d\u89d2\u7dda\u7d04\u675f\u7684\u8655\u7406\u65b9\u5f0f\u985e\u4f3c\u3002\u5c0d\u65bc\u5217\u7d04\u675f\uff0c\u6211\u5011\u5229\u7528\u4e00\u500b\u9663\u5217\u4f86\u8a18\u9304\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u5f9e\u800c\u6307\u793a\u9078\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5c0d\u65bc\u5c0d\u89d2\u7dda\u7d04\u675f\uff0c\u6211\u5011\u85c9\u52a9\u5169\u500b\u9663\u5217\u4f86\u5206\u5225\u8a18\u9304\u8a72\u4e3b\u3001\u6b21\u5c0d\u89d2\u7dda\u4e0a\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96e3\u9ede\u5728\u65bc\u627e\u8655\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5c0d\u89d2\u7dda\u4e0a\u683c\u5b50\u6eff\u8db3\u7684\u884c\u5217\u7d22\u5f15\u898f\u5f8b\u3002</li> </ul>"},{"location":"chapter_backtracking/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u600e\u9ebc\u7406\u89e3\u56de\u6eaf\u548c\u905e\u8ff4\u7684\u95dc\u4fc2\uff1f</p> <p>\u7e3d\u7684\u4f86\u770b\uff0c\u56de\u6eaf\u662f\u4e00\u7a2e\u201c\u6f14\u7b97\u6cd5\u7b56\u7565\u201d\uff0c\u800c\u905e\u8ff4\u66f4\u50cf\u662f\u4e00\u500b\u201c\u5de5\u5177\u201d\u3002</p> <ul> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u5be6\u73fe\u3002\u7136\u800c\uff0c\u56de\u6eaf\u662f\u905e\u8ff4\u7684\u61c9\u7528\u5834\u666f\u4e4b\u4e00\uff0c\u662f\u905e\u8ff4\u5728\u641c\u5c0b\u554f\u984c\u4e2d\u7684\u61c9\u7528\u3002</li> <li>\u905e\u8ff4\u7684\u7d50\u69cb\u9ad4\u73fe\u4e86\u201c\u5b50\u554f\u984c\u5206\u89e3\u201d\u7684\u89e3\u984c\u7bc4\u5f0f\uff0c\u5e38\u7528\u65bc\u89e3\u6c7a\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52d5\u614b\u898f\u5283\uff08\u8a18\u61b6\u5316\u905e\u8ff4\uff09\u7b49\u554f\u984c\u3002</li> </ul>"},{"location":"chapter_computational_complexity/","title":"\u7b2c 2 \u7ae0 \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<p>Abstract</p> <p>\u8907\u96dc\u5ea6\u5206\u6790\u7336\u5982\u6d69\u701a\u7684\u6f14\u7b97\u6cd5\u5b87\u5b99\u4e2d\u7684\u6642\u7a7a\u56ae\u5c0e\u3002</p> <p>\u5b83\u5e36\u9818\u6211\u5011\u5728\u6642\u9593\u8207\u7a7a\u9593\u9019\u5169\u500b\u7dad\u5ea6\u4e0a\u6df1\u5165\u63a2\u7d22\uff0c\u5c0b\u627e\u66f4\u512a\u96c5\u7684\u89e3\u6c7a\u65b9\u6848\u3002</p>"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>2.1 \u00a0 \u6f14\u7b97\u6cd5\u6548\u7387\u8a55\u4f30</li> <li>2.2 \u00a0 \u8fed\u4ee3\u8207\u905e\u8ff4</li> <li>2.3 \u00a0 \u6642\u9593\u8907\u96dc\u5ea6</li> <li>2.4 \u00a0 \u7a7a\u9593\u8907\u96dc\u5ea6</li> <li>2.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/","title":"2.2 \u00a0 \u8fed\u4ee3\u8207\u905e\u8ff4","text":"<p>\u5728\u6f14\u7b97\u6cd5\u4e2d\uff0c\u91cd\u8907\u57f7\u884c\u67d0\u500b\u4efb\u52d9\u662f\u5f88\u5e38\u898b\u7684\uff0c\u5b83\u8207\u8907\u96dc\u5ea6\u5206\u6790\u606f\u606f\u76f8\u95dc\u3002\u56e0\u6b64\uff0c\u5728\u4ecb\u7d39\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u4e4b\u524d\uff0c\u6211\u5011\u5148\u4f86\u4e86\u89e3\u5982\u4f55\u5728\u7a0b\u5f0f\u4e2d\u5be6\u73fe\u91cd\u8907\u57f7\u884c\u4efb\u52d9\uff0c\u5373\u5169\u7a2e\u57fa\u672c\u7684\u7a0b\u5f0f\u63a7\u5236\u7d50\u69cb\uff1a\u8fed\u4ee3\u3001\u905e\u8ff4\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","title":"2.2.1 \u00a0 \u8fed\u4ee3","text":"<p>\u8fed\u4ee3\uff08iteration\uff09\u662f\u4e00\u7a2e\u91cd\u8907\u57f7\u884c\u67d0\u500b\u4efb\u52d9\u7684\u63a7\u5236\u7d50\u69cb\u3002\u5728\u8fed\u4ee3\u4e2d\uff0c\u7a0b\u5f0f\u6703\u5728\u6eff\u8db3\u4e00\u5b9a\u7684\u689d\u4ef6\u4e0b\u91cd\u8907\u57f7\u884c\u67d0\u6bb5\u7a0b\u5f0f\u78bc\uff0c\u76f4\u5230\u9019\u500b\u689d\u4ef6\u4e0d\u518d\u6eff\u8db3\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","title":"1. \u00a0 for \u8ff4\u5708","text":"<p><code>for</code> \u8ff4\u5708\u662f\u6700\u5e38\u898b\u7684\u8fed\u4ee3\u5f62\u5f0f\u4e4b\u4e00\uff0c\u9069\u5408\u5728\u9810\u5148\u77e5\u9053\u8fed\u4ee3\u6b21\u6578\u6642\u4f7f\u7528\u3002</p> <p>\u4ee5\u4e0b\u51fd\u5f0f\u57fa\u65bc <code>for</code> \u8ff4\u5708\u5be6\u73fe\u4e86\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff0c\u6c42\u548c\u7d50\u679c\u4f7f\u7528\u8b8a\u6578 <code>res</code> \u8a18\u9304\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0cPython \u4e2d <code>range(a, b)</code> \u5c0d\u61c9\u7684\u5340\u9593\u662f\u201c\u5de6\u9589\u53f3\u958b\u201d\u7684\uff0c\u5c0d\u61c9\u7684\u8d70\u8a2a\u7bc4\u570d\u70ba \\(a, a + 1, \\dots, b-1\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for \u8ff4\u5708\"\"\"\n    res = 0\n    # \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for \u8ff4\u5708 */\nint forLoop(int n) {\n    int res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for \u8ff4\u5708 */\nint forLoop(int n) {\n    int res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for \u8ff4\u5708 */\nint ForLoop(int n) {\n    int res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for \u8ff4\u5708 */\nfunc forLoop(n int) int {\n    res := 0\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for \u8ff4\u5708 */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for \u8ff4\u5708 */\nfunction forLoop(n) {\n    let res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for \u8ff4\u5708 */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for \u8ff4\u5708 */\nint forLoop(int n) {\n  int res = 0;\n  // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for \u8ff4\u5708 */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for \u8ff4\u5708 */\nint forLoop(int n) {\n    int res = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for \u8ff4\u5708 */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### for \u8ff4\u5708 ###\ndef for_loop(n)\n  res = 0\n\n  # \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n  for i in 1..n\n    res += i\n  end\n\n  res\nend\n</code></pre> iteration.zig<pre><code>// for \u8ff4\u5708\nfn forLoop(n: usize) i32 {\n    var res: i32 = 0;\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        res = res + @as(i32, @intCast(i));\n    }\n    return res;\n} \n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-1 \u662f\u8a72\u6c42\u548c\u51fd\u5f0f\u7684\u6d41\u7a0b\u6846\u5716\u3002</p> <p></p> <p> \u5716 2-1 \u00a0 \u6c42\u548c\u51fd\u5f0f\u7684\u6d41\u7a0b\u6846\u5716 </p> <p>\u6b64\u6c42\u548c\u51fd\u5f0f\u7684\u64cd\u4f5c\u6578\u91cf\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8aaa\u6210\u201c\u7dda\u6027\u95dc\u4fc2\u201d\u3002\u5be6\u969b\u4e0a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u63cf\u8ff0\u7684\u5c31\u662f\u9019\u500b\u201c\u7dda\u6027\u95dc\u4fc2\u201d\u3002\u76f8\u95dc\u5167\u5bb9\u5c07\u6703\u5728\u4e0b\u4e00\u7bc0\u4e2d\u8a73\u7d30\u4ecb\u7d39\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","title":"2. \u00a0 while \u8ff4\u5708","text":"<p>\u8207 <code>for</code> \u8ff4\u5708\u985e\u4f3c\uff0c<code>while</code> \u8ff4\u5708\u4e5f\u662f\u4e00\u7a2e\u5be6\u73fe\u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5728 <code>while</code> \u8ff4\u5708\u4e2d\uff0c\u7a0b\u5f0f\u6bcf\u8f2a\u90fd\u6703\u5148\u6aa2\u67e5\u689d\u4ef6\uff0c\u5982\u679c\u689d\u4ef6\u70ba\u771f\uff0c\u5247\u7e7c\u7e8c\u57f7\u884c\uff0c\u5426\u5247\u5c31\u7d50\u675f\u8ff4\u5708\u3002</p> <p>\u4e0b\u9762\u6211\u5011\u7528 <code>while</code> \u8ff4\u5708\u4f86\u5be6\u73fe\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while \u8ff4\u5708\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    # \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n:\n        res += i\n        i += 1  # \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u8ff4\u5708 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u8ff4\u5708 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u8ff4\u5708 */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u8ff4\u5708 */\nfunc whileLoop(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    i := 1\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u8ff4\u5708 */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i\n        i += 1 // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u8ff4\u5708 */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u8ff4\u5708 */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u8ff4\u5708 */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n  // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n  while (i &lt;= n) {\n    res += i;\n    i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u8ff4\u5708 */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i;\n        i += 1; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u8ff4\u5708 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u8ff4\u5708 */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i\n        i++ // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while \u8ff4\u5708 ###\ndef while_loop(n)\n  res = 0\n  i = 1 # \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n\n  # \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n  while i &lt;= n\n    res += i\n    i += 1 # \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n  end\n\n  res\nend\n</code></pre> iteration.zig<pre><code>// while \u8ff4\u5708\nfn whileLoop(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += @intCast(i);\n        i += 1;\n    }\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p><code>while</code> \u8ff4\u5708\u6bd4 <code>for</code> \u8ff4\u5708\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\u3002\u5728 <code>while</code> \u8ff4\u5708\u4e2d\uff0c\u6211\u5011\u53ef\u4ee5\u81ea\u7531\u5730\u8a2d\u8a08\u689d\u4ef6\u8b8a\u6578\u7684\u521d\u59cb\u5316\u548c\u66f4\u65b0\u6b65\u9a5f\u3002</p> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u689d\u4ef6\u8b8a\u6578 \\(i\\) \u6bcf\u8f2a\u9032\u884c\u5169\u6b21\u66f4\u65b0\uff0c\u9019\u7a2e\u60c5\u6cc1\u5c31\u4e0d\u592a\u65b9\u4fbf\u7528 <code>for</code> \u8ff4\u5708\u5be6\u73fe\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    # \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n:\n        res += i\n        # \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    i := 1\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n  // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n  while (i &lt;= n) {\n    res += i;\n    // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09 */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09###\ndef while_loop_ii(n)\n  res = 0\n  i = 1 # \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n\n  # \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n  while i &lt;= n\n    res += i\n    # \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n    i += 1\n    i *= 2\n  end\n\n  res\nend\n</code></pre> iteration.zig<pre><code>//  while \u8ff4\u5708\uff08\u5169\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u689d\u4ef6\u8b8a\u6578\n    // \u8ff4\u5708\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += @intCast(i);\n        // \u66f4\u65b0\u689d\u4ef6\u8b8a\u6578\n        i += 1;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u7e3d\u7684\u4f86\u8aaa\uff0c<code>for</code> \u8ff4\u5708\u7684\u7a0b\u5f0f\u78bc\u66f4\u52a0\u7dca\u6e4a\uff0c<code>while</code> \u8ff4\u5708\u66f4\u52a0\u9748\u6d3b\uff0c\u5169\u8005\u90fd\u53ef\u4ee5\u5be6\u73fe\u8fed\u4ee3\u7d50\u69cb\u3002\u9078\u64c7\u4f7f\u7528\u54ea\u4e00\u500b\u61c9\u8a72\u6839\u64da\u7279\u5b9a\u554f\u984c\u7684\u9700\u6c42\u4f86\u6c7a\u5b9a\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","title":"3. \u00a0 \u5de2\u72c0\u8ff4\u5708","text":"<p>\u6211\u5011\u53ef\u4ee5\u5728\u4e00\u500b\u8ff4\u5708\u7d50\u69cb\u5167\u5de2\u72c0\u53e6\u4e00\u500b\u8ff4\u5708\u7d50\u69cb\uff0c\u4e0b\u9762\u4ee5 <code>for</code> \u8ff4\u5708\u70ba\u4f8b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"\u96d9\u5c64 for \u8ff4\u5708\"\"\"\n    res = \"\"\n    # \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfunction nestedForLoop(n) {\n    let res = '';\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nchar *nestedForLoop(int n) {\n    // n * n \u70ba\u5c0d\u61c9\u9ede\u6578\u91cf\uff0c\"(i, j), \" \u5c0d\u61c9\u5b57\u4e32\u9577\u6700\u5927\u70ba 6+10*2\uff0c\u52a0\u4e0a\u6700\u5f8c\u4e00\u500b\u7a7a\u5b57\u5143 \\0 \u7684\u984d\u5916\u7a7a\u9593\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* \u96d9\u5c64 for \u8ff4\u5708 */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>### \u96d9\u5c64 for \u8ff4\u5708 ###\ndef nested_for_loop(n)\n  res = \"\"\n\n  # \u8ff4\u5708 i = 1, 2, ..., n-1, n\n  for i in 1..n\n    # \u8ff4\u5708 j = 1, 2, ..., n-1, n\n    for j in 1..n\n      res += \"(#{i}, #{j}), \"\n    end\n  end\n\n  res\nend\n</code></pre> iteration.zig<pre><code>// \u96d9\u5c64 for \u8ff4\u5708\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n    var res = std.ArrayList(u8).init(allocator);\n    defer res.deinit();\n    var buffer: [20]u8 = undefined;\n    // \u8ff4\u5708 i = 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        // \u8ff4\u5708 j = 1, 2, ..., n-1, n\n        for (1..n+1) |j| {\n            var _str = try std.fmt.bufPrint(&amp;buffer, \"({d}, {d}), \", .{i, j});\n            try res.appendSlice(_str);\n        }\n    }\n    return res.toOwnedSlice();\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-2 \u662f\u8a72\u5de2\u72c0\u8ff4\u5708\u7684\u6d41\u7a0b\u6846\u5716\u3002</p> <p></p> <p> \u5716 2-2 \u00a0 \u5de2\u72c0\u8ff4\u5708\u7684\u6d41\u7a0b\u6846\u5716 </p> <p>\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u51fd\u5f0f\u7684\u64cd\u4f5c\u6578\u91cf\u8207 \\(n^2\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8aaa\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u548c\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u6210\u201c\u5e73\u65b9\u95dc\u4fc2\u201d\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u7e7c\u7e8c\u65b0\u589e\u5de2\u72c0\u8ff4\u5708\uff0c\u6bcf\u4e00\u6b21\u5de2\u72c0\u90fd\u662f\u4e00\u6b21\u201c\u5347\u7dad\u201d\uff0c\u5c07\u6703\u4f7f\u6642\u9593\u8907\u96dc\u5ea6\u63d0\u9ad8\u81f3\u201c\u7acb\u65b9\u95dc\u4fc2\u201d\u201c\u56db\u6b21\u65b9\u95dc\u4fc2\u201d\uff0c\u4ee5\u6b64\u985e\u63a8\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","title":"2.2.2 \u00a0 \u905e\u8ff4","text":"<p>\u905e\u8ff4\uff08recursion\uff09\u662f\u4e00\u7a2e\u6f14\u7b97\u6cd5\u7b56\u7565\uff0c\u900f\u904e\u51fd\u5f0f\u547c\u53eb\u81ea\u8eab\u4f86\u89e3\u6c7a\u554f\u984c\u3002\u5b83\u4e3b\u8981\u5305\u542b\u5169\u500b\u968e\u6bb5\u3002</p> <ol> <li>\u905e\uff1a\u7a0b\u5f0f\u4e0d\u65b7\u6df1\u5165\u5730\u547c\u53eb\u81ea\u8eab\uff0c\u901a\u5e38\u50b3\u5165\u66f4\u5c0f\u6216\u66f4\u7c21\u5316\u7684\u53c3\u6578\uff0c\u76f4\u5230\u9054\u5230\u201c\u7d42\u6b62\u689d\u4ef6\u201d\u3002</li> <li>\u8ff4\uff1a\u89f8\u767c\u201c\u7d42\u6b62\u689d\u4ef6\u201d\u5f8c\uff0c\u7a0b\u5f0f\u5f9e\u6700\u6df1\u5c64\u7684\u905e\u8ff4\u51fd\u5f0f\u958b\u59cb\u9010\u5c64\u8fd4\u56de\uff0c\u532f\u805a\u6bcf\u4e00\u5c64\u7684\u7d50\u679c\u3002</li> </ol> <p>\u800c\u5f9e\u5be6\u73fe\u7684\u89d2\u5ea6\u770b\uff0c\u905e\u8ff4\u7a0b\u5f0f\u78bc\u4e3b\u8981\u5305\u542b\u4e09\u500b\u8981\u7d20\u3002</p> <ol> <li>\u7d42\u6b62\u689d\u4ef6\uff1a\u7528\u65bc\u6c7a\u5b9a\u4ec0\u9ebc\u6642\u5019\u7531\u201c\u905e\u201d\u8f49\u201c\u8ff4\u201d\u3002</li> <li>\u905e\u8ff4\u547c\u53eb\uff1a\u5c0d\u61c9\u201c\u905e\u201d\uff0c\u51fd\u5f0f\u547c\u53eb\u81ea\u8eab\uff0c\u901a\u5e38\u8f38\u5165\u66f4\u5c0f\u6216\u66f4\u7c21\u5316\u7684\u53c3\u6578\u3002</li> <li>\u8fd4\u56de\u7d50\u679c\uff1a\u5c0d\u61c9\u201c\u8ff4\u201d\uff0c\u5c07\u7576\u524d\u905e\u8ff4\u5c64\u7d1a\u7684\u7d50\u679c\u8fd4\u56de\u81f3\u4e0a\u4e00\u5c64\u3002</li> </ol> <p>\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u53ea\u9700\u547c\u53eb\u51fd\u5f0f <code>recur(n)</code>  \uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210 \\(1 + 2 + \\dots + n\\) \u7684\u8a08\u7b97\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"\u905e\u8ff4\"\"\"\n    # \u7d42\u6b62\u689d\u4ef6\n    if n == 1:\n        return 1\n    # \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    res = recur(n - 1)\n    # \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* \u905e\u8ff4 */\nint recur(int n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    int res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* \u905e\u8ff4 */\nint recur(int n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    int res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* \u905e\u8ff4 */\nint Recur(int n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    int res = Recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* \u905e\u8ff4 */\nfunc recur(n int) int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    res := recur(n - 1)\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* \u905e\u8ff4 */\nfunc recur(n: Int) -&gt; Int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    let res = recur(n: n - 1)\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* \u905e\u8ff4 */\nfunction recur(n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n === 1) return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    const res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* \u905e\u8ff4 */\nfunction recur(n: number): number {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n === 1) return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    const res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* \u905e\u8ff4 */\nint recur(int n) {\n  // \u7d42\u6b62\u689d\u4ef6\n  if (n == 1) return 1;\n  // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n  int res = recur(n - 1);\n  // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* \u905e\u8ff4 */\nfn recur(n: i32) -&gt; i32 {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 1 {\n        return 1;\n    }\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    let res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* \u905e\u8ff4 */\nint recur(int n) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    int res = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* \u905e\u8ff4 */\nfun recur(n: Int): Int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1)\n        return 1\n    // \u905e: \u905e\u8ff4\u547c\u53eb\n    val res = recur(n - 1)\n    // \u8ff4: \u8fd4\u56de\u7d50\u679c\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>### \u905e\u8ff4 ###\ndef recur(n)\n  # \u7d42\u6b62\u689d\u4ef6\n  return 1 if n == 1\n  # \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n  res = recur(n - 1)\n  # \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n  n + res\nend\n</code></pre> recursion.zig<pre><code>// \u905e\u8ff4\u51fd\u5f0f\nfn recur(n: i32) i32 {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 1) {\n        return 1;\n    }\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    var res: i32 = recur(n - 1);\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    return n + res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-3 \u5c55\u793a\u4e86\u8a72\u51fd\u5f0f\u7684\u905e\u8ff4\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 2-3 \u00a0 \u6c42\u548c\u51fd\u5f0f\u7684\u905e\u8ff4\u904e\u7a0b </p> <p>\u96d6\u7136\u5f9e\u8a08\u7b97\u89d2\u5ea6\u770b\uff0c\u8fed\u4ee3\u8207\u905e\u8ff4\u53ef\u4ee5\u5f97\u5230\u76f8\u540c\u7684\u7d50\u679c\uff0c\u4f46\u5b83\u5011\u4ee3\u8868\u4e86\u5169\u7a2e\u5b8c\u5168\u4e0d\u540c\u7684\u601d\u8003\u548c\u89e3\u6c7a\u554f\u984c\u7684\u7bc4\u5f0f\u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u5730\u89e3\u6c7a\u554f\u984c\u3002\u5f9e\u6700\u57fa\u790e\u7684\u6b65\u9a5f\u958b\u59cb\uff0c\u7136\u5f8c\u4e0d\u65b7\u91cd\u8907\u6216\u7d2f\u52a0\u9019\u4e9b\u6b65\u9a5f\uff0c\u76f4\u5230\u4efb\u52d9\u5b8c\u6210\u3002</li> <li>\u905e\u8ff4\uff1a\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u5730\u89e3\u6c7a\u554f\u984c\u3002\u5c07\u539f\u554f\u984c\u5206\u89e3\u70ba\u66f4\u5c0f\u7684\u5b50\u554f\u984c\uff0c\u9019\u4e9b\u5b50\u554f\u984c\u548c\u539f\u554f\u984c\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\u3002\u63a5\u4e0b\u4f86\u5c07\u5b50\u554f\u984c\u7e7c\u7e8c\u5206\u89e3\u70ba\u66f4\u5c0f\u7684\u5b50\u554f\u984c\uff0c\u76f4\u5230\u57fa\u672c\u60c5\u6cc1\u6642\u505c\u6b62\uff08\u57fa\u672c\u60c5\u6cc1\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff09\u3002</li> </ul> <p>\u4ee5\u4e0a\u8ff0\u6c42\u548c\u51fd\u5f0f\u70ba\u4f8b\uff0c\u8a2d\u554f\u984c \\(f(n) = 1 + 2 + \\dots + n\\) \u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u5728\u8ff4\u5708\u4e2d\u6a21\u64ec\u6c42\u548c\u904e\u7a0b\uff0c\u5f9e \\(1\\) \u8d70\u8a2a\u5230 \\(n\\) \uff0c\u6bcf\u8f2a\u57f7\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5373\u53ef\u6c42\u5f97 \\(f(n)\\) \u3002</li> <li>\u905e\u8ff4\uff1a\u5c07\u554f\u984c\u5206\u89e3\u70ba\u5b50\u554f\u984c \\(f(n) = n + f(n-1)\\) \uff0c\u4e0d\u65b7\uff08\u905e\u8ff4\u5730\uff09\u5206\u89e3\u4e0b\u53bb\uff0c\u76f4\u81f3\u57fa\u672c\u60c5\u6cc1 \\(f(1) = 1\\) \u6642\u7d42\u6b62\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1","title":"1. \u00a0 \u547c\u53eb\u5806\u758a","text":"<p>\u905e\u8ff4\u51fd\u5f0f\u6bcf\u6b21\u547c\u53eb\u81ea\u8eab\u6642\uff0c\u7cfb\u7d71\u90fd\u6703\u70ba\u65b0\u958b\u555f\u7684\u51fd\u5f0f\u5206\u914d\u8a18\u61b6\u9ad4\uff0c\u4ee5\u5132\u5b58\u5340\u57df\u6027\u8b8a\u6578\u3001\u547c\u53eb\u4f4d\u5740\u548c\u5176\u4ed6\u8cc7\u8a0a\u7b49\u3002\u9019\u5c07\u5c0e\u81f4\u5169\u65b9\u9762\u7684\u7d50\u679c\u3002</p> <ul> <li>\u51fd\u5f0f\u7684\u4e0a\u4e0b\u6587\u8cc7\u6599\u90fd\u5132\u5b58\u5728\u7a31\u70ba\u201c\u5806\u758a\u5e40\u7a7a\u9593\u201d\u7684\u8a18\u61b6\u9ad4\u5340\u57df\u4e2d\uff0c\u76f4\u81f3\u51fd\u5f0f\u8fd4\u56de\u5f8c\u624d\u6703\u88ab\u91cb\u653e\u3002\u56e0\u6b64\uff0c\u905e\u8ff4\u901a\u5e38\u6bd4\u8fed\u4ee3\u66f4\u52a0\u8017\u8cbb\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> <li>\u905e\u8ff4\u547c\u53eb\u51fd\u5f0f\u6703\u7522\u751f\u984d\u5916\u7684\u958b\u92b7\u3002\u56e0\u6b64\u905e\u8ff4\u901a\u5e38\u6bd4\u8ff4\u5708\u7684\u6642\u9593\u6548\u7387\u66f4\u4f4e\u3002</li> </ul> <p>\u5982\u5716 2-4 \u6240\u793a\uff0c\u5728\u89f8\u767c\u7d42\u6b62\u689d\u4ef6\u524d\uff0c\u540c\u6642\u5b58\u5728 \\(n\\) \u500b\u672a\u8fd4\u56de\u7684\u905e\u8ff4\u51fd\u5f0f\uff0c\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \u3002</p> <p></p> <p> \u5716 2-4 \u00a0 \u905e\u8ff4\u547c\u53eb\u6df1\u5ea6 </p> <p>\u5728\u5be6\u969b\u4e2d\uff0c\u7a0b\u5f0f\u8a9e\u8a00\u5141\u8a31\u7684\u905e\u8ff4\u6df1\u5ea6\u901a\u5e38\u662f\u6709\u9650\u7684\uff0c\u904e\u6df1\u7684\u905e\u8ff4\u53ef\u80fd\u5c0e\u81f4\u5806\u758a\u6ea2\u4f4d\u932f\u8aa4\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","title":"2. \u00a0 \u5c3e\u905e\u8ff4","text":"<p>\u6709\u8da3\u7684\u662f\uff0c\u5982\u679c\u51fd\u5f0f\u5728\u8fd4\u56de\u524d\u7684\u6700\u5f8c\u4e00\u6b65\u624d\u9032\u884c\u905e\u8ff4\u547c\u53eb\uff0c\u5247\u8a72\u51fd\u5f0f\u53ef\u4ee5\u88ab\u7de8\u8b6f\u5668\u6216\u76f4\u8b6f\u5668\u6700\u4f73\u5316\uff0c\u4f7f\u5176\u5728\u7a7a\u9593\u6548\u7387\u4e0a\u8207\u8fed\u4ee3\u76f8\u7576\u3002\u9019\u7a2e\u60c5\u6cc1\u88ab\u7a31\u70ba\u5c3e\u905e\u8ff4\uff08tail recursion\uff09\u3002</p> <ul> <li>\u666e\u901a\u905e\u8ff4\uff1a\u7576\u51fd\u5f0f\u8fd4\u56de\u5230\u4e0a\u4e00\u5c64\u7d1a\u7684\u51fd\u5f0f\u5f8c\uff0c\u9700\u8981\u7e7c\u7e8c\u57f7\u884c\u7a0b\u5f0f\u78bc\uff0c\u56e0\u6b64\u7cfb\u7d71\u9700\u8981\u5132\u5b58\u4e0a\u4e00\u5c64\u547c\u53eb\u7684\u4e0a\u4e0b\u6587\u3002</li> <li>\u5c3e\u905e\u8ff4\uff1a\u905e\u8ff4\u547c\u53eb\u662f\u51fd\u5f0f\u8fd4\u56de\u524d\u7684\u6700\u5f8c\u4e00\u500b\u64cd\u4f5c\uff0c\u9019\u610f\u5473\u8457\u51fd\u5f0f\u8fd4\u56de\u5230\u4e0a\u4e00\u5c64\u7d1a\u5f8c\uff0c\u7121\u9808\u7e7c\u7e8c\u57f7\u884c\u5176\u4ed6\u64cd\u4f5c\uff0c\u56e0\u6b64\u7cfb\u7d71\u7121\u9808\u5132\u5b58\u4e0a\u4e00\u5c64\u51fd\u5f0f\u7684\u4e0a\u4e0b\u6587\u3002</li> </ul> <p>\u4ee5\u8a08\u7b97 \\(1 + 2 + \\dots + n\\) \u70ba\u4f8b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u7d50\u679c\u8b8a\u6578 <code>res</code> \u8a2d\u70ba\u51fd\u5f0f\u53c3\u6578\uff0c\u5f9e\u800c\u5be6\u73fe\u5c3e\u905e\u8ff4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"\u5c3e\u905e\u8ff4\"\"\"\n    # \u7d42\u6b62\u689d\u4ef6\n    if n == 0:\n        return res\n    # \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* \u5c3e\u905e\u8ff4 */\nint tailRecur(int n, int res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* \u5c3e\u905e\u8ff4 */\nint tailRecur(int n, int res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* \u5c3e\u905e\u8ff4 */\nint TailRecur(int n, int res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* \u5c3e\u905e\u8ff4 */\nfunc tailRecur(n int, res int) int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* \u5c3e\u905e\u8ff4 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* \u5c3e\u905e\u8ff4 */\nfunction tailRecur(n, res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* \u5c3e\u905e\u8ff4 */\nfunction tailRecur(n: number, res: number): number {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* \u5c3e\u905e\u8ff4 */\nint tailRecur(int n, int res) {\n  // \u7d42\u6b62\u689d\u4ef6\n  if (n == 0) return res;\n  // \u5c3e\u905e\u8ff4\u547c\u53eb\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* \u5c3e\u905e\u8ff4 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // \u7d42\u6b62\u689d\u4ef6\n    if n == 0 {\n        return res;\n    }\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* \u5c3e\u905e\u8ff4 */\nint tailRecur(int n, int res) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* \u5c3e\u905e\u8ff4 */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // \u65b0\u589e tailrec \u95dc\u9375\u8a5e\uff0c\u4ee5\u958b\u555f\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0)\n        return res\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>### \u5c3e\u905e\u8ff4 ###\ndef tail_recur(n, res)\n  # \u7d42\u6b62\u689d\u4ef6\n  return res if n == 0\n  # \u5c3e\u905e\u8ff4\u547c\u53eb\n  tail_recur(n - 1, res + n)\nend\n</code></pre> recursion.zig<pre><code>// \u5c3e\u905e\u8ff4\u51fd\u5f0f\nfn tailRecur(n: i32, res: i32) i32 {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (n == 0) {\n        return res;\n    }\n    // \u5c3e\u905e\u8ff4\u547c\u53eb\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5c3e\u905e\u8ff4\u7684\u57f7\u884c\u904e\u7a0b\u5982\u5716 2-5 \u6240\u793a\u3002\u5c0d\u6bd4\u666e\u901a\u905e\u8ff4\u548c\u5c3e\u905e\u8ff4\uff0c\u5169\u8005\u7684\u6c42\u548c\u64cd\u4f5c\u7684\u57f7\u884c\u9ede\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li>\u666e\u901a\u905e\u8ff4\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u8ff4\u201d\u7684\u904e\u7a0b\u4e2d\u57f7\u884c\u7684\uff0c\u6bcf\u5c64\u8fd4\u56de\u5f8c\u90fd\u8981\u518d\u57f7\u884c\u4e00\u6b21\u6c42\u548c\u64cd\u4f5c\u3002</li> <li>\u5c3e\u905e\u8ff4\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u905e\u201d\u7684\u904e\u7a0b\u4e2d\u57f7\u884c\u7684\uff0c\u201c\u8ff4\u201d\u7684\u904e\u7a0b\u53ea\u9700\u5c64\u5c64\u8fd4\u56de\u3002</li> </ul> <p></p> <p> \u5716 2-5 \u00a0 \u5c3e\u905e\u8ff4\u904e\u7a0b </p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u8a31\u591a\u7de8\u8b6f\u5668\u6216\u76f4\u8b6f\u5668\u4e26\u4e0d\u652f\u6301\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\u3002\u4f8b\u5982\uff0cPython \u9810\u8a2d\u4e0d\u652f\u6301\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff0c\u56e0\u6b64\u5373\u4f7f\u51fd\u5f0f\u662f\u5c3e\u905e\u8ff4\u5f62\u5f0f\uff0c\u4ecd\u7136\u53ef\u80fd\u6703\u9047\u5230\u5806\u758a\u6ea2\u4f4d\u554f\u984c\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","title":"3. \u00a0 \u905e\u8ff4\u6a39","text":"<p>\u7576\u8655\u7406\u8207\u201c\u5206\u6cbb\u201d\u76f8\u95dc\u7684\u6f14\u7b97\u6cd5\u554f\u984c\u6642\uff0c\u905e\u8ff4\u5f80\u5f80\u6bd4\u8fed\u4ee3\u7684\u601d\u8def\u66f4\u52a0\u76f4\u89c0\u3001\u7a0b\u5f0f\u78bc\u66f4\u52a0\u6613\u8b80\u3002\u4ee5\u201c\u8cbb\u6ce2\u90a3\u5951\u6578\u5217\u201d\u70ba\u4f8b\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u8cbb\u6ce2\u90a3\u5951\u6578\u5217 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) \uff0c\u6c42\u8a72\u6578\u5217\u7684\u7b2c \\(n\\) \u500b\u6578\u5b57\u3002</p> <p>\u8a2d\u8cbb\u6ce2\u90a3\u5951\u6578\u5217\u7684\u7b2c \\(n\\) \u500b\u6578\u5b57\u70ba \\(f(n)\\) \uff0c\u6613\u5f97\u5169\u500b\u7d50\u8ad6\u3002</p> <ul> <li>\u6578\u5217\u7684\u524d\u5169\u500b\u6578\u5b57\u70ba \\(f(1) = 0\\) \u548c \\(f(2) = 1\\) \u3002</li> <li>\u6578\u5217\u4e2d\u7684\u6bcf\u500b\u6578\u5b57\u662f\u524d\u5169\u500b\u6578\u5b57\u7684\u548c\uff0c\u5373 \\(f(n) = f(n - 1) + f(n - 2)\\) \u3002</li> </ul> <p>\u6309\u7167\u905e\u63a8\u95dc\u4fc2\u9032\u884c\u905e\u8ff4\u547c\u53eb\uff0c\u5c07\u524d\u5169\u500b\u6578\u5b57\u4f5c\u70ba\u7d42\u6b62\u689d\u4ef6\uff0c\u4fbf\u53ef\u5beb\u51fa\u905e\u8ff4\u7a0b\u5f0f\u78bc\u3002\u547c\u53eb <code>fib(n)</code> \u5373\u53ef\u5f97\u5230\u8cbb\u6ce2\u90a3\u5951\u6578\u5217\u7684\u7b2c \\(n\\) \u500b\u6578\u5b57\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"\u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4\"\"\"\n    # \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # \u8fd4\u56de\u7d50\u679c f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint fib(int n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint fib(int n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint Fib(int n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfunc fib(n int) int {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfunc fib(n: Int) -&gt; Int {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfunction fib(n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfunction fib(n: number): number {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint fib(int n) {\n  // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // \u8fd4\u56de\u7d50\u679c f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfn fib(n: i32) -&gt; i32 {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nint fib(int n) {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 */\nfun fib(n: Int): Int {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res\n}\n</code></pre> recursion.rb<pre><code>### \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\uff1a\u905e\u8ff4 ###\ndef fib(n)\n  # \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n  return n - 1 if n == 1 || n == 2\n  # \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n  res = fib(n - 1) + fib(n - 2)\n  # \u8fd4\u56de\u7d50\u679c f(n)\n  res\nend\n</code></pre> recursion.zig<pre><code>// \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\nfn fib(n: i32) i32 {\n    // \u7d42\u6b62\u689d\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 or n == 2) {\n        return n - 1;\n    }\n    // \u905e\u8ff4\u547c\u53eb f(n) = f(n-1) + f(n-2)\n    var res: i32 = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7d50\u679c f(n)\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u5728\u51fd\u5f0f\u5167\u905e\u8ff4\u547c\u53eb\u4e86\u5169\u500b\u51fd\u5f0f\uff0c\u9019\u610f\u5473\u8457\u5f9e\u4e00\u500b\u547c\u53eb\u7522\u751f\u4e86\u5169\u500b\u547c\u53eb\u5206\u652f\u3002\u5982\u5716 2-6 \u6240\u793a\uff0c\u9019\u6a23\u4e0d\u65b7\u905e\u8ff4\u547c\u53eb\u4e0b\u53bb\uff0c\u6700\u7d42\u5c07\u7522\u751f\u4e00\u68f5\u5c64\u6578\u70ba \\(n\\) \u7684\u905e\u8ff4\u6a39\uff08recursion tree\uff09\u3002</p> <p></p> <p> \u5716 2-6 \u00a0 \u8cbb\u6ce2\u90a3\u5951\u6578\u5217\u7684\u905e\u8ff4\u6a39 </p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u905e\u8ff4\u9ad4\u73fe\u4e86\u201c\u5c07\u554f\u984c\u5206\u89e3\u70ba\u66f4\u5c0f\u5b50\u554f\u984c\u201d\u7684\u601d\u7dad\u7bc4\u5f0f\uff0c\u9019\u7a2e\u5206\u6cbb\u7b56\u7565\u81f3\u95dc\u91cd\u8981\u3002</p> <ul> <li>\u5f9e\u6f14\u7b97\u6cd5\u89d2\u5ea6\u770b\uff0c\u641c\u5c0b\u3001\u6392\u5e8f\u3001\u56de\u6eaf\u3001\u5206\u6cbb\u3001\u52d5\u614b\u898f\u5283\u7b49\u8a31\u591a\u91cd\u8981\u6f14\u7b97\u6cd5\u7b56\u7565\u76f4\u63a5\u6216\u9593\u63a5\u5730\u61c9\u7528\u4e86\u9019\u7a2e\u601d\u7dad\u65b9\u5f0f\u3002</li> <li>\u5f9e\u8cc7\u6599\u7d50\u69cb\u89d2\u5ea6\u770b\uff0c\u905e\u8ff4\u5929\u7136\u9069\u5408\u8655\u7406\u93c8\u7d50\u4e32\u5217\u3001\u6a39\u548c\u5716\u7684\u76f8\u95dc\u554f\u984c\uff0c\u56e0\u70ba\u5b83\u5011\u975e\u5e38\u9069\u5408\u7528\u5206\u6cbb\u601d\u60f3\u9032\u884c\u5206\u6790\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#223","title":"2.2.3 \u00a0 \u5169\u8005\u5c0d\u6bd4","text":"<p>\u7e3d\u7d50\u4ee5\u4e0a\u5167\u5bb9\uff0c\u5982\u8868 2-1 \u6240\u793a\uff0c\u8fed\u4ee3\u548c\u905e\u8ff4\u5728\u5be6\u73fe\u3001\u6548\u80fd\u548c\u9069\u7528\u6027\u4e0a\u6709\u6240\u4e0d\u540c\u3002</p> <p> \u8868 2-1 \u00a0 \u8fed\u4ee3\u8207\u905e\u8ff4\u7279\u9ede\u5c0d\u6bd4 </p> \u8fed\u4ee3 \u905e\u8ff4 \u5be6\u73fe\u65b9\u5f0f \u8ff4\u5708\u7d50\u69cb \u51fd\u5f0f\u547c\u53eb\u81ea\u8eab \u6642\u9593\u6548\u7387 \u6548\u7387\u901a\u5e38\u8f03\u9ad8\uff0c\u7121\u51fd\u5f0f\u547c\u53eb\u958b\u92b7 \u6bcf\u6b21\u51fd\u5f0f\u547c\u53eb\u90fd\u6703\u7522\u751f\u958b\u92b7 \u8a18\u61b6\u9ad4\u4f7f\u7528 \u901a\u5e38\u4f7f\u7528\u56fa\u5b9a\u5927\u5c0f\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593 \u7d2f\u7a4d\u51fd\u5f0f\u547c\u53eb\u53ef\u80fd\u4f7f\u7528\u5927\u91cf\u7684\u5806\u758a\u5e40\u7a7a\u9593 \u9069\u7528\u554f\u984c \u9069\u7528\u65bc\u7c21\u55ae\u8ff4\u5708\u4efb\u52d9\uff0c\u7a0b\u5f0f\u78bc\u76f4\u89c0\u3001\u53ef\u8b80\u6027\u597d \u9069\u7528\u65bc\u5b50\u554f\u984c\u5206\u89e3\uff0c\u5982\u6a39\u3001\u5716\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u7b49\uff0c\u7a0b\u5f0f\u78bc\u7d50\u69cb\u7c21\u6f54\u3001\u6e05\u6670 <p>Tip</p> <p>\u5982\u679c\u611f\u89ba\u4ee5\u4e0b\u5167\u5bb9\u7406\u89e3\u56f0\u96e3\uff0c\u53ef\u4ee5\u5728\u8b80\u5b8c\u201c\u5806\u758a\u201d\u7ae0\u7bc0\u5f8c\u518d\u4f86\u8907\u7fd2\u3002</p> <p>\u90a3\u9ebc\uff0c\u8fed\u4ee3\u548c\u905e\u8ff4\u5177\u6709\u4ec0\u9ebc\u5167\u5728\u95dc\u806f\u5462\uff1f\u4ee5\u4e0a\u8ff0\u905e\u8ff4\u51fd\u5f0f\u70ba\u4f8b\uff0c\u6c42\u548c\u64cd\u4f5c\u5728\u905e\u8ff4\u7684\u201c\u8ff4\u201d\u968e\u6bb5\u9032\u884c\u3002\u9019\u610f\u5473\u8457\u6700\u521d\u88ab\u547c\u53eb\u7684\u51fd\u5f0f\u5be6\u969b\u4e0a\u662f\u6700\u5f8c\u5b8c\u6210\u5176\u6c42\u548c\u64cd\u4f5c\u7684\uff0c\u9019\u7a2e\u5de5\u4f5c\u6a5f\u5236\u8207\u5806\u758a\u7684\u201c\u5148\u5165\u5f8c\u51fa\u201d\u539f\u5247\u7570\u66f2\u540c\u5de5\u3002</p> <p>\u4e8b\u5be6\u4e0a\uff0c\u201c\u547c\u53eb\u5806\u758a\u201d\u548c\u201c\u5806\u758a\u5e40\u7a7a\u9593\u201d\u9019\u985e\u905e\u8ff4\u8853\u8a9e\u5df2\u7d93\u6697\u793a\u4e86\u905e\u8ff4\u8207\u5806\u758a\u4e4b\u9593\u7684\u5bc6\u5207\u95dc\u4fc2\u3002</p> <ol> <li>\u905e\uff1a\u7576\u51fd\u5f0f\u88ab\u547c\u53eb\u6642\uff0c\u7cfb\u7d71\u6703\u5728\u201c\u547c\u53eb\u5806\u758a\u201d\u4e0a\u70ba\u8a72\u51fd\u5f0f\u5206\u914d\u65b0\u7684\u5806\u758a\u5e40\uff0c\u7528\u65bc\u5132\u5b58\u51fd\u5f0f\u7684\u5340\u57df\u6027\u8b8a\u6578\u3001\u53c3\u6578\u3001\u8fd4\u56de\u4f4d\u5740\u7b49\u8cc7\u6599\u3002</li> <li>\u8ff4\uff1a\u7576\u51fd\u5f0f\u5b8c\u6210\u57f7\u884c\u4e26\u8fd4\u56de\u6642\uff0c\u5c0d\u61c9\u7684\u5806\u758a\u5e40\u6703\u88ab\u5f9e\u201c\u547c\u53eb\u5806\u758a\u201d\u4e0a\u79fb\u9664\uff0c\u6062\u5fa9\u4e4b\u524d\u51fd\u5f0f\u7684\u57f7\u884c\u74b0\u5883\u3002</li> </ol> <p>\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u547c\u53eb\u5806\u758a\u7684\u884c\u70ba\uff0c\u5f9e\u800c\u5c07\u905e\u8ff4\u8f49\u5316\u70ba\u8fed\u4ee3\u5f62\u5f0f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4\"\"\"\n    # \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    stack = []\n    res = 0\n    # \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for i in range(n, 0, -1):\n        # \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.append(i)\n    # \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while stack:\n        # \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (int i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (!stack.empty()) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (int i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (!stack.isEmpty()) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint ForLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (int i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.Push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (stack.Count &gt; 0) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfunc forLoopRecur(n int) int {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    stack := list.New()\n    res := 0\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for i := n; i &gt; 0; i-- {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.PushBack(i)\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    for stack.Len() != 0 {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    var stack: [Int] = []\n    var res = 0\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for i in (1 ... n).reversed() {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.append(i)\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while !stack.isEmpty {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfunction forLoopRecur(n) {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    const stack = [];\n    let res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (let i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (stack.length) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfunction forLoopRecur(n: number): number {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a \n    const stack: number[] = [];\n    let res: number = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (let i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (stack.length) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint forLoopRecur(int n) {\n  // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n  for (int i = n; i &gt; 0; i--) {\n    // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n    stack.add(i);\n  }\n  // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n  while (!stack.isEmpty) {\n    // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for i in (1..=n).rev() {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i);\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while !stack.is_empty() {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nint forLoopRecur(int n) {\n    int stack[1000]; // \u85c9\u52a9\u4e00\u500b\u5927\u9663\u5217\u4f86\u6a21\u64ec\u5806\u758a\n    int top = -1;    // \u5806\u758a\u9802\u7d22\u5f15\n    int res = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    for (int i = n; i &gt; 0; i--) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack[1 + top++] = i;\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    while (top &gt;= 0) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 */\nfun forLoopRecur(n: Int): Int {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // \u905e: \u905e\u8ff4\u547c\u53eb\n    for (i in n downTo 0) {\n        // \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n        stack.push(i)\n    }\n    // \u8ff4: \u8fd4\u56de\u7d50\u679c\n    while (stack.isNotEmpty()) {\n        // \u900f\u904e\u201c\u51fa\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u8ff4\u201d\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>### \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4 ###\ndef for_loop_recur(n)\n  # \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n  stack = []\n  res = 0\n\n  # \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n  for i in n.downto(0)\n    # \u900f\u904e\u201c\u5165\u5806\u758a\u64cd\u4f5c\u201d\u6a21\u64ec\u201c\u905e\u201d\n    stack &lt;&lt; i\n  end\n  # \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n  while !stack.empty?\n    res += stack.pop\n  end\n\n  # res = 1+2+3+...+n\n  res\nend\n</code></pre> recursion.zig<pre><code>// \u4f7f\u7528\u8fed\u4ee3\u6a21\u64ec\u905e\u8ff4\nfn forLoopRecur(comptime n: i32) i32 {\n    // \u4f7f\u7528\u4e00\u500b\u986f\u5f0f\u7684\u5806\u758a\u4f86\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\n    var stack: [n]i32 = undefined;\n    var res: i32 = 0;\n    // \u905e\uff1a\u905e\u8ff4\u547c\u53eb\n    var i: usize = n;\n    while (i &gt; 0) {\n        stack[i - 1] = @intCast(i);\n        i -= 1;\n    }\n    // \u8ff4\uff1a\u8fd4\u56de\u7d50\u679c\n    var index: usize = n;\n    while (index &gt; 0) {\n        index -= 1;\n        res += stack[index];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\uff0c\u7576\u905e\u8ff4\u8f49\u5316\u70ba\u8fed\u4ee3\u5f8c\uff0c\u7a0b\u5f0f\u78bc\u8b8a\u5f97\u66f4\u52a0\u8907\u96dc\u4e86\u3002\u5118\u7ba1\u8fed\u4ee3\u548c\u905e\u8ff4\u5728\u5f88\u591a\u60c5\u6cc1\u4e0b\u53ef\u4ee5\u4e92\u76f8\u8f49\u5316\uff0c\u4f46\u4e0d\u4e00\u5b9a\u503c\u5f97\u9019\u6a23\u505a\uff0c\u6709\u4ee5\u4e0b\u5169\u9ede\u539f\u56e0\u3002</p> <ul> <li>\u8f49\u5316\u5f8c\u7684\u7a0b\u5f0f\u78bc\u53ef\u80fd\u66f4\u52a0\u96e3\u4ee5\u7406\u89e3\uff0c\u53ef\u8b80\u6027\u66f4\u5dee\u3002</li> <li>\u5c0d\u65bc\u67d0\u4e9b\u8907\u96dc\u554f\u984c\uff0c\u6a21\u64ec\u7cfb\u7d71\u547c\u53eb\u5806\u758a\u7684\u884c\u70ba\u53ef\u80fd\u975e\u5e38\u56f0\u96e3\u3002</li> </ul> <p>\u7e3d\u4e4b\uff0c\u9078\u64c7\u8fed\u4ee3\u9084\u662f\u905e\u8ff4\u53d6\u6c7a\u65bc\u7279\u5b9a\u554f\u984c\u7684\u6027\u8cea\u3002\u5728\u7a0b\u5f0f\u8a2d\u8a08\u5be6\u8e10\u4e2d\uff0c\u6b0a\u8861\u5169\u8005\u7684\u512a\u52a3\u4e26\u6839\u64da\u60c5\u5883\u9078\u64c7\u5408\u9069\u7684\u65b9\u6cd5\u81f3\u95dc\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 \u6f14\u7b97\u6cd5\u6548\u7387\u8a55\u4f30","text":"<p>\u5728\u6f14\u7b97\u6cd5\u8a2d\u8a08\u4e2d\uff0c\u6211\u5011\u5148\u5f8c\u8ffd\u6c42\u4ee5\u4e0b\u5169\u500b\u5c64\u9762\u7684\u76ee\u6a19\u3002</p> <ol> <li>\u627e\u5230\u554f\u984c\u89e3\u6cd5\uff1a\u6f14\u7b97\u6cd5\u9700\u8981\u5728\u898f\u5b9a\u7684\u8f38\u5165\u7bc4\u570d\u5167\u53ef\u9760\u5730\u6c42\u5f97\u554f\u984c\u7684\u6b63\u78ba\u89e3\u3002</li> <li>\u5c0b\u6c42\u6700\u512a\u89e3\u6cd5\uff1a\u540c\u4e00\u500b\u554f\u984c\u53ef\u80fd\u5b58\u5728\u591a\u7a2e\u89e3\u6cd5\uff0c\u6211\u5011\u5e0c\u671b\u627e\u5230\u5118\u53ef\u80fd\u9ad8\u6548\u7684\u6f14\u7b97\u6cd5\u3002</li> </ol> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u5728\u80fd\u5920\u89e3\u6c7a\u554f\u984c\u7684\u524d\u63d0\u4e0b\uff0c\u6f14\u7b97\u6cd5\u6548\u7387\u5df2\u6210\u70ba\u8861\u91cf\u6f14\u7b97\u6cd5\u512a\u52a3\u7684\u4e3b\u8981\u8a55\u50f9\u6307\u6a19\uff0c\u5b83\u5305\u62ec\u4ee5\u4e0b\u5169\u500b\u7dad\u5ea6\u3002</p> <ul> <li>\u6642\u9593\u6548\u7387\uff1a\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u7684\u9577\u77ed\u3002</li> <li>\u7a7a\u9593\u6548\u7387\uff1a\u6f14\u7b97\u6cd5\u4f54\u7528\u8a18\u61b6\u9ad4\u7a7a\u9593\u7684\u5927\u5c0f\u3002</li> </ul> <p>\u7c21\u800c\u8a00\u4e4b\uff0c\u6211\u5011\u7684\u76ee\u6a19\u662f\u8a2d\u8a08\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u3002\u800c\u6709\u6548\u5730\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\u81f3\u95dc\u91cd\u8981\uff0c\u56e0\u70ba\u53ea\u6709\u9019\u6a23\uff0c\u6211\u5011\u624d\u80fd\u5c07\u5404\u7a2e\u6f14\u7b97\u6cd5\u9032\u884c\u5c0d\u6bd4\uff0c\u9032\u800c\u6307\u5c0e\u6f14\u7b97\u6cd5\u8a2d\u8a08\u8207\u6700\u4f73\u5316\u904e\u7a0b\u3002</p> <p>\u6548\u7387\u8a55\u4f30\u65b9\u6cd5\u4e3b\u8981\u5206\u70ba\u5169\u7a2e\uff1a\u5be6\u969b\u6e2c\u8a66\u3001\u7406\u8ad6\u4f30\u7b97\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1 \u00a0 \u5be6\u969b\u6e2c\u8a66","text":"<p>\u5047\u8a2d\u6211\u5011\u73fe\u5728\u6709\u6f14\u7b97\u6cd5 <code>A</code> \u548c\u6f14\u7b97\u6cd5 <code>B</code> \uff0c\u5b83\u5011\u90fd\u80fd\u89e3\u6c7a\u540c\u4e00\u554f\u984c\uff0c\u73fe\u5728\u9700\u8981\u5c0d\u6bd4\u9019\u5169\u500b\u6f14\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u627e\u4e00\u81fa\u8a08\u7b97\u6a5f\uff0c\u57f7\u884c\u9019\u5169\u500b\u6f14\u7b97\u6cd5\uff0c\u4e26\u76e3\u63a7\u8a18\u9304\u5b83\u5011\u7684\u57f7\u884c\u6642\u9593\u548c\u8a18\u61b6\u9ad4\u4f54\u7528\u60c5\u6cc1\u3002\u9019\u7a2e\u8a55\u4f30\u65b9\u5f0f\u80fd\u5920\u53cd\u6620\u771f\u5be6\u60c5\u6cc1\uff0c\u4f46\u4e5f\u5b58\u5728\u8f03\u5927\u7684\u4fb7\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u96e3\u4ee5\u6392\u9664\u6e2c\u8a66\u74b0\u5883\u7684\u5e72\u64fe\u56e0\u7d20\u3002\u786c\u9ad4\u914d\u7f6e\u6703\u5f71\u97ff\u6f14\u7b97\u6cd5\u7684\u6548\u80fd\u8868\u73fe\u3002\u6bd4\u5982\u4e00\u500b\u6f14\u7b97\u6cd5\u7684\u4e26\u884c\u5ea6\u8f03\u9ad8\uff0c\u90a3\u9ebc\u5b83\u5c31\u66f4\u9069\u5408\u5728\u591a\u6838 CPU \u4e0a\u57f7\u884c\uff0c\u4e00\u500b\u6f14\u7b97\u6cd5\u7684\u8a18\u61b6\u9ad4\u64cd\u4f5c\u5bc6\u96c6\uff0c\u90a3\u9ebc\u5b83\u5728\u9ad8\u6548\u80fd\u8a18\u61b6\u9ad4\u4e0a\u7684\u8868\u73fe\u5c31\u6703\u66f4\u597d\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u6f14\u7b97\u6cd5\u5728\u4e0d\u540c\u7684\u6a5f\u5668\u4e0a\u7684\u6e2c\u8a66\u7d50\u679c\u53ef\u80fd\u662f\u4e0d\u4e00\u81f4\u7684\u3002\u9019\u610f\u5473\u8457\u6211\u5011\u9700\u8981\u5728\u5404\u7a2e\u6a5f\u5668\u4e0a\u9032\u884c\u6e2c\u8a66\uff0c\u7d71\u8a08\u5e73\u5747\u6548\u7387\uff0c\u800c\u9019\u662f\u4e0d\u73fe\u5be6\u7684\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5c55\u958b\u5b8c\u6574\u6e2c\u8a66\u975e\u5e38\u8017\u8cbb\u8cc7\u6e90\u3002\u96a8\u8457\u8f38\u5165\u8cc7\u6599\u91cf\u7684\u8b8a\u5316\uff0c\u6f14\u7b97\u6cd5\u6703\u8868\u73fe\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5728\u8f38\u5165\u8cc7\u6599\u91cf\u8f03\u5c0f\u6642\uff0c\u6f14\u7b97\u6cd5 <code>A</code> \u7684\u57f7\u884c\u6642\u9593\u6bd4\u6f14\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u800c\u5728\u8f38\u5165\u8cc7\u6599\u91cf\u8f03\u5927\u6642\uff0c\u6e2c\u8a66\u7d50\u679c\u53ef\u80fd\u6070\u6070\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u70ba\u4e86\u5f97\u5230\u6709\u8aaa\u670d\u529b\u7684\u7d50\u8ad6\uff0c\u6211\u5011\u9700\u8981\u6e2c\u8a66\u5404\u7a2e\u898f\u6a21\u7684\u8f38\u5165\u8cc7\u6599\uff0c\u800c\u9019\u9700\u8981\u8017\u8cbb\u5927\u91cf\u7684\u8a08\u7b97\u8cc7\u6e90\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2 \u00a0 \u7406\u8ad6\u4f30\u7b97","text":"<p>\u7531\u65bc\u5be6\u969b\u6e2c\u8a66\u5177\u6709\u8f03\u5927\u7684\u4fb7\u9650\u6027\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u8003\u616e\u50c5\u900f\u904e\u4e00\u4e9b\u8a08\u7b97\u4f86\u8a55\u4f30\u6f14\u7b97\u6cd5\u7684\u6548\u7387\u3002\u9019\u7a2e\u4f30\u7b97\u65b9\u6cd5\u88ab\u7a31\u70ba\u6f38\u8fd1\u8907\u96dc\u5ea6\u5206\u6790\uff08asymptotic complexity analysis\uff09\uff0c\u7c21\u7a31\u8907\u96dc\u5ea6\u5206\u6790\u3002</p> <p>\u8907\u96dc\u5ea6\u5206\u6790\u80fd\u5920\u9ad4\u73fe\u6f14\u7b97\u6cd5\u57f7\u884c\u6240\u9700\u7684\u6642\u9593\u548c\u7a7a\u9593\u8cc7\u6e90\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u4e4b\u9593\u7684\u95dc\u4fc2\u3002\u5b83\u63cf\u8ff0\u4e86\u96a8\u8457\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u7684\u589e\u52a0\uff0c\u6f14\u7b97\u6cd5\u57f7\u884c\u6240\u9700\u6642\u9593\u548c\u7a7a\u9593\u7684\u589e\u9577\u8da8\u52e2\u3002\u9019\u500b\u5b9a\u7fa9\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5176\u5206\u70ba\u4e09\u500b\u91cd\u9ede\u4f86\u7406\u89e3\u3002</p> <ul> <li>\u201c\u6642\u9593\u548c\u7a7a\u9593\u8cc7\u6e90\u201d\u5206\u5225\u5c0d\u61c9\u6642\u9593\u8907\u96dc\u5ea6\uff08time complexity\uff09\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\uff08space complexity\uff09\u3002</li> <li>\u201c\u96a8\u8457\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u7684\u589e\u52a0\u201d\u610f\u5473\u8457\u8907\u96dc\u5ea6\u53cd\u6620\u4e86\u6f14\u7b97\u6cd5\u57f7\u884c\u6548\u7387\u8207\u8f38\u5165\u8cc7\u6599\u9ad4\u91cf\u4e4b\u9593\u7684\u95dc\u4fc2\u3002</li> <li>\u201c\u6642\u9593\u548c\u7a7a\u9593\u7684\u589e\u9577\u8da8\u52e2\u201d\u8868\u793a\u8907\u96dc\u5ea6\u5206\u6790\u95dc\u6ce8\u7684\u4e0d\u662f\u57f7\u884c\u6642\u9593\u6216\u4f54\u7528\u7a7a\u9593\u7684\u5177\u9ad4\u503c\uff0c\u800c\u662f\u6642\u9593\u6216\u7a7a\u9593\u589e\u9577\u7684\u201c\u5feb\u6162\u201d\u3002</li> </ul> <p>\u8907\u96dc\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5be6\u969b\u6e2c\u8a66\u65b9\u6cd5\u7684\u5f0a\u7aef\uff0c\u9ad4\u73fe\u5728\u4ee5\u4e0b\u5e7e\u500b\u65b9\u9762\u3002</p> <ul> <li>\u5b83\u7121\u9700\u5be6\u969b\u57f7\u884c\u7a0b\u5f0f\u78bc\uff0c\u66f4\u52a0\u7da0\u8272\u7bc0\u80fd\u3002</li> <li>\u5b83\u7368\u7acb\u65bc\u6e2c\u8a66\u74b0\u5883\uff0c\u5206\u6790\u7d50\u679c\u9069\u7528\u65bc\u6240\u6709\u57f7\u884c\u5e73\u81fa\u3002</li> <li>\u5b83\u53ef\u4ee5\u9ad4\u73fe\u4e0d\u540c\u8cc7\u6599\u91cf\u4e0b\u7684\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u8cc7\u6599\u91cf\u4e0b\u7684\u6f14\u7b97\u6cd5\u6548\u80fd\u3002</li> </ul> <p>Tip</p> <p>\u5982\u679c\u4f60\u4ecd\u5c0d\u8907\u96dc\u5ea6\u7684\u6982\u5ff5\u611f\u5230\u56f0\u60d1\uff0c\u7121\u9808\u64d4\u5fc3\uff0c\u6211\u5011\u6703\u5728\u5f8c\u7e8c\u7ae0\u7bc0\u4e2d\u8a73\u7d30\u4ecb\u7d39\u3002</p> <p>\u8907\u96dc\u5ea6\u5206\u6790\u70ba\u6211\u5011\u63d0\u4f9b\u4e86\u4e00\u628a\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6a19\u5c3a\u201d\uff0c\u4f7f\u6211\u5011\u53ef\u4ee5\u8861\u91cf\u57f7\u884c\u67d0\u500b\u6f14\u7b97\u6cd5\u6240\u9700\u7684\u6642\u9593\u548c\u7a7a\u9593\u8cc7\u6e90\uff0c\u5c0d\u6bd4\u4e0d\u540c\u6f14\u7b97\u6cd5\u4e4b\u9593\u7684\u6548\u7387\u3002</p> <p>\u8907\u96dc\u5ea6\u662f\u500b\u6578\u5b78\u6982\u5ff5\uff0c\u5c0d\u65bc\u521d\u5b78\u8005\u53ef\u80fd\u6bd4\u8f03\u62bd\u8c61\uff0c\u5b78\u7fd2\u96e3\u5ea6\u76f8\u5c0d\u8f03\u9ad8\u3002\u5f9e\u9019\u500b\u89d2\u5ea6\u770b\uff0c\u8907\u96dc\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9069\u5408\u4f5c\u70ba\u6700\u5148\u4ecb\u7d39\u7684\u5167\u5bb9\u3002\u7136\u800c\uff0c\u7576\u6211\u5011\u8a0e\u8ad6\u67d0\u500b\u8cc7\u6599\u7d50\u69cb\u6216\u6f14\u7b97\u6cd5\u7684\u7279\u9ede\u6642\uff0c\u96e3\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u57f7\u884c\u901f\u5ea6\u548c\u7a7a\u9593\u4f7f\u7528\u60c5\u6cc1\u3002</p> <p>\u7d9c\u4e0a\u6240\u8ff0\uff0c\u5efa\u8b70\u4f60\u5728\u6df1\u5165\u5b78\u7fd2\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u4e4b\u524d\uff0c\u5148\u5c0d\u8907\u96dc\u5ea6\u5206\u6790\u5efa\u7acb\u521d\u6b65\u7684\u77ad\u89e3\uff0c\u4ee5\u4fbf\u80fd\u5920\u5b8c\u6210\u7c21\u55ae\u6f14\u7b97\u6cd5\u7684\u8907\u96dc\u5ea6\u5206\u6790\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 \u7a7a\u9593\u8907\u96dc\u5ea6","text":"<p>\u7a7a\u9593\u8907\u96dc\u5ea6\uff08space complexity\uff09\u7528\u65bc\u8861\u91cf\u6f14\u7b97\u6cd5\u4f54\u7528\u8a18\u61b6\u9ad4\u7a7a\u9593\u96a8\u8457\u8cc7\u6599\u91cf\u8b8a\u5927\u6642\u7684\u589e\u9577\u8da8\u52e2\u3002\u9019\u500b\u6982\u5ff5\u8207\u6642\u9593\u8907\u96dc\u5ea6\u975e\u5e38\u985e\u4f3c\uff0c\u53ea\u9700\u5c07\u201c\u57f7\u884c\u6642\u9593\u201d\u66ff\u63db\u70ba\u201c\u4f54\u7528\u8a18\u61b6\u9ad4\u7a7a\u9593\u201d\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#241","title":"2.4.1 \u00a0 \u6f14\u7b97\u6cd5\u76f8\u95dc\u7a7a\u9593","text":"<p>\u6f14\u7b97\u6cd5\u5728\u57f7\u884c\u904e\u7a0b\u4e2d\u4f7f\u7528\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u5e7e\u7a2e\u3002</p> <ul> <li>\u8f38\u5165\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u6f14\u7b97\u6cd5\u7684\u8f38\u5165\u8cc7\u6599\u3002</li> <li>\u66ab\u5b58\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u6f14\u7b97\u6cd5\u5728\u57f7\u884c\u904e\u7a0b\u4e2d\u7684\u8b8a\u6578\u3001\u7269\u4ef6\u3001\u51fd\u5f0f\u4e0a\u4e0b\u6587\u7b49\u8cc7\u6599\u3002</li> <li>\u8f38\u51fa\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u6f14\u7b97\u6cd5\u7684\u8f38\u51fa\u8cc7\u6599\u3002</li> </ul> <p>\u4e00\u822c\u60c5\u6cc1\u4e0b\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u7684\u7d71\u8a08\u7bc4\u570d\u662f\u201c\u66ab\u5b58\u7a7a\u9593\u201d\u52a0\u4e0a\u201c\u8f38\u51fa\u7a7a\u9593\u201d\u3002</p> <p>\u66ab\u5b58\u7a7a\u9593\u53ef\u4ee5\u9032\u4e00\u6b65\u5283\u5206\u70ba\u4e09\u500b\u90e8\u5206\u3002</p> <ul> <li>\u66ab\u5b58\u8cc7\u6599\uff1a\u7528\u65bc\u5132\u5b58\u6f14\u7b97\u6cd5\u57f7\u884c\u904e\u7a0b\u4e2d\u7684\u5404\u7a2e\u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u7b49\u3002</li> <li>\u5806\u758a\u5e40\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u547c\u53eb\u51fd\u5f0f\u7684\u4e0a\u4e0b\u6587\u8cc7\u6599\u3002\u7cfb\u7d71\u5728\u6bcf\u6b21\u547c\u53eb\u51fd\u5f0f\u6642\u90fd\u6703\u5728\u5806\u758a\u9802\u90e8\u5efa\u7acb\u4e00\u500b\u5806\u758a\u5e40\uff0c\u51fd\u5f0f\u8fd4\u56de\u5f8c\uff0c\u5806\u758a\u5e40\u7a7a\u9593\u6703\u88ab\u91cb\u653e\u3002</li> <li>\u6307\u4ee4\u7a7a\u9593\uff1a\u7528\u65bc\u5132\u5b58\u7de8\u8b6f\u5f8c\u7684\u7a0b\u5f0f\u6307\u4ee4\uff0c\u5728\u5be6\u969b\u7d71\u8a08\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8a08\u3002</li> </ul> <p>\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5f0f\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u6642\uff0c\u6211\u5011\u901a\u5e38\u7d71\u8a08\u66ab\u5b58\u8cc7\u6599\u3001\u5806\u758a\u5e40\u7a7a\u9593\u548c\u8f38\u51fa\u8cc7\u6599\u4e09\u90e8\u5206\uff0c\u5982\u5716 2-15 \u6240\u793a\u3002</p> <p></p> <p> \u5716 2-15 \u00a0 \u6f14\u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u95dc\u7a7a\u9593 </p> <p>\u76f8\u95dc\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class Node:\n    \"\"\"\u985e\u5225\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # \u7bc0\u9ede\u503c\n        self.next: Node | None = None  # \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n\ndef function() -&gt; int:\n    \"\"\"\u51fd\u5f0f\"\"\"\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n\ndef algorithm(n) -&gt; int:  # \u8f38\u5165\u8cc7\u6599\n    A = 0                 # \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff0c\u4e00\u822c\u7528\u5927\u5beb\u5b57\u6bcd\u8868\u793a\uff09\n    b = 0                 # \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    node = Node(0)        # \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    c = function()        # \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return A + b + c      # \u8f38\u51fa\u8cc7\u6599\n</code></pre> <pre><code>/* \u7d50\u69cb\u9ad4 */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* \u51fd\u5f0f */\nint func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f38\u5165\u8cc7\u6599\n    const int a = 0;          // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    int b = 0;                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    Node* node = new Node(0); // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    int c = func();           // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* \u51fd\u5f0f */\nint function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f38\u5165\u8cc7\u6599\n    final int a = 0;          // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    int b = 0;                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    Node node = new Node(0);  // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    int c = function();       // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* \u51fd\u5f0f */\nint Function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint Algorithm(int n) {        // \u8f38\u5165\u8cc7\u6599\n    const int a = 0;          // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    int b = 0;                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    Node node = new(0);       // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    int c = Function();       // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u7d50\u69cb\u9ad4 */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* \u5efa\u7acb node \u7d50\u69cb\u9ad4  */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* \u51fd\u5f0f */\nfunc function() int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n int) int { // \u8f38\u5165\u8cc7\u6599\n    const a = 0             // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    b := 0                  // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    newNode(0)              // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    c := function()         // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c        // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* \u51fd\u5f0f */\nfunc function() -&gt; Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // \u8f38\u5165\u8cc7\u6599\n    let a = 0             // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    var b = 0             // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    let node = Node(x: 0) // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    let c = function()    // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c      // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // \u7bc0\u9ede\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u5f0f */\nfunction constFunc() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n) {       // \u8f38\u5165\u8cc7\u6599\n    const a = 0;              // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    let b = 0;                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    const node = new Node(0); // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    const c = constFunc();    // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // \u7bc0\u9ede\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u5f0f */\nfunction constFunc(): number {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n: number): number { // \u8f38\u5165\u8cc7\u6599\n    const a = 0;                        // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    let b = 0;                          // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    const node = new Node(0);           // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    const c = constFunc();              // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;                   // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* \u51fd\u5f0f */\nint function() {\n  // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n  return 0;\n}\n\nint algorithm(int n) {  // \u8f38\u5165\u8cc7\u6599\n  const int a = 0;      // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n  int b = 0;            // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n  Node node = Node(0);  // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n  int c = function();   // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n  return a + b + c;     // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u7d50\u69cb\u9ad4 */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* \u5efa\u7acb Node \u7d50\u69cb\u9ad4 */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* \u51fd\u5f0f */\nfn function() -&gt; i32 {      \n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // \u8f38\u5165\u8cc7\u6599\n    const a: i32 = 0;               // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    let mut b = 0;                  // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    let node = Node::new(0);        // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    let c = function();             // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;               // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u51fd\u5f0f */\nint func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) { // \u8f38\u5165\u8cc7\u6599\n    const int a = 0;   // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    int b = 0;         // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    int c = func();    // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c;  // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>/* \u985e\u5225 */\nclass Node(var _val: Int) {\n    var next: Node? = null\n}\n\n/* \u51fd\u5f0f */\nfun function(): Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfun algorithm(n: Int): Int { // \u8f38\u5165\u8cc7\u6599\n    val a = 0                // \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    var b = 0                // \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    val node = Node(0)       // \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    val c = function()       // \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    return a + b + c         // \u8f38\u51fa\u8cc7\u6599\n}\n</code></pre> <pre><code>### \u985e\u5225 ###\nclass Node\n    attr_accessor :val      # \u7bc0\u9ede\u503c\n    attr_accessor :next     # \u6307\u5411\u4e0b\u4e00\u7bc0\u9ede\u7684\u5f15\u7528\n\n    def initialize(x)\n        @val = x\n    end\nend\n\n### \u51fd\u5f0f ###\ndef function\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    0\nend\n\n### \u6f14\u7b97\u6cd5 ###\ndef algorithm(n)        # \u8f38\u5165\u8cc7\u6599\n    a = 0               # \u66ab\u5b58\u8cc7\u6599\uff08\u5e38\u6578\uff09\n    b = 0               # \u66ab\u5b58\u8cc7\u6599\uff08\u8b8a\u6578\uff09\n    node = Node.new(0)  # \u66ab\u5b58\u8cc7\u6599\uff08\u7269\u4ef6\uff09\n    c = function        # \u5806\u758a\u5e40\u7a7a\u9593\uff08\u547c\u53eb\u51fd\u5f0f\uff09\n    a + b + c           # \u8f38\u51fa\u8cc7\u6599\nend\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#242","title":"2.4.2 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u7a7a\u9593\u8907\u96dc\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u8207\u6642\u9593\u8907\u96dc\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u9700\u5c07\u7d71\u8a08\u7269\u4ef6\u5f9e\u201c\u64cd\u4f5c\u6578\u91cf\u201d\u8f49\u70ba\u201c\u4f7f\u7528\u7a7a\u9593\u5927\u5c0f\u201d\u3002</p> <p>\u800c\u8207\u6642\u9593\u8907\u96dc\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u5011\u901a\u5e38\u53ea\u95dc\u6ce8\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u3002\u9019\u662f\u56e0\u70ba\u8a18\u61b6\u9ad4\u7a7a\u9593\u662f\u4e00\u9805\u786c\u6027\u8981\u6c42\uff0c\u6211\u5011\u5fc5\u9808\u78ba\u4fdd\u5728\u6240\u6709\u8f38\u5165\u8cc7\u6599\u4e0b\u90fd\u6709\u8db3\u5920\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u9810\u7559\u3002</p> <p>\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u5169\u5c64\u542b\u7fa9\u3002</p> <ol> <li>\u4ee5\u6700\u5dee\u8f38\u5165\u8cc7\u6599\u70ba\u6e96\uff1a\u7576 \\(n &lt; 10\\) \u6642\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \uff1b\u4f46\u7576 \\(n &gt; 10\\) \u6642\uff0c\u521d\u59cb\u5316\u7684\u9663\u5217 <code>nums</code> \u4f54\u7528 \\(O(n)\\) \u7a7a\u9593\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</li> <li>\u4ee5\u6f14\u7b97\u6cd5\u57f7\u884c\u4e2d\u7684\u5cf0\u503c\u8a18\u61b6\u9ad4\u70ba\u6e96\uff1a\u4f8b\u5982\uff0c\u7a0b\u5f0f\u5728\u57f7\u884c\u6700\u5f8c\u4e00\u884c\u4e4b\u524d\uff0c\u4f54\u7528 \\(O(1)\\) \u7a7a\u9593\uff1b\u7576\u521d\u59cb\u5316\u9663\u5217 <code>nums</code> \u6642\uff0c\u7a0b\u5f0f\u4f54\u7528 \\(O(n)\\) \u7a7a\u9593\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    val a = 0                    // O(1)\n    val b = IntArray(10000)      // O(1)\n    if (n &gt; 10) {\n        val nums = IntArray(n)   // O(n)\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 0                           # O(1)\n    b = Array.new(10000)            # O(1)\n    nums = Array.new(n) if n &gt; 10   # O(n)\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u5728\u905e\u8ff4\u51fd\u5f0f\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7d71\u8a08\u5806\u758a\u5e40\u7a7a\u9593\u3002\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def function() -&gt; int:\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef loop(n: int):\n    \"\"\"\u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1)\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"\u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  return recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfn recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>fun function(): Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n/* \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) */\nfun loop(n: Int) {\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n/* \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) */\nfun recur(n: Int) {\n    if (n == 1) return\n    return recur(n - 1)\n}\n</code></pre> <pre><code>def function\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    0\nend\n\n### \u8ff4\u5708\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(1) ###\ndef loop(n)\n    (0...n).each { function }\nend\n\n### \u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n) ###\ndef recur(n)\n    return if n == 1\n    recur(n - 1)\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u51fd\u5f0f <code>loop()</code> \u548c <code>recur()</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n)\\) \uff0c\u4f46\u7a7a\u9593\u8907\u96dc\u5ea6\u4e0d\u540c\u3002</p> <ul> <li>\u51fd\u5f0f <code>loop()</code> \u5728\u8ff4\u5708\u4e2d\u547c\u53eb\u4e86 \\(n\\) \u6b21 <code>function()</code> \uff0c\u6bcf\u8f2a\u4e2d\u7684 <code>function()</code> \u90fd\u8fd4\u56de\u4e26\u91cb\u653e\u4e86\u5806\u758a\u5e40\u7a7a\u9593\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u4ecd\u70ba \\(O(1)\\) \u3002</li> <li>\u905e\u8ff4\u51fd\u5f0f <code>recur()</code> \u5728\u57f7\u884c\u904e\u7a0b\u4e2d\u6703\u540c\u6642\u5b58\u5728 \\(n\\) \u500b\u672a\u8fd4\u56de\u7684 <code>recur()</code> \uff0c\u5f9e\u800c\u4f54\u7528 \\(O(n)\\) \u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002</li> </ul>"},{"location":"chapter_computational_complexity/space_complexity/#243","title":"2.4.3 \u00a0 \u5e38\u898b\u578b\u5225","text":"<p>\u8a2d\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff0c\u5716 2-16 \u5c55\u793a\u4e86\u5e38\u898b\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u578b\u5225\uff08\u5f9e\u4f4e\u5230\u9ad8\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline \\text{\u5e38\u6578\u968e} &lt; \\text{\u5c0d\u6578\u968e} &lt; \\text{\u7dda\u6027\u968e} &lt; \\text{\u5e73\u65b9\u968e} &lt; \\text{\u6307\u6578\u968e} \\end{aligned} \\] <p></p> <p> \u5716 2-16 \u00a0 \u5e38\u898b\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u578b\u5225 </p>"},{"location":"chapter_computational_complexity/space_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6578\u968e \\(O(1)\\)","text":"<p>\u5e38\u6578\u968e\u5e38\u898b\u65bc\u6578\u91cf\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7121\u95dc\u7684\u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u8ff4\u5708\u4e2d\u521d\u59cb\u5316\u8b8a\u6578\u6216\u547c\u53eb\u51fd\u5f0f\u800c\u4f54\u7528\u7684\u8a18\u61b6\u9ad4\uff0c\u5728\u9032\u5165\u4e0b\u4e00\u8ff4\u5708\u5f8c\u5c31\u6703\u88ab\u91cb\u653e\uff0c\u56e0\u6b64\u4e0d\u6703\u7d2f\u7a4d\u4f54\u7528\u7a7a\u9593\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u4ecd\u70ba \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"\u51fd\u5f0f\"\"\"\n    # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef constant(n: int):\n    \"\"\"\u5e38\u6578\u968e\"\"\"\n    # \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for _ in range(n):\n        c = 0\n    # \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* \u51fd\u5f0f */\nint func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid constant(int n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u51fd\u5f0f */\nint function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid constant(int n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u51fd\u5f0f */\nint Function() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid Constant(int n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u51fd\u5f0f */\nfunc function() int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\n/* \u5e38\u6578\u968e */\nfunc spaceConstant(n int) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* \u51fd\u5f0f */\n@discardableResult\nfunc function() -&gt; Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6578\u968e */\nfunc constant(n: Int) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* \u51fd\u5f0f */\nfunction constFunc() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nfunction constant(n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u51fd\u5f0f */\nfunction constFunc(): number {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nfunction constant(n: number): void {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u51fd\u5f0f */\nint function() {\n  // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid constant(int n) {\n  // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u51fd\u5f0f */\nfn function() -&gt; i32 {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\n#[allow(unused)]\nfn constant(n: i32) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for i in 0..n {\n        let c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u51fd\u5f0f */\nint func() {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6578\u968e */\nvoid constant(int n) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u51fd\u5f0f */\nfun function(): Int {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6578\u968e */\nfun constant(n: Int) {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### \u51fd\u5f0f ###\ndef function\n  # \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n  0\nend\n\n### \u5e38\u6578\u968e ###\ndef constant(n)\n  # \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n  a = 0\n  nums = [0] * 10000\n  node = ListNode.new\n\n  # \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n  (0...n).each { c = 0 }\n  # \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n  (0...n).each { function }\nend\n</code></pre> space_complexity.zig<pre><code>// \u51fd\u5f0f\nfn function() i32 {\n    // \u57f7\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n// \u5e38\u6578\u968e\nfn constant(n: i32) void {\n    // \u5e38\u6578\u3001\u8b8a\u6578\u3001\u7269\u4ef6\u4f54\u7528 O(1) \u7a7a\u9593\n    const a: i32 = 0;\n    var b: i32 = 0;\n    var nums = [_]i32{0}**10000;\n    var node = inc.ListNode(i32){.val = 0};\n    var i: i32 = 0;\n    // \u8ff4\u5708\u4e2d\u7684\u8b8a\u6578\u4f54\u7528 O(1) \u7a7a\u9593\n    while (i &lt; n) : (i += 1) {\n        var c: i32 = 0;\n        _ = c;\n    }\n    // \u8ff4\u5708\u4e2d\u7684\u51fd\u5f0f\u4f54\u7528 O(1) \u7a7a\u9593\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        _ = function();\n    }\n    _ = a;\n    _ = b;\n    _ = nums;\n    _ = node;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_computational_complexity/space_complexity/#2-on","title":"2. \u00a0 \u7dda\u6027\u968e \\(O(n)\\)","text":"<p>\u7dda\u6027\u968e\u5e38\u898b\u65bc\u5143\u7d20\u6578\u91cf\u8207 \\(n\\) \u6210\u6b63\u6bd4\u7684\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\u7b49\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"\u7dda\u6027\u968e\"\"\"\n    # \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    nums = [0] * n\n    # \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* \u7dda\u6027\u968e */\nvoid linear(int n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    vector&lt;int&gt; nums(n);\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u7dda\u6027\u968e */\nvoid linear(int n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    int[] nums = new int[n];\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7dda\u6027\u968e */\nvoid Linear(int n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    int[] nums = new int[n];\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u7dda\u6027\u968e */\nfunc spaceLinear(n int) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    _ = make([]int, n)\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7dda\u6027\u968e */\nfunc linear(n: Int) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    let nums = Array(repeating: 0, count: n)\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* \u7dda\u6027\u968e */\nfunction linear(n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    const nums = new Array(n);\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7dda\u6027\u968e */\nfunction linear(n: number): void {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    const nums = new Array(n);\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7dda\u6027\u968e */\nvoid linear(int n) {\n  // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7dda\u6027\u968e */\n#[allow(unused)]\nfn linear(n: i32) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    let mut nums = vec![0; n as usize];\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u96dc\u6e4a\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u65bc uthash.h \u5be6\u73fe\n} HashTable;\n\n/* \u7dda\u6027\u968e */\nvoid linear(int n) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // \u8a18\u61b6\u9ad4\u91cb\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // \u8a18\u61b6\u9ad4\u91cb\u653e\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7dda\u6027\u968e */\nfun linear(n: Int) {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    val nums = Array(n) { 0 }\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### \u7dda\u6027\u968e ###\ndef linear(n)\n  # \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n  nums = Array.new(n, 0)\n\n  # \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n  hmap = {}\n  for i in 0...n\n    hmap[i] = i.to_s\n  end\nend\n</code></pre> space_complexity.zig<pre><code>// \u7dda\u6027\u968e\nfn linear(comptime n: i32) !void {\n    // \u9577\u5ea6\u70ba n \u7684\u9663\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    var nums = [_]i32{0}**n;\n    // \u9577\u5ea6\u70ba n \u7684\u4e32\u5217\u4f54\u7528 O(n) \u7a7a\u9593\n    var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        try nodes.append(i);\n    }\n    // \u9577\u5ea6\u70ba n \u7684\u96dc\u6e4a\u8868\u4f54\u7528 O(n) \u7a7a\u9593\n    var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n    defer map.deinit();\n    var j: i32 = 0;\n    while (j &lt; n) : (j += 1) {\n        const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n        defer std.heap.page_allocator.free(string);\n        try map.put(i, string);\n    }\n    _ = nums;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 2-17 \u6240\u793a\uff0c\u6b64\u51fd\u5f0f\u7684\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u5373\u540c\u6642\u5b58\u5728 \\(n\\) \u500b\u672a\u8fd4\u56de\u7684 <code>linear_recur()</code> \u51fd\u5f0f\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u5806\u758a\u5e40\u7a7a\u9593\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"\u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    print(\"\u905e\u8ff4 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"\u905e\u8ff4 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid linearRecur(int n) {\n    System.out.println(\"\u905e\u8ff4 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"\u905e\u8ff4 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"\u905e\u8ff4 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc linearRecur(n: Int) {\n    print(\"\u905e\u8ff4 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction linearRecur(n) {\n    console.log(`\u905e\u8ff4 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction linearRecur(n: number): void {\n    console.log(`\u905e\u8ff4 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid linearRecur(int n) {\n  print('\u905e\u8ff4 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn linear_recur(n: i32) {\n    println!(\"\u905e\u8ff4 n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nvoid linearRecur(int n) {\n    printf(\"\u905e\u8ff4 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfun linearRecur(n: Int) {\n    println(\"\u905e\u8ff4 n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef linear_recur(n)\n  puts \"\u905e\u8ff4 n = #{n}\"\n  return if n == 1\n  linear_recur(n - 1)\nend\n</code></pre> space_complexity.zig<pre><code>// \u7dda\u6027\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn linearRecur(comptime n: i32) void {\n    std.debug.print(\"\u905e\u8ff4 n = {}\\n\", .{n});\n    if (n == 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-17 \u00a0 \u905e\u8ff4\u51fd\u5f0f\u7522\u751f\u7684\u7dda\u6027\u968e\u7a7a\u9593\u8907\u96dc\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u968e \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u968e\u5e38\u898b\u65bc\u77e9\u9663\u548c\u5716\uff0c\u5143\u7d20\u6578\u91cf\u8207 \\(n\\) \u6210\u5e73\u65b9\u95dc\u4fc2\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"\u5e73\u65b9\u968e\"\"\"\n    # \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u968e */\nvoid quadratic(int n) {\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u968e */\nvoid quadratic(int n) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    int[][] numMatrix = new int[n][n];\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u968e */\nvoid Quadratic(int n) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    int[,] numMatrix = new int[n, n];\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u968e */\nfunc spaceQuadratic(n int) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u968e */\nfunc quadratic(n: Int) {\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u968e */\nfunction quadratic(n) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u968e */\nfunction quadratic(n: number): void {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u968e */\nvoid quadratic(int n) {\n  // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u968e */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u968e */\nvoid quadratic(int n) {\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // \u8a18\u61b6\u9ad4\u91cb\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u968e */\nfun quadratic(n: Int) {\n    // \u77e9\u9663\u4f54\u7528 O(n^2) \u7a7a\u9593\n    val numMatrix = arrayOfNulls&lt;Array&lt;Int&gt;?&gt;(n)\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    val numList = mutableListOf&lt;MutableList&lt;Int&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### \u5e73\u65b9\u968e ###\ndef quadratic(n)\n  # \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n  Array.new(n) { Array.new(n, 0) }\nend\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u968e\nfn quadratic(n: i32) !void {\n    // \u4e8c\u7dad\u4e32\u5217\u4f54\u7528 O(n^2) \u7a7a\u9593\n    var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n        defer tmp.deinit();\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            try tmp.append(0);\n        }\n        try nodes.append(tmp);\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 2-18 \u6240\u793a\uff0c\u8a72\u51fd\u5f0f\u7684\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u5728\u6bcf\u500b\u905e\u8ff4\u51fd\u5f0f\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u500b\u9663\u5217\uff0c\u9577\u5ea6\u5206\u5225\u70ba \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u5e73\u5747\u9577\u5ea6\u70ba \\(n / 2\\) \uff0c\u56e0\u6b64\u7e3d\u9ad4\u4f54\u7528 \\(O(n^2)\\) \u7a7a\u9593\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    if n &lt;= 0:\n        return 0\n    # \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"\u905e\u8ff4 n = \" &lt;&lt; n &lt;&lt; \" \u4e2d\u7684 nums \u9577\u5ea6 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"\u905e\u8ff4 n = \" + n + \" \u4e2d\u7684 nums \u9577\u5ea6 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"\u905e\u8ff4 n = \" + n + \" \u4e2d\u7684 nums \u9577\u5ea6 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"\u905e\u8ff4 n = %d \u4e2d\u7684 nums \u9577\u5ea6 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"\u905e\u8ff4 n = \\(n) \u4e2d\u7684 nums \u9577\u5ea6 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u905e\u8ff4 n = ${n} \u4e2d\u7684 nums \u9577\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u905e\u8ff4 n = ${n} \u4e2d\u7684 nums \u9577\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('\u905e\u8ff4 n = $n \u4e2d\u7684 nums \u9577\u5ea6 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"\u905e\u8ff4 n = {} \u4e2d\u7684 nums \u9577\u5ea6 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"\u905e\u8ff4 n = %d \u4e2d\u7684 nums \u9577\u5ea6 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n    val nums = Array(n) { 0 }\n    println(\"\u905e\u8ff4 n = $n \u4e2d\u7684 nums \u9577\u5ea6 = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef quadratic_recur(n)\n  return 0 unless n &gt; 0\n\n  # \u9663\u5217 nums \u9577\u5ea6\u70ba n, n-1, ..., 2, 1\n  nums = Array.new(n, 0)\n  quadratic_recur(n - 1)\nend\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n    if (n &lt;= 0) return 0;\n    var nums = [_]i32{0}**n;\n    std.debug.print(\"\u905e\u8ff4 n = {} \u4e2d\u7684 nums \u9577\u5ea6 = {}\\n\", .{n, nums.len});\n    return quadraticRecur(n - 1);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-18 \u00a0 \u905e\u8ff4\u51fd\u5f0f\u7522\u751f\u7684\u5e73\u65b9\u968e\u7a7a\u9593\u8907\u96dc\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6578\u968e \\(O(2^n)\\)","text":"<p>\u6307\u6578\u968e\u5e38\u898b\u65bc\u4e8c\u5143\u6a39\u3002\u89c0\u5bdf\u5716 2-19 \uff0c\u5c64\u6578\u70ba \\(n\\) \u7684\u201c\u6eff\u4e8c\u5143\u6a39\u201d\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\(2^n - 1\\) \uff0c\u4f54\u7528 \\(O(2^n)\\) \u7a7a\u9593\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"\u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09 */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>### \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09###\ndef build_tree(n)\n  return if n == 0\n\n  TreeNode.new.tap do |root|\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n  end\nend\n</code></pre> space_complexity.zig<pre><code>// \u6307\u6578\u968e\uff08\u5efa\u7acb\u6eff\u4e8c\u5143\u6a39\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n    if (n == 0) return null;\n    const root = try mem_allocator.create(inc.TreeNode(i32));\n    root.init(0);\n    root.left = try buildTree(mem_allocator, n - 1);\n    root.right = try buildTree(mem_allocator, n - 1);\n    return root;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-19 \u00a0 \u6eff\u4e8c\u5143\u6a39\u7522\u751f\u7684\u6307\u6578\u968e\u7a7a\u9593\u8907\u96dc\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","title":"5. \u00a0 \u5c0d\u6578\u968e \\(O(\\log n)\\)","text":"<p>\u5c0d\u6578\u968e\u5e38\u898b\u65bc\u5206\u6cbb\u6f14\u7b97\u6cd5\u3002\u4f8b\u5982\u5408\u4f75\u6392\u5e8f\uff0c\u8f38\u5165\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217\uff0c\u6bcf\u8f2a\u905e\u8ff4\u5c07\u9663\u5217\u5f9e\u4e2d\u9ede\u8655\u5283\u5206\u70ba\u5169\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u70ba \\(\\log n\\) \u7684\u905e\u8ff4\u6a39\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002</p> <p>\u518d\u4f8b\u5982\u5c07\u6578\u5b57\u8f49\u5316\u70ba\u5b57\u4e32\uff0c\u8f38\u5165\u4e00\u500b\u6b63\u6574\u6578 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6578\u70ba \\(\\lfloor \\log_{10} n \\rfloor + 1\\) \uff0c\u5373\u5c0d\u61c9\u5b57\u4e32\u9577\u5ea6\u70ba \\(\\lfloor \\log_{10} n \\rfloor + 1\\) \uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log_{10} n + 1) = O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#244","title":"2.4.4 \u00a0 \u6b0a\u8861\u6642\u9593\u8207\u7a7a\u9593","text":"<p>\u7406\u60f3\u60c5\u6cc1\u4e0b\uff0c\u6211\u5011\u5e0c\u671b\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u90fd\u80fd\u9054\u5230\u6700\u512a\u3002\u7136\u800c\u5728\u5be6\u969b\u60c5\u6cc1\u4e2d\uff0c\u540c\u6642\u6700\u4f73\u5316\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u975e\u5e38\u56f0\u96e3\u3002</p> <p>\u964d\u4f4e\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba\u4ee3\u50f9\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u5011\u5c07\u72a7\u7272\u8a18\u61b6\u9ad4\u7a7a\u9593\u4f86\u63d0\u5347\u6f14\u7b97\u6cd5\u57f7\u884c\u901f\u5ea6\u7684\u601d\u8def\u7a31\u70ba\u201c\u4ee5\u7a7a\u9593\u63db\u6642\u9593\u201d\uff1b\u53cd\u4e4b\uff0c\u5247\u7a31\u70ba\u201c\u4ee5\u6642\u9593\u63db\u7a7a\u9593\u201d\u3002</p> <p>\u9078\u64c7\u54ea\u7a2e\u601d\u8def\u53d6\u6c7a\u65bc\u6211\u5011\u66f4\u770b\u91cd\u54ea\u500b\u65b9\u9762\u3002\u5728\u5927\u591a\u6578\u60c5\u6cc1\u4e0b\uff0c\u6642\u9593\u6bd4\u7a7a\u9593\u66f4\u5bf6\u8cb4\uff0c\u56e0\u6b64\u201c\u4ee5\u7a7a\u9593\u63db\u6642\u9593\u201d\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u7576\u7136\uff0c\u5728\u8cc7\u6599\u91cf\u5f88\u5927\u7684\u60c5\u6cc1\u4e0b\uff0c\u63a7\u5236\u7a7a\u9593\u8907\u96dc\u5ea6\u4e5f\u975e\u5e38\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_computational_complexity/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<p>\u6f14\u7b97\u6cd5\u6548\u7387\u8a55\u4f30</p> <ul> <li>\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u662f\u8861\u91cf\u6f14\u7b97\u6cd5\u512a\u52a3\u7684\u5169\u500b\u4e3b\u8981\u8a55\u50f9\u6307\u6a19\u3002</li> <li>\u6211\u5011\u53ef\u4ee5\u900f\u904e\u5be6\u969b\u6e2c\u8a66\u4f86\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96e3\u4ee5\u6d88\u9664\u6e2c\u8a66\u74b0\u5883\u7684\u5f71\u97ff\uff0c\u4e14\u6703\u8017\u8cbb\u5927\u91cf\u8a08\u7b97\u8cc7\u6e90\u3002</li> <li>\u8907\u96dc\u5ea6\u5206\u6790\u53ef\u4ee5\u6d88\u9664\u5be6\u969b\u6e2c\u8a66\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7d50\u679c\u9069\u7528\u65bc\u6240\u6709\u57f7\u884c\u5e73\u81fa\uff0c\u4e26\u4e14\u80fd\u5920\u63ed\u793a\u6f14\u7b97\u6cd5\u5728\u4e0d\u540c\u8cc7\u6599\u898f\u6a21\u4e0b\u7684\u6548\u7387\u3002</li> </ul> <p>\u6642\u9593\u8907\u96dc\u5ea6</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u7528\u65bc\u8861\u91cf\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u96a8\u8cc7\u6599\u91cf\u589e\u9577\u7684\u8da8\u52e2\uff0c\u53ef\u4ee5\u6709\u6548\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u6cc1\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f38\u5165\u7684\u8cc7\u6599\u91cf\u8f03\u5c0f\u6216\u6642\u9593\u8907\u96dc\u5ea6\u76f8\u540c\u6642\uff0c\u7121\u6cd5\u7cbe\u78ba\u5c0d\u6bd4\u6f14\u7b97\u6cd5\u6548\u7387\u7684\u512a\u52a3\u3002</li> <li>\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u865f\u8868\u793a\uff0c\u5c0d\u61c9\u51fd\u5f0f\u6f38\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u7576 \\(n\\) \u8da8\u5411\u6b63\u7121\u7aae\u6642\uff0c\u64cd\u4f5c\u6578\u91cf \\(T(n)\\) \u7684\u589e\u9577\u7d1a\u5225\u3002</li> <li>\u63a8\u7b97\u6642\u9593\u8907\u96dc\u5ea6\u5206\u70ba\u5169\u6b65\uff0c\u9996\u5148\u7d71\u8a08\u64cd\u4f5c\u6578\u91cf\uff0c\u7136\u5f8c\u5224\u65b7\u6f38\u8fd1\u4e0a\u754c\u3002</li> <li>\u5e38\u898b\u6642\u9593\u8907\u96dc\u5ea6\u5f9e\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n \\log n)\\)\u3001\\(O(n^2)\\)\u3001\\(O(2^n)\\) \u548c \\(O(n!)\\) \u7b49\u3002</li> <li>\u67d0\u4e9b\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u8207\u8f38\u5165\u8cc7\u6599\u7684\u5206\u4f48\u6709\u95dc\u3002\u6642\u9593\u8907\u96dc\u5ea6\u5206\u70ba\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u5e7e\u4e4e\u4e0d\u7528\uff0c\u56e0\u70ba\u8f38\u5165\u8cc7\u6599\u4e00\u822c\u9700\u8981\u6eff\u8db3\u56b4\u683c\u689d\u4ef6\u624d\u80fd\u9054\u5230\u6700\u4f73\u60c5\u6cc1\u3002</li> <li>\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u53cd\u6620\u6f14\u7b97\u6cd5\u5728\u96a8\u6a5f\u8cc7\u6599\u8f38\u5165\u4e0b\u7684\u57f7\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5be6\u969b\u61c9\u7528\u4e2d\u7684\u6f14\u7b97\u6cd5\u6548\u80fd\u3002\u8a08\u7b97\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u9700\u8981\u7d71\u8a08\u8f38\u5165\u8cc7\u6599\u5206\u4f48\u4ee5\u53ca\u7d9c\u5408\u5f8c\u7684\u6578\u5b78\u671f\u671b\u3002</li> </ul> <p>\u7a7a\u9593\u8907\u96dc\u5ea6</p> <ul> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u7684\u4f5c\u7528\u985e\u4f3c\u65bc\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u7528\u65bc\u8861\u91cf\u6f14\u7b97\u6cd5\u4f54\u7528\u8a18\u61b6\u9ad4\u7a7a\u9593\u96a8\u8cc7\u6599\u91cf\u589e\u9577\u7684\u8da8\u52e2\u3002</li> <li>\u6f14\u7b97\u6cd5\u57f7\u884c\u904e\u7a0b\u4e2d\u7684\u76f8\u95dc\u8a18\u61b6\u9ad4\u7a7a\u9593\u53ef\u5206\u70ba\u8f38\u5165\u7a7a\u9593\u3001\u66ab\u5b58\u7a7a\u9593\u3001\u8f38\u51fa\u7a7a\u9593\u3002\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\u8f38\u5165\u7a7a\u9593\u4e0d\u7d0d\u5165\u7a7a\u9593\u8907\u96dc\u5ea6\u8a08\u7b97\u3002\u66ab\u5b58\u7a7a\u9593\u53ef\u5206\u70ba\u66ab\u5b58\u8cc7\u6599\u3001\u5806\u758a\u5e40\u7a7a\u9593\u548c\u6307\u4ee4\u7a7a\u9593\uff0c\u5176\u4e2d\u5806\u758a\u5e40\u7a7a\u9593\u901a\u5e38\u50c5\u5728\u905e\u8ff4\u51fd\u5f0f\u4e2d\u5f71\u97ff\u7a7a\u9593\u8907\u96dc\u5ea6\u3002</li> <li>\u6211\u5011\u901a\u5e38\u53ea\u95dc\u6ce8\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\uff0c\u5373\u7d71\u8a08\u6f14\u7b97\u6cd5\u5728\u6700\u5dee\u8f38\u5165\u8cc7\u6599\u548c\u6700\u5dee\u57f7\u884c\u6642\u523b\u4e0b\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u3002</li> <li>\u5e38\u898b\u7a7a\u9593\u8907\u96dc\u5ea6\u5f9e\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n^2)\\) \u548c \\(O(2^n)\\) \u7b49\u3002</li> </ul>"},{"location":"chapter_computational_complexity/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u5c3e\u905e\u8ff4\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u662f \\(O(1)\\) \u55ce\uff1f</p> <p>\u7406\u8ad6\u4e0a\uff0c\u5c3e\u905e\u8ff4\u51fd\u5f0f\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u6700\u4f73\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u904e\u7d55\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\uff08\u4f8b\u5982 Java\u3001Python\u3001C++\u3001Go\u3001C# \u7b49\uff09\u4e0d\u652f\u6301\u81ea\u52d5\u6700\u4f73\u5316\u5c3e\u905e\u8ff4\uff0c\u56e0\u6b64\u901a\u5e38\u8a8d\u70ba\u7a7a\u9593\u8907\u96dc\u5ea6\u662f \\(O(n)\\) \u3002</p> <p>Q\uff1a\u51fd\u5f0f\u548c\u65b9\u6cd5\u9019\u5169\u500b\u8853\u8a9e\u7684\u5340\u5225\u662f\u4ec0\u9ebc\uff1f</p> <p>\u51fd\u5f0f\uff08function\uff09\u53ef\u4ee5\u88ab\u7368\u7acb\u57f7\u884c\uff0c\u6240\u6709\u53c3\u6578\u90fd\u4ee5\u986f\u5f0f\u50b3\u905e\u3002\u65b9\u6cd5\uff08method\uff09\u8207\u4e00\u500b\u7269\u4ef6\u95dc\u806f\uff0c\u88ab\u96b1\u5f0f\u50b3\u905e\u7d66\u547c\u53eb\u5b83\u7684\u7269\u4ef6\uff0c\u80fd\u5920\u5c0d\u985e\u5225\u7684\u4f8b\u9805\u4e2d\u5305\u542b\u7684\u8cc7\u6599\u9032\u884c\u64cd\u4f5c\u3002</p> <p>\u4e0b\u9762\u4ee5\u5e7e\u7a2e\u5e38\u898b\u7684\u7a0b\u5f0f\u8a9e\u8a00\u70ba\u4f8b\u4f86\u8aaa\u660e\u3002</p> <ul> <li>C \u8a9e\u8a00\u662f\u7a0b\u5e8f\u5f0f\u7a0b\u5f0f\u8a2d\u8a08\u8a9e\u8a00\uff0c\u6c92\u6709\u7269\u4ef6\u5c0e\u5411\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u53ea\u6709\u51fd\u5f0f\u3002\u4f46\u6211\u5011\u53ef\u4ee5\u900f\u904e\u5efa\u7acb\u7d50\u69cb\u9ad4\uff08struct\uff09\u4f86\u6a21\u64ec\u7269\u4ef6\u5c0e\u5411\u7a0b\u5f0f\u8a2d\u8a08\uff0c\u8207\u7d50\u69cb\u9ad4\u76f8\u95dc\u806f\u7684\u51fd\u5f0f\u5c31\u76f8\u7576\u65bc\u5176\u4ed6\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u7684\u65b9\u6cd5\u3002</li> <li>Java \u548c C# \u662f\u7269\u4ef6\u5c0e\u5411\u7684\u7a0b\u5f0f\u8a9e\u8a00\uff0c\u7a0b\u5f0f\u78bc\u584a\uff08\u65b9\u6cd5\uff09\u901a\u5e38\u4f5c\u70ba\u67d0\u500b\u985e\u5225\u7684\u4e00\u90e8\u5206\u3002\u975c\u614b\u65b9\u6cd5\u7684\u884c\u70ba\u985e\u4f3c\u65bc\u51fd\u5f0f\uff0c\u56e0\u70ba\u5b83\u88ab\u7e6b\u7d50\u5728\u985e\u5225\u4e0a\uff0c\u4e0d\u80fd\u8a2a\u554f\u7279\u5b9a\u7684\u4f8b\u9805\u8b8a\u6578\u3002</li> <li>C++ \u548c Python \u65e2\u652f\u6301\u7a0b\u5e8f\u5f0f\u7a0b\u5f0f\u8a2d\u8a08\uff08\u51fd\u5f0f\uff09\uff0c\u4e5f\u652f\u6301\u7269\u4ef6\u5c0e\u5411\u7a0b\u5f0f\u8a2d\u8a08\uff08\u65b9\u6cd5\uff09\u3002</li> </ul> <p>Q\uff1a\u5716\u89e3\u201c\u5e38\u898b\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u578b\u5225\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u4f54\u7528\u7a7a\u9593\u7684\u7d55\u5c0d\u5927\u5c0f\uff1f</p> <p>\u4e0d\u662f\uff0c\u8a72\u5716\u5c55\u793a\u7684\u662f\u7a7a\u9593\u8907\u96dc\u5ea6\uff0c\u5176\u53cd\u6620\u7684\u662f\u589e\u9577\u8da8\u52e2\uff0c\u800c\u4e0d\u662f\u4f54\u7528\u7a7a\u9593\u7684\u7d55\u5c0d\u5927\u5c0f\u3002</p> <p>\u5047\u8a2d\u53d6 \\(n = 8\\) \uff0c\u4f60\u53ef\u80fd\u6703\u767c\u73fe\u6bcf\u689d\u66f2\u7dda\u7684\u503c\u8207\u51fd\u5f0f\u5c0d\u61c9\u4e0d\u4e0a\u3002\u9019\u662f\u56e0\u70ba\u6bcf\u689d\u66f2\u7dda\u90fd\u5305\u542b\u4e00\u500b\u5e38\u6578\u9805\uff0c\u7528\u65bc\u5c07\u53d6\u503c\u7bc4\u570d\u58d3\u7e2e\u5230\u4e00\u500b\u8996\u89ba\u8212\u9069\u7684\u7bc4\u570d\u5167\u3002</p> <p>\u5728\u5be6\u969b\u4e2d\uff0c\u56e0\u70ba\u6211\u5011\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u500b\u65b9\u6cd5\u7684\u201c\u5e38\u6578\u9805\u201d\u8907\u96dc\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u7121\u6cd5\u50c5\u6191\u8907\u96dc\u5ea6\u4f86\u9078\u64c7 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u512a\u89e3\u6cd5\u3002\u4f46\u5c0d\u65bc \\(n = 8^5\\) \u5c31\u5f88\u597d\u9078\u4e86\uff0c\u9019\u6642\u589e\u9577\u8da8\u52e2\u5df2\u7d93\u4f54\u4e3b\u5c0e\u4e86\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 \u6642\u9593\u8907\u96dc\u5ea6","text":"<p>\u57f7\u884c\u6642\u9593\u53ef\u4ee5\u76f4\u89c0\u4e14\u6e96\u78ba\u5730\u53cd\u6620\u6f14\u7b97\u6cd5\u7684\u6548\u7387\u3002\u5982\u679c\u6211\u5011\u60f3\u6e96\u78ba\u9810\u4f30\u4e00\u6bb5\u7a0b\u5f0f\u78bc\u7684\u57f7\u884c\u6642\u9593\uff0c\u61c9\u8a72\u5982\u4f55\u64cd\u4f5c\u5462\uff1f</p> <ol> <li>\u78ba\u5b9a\u57f7\u884c\u5e73\u81fa\uff0c\u5305\u62ec\u786c\u9ad4\u914d\u7f6e\u3001\u7a0b\u5f0f\u8a9e\u8a00\u3001\u7cfb\u7d71\u74b0\u5883\u7b49\uff0c\u9019\u4e9b\u56e0\u7d20\u90fd\u6703\u5f71\u97ff\u7a0b\u5f0f\u78bc\u7684\u57f7\u884c\u6548\u7387\u3002</li> <li>\u8a55\u4f30\u5404\u7a2e\u8a08\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u57f7\u884c\u6642\u9593\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c <code>+</code> \u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c <code>*</code> \u9700\u8981 10 ns \uff0c\u5217\u5370\u64cd\u4f5c <code>print()</code> \u9700\u8981 5 ns \u7b49\u3002</li> <li>\u7d71\u8a08\u7a0b\u5f0f\u78bc\u4e2d\u6240\u6709\u7684\u8a08\u7b97\u64cd\u4f5c\uff0c\u4e26\u5c07\u6240\u6709\u64cd\u4f5c\u7684\u57f7\u884c\u6642\u9593\u6c42\u548c\uff0c\u5f9e\u800c\u5f97\u5230\u57f7\u884c\u6642\u9593\u3002</li> </ol> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # \u8ff4\u5708 n \u6b21\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // \u8ff4\u5708 n \u6b21\n  for (int i = 0; i &lt; n; i++) { // 1 ns\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for _ in 0..n {     // 1 ns\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // 1 ns\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfun algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (i in 0..&lt;n) {  // 1 ns\n        println(0)      // 5 ns\n    }\n}\n</code></pre> <pre><code># \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\ndef algorithm(n)\n    a = 2       # 1 ns\n    a = a + 1   # 1 ns\n    a = a * 2   # 10 ns\n    # \u8ff4\u5708 n \u6b21\n    (0...n).each do # 1 ns\n        puts 0      # 5 ns\n    end\nend\n</code></pre> <pre><code>// \u5728\u67d0\u57f7\u884c\u5e73\u81fa\u4e0b\nfn algorithm(n: usize) void {\n    var a: i32 = 2; // 1 ns\n    a += 1; // 1 ns\n    a *= 2; // 10 ns\n    // \u8ff4\u5708 n \u6b21\n    for (0..n) |_| { // 1 ns\n        std.debug.print(\"{}\\n\", .{0}); // 5 ns\n    }\n}\n</code></pre> <p>\u6839\u64da\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u6f14\u7b97\u6cd5\u7684\u57f7\u884c\u6642\u9593\u70ba \\((6n + 12)\\) ns \uff1a</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>\u4f46\u5be6\u969b\u4e0a\uff0c\u7d71\u8a08\u6f14\u7b97\u6cd5\u7684\u57f7\u884c\u6642\u9593\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73fe\u5be6\u3002\u9996\u5148\uff0c\u6211\u5011\u4e0d\u5e0c\u671b\u5c07\u9810\u4f30\u6642\u9593\u548c\u57f7\u884c\u5e73\u81fa\u7e6b\u7d50\uff0c\u56e0\u70ba\u6f14\u7b97\u6cd5\u9700\u8981\u5728\u5404\u7a2e\u4e0d\u540c\u7684\u5e73\u81fa\u4e0a\u57f7\u884c\u3002\u5176\u6b21\uff0c\u6211\u5011\u5f88\u96e3\u7372\u77e5\u6bcf\u7a2e\u64cd\u4f5c\u7684\u57f7\u884c\u6642\u9593\uff0c\u9019\u7d66\u9810\u4f30\u904e\u7a0b\u5e36\u4f86\u4e86\u6975\u5927\u7684\u96e3\u5ea6\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#231","title":"2.3.1 \u00a0 \u7d71\u8a08\u6642\u9593\u589e\u9577\u8da8\u52e2","text":"<p>\u6642\u9593\u8907\u96dc\u5ea6\u5206\u6790\u7d71\u8a08\u7684\u4e0d\u662f\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\uff0c\u800c\u662f\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u96a8\u8457\u8cc7\u6599\u91cf\u8b8a\u5927\u6642\u7684\u589e\u9577\u8da8\u52e2\u3002</p> <p>\u201c\u6642\u9593\u589e\u9577\u8da8\u52e2\u201d\u9019\u500b\u6982\u5ff5\u6bd4\u8f03\u62bd\u8c61\uff0c\u6211\u5011\u900f\u904e\u4e00\u500b\u4f8b\u5b50\u4f86\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8a2d\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff0c\u7d66\u5b9a\u4e09\u500b\u6f14\u7b97\u6cd5 <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\ndef algorithm_A(n: int):\n    print(0)\n# \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithmA(int n) {\n  print(0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfun algoritm_A(n: Int) {\n    println(0)\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfun algorithm_B(n: Int) {\n    for (i in 0..&lt;n){\n        println(0)\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfun algorithm_C(n: Int) {\n    for (i in 0..&lt;1000000) {\n        println(0)\n    }\n}\n</code></pre> <pre><code># \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\ndef algorithm_A(n)\n    puts 0\nend\n\n# \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\ndef algorithm_B(n)\n    (0...n).each { puts 0 }\nend\n\n# \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\ndef algorithm_C(n)\n    (0...1_000_000).each { puts 0 }\nend\n</code></pre> <pre><code>// \u6f14\u7b97\u6cd5 A \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfn algorithm_A(n: usize) void {\n    _ = n;\n    std.debug.print(\"{}\\n\", .{0});\n}\n// \u6f14\u7b97\u6cd5 B \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u7dda\u6027\u968e\nfn algorithm_B(n: i32) void {\n    for (0..n) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n// \u6f14\u7b97\u6cd5 C \u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u5e38\u6578\u968e\nfn algorithm_C(n: i32) void {\n    _ = n;\n    for (0..1000000) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n</code></pre> <p>\u5716 2-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4e09\u500b\u6f14\u7b97\u6cd5\u51fd\u5f0f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <ul> <li>\u6f14\u7b97\u6cd5 <code>A</code> \u53ea\u6709 \\(1\\) \u500b\u5217\u5370\u64cd\u4f5c\uff0c\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u4e0d\u96a8\u8457 \\(n\\) \u589e\u5927\u800c\u589e\u9577\u3002\u6211\u5011\u7a31\u6b64\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba\u201c\u5e38\u6578\u968e\u201d\u3002</li> <li>\u6f14\u7b97\u6cd5 <code>B</code> \u4e2d\u7684\u5217\u5370\u64cd\u4f5c\u9700\u8981\u8ff4\u5708 \\(n\\) \u6b21\uff0c\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u96a8\u8457 \\(n\\) \u589e\u5927\u5448\u7dda\u6027\u589e\u9577\u3002\u6b64\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u88ab\u7a31\u70ba\u201c\u7dda\u6027\u968e\u201d\u3002</li> <li>\u6f14\u7b97\u6cd5 <code>C</code> \u4e2d\u7684\u5217\u5370\u64cd\u4f5c\u9700\u8981\u8ff4\u5708 \\(1000000\\) \u6b21\uff0c\u96d6\u7136\u57f7\u884c\u6642\u9593\u5f88\u9577\uff0c\u4f46\u5b83\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7121\u95dc\u3002\u56e0\u6b64 <code>C</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u548c <code>A</code> \u76f8\u540c\uff0c\u4ecd\u70ba\u201c\u5e38\u6578\u968e\u201d\u3002</li> </ul> <p></p> <p> \u5716 2-7 \u00a0 \u6f14\u7b97\u6cd5 A\u3001B \u548c C \u7684\u6642\u9593\u589e\u9577\u8da8\u52e2 </p> <p>\u76f8\u8f03\u65bc\u76f4\u63a5\u7d71\u8a08\u6f14\u7b97\u6cd5\u7684\u57f7\u884c\u6642\u9593\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u7279\u9ede\u5462\uff1f</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u80fd\u5920\u6709\u6548\u8a55\u4f30\u6f14\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u6f14\u7b97\u6cd5 <code>B</code> \u7684\u57f7\u884c\u6642\u9593\u5448\u7dda\u6027\u589e\u9577\uff0c\u5728 \\(n &gt; 1\\) \u6642\u6bd4\u6f14\u7b97\u6cd5 <code>A</code> \u66f4\u6162\uff0c\u5728 \\(n &gt; 1000000\\) \u6642\u6bd4\u6f14\u7b97\u6cd5 <code>C</code> \u66f4\u6162\u3002\u4e8b\u5be6\u4e0a\uff0c\u53ea\u8981\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u8db3\u5920\u5927\uff0c\u8907\u96dc\u5ea6\u70ba\u201c\u5e38\u6578\u968e\u201d\u7684\u6f14\u7b97\u6cd5\u4e00\u5b9a\u512a\u65bc\u201c\u7dda\u6027\u968e\u201d\u7684\u6f14\u7b97\u6cd5\uff0c\u9019\u6b63\u662f\u6642\u9593\u589e\u9577\u8da8\u52e2\u7684\u542b\u7fa9\u3002</li> <li>\u6642\u9593\u8907\u96dc\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7c21\u4fbf\u3002\u986f\u7136\uff0c\u57f7\u884c\u5e73\u81fa\u548c\u8a08\u7b97\u64cd\u4f5c\u578b\u5225\u90fd\u8207\u6f14\u7b97\u6cd5\u57f7\u884c\u6642\u9593\u7684\u589e\u9577\u8da8\u52e2\u7121\u95dc\u3002\u56e0\u6b64\u5728\u6642\u9593\u8907\u96dc\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u5011\u53ef\u4ee5\u7c21\u55ae\u5730\u5c07\u6240\u6709\u8a08\u7b97\u64cd\u4f5c\u7684\u57f7\u884c\u6642\u9593\u8996\u70ba\u76f8\u540c\u7684\u201c\u55ae\u4f4d\u6642\u9593\u201d\uff0c\u5f9e\u800c\u5c07\u201c\u8a08\u7b97\u64cd\u4f5c\u57f7\u884c\u6642\u9593\u7d71\u8a08\u201d\u7c21\u5316\u70ba\u201c\u8a08\u7b97\u64cd\u4f5c\u6578\u91cf\u7d71\u8a08\u201d\uff0c\u9019\u6a23\u4e00\u4f86\u4f30\u7b97\u96e3\u5ea6\u5c31\u5927\u5927\u964d\u4f4e\u4e86\u3002</li> <li>\u6642\u9593\u8907\u96dc\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u4fb7\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5118\u7ba1\u6f14\u7b97\u6cd5 <code>A</code> \u548c <code>C</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u76f8\u540c\uff0c\u4f46\u5be6\u969b\u57f7\u884c\u6642\u9593\u5dee\u5225\u5f88\u5927\u3002\u540c\u6a23\uff0c\u5118\u7ba1\u6f14\u7b97\u6cd5 <code>B</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u6bd4 <code>C</code> \u9ad8\uff0c\u4f46\u5728\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u8f03\u5c0f\u6642\uff0c\u6f14\u7b97\u6cd5 <code>B</code> \u660e\u986f\u512a\u65bc\u6f14\u7b97\u6cd5 <code>C</code> \u3002\u5c0d\u65bc\u6b64\u985e\u60c5\u6cc1\uff0c\u6211\u5011\u6642\u5e38\u96e3\u4ee5\u50c5\u6191\u6642\u9593\u8907\u96dc\u5ea6\u5224\u65b7\u6f14\u7b97\u6cd5\u6548\u7387\u7684\u9ad8\u4f4e\u3002\u7576\u7136\uff0c\u5118\u7ba1\u5b58\u5728\u4e0a\u8ff0\u554f\u984c\uff0c\u8907\u96dc\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8a55\u5224\u6f14\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002</li> </ul>"},{"location":"chapter_computational_complexity/time_complexity/#232","title":"2.3.2 \u00a0 \u51fd\u5f0f\u6f38\u8fd1\u4e0a\u754c","text":"<p>\u7d66\u5b9a\u4e00\u500b\u8f38\u5165\u5927\u5c0f\u70ba \\(n\\) \u7684\u51fd\u5f0f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # \u8ff4\u5708 n \u6b21\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // \u8ff4\u5708 n \u6b21\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u8ff4\u5708 n \u6b21\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u8ff4\u5708 n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u8ff4\u5708 n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // \u8ff4\u5708 n \u6b21\n  for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // \u8ff4\u5708 n \u6b21\n    for _ in 0..n { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u8ff4\u5708 n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        printf(\"%d\", 0);            // +1\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u8ff4\u5708 n \u6b21\n    for (i in 0..&lt;n) { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        println(0) // +1\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +1\n    a = a + 1   # +1\n    a = a * 2   # +1\n    # \u8ff4\u5708 n \u6b21\n    (0...n).each do # +1\n        puts 0      # +1\n    end\nend\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u8ff4\u5708 n \u6b21\n    for (0..n) |_| { // +1\uff08\u6bcf\u8f2a\u90fd\u57f7\u884c i ++\uff09\n        std.debug.print(\"{}\\n\", .{0}); // +1\n    }\n}\n</code></pre> <p>\u8a2d\u6f14\u7b97\u6cd5\u7684\u64cd\u4f5c\u6578\u91cf\u662f\u4e00\u500b\u95dc\u65bc\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7684\u51fd\u5f0f\uff0c\u8a18\u70ba \\(T(n)\\) \uff0c\u5247\u4ee5\u4e0a\u51fd\u5f0f\u7684\u64cd\u4f5c\u6578\u91cf\u70ba\uff1a</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u5f0f\uff0c\u8aaa\u660e\u5176\u57f7\u884c\u6642\u9593\u7684\u589e\u9577\u8da8\u52e2\u662f\u7dda\u6027\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f\u7dda\u6027\u968e\u3002</p> <p>\u6211\u5011\u5c07\u7dda\u6027\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6\u8a18\u70ba \\(O(n)\\) \uff0c\u9019\u500b\u6578\u5b78\u7b26\u865f\u7a31\u70ba\u5927 \\(O\\) \u8a18\u865f\uff08big-\\(O\\) notation\uff09\uff0c\u8868\u793a\u51fd\u5f0f \\(T(n)\\) \u7684\u6f38\u8fd1\u4e0a\u754c\uff08asymptotic upper bound\uff09\u3002</p> <p>\u6642\u9593\u8907\u96dc\u5ea6\u5206\u6790\u672c\u8cea\u4e0a\u662f\u8a08\u7b97\u201c\u64cd\u4f5c\u6578\u91cf \\(T(n)\\)\u201d\u7684\u6f38\u8fd1\u4e0a\u754c\uff0c\u5b83\u5177\u6709\u660e\u78ba\u7684\u6578\u5b78\u5b9a\u7fa9\u3002</p> <p>\u51fd\u5f0f\u6f38\u8fd1\u4e0a\u754c</p> <p>\u82e5\u5b58\u5728\u6b63\u5be6\u6578 \\(c\\) \u548c\u5be6\u6578 \\(n_0\\) \uff0c\u4f7f\u5f97\u5c0d\u65bc\u6240\u6709\u7684 \\(n &gt; n_0\\) \uff0c\u5747\u6709 \\(T(n) \\leq c \\cdot f(n)\\) \uff0c\u5247\u53ef\u8a8d\u70ba \\(f(n)\\) \u7d66\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u500b\u6f38\u8fd1\u4e0a\u754c\uff0c\u8a18\u70ba \\(T(n) = O(f(n))\\) \u3002</p> <p>\u5982\u5716 2-8 \u6240\u793a\uff0c\u8a08\u7b97\u6f38\u8fd1\u4e0a\u754c\u5c31\u662f\u5c0b\u627e\u4e00\u500b\u51fd\u5f0f \\(f(n)\\) \uff0c\u4f7f\u5f97\u7576 \\(n\\) \u8da8\u5411\u65bc\u7121\u7aae\u5927\u6642\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u8655\u65bc\u76f8\u540c\u7684\u589e\u9577\u7d1a\u5225\uff0c\u50c5\u76f8\u5dee\u4e00\u500b\u5e38\u6578\u9805 \\(c\\) \u7684\u500d\u6578\u3002</p> <p></p> <p> \u5716 2-8 \u00a0 \u51fd\u5f0f\u7684\u6f38\u8fd1\u4e0a\u754c </p>"},{"location":"chapter_computational_complexity/time_complexity/#233","title":"2.3.3 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u6f38\u8fd1\u4e0a\u754c\u7684\u6578\u5b78\u5473\u5152\u6709\u9ede\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89ba\u6c92\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u7121\u9808\u64d4\u5fc3\u3002\u6211\u5011\u53ef\u4ee5\u5148\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u5728\u4e0d\u65b7\u7684\u5be6\u8e10\u4e2d\uff0c\u5c31\u53ef\u4ee5\u9010\u6f38\u9818\u609f\u5176\u6578\u5b78\u610f\u7fa9\u3002</p> <p>\u6839\u64da\u5b9a\u7fa9\uff0c\u78ba\u5b9a \\(f(n)\\) \u4e4b\u5f8c\uff0c\u6211\u5011\u4fbf\u53ef\u5f97\u5230\u6642\u9593\u8907\u96dc\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u9ebc\u5982\u4f55\u78ba\u5b9a\u6f38\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u7e3d\u9ad4\u5206\u70ba\u5169\u6b65\uff1a\u9996\u5148\u7d71\u8a08\u64cd\u4f5c\u6578\u91cf\uff0c\u7136\u5f8c\u5224\u65b7\u6f38\u8fd1\u4e0a\u754c\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1. \u00a0 \u7b2c\u4e00\u6b65\uff1a\u7d71\u8a08\u64cd\u4f5c\u6578\u91cf","text":"<p>\u91dd\u5c0d\u7a0b\u5f0f\u78bc\uff0c\u9010\u884c\u5f9e\u4e0a\u5230\u4e0b\u8a08\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u65bc\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6578\u9805 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6578\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u7a2e\u4fc2\u6578\u3001\u5e38\u6578\u9805\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002\u6839\u64da\u6b64\u539f\u5247\uff0c\u53ef\u4ee5\u7e3d\u7d50\u51fa\u4ee5\u4e0b\u8a08\u6578\u7c21\u5316\u6280\u5de7\u3002</p> <ol> <li>\u5ffd\u7565 \\(T(n)\\) \u4e2d\u7684\u5e38\u6578\u9805\u3002\u56e0\u70ba\u5b83\u5011\u90fd\u8207 \\(n\\) \u7121\u95dc\uff0c\u6240\u4ee5\u5c0d\u6642\u9593\u8907\u96dc\u5ea6\u4e0d\u7522\u751f\u5f71\u97ff\u3002</li> <li>\u7701\u7565\u6240\u6709\u4fc2\u6578\u3002\u4f8b\u5982\uff0c\u8ff4\u5708 \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7c21\u5316\u8a18\u70ba \\(n\\) \u6b21\uff0c\u56e0\u70ba \\(n\\) \u524d\u9762\u7684\u4fc2\u6578\u5c0d\u6642\u9593\u8907\u96dc\u5ea6\u6c92\u6709\u5f71\u97ff\u3002</li> <li>\u8ff4\u5708\u5de2\u72c0\u6642\u4f7f\u7528\u4e58\u6cd5\u3002\u7e3d\u64cd\u4f5c\u6578\u91cf\u7b49\u65bc\u5916\u5c64\u8ff4\u5708\u548c\u5167\u5c64\u8ff4\u5708\u64cd\u4f5c\u6578\u91cf\u4e4b\u7a4d\uff0c\u6bcf\u4e00\u5c64\u8ff4\u5708\u4f9d\u7136\u53ef\u4ee5\u5206\u5225\u5957\u7528\u7b2c <code>1.</code> \u9ede\u548c\u7b2c <code>2.</code> \u9ede\u7684\u6280\u5de7\u3002</li> </ol> <p>\u7d66\u5b9a\u4e00\u500b\u51fd\u5f0f\uff0c\u6211\u5011\u53ef\u4ee5\u7528\u4e0a\u8ff0\u6280\u5de7\u4f86\u7d71\u8a08\u64cd\u4f5c\u6578\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 1      # +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  # +0\uff08\u6280\u5de7 1\uff09\n    # +n\uff08\u6280\u5de7 2\uff09\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n\uff08\u6280\u5de7 3\uff09\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n  a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n  // +n\uff08\u6280\u5de7 2\uff09\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n\uff08\u6280\u5de7 3\uff09\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1   // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n   // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (i in 0..&lt;5 * n + 1) {\n        println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (i in 0..&lt;2 * n) {\n        for (j in 0..&lt;n + 1) {\n            println(0)\n        }\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +0\uff08\u6280\u5de7 1\uff09\n    a = a + n   # +0\uff08\u6280\u5de7 1\uff09\n    # +n\uff08\u6280\u5de7 2\uff09\n    (0...(5 * n + 1)).each do { puts 0 }\n    # +n*n\uff08\u6280\u5de7 3\uff09\n    (0...(2 * n)).each do\n        (0...(n + 1)).each do { puts 0 }\n    end\nend\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + @as(i32, @intCast(n));        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for(0..(5 * n + 1)) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for(0..(2 * n)) |_| {\n        for(0..(n + 1)) |_| {\n            std.debug.print(\"{}\\n\", .{0});\n        }\n    }\n}\n</code></pre> <p>\u4ee5\u4e0b\u516c\u5f0f\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u5f8c\u7684\u7d71\u8a08\u7d50\u679c\uff0c\u5169\u8005\u63a8\u7b97\u51fa\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n^2)\\) \u3002</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{\u5b8c\u6574\u7d71\u8a08 (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{\u5077\u61f6\u7d71\u8a08 (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2. \u00a0 \u7b2c\u4e8c\u6b65\uff1a\u5224\u65b7\u6f38\u8fd1\u4e0a\u754c","text":"<p>\u6642\u9593\u8907\u96dc\u5ea6\u7531 \\(T(n)\\) \u4e2d\u6700\u9ad8\u968e\u7684\u9805\u4f86\u6c7a\u5b9a\u3002\u9019\u662f\u56e0\u70ba\u5728 \\(n\\) \u8da8\u65bc\u7121\u7aae\u5927\u6642\uff0c\u6700\u9ad8\u968e\u7684\u9805\u5c07\u767c\u63ee\u4e3b\u5c0e\u4f5c\u7528\uff0c\u5176\u4ed6\u9805\u7684\u5f71\u97ff\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002</p> <p>\u8868 2-2 \u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u8a87\u5f35\u7684\u503c\u662f\u70ba\u4e86\u5f37\u8abf\u201c\u4fc2\u6578\u7121\u6cd5\u64bc\u52d5\u968e\u6578\u201d\u9019\u4e00\u7d50\u8ad6\u3002\u7576 \\(n\\) \u8da8\u65bc\u7121\u7aae\u5927\u6642\uff0c\u9019\u4e9b\u5e38\u6578\u8b8a\u5f97\u7121\u8db3\u8f15\u91cd\u3002</p> <p> \u8868 2-2 \u00a0 \u4e0d\u540c\u64cd\u4f5c\u6578\u91cf\u5c0d\u61c9\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> \u64cd\u4f5c\u6578\u91cf \\(T(n)\\) \u6642\u9593\u8907\u96dc\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234","title":"2.3.4 \u00a0 \u5e38\u898b\u578b\u5225","text":"<p>\u8a2d\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff0c\u5e38\u898b\u7684\u6642\u9593\u8907\u96dc\u5ea6\u578b\u5225\u5982\u5716 2-9 \u6240\u793a\uff08\u6309\u7167\u5f9e\u4f4e\u5230\u9ad8\u7684\u9806\u5e8f\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline \\text{\u5e38\u6578\u968e} &lt; \\text{\u5c0d\u6578\u968e} &lt; \\text{\u7dda\u6027\u968e} &lt; \\text{\u7dda\u6027\u5c0d\u6578\u968e} &lt; \\text{\u5e73\u65b9\u968e} &lt; \\text{\u6307\u6578\u968e} &lt; \\text{\u968e\u4e58\u968e} \\end{aligned} \\] <p></p> <p> \u5716 2-9 \u00a0 \u5e38\u898b\u7684\u6642\u9593\u8907\u96dc\u5ea6\u578b\u5225 </p>"},{"location":"chapter_computational_complexity/time_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6578\u968e \\(O(1)\\)","text":"<p>\u5e38\u6578\u968e\u7684\u64cd\u4f5c\u6578\u91cf\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7121\u95dc\uff0c\u5373\u4e0d\u96a8\u8457 \\(n\\) \u7684\u8b8a\u5316\u800c\u8b8a\u5316\u3002</p> <p>\u5728\u4ee5\u4e0b\u51fd\u5f0f\u4e2d\uff0c\u5118\u7ba1\u64cd\u4f5c\u6578\u91cf <code>size</code> \u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u65bc\u5176\u8207\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u7121\u95dc\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u4ecd\u70ba \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"\u5e38\u6578\u968e\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e38\u6578\u968e */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e38\u6578\u968e */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e38\u6578\u968e */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e38\u6578\u968e */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e38\u6578\u968e */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e38\u6578\u968e */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e38\u6578\u968e */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e38\u6578\u968e */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e38\u6578\u968e */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e38\u6578\u968e */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    int i = 0;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e38\u6578\u968e */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 100000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u5e38\u6578\u968e ###\ndef constant(n)\n  count = 0\n  size = 100000\n\n  (0...size).each { count += 1 }\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u5e38\u6578\u968e\nfn constant(n: i32) i32 {\n    _ = n;\n    var count: i32 = 0;\n    const size: i32 = 100_000;\n    var i: i32 = 0;\n    while(i&lt;size) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#2-on","title":"2. \u00a0 \u7dda\u6027\u968e \\(O(n)\\)","text":"<p>\u7dda\u6027\u968e\u7684\u64cd\u4f5c\u6578\u91cf\u76f8\u5c0d\u65bc\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u4ee5\u7dda\u6027\u7d1a\u5225\u589e\u9577\u3002\u7dda\u6027\u968e\u901a\u5e38\u51fa\u73fe\u5728\u55ae\u5c64\u8ff4\u5708\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"\u7dda\u6027\u968e\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7dda\u6027\u968e */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7dda\u6027\u968e */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7dda\u6027\u968e */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7dda\u6027\u968e */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7dda\u6027\u968e */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7dda\u6027\u968e */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7dda\u6027\u968e */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7dda\u6027\u968e */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7dda\u6027\u968e */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7dda\u6027\u968e */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7dda\u6027\u968e */\nfun linear(n: Int): Int {\n    var count = 0\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u7dda\u6027\u968e ###\ndef linear(n)\n  count = 0\n  (0...n).each { count += 1 }\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u7dda\u6027\u968e\nfn linear(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u8d70\u8a2a\u9663\u5217\u548c\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u7b49\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u70ba\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u7684\u9577\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"\u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09\"\"\"\n    count = 0\n    # \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09 */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09###\ndef array_traversal(nums)\n  count = 0\n\n  # \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n  for num in nums\n    count += 1\n  end\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u7dda\u6027\u968e\uff08\u8d70\u8a2a\u9663\u5217\uff09\nfn arrayTraversal(nums: []i32) i32 {\n    var count: i32 = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u9663\u5217\u9577\u5ea6\u6210\u6b63\u6bd4\n    for (nums) |_| {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u9700\u6839\u64da\u8f38\u5165\u8cc7\u6599\u7684\u578b\u5225\u4f86\u5177\u9ad4\u78ba\u5b9a\u3002\u6bd4\u5982\u5728\u7b2c\u4e00\u500b\u793a\u4f8b\u4e2d\uff0c\u8b8a\u6578 \\(n\\) \u70ba\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\uff1b\u5728\u7b2c\u4e8c\u500b\u793a\u4f8b\u4e2d\uff0c\u9663\u5217\u9577\u5ea6 \\(n\\) \u70ba\u8cc7\u6599\u5927\u5c0f\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u968e \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u968e\u7684\u64cd\u4f5c\u6578\u91cf\u76f8\u5c0d\u65bc\u8f38\u5165\u8cc7\u6599\u5927\u5c0f \\(n\\) \u4ee5\u5e73\u65b9\u7d1a\u5225\u589e\u9577\u3002\u5e73\u65b9\u968e\u901a\u5e38\u51fa\u73fe\u5728\u5de2\u72c0\u8ff4\u5708\u4e2d\uff0c\u5916\u5c64\u8ff4\u5708\u548c\u5167\u5c64\u8ff4\u5708\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n)\\) \uff0c\u56e0\u6b64\u7e3d\u9ad4\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u968e\"\"\"\n    count = 0\n    # \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u968e */\nint quadratic(int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u968e */\nint quadratic(int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u968e */\nint Quadratic(int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u968e */\nfunc quadratic(n int) int {\n    count := 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u968e */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u968e */\nfunction quadratic(n) {\n    let count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u968e */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u968e */\nint quadratic(int n) {\n  int count = 0;\n  // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u968e */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u968e */\nint quadratic(int n) {\n    int count = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u968e */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u5e73\u65b9\u968e ###\ndef quadratic(n)\n  count = 0\n\n  # \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n  for i in 0...n\n    for j in 0...n\n      count += 1\n    end\n  end\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u968e\nfn quadratic(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u8ff4\u5708\u6b21\u6578\u8207\u8cc7\u6599\u5927\u5c0f n \u6210\u5e73\u65b9\u95dc\u4fc2\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            count += 1;\n        }\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-10 \u5c0d\u6bd4\u4e86\u5e38\u6578\u968e\u3001\u7dda\u6027\u968e\u548c\u5e73\u65b9\u968e\u4e09\u7a2e\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <p></p> <p> \u5716 2-10 \u00a0 \u5e38\u6578\u968e\u3001\u7dda\u6027\u968e\u548c\u5e73\u65b9\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u4ee5\u6ce1\u6cab\u6392\u5e8f\u70ba\u4f8b\uff0c\u5916\u5c64\u8ff4\u5708\u57f7\u884c \\(n - 1\\) \u6b21\uff0c\u5167\u5c64\u8ff4\u5708\u57f7\u884c \\(n-1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u5e73\u5747\u70ba \\(n / 2\\) \u6b21\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O((n - 1) n / 2) = O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"\u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09\"\"\"\n    count = 0  # \u8a08\u6578\u5668\n    # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n    int count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n    int count = 0;  // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n    count := 0 // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n    let count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // \u8a08\u6578\u5668\n  // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // \u8a08\u6578\u5668\n\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09 */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0 // \u8a08\u6578\u5668\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                count += 3 // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09###\ndef bubble_sort(nums)\n  count = 0  # \u8a08\u6578\u5668\n\n  # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for i in (nums.length - 1).downto(0)\n    # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        tmp = nums[j]\n        nums[j] = nums[j + 1]\n        nums[j + 1] = tmp\n        count += 3 # \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n      end\n    end\n  end\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u968e\uff08\u6ce1\u6cab\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n    var count: i32 = 0;  // \u8a08\u6578\u5668 \n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3;  // \u5143\u7d20\u4ea4\u63db\u5305\u542b 3 \u500b\u55ae\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6578\u968e \\(O(2^n)\\)","text":"<p>\u751f\u7269\u5b78\u7684\u201c\u7d30\u80de\u5206\u88c2\u201d\u662f\u6307\u6578\u968e\u589e\u9577\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72c0\u614b\u70ba \\(1\\) \u500b\u7d30\u80de\uff0c\u5206\u88c2\u4e00\u8f2a\u5f8c\u8b8a\u70ba \\(2\\) \u500b\uff0c\u5206\u88c2\u5169\u8f2a\u5f8c\u8b8a\u70ba \\(4\\) \u500b\uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f2a\u5f8c\u6709 \\(2^n\\) \u500b\u7d30\u80de\u3002</p> <p>\u5716 2-11 \u548c\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6a21\u64ec\u4e86\u7d30\u80de\u5206\u88c2\u7684\u904e\u7a0b\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^n)\\) \u3002\u8acb\u6ce8\u610f\uff0c\u8f38\u5165 \\(n\\) \u8868\u793a\u5206\u88c2\u8f2a\u6578\uff0c\u8fd4\u56de\u503c <code>count</code> \u8868\u793a\u7e3d\u5206\u88c2\u6b21\u6578\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"\u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09\"\"\"\n    count = 0\n    base = 1\n    # \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfun exponential(n: Int): Int {\n    var count = 0\n    var base = 1\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09###\ndef exponential(n)\n  count, base = 0, 1\n\n  # \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  (0...n).each do\n    (0...base).each { count += 1 }\n    base *= 2\n  end\n\n  # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09\nfn exponential(n: i32) i32 {\n    var count: i32 = 0;\n    var bas: i32 = 1;\n    var i: i32 = 0;\n    // \u7d30\u80de\u6bcf\u8f2a\u4e00\u5206\u70ba\u4e8c\uff0c\u5f62\u6210\u6578\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; bas) : (j += 1) {\n            count += 1;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-11 \u00a0 \u6307\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u5728\u5be6\u969b\u6f14\u7b97\u6cd5\u4e2d\uff0c\u6307\u6578\u968e\u5e38\u51fa\u73fe\u65bc\u905e\u8ff4\u51fd\u5f0f\u4e2d\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u5176\u905e\u8ff4\u5730\u4e00\u5206\u70ba\u4e8c\uff0c\u7d93\u904e \\(n\\) \u6b21\u5206\u88c2\u5f8c\u505c\u6b62\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"\u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef exp_recur(n)\n  return 1 if n == 1\n  exp_recur(n - 1) + exp_recur(n - 1) + 1\nend\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn expRecur(n: i32) i32 {\n    if (n == 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6307\u6578\u968e\u589e\u9577\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u7aae\u8209\u6cd5\uff08\u66b4\u529b\u641c\u5c0b\u3001\u56de\u6eaf\u7b49\uff09\u4e2d\u6bd4\u8f03\u5e38\u898b\u3002\u5c0d\u65bc\u8cc7\u6599\u898f\u6a21\u8f03\u5927\u7684\u554f\u984c\uff0c\u6307\u6578\u968e\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff0c\u901a\u5e38\u9700\u8981\u4f7f\u7528\u52d5\u614b\u898f\u5283\u6216\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7b49\u4f86\u89e3\u6c7a\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","title":"5. \u00a0 \u5c0d\u6578\u968e \\(O(\\log n)\\)","text":"<p>\u8207\u6307\u6578\u968e\u76f8\u53cd\uff0c\u5c0d\u6578\u968e\u53cd\u6620\u4e86\u201c\u6bcf\u8f2a\u7e2e\u6e1b\u5230\u4e00\u534a\u201d\u7684\u60c5\u6cc1\u3002\u8a2d\u8f38\u5165\u8cc7\u6599\u5927\u5c0f\u70ba \\(n\\) \uff0c\u7531\u65bc\u6bcf\u8f2a\u7e2e\u6e1b\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u8ff4\u5708\u6b21\u6578\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u5f0f\u3002</p> <p>\u5716 2-12 \u548c\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6a21\u64ec\u4e86\u201c\u6bcf\u8f2a\u7e2e\u6e1b\u5230\u4e00\u534a\u201d\u7684\u904e\u7a0b\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log_2 n)\\) \uff0c\u7c21\u8a18\u70ba \\(O(\\log n)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"\u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09 */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09###\ndef logarithmic(n)\n  count = 0\n\n  while n &gt; 1\n    n /= 2\n    count += 1\n  end\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u5c0d\u6578\u968e\uff08\u8ff4\u5708\u5be6\u73fe\uff09\nfn logarithmic(n: i32) i32 {\n    var count: i32 = 0;\n    var n_var = n;\n    while (n_var &gt; 1)\n    {\n        n_var = n_var / 2;\n        count +=1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-12 \u00a0 \u5c0d\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u8207\u6307\u6578\u968e\u985e\u4f3c\uff0c\u5c0d\u6578\u968e\u4e5f\u5e38\u51fa\u73fe\u65bc\u905e\u8ff4\u51fd\u5f0f\u4e2d\u3002\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u5f62\u6210\u4e86\u4e00\u68f5\u9ad8\u5ea6\u70ba \\(\\log_2 n\\) \u7684\u905e\u8ff4\u6a39\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"\u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef log_recur(n)\n  return 0 unless n &gt; 1\n  log_recur(n / 2) + 1\nend\n</code></pre> time_complexity.zig<pre><code>// \u5c0d\u6578\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn logRecur(n: i32) i32 {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5c0d\u6578\u968e\u5e38\u51fa\u73fe\u65bc\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6f14\u7b97\u6cd5\u4e2d\uff0c\u9ad4\u73fe\u4e86\u201c\u4e00\u5206\u70ba\u591a\u201d\u548c\u201c\u5316\u7e41\u70ba\u7c21\u201d\u7684\u6f14\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u589e\u9577\u7de9\u6162\uff0c\u662f\u50c5\u6b21\u65bc\u5e38\u6578\u968e\u7684\u7406\u60f3\u7684\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <p>\\(O(\\log n)\\) \u7684\u5e95\u6578\u662f\u591a\u5c11\uff1f</p> <p>\u6e96\u78ba\u4f86\u8aaa\uff0c\u201c\u4e00\u5206\u70ba \\(m\\)\u201d\u5c0d\u61c9\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(\\log_m n)\\) \u3002\u800c\u900f\u904e\u5c0d\u6578\u63db\u5e95\u516c\u5f0f\uff0c\u6211\u5011\u53ef\u4ee5\u5f97\u5230\u5177\u6709\u4e0d\u540c\u5e95\u6578\u3001\u76f8\u7b49\u7684\u6642\u9593\u8907\u96dc\u5ea6\uff1a</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u5e95\u6578 \\(m\\) \u53ef\u4ee5\u5728\u4e0d\u5f71\u97ff\u8907\u96dc\u5ea6\u7684\u524d\u63d0\u4e0b\u8f49\u63db\u3002\u56e0\u6b64\u6211\u5011\u901a\u5e38\u6703\u7701\u7565\u5e95\u6578 \\(m\\) \uff0c\u5c07\u5c0d\u6578\u968e\u76f4\u63a5\u8a18\u70ba \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","title":"6. \u00a0 \u7dda\u6027\u5c0d\u6578\u968e \\(O(n \\log n)\\)","text":"<p>\u7dda\u6027\u5c0d\u6578\u968e\u5e38\u51fa\u73fe\u65bc\u5de2\u72c0\u8ff4\u5708\u4e2d\uff0c\u5169\u5c64\u8ff4\u5708\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5206\u5225\u70ba \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002\u76f8\u95dc\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"\u7dda\u6027\u5c0d\u6578\u968e\"\"\"\n    if n &lt;= 1:\n        return 1\n    # \u4e00\u5206\u70ba\u4e8c\uff0c\u5b50\u554f\u984c\u7684\u898f\u6a21\u6e1b\u5c0f\u4e00\u534a\n    count = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    # \u7576\u524d\u5b50\u554f\u984c\u5305\u542b n \u500b\u64cd\u4f5c\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7dda\u6027\u5c0d\u6578\u968e */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u7dda\u6027\u5c0d\u6578\u968e ###\ndef linear_log_recur(n)\n  return 1 unless n &gt; 1\n\n  count = linear_log_recur(n / 2) + linear_log_recur(n / 2)\n  (0...n).each { count += 1 }\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u7dda\u6027\u5c0d\u6578\u968e\nfn linearLogRecur(n: i32) i32 {\n    if (n &lt;= 1) return 1;\n    var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 2-13 \u5c55\u793a\u4e86\u7dda\u6027\u5c0d\u6578\u968e\u7684\u751f\u6210\u65b9\u5f0f\u3002\u4e8c\u5143\u6a39\u7684\u6bcf\u4e00\u5c64\u7684\u64cd\u4f5c\u7e3d\u6578\u90fd\u70ba \\(n\\) \uff0c\u6a39\u5171\u6709 \\(\\log_2 n + 1\\) \u5c64\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</p> <p></p> <p> \u5716 2-13 \u00a0 \u7dda\u6027\u5c0d\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u4e3b\u6d41\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u70ba \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5408\u4f75\u6392\u5e8f\u3001\u5806\u7a4d\u6392\u5e8f\u7b49\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#7-on","title":"7. \u00a0 \u968e\u4e58\u968e \\(O(n!)\\)","text":"<p>\u968e\u4e58\u968e\u5c0d\u61c9\u6578\u5b78\u4e0a\u7684\u201c\u5168\u6392\u5217\u201d\u554f\u984c\u3002\u7d66\u5b9a \\(n\\) \u500b\u4e92\u4e0d\u91cd\u8907\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6578\u91cf\u70ba\uff1a</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>\u968e\u4e58\u901a\u5e38\u4f7f\u7528\u905e\u8ff4\u5be6\u73fe\u3002\u5982\u5716 2-14 \u548c\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\uff0c\u7b2c\u4e00\u5c64\u5206\u88c2\u51fa \\(n\\) \u500b\uff0c\u7b2c\u4e8c\u5c64\u5206\u88c2\u51fa \\(n - 1\\) \u500b\uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c64\u6642\u505c\u6b62\u5206\u88c2\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"\u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09 */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09###\ndef factorial_recur(n)\n  return 1 if n == 0\n\n  count = 0\n  # \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n  (0...n).each { count += factorial_recur(n - 1) }\n\n  count\nend\n</code></pre> time_complexity.zig<pre><code>// \u968e\u4e58\u968e\uff08\u905e\u8ff4\u5be6\u73fe\uff09\nfn factorialRecur(n: i32) i32 {\n    if (n == 0) return 1;\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u5f9e 1 \u500b\u5206\u88c2\u51fa n \u500b\n    while (i &lt; n) : (i += 1) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 2-14 \u00a0 \u968e\u4e58\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6 </p> <p>\u8acb\u6ce8\u610f\uff0c\u56e0\u70ba\u7576 \\(n \\geq 4\\) \u6642\u6046\u6709 \\(n! &gt; 2^n\\) \uff0c\u6240\u4ee5\u968e\u4e58\u968e\u6bd4\u6307\u6578\u968e\u589e\u9577\u5f97\u66f4\u5feb\uff0c\u5728 \\(n\\) \u8f03\u5927\u6642\u4e5f\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#235","title":"2.3.5 \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6","text":"<p>\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u6548\u7387\u5f80\u5f80\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u8207\u8f38\u5165\u8cc7\u6599\u7684\u5206\u4f48\u6709\u95dc\u3002\u5047\u8a2d\u8f38\u5165\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217 <code>nums</code> \uff0c\u5176\u4e2d <code>nums</code> \u7531\u5f9e \\(1\\) \u81f3 \\(n\\) \u7684\u6578\u5b57\u7d44\u6210\uff0c\u6bcf\u500b\u6578\u5b57\u53ea\u51fa\u73fe\u4e00\u6b21\uff1b\u4f46\u5143\u7d20\u9806\u5e8f\u662f\u96a8\u6a5f\u6253\u4e82\u7684\uff0c\u4efb\u52d9\u76ee\u6a19\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u5011\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7d50\u8ad6\u3002</p> <ul> <li>\u7576 <code>nums = [?, ?, ..., 1]</code> \uff0c\u5373\u7576\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u6642\uff0c\u9700\u8981\u5b8c\u6574\u8d70\u8a2a\u9663\u5217\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\) \u3002</li> <li>\u7576 <code>nums = [1, ?, ?, ...]</code> \uff0c\u5373\u7576\u9996\u500b\u5143\u7d20\u70ba \\(1\\) \u6642\uff0c\u7121\u8ad6\u9663\u5217\u591a\u9577\u90fd\u4e0d\u9700\u8981\u7e7c\u7e8c\u8d70\u8a2a\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 \\(\\Omega(1)\\) \u3002</li> </ul> <p>\u201c\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u201d\u5c0d\u61c9\u51fd\u5f0f\u6f38\u8fd1\u4e0a\u754c\uff0c\u4f7f\u7528\u5927 \\(O\\) \u8a18\u865f\u8868\u793a\u3002\u76f8\u61c9\u5730\uff0c\u201c\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u201d\u5c0d\u61c9\u51fd\u5f0f\u6f38\u8fd1\u4e0b\u754c\uff0c\u7528 \\(\\Omega\\) \u8a18\u865f\u8868\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"\u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba: 1, 2, ..., n \uff0c\u9806\u5e8f\u88ab\u6253\u4e82\"\"\"\n    # \u751f\u6210\u9663\u5217 nums =: 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"\u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n    for i in range(len(nums)):\n        # \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        # \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u4f7f\u7528\u7cfb\u7d71\u6642\u9593\u751f\u6210\u96a8\u6a5f\u7a2e\u5b50\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    nums.shuffle()\n    return nums\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n  nums.shuffle();\n\n  return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n    // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nint *randomNumbers(int n) {\n    // \u5206\u914d\u5806\u7a4d\u5340\u8a18\u61b6\u9ad4\uff08\u5efa\u7acb\u4e00\u7dad\u53ef\u8b8a\u9577\u9663\u5217\uff1a\u9663\u5217\u4e2d\u5143\u7d20\u6578\u91cf\u70ba n \uff0c\u5143\u7d20\u578b\u5225\u70ba int \uff09\n    int *nums = (int *)malloc(n * sizeof(int));\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82 */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    nums.shuffle()\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = nums[i]\n    }\n    return res\n}\n\n/* \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>### \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba: 1, 2, ..., n \uff0c\u9806\u5e8f\u88ab\u6253\u4e82 ###\ndef random_numbers(n)\n  # \u751f\u6210\u9663\u5217 nums =: 1, 2, 3, ..., n\n  nums = Array.new(n) { |i| i + 1 }\n  # \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n  nums.shuffle!\nend\n\n### \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15 ###\ndef find_one(nums)\n  for i in 0...nums.length\n    # \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n    # \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n    return i if nums[i] == 1\n  end\n\n  -1\nend\n</code></pre> worst_best_time_complexity.zig<pre><code>// \u751f\u6210\u4e00\u500b\u9663\u5217\uff0c\u5143\u7d20\u70ba { 1, 2, ..., n }\uff0c\u9806\u5e8f\u88ab\u6253\u4e82\nfn randomNumbers(comptime n: usize) [n]i32 {\n    var nums: [n]i32 = undefined;\n    // \u751f\u6210\u9663\u5217 nums = { 1, 2, 3, ..., n }\n    for (&amp;nums, 0..) |*num, i| {\n        num.* = @as(i32, @intCast(i)) + 1;\n    }\n    // \u96a8\u6a5f\u6253\u4e82\u9663\u5217\u5143\u7d20\n    const rand = std.crypto.random;\n    rand.shuffle(i32, &amp;nums);\n    return nums;\n}\n\n// \u67e5\u8a62\u9663\u5217 nums \u4e2d\u6578\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n    for (nums, 0..) |num, i| {\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u982d\u90e8\u6642\uff0c\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 O(1)\n        // \u7576\u5143\u7d20 1 \u5728\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u9054\u5230\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6 O(n)\n        if (num == 1) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u6211\u5011\u5728\u5be6\u969b\u4e2d\u5f88\u5c11\u4f7f\u7528\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u56e0\u70ba\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6a5f\u7387\u4e0b\u624d\u80fd\u9054\u5230\uff0c\u53ef\u80fd\u6703\u5e36\u4f86\u4e00\u5b9a\u7684\u8aa4\u5c0e\u6027\u3002\u800c\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u66f4\u70ba\u5be6\u7528\uff0c\u56e0\u70ba\u5b83\u7d66\u51fa\u4e86\u4e00\u500b\u6548\u7387\u5b89\u5168\u503c\uff0c\u8b93\u6211\u5011\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u6f14\u7b97\u6cd5\u3002</p> <p>\u5f9e\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u548c\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u53ea\u51fa\u73fe\u65bc\u201c\u7279\u6b8a\u7684\u8cc7\u6599\u5206\u4f48\u201d\uff0c\u9019\u4e9b\u60c5\u6cc1\u7684\u51fa\u73fe\u6a5f\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u4e26\u4e0d\u80fd\u771f\u5be6\u5730\u53cd\u6620\u6f14\u7b97\u6cd5\u57f7\u884c\u6548\u7387\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u9ad4\u73fe\u6f14\u7b97\u6cd5\u5728\u96a8\u6a5f\u8f38\u5165\u8cc7\u6599\u4e0b\u7684\u57f7\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8a18\u865f\u4f86\u8868\u793a\u3002</p> <p>\u5c0d\u65bc\u90e8\u5206\u6f14\u7b97\u6cd5\uff0c\u6211\u5011\u53ef\u4ee5\u7c21\u55ae\u5730\u63a8\u7b97\u51fa\u96a8\u6a5f\u8cc7\u6599\u5206\u4f48\u4e0b\u7684\u5e73\u5747\u60c5\u6cc1\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u65bc\u8f38\u5165\u9663\u5217\u662f\u88ab\u6253\u4e82\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73fe\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6a5f\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u9ebc\u6f14\u7b97\u6cd5\u7684\u5e73\u5747\u8ff4\u5708\u6b21\u6578\u5c31\u662f\u9663\u5217\u9577\u5ea6\u7684\u4e00\u534a \\(n / 2\\) \uff0c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(\\Theta(n / 2) = \\Theta(n)\\) \u3002</p> <p>\u4f46\u5c0d\u65bc\u8f03\u70ba\u8907\u96dc\u7684\u6f14\u7b97\u6cd5\uff0c\u8a08\u7b97\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u5f80\u5f80\u6bd4\u8f03\u56f0\u96e3\uff0c\u56e0\u70ba\u5f88\u96e3\u5206\u6790\u51fa\u5728\u8cc7\u6599\u5206\u4f48\u4e0b\u7684\u6574\u9ad4\u6578\u5b78\u671f\u671b\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u6211\u5011\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u4f5c\u70ba\u6f14\u7b97\u6cd5\u6548\u7387\u7684\u8a55\u5224\u6a19\u6e96\u3002</p> <p>\u70ba\u4ec0\u9ebc\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u865f\uff1f</p> <p>\u53ef\u80fd\u7531\u65bc \\(O\\) \u7b26\u865f\u904e\u65bc\u6717\u6717\u4e0a\u53e3\uff0c\u56e0\u6b64\u6211\u5011\u5e38\u5e38\u4f7f\u7528\u5b83\u4f86\u8868\u793a\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u3002\u4f46\u5f9e\u56b4\u683c\u610f\u7fa9\u4e0a\u8b1b\uff0c\u9019\u7a2e\u505a\u6cd5\u4e26\u4e0d\u898f\u7bc4\u3002\u5728\u672c\u66f8\u548c\u5176\u4ed6\u8cc7\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u985e\u4f3c\u201c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8acb\u5c07\u5176\u76f4\u63a5\u7406\u89e3\u70ba \\(\\Theta(n)\\) \u3002</p>"},{"location":"chapter_data_structure/","title":"\u7b2c 3 \u7ae0 \u00a0 \u8cc7\u6599\u7d50\u69cb","text":"<p>Abstract</p> <p>\u8cc7\u6599\u7d50\u69cb\u5982\u540c\u4e00\u526f\u7a69\u56fa\u800c\u591a\u6a23\u7684\u6846\u67b6\u3002</p> <p>\u5b83\u70ba\u8cc7\u6599\u7684\u6709\u5e8f\u7d44\u7e54\u63d0\u4f9b\u4e86\u85cd\u5716\uff0c\u6f14\u7b97\u6cd5\u5f97\u4ee5\u5728\u6b64\u57fa\u790e\u4e0a\u751f\u52d5\u8d77\u4f86\u3002</p>"},{"location":"chapter_data_structure/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>3.1 \u00a0 \u8cc7\u6599\u7d50\u69cb\u5206\u985e</li> <li>3.2 \u00a0 \u57fa\u672c\u8cc7\u6599\u578b\u5225</li> <li>3.3 \u00a0 \u6578\u5b57\u7de8\u78bc *</li> <li>3.4 \u00a0 \u5b57\u5143\u7de8\u78bc *</li> <li>3.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_data_structure/basic_data_types/","title":"3.2 \u00a0 \u57fa\u672c\u8cc7\u6599\u578b\u5225","text":"<p>\u7576\u8ac7\u53ca\u8a08\u7b97\u6a5f\u4e2d\u7684\u8cc7\u6599\u6642\uff0c\u6211\u5011\u6703\u60f3\u5230\u6587\u5b57\u3001\u5716\u7247\u3001\u5f71\u7247\u3001\u8a9e\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u7a2e\u5f62\u5f0f\u3002\u5118\u7ba1\u9019\u4e9b\u8cc7\u6599\u7684\u7d44\u7e54\u5f62\u5f0f\u5404\u7570\uff0c\u4f46\u5b83\u5011\u90fd\u7531\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u69cb\u6210\u3002</p> <p>\u57fa\u672c\u8cc7\u6599\u578b\u5225\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u9032\u884c\u904b\u7b97\u7684\u578b\u5225\uff0c\u5728\u6f14\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\uff0c\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u5e7e\u7a2e\u3002</p> <ul> <li>\u6574\u6578\u578b\u5225 <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \u3002</li> <li>\u6d6e\u9ede\u6578\u578b\u5225 <code>float</code>\u3001<code>double</code> \uff0c\u7528\u65bc\u8868\u793a\u5c0f\u6578\u3002</li> <li>\u5b57\u5143\u578b\u5225 <code>char</code> \uff0c\u7528\u65bc\u8868\u793a\u5404\u7a2e\u8a9e\u8a00\u7684\u5b57\u6bcd\u3001\u6a19\u9ede\u7b26\u865f\u751a\u81f3\u8868\u60c5\u7b26\u865f\u7b49\u3002</li> <li>\u5e03\u6797\u578b\u5225 <code>bool</code> \uff0c\u7528\u65bc\u8868\u793a\u201c\u662f\u201d\u8207\u201c\u5426\u201d\u5224\u65b7\u3002</li> </ul> <p>\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4ee5\u4e8c\u9032\u4f4d\u5236\u7684\u5f62\u5f0f\u5132\u5b58\u5728\u8a08\u7b97\u6a5f\u4e2d\u3002\u4e00\u500b\u4e8c\u9032\u4f4d\u5236\u4f4d\u5373\u70ba \\(1\\) \u4f4d\u5143\u3002\u5728\u7d55\u5927\u591a\u6578\u73fe\u4ee3\u4f5c\u696d\u7cfb\u7d71\u4e2d\uff0c\\(1\\) \u4f4d\u5143\u7d44\uff08byte\uff09\u7531 \\(8\\) \u4f4d\u5143\uff08bit\uff09\u7d44\u6210\u3002</p> <p>\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u53d6\u503c\u7bc4\u570d\u53d6\u6c7a\u65bc\u5176\u4f54\u7528\u7684\u7a7a\u9593\u5927\u5c0f\u3002\u4e0b\u9762\u4ee5 Java \u70ba\u4f8b\u3002</p> <ul> <li>\u6574\u6578\u578b\u5225 <code>byte</code> \u4f54\u7528 \\(1\\) \u4f4d\u5143\u7d44 = \\(8\\) \u4f4d\u5143 \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u500b\u6578\u5b57\u3002</li> <li>\u6574\u6578\u578b\u5225 <code>int</code> \u4f54\u7528 \\(4\\) \u4f4d\u5143\u7d44 = \\(32\\) \u4f4d\u5143 \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u500b\u6578\u5b57\u3002</li> </ul> <p>\u8868 3-1 \u5217\u8209\u4e86 Java \u4e2d\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u4f54\u7528\u7a7a\u9593\u3001\u53d6\u503c\u7bc4\u570d\u548c\u9810\u8a2d\u503c\u3002\u6b64\u8868\u683c\u7121\u9808\u6b7b\u8a18\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u6642\u53ef\u4ee5\u900f\u904e\u67e5\u8868\u4f86\u56de\u61b6\u3002</p> <p> \u8868 3-1 \u00a0 \u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u4f54\u7528\u7a7a\u9593\u548c\u53d6\u503c\u7bc4\u570d </p> \u578b\u5225 \u7b26\u865f \u4f54\u7528\u7a7a\u9593 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9810\u8a2d\u503c \u6574\u6578 <code>byte</code> 1 \u4f4d\u5143\u7d44 \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2 \u4f4d\u5143\u7d44 \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4 \u4f4d\u5143\u7d44 \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8 \u4f4d\u5143\u7d44 \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u9ede\u6578 <code>float</code> 4 \u4f4d\u5143\u7d44 \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 \u4f4d\u5143\u7d44 \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u5143 <code>char</code> 2 \u4f4d\u5143\u7d44 \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u6797 <code>bool</code> 1 \u4f4d\u5143\u7d44 \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>\u8acb\u6ce8\u610f\uff0c\u8868 3-1 \u91dd\u5c0d\u7684\u662f Java \u7684\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u60c5\u6cc1\u3002\u6bcf\u7a2e\u7a0b\u5f0f\u8a9e\u8a00\u90fd\u6709\u5404\u81ea\u7684\u8cc7\u6599\u578b\u5225\u5b9a\u7fa9\uff0c\u5b83\u5011\u7684\u4f54\u7528\u7a7a\u9593\u3001\u53d6\u503c\u7bc4\u570d\u548c\u9810\u8a2d\u503c\u53ef\u80fd\u6703\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u5728 Python \u4e2d\uff0c\u6574\u6578\u578b\u5225 <code>int</code> \u53ef\u4ee5\u662f\u4efb\u610f\u5927\u5c0f\uff0c\u53ea\u53d7\u9650\u65bc\u53ef\u7528\u8a18\u61b6\u9ad4\uff1b\u6d6e\u9ede\u6578 <code>float</code> \u662f\u96d9\u7cbe\u5ea6 64 \u4f4d\uff1b\u6c92\u6709 <code>char</code> \u578b\u5225\uff0c\u55ae\u500b\u5b57\u5143\u5be6\u969b\u4e0a\u662f\u9577\u5ea6\u70ba 1 \u7684\u5b57\u4e32 <code>str</code> \u3002</li> <li>C \u548c C++ \u672a\u660e\u78ba\u898f\u5b9a\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u5927\u5c0f\uff0c\u800c\u56e0\u5be6\u73fe\u548c\u5e73\u81fa\u5404\u7570\u3002\u8868 3-1 \u9075\u5faa LP64 \u8cc7\u6599\u6a21\u578b\uff0c\u5176\u7528\u65bc\u5305\u62ec Linux \u548c macOS \u5728\u5167\u7684 Unix 64 \u4f4d\u4f5c\u696d\u7cfb\u7d71\u3002</li> <li>\u5b57\u5143 <code>char</code> \u7684\u5927\u5c0f\u5728 C \u548c C++ \u4e2d\u70ba 1 \u4f4d\u5143\u7d44\uff0c\u5728\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u53d6\u6c7a\u65bc\u7279\u5b9a\u7684\u5b57\u5143\u7de8\u78bc\u65b9\u6cd5\uff0c\u8a73\u898b\u201c\u5b57\u5143\u7de8\u78bc\u201d\u7ae0\u7bc0\u3002</li> <li>\u5373\u4f7f\u8868\u793a\u5e03\u6797\u91cf\u50c5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u8a18\u61b6\u9ad4\u4e2d\u901a\u5e38\u4e5f\u5132\u5b58\u70ba 1 \u4f4d\u5143\u7d44\u3002\u9019\u662f\u56e0\u70ba\u73fe\u4ee3\u8a08\u7b97\u6a5f CPU \u901a\u5e38\u5c07 1 \u4f4d\u5143\u7d44\u4f5c\u70ba\u6700\u5c0f\u5b9a\u5740\u8a18\u61b6\u9ad4\u55ae\u5143\u3002</li> </ul> <p>\u90a3\u9ebc\uff0c\u57fa\u672c\u8cc7\u6599\u578b\u5225\u8207\u8cc7\u6599\u7d50\u69cb\u4e4b\u9593\u6709\u4ec0\u9ebc\u95dc\u806f\u5462\uff1f\u6211\u5011\u77e5\u9053\uff0c\u8cc7\u6599\u7d50\u69cb\u662f\u5728\u8a08\u7b97\u6a5f\u4e2d\u7d44\u7e54\u8207\u5132\u5b58\u8cc7\u6599\u7684\u65b9\u5f0f\u3002\u9019\u53e5\u8a71\u7684\u4e3b\u8a9e\u662f\u201c\u7d50\u69cb\u201d\u800c\u975e\u201c\u8cc7\u6599\u201d\u3002</p> <p>\u5982\u679c\u60f3\u8868\u793a\u201c\u4e00\u6392\u6578\u5b57\u201d\uff0c\u6211\u5011\u81ea\u7136\u6703\u60f3\u5230\u4f7f\u7528\u9663\u5217\u3002\u9019\u662f\u56e0\u70ba\u9663\u5217\u7684\u7dda\u6027\u7d50\u69cb\u53ef\u4ee5\u8868\u793a\u6578\u5b57\u7684\u76f8\u9130\u95dc\u4fc2\u548c\u9806\u5e8f\u95dc\u4fc2\uff0c\u4f46\u81f3\u65bc\u5132\u5b58\u7684\u5167\u5bb9\u662f\u6574\u6578 <code>int</code>\u3001\u5c0f\u6578 <code>float</code> \u9084\u662f\u5b57\u5143 <code>char</code> \uff0c\u5247\u8207\u201c\u8cc7\u6599\u7d50\u69cb\u201d\u7121\u95dc\u3002</p> <p>\u63db\u53e5\u8a71\u8aaa\uff0c\u57fa\u672c\u8cc7\u6599\u578b\u5225\u63d0\u4f9b\u4e86\u8cc7\u6599\u7684\u201c\u5167\u5bb9\u578b\u5225\u201d\uff0c\u800c\u8cc7\u6599\u7d50\u69cb\u63d0\u4f9b\u4e86\u8cc7\u6599\u7684\u201c\u7d44\u7e54\u65b9\u5f0f\u201d\u3002\u4f8b\u5982\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u7528\u76f8\u540c\u7684\u8cc7\u6599\u7d50\u69cb\uff08\u9663\u5217\uff09\u4f86\u5132\u5b58\u8207\u8868\u793a\u4e0d\u540c\u7684\u57fa\u672c\u8cc7\u6599\u578b\u5225\uff0c\u5305\u62ec <code>int</code>\u3001<code>float</code>\u3001<code>char</code>\u3001<code>bool</code> \u7b49\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python \u7684\u5b57\u5143\u5be6\u969b\u4e0a\u662f\u9577\u5ea6\u70ba 1 \u7684\u5b57\u4e32\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python \u7684\u4e32\u5217\u53ef\u4ee5\u81ea\u7531\u5132\u5b58\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u548c\u7269\u4ef6\u5f15\u7528\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript \u7684\u9663\u5217\u53ef\u4ee5\u81ea\u7531\u5132\u5b58\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u548c\u7269\u4ef6\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0; 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u4f86\u521d\u59cb\u5316\u9663\u5217\nval numbers = IntArray(5)\nval decinals = FloatArray(5)\nval characters = CharArray(5)\nval bools = BooleanArray(5)\n</code></pre> <pre><code># Ruby \u7684\u4e32\u5217\u53ef\u4ee5\u81ea\u7531\u5132\u5b58\u5404\u7a2e\u57fa\u672c\u8cc7\u6599\u578b\u5225\u548c\u7269\u4ef6\u5f15\u7528\ndata = [0, 0.0, 'a', false, ListNode(0)]\n</code></pre> <pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 \u5b57\u5143\u7de8\u78bc *","text":"<p>\u5728\u8a08\u7b97\u6a5f\u4e2d\uff0c\u6240\u6709\u8cc7\u6599\u90fd\u662f\u4ee5\u4e8c\u9032\u4f4d\u5236\u6578\u7684\u5f62\u5f0f\u5132\u5b58\u7684\uff0c\u5b57\u5143 <code>char</code> \u4e5f\u4e0d\u4f8b\u5916\u3002\u70ba\u4e86\u8868\u793a\u5b57\u5143\uff0c\u6211\u5011\u9700\u8981\u5efa\u7acb\u4e00\u5957\u201c\u5b57\u7b26\u96c6\u201d\uff0c\u898f\u5b9a\u6bcf\u500b\u5b57\u5143\u548c\u4e8c\u9032\u4f4d\u5236\u6578\u4e4b\u9593\u7684\u4e00\u4e00\u5c0d\u61c9\u95dc\u4fc2\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u5f8c\uff0c\u8a08\u7b97\u6a5f\u5c31\u53ef\u4ee5\u900f\u904e\u67e5\u8868\u5b8c\u6210\u4e8c\u9032\u4f4d\u5236\u6578\u5230\u5b57\u5143\u7684\u8f49\u63db\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1 \u00a0 ASCII \u5b57\u7b26\u96c6","text":"<p>ASCII \u78bc\u662f\u6700\u65e9\u51fa\u73fe\u7684\u5b57\u7b26\u96c6\uff0c\u5176\u5168\u7a31\u70ba American Standard Code for Information Interchange\uff08\u7f8e\u570b\u6a19\u6e96\u8cc7\u8a0a\u4ea4\u63db\u7a0b\u5f0f\u78bc\uff09\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u9032\u4f4d\u5236\u6578\uff08\u4e00\u500b\u4f4d\u5143\u7d44\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u500b\u5b57\u5143\uff0c\u6700\u591a\u80fd\u5920\u8868\u793a 128 \u500b\u4e0d\u540c\u7684\u5b57\u5143\u3002\u5982\u5716 3-6 \u6240\u793a\uff0cASCII \u78bc\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5beb\u3001\u6578\u5b57 0 ~ 9\u3001\u4e00\u4e9b\u6a19\u9ede\u7b26\u865f\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u5143\uff08\u5982\u63db\u884c\u7b26\u548c\u88fd\u8868\u7b26\uff09\u3002</p> <p></p> <p> \u5716 3-6 \u00a0 ASCII \u78bc </p> <p>\u7136\u800c\uff0cASCII \u78bc\u50c5\u80fd\u5920\u8868\u793a\u82f1\u6587\u3002\u96a8\u8457\u8a08\u7b97\u6a5f\u7684\u5168\u7403\u5316\uff0c\u8a95\u751f\u4e86\u4e00\u7a2e\u80fd\u5920\u8868\u793a\u66f4\u591a\u8a9e\u8a00\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u790e\u4e0a\u64f4\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u5920\u8868\u793a 256 \u500b\u4e0d\u540c\u7684\u5b57\u5143\u3002</p> <p>\u5728\u4e16\u754c\u7bc4\u570d\u5167\uff0c\u9678\u7e8c\u51fa\u73fe\u4e86\u4e00\u6279\u9069\u7528\u65bc\u4e0d\u540c\u5730\u5340\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u9019\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u500b\u5b57\u5143\u7d71\u4e00\u70ba ASCII \u78bc\uff0c\u5f8c 128 \u500b\u5b57\u5143\u5b9a\u7fa9\u4e0d\u540c\uff0c\u4ee5\u9069\u61c9\u4e0d\u540c\u8a9e\u8a00\u7684\u9700\u6c42\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2 \u00a0 GBK \u5b57\u7b26\u96c6","text":"<p>\u5f8c\u4f86\u4eba\u5011\u767c\u73fe\uff0cEASCII \u78bc\u4ecd\u7136\u7121\u6cd5\u6eff\u8db3\u8a31\u591a\u8a9e\u8a00\u7684\u5b57\u5143\u6578\u91cf\u8981\u6c42\u3002\u6bd4\u5982\u6f22\u5b57\u6709\u8fd1\u5341\u842c\u500b\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u5e7e\u5343\u500b\u3002\u4e2d\u570b\u570b\u5bb6\u6a19\u6e96\u7e3d\u5c40\u65bc 1980 \u5e74\u91cb\u51fa\u4e86 GB2312 \u5b57\u7b26\u96c6\uff0c\u5176\u6536\u9304\u4e86 6763 \u500b\u6f22\u5b57\uff0c\u57fa\u672c\u6eff\u8db3\u4e86\u6f22\u5b57\u7684\u8a08\u7b97\u6a5f\u8655\u7406\u9700\u8981\u3002</p> <p>\u7136\u800c\uff0cGB2312 \u7121\u6cd5\u8655\u7406\u90e8\u5206\u7f55\u898b\u5b57\u548c\u7e41\u9ad4\u5b57\u3002GBK \u5b57\u7b26\u96c6\u662f\u5728 GB2312 \u7684\u57fa\u790e\u4e0a\u64f4\u5c55\u5f97\u5230\u7684\uff0c\u5b83\u5171\u6536\u9304\u4e86 21886 \u500b\u6f22\u5b57\u3002\u5728 GBK \u7684\u7de8\u78bc\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u5143\u4f7f\u7528\u4e00\u500b\u4f4d\u5143\u7d44\u8868\u793a\uff0c\u6f22\u5b57\u4f7f\u7528\u5169\u500b\u4f4d\u5143\u7d44\u8868\u793a\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3 \u00a0 Unicode \u5b57\u7b26\u96c6","text":"<p>\u96a8\u8457\u8a08\u7b97\u6a5f\u6280\u8853\u7684\u84ec\u52c3\u767c\u5c55\uff0c\u5b57\u7b26\u96c6\u8207\u7de8\u78bc\u6a19\u6e96\u767e\u82b1\u9f4a\u653e\uff0c\u800c\u9019\u5e36\u4f86\u4e86\u8a31\u591a\u554f\u984c\u3002\u4e00\u65b9\u9762\uff0c\u9019\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u7fa9\u4e86\u7279\u5b9a\u8a9e\u8a00\u7684\u5b57\u5143\uff0c\u7121\u6cd5\u5728\u591a\u8a9e\u8a00\u74b0\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u7a2e\u8a9e\u8a00\u5b58\u5728\u591a\u7a2e\u5b57\u7b26\u96c6\u6a19\u6e96\uff0c\u5982\u679c\u5169\u81fa\u8a08\u7b97\u6a5f\u4f7f\u7528\u7684\u662f\u4e0d\u540c\u7684\u7de8\u78bc\u6a19\u6e96\uff0c\u5247\u5728\u8cc7\u8a0a\u50b3\u905e\u6642\u5c31\u6703\u51fa\u73fe\u4e82\u78bc\u3002</p> <p>\u90a3\u500b\u6642\u4ee3\u7684\u7814\u7a76\u4eba\u54e1\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u500b\u8db3\u5920\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c07\u4e16\u754c\u7bc4\u570d\u5167\u7684\u6240\u6709\u8a9e\u8a00\u548c\u7b26\u865f\u90fd\u6536\u9304\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u6c7a\u8de8\u8a9e\u8a00\u74b0\u5883\u548c\u4e82\u78bc\u554f\u984c\u4e86\u55ce\uff1f\u5728\u9019\u7a2e\u60f3\u6cd5\u7684\u9a45\u52d5\u4e0b\uff0c\u4e00\u500b\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u61c9\u904b\u800c\u751f\u3002</p> <p>Unicode \u7684\u4e2d\u6587\u540d\u7a31\u70ba\u201c\u7d71\u4e00\u78bc\u201d\uff0c\u7406\u8ad6\u4e0a\u80fd\u5bb9\u7d0d 100 \u591a\u842c\u500b\u5b57\u5143\u3002\u5b83\u81f4\u529b\u65bc\u5c07\u5168\u7403\u7bc4\u570d\u5167\u7684\u5b57\u5143\u7d0d\u5165\u7d71\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u7a2e\u901a\u7528\u7684\u5b57\u7b26\u96c6\u4f86\u8655\u7406\u548c\u986f\u793a\u5404\u7a2e\u8a9e\u8a00\u6587\u5b57\uff0c\u6e1b\u5c11\u56e0\u70ba\u7de8\u78bc\u6a19\u6e96\u4e0d\u540c\u800c\u7522\u751f\u7684\u4e82\u78bc\u554f\u984c\u3002</p> <p>\u81ea 1991 \u5e74\u91cb\u51fa\u4ee5\u4f86\uff0cUnicode \u4e0d\u65b7\u64f4\u5145\u65b0\u7684\u8a9e\u8a00\u8207\u5b57\u5143\u3002\u622a\u81f3 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7d93\u5305\u542b 149186 \u500b\u5b57\u5143\uff0c\u5305\u62ec\u5404\u7a2e\u8a9e\u8a00\u7684\u5b57\u5143\u3001\u7b26\u865f\u751a\u81f3\u8868\u60c5\u7b26\u865f\u7b49\u3002\u5728\u9f90\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u5143\u4f54\u7528 2 \u4f4d\u5143\u7d44\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u5143\u4f54\u7528 3 \u4f4d\u5143\u7d44\u751a\u81f3 4 \u4f4d\u5143\u7d44\u3002</p> <p>Unicode \u662f\u4e00\u7a2e\u901a\u7528\u5b57\u7b26\u96c6\uff0c\u672c\u8cea\u4e0a\u662f\u7d66\u6bcf\u500b\u5b57\u5143\u5206\u914d\u4e00\u500b\u7de8\u865f\uff08\u7a31\u70ba\u201c\u78bc\u9ede\u201d\uff09\uff0c\u4f46\u5b83\u4e26\u6c92\u6709\u898f\u5b9a\u5728\u8a08\u7b97\u6a5f\u4e2d\u5982\u4f55\u5132\u5b58\u9019\u4e9b\u5b57\u5143\u78bc\u9ede\u3002\u6211\u5011\u4e0d\u7981\u6703\u554f\uff1a\u7576\u591a\u7a2e\u9577\u5ea6\u7684 Unicode \u78bc\u9ede\u540c\u6642\u51fa\u73fe\u5728\u4e00\u500b\u6587\u5b57\u4e2d\u6642\uff0c\u7cfb\u7d71\u5982\u4f55\u89e3\u6790\u5b57\u5143\uff1f\u4f8b\u5982\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba 2 \u4f4d\u5143\u7d44\u7684\u7de8\u78bc\uff0c\u7cfb\u7d71\u5982\u4f55\u78ba\u8a8d\u5b83\u662f\u4e00\u500b 2 \u4f4d\u5143\u7d44\u7684\u5b57\u5143\u9084\u662f\u5169\u500b 1 \u4f4d\u5143\u7d44\u7684\u5b57\u5143\uff1f</p> <p>\u5c0d\u65bc\u4ee5\u4e0a\u554f\u984c\uff0c\u4e00\u7a2e\u76f4\u63a5\u7684\u89e3\u6c7a\u65b9\u6848\u662f\u5c07\u6240\u6709\u5b57\u5143\u5132\u5b58\u70ba\u7b49\u9577\u7684\u7de8\u78bc\u3002\u5982\u5716 3-7 \u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u500b\u5b57\u5143\u4f54\u7528 1 \u4f4d\u5143\u7d44\uff0c\u201c\u6f14\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u500b\u5b57\u5143\u4f54\u7528 2 \u4f4d\u5143\u7d44\u3002\u6211\u5011\u53ef\u4ee5\u900f\u904e\u9ad8\u4f4d\u586b 0 \u5c07\u201cHello \u6f14\u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u5143\u90fd\u7de8\u78bc\u70ba 2 \u4f4d\u5143\u7d44\u9577\u5ea6\u3002\u9019\u6a23\u7cfb\u7d71\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u4f4d\u5143\u7d44\u89e3\u6790\u4e00\u500b\u5b57\u5143\uff0c\u6062\u5fa9\u9019\u500b\u77ed\u8a9e\u7684\u5167\u5bb9\u4e86\u3002</p> <p></p> <p> \u5716 3-7 \u00a0 Unicode \u7de8\u78bc\u793a\u4f8b </p> <p>\u7136\u800c ASCII \u78bc\u5df2\u7d93\u5411\u6211\u5011\u8b49\u660e\uff0c\u7de8\u78bc\u82f1\u6587\u53ea\u9700 1 \u4f4d\u5143\u7d44\u3002\u82e5\u63a1\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u5b57\u4f54\u7528\u7a7a\u9593\u7684\u5927\u5c0f\u5c07\u6703\u662f ASCII \u7de8\u78bc\u4e0b\u7684\u5169\u500d\uff0c\u975e\u5e38\u6d6a\u8cbb\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002\u56e0\u6b64\uff0c\u6211\u5011\u9700\u8981\u4e00\u7a2e\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7de8\u78bc\u65b9\u6cd5\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4 \u00a0 UTF-8 \u7de8\u78bc","text":"<p>\u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u70ba\u570b\u969b\u4e0a\u4f7f\u7528\u6700\u5ee3\u6cdb\u7684 Unicode \u7de8\u78bc\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u7a2e\u53ef\u8b8a\u9577\u5ea6\u7684\u7de8\u78bc\uff0c\u4f7f\u7528 1 \u5230 4 \u4f4d\u5143\u7d44\u4f86\u8868\u793a\u4e00\u500b\u5b57\u5143\uff0c\u6839\u64da\u5b57\u5143\u7684\u8907\u96dc\u6027\u800c\u8b8a\u3002ASCII \u5b57\u5143\u53ea\u9700 1 \u4f4d\u5143\u7d44\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u81d8\u5b57\u6bcd\u9700\u8981 2 \u4f4d\u5143\u7d44\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u5143\u9700\u8981 3 \u4f4d\u5143\u7d44\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u5143\u9700\u8981 4 \u4f4d\u5143\u7d44\u3002</p> <p>UTF-8 \u7684\u7de8\u78bc\u898f\u5247\u4e26\u4e0d\u8907\u96dc\uff0c\u5206\u70ba\u4ee5\u4e0b\u5169\u7a2e\u60c5\u6cc1\u3002</p> <ul> <li>\u5c0d\u65bc\u9577\u5ea6\u70ba 1 \u4f4d\u5143\u7d44\u7684\u5b57\u5143\uff0c\u5c07\u6700\u9ad8\u4f4d\u8a2d\u5b9a\u70ba \\(0\\) \uff0c\u5176\u9918 7 \u4f4d\u8a2d\u5b9a\u70ba Unicode \u78bc\u9ede\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u5143\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u4f54\u64da\u4e86\u524d 128 \u500b\u78bc\u9ede\u3002\u4e5f\u5c31\u662f\u8aaa\uff0cUTF-8 \u7de8\u78bc\u53ef\u4ee5\u5411\u4e0b\u76f8\u5bb9 ASCII \u78bc\u3002\u9019\u610f\u5473\u8457\u6211\u5011\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u4f86\u89e3\u6790\u5e74\u4ee3\u4e45\u9060\u7684 ASCII \u78bc\u6587\u5b57\u3002</li> <li>\u5c0d\u65bc\u9577\u5ea6\u70ba \\(n\\) \u4f4d\u5143\u7d44\u7684\u5b57\u5143\uff08\u5176\u4e2d \\(n &gt; 1\\)\uff09\uff0c\u5c07\u9996\u500b\u4f4d\u5143\u7d44\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8a2d\u5b9a\u70ba \\(1\\) \uff0c\u7b2c \\(n + 1\\) \u4f4d\u8a2d\u5b9a\u70ba \\(0\\) \uff1b\u5f9e\u7b2c\u4e8c\u500b\u4f4d\u5143\u7d44\u958b\u59cb\uff0c\u5c07\u6bcf\u500b\u4f4d\u5143\u7d44\u7684\u9ad8 2 \u4f4d\u90fd\u8a2d\u5b9a\u70ba \\(10\\) \uff1b\u5176\u9918\u6240\u6709\u4f4d\u7528\u65bc\u586b\u5145\u5b57\u5143\u7684 Unicode \u78bc\u9ede\u3002</li> </ul> <p>\u5716 3-8 \u5c55\u793a\u4e86\u201cHello\u6f14\u7b97\u6cd5\u201d\u5c0d\u61c9\u7684 UTF-8 \u7de8\u78bc\u3002\u89c0\u5bdf\u767c\u73fe\uff0c\u7531\u65bc\u6700\u9ad8 \\(n\\) \u4f4d\u90fd\u8a2d\u5b9a\u70ba \\(1\\) \uff0c\u56e0\u6b64\u7cfb\u7d71\u53ef\u4ee5\u900f\u904e\u8b80\u53d6\u6700\u9ad8\u4f4d \\(1\\) \u7684\u500b\u6578\u4f86\u89e3\u6790\u51fa\u5b57\u5143\u7684\u9577\u5ea6\u70ba \\(n\\) \u3002</p> <p>\u4f46\u70ba\u4ec0\u9ebc\u8981\u5c07\u5176\u9918\u6240\u6709\u4f4d\u5143\u7d44\u7684\u9ad8 2 \u4f4d\u90fd\u8a2d\u5b9a\u70ba \\(10\\) \u5462\uff1f\u5be6\u969b\u4e0a\uff0c\u9019\u500b \\(10\\) \u80fd\u5920\u8d77\u5230\u6821\u9a57\u7b26\u7684\u4f5c\u7528\u3002\u5047\u8a2d\u7cfb\u7d71\u5f9e\u4e00\u500b\u932f\u8aa4\u7684\u4f4d\u5143\u7d44\u958b\u59cb\u89e3\u6790\u6587\u5b57\uff0c\u4f4d\u5143\u7d44\u982d\u90e8\u7684 \\(10\\) \u80fd\u5920\u5e6b\u52a9\u7cfb\u7d71\u5feb\u901f\u5224\u65b7\u51fa\u7570\u5e38\u3002</p> <p>\u4e4b\u6240\u4ee5\u5c07 \\(10\\) \u7576\u4f5c\u6821\u9a57\u7b26\uff0c\u662f\u56e0\u70ba\u5728 UTF-8 \u7de8\u78bc\u898f\u5247\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u5143\u7684\u6700\u9ad8\u5169\u4f4d\u662f \\(10\\) \u3002\u9019\u500b\u7d50\u8ad6\u53ef\u4ee5\u7528\u53cd\u8b49\u6cd5\u4f86\u8b49\u660e\uff1a\u5047\u8a2d\u4e00\u500b\u5b57\u5143\u7684\u6700\u9ad8\u5169\u4f4d\u662f \\(10\\) \uff0c\u8aaa\u660e\u8a72\u5b57\u5143\u7684\u9577\u5ea6\u70ba \\(1\\) \uff0c\u5c0d\u61c9 ASCII \u78bc\u3002\u800c ASCII \u78bc\u7684\u6700\u9ad8\u4f4d\u61c9\u8a72\u662f \\(0\\) \uff0c\u8207\u5047\u8a2d\u77db\u76fe\u3002</p> <p></p> <p> \u5716 3-8 \u00a0 UTF-8 \u7de8\u78bc\u793a\u4f8b </p> <p>\u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u898b\u7684\u7de8\u78bc\u65b9\u5f0f\u9084\u5305\u62ec\u4ee5\u4e0b\u5169\u7a2e\u3002</p> <ul> <li>UTF-16 \u7de8\u78bc\uff1a\u4f7f\u7528 2 \u6216 4 \u4f4d\u5143\u7d44\u4f86\u8868\u793a\u4e00\u500b\u5b57\u5143\u3002\u6240\u6709\u7684 ASCII \u5b57\u5143\u548c\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u5143\uff0c\u90fd\u7528 2 \u4f4d\u5143\u7d44\u8868\u793a\uff1b\u5c11\u6578\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u4f4d\u5143\u7d44\u8868\u793a\u3002\u5c0d\u65bc 2 \u4f4d\u5143\u7d44\u7684\u5b57\u5143\uff0cUTF-16 \u7de8\u78bc\u8207 Unicode \u78bc\u9ede\u76f8\u7b49\u3002</li> <li>UTF-32 \u7de8\u78bc\uff1a\u6bcf\u500b\u5b57\u5143\u90fd\u4f7f\u7528 4 \u4f4d\u5143\u7d44\u3002\u9019\u610f\u5473\u8457 UTF-32 \u6bd4 UTF-8 \u548c UTF-16 \u66f4\u4f54\u7528\u7a7a\u9593\uff0c\u7279\u5225\u662f\u5c0d\u65bc ASCII \u5b57\u5143\u4f54\u6bd4\u8f03\u9ad8\u7684\u6587\u5b57\u3002</li> </ul> <p>\u5f9e\u5132\u5b58\u7a7a\u9593\u4f54\u7528\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u5143\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u70ba\u5b83\u50c5\u9700 1 \u4f4d\u5143\u7d44\uff1b\u4f7f\u7528 UTF-16 \u7de8\u78bc\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u5143\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u6703\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u70ba\u5b83\u50c5\u9700 2 \u4f4d\u5143\u7d44\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u4f4d\u5143\u7d44\u3002</p> <p>\u5f9e\u76f8\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8a31\u591a\u5de5\u5177\u548c\u5eab\u512a\u5148\u652f\u6301 UTF-8 \u3002</p>"},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5 \u00a0 \u7a0b\u5f0f\u8a9e\u8a00\u7684\u5b57\u5143\u7de8\u78bc","text":"<p>\u5c0d\u65bc\u4ee5\u5f80\u7684\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\uff0c\u7a0b\u5f0f\u57f7\u884c\u4e2d\u7684\u5b57\u4e32\u90fd\u63a1\u7528 UTF-16 \u6216 UTF-32 \u9019\u985e\u7b49\u9577\u7de8\u78bc\u3002\u5728\u7b49\u9577\u7de8\u78bc\u4e0b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5b57\u4e32\u770b\u4f5c\u9663\u5217\u4f86\u8655\u7406\uff0c\u9019\u7a2e\u505a\u6cd5\u5177\u6709\u4ee5\u4e0b\u512a\u9ede\u3002</p> <ul> <li>\u96a8\u6a5f\u8a2a\u554f\uff1aUTF-16 \u7de8\u78bc\u7684\u5b57\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u9032\u884c\u96a8\u6a5f\u8a2a\u554f\u3002UTF-8 \u662f\u4e00\u7a2e\u8b8a\u9577\u7de8\u78bc\uff0c\u8981\u60f3\u627e\u5230\u7b2c \\(i\\) \u500b\u5b57\u5143\uff0c\u6211\u5011\u9700\u8981\u5f9e\u5b57\u4e32\u7684\u958b\u59cb\u8655\u8d70\u8a2a\u5230\u7b2c \\(i\\) \u500b\u5b57\u5143\uff0c\u9019\u9700\u8981 \\(O(n)\\) \u7684\u6642\u9593\u3002</li> <li>\u5b57\u5143\u8a08\u6578\uff1a\u8207\u96a8\u6a5f\u8a2a\u554f\u985e\u4f3c\uff0c\u8a08\u7b97 UTF-16 \u7de8\u78bc\u7684\u5b57\u4e32\u7684\u9577\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8a08\u7b97 UTF-8 \u7de8\u78bc\u7684\u5b57\u4e32\u7684\u9577\u5ea6\u9700\u8981\u8d70\u8a2a\u6574\u500b\u5b57\u4e32\u3002</li> <li>\u5b57\u4e32\u64cd\u4f5c\uff1a\u5728 UTF-16 \u7de8\u78bc\u7684\u5b57\u4e32\u4e0a\uff0c\u5f88\u591a\u5b57\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u9023\u7dda\u3001\u63d2\u5165\u3001\u522a\u9664\u7b49\uff09\u66f4\u5bb9\u6613\u9032\u884c\u3002\u5728 UTF-8 \u7de8\u78bc\u7684\u5b57\u4e32\u4e0a\uff0c\u9032\u884c\u9019\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u984d\u5916\u7684\u8a08\u7b97\uff0c\u4ee5\u78ba\u4fdd\u4e0d\u6703\u7522\u751f\u7121\u6548\u7684 UTF-8 \u7de8\u78bc\u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u7a0b\u5f0f\u8a9e\u8a00\u7684\u5b57\u5143\u7de8\u78bc\u65b9\u6848\u8a2d\u8a08\u662f\u4e00\u500b\u5f88\u6709\u8da3\u7684\u8a71\u984c\uff0c\u6d89\u53ca\u8a31\u591a\u56e0\u7d20\u3002</p> <ul> <li>Java \u7684 <code>String</code> \u578b\u5225\u4f7f\u7528 UTF-16 \u7de8\u78bc\uff0c\u6bcf\u500b\u5b57\u5143\u4f54\u7528 2 \u4f4d\u5143\u7d44\u3002\u9019\u662f\u56e0\u70ba Java \u8a9e\u8a00\u8a2d\u8a08\u4e4b\u521d\uff0c\u4eba\u5011\u8a8d\u70ba 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u5143\u3002\u7136\u800c\uff0c\u9019\u662f\u4e00\u500b\u4e0d\u6b63\u78ba\u7684\u5224\u65b7\u3002\u5f8c\u4f86 Unicode \u898f\u7bc4\u64f4\u5c55\u5230\u4e86\u8d85\u904e 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u5143\u73fe\u5728\u53ef\u80fd\u7531\u4e00\u5c0d 16 \u4f4d\u7684\u503c\uff08\u7a31\u70ba\u201c\u4ee3\u7406\u5c0d\u201d\uff09\u8868\u793a\u3002</li> <li>JavaScript \u548c TypeScript \u7684\u5b57\u4e32\u4f7f\u7528 UTF-16 \u7de8\u78bc\u7684\u539f\u56e0\u8207 Java \u985e\u4f3c\u3002\u7576 1995 \u5e74 Netscape \u516c\u53f8\u9996\u6b21\u63a8\u51fa JavaScript \u8a9e\u8a00\u6642\uff0cUnicode \u9084\u8655\u65bc\u767c\u5c55\u65e9\u671f\uff0c\u90a3\u6642\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7de8\u78bc\u5c31\u8db3\u4ee5\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u5143\u4e86\u3002</li> <li>C# \u4f7f\u7528 UTF-16 \u7de8\u78bc\uff0c\u4e3b\u8981\u662f\u56e0\u70ba .NET \u5e73\u81fa\u662f\u7531 Microsoft \u8a2d\u8a08\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u8853\uff08\u5305\u62ec Windows \u4f5c\u696d\u7cfb\u7d71\uff09\u90fd\u5ee3\u6cdb\u4f7f\u7528 UTF-16 \u7de8\u78bc\u3002</li> </ul> <p>\u7531\u65bc\u4ee5\u4e0a\u7a0b\u5f0f\u8a9e\u8a00\u5c0d\u5b57\u5143\u6578\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u5011\u4e0d\u5f97\u4e0d\u63a1\u53d6\u201c\u4ee3\u7406\u5c0d\u201d\u7684\u65b9\u5f0f\u4f86\u8868\u793a\u8d85\u904e 16 \u4f4d\u9577\u5ea6\u7684 Unicode \u5b57\u5143\u3002\u9019\u662f\u4e00\u500b\u4e0d\u5f97\u5df2\u70ba\u4e4b\u7684\u7121\u5948\u4e4b\u8209\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5c0d\u7684\u5b57\u4e32\u4e2d\uff0c\u4e00\u500b\u5b57\u5143\u53ef\u80fd\u4f54\u7528 2 \u4f4d\u5143\u7d44\u6216 4 \u4f4d\u5143\u7d44\uff0c\u5f9e\u800c\u55aa\u5931\u4e86\u7b49\u9577\u7de8\u78bc\u7684\u512a\u52e2\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u8655\u7406\u4ee3\u7406\u5c0d\u9700\u8981\u984d\u5916\u589e\u52a0\u7a0b\u5f0f\u78bc\uff0c\u9019\u63d0\u9ad8\u4e86\u7a0b\u5f0f\u8a2d\u8a08\u7684\u8907\u96dc\u6027\u548c\u9664\u932f\u96e3\u5ea6\u3002</p> <p>\u51fa\u65bc\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7a0b\u5f0f\u8a9e\u8a00\u63d0\u51fa\u4e86\u4e00\u4e9b\u4e0d\u540c\u7684\u7de8\u78bc\u65b9\u6848\u3002</p> <ul> <li>Python \u4e2d\u7684 <code>str</code> \u4f7f\u7528 Unicode \u7de8\u78bc\uff0c\u4e26\u63a1\u7528\u4e00\u7a2e\u9748\u6d3b\u7684\u5b57\u4e32\u8868\u793a\uff0c\u5132\u5b58\u7684\u5b57\u5143\u9577\u5ea6\u53d6\u6c7a\u65bc\u5b57\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u78bc\u9ede\u3002\u82e5\u5b57\u4e32\u4e2d\u5168\u90e8\u662f ASCII \u5b57\u5143\uff0c\u5247\u6bcf\u500b\u5b57\u5143\u4f54\u7528 1 \u4f4d\u5143\u7d44\uff1b\u5982\u679c\u6709\u5b57\u5143\u8d85\u51fa\u4e86 ASCII \u7bc4\u570d\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8a9e\u8a00\u5e73\u9762\uff08BMP\uff09\u5167\uff0c\u5247\u6bcf\u500b\u5b57\u5143\u4f54\u7528 2 \u4f4d\u5143\u7d44\uff1b\u5982\u679c\u6709\u8d85\u51fa BMP \u7684\u5b57\u5143\uff0c\u5247\u6bcf\u500b\u5b57\u5143\u4f54\u7528 4 \u4f4d\u5143\u7d44\u3002</li> <li>Go \u8a9e\u8a00\u7684 <code>string</code> \u578b\u5225\u5728\u5167\u90e8\u4f7f\u7528 UTF-8 \u7de8\u78bc\u3002Go \u8a9e\u8a00\u9084\u63d0\u4f9b\u4e86 <code>rune</code> \u578b\u5225\uff0c\u5b83\u7528\u65bc\u8868\u793a\u55ae\u500b Unicode \u78bc\u9ede\u3002</li> <li>Rust \u8a9e\u8a00\u7684 <code>str</code> \u548c <code>String</code> \u578b\u5225\u5728\u5167\u90e8\u4f7f\u7528 UTF-8 \u7de8\u78bc\u3002Rust \u4e5f\u63d0\u4f9b\u4e86 <code>char</code> \u578b\u5225\uff0c\u7528\u65bc\u8868\u793a\u55ae\u500b Unicode \u78bc\u9ede\u3002</li> </ul> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8a0e\u8ad6\u7684\u90fd\u662f\u5b57\u4e32\u5728\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u7684\u5132\u5b58\u65b9\u5f0f\uff0c\u9019\u548c\u5b57\u4e32\u5982\u4f55\u5728\u6a94\u6848\u4e2d\u5132\u5b58\u6216\u5728\u7db2\u8def\u4e2d\u50b3\u8f38\u662f\u4e0d\u540c\u7684\u554f\u984c\u3002\u5728\u6a94\u6848\u5132\u5b58\u6216\u7db2\u8def\u50b3\u8f38\u4e2d\uff0c\u6211\u5011\u901a\u5e38\u6703\u5c07\u5b57\u4e32\u7de8\u78bc\u70ba UTF-8 \u683c\u5f0f\uff0c\u4ee5\u9054\u5230\u6700\u512a\u7684\u76f8\u5bb9\u6027\u548c\u7a7a\u9593\u6548\u7387\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 \u8cc7\u6599\u7d50\u69cb\u5206\u985e","text":"<p>\u5e38\u898b\u7684\u8cc7\u6599\u7d50\u69cb\u5305\u62ec\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u6a39\u3001\u5806\u7a4d\u3001\u5716\uff0c\u5b83\u5011\u53ef\u4ee5\u5f9e\u201c\u908f\u8f2f\u7d50\u69cb\u201d\u548c\u201c\u7269\u7406\u7d50\u69cb\u201d\u5169\u500b\u7dad\u5ea6\u9032\u884c\u5206\u985e\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1 \u00a0 \u908f\u8f2f\u7d50\u69cb\uff1a\u7dda\u6027\u8207\u975e\u7dda\u6027","text":"<p>\u908f\u8f2f\u7d50\u69cb\u63ed\u793a\u4e86\u8cc7\u6599\u5143\u7d20\u4e4b\u9593\u7684\u908f\u8f2f\u95dc\u4fc2\u3002\u5728\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u4e2d\uff0c\u8cc7\u6599\u6309\u7167\u4e00\u5b9a\u9806\u5e8f\u6392\u5217\uff0c\u9ad4\u73fe\u4e86\u8cc7\u6599\u4e4b\u9593\u7684\u7dda\u6027\u95dc\u4fc2\uff1b\u800c\u5728\u6a39\u4e2d\uff0c\u8cc7\u6599\u5f9e\u9802\u90e8\u5411\u4e0b\u6309\u5c64\u6b21\u6392\u5217\uff0c\u8868\u73fe\u51fa\u201c\u7956\u5148\u201d\u8207\u201c\u5f8c\u4ee3\u201d\u4e4b\u9593\u7684\u6d3e\u751f\u95dc\u4fc2\uff1b\u5716\u5247\u7531\u7bc0\u9ede\u548c\u908a\u69cb\u6210\uff0c\u53cd\u6620\u4e86\u8907\u96dc\u7684\u7db2\u8def\u95dc\u4fc2\u3002</p> <p>\u5982\u5716 3-1 \u6240\u793a\uff0c\u908f\u8f2f\u7d50\u69cb\u53ef\u5206\u70ba\u201c\u7dda\u6027\u201d\u548c\u201c\u975e\u7dda\u6027\u201d\u5169\u5927\u985e\u3002\u7dda\u6027\u7d50\u69cb\u6bd4\u8f03\u76f4\u89c0\uff0c\u6307\u8cc7\u6599\u5728\u908f\u8f2f\u95dc\u4fc2\u4e0a\u5448\u7dda\u6027\u6392\u5217\uff1b\u975e\u7dda\u6027\u7d50\u69cb\u5247\u76f8\u53cd\uff0c\u5448\u975e\u7dda\u6027\u6392\u5217\u3002</p> <ul> <li>\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff1a\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\uff0c\u5143\u7d20\u4e4b\u9593\u662f\u4e00\u5c0d\u4e00\u7684\u9806\u5e8f\u95dc\u4fc2\u3002</li> <li>\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff1a\u6a39\u3001\u5806\u7a4d\u3001\u5716\u3001\u96dc\u6e4a\u8868\u3002</li> </ul> <p>\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u53ef\u4ee5\u9032\u4e00\u6b65\u5283\u5206\u70ba\u6a39\u5f62\u7d50\u69cb\u548c\u7db2\u72c0\u7d50\u69cb\u3002</p> <ul> <li>\u6a39\u5f62\u7d50\u69cb\uff1a\u6a39\u3001\u5806\u7a4d\u3001\u96dc\u6e4a\u8868\uff0c\u5143\u7d20\u4e4b\u9593\u662f\u4e00\u5c0d\u591a\u7684\u95dc\u4fc2\u3002</li> <li>\u7db2\u72c0\u7d50\u69cb\uff1a\u5716\uff0c\u5143\u7d20\u4e4b\u9593\u662f\u591a\u5c0d\u591a\u7684\u95dc\u4fc2\u3002</li> </ul> <p></p> <p> \u5716 3-1 \u00a0 \u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u8207\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb </p>"},{"location":"chapter_data_structure/classification_of_data_structure/#312","title":"3.1.2 \u00a0 \u7269\u7406\u7d50\u69cb\uff1a\u9023\u7e8c\u8207\u5206\u6563","text":"<p>\u7576\u6f14\u7b97\u6cd5\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u6b63\u5728\u8655\u7406\u7684\u8cc7\u6599\u4e3b\u8981\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u4e2d\u3002\u5716 3-2 \u5c55\u793a\u4e86\u4e00\u500b\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u689d\uff0c\u5176\u4e2d\u6bcf\u500b\u9ed1\u8272\u65b9\u584a\u90fd\u5305\u542b\u4e00\u584a\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002\u6211\u5011\u53ef\u4ee5\u5c07\u8a18\u61b6\u9ad4\u60f3\u8c61\u6210\u4e00\u500b\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u500b\u55ae\u5143\u683c\u90fd\u53ef\u4ee5\u5132\u5b58\u4e00\u5b9a\u5927\u5c0f\u7684\u8cc7\u6599\u3002</p> <p>\u7cfb\u7d71\u900f\u904e\u8a18\u61b6\u9ad4\u4f4d\u5740\u4f86\u8a2a\u554f\u76ee\u6a19\u4f4d\u7f6e\u7684\u8cc7\u6599\u3002\u5982\u5716 3-2 \u6240\u793a\uff0c\u8a08\u7b97\u6a5f\u6839\u64da\u7279\u5b9a\u898f\u5247\u70ba\u8868\u683c\u4e2d\u7684\u6bcf\u500b\u55ae\u5143\u683c\u5206\u914d\u7de8\u865f\uff0c\u78ba\u4fdd\u6bcf\u500b\u8a18\u61b6\u9ad4\u7a7a\u9593\u90fd\u6709\u552f\u4e00\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u3002\u6709\u4e86\u9019\u4e9b\u4f4d\u5740\uff0c\u7a0b\u5f0f\u4fbf\u53ef\u4ee5\u8a2a\u554f\u8a18\u61b6\u9ad4\u4e2d\u7684\u8cc7\u6599\u3002</p> <p></p> <p> \u5716 3-2 \u00a0 \u8a18\u61b6\u9ad4\u689d\u3001\u8a18\u61b6\u9ad4\u7a7a\u9593\u3001\u8a18\u61b6\u9ad4\u4f4d\u5740 </p> <p>Tip</p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u5c07\u8a18\u61b6\u9ad4\u6bd4\u4f5c Excel \u8868\u683c\u662f\u4e00\u500b\u7c21\u5316\u7684\u985e\u6bd4\uff0c\u5be6\u969b\u8a18\u61b6\u9ad4\u7684\u5de5\u4f5c\u6a5f\u5236\u6bd4\u8f03\u8907\u96dc\uff0c\u6d89\u53ca\u4f4d\u5740\u7a7a\u9593\u3001\u8a18\u61b6\u9ad4\u7ba1\u7406\u3001\u5feb\u53d6\u6a5f\u5236\u3001\u865b\u64ec\u8a18\u61b6\u9ad4\u548c\u7269\u7406\u8a18\u61b6\u9ad4\u7b49\u6982\u5ff5\u3002</p> <p>\u8a18\u61b6\u9ad4\u662f\u6240\u6709\u7a0b\u5f0f\u7684\u5171\u4eab\u8cc7\u6e90\uff0c\u7576\u67d0\u584a\u8a18\u61b6\u9ad4\u88ab\u67d0\u500b\u7a0b\u5f0f\u4f54\u7528\u6642\uff0c\u5247\u901a\u5e38\u7121\u6cd5\u88ab\u5176\u4ed6\u7a0b\u5f0f\u540c\u6642\u4f7f\u7528\u4e86\u3002\u56e0\u6b64\u5728\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u8a2d\u8a08\u4e2d\uff0c\u8a18\u61b6\u9ad4\u8cc7\u6e90\u662f\u4e00\u500b\u91cd\u8981\u7684\u8003\u616e\u56e0\u7d20\u3002\u6bd4\u5982\uff0c\u6f14\u7b97\u6cd5\u6240\u4f54\u7528\u7684\u8a18\u61b6\u9ad4\u5cf0\u503c\u4e0d\u61c9\u8d85\u904e\u7cfb\u7d71\u5269\u9918\u7a7a\u9592\u8a18\u61b6\u9ad4\uff1b\u5982\u679c\u7f3a\u5c11\u9023\u7e8c\u5927\u584a\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\uff0c\u90a3\u9ebc\u6240\u9078\u7528\u7684\u8cc7\u6599\u7d50\u69cb\u5fc5\u9808\u80fd\u5920\u5132\u5b58\u5728\u5206\u6563\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u5167\u3002</p> <p>\u5982\u5716 3-3 \u6240\u793a\uff0c\u7269\u7406\u7d50\u69cb\u53cd\u6620\u4e86\u8cc7\u6599\u5728\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u4e2d\u7684\u5132\u5b58\u65b9\u5f0f\uff0c\u53ef\u5206\u70ba\u9023\u7e8c\u7a7a\u9593\u5132\u5b58\uff08\u9663\u5217\uff09\u548c\u5206\u6563\u7a7a\u9593\u5132\u5b58\uff08\u93c8\u7d50\u4e32\u5217\uff09\u3002\u7269\u7406\u7d50\u69cb\u5f9e\u5e95\u5c64\u6c7a\u5b9a\u4e86\u8cc7\u6599\u7684\u8a2a\u554f\u3001\u66f4\u65b0\u3001\u589e\u522a\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u5169\u7a2e\u7269\u7406\u7d50\u69cb\u5728\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u65b9\u9762\u5448\u73fe\u51fa\u4e92\u88dc\u7684\u7279\u9ede\u3002</p> <p></p> <p> \u5716 3-3 \u00a0 \u9023\u7e8c\u7a7a\u9593\u5132\u5b58\u8207\u5206\u6563\u7a7a\u9593\u5132\u5b58 </p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u6240\u6709\u8cc7\u6599\u7d50\u69cb\u90fd\u662f\u57fa\u65bc\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u6216\u4e8c\u8005\u7684\u7d44\u5408\u5be6\u73fe\u7684\u3002\u4f8b\u5982\uff0c\u5806\u758a\u548c\u4f47\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u9663\u5217\u5be6\u73fe\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\uff1b\u800c\u96dc\u6e4a\u8868\u7684\u5be6\u73fe\u53ef\u80fd\u540c\u6642\u5305\u542b\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u3002</p> <ul> <li>\u57fa\u65bc\u9663\u5217\u53ef\u5be6\u73fe\uff1a\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u6a39\u3001\u5806\u7a4d\u3001\u5716\u3001\u77e9\u9663\u3001\u5f35\u91cf\uff08\u7dad\u5ea6 \\(\\geq 3\\) \u7684\u9663\u5217\uff09\u7b49\u3002</li> <li>\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u53ef\u5be6\u73fe\uff1a\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u6a39\u3001\u5806\u7a4d\u3001\u5716\u7b49\u3002</li> </ul> <p>\u93c8\u7d50\u4e32\u5217\u5728\u521d\u59cb\u5316\u5f8c\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5f0f\u57f7\u884c\u904e\u7a0b\u4e2d\u5c0d\u5176\u9577\u5ea6\u9032\u884c\u8abf\u6574\uff0c\u56e0\u6b64\u4e5f\u7a31\u201c\u52d5\u614b\u8cc7\u6599\u7d50\u69cb\u201d\u3002\u9663\u5217\u5728\u521d\u59cb\u5316\u5f8c\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u4e5f\u7a31\u201c\u975c\u614b\u8cc7\u6599\u7d50\u69cb\u201d\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u9663\u5217\u53ef\u900f\u904e\u91cd\u65b0\u5206\u914d\u8a18\u61b6\u9ad4\u5be6\u73fe\u9577\u5ea6\u8b8a\u5316\uff0c\u5f9e\u800c\u5177\u5099\u4e00\u5b9a\u7684\u201c\u52d5\u614b\u6027\u201d\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u611f\u89ba\u7269\u7406\u7d50\u69cb\u7406\u89e3\u8d77\u4f86\u6709\u56f0\u96e3\uff0c\u5efa\u8b70\u5148\u95b1\u8b80\u4e0b\u4e00\u7ae0\uff0c\u7136\u5f8c\u518d\u56de\u9867\u672c\u7bc0\u5167\u5bb9\u3002</p>"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 \u6578\u5b57\u7de8\u78bc *","text":"<p>Tip</p> <p>\u5728\u672c\u66f8\u4e2d\uff0c\u6a19\u984c\u5e36\u6709 * \u7b26\u865f\u7684\u662f\u9078\u8b80\u7ae0\u7bc0\u3002\u5982\u679c\u4f60\u6642\u9593\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96e3\uff0c\u53ef\u4ee5\u5148\u8df3\u904e\uff0c\u7b49\u5b78\u5b8c\u5fc5\u8b80\u7ae0\u7bc0\u5f8c\u518d\u55ae\u7368\u653b\u514b\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1 \u00a0 \u539f\u78bc\u3001\u4e00\u88dc\u6578\u548c\u4e8c\u88dc\u6578","text":"<p>\u5728\u4e0a\u4e00\u7bc0\u7684\u8868\u683c\u4e2d\u6211\u5011\u767c\u73fe\uff0c\u6240\u6709\u6574\u6578\u578b\u5225\u80fd\u5920\u8868\u793a\u7684\u8ca0\u6578\u90fd\u6bd4\u6b63\u6578\u591a\u4e00\u500b\uff0c\u4f8b\u5982 <code>byte</code> \u7684\u53d6\u503c\u7bc4\u570d\u662f \\([-128, 127]\\) \u3002\u9019\u500b\u73fe\u8c61\u6bd4\u8f03\u53cd\u76f4\u89ba\uff0c\u5b83\u7684\u5167\u5728\u539f\u56e0\u6d89\u53ca\u539f\u78bc\u3001\u4e00\u88dc\u6578\u3001\u4e8c\u88dc\u6578\u7684\u76f8\u95dc\u77e5\u8b58\u3002</p> <p>\u9996\u5148\u9700\u8981\u6307\u51fa\uff0c\u6578\u5b57\u662f\u4ee5\u201c\u4e8c\u88dc\u6578\u201d\u7684\u5f62\u5f0f\u5132\u5b58\u5728\u8a08\u7b97\u6a5f\u4e2d\u7684\u3002\u5728\u5206\u6790\u9019\u6a23\u505a\u7684\u539f\u56e0\u4e4b\u524d\uff0c\u9996\u5148\u7d66\u51fa\u4e09\u8005\u7684\u5b9a\u7fa9\u3002</p> <ul> <li>\u539f\u78bc\uff1a\u6211\u5011\u5c07\u6578\u5b57\u7684\u4e8c\u9032\u4f4d\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u8996\u70ba\u7b26\u865f\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6578\uff0c\\(1\\) \u8868\u793a\u8ca0\u6578\uff0c\u5176\u9918\u4f4d\u8868\u793a\u6578\u5b57\u7684\u503c\u3002</li> <li>\u4e00\u88dc\u6578\uff1a\u6b63\u6578\u7684\u4e00\u88dc\u6578\u8207\u5176\u539f\u78bc\u76f8\u540c\uff0c\u8ca0\u6578\u7684\u4e00\u88dc\u6578\u662f\u5c0d\u5176\u539f\u78bc\u9664\u7b26\u865f\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002</li> <li>\u4e8c\u88dc\u6578\uff1a\u6b63\u6578\u7684\u4e8c\u88dc\u6578\u8207\u5176\u539f\u78bc\u76f8\u540c\uff0c\u8ca0\u6578\u7684\u4e8c\u88dc\u6578\u662f\u5728\u5176\u4e00\u88dc\u6578\u7684\u57fa\u790e\u4e0a\u52a0 \\(1\\) \u3002</li> </ul> <p>\u5716 3-4 \u5c55\u793a\u4e86\u539f\u78bc\u3001\u4e00\u88dc\u6578\u548c\u4e8c\u88dc\u6578\u4e4b\u9593\u7684\u8f49\u63db\u65b9\u6cd5\u3002</p> <p></p> <p> \u5716 3-4 \u00a0 \u539f\u78bc\u3001\u4e00\u88dc\u6578\u8207\u4e8c\u88dc\u6578\u4e4b\u9593\u7684\u76f8\u4e92\u8f49\u63db </p> <p>\u539f\u78bc\uff08sign-magnitude\uff09\u96d6\u7136\u6700\u76f4\u89c0\uff0c\u4f46\u5b58\u5728\u4e00\u4e9b\u4fb7\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u8ca0\u6578\u7684\u539f\u78bc\u4e0d\u80fd\u76f4\u63a5\u7528\u65bc\u904b\u7b97\u3002\u4f8b\u5982\u5728\u539f\u78bc\u4e0b\u8a08\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7d50\u679c\u662f \\(-3\\) \uff0c\u9019\u986f\u7136\u662f\u4e0d\u5c0d\u7684\u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>\u70ba\u4e86\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u8a08\u7b97\u6a5f\u5f15\u5165\u4e86\u4e00\u88dc\u6578\uff081's complement\uff09\u3002\u5982\u679c\u6211\u5011\u5148\u5c07\u539f\u78bc\u8f49\u63db\u70ba\u4e00\u88dc\u6578\uff0c\u4e26\u5728\u4e00\u88dc\u6578\u4e0b\u8a08\u7b97 \\(1 + (-2)\\) \uff0c\u6700\u5f8c\u5c07\u7d50\u679c\u5f9e\u4e00\u88dc\u6578\u8f49\u63db\u56de\u539f\u78bc\uff0c\u5247\u53ef\u5f97\u5230\u6b63\u78ba\u7d50\u679c \\(-1\\) \u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(\u539f\u78bc)} + 1000 \\; 0010 \\; \\text{(\u539f\u78bc)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(\u4e00\u88dc\u6578)} + 1111  \\; 1101 \\; \\text{(\u4e00\u88dc\u6578)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(\u4e00\u88dc\u6578)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u539f\u78bc)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>\u53e6\u4e00\u65b9\u9762\uff0c\u6578\u5b57\u96f6\u7684\u539f\u78bc\u6709 \\(+0\\) \u548c \\(-0\\) \u5169\u7a2e\u8868\u793a\u65b9\u5f0f\u3002\u9019\u610f\u5473\u8457\u6578\u5b57\u96f6\u5c0d\u61c9\u5169\u500b\u4e0d\u540c\u7684\u4e8c\u9032\u4f4d\u5236\u7de8\u78bc\uff0c\u9019\u53ef\u80fd\u6703\u5e36\u4f86\u6b67\u7fa9\u3002\u6bd4\u5982\u5728\u689d\u4ef6\u5224\u65b7\u4e2d\uff0c\u5982\u679c\u6c92\u6709\u5340\u5206\u6b63\u96f6\u548c\u8ca0\u96f6\uff0c\u5247\u53ef\u80fd\u6703\u5c0e\u81f4\u5224\u65b7\u7d50\u679c\u51fa\u932f\u3002\u800c\u5982\u679c\u6211\u5011\u60f3\u8655\u7406\u6b63\u96f6\u548c\u8ca0\u96f6\u6b67\u7fa9\uff0c\u5247\u9700\u8981\u5f15\u5165\u984d\u5916\u7684\u5224\u65b7\u64cd\u4f5c\uff0c\u9019\u53ef\u80fd\u6703\u964d\u4f4e\u8a08\u7b97\u6a5f\u7684\u904b\u7b97\u6548\u7387\u3002</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>\u8207\u539f\u78bc\u4e00\u6a23\uff0c\u4e00\u88dc\u6578\u4e5f\u5b58\u5728\u6b63\u8ca0\u96f6\u6b67\u7fa9\u554f\u984c\uff0c\u56e0\u6b64\u8a08\u7b97\u6a5f\u9032\u4e00\u6b65\u5f15\u5165\u4e86\u4e8c\u88dc\u6578\uff082's complement\uff09\u3002\u6211\u5011\u5148\u4f86\u89c0\u5bdf\u4e00\u4e0b\u8ca0\u96f6\u7684\u539f\u78bc\u3001\u4e00\u88dc\u6578\u3001\u4e8c\u88dc\u6578\u7684\u8f49\u63db\u904e\u7a0b\uff1a</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(\u539f\u78bc)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(\u4e00\u88dc\u6578)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(\u4e8c\u88dc\u6578)} \\newline \\end{aligned} \\] <p>\u5728\u8ca0\u96f6\u7684\u4e00\u88dc\u6578\u57fa\u790e\u4e0a\u52a0 \\(1\\) \u6703\u7522\u751f\u9032\u4f4d\uff0c\u4f46 <code>byte</code> \u578b\u5225\u7684\u9577\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u4f4d\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u6703\u88ab\u6368\u68c4\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u8ca0\u96f6\u7684\u4e8c\u88dc\u6578\u70ba \\(0000 \\; 0000\\) \uff0c\u8207\u6b63\u96f6\u7684\u4e8c\u88dc\u6578\u76f8\u540c\u3002\u9019\u610f\u5473\u8457\u5728\u4e8c\u88dc\u6578\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u500b\u96f6\uff0c\u6b63\u8ca0\u96f6\u6b67\u7fa9\u5f9e\u800c\u5f97\u5230\u89e3\u6c7a\u3002</p> <p>\u9084\u5269\u6700\u5f8c\u4e00\u500b\u7591\u60d1\uff1a<code>byte</code> \u578b\u5225\u7684\u53d6\u503c\u7bc4\u570d\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u4f86\u7684\u4e00\u500b\u8ca0\u6578 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u5011\u6ce8\u610f\u5230\uff0c\u5340\u9593 \\([-127, +127]\\) \u5167\u7684\u6240\u6709\u6574\u6578\u90fd\u6709\u5c0d\u61c9\u7684\u539f\u78bc\u3001\u4e00\u88dc\u6578\u548c\u4e8c\u88dc\u6578\uff0c\u4e26\u4e14\u539f\u78bc\u548c\u4e8c\u88dc\u6578\u4e4b\u9593\u53ef\u4ee5\u4e92\u76f8\u8f49\u63db\u3002</p> <p>\u7136\u800c\uff0c\u4e8c\u88dc\u6578 \\(1000 \\; 0000\\) \u662f\u4e00\u500b\u4f8b\u5916\uff0c\u5b83\u4e26\u6c92\u6709\u5c0d\u61c9\u7684\u539f\u78bc\u3002\u6839\u64da\u8f49\u63db\u65b9\u6cd5\uff0c\u6211\u5011\u5f97\u5230\u8a72\u4e8c\u88dc\u6578\u7684\u539f\u78bc\u70ba \\(0000 \\; 0000\\) \u3002\u9019\u986f\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u70ba\u8a72\u539f\u78bc\u8868\u793a\u6578\u5b57 \\(0\\) \uff0c\u5b83\u7684\u4e8c\u88dc\u6578\u61c9\u8a72\u662f\u81ea\u8eab\u3002\u8a08\u7b97\u6a5f\u898f\u5b9a\u9019\u500b\u7279\u6b8a\u7684\u4e8c\u88dc\u6578 \\(1000 \\; 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5be6\u969b\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u4e8c\u88dc\u6578\u4e0b\u7684\u8a08\u7b97\u7d50\u679c\u5c31\u662f \\(-128\\) \u3002</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(\u539f\u78bc)} + 1000 \\; 0001 \\; \\text{(\u539f\u78bc)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u4e00\u88dc\u6578)} + 1111  \\; 1110 \\; \\text{(\u4e00\u88dc\u6578)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u4e8c\u88dc\u6578)} + 1111  \\; 1111 \\; \\text{(\u4e8c\u88dc\u6578)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u4e8c\u88dc\u6578)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>\u4f60\u53ef\u80fd\u5df2\u7d93\u767c\u73fe\u4e86\uff0c\u4e0a\u8ff0\u6240\u6709\u8a08\u7b97\u90fd\u662f\u52a0\u6cd5\u904b\u7b97\u3002\u9019\u6697\u793a\u8457\u4e00\u500b\u91cd\u8981\u4e8b\u5be6\uff1a\u8a08\u7b97\u6a5f\u5167\u90e8\u7684\u786c\u9ad4\u96fb\u8def\u4e3b\u8981\u662f\u57fa\u65bc\u52a0\u6cd5\u904b\u7b97\u8a2d\u8a08\u7684\u3002\u9019\u662f\u56e0\u70ba\u52a0\u6cd5\u904b\u7b97\u76f8\u5c0d\u65bc\u5176\u4ed6\u904b\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u6e1b\u6cd5\uff09\u4f86\u8aaa\uff0c\u786c\u9ad4\u5be6\u73fe\u8d77\u4f86\u66f4\u7c21\u55ae\uff0c\u66f4\u5bb9\u6613\u9032\u884c\u4e26\u884c\u5316\u8655\u7406\uff0c\u904b\u7b97\u901f\u5ea6\u66f4\u5feb\u3002</p> <p>\u8acb\u6ce8\u610f\uff0c\u9019\u4e26\u4e0d\u610f\u5473\u8457\u8a08\u7b97\u6a5f\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u900f\u904e\u5c07\u52a0\u6cd5\u8207\u4e00\u4e9b\u57fa\u672c\u908f\u8f2f\u904b\u7b97\u7d50\u5408\uff0c\u8a08\u7b97\u6a5f\u80fd\u5920\u5be6\u73fe\u5404\u7a2e\u5176\u4ed6\u7684\u6578\u5b78\u904b\u7b97\u3002\u4f8b\u5982\uff0c\u8a08\u7b97\u6e1b\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f49\u63db\u70ba\u8a08\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8a08\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f49\u63db\u70ba\u8a08\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u6e1b\u6cd5\u3002</p> <p>\u73fe\u5728\u6211\u5011\u53ef\u4ee5\u7e3d\u7d50\u51fa\u8a08\u7b97\u6a5f\u4f7f\u7528\u4e8c\u88dc\u6578\u7684\u539f\u56e0\uff1a\u57fa\u65bc\u4e8c\u88dc\u6578\u8868\u793a\uff0c\u8a08\u7b97\u6a5f\u53ef\u4ee5\u7528\u540c\u6a23\u7684\u96fb\u8def\u548c\u64cd\u4f5c\u4f86\u8655\u7406\u6b63\u6578\u548c\u8ca0\u6578\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8a2d\u8a08\u7279\u6b8a\u7684\u786c\u9ad4\u96fb\u8def\u4f86\u8655\u7406\u6e1b\u6cd5\uff0c\u4e26\u4e14\u7121\u9808\u7279\u5225\u8655\u7406\u6b63\u8ca0\u96f6\u7684\u6b67\u7fa9\u554f\u984c\u3002\u9019\u5927\u5927\u7c21\u5316\u4e86\u786c\u9ad4\u8a2d\u8a08\uff0c\u63d0\u9ad8\u4e86\u904b\u7b97\u6548\u7387\u3002</p> <p>\u4e8c\u88dc\u6578\u7684\u8a2d\u8a08\u975e\u5e38\u7cbe\u5999\uff0c\u56e0\u7bc7\u5e45\u95dc\u4fc2\u6211\u5011\u5c31\u5148\u4ecb\u7d39\u5230\u9019\u88e1\uff0c\u5efa\u8b70\u6709\u8208\u8da3\u7684\u8b80\u8005\u9032\u4e00\u6b65\u6df1\u5165\u77ad\u89e3\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2 \u00a0 \u6d6e\u9ede\u6578\u7de8\u78bc","text":"<p>\u7d30\u5fc3\u7684\u4f60\u53ef\u80fd\u6703\u767c\u73fe\uff1a<code>int</code> \u548c <code>float</code> \u9577\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 \u4f4d\u5143\u7d44 \uff0c\u4f46\u70ba\u4ec0\u9ebc <code>float</code> \u7684\u53d6\u503c\u7bc4\u570d\u9060\u5927\u65bc <code>int</code> \uff1f\u9019\u975e\u5e38\u53cd\u76f4\u89ba\uff0c\u56e0\u70ba\u6309\u7406\u8aaa <code>float</code> \u9700\u8981\u8868\u793a\u5c0f\u6578\uff0c\u53d6\u503c\u7bc4\u570d\u61c9\u8a72\u8b8a\u5c0f\u624d\u5c0d\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u9019\u662f\u56e0\u70ba\u6d6e\u9ede\u6578 <code>float</code> \u63a1\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u8a18\u4e00\u500b 32 \u4f4d\u5143\u9577\u5ea6\u7684\u4e8c\u9032\u4f4d\u5236\u6578\u70ba\uff1a</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>\u6839\u64da IEEE 754 \u6a19\u6e96\uff0c32-bit \u9577\u5ea6\u7684 <code>float</code> \u7531\u4ee5\u4e0b\u4e09\u500b\u90e8\u5206\u69cb\u6210\u3002</p> <ul> <li>\u7b26\u865f\u4f4d \\(\\mathrm{S}\\) \uff1a\u4f54 1 \u4f4d \uff0c\u5c0d\u61c9 \\(b_{31}\\) \u3002</li> <li>\u6307\u6578\u4f4d \\(\\mathrm{E}\\) \uff1a\u4f54 8 \u4f4d \uff0c\u5c0d\u61c9 \\(b_{30} b_{29} \\ldots b_{23}\\) \u3002</li> <li>\u5206\u6578\u4f4d \\(\\mathrm{N}\\) \uff1a\u4f54 23 \u4f4d \uff0c\u5c0d\u61c9 \\(b_{22} b_{21} \\ldots b_0\\) \u3002</li> </ul> <p>\u4e8c\u9032\u4f4d\u5236\u6578 <code>float</code> \u5c0d\u61c9\u503c\u7684\u8a08\u7b97\u65b9\u6cd5\u70ba\uff1a</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>\u8f49\u5316\u5230\u5341\u9032\u4f4d\u5236\u4e0b\u7684\u8a08\u7b97\u516c\u5f0f\u70ba\uff1a</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>\u5176\u4e2d\u5404\u9805\u7684\u53d6\u503c\u7bc4\u570d\u70ba\uff1a</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> \u5716 3-5 \u00a0 IEEE 754 \u6a19\u6e96\u4e0b\u7684 float \u7684\u8a08\u7b97\u793a\u4f8b </p> <p>\u89c0\u5bdf\u5716 3-5 \uff0c\u7d66\u5b9a\u4e00\u500b\u793a\u4f8b\u8cc7\u6599 \\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u5247\u6709\uff1a</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>\u73fe\u5728\u6211\u5011\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u554f\u984c\uff1a<code>float</code> \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6578\u4f4d\uff0c\u5c0e\u81f4\u5176\u53d6\u503c\u7bc4\u570d\u9060\u5927\u65bc <code>int</code> \u3002\u6839\u64da\u4ee5\u4e0a\u8a08\u7b97\uff0c<code>float</code> \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6578\u70ba \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u63db\u7b26\u865f\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8ca0\u6578\u3002</p> <p>\u5118\u7ba1\u6d6e\u9ede\u6578 <code>float</code> \u64f4\u5c55\u4e86\u53d6\u503c\u7bc4\u570d\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u72a7\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6578\u578b\u5225 <code>int</code> \u5c07\u5168\u90e8 32 \u4f4d\u5143\u7528\u65bc\u8868\u793a\u6578\u5b57\uff0c\u6578\u5b57\u662f\u5747\u52fb\u5206\u4f48\u7684\uff1b\u800c\u7531\u65bc\u6307\u6578\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u9ede\u6578 <code>float</code> \u7684\u6578\u503c\u8d8a\u5927\uff0c\u76f8\u9130\u5169\u500b\u6578\u5b57\u4e4b\u9593\u7684\u5dee\u503c\u5c31\u6703\u8da8\u5411\u8d8a\u5927\u3002</p> <p>\u5982\u8868 3-2 \u6240\u793a\uff0c\u6307\u6578\u4f4d \\(\\mathrm{E} = 0\\) \u548c \\(\\mathrm{E} = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u7fa9\uff0c\u7528\u65bc\u8868\u793a\u96f6\u3001\u7121\u7aae\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002</p> <p> \u8868 3-2 \u00a0 \u6307\u6578\u4f4d\u542b\u7fa9 </p> \u6307\u6578\u4f4d E \u5206\u6578\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6578\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8a08\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u898f\u6578 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u898f\u6578 \u6b63\u898f\u6578 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u6b21\u6b63\u898f\u6578\u986f\u8457\u63d0\u5347\u4e86\u6d6e\u9ede\u6578\u7684\u7cbe\u5ea6\u3002\u6700\u5c0f\u6b63\u6b63\u898f\u6578\u70ba \\(2^{-126}\\) \uff0c\u6700\u5c0f\u6b63\u6b21\u6b63\u898f\u6578\u70ba \\(2^{-126} \\times 2^{-23}\\) \u3002</p> <p>\u96d9\u7cbe\u5ea6 <code>double</code> \u4e5f\u63a1\u7528\u985e\u4f3c\u65bc <code>float</code> \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u505a\u8d05\u8ff0\u3002</p>"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_data_structure/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u8cc7\u6599\u7d50\u69cb\u53ef\u4ee5\u5f9e\u908f\u8f2f\u7d50\u69cb\u548c\u7269\u7406\u7d50\u69cb\u5169\u500b\u89d2\u5ea6\u9032\u884c\u5206\u985e\u3002\u908f\u8f2f\u7d50\u69cb\u63cf\u8ff0\u4e86\u8cc7\u6599\u5143\u7d20\u4e4b\u9593\u7684\u908f\u8f2f\u95dc\u4fc2\uff0c\u800c\u7269\u7406\u7d50\u69cb\u63cf\u8ff0\u4e86\u8cc7\u6599\u5728\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u4e2d\u7684\u5132\u5b58\u65b9\u5f0f\u3002</li> <li>\u5e38\u898b\u7684\u908f\u8f2f\u7d50\u69cb\u5305\u62ec\u7dda\u6027\u3001\u6a39\u72c0\u548c\u7db2\u72c0\u7b49\u3002\u901a\u5e38\u6211\u5011\u6839\u64da\u908f\u8f2f\u7d50\u69cb\u5c07\u8cc7\u6599\u7d50\u69cb\u5206\u70ba\u7dda\u6027\uff08\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\uff09\u548c\u975e\u7dda\u6027\uff08\u6a39\u3001\u5716\u3001\u5806\u7a4d\uff09\u5169\u7a2e\u3002\u96dc\u6e4a\u8868\u7684\u5be6\u73fe\u53ef\u80fd\u540c\u6642\u5305\u542b\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u548c\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u7576\u7a0b\u5f0f\u57f7\u884c\u6642\uff0c\u8cc7\u6599\u88ab\u5132\u5b58\u5728\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u4e2d\u3002\u6bcf\u500b\u8a18\u61b6\u9ad4\u7a7a\u9593\u90fd\u64c1\u6709\u5c0d\u61c9\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\uff0c\u7a0b\u5f0f\u900f\u904e\u9019\u4e9b\u8a18\u61b6\u9ad4\u4f4d\u5740\u8a2a\u554f\u8cc7\u6599\u3002</li> <li>\u7269\u7406\u7d50\u69cb\u4e3b\u8981\u5206\u70ba\u9023\u7e8c\u7a7a\u9593\u5132\u5b58\uff08\u9663\u5217\uff09\u548c\u5206\u6563\u7a7a\u9593\u5132\u5b58\uff08\u93c8\u7d50\u4e32\u5217\uff09\u3002\u6240\u6709\u8cc7\u6599\u7d50\u69cb\u90fd\u662f\u7531\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u6216\u5169\u8005\u7684\u7d44\u5408\u5be6\u73fe\u7684\u3002</li> <li>\u8a08\u7b97\u6a5f\u4e2d\u7684\u57fa\u672c\u8cc7\u6599\u578b\u5225\u5305\u62ec\u6574\u6578 <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \uff0c\u6d6e\u9ede\u6578 <code>float</code>\u3001<code>double</code> \uff0c\u5b57\u5143 <code>char</code> \u548c\u5e03\u6797 <code>bool</code> \u3002\u5b83\u5011\u7684\u53d6\u503c\u7bc4\u570d\u53d6\u6c7a\u65bc\u4f54\u7528\u7a7a\u9593\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002</li> <li>\u539f\u78bc\u3001\u4e00\u88dc\u6578\u548c\u4e8c\u88dc\u6578\u662f\u5728\u8a08\u7b97\u6a5f\u4e2d\u7de8\u78bc\u6578\u5b57\u7684\u4e09\u7a2e\u65b9\u6cd5\uff0c\u5b83\u5011\u4e4b\u9593\u53ef\u4ee5\u76f8\u4e92\u8f49\u63db\u3002\u6574\u6578\u7684\u539f\u78bc\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u865f\u4f4d\uff0c\u5176\u9918\u4f4d\u662f\u6578\u5b57\u7684\u503c\u3002</li> <li>\u6574\u6578\u5728\u8a08\u7b97\u6a5f\u4e2d\u662f\u4ee5\u4e8c\u88dc\u6578\u7684\u5f62\u5f0f\u5132\u5b58\u7684\u3002\u5728\u4e8c\u88dc\u6578\u8868\u793a\u4e0b\uff0c\u8a08\u7b97\u6a5f\u53ef\u4ee5\u5c0d\u6b63\u6578\u548c\u8ca0\u6578\u7684\u52a0\u6cd5\u4e00\u8996\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u70ba\u6e1b\u6cd5\u64cd\u4f5c\u55ae\u7368\u8a2d\u8a08\u7279\u6b8a\u7684\u786c\u9ad4\u96fb\u8def\uff0c\u4e26\u4e14\u4e0d\u5b58\u5728\u6b63\u8ca0\u96f6\u6b67\u7fa9\u7684\u554f\u984c\u3002</li> <li>\u6d6e\u9ede\u6578\u7684\u7de8\u78bc\u7531 1 \u4f4d\u7b26\u865f\u4f4d\u30018 \u4f4d\u6307\u6578\u4f4d\u548c 23 \u4f4d\u5206\u6578\u4f4d\u69cb\u6210\u3002\u7531\u65bc\u5b58\u5728\u6307\u6578\u4f4d\uff0c\u56e0\u6b64\u6d6e\u9ede\u6578\u7684\u53d6\u503c\u7bc4\u570d\u9060\u5927\u65bc\u6574\u6578\uff0c\u4ee3\u50f9\u662f\u72a7\u7272\u4e86\u7cbe\u5ea6\u3002</li> <li>ASCII \u78bc\u662f\u6700\u65e9\u51fa\u73fe\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u9577\u5ea6\u70ba 1 \u4f4d\u5143\u7d44\uff0c\u5171\u6536\u9304 127 \u500b\u5b57\u5143\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u9304\u5169\u842c\u591a\u500b\u6f22\u5b57\u3002Unicode \u81f4\u529b\u65bc\u63d0\u4f9b\u4e00\u500b\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6a19\u6e96\uff0c\u6536\u9304\u4e16\u754c\u4e0a\u5404\u7a2e\u8a9e\u8a00\u7684\u5b57\u5143\uff0c\u5f9e\u800c\u89e3\u6c7a\u7531\u65bc\u5b57\u5143\u7de8\u78bc\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5c0e\u81f4\u7684\u4e82\u78bc\u554f\u984c\u3002</li> <li>UTF-8 \u662f\u6700\u53d7\u6b61\u8fce\u7684 Unicode \u7de8\u78bc\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u7a2e\u8b8a\u9577\u7684\u7de8\u78bc\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u64f4\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5132\u5b58\u7a7a\u9593\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u9577\u7684\u7de8\u78bc\u65b9\u6cd5\u3002\u5728\u7de8\u78bc\u4e2d\u6587\u6642\uff0cUTF-16 \u4f54\u7528\u7684\u7a7a\u9593\u6bd4 UTF-8 \u66f4\u5c0f\u3002Java \u548c C# \u7b49\u7a0b\u5f0f\u8a9e\u8a00\u9810\u8a2d\u4f7f\u7528 UTF-16 \u7de8\u78bc\u3002</li> </ul>"},{"location":"chapter_data_structure/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u70ba\u4ec0\u9ebc\u96dc\u6e4a\u8868\u540c\u6642\u5305\u542b\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u548c\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff1f</p> <p>\u96dc\u6e4a\u8868\u5e95\u5c64\u662f\u9663\u5217\uff0c\u800c\u70ba\u4e86\u89e3\u6c7a\u96dc\u6e4a\u885d\u7a81\uff0c\u6211\u5011\u53ef\u80fd\u6703\u4f7f\u7528\u201c\u93c8\u5f0f\u4f4d\u5740\u201d\uff08\u5f8c\u7e8c\u201c\u96dc\u6e4a\u885d\u7a81\u201d\u7ae0\u7bc0\u6703\u8b1b\uff09\uff1a\u9663\u5217\u4e2d\u6bcf\u500b\u6876\u6307\u5411\u4e00\u500b\u93c8\u7d50\u4e32\u5217\uff0c\u7576\u93c8\u7d50\u4e32\u5217\u9577\u5ea6\u8d85\u904e\u4e00\u5b9a\u95be\u503c\u6642\uff0c\u53c8\u53ef\u80fd\u88ab\u8f49\u5316\u70ba\u6a39\uff08\u901a\u5e38\u70ba\u7d05\u9ed1\u6a39\uff09\u3002</p> <p>\u5f9e\u5132\u5b58\u7684\u89d2\u5ea6\u4f86\u770b\uff0c\u96dc\u6e4a\u8868\u7684\u5e95\u5c64\u662f\u9663\u5217\uff0c\u5176\u4e2d\u6bcf\u4e00\u500b\u6876\u69fd\u4f4d\u53ef\u80fd\u5305\u542b\u4e00\u500b\u503c\uff0c\u4e5f\u53ef\u80fd\u5305\u542b\u4e00\u500b\u93c8\u7d50\u4e32\u5217\u6216\u4e00\u68f5\u6a39\u3002\u56e0\u6b64\uff0c\u96dc\u6e4a\u8868\u53ef\u80fd\u540c\u6642\u5305\u542b\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff08\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\uff09\u548c\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff08\u6a39\uff09\u3002</p> <p>Q\uff1a<code>char</code> \u578b\u5225\u7684\u9577\u5ea6\u662f 1 \u4f4d\u5143\u7d44\u55ce\uff1f</p> <p><code>char</code> \u578b\u5225\u7684\u9577\u5ea6\u7531\u7a0b\u5f0f\u8a9e\u8a00\u63a1\u7528\u7684\u7de8\u78bc\u65b9\u6cd5\u6c7a\u5b9a\u3002\u4f8b\u5982\uff0cJava\u3001JavaScript\u3001TypeScript\u3001C# \u90fd\u63a1\u7528 UTF-16 \u7de8\u78bc\uff08\u5132\u5b58 Unicode \u78bc\u9ede\uff09\uff0c\u56e0\u6b64 <code>char</code> \u578b\u5225\u7684\u9577\u5ea6\u70ba 2 \u4f4d\u5143\u7d44\u3002</p> <p>Q\uff1a\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u8cc7\u6599\u7d50\u69cb\u4e5f\u7a31\u201c\u975c\u614b\u8cc7\u6599\u7d50\u69cb\u201d \u662f\u5426\u6709\u6b67\u7fa9\uff1f\u5806\u758a\u4e5f\u53ef\u4ee5\u9032\u884c\u51fa\u5806\u758a\u548c\u5165\u5806\u758a\u7b49\u64cd\u4f5c\uff0c\u9019\u4e9b\u64cd\u4f5c\u90fd\u662f\u201c\u52d5\u614b\u201d\u7684\u3002</p> <p>\u5806\u758a\u78ba\u5be6\u53ef\u4ee5\u5be6\u73fe\u52d5\u614b\u7684\u8cc7\u6599\u64cd\u4f5c\uff0c\u4f46\u8cc7\u6599\u7d50\u69cb\u4ecd\u7136\u662f\u201c\u975c\u614b\u201d\uff08\u9577\u5ea6\u4e0d\u53ef\u8b8a\uff09\u7684\u3002\u5118\u7ba1\u57fa\u65bc\u9663\u5217\u7684\u8cc7\u6599\u7d50\u69cb\u53ef\u4ee5\u52d5\u614b\u5730\u65b0\u589e\u6216\u522a\u9664\u5143\u7d20\uff0c\u4f46\u5b83\u5011\u7684\u5bb9\u91cf\u662f\u56fa\u5b9a\u7684\u3002\u5982\u679c\u8cc7\u6599\u91cf\u8d85\u51fa\u4e86\u9810\u5206\u914d\u7684\u5927\u5c0f\uff0c\u5c31\u9700\u8981\u5efa\u7acb\u4e00\u500b\u65b0\u7684\u66f4\u5927\u7684\u9663\u5217\uff0c\u4e26\u5c07\u820a\u9663\u5217\u7684\u5167\u5bb9\u8907\u88fd\u5230\u65b0\u9663\u5217\u4e2d\u3002</p> <p>Q\uff1a\u5728\u69cb\u5efa\u5806\u758a\uff08\u4f47\u5217\uff09\u7684\u6642\u5019\uff0c\u672a\u6307\u5b9a\u5b83\u7684\u5927\u5c0f\uff0c\u70ba\u4ec0\u9ebc\u5b83\u5011\u662f\u201c\u975c\u614b\u8cc7\u6599\u7d50\u69cb\u201d\u5462\uff1f</p> <p>\u5728\u9ad8\u968e\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\uff0c\u6211\u5011\u7121\u9808\u4eba\u5de5\u6307\u5b9a\u5806\u758a\uff08\u4f47\u5217\uff09\u7684\u521d\u59cb\u5bb9\u91cf\uff0c\u9019\u500b\u5de5\u4f5c\u7531\u985e\u5225\u5167\u90e8\u81ea\u52d5\u5b8c\u6210\u3002\u4f8b\u5982\uff0cJava \u7684 <code>ArrayList</code> \u7684\u521d\u59cb\u5bb9\u91cf\u901a\u5e38\u70ba 10\u3002\u53e6\u5916\uff0c\u64f4\u5bb9\u64cd\u4f5c\u4e5f\u662f\u81ea\u52d5\u5be6\u73fe\u7684\u3002\u8a73\u898b\u5f8c\u7e8c\u7684\u201c\u4e32\u5217\u201d\u7ae0\u7bc0\u3002</p> <p>Q\uff1a\u539f\u78bc\u8f49\u4e8c\u88dc\u6578\u7684\u65b9\u6cd5\u662f\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\uff0c\u90a3\u9ebc\u4e8c\u88dc\u6578\u8f49\u539f\u78bc\u61c9\u8a72\u662f\u9006\u904b\u7b97\u201c\u5148\u6e1b 1 \u5f8c\u53d6\u53cd\u201d\uff0c\u800c\u4e8c\u88dc\u6578\u8f49\u539f\u78bc\u4e5f\u4e00\u6a23\u53ef\u4ee5\u900f\u904e\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u5f97\u5230\uff0c\u9019\u662f\u70ba\u4ec0\u9ebc\u5462\uff1f</p> <p>\u9019\u662f\u56e0\u70ba\u539f\u78bc\u548c\u4e8c\u88dc\u6578\u7684\u76f8\u4e92\u8f49\u63db\u5be6\u969b\u4e0a\u662f\u8a08\u7b97\u201c\u88dc\u6578\u201d\u7684\u904e\u7a0b\u3002\u6211\u5011\u5148\u7d66\u51fa\u88dc\u6578\u7684\u5b9a\u7fa9\uff1a\u5047\u8a2d \\(a + b = c\\) \uff0c\u90a3\u9ebc\u6211\u5011\u7a31 \\(a\\) \u662f \\(b\\) \u5230 \\(c\\) \u7684\u88dc\u6578\uff0c\u53cd\u4e4b\u4e5f\u7a31 \\(b\\) \u662f \\(a\\) \u5230 \\(c\\) \u7684\u88dc\u6578\u3002</p> <p>\u7d66\u5b9a\u4e00\u500b \\(n = 4\\) \u4f4d\u9577\u5ea6\u7684\u4e8c\u9032\u4f4d\u5236\u6578 \\(0010\\) \uff0c\u5982\u679c\u5c07\u9019\u500b\u6578\u5b57\u770b\u4f5c\u539f\u78bc\uff08\u4e0d\u8003\u616e\u7b26\u865f\u4f4d\uff09\uff0c\u90a3\u9ebc\u5b83\u7684\u4e8c\u88dc\u6578\u9700\u900f\u904e\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u5f97\u5230\uff1a</p> \\[ 0010 \\rightarrow 1101 \\rightarrow 1110 \\] <p>\u6211\u5011\u6703\u767c\u73fe\uff0c\u539f\u78bc\u548c\u4e8c\u88dc\u6578\u7684\u548c\u662f \\(0010 + 1110 = 10000\\) \uff0c\u4e5f\u5c31\u662f\u8aaa\uff0c\u4e8c\u88dc\u6578 \\(1110\\) \u662f\u539f\u78bc \\(0010\\) \u5230 \\(10000\\) \u7684\u201c\u88dc\u6578\u201d\u3002\u9019\u610f\u5473\u8457\u4e0a\u8ff0\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u5be6\u969b\u4e0a\u662f\u8a08\u7b97\u5230 \\(10000\\) \u7684\u88dc\u6578\u7684\u904e\u7a0b\u3002</p> <p>\u90a3\u9ebc\uff0c\u4e8c\u88dc\u6578 \\(1110\\) \u5230 \\(10000\\) \u7684\u201c\u88dc\u6578\u201d\u662f\u591a\u5c11\u5462\uff1f\u6211\u5011\u4f9d\u7136\u53ef\u4ee5\u7528\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u5f97\u5230\u5b83\uff1a</p> \\[ 1110 \\rightarrow 0001 \\rightarrow 0010 \\] <p>\u63db\u53e5\u8a71\u8aaa\uff0c\u539f\u78bc\u548c\u4e8c\u88dc\u6578\u4e92\u70ba\u5c0d\u65b9\u5230 \\(10000\\) \u7684\u201c\u88dc\u6578\u201d\uff0c\u56e0\u6b64\u201c\u539f\u78bc\u8f49\u4e8c\u88dc\u6578\u201d\u548c\u201c\u4e8c\u88dc\u6578\u8f49\u539f\u78bc\u201d\u53ef\u4ee5\u7528\u76f8\u540c\u7684\u64cd\u4f5c\uff08\u5148\u53d6\u53cd\u5f8c\u52a0 1 \uff09\u5be6\u73fe\u3002</p> <p>\u7576\u7136\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u7528\u9006\u904b\u7b97\u4f86\u6c42\u4e8c\u88dc\u6578 \\(1110\\) \u7684\u539f\u78bc\uff0c\u5373\u201c\u5148\u6e1b 1 \u5f8c\u53d6\u53cd\u201d\uff1a</p> \\[ 1110 \\rightarrow 1101 \\rightarrow 0010 \\] <p>\u7e3d\u7d50\u4f86\u770b\uff0c\u201c\u5148\u53d6\u53cd\u5f8c\u52a0 1\u201d\u548c\u201c\u5148\u6e1b 1 \u5f8c\u53d6\u53cd\u201d\u9019\u5169\u7a2e\u904b\u7b97\u90fd\u662f\u5728\u8a08\u7b97\u5230 \\(10000\\) \u7684\u88dc\u6578\uff0c\u5b83\u5011\u662f\u7b49\u50f9\u7684\u3002</p> <p>\u672c\u8cea\u4e0a\u770b\uff0c\u201c\u53d6\u53cd\u201d\u64cd\u4f5c\u5be6\u969b\u4e0a\u662f\u6c42\u5230 \\(1111\\) \u7684\u88dc\u6578\uff08\u56e0\u70ba\u6046\u6709 <code>\u539f\u78bc + \u4e00\u88dc\u6578 = 1111</code>\uff09\uff1b\u800c\u5728\u4e00\u88dc\u6578\u57fa\u790e\u4e0a\u518d\u52a0 1 \u5f97\u5230\u7684\u4e8c\u88dc\u6578\uff0c\u5c31\u662f\u5230 \\(10000\\) \u7684\u88dc\u6578\u3002</p> <p>\u4e0a\u8ff0\u4ee5 \\(n = 4\\) \u70ba\u4f8b\uff0c\u5176\u53ef\u88ab\u63a8\u5ee3\u81f3\u4efb\u610f\u4f4d\u6578\u7684\u4e8c\u9032\u4f4d\u5236\u6578\u3002</p>"},{"location":"chapter_divide_and_conquer/","title":"\u7b2c 12 \u7ae0 \u00a0 \u5206\u6cbb","text":"<p>Abstract</p> <p>\u96e3\u984c\u88ab\u9010\u5c64\u62c6\u89e3\uff0c\u6bcf\u4e00\u6b21\u7684\u62c6\u89e3\u90fd\u4f7f\u5b83\u8b8a\u5f97\u66f4\u70ba\u7c21\u55ae\u3002</p> <p>\u5206\u800c\u6cbb\u4e4b\u63ed\u793a\u4e86\u4e00\u500b\u91cd\u8981\u7684\u4e8b\u5be6\uff1a\u5f9e\u7c21\u55ae\u505a\u8d77\uff0c\u4e00\u5207\u90fd\u4e0d\u518d\u8907\u96dc\u3002</p>"},{"location":"chapter_divide_and_conquer/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>12.1 \u00a0 \u5206\u6cbb\u6f14\u7b97\u6cd5</li> <li>12.2 \u00a0 \u5206\u6cbb\u641c\u5c0b\u7b56\u7565</li> <li>12.3 \u00a0 \u69cb\u5efa\u6a39\u554f\u984c</li> <li>12.4 \u00a0 \u6cb3\u5167\u5854\u554f\u984c</li> <li>12.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_divide_and_conquer/binary_search_recur/","title":"12.2 \u00a0 \u5206\u6cbb\u641c\u5c0b\u7b56\u7565","text":"<p>\u6211\u5011\u5df2\u7d93\u5b78\u904e\uff0c\u641c\u5c0b\u6f14\u7b97\u6cd5\u5206\u70ba\u5169\u5927\u985e\u3002</p> <ul> <li>\u66b4\u529b\u641c\u5c0b\uff1a\u5b83\u900f\u904e\u8d70\u8a2a\u8cc7\u6599\u7d50\u69cb\u5be6\u73fe\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</li> <li>\u81ea\u9069\u61c9\u641c\u5c0b\uff1a\u5b83\u5229\u7528\u7279\u6709\u7684\u8cc7\u6599\u7d44\u7e54\u5f62\u5f0f\u6216\u5148\u9a57\u8cc7\u8a0a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \u7684\u641c\u5c0b\u6f14\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u5be6\u73fe\u7684\uff0c\u4f8b\u5982\u4e8c\u5206\u641c\u5c0b\u548c\u6a39\u3002</p> <ul> <li>\u4e8c\u5206\u641c\u5c0b\u7684\u6bcf\u4e00\u6b65\u90fd\u5c07\u554f\u984c\uff08\u5728\u9663\u5217\u4e2d\u641c\u7d22\u76ee\u6a19\u5143\u7d20\uff09\u5206\u89e3\u70ba\u4e00\u500b\u5c0f\u554f\u984c\uff08\u5728\u9663\u5217\u7684\u4e00\u534a\u4e2d\u641c\u7d22\u76ee\u6a19\u5143\u7d20\uff09\uff0c\u9019\u500b\u904e\u7a0b\u4e00\u76f4\u6301\u7e8c\u5230\u9663\u5217\u70ba\u7a7a\u6216\u627e\u5230\u76ee\u6a19\u5143\u7d20\u70ba\u6b62\u3002</li> <li>\u6a39\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4ee3\u8868\uff0c\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u3001AVL \u6a39\u3001\u5806\u7a4d\u7b49\u8cc7\u6599\u7d50\u69cb\u4e2d\uff0c\u5404\u7a2e\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u7686\u70ba \\(O(\\log n)\\) \u3002</li> </ul> <p>\u4e8c\u5206\u641c\u5c0b\u7684\u5206\u6cbb\u7b56\u7565\u5982\u4e0b\u6240\u793a\u3002</p> <ul> <li>\u554f\u984c\u53ef\u4ee5\u5206\u89e3\uff1a\u4e8c\u5206\u641c\u5c0b\u905e\u8ff4\u5730\u5c07\u539f\u554f\u984c\uff08\u5728\u9663\u5217\u4e2d\u9032\u884c\u67e5\u8a62\uff09\u5206\u89e3\u70ba\u5b50\u554f\u984c\uff08\u5728\u9663\u5217\u7684\u4e00\u534a\u4e2d\u9032\u884c\u67e5\u8a62\uff09\uff0c\u9019\u662f\u900f\u904e\u6bd4\u8f03\u4e2d\u9593\u5143\u7d20\u548c\u76ee\u6a19\u5143\u7d20\u4f86\u5be6\u73fe\u7684\u3002</li> <li>\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff1a\u5728\u4e8c\u5206\u641c\u5c0b\u4e2d\uff0c\u6bcf\u8f2a\u53ea\u8655\u7406\u4e00\u500b\u5b50\u554f\u984c\uff0c\u5b83\u4e0d\u53d7\u5176\u4ed6\u5b50\u554f\u984c\u7684\u5f71\u97ff\u3002</li> <li>\u5b50\u554f\u984c\u7684\u89e3\u7121\u9808\u5408\u4f75\uff1a\u4e8c\u5206\u641c\u5c0b\u65e8\u5728\u67e5\u8a62\u4e00\u500b\u7279\u5b9a\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5c07\u5b50\u554f\u984c\u7684\u89e3\u9032\u884c\u5408\u4f75\u3002\u7576\u5b50\u554f\u984c\u5f97\u5230\u89e3\u6c7a\u6642\uff0c\u539f\u554f\u984c\u4e5f\u6703\u540c\u6642\u5f97\u5230\u89e3\u6c7a\u3002</li> </ul> <p>\u5206\u6cbb\u80fd\u5920\u63d0\u5347\u641c\u5c0b\u6548\u7387\uff0c\u672c\u8cea\u4e0a\u662f\u56e0\u70ba\u66b4\u529b\u641c\u5c0b\u6bcf\u8f2a\u53ea\u80fd\u6392\u9664\u4e00\u500b\u9078\u9805\uff0c\u800c\u5206\u6cbb\u641c\u5c0b\u6bcf\u8f2a\u53ef\u4ee5\u6392\u9664\u4e00\u534a\u9078\u9805\u3002</p>"},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","title":"1. \u00a0 \u57fa\u65bc\u5206\u6cbb\u5be6\u73fe\u4e8c\u5206\u641c\u5c0b","text":"<p>\u5728\u4e4b\u524d\u7684\u7ae0\u7bc0\u4e2d\uff0c\u4e8c\u5206\u641c\u5c0b\u662f\u57fa\u65bc\u905e\u63a8\uff08\u8fed\u4ee3\uff09\u5be6\u73fe\u7684\u3002\u73fe\u5728\u6211\u5011\u57fa\u65bc\u5206\u6cbb\uff08\u905e\u8ff4\uff09\u4f86\u5be6\u73fe\u5b83\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u6709\u5e8f\u9663\u5217 <code>nums</code> \uff0c\u5176\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\uff0c\u8acb\u67e5\u8a62\u5143\u7d20 <code>target</code> \u3002</p> <p>\u5f9e\u5206\u6cbb\u89d2\u5ea6\uff0c\u6211\u5011\u5c07\u641c\u5c0b\u5340\u9593 \\([i, j]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\u8a18\u70ba \\(f(i, j)\\) \u3002</p> <p>\u4ee5\u539f\u554f\u984c \\(f(0, n-1)\\) \u70ba\u8d77\u59cb\u9ede\uff0c\u900f\u904e\u4ee5\u4e0b\u6b65\u9a5f\u9032\u884c\u4e8c\u5206\u641c\u5c0b\u3002</p> <ol> <li>\u8a08\u7b97\u641c\u5c0b\u5340\u9593 \\([i, j]\\) \u7684\u4e2d\u9ede \\(m\\) \uff0c\u6839\u64da\u5b83\u6392\u9664\u4e00\u534a\u641c\u5c0b\u5340\u9593\u3002</li> <li>\u905e\u8ff4\u6c42\u89e3\u898f\u6a21\u6e1b\u5c0f\u4e00\u534a\u7684\u5b50\u554f\u984c\uff0c\u53ef\u80fd\u70ba \\(f(i, m-1)\\) \u6216 \\(f(m+1, j)\\) \u3002</li> <li>\u8ff4\u5708\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\uff0c\u76f4\u81f3\u627e\u5230 <code>target</code> \u6216\u5340\u9593\u70ba\u7a7a\u6642\u8fd4\u56de\u3002</li> </ol> <p>\u5716 12-4 \u5c55\u793a\u4e86\u5728\u9663\u5217\u4e2d\u4e8c\u5206\u641c\u5c0b\u5143\u7d20 \\(6\\) \u7684\u5206\u6cbb\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 12-4 \u00a0 \u4e8c\u5206\u641c\u5c0b\u7684\u5206\u6cbb\u904e\u7a0b </p> <p>\u5728\u5be6\u73fe\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u6211\u5011\u5ba3\u544a\u4e00\u500b\u905e\u8ff4\u51fd\u5f0f <code>dfs()</code> \u4f86\u6c42\u89e3\u554f\u984c \\(f(i, j)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j)\"\"\"\n    # \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if i &gt; j:\n        return -1\n    # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1)\n    else:\n        # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\"\"\"\n    n = len(nums)\n    # \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // \u5982\u679c\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u6c92\u6709\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    //    \u8a08\u7b97\u7d22\u5f15\u4e2d\u9ede\n    m := i + ((j - i) &gt;&gt; 1)\n    //\u5224\u65b7\u4e2d\u9ede\u8207\u76ee\u6a19\u5143\u7d20\u5927\u5c0f\n    if nums[m] &lt; target {\n        // \u5c0f\u65bc\u5247\u905e\u8ff4\u53f3\u534a\u9663\u5217\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // \u5927\u65bc\u5247\u905e\u8ff4\u5de6\u534a\u9663\u5217\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m-1)\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    dfs(nums: nums, target: target, i: nums.startIndex, j: nums.endIndex - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n  if (i &gt; j) {\n    return -1;\n  }\n  // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return m;\n  }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // \u6c42\u89e3\u554f\u984c f(0, n-1)\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if i &gt; j {\n        return -1;\n    }\n    let m: i32 = i + (j - i) / 2;\n    if nums[m as usize] &lt; target {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) */\nfun dfs(\n    nums: IntArray,\n    target: Int,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1\n    }\n    // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    val m = (i + j) / 2\n    return if (nums[m] &lt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n        dfs(nums, target, m + 1, j)\n    } else if (nums[m] &gt; target) {\n        // \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n        dfs(nums, target, i, m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        m\n    }\n}\n\n/* \u4e8c\u5206\u641c\u5c0b */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    val n = nums.size\n    // \u6c42\u89e3\u554f\u984c f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\uff1a\u554f\u984c f(i, j) ###\ndef dfs(nums, target, i, j)\n  # \u82e5\u5340\u9593\u70ba\u7a7a\uff0c\u4ee3\u8868\u7121\u76ee\u6a19\u5143\u7d20\uff0c\u5247\u8fd4\u56de -1\n  return -1 if i &gt; j\n\n  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n  m = (i + j) / 2\n\n  if nums[m] &lt; target\n    # \u905e\u8ff4\u5b50\u554f\u984c f(m+1, j)\n    return dfs(nums, target, m + 1, j)\n  elsif nums[m] &gt; target\n    # \u905e\u8ff4\u5b50\u554f\u984c f(i, m-1)\n    return dfs(nums, target, i, m - 1)\n  else\n    # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return m\n  end\nend\n\n### \u4e8c\u5206\u641c\u5c0b ###\ndef binary_search(nums, target)\n  n = nums.length\n  # \u6c42\u89e3\u554f\u984c f(0, n-1)\n  dfs(nums, target, 0, n - 1)\nend\n</code></pre> binary_search_recur.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{binarySearch}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","title":"12.3 \u00a0 \u69cb\u5efa\u4e8c\u5143\u6a39\u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u68f5\u4e8c\u5143\u6a39\u7684\u524d\u5e8f\u8d70\u8a2a <code>preorder</code> \u548c\u4e2d\u5e8f\u8d70\u8a2a <code>inorder</code> \uff0c\u8acb\u5f9e\u4e2d\u69cb\u5efa\u4e8c\u5143\u6a39\uff0c\u8fd4\u56de\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede\u3002\u5047\u8a2d\u4e8c\u5143\u6a39\u4e2d\u6c92\u6709\u503c\u91cd\u8907\u7684\u7bc0\u9ede\uff08\u5982\u5716 12-5 \u6240\u793a\uff09\u3002</p> <p></p> <p> \u5716 12-5 \u00a0 \u69cb\u5efa\u4e8c\u5143\u6a39\u7684\u793a\u4f8b\u8cc7\u6599 </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","title":"1. \u00a0 \u5224\u65b7\u662f\u5426\u70ba\u5206\u6cbb\u554f\u984c","text":"<p>\u539f\u554f\u984c\u5b9a\u7fa9\u70ba\u5f9e <code>preorder</code> \u548c <code>inorder</code> \u69cb\u5efa\u4e8c\u5143\u6a39\uff0c\u662f\u4e00\u500b\u5178\u578b\u7684\u5206\u6cbb\u554f\u984c\u3002</p> <ul> <li>\u554f\u984c\u53ef\u4ee5\u5206\u89e3\uff1a\u5f9e\u5206\u6cbb\u7684\u89d2\u5ea6\u5207\u5165\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u539f\u554f\u984c\u5283\u5206\u70ba\u5169\u500b\u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\u3001\u69cb\u5efa\u53f3\u5b50\u6a39\uff0c\u52a0\u4e0a\u4e00\u6b65\u64cd\u4f5c\uff1a\u521d\u59cb\u5316\u6839\u7bc0\u9ede\u3002\u800c\u5c0d\u65bc\u6bcf\u68f5\u5b50\u6a39\uff08\u5b50\u554f\u984c\uff09\uff0c\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u8907\u7528\u4ee5\u4e0a\u5283\u5206\u65b9\u6cd5\uff0c\u5c07\u5176\u5283\u5206\u70ba\u66f4\u5c0f\u7684\u5b50\u6a39\uff08\u5b50\u554f\u984c\uff09\uff0c\u76f4\u81f3\u9054\u5230\u6700\u5c0f\u5b50\u554f\u984c\uff08\u7a7a\u5b50\u6a39\uff09\u6642\u7d42\u6b62\u3002</li> <li>\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff1a\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\u662f\u76f8\u4e92\u7368\u7acb\u7684\uff0c\u5b83\u5011\u4e4b\u9593\u6c92\u6709\u4ea4\u96c6\u3002\u5728\u69cb\u5efa\u5de6\u5b50\u6a39\u6642\uff0c\u6211\u5011\u53ea\u9700\u95dc\u6ce8\u4e2d\u5e8f\u8d70\u8a2a\u548c\u524d\u5e8f\u8d70\u8a2a\u4e2d\u8207\u5de6\u5b50\u6a39\u5c0d\u61c9\u7684\u90e8\u5206\u3002\u53f3\u5b50\u6a39\u540c\u7406\u3002</li> <li>\u5b50\u554f\u984c\u7684\u89e3\u53ef\u4ee5\u5408\u4f75\uff1a\u4e00\u65e6\u5f97\u5230\u4e86\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\uff08\u5b50\u554f\u984c\u7684\u89e3\uff09\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5c07\u5b83\u5011\u9023\u7d50\u5230\u6839\u7bc0\u9ede\u4e0a\uff0c\u5f97\u5230\u539f\u554f\u984c\u7684\u89e3\u3002</li> </ul>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2","title":"2. \u00a0 \u5982\u4f55\u5283\u5206\u5b50\u6a39","text":"<p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u9019\u9053\u984c\u53ef\u4ee5\u4f7f\u7528\u5206\u6cbb\u4f86\u6c42\u89e3\uff0c\u4f46\u5982\u4f55\u900f\u904e\u524d\u5e8f\u8d70\u8a2a <code>preorder</code> \u548c\u4e2d\u5e8f\u8d70\u8a2a <code>inorder</code> \u4f86\u5283\u5206\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\u5462\uff1f</p> <p>\u6839\u64da\u5b9a\u7fa9\uff0c<code>preorder</code> \u548c <code>inorder</code> \u90fd\u53ef\u4ee5\u5283\u5206\u70ba\u4e09\u500b\u90e8\u5206\u3002</p> <ul> <li>\u524d\u5e8f\u8d70\u8a2a\uff1a<code>[ \u6839\u7bc0\u9ede | \u5de6\u5b50\u6a39 | \u53f3\u5b50\u6a39 ]</code> \uff0c\u4f8b\u5982\u5716 12-5 \u7684\u6a39\u5c0d\u61c9 <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> <li>\u4e2d\u5e8f\u8d70\u8a2a\uff1a<code>[ \u5de6\u5b50\u6a39 | \u6839\u7bc0\u9ede \uff5c \u53f3\u5b50\u6a39 ]</code> \uff0c\u4f8b\u5982\u5716 12-5 \u7684\u6a39\u5c0d\u61c9 <code>[ 9 | 3 | 1 2 7 ]</code> \u3002</li> </ul> <p>\u4ee5\u4e0a\u5716\u8cc7\u6599\u70ba\u4f8b\uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u5716 12-6 \u6240\u793a\u7684\u6b65\u9a5f\u5f97\u5230\u5283\u5206\u7d50\u679c\u3002</p> <ol> <li>\u524d\u5e8f\u8d70\u8a2a\u7684\u9996\u5143\u7d20 3 \u662f\u6839\u7bc0\u9ede\u7684\u503c\u3002</li> <li>\u67e5\u8a62\u6839\u7bc0\u9ede 3 \u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u5229\u7528\u8a72\u7d22\u5f15\u53ef\u5c07 <code>inorder</code> \u5283\u5206\u70ba <code>[ 9 | 3 \uff5c 1 2 7 ]</code> \u3002</li> <li>\u6839\u64da <code>inorder</code> \u7684\u5283\u5206\u7d50\u679c\uff0c\u6613\u5f97\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u5206\u5225\u70ba 1 \u548c 3 \uff0c\u5f9e\u800c\u53ef\u5c07 <code>preorder</code> \u5283\u5206\u70ba <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> </ol> <p></p> <p> \u5716 12-6 \u00a0 \u5728\u524d\u5e8f\u8d70\u8a2a\u548c\u4e2d\u5e8f\u8d70\u8a2a\u4e2d\u5283\u5206\u5b50\u6a39 </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","title":"3. \u00a0 \u57fa\u65bc\u8b8a\u6578\u63cf\u8ff0\u5b50\u6a39\u5340\u9593","text":"<p>\u6839\u64da\u4ee5\u4e0a\u5283\u5206\u65b9\u6cd5\uff0c\u6211\u5011\u5df2\u7d93\u5f97\u5230\u6839\u7bc0\u9ede\u3001\u5de6\u5b50\u6a39\u3001\u53f3\u5b50\u6a39\u5728 <code>preorder</code> \u548c <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u5340\u9593\u3002\u800c\u70ba\u4e86\u63cf\u8ff0\u9019\u4e9b\u7d22\u5f15\u5340\u9593\uff0c\u6211\u5011\u9700\u8981\u85c9\u52a9\u5e7e\u500b\u6307\u6a19\u8b8a\u6578\u3002</p> <ul> <li>\u5c07\u7576\u524d\u6a39\u7684\u6839\u7bc0\u9ede\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\u8a18\u70ba \\(i\\) \u3002</li> <li>\u5c07\u7576\u524d\u6a39\u7684\u6839\u7bc0\u9ede\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u8a18\u70ba \\(m\\) \u3002</li> <li>\u5c07\u7576\u524d\u6a39\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u5340\u9593\u8a18\u70ba \\([l, r]\\) \u3002</li> </ul> <p>\u5982\u8868 12-1 \u6240\u793a\uff0c\u900f\u904e\u4ee5\u4e0a\u8b8a\u6578\u5373\u53ef\u8868\u793a\u6839\u7bc0\u9ede\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u4ee5\u53ca\u5b50\u6a39\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u5340\u9593\u3002</p> <p> \u8868 12-1 \u00a0 \u6839\u7bc0\u9ede\u548c\u5b50\u6a39\u5728\u524d\u5e8f\u8d70\u8a2a\u548c\u4e2d\u5e8f\u8d70\u8a2a\u4e2d\u7684\u7d22\u5f15 </p> \u6839\u7bc0\u9ede\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15 \u5b50\u6a39\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u5340\u9593 \u7576\u524d\u6a39 \\(i\\) \\([l, r]\\) \u5de6\u5b50\u6a39 \\(i + 1\\) \\([l, m-1]\\) \u53f3\u5b50\u6a39 \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>\u8acb\u6ce8\u610f\uff0c\u53f3\u5b50\u6a39\u6839\u7bc0\u9ede\u7d22\u5f15\u4e2d\u7684 \\((m-l)\\) \u7684\u542b\u7fa9\u662f\u201c\u5de6\u5b50\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u201d\uff0c\u5efa\u8b70\u7d50\u5408\u5716 12-7 \u7406\u89e3\u3002</p> <p></p> <p> \u5716 12-7 \u00a0 \u6839\u7bc0\u9ede\u548c\u5de6\u53f3\u5b50\u6a39\u7684\u7d22\u5f15\u5340\u9593\u8868\u793a </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","title":"4. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u70ba\u4e86\u63d0\u5347\u67e5\u8a62 \\(m\\) \u7684\u6548\u7387\uff0c\u6211\u5011\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u8868 <code>hmap</code> \u4f86\u5132\u5b58\u9663\u5217 <code>inorder</code> \u4e2d\u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"\u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb\"\"\"\n    # \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if r - l &lt; 0:\n        return None\n    # \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    root = TreeNode(preorder[i])\n    # \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    m = inorder_map[preorder[i]]\n    # \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"\u69cb\u5efa\u4e8c\u5143\u6a39\"\"\"\n    # \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    TreeNode *root = new TreeNode(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    TreeNode root = new TreeNode(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    int m = inorderMap.get(preorder[i]);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    TreeNode root = new(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if r-l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    root := NewTreeNode(preorder[i])\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    m := inorderMap[preorder[i]]\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if r - l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    let root = TreeNode(x: preorder[i])\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    let m = inorderMap[preorder[i]]!\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: inorder.startIndex, l: inorder.startIndex, r: inorder.endIndex - 1)\n}\n</code></pre> build_tree.js<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    const root = new TreeNode(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfunction buildTree(preorder, inorder) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n  TreeNode? root = TreeNode(preorder[i]);\n  // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n  int m = inorderMap[preorder[i]]!;\n  // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // \u8fd4\u56de\u6839\u7bc0\u9ede\n  return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfn dfs(\n    preorder: &amp;[i32],\n    inorder_map: &amp;HashMap&lt;i32, i32&gt;,\n    i: i32,\n    l: i32,\n    r: i32,\n) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if r - l &lt; 0 {\n        return None;\n    }\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    let root = TreeNode::new(preorder[i as usize]);\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    Some(root)\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root;\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.kt<pre><code>/* \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb */\nfun dfs(\n    preorder: IntArray,\n    inorderMap: Map&lt;Int?, Int?&gt;,\n    i: Int,\n    l: Int,\n    r: Int\n): TreeNode? {\n    // \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n    if (r - l &lt; 0) return null\n    // \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    val root = TreeNode(preorder[i])\n    // \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n    val m = inorderMap[preorder[i]]!!\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1)\n    // \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r)\n    // \u8fd4\u56de\u6839\u7bc0\u9ede\n    return root\n}\n\n/* \u69cb\u5efa\u4e8c\u5143\u6a39 */\nfun buildTree(preorder: IntArray, inorder: IntArray): TreeNode? {\n    // \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n    val inorderMap = HashMap&lt;Int?, Int?&gt;()\n    for (i in inorder.indices) {\n        inorderMap[inorder[i]] = i\n    }\n    val root = dfs(preorder, inorderMap, 0, 0, inorder.size - 1)\n    return root\n}\n</code></pre> build_tree.rb<pre><code>### \u69cb\u5efa\u4e8c\u5143\u6a39\uff1a\u5206\u6cbb ###\ndef dfs(preorder, inorder_map, i, l, r)\n  # \u5b50\u6a39\u5340\u9593\u70ba\u7a7a\u6642\u7d42\u6b62\n  return if r - l &lt; 0\n\n  # \u521d\u59cb\u5316\u6839\u7bc0\u9ede\n  root = TreeNode.new(preorder[i])\n  # \u67e5\u8a62 m \uff0c\u5f9e\u800c\u5283\u5206\u5de6\u53f3\u5b50\u6a39\n  m = inorder_map[preorder[i]]\n  # \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u5de6\u5b50\u6a39\n  root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n  # \u5b50\u554f\u984c\uff1a\u69cb\u5efa\u53f3\u5b50\u6a39\n  root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n\n  # \u8fd4\u56de\u6839\u7bc0\u9ede\n  root\nend\n\n### \u69cb\u5efa\u4e8c\u5143\u6a39 ###\ndef build_tree(preorder, inorder)\n  # \u521d\u59cb\u5316\u96dc\u6e4a\u8868\uff0c\u5132\u5b58 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\n  inorder_map = {}\n  inorder.each_with_index { |val, i| inorder_map[val] = i }\n  dfs(preorder, inorder_map, 0, 0, inorder.length - 1)\nend\n</code></pre> build_tree.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{buildTree}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 12-8 \u5c55\u793a\u4e86\u69cb\u5efa\u4e8c\u5143\u6a39\u7684\u905e\u8ff4\u904e\u7a0b\uff0c\u5404\u500b\u7bc0\u9ede\u662f\u5728\u5411\u4e0b\u201c\u905e\u201d\u7684\u904e\u7a0b\u4e2d\u5efa\u7acb\u7684\uff0c\u800c\u5404\u689d\u908a\uff08\u5f15\u7528\uff09\u662f\u5728\u5411\u4e0a\u201c\u8ff4\u201d\u7684\u904e\u7a0b\u4e2d\u5efa\u7acb\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 12-8 \u00a0 \u69cb\u5efa\u4e8c\u5143\u6a39\u7684\u905e\u8ff4\u904e\u7a0b </p> <p>\u6bcf\u500b\u905e\u8ff4\u51fd\u5f0f\u5167\u7684\u524d\u5e8f\u8d70\u8a2a <code>preorder</code> \u548c\u4e2d\u5e8f\u8d70\u8a2a <code>inorder</code> \u7684\u5283\u5206\u7d50\u679c\u5982\u5716 12-9 \u6240\u793a\u3002</p> <p></p> <p> \u5716 12-9 \u00a0 \u6bcf\u500b\u905e\u8ff4\u51fd\u5f0f\u4e2d\u7684\u5283\u5206\u7d50\u679c </p> <p>\u8a2d\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\(n\\) \uff0c\u521d\u59cb\u5316\u6bcf\u4e00\u500b\u7bc0\u9ede\uff08\u57f7\u884c\u4e00\u500b\u905e\u8ff4\u51fd\u5f0f <code>dfs()</code> \uff09\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002\u56e0\u6b64\u7e3d\u9ad4\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> <p>\u96dc\u6e4a\u8868\u5132\u5b58 <code>inorder</code> \u5143\u7d20\u5230\u7d22\u5f15\u7684\u5c0d\u6620\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u5373\u4e8c\u5143\u6a39\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u6642\uff0c\u905e\u8ff4\u6df1\u5ea6\u9054\u5230 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002\u56e0\u6b64\u7e3d\u9ad4\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/","title":"12.1 \u00a0 \u5206\u6cbb\u6f14\u7b97\u6cd5","text":"<p>\u5206\u6cbb\uff08divide and conquer\uff09\uff0c\u5168\u7a31\u5206\u800c\u6cbb\u4e4b\uff0c\u662f\u4e00\u7a2e\u975e\u5e38\u91cd\u8981\u4e14\u5e38\u898b\u7684\u6f14\u7b97\u6cd5\u7b56\u7565\u3002\u5206\u6cbb\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u5be6\u73fe\uff0c\u5305\u62ec\u201c\u5206\u201d\u548c\u201c\u6cbb\u201d\u5169\u500b\u6b65\u9a5f\u3002</p> <ol> <li>\u5206\uff08\u5283\u5206\u968e\u6bb5\uff09\uff1a\u905e\u8ff4\u5730\u5c07\u539f\u554f\u984c\u5206\u89e3\u70ba\u5169\u500b\u6216\u591a\u500b\u5b50\u554f\u984c\uff0c\u76f4\u81f3\u5230\u9054\u6700\u5c0f\u5b50\u554f\u984c\u6642\u7d42\u6b62\u3002</li> <li>\u6cbb\uff08\u5408\u4f75\u968e\u6bb5\uff09\uff1a\u5f9e\u5df2\u77e5\u89e3\u7684\u6700\u5c0f\u5b50\u554f\u984c\u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5730\u5c07\u5b50\u554f\u984c\u7684\u89e3\u9032\u884c\u5408\u4f75\uff0c\u5f9e\u800c\u69cb\u5efa\u51fa\u539f\u554f\u984c\u7684\u89e3\u3002</li> </ol> <p>\u5982\u5716 12-1 \u6240\u793a\uff0c\u201c\u5408\u4f75\u6392\u5e8f\u201d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u61c9\u7528\u4e4b\u4e00\u3002</p> <ol> <li>\u5206\uff1a\u905e\u8ff4\u5730\u5c07\u539f\u9663\u5217\uff08\u539f\u554f\u984c\uff09\u5283\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff08\u5b50\u554f\u984c\uff09\uff0c\u76f4\u5230\u5b50\u9663\u5217\u53ea\u5269\u4e00\u500b\u5143\u7d20\uff08\u6700\u5c0f\u5b50\u554f\u984c\uff09\u3002</li> <li>\u6cbb\uff1a\u5f9e\u5e95\u81f3\u9802\u5730\u5c07\u6709\u5e8f\u7684\u5b50\u9663\u5217\uff08\u5b50\u554f\u984c\u7684\u89e3\uff09\u9032\u884c\u5408\u4f75\uff0c\u5f9e\u800c\u5f97\u5230\u6709\u5e8f\u7684\u539f\u9663\u5217\uff08\u539f\u554f\u984c\u7684\u89e3\uff09\u3002</li> </ol> <p></p> <p> \u5716 12-1 \u00a0 \u5408\u4f75\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","title":"12.1.1 \u00a0 \u5982\u4f55\u5224\u65b7\u5206\u6cbb\u554f\u984c","text":"<p>\u4e00\u500b\u554f\u984c\u662f\u5426\u9069\u5408\u4f7f\u7528\u5206\u6cbb\u89e3\u6c7a\uff0c\u901a\u5e38\u53ef\u4ee5\u53c3\u8003\u4ee5\u4e0b\u5e7e\u500b\u5224\u65b7\u4f9d\u64da\u3002</p> <ol> <li>\u554f\u984c\u53ef\u4ee5\u5206\u89e3\uff1a\u539f\u554f\u984c\u53ef\u4ee5\u5206\u89e3\u6210\u898f\u6a21\u66f4\u5c0f\u3001\u985e\u4f3c\u7684\u5b50\u554f\u984c\uff0c\u4ee5\u53ca\u80fd\u5920\u4ee5\u76f8\u540c\u65b9\u5f0f\u905e\u8ff4\u5730\u9032\u884c\u5283\u5206\u3002</li> <li>\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff1a\u5b50\u554f\u984c\u4e4b\u9593\u6c92\u6709\u91cd\u758a\uff0c\u4e92\u4e0d\u4f9d\u8cf4\uff0c\u53ef\u4ee5\u7368\u7acb\u89e3\u6c7a\u3002</li> <li>\u5b50\u554f\u984c\u7684\u89e3\u53ef\u4ee5\u5408\u4f75\uff1a\u539f\u554f\u984c\u7684\u89e3\u900f\u904e\u5408\u4f75\u5b50\u554f\u984c\u7684\u89e3\u5f97\u4f86\u3002</li> </ol> <p>\u986f\u7136\uff0c\u5408\u4f75\u6392\u5e8f\u6eff\u8db3\u4ee5\u4e0a\u4e09\u500b\u5224\u65b7\u4f9d\u64da\u3002</p> <ol> <li>\u554f\u984c\u53ef\u4ee5\u5206\u89e3\uff1a\u905e\u8ff4\u5730\u5c07\u9663\u5217\uff08\u539f\u554f\u984c\uff09\u5283\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff08\u5b50\u554f\u984c\uff09\u3002</li> <li>\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff1a\u6bcf\u500b\u5b50\u9663\u5217\u90fd\u53ef\u4ee5\u7368\u7acb\u5730\u9032\u884c\u6392\u5e8f\uff08\u5b50\u554f\u984c\u53ef\u4ee5\u7368\u7acb\u9032\u884c\u6c42\u89e3\uff09\u3002</li> <li>\u5b50\u554f\u984c\u7684\u89e3\u53ef\u4ee5\u5408\u4f75\uff1a\u5169\u500b\u6709\u5e8f\u5b50\u9663\u5217\uff08\u5b50\u554f\u984c\u7684\u89e3\uff09\u53ef\u4ee5\u5408\u4f75\u70ba\u4e00\u500b\u6709\u5e8f\u9663\u5217\uff08\u539f\u554f\u984c\u7684\u89e3\uff09\u3002</li> </ol>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212","title":"12.1.2 \u00a0 \u900f\u904e\u5206\u6cbb\u63d0\u5347\u6548\u7387","text":"<p>\u5206\u6cbb\u4e0d\u50c5\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u6c7a\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u5f80\u5f80\u9084\u53ef\u4ee5\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\u3002\u5728\u6392\u5e8f\u6f14\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u3001\u5408\u4f75\u6392\u5e8f\u3001\u5806\u7a4d\u6392\u5e8f\u76f8\u8f03\u65bc\u9078\u64c7\u3001\u5192\u6ce1\u3001\u63d2\u5165\u6392\u5e8f\u66f4\u5feb\uff0c\u5c31\u662f\u56e0\u70ba\u5b83\u5011\u61c9\u7528\u4e86\u5206\u6cbb\u7b56\u7565\u3002</p> <p>\u90a3\u9ebc\uff0c\u6211\u5011\u4e0d\u7981\u767c\u554f\uff1a\u70ba\u4ec0\u9ebc\u5206\u6cbb\u53ef\u4ee5\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u5176\u5e95\u5c64\u908f\u8f2f\u662f\u4ec0\u9ebc\uff1f\u63db\u53e5\u8a71\u8aaa\uff0c\u5c07\u5927\u554f\u984c\u5206\u89e3\u70ba\u591a\u500b\u5b50\u554f\u984c\u3001\u89e3\u6c7a\u5b50\u554f\u984c\u3001\u5c07\u5b50\u554f\u984c\u7684\u89e3\u5408\u4f75\u70ba\u539f\u554f\u984c\u7684\u89e3\uff0c\u9019\u5e7e\u6b65\u7684\u6548\u7387\u70ba\u4ec0\u9ebc\u6bd4\u76f4\u63a5\u89e3\u6c7a\u539f\u554f\u984c\u7684\u6548\u7387\u66f4\u9ad8\uff1f\u9019\u500b\u554f\u984c\u53ef\u4ee5\u5f9e\u64cd\u4f5c\u6578\u91cf\u548c\u5e73\u884c\u8a08\u7b97\u5169\u65b9\u9762\u4f86\u8a0e\u8ad6\u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","title":"1. \u00a0 \u64cd\u4f5c\u6578\u91cf\u6700\u4f73\u5316","text":"<p>\u4ee5\u201c\u6ce1\u6cab\u6392\u5e8f\u201d\u70ba\u4f8b\uff0c\u5176\u8655\u7406\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217\u9700\u8981 \\(O(n^2)\\) \u6642\u9593\u3002\u5047\u8a2d\u6211\u5011\u6309\u7167\u5716 12-2 \u6240\u793a\u7684\u65b9\u5f0f\uff0c\u5c07\u9663\u5217\u5f9e\u4e2d\u9ede\u8655\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff0c\u5247\u5283\u5206\u9700\u8981 \\(O(n)\\) \u6642\u9593\uff0c\u6392\u5e8f\u6bcf\u500b\u5b50\u9663\u5217\u9700\u8981 \\(O((n / 2)^2)\\) \u6642\u9593\uff0c\u5408\u4f75\u5169\u500b\u5b50\u9663\u5217\u9700\u8981 \\(O(n)\\) \u6642\u9593\uff0c\u7e3d\u9ad4\u6642\u9593\u8907\u96dc\u5ea6\u70ba\uff1a</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> \u5716 12-2 \u00a0 \u5283\u5206\u9663\u5217\u524d\u5f8c\u7684\u6ce1\u6cab\u6392\u5e8f </p> <p>\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u8a08\u7b97\u4ee5\u4e0b\u4e0d\u7b49\u5f0f\uff0c\u5176\u5de6\u908a\u548c\u53f3\u908a\u5206\u5225\u70ba\u5283\u5206\u524d\u548c\u5283\u5206\u5f8c\u7684\u64cd\u4f5c\u7e3d\u6578\uff1a</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>\u9019\u610f\u5473\u8457\u7576 \\(n &gt; 4\\) \u6642\uff0c\u5283\u5206\u5f8c\u7684\u64cd\u4f5c\u6578\u91cf\u66f4\u5c11\uff0c\u6392\u5e8f\u6548\u7387\u61c9\u8a72\u66f4\u9ad8\u3002\u8acb\u6ce8\u610f\uff0c\u5283\u5206\u5f8c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4ecd\u7136\u662f\u5e73\u65b9\u968e \\(O(n^2)\\) \uff0c\u53ea\u662f\u8907\u96dc\u5ea6\u4e2d\u7684\u5e38\u6578\u9805\u8b8a\u5c0f\u4e86\u3002</p> <p>\u9032\u4e00\u6b65\u60f3\uff0c\u5982\u679c\u6211\u5011\u628a\u5b50\u9663\u5217\u4e0d\u65b7\u5730\u518d\u5f9e\u4e2d\u9ede\u8655\u5283\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff0c\u76f4\u81f3\u5b50\u9663\u5217\u53ea\u5269\u4e00\u500b\u5143\u7d20\u6642\u505c\u6b62\u5283\u5206\u5462\uff1f\u9019\u7a2e\u601d\u8def\u5be6\u969b\u4e0a\u5c31\u662f\u201c\u5408\u4f75\u6392\u5e8f\u201d\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</p> <p>\u518d\u601d\u8003\uff0c\u5982\u679c\u6211\u5011\u591a\u8a2d\u5b9a\u5e7e\u500b\u5283\u5206\u9ede\uff0c\u5c07\u539f\u9663\u5217\u5e73\u5747\u5283\u5206\u70ba \\(k\\) \u500b\u5b50\u9663\u5217\u5462\uff1f\u9019\u7a2e\u60c5\u6cc1\u8207\u201c\u6876\u6392\u5e8f\u201d\u975e\u5e38\u985e\u4f3c\uff0c\u5b83\u975e\u5e38\u9069\u5408\u6392\u5e8f\u6d77\u91cf\u8cc7\u6599\uff0c\u7406\u8ad6\u4e0a\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u9054\u5230 \\(O(n + k)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","title":"2. \u00a0 \u5e73\u884c\u8a08\u7b97\u6700\u4f73\u5316","text":"<p>\u6211\u5011\u77e5\u9053\uff0c\u5206\u6cbb\u751f\u6210\u7684\u5b50\u554f\u984c\u662f\u76f8\u4e92\u7368\u7acb\u7684\uff0c\u56e0\u6b64\u901a\u5e38\u53ef\u4ee5\u4e26\u884c\u89e3\u6c7a\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u5206\u6cbb\u4e0d\u50c5\u53ef\u4ee5\u964d\u4f4e\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u9084\u6709\u5229\u65bc\u4f5c\u696d\u7cfb\u7d71\u7684\u4e26\u884c\u6700\u4f73\u5316\u3002</p> <p>\u4e26\u884c\u6700\u4f73\u5316\u5728\u591a\u6838\u6216\u591a\u8655\u7406\u5668\u7684\u74b0\u5883\u4e2d\u5c24\u5176\u6709\u6548\uff0c\u56e0\u70ba\u7cfb\u7d71\u53ef\u4ee5\u540c\u6642\u8655\u7406\u591a\u500b\u5b50\u554f\u984c\uff0c\u66f4\u52a0\u5145\u5206\u5730\u5229\u7528\u8a08\u7b97\u8cc7\u6e90\uff0c\u5f9e\u800c\u986f\u8457\u6e1b\u5c11\u7e3d\u9ad4\u7684\u57f7\u884c\u6642\u9593\u3002</p> <p>\u6bd4\u5982\u5728\u5716 12-3 \u6240\u793a\u7684\u201c\u6876\u6392\u5e8f\u201d\u4e2d\uff0c\u6211\u5011\u5c07\u6d77\u91cf\u7684\u8cc7\u6599\u5e73\u5747\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\uff0c\u5247\u53ef\u5c07\u6240\u6709\u6876\u7684\u6392\u5e8f\u4efb\u52d9\u5206\u6563\u5230\u5404\u500b\u8a08\u7b97\u55ae\u5143\uff0c\u5b8c\u6210\u5f8c\u518d\u5408\u4f75\u7d50\u679c\u3002</p> <p></p> <p> \u5716 12-3 \u00a0 \u6876\u6392\u5e8f\u7684\u5e73\u884c\u8a08\u7b97 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","title":"12.1.3 \u00a0 \u5206\u6cbb\u5e38\u898b\u61c9\u7528","text":"<p>\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u53ef\u4ee5\u7528\u4f86\u89e3\u6c7a\u8a31\u591a\u7d93\u5178\u6f14\u7b97\u6cd5\u554f\u984c\u3002</p> <ul> <li>\u5c0b\u627e\u6700\u8fd1\u9ede\u5c0d\uff1a\u8a72\u6f14\u7b97\u6cd5\u9996\u5148\u5c07\u9ede\u96c6\u5206\u6210\u5169\u90e8\u5206\uff0c\u7136\u5f8c\u5206\u5225\u627e\u51fa\u5169\u90e8\u5206\u4e2d\u7684\u6700\u8fd1\u9ede\u5c0d\uff0c\u6700\u5f8c\u627e\u51fa\u8de8\u8d8a\u5169\u90e8\u5206\u7684\u6700\u8fd1\u9ede\u5c0d\u3002</li> <li>\u5927\u6574\u6578\u4e58\u6cd5\uff1a\u4f8b\u5982 Karatsuba \u6f14\u7b97\u6cd5\uff0c\u5b83\u5c07\u5927\u6574\u6578\u4e58\u6cd5\u5206\u89e3\u70ba\u5e7e\u500b\u8f03\u5c0f\u7684\u6574\u6578\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u77e9\u9663\u4e58\u6cd5\uff1a\u4f8b\u5982 Strassen \u6f14\u7b97\u6cd5\uff0c\u5b83\u5c07\u5927\u77e9\u9663\u4e58\u6cd5\u5206\u89e3\u70ba\u591a\u500b\u5c0f\u77e9\u9663\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u6cb3\u5167\u5854\u554f\u984c\uff1a\u6cb3\u5167\u5854\u554f\u984c\u53ef\u4ee5\u900f\u904e\u905e\u8ff4\u89e3\u6c7a\uff0c\u9019\u662f\u5178\u578b\u7684\u5206\u6cbb\u7b56\u7565\u61c9\u7528\u3002</li> <li>\u6c42\u89e3\u9006\u5e8f\u5c0d\uff1a\u5728\u4e00\u500b\u5e8f\u5217\u4e2d\uff0c\u5982\u679c\u524d\u9762\u7684\u6578\u5b57\u5927\u65bc\u5f8c\u9762\u7684\u6578\u5b57\uff0c\u90a3\u9ebc\u9019\u5169\u500b\u6578\u5b57\u69cb\u6210\u4e00\u500b\u9006\u5e8f\u5c0d\u3002\u6c42\u89e3\u9006\u5e8f\u5c0d\u554f\u984c\u53ef\u4ee5\u5229\u7528\u5206\u6cbb\u7684\u601d\u60f3\uff0c\u85c9\u52a9\u5408\u4f75\u6392\u5e8f\u9032\u884c\u6c42\u89e3\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5728\u6f14\u7b97\u6cd5\u548c\u8cc7\u6599\u7d50\u69cb\u7684\u8a2d\u8a08\u4e2d\u61c9\u7528\u5f97\u975e\u5e38\u5ee3\u6cdb\u3002</p> <ul> <li>\u4e8c\u5206\u641c\u5c0b\uff1a\u4e8c\u5206\u641c\u5c0b\u662f\u5c07\u6709\u5e8f\u9663\u5217\u5f9e\u4e2d\u9ede\u7d22\u5f15\u8655\u5206\u70ba\u5169\u90e8\u5206\uff0c\u7136\u5f8c\u6839\u64da\u76ee\u6a19\u503c\u8207\u4e2d\u9593\u5143\u7d20\u503c\u6bd4\u8f03\u7d50\u679c\uff0c\u6c7a\u5b9a\u6392\u9664\u54ea\u4e00\u534a\u5340\u9593\uff0c\u4e26\u5728\u5269\u9918\u5340\u9593\u57f7\u884c\u76f8\u540c\u7684\u4e8c\u5206\u64cd\u4f5c\u3002</li> <li>\u5408\u4f75\u6392\u5e8f\uff1a\u672c\u7bc0\u958b\u982d\u5df2\u4ecb\u7d39\uff0c\u4e0d\u518d\u8d05\u8ff0\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\uff1a\u5feb\u901f\u6392\u5e8f\u662f\u9078\u53d6\u4e00\u500b\u57fa\u6e96\u503c\uff0c\u7136\u5f8c\u628a\u9663\u5217\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\uff0c\u4e00\u500b\u5b50\u9663\u5217\u7684\u5143\u7d20\u6bd4\u57fa\u6e96\u503c\u5c0f\uff0c\u53e6\u4e00\u5b50\u9663\u5217\u7684\u5143\u7d20\u6bd4\u57fa\u6e96\u503c\u5927\uff0c\u518d\u5c0d\u9019\u5169\u90e8\u5206\u9032\u884c\u76f8\u540c\u7684\u5283\u5206\u64cd\u4f5c\uff0c\u76f4\u81f3\u5b50\u9663\u5217\u53ea\u5269\u4e0b\u4e00\u500b\u5143\u7d20\u3002</li> <li>\u6876\u6392\u5e8f\uff1a\u6876\u6392\u5e8f\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c07\u8cc7\u6599\u5206\u6563\u5230\u591a\u500b\u6876\uff0c\u7136\u5f8c\u5c0d\u6bcf\u500b\u6876\u5167\u7684\u5143\u7d20\u9032\u884c\u6392\u5e8f\uff0c\u6700\u5f8c\u5c07\u5404\u500b\u6876\u7684\u5143\u7d20\u4f9d\u6b21\u53d6\u51fa\uff0c\u5f9e\u800c\u5f97\u5230\u4e00\u500b\u6709\u5e8f\u9663\u5217\u3002</li> <li>\u6a39\uff1a\u4f8b\u5982\u4e8c\u5143\u641c\u5c0b\u6a39\u3001AVL \u6a39\u3001\u7d05\u9ed1\u6a39\u3001B \u6a39\u3001B+ \u6a39\u7b49\uff0c\u5b83\u5011\u7684\u67e5\u8a62\u3001\u63d2\u5165\u548c\u522a\u9664\u7b49\u64cd\u4f5c\u90fd\u53ef\u4ee5\u8996\u70ba\u5206\u6cbb\u7b56\u7565\u7684\u61c9\u7528\u3002</li> <li>\u5806\u7a4d\uff1a\u5806\u7a4d\u662f\u4e00\u7a2e\u7279\u6b8a\u7684\u5b8c\u5168\u4e8c\u5143\u6a39\uff0c\u5176\u5404\u7a2e\u64cd\u4f5c\uff0c\u5982\u63d2\u5165\u3001\u522a\u9664\u548c\u5806\u7a4d\u5316\uff0c\u5be6\u969b\u4e0a\u90fd\u96b1\u542b\u4e86\u5206\u6cbb\u7684\u601d\u60f3\u3002</li> <li>\u96dc\u6e4a\u8868\uff1a\u96d6\u7136\u96dc\u6e4a\u8868\u4e26\u4e0d\u76f4\u63a5\u61c9\u7528\u5206\u6cbb\uff0c\u4f46\u67d0\u4e9b\u96dc\u6e4a\u885d\u7a81\u89e3\u6c7a\u65b9\u6848\u9593\u63a5\u61c9\u7528\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u4f8b\u5982\uff0c\u93c8\u5f0f\u4f4d\u5740\u4e2d\u7684\u9577\u93c8\u7d50\u4e32\u5217\u6703\u88ab\u8f49\u5316\u70ba\u7d05\u9ed1\u6a39\uff0c\u4ee5\u63d0\u5347\u67e5\u8a62\u6548\u7387\u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u51fa\uff0c\u5206\u6cbb\u662f\u4e00\u7a2e\u201c\u6f64\u7269\u7d30\u7121\u8072\u201d\u7684\u6f14\u7b97\u6cd5\u601d\u60f3\uff0c\u96b1\u542b\u5728\u5404\u7a2e\u6f14\u7b97\u6cd5\u8207\u8cc7\u6599\u7d50\u69cb\u4e4b\u4e2d\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/","title":"12.4 \u00a0 \u6cb3\u5167\u5854\u554f\u984c","text":"<p>\u5728\u5408\u4f75\u6392\u5e8f\u548c\u69cb\u5efa\u4e8c\u5143\u6a39\u4e2d\uff0c\u6211\u5011\u90fd\u662f\u5c07\u539f\u554f\u984c\u5206\u89e3\u70ba\u5169\u500b\u898f\u6a21\u70ba\u539f\u554f\u984c\u4e00\u534a\u7684\u5b50\u554f\u984c\u3002\u7136\u800c\u5c0d\u65bc\u6cb3\u5167\u5854\u554f\u984c\uff0c\u6211\u5011\u63a1\u7528\u4e0d\u540c\u7684\u5206\u89e3\u7b56\u7565\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e09\u6839\u67f1\u5b50\uff0c\u8a18\u70ba <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \u3002\u8d77\u59cb\u72c0\u614b\u4e0b\uff0c\u67f1\u5b50 <code>A</code> \u4e0a\u5957\u8457 \\(n\\) \u500b\u5713\u76e4\uff0c\u5b83\u5011\u5f9e\u4e0a\u5230\u4e0b\u6309\u7167\u5f9e\u5c0f\u5230\u5927\u7684\u9806\u5e8f\u6392\u5217\u3002\u6211\u5011\u7684\u4efb\u52d9\u662f\u8981\u628a\u9019 \\(n\\) \u500b\u5713\u76e4\u79fb\u5230\u67f1\u5b50 <code>C</code> \u4e0a\uff0c\u4e26\u4fdd\u6301\u5b83\u5011\u7684\u539f\u6709\u9806\u5e8f\u4e0d\u8b8a\uff08\u5982\u5716 12-10 \u6240\u793a\uff09\u3002\u5728\u79fb\u52d5\u5713\u76e4\u7684\u904e\u7a0b\u4e2d\uff0c\u9700\u8981\u9075\u5b88\u4ee5\u4e0b\u898f\u5247\u3002</p> <ol> <li>\u5713\u76e4\u53ea\u80fd\u5f9e\u4e00\u6839\u67f1\u5b50\u9802\u90e8\u62ff\u51fa\uff0c\u5f9e\u53e6\u4e00\u6839\u67f1\u5b50\u9802\u90e8\u653e\u5165\u3002</li> <li>\u6bcf\u6b21\u53ea\u80fd\u79fb\u52d5\u4e00\u500b\u5713\u76e4\u3002</li> <li>\u5c0f\u5713\u76e4\u5fc5\u9808\u6642\u523b\u4f4d\u65bc\u5927\u5713\u76e4\u4e4b\u4e0a\u3002</li> </ol> <p></p> <p> \u5716 12-10 \u00a0 \u6cb3\u5167\u5854\u554f\u984c\u793a\u4f8b </p> <p>\u6211\u5011\u5c07\u898f\u6a21\u70ba \\(i\\) \u7684\u6cb3\u5167\u5854\u554f\u984c\u8a18\u4f5c \\(f(i)\\) \u3002\u4f8b\u5982 \\(f(3)\\) \u4ee3\u8868\u5c07 \\(3\\) \u500b\u5713\u76e4\u5f9e <code>A</code> \u79fb\u52d5\u81f3 <code>C</code> \u7684\u6cb3\u5167\u5854\u554f\u984c\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#1","title":"1. \u00a0 \u8003\u616e\u57fa\u672c\u60c5\u6cc1","text":"<p>\u5982\u5716 12-11 \u6240\u793a\uff0c\u5c0d\u65bc\u554f\u984c \\(f(1)\\) \uff0c\u5373\u7576\u53ea\u6709\u4e00\u500b\u5713\u76e4\u6642\uff0c\u6211\u5011\u5c07\u5b83\u76f4\u63a5\u5f9e <code>A</code> \u79fb\u52d5\u81f3 <code>C</code> \u5373\u53ef\u3002</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> \u5716 12-11 \u00a0 \u898f\u6a21\u70ba 1 \u7684\u554f\u984c\u7684\u89e3 </p> <p>\u5982\u5716 12-12 \u6240\u793a\uff0c\u5c0d\u65bc\u554f\u984c \\(f(2)\\) \uff0c\u5373\u7576\u6709\u5169\u500b\u5713\u76e4\u6642\uff0c\u7531\u65bc\u8981\u6642\u523b\u6eff\u8db3\u5c0f\u5713\u76e4\u5728\u5927\u5713\u76e4\u4e4b\u4e0a\uff0c\u56e0\u6b64\u9700\u8981\u85c9\u52a9 <code>B</code> \u4f86\u5b8c\u6210\u79fb\u52d5\u3002</p> <ol> <li>\u5148\u5c07\u4e0a\u9762\u7684\u5c0f\u5713\u76e4\u5f9e <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u518d\u5c07\u5927\u5713\u76e4\u5f9e <code>A</code> \u79fb\u81f3 <code>C</code> \u3002</li> <li>\u6700\u5f8c\u5c07\u5c0f\u5713\u76e4\u5f9e <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 12-12 \u00a0 \u898f\u6a21\u70ba 2 \u7684\u554f\u984c\u7684\u89e3 </p> <p>\u89e3\u6c7a\u554f\u984c \\(f(2)\\) \u7684\u904e\u7a0b\u53ef\u7e3d\u7d50\u70ba\uff1a\u5c07\u5169\u500b\u5713\u76e4\u85c9\u52a9 <code>B</code> \u5f9e <code>A</code> \u79fb\u81f3 <code>C</code> \u3002\u5176\u4e2d\uff0c<code>C</code> \u7a31\u70ba\u76ee\u6a19\u67f1\u3001<code>B</code> \u7a31\u70ba\u7de9\u885d\u67f1\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#2","title":"2. \u00a0 \u5b50\u554f\u984c\u5206\u89e3","text":"<p>\u5c0d\u65bc\u554f\u984c \\(f(3)\\) \uff0c\u5373\u7576\u6709\u4e09\u500b\u5713\u76e4\u6642\uff0c\u60c5\u6cc1\u8b8a\u5f97\u7a0d\u5fae\u8907\u96dc\u4e86\u4e00\u4e9b\u3002</p> <p>\u56e0\u70ba\u5df2\u77e5 \\(f(1)\\) \u548c \\(f(2)\\) \u7684\u89e3\uff0c\u6240\u4ee5\u6211\u5011\u53ef\u5f9e\u5206\u6cbb\u89d2\u5ea6\u601d\u8003\uff0c\u5c07 <code>A</code> \u9802\u90e8\u7684\u5169\u500b\u5713\u76e4\u770b\u4f5c\u4e00\u500b\u6574\u9ad4\uff0c\u57f7\u884c\u5716 12-13 \u6240\u793a\u7684\u6b65\u9a5f\u3002\u9019\u6a23\u4e09\u500b\u5713\u76e4\u5c31\u88ab\u9806\u5229\u5730\u5f9e <code>A</code> \u79fb\u81f3 <code>C</code> \u4e86\u3002</p> <ol> <li>\u4ee4 <code>B</code> \u70ba\u76ee\u6a19\u67f1\u3001<code>C</code> \u70ba\u7de9\u885d\u67f1\uff0c\u5c07\u5169\u500b\u5713\u76e4\u5f9e <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c07 <code>A</code> \u4e2d\u5269\u9918\u7684\u4e00\u500b\u5713\u76e4\u5f9e <code>A</code> \u76f4\u63a5\u79fb\u52d5\u81f3 <code>C</code> \u3002</li> <li>\u4ee4 <code>C</code> \u70ba\u76ee\u6a19\u67f1\u3001<code>A</code> \u70ba\u7de9\u885d\u67f1\uff0c\u5c07\u5169\u500b\u5713\u76e4\u5f9e <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 12-13 \u00a0 \u898f\u6a21\u70ba 3 \u7684\u554f\u984c\u7684\u89e3 </p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u6211\u5011\u5c07\u554f\u984c \\(f(3)\\) \u5283\u5206\u70ba\u5169\u500b\u5b50\u554f\u984c \\(f(2)\\) \u548c\u4e00\u500b\u5b50\u554f\u984c \\(f(1)\\) \u3002\u6309\u9806\u5e8f\u89e3\u6c7a\u9019\u4e09\u500b\u5b50\u554f\u984c\u4e4b\u5f8c\uff0c\u539f\u554f\u984c\u96a8\u4e4b\u5f97\u5230\u89e3\u6c7a\u3002\u9019\u8aaa\u660e\u5b50\u554f\u984c\u662f\u7368\u7acb\u7684\uff0c\u800c\u4e14\u89e3\u53ef\u4ee5\u5408\u4f75\u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u5011\u53ef\u7e3d\u7d50\u51fa\u5716 12-14 \u6240\u793a\u7684\u89e3\u6c7a\u6cb3\u5167\u5854\u554f\u984c\u7684\u5206\u6cbb\u7b56\u7565\uff1a\u5c07\u539f\u554f\u984c \\(f(n)\\) \u5283\u5206\u70ba\u5169\u500b\u5b50\u554f\u984c \\(f(n-1)\\) \u548c\u4e00\u500b\u5b50\u554f\u984c \\(f(1)\\) \uff0c\u4e26\u6309\u7167\u4ee5\u4e0b\u9806\u5e8f\u89e3\u6c7a\u9019\u4e09\u500b\u5b50\u554f\u984c\u3002</p> <ol> <li>\u5c07 \\(n-1\\) \u500b\u5713\u76e4\u85c9\u52a9 <code>C</code> \u5f9e <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c07\u5269\u9918 \\(1\\) \u500b\u5713\u76e4\u5f9e <code>A</code> \u76f4\u63a5\u79fb\u81f3 <code>C</code> \u3002</li> <li>\u5c07 \\(n-1\\) \u500b\u5713\u76e4\u85c9\u52a9 <code>A</code> \u5f9e <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> <p>\u5c0d\u65bc\u9019\u5169\u500b\u5b50\u554f\u984c \\(f(n-1)\\) \uff0c\u53ef\u4ee5\u900f\u904e\u76f8\u540c\u7684\u65b9\u5f0f\u9032\u884c\u905e\u8ff4\u5283\u5206\uff0c\u76f4\u81f3\u9054\u5230\u6700\u5c0f\u5b50\u554f\u984c \\(f(1)\\) \u3002\u800c \\(f(1)\\) \u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u53ea\u9700\u4e00\u6b21\u79fb\u52d5\u64cd\u4f5c\u5373\u53ef\u3002</p> <p></p> <p> \u5716 12-14 \u00a0 \u89e3\u6c7a\u6cb3\u5167\u5854\u554f\u984c\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#3","title":"3. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5728\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u6211\u5011\u5ba3\u544a\u4e00\u500b\u905e\u8ff4\u51fd\u5f0f <code>dfs(i, src, buf, tar)</code> \uff0c\u5b83\u7684\u4f5c\u7528\u662f\u5c07\u67f1 <code>src</code> \u9802\u90e8\u7684 \\(i\\) \u500b\u5713\u76e4\u85c9\u52a9\u7de9\u885d\u67f1 <code>buf</code> \u79fb\u52d5\u81f3\u76ee\u6a19\u67f1 <code>tar</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"\u79fb\u52d5\u4e00\u500b\u5713\u76e4\"\"\"\n    # \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    pan = src.pop()\n    # \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"\u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i)\"\"\"\n    # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if i == 1:\n        move(src, tar)\n        return\n    # \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    # \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar)\n    # \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"\u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c\"\"\"\n    n = len(A)\n    # \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    int pan = src.back();\n    src.pop_back();\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.push_back(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    Integer pan = src.remove(src.size() - 1);\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.Add(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    DFS(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    Move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    DFS(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfunc move(src, tar *list.List) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    pan := src.Back()\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.PushBack(pan.Value)\n    // \u79fb\u9664 src \u9802\u90e8\u5713\u76e4\n    src.Remove(pan)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfsHanota(i-1, src, tar, buf)\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    let pan = src.popLast()!\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.append(pan)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src: &amp;src, tar: &amp;tar)\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // \u4e32\u5217\u5c3e\u90e8\u662f\u67f1\u5b50\u9802\u90e8\n    // \u5c07 src \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfunction move(src, tar) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    const pan = src.pop();\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfunction dfs(i, src, buf, tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfunction move(src: number[], tar: number[]): void {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    const pan = src.pop();\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n  int pan = src.removeLast();\n  // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n  tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n  dfs(i - 1, src, tar, buf);\n  // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n  move(src, tar);\n  // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n  dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    let pan = src.pop().unwrap();\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move_pan(src, tar);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, srcSize, tar, tarSize);\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.kt<pre><code>/* \u79fb\u52d5\u4e00\u500b\u5713\u76e4 */\nfun move(src: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n    val pan = src.removeAt(src.size - 1)\n    // \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n    tar.add(pan)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) */\nfun dfs(i: Int, src: MutableList&lt;Int&gt;, buf: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    // \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c */\nfun solveHanota(A: MutableList&lt;Int&gt;, B: MutableList&lt;Int&gt;, C: MutableList&lt;Int&gt;) {\n    val n = A.size\n    // \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n}\n</code></pre> hanota.rb<pre><code>### \u79fb\u52d5\u4e00\u500b\u5713\u76e4 ###\ndef move(src, tar)\n  # \u5f9e src \u9802\u90e8\u62ff\u51fa\u4e00\u500b\u5713\u76e4\n  pan = src.pop\n  # \u5c07\u5713\u76e4\u653e\u5165 tar \u9802\u90e8\n  tar &lt;&lt; pan\nend\n\n### \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c f(i) ###\ndef dfs(i, src, buf, tar)\n  # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u500b\u5713\u76e4\uff0c\u5247\u76f4\u63a5\u5c07\u5176\u79fb\u5230 tar\n  if i == 1\n    move(src, tar)\n    return\n  end\n\n  # \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 src \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 tar \u79fb\u5230 buf\n  dfs(i - 1, src, tar, buf)\n  # \u5b50\u554f\u984c f(1) \uff1a\u5c07 src \u5269\u9918\u4e00\u500b\u5713\u76e4\u79fb\u5230 tar\n  move(src, tar)\n  # \u5b50\u554f\u984c f(i-1) \uff1a\u5c07 buf \u9802\u90e8 i-1 \u500b\u5713\u76e4\u85c9\u52a9 src \u79fb\u5230 tar\n  dfs(i - 1, buf, src, tar)\nend\n\n### \u6c42\u89e3\u6cb3\u5167\u5854\u554f\u984c ###\ndef solve_hanota(_A, _B, _C)\n  n = _A.length\n  # \u5c07 A \u9802\u90e8 n \u500b\u5713\u76e4\u85c9\u52a9 B \u79fb\u5230 C\n  dfs(n, _A, _B, _C)\nend\n</code></pre> hanota.zig<pre><code>[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solveHanota}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 12-15 \u6240\u793a\uff0c\u6cb3\u5167\u5854\u554f\u984c\u5f62\u6210\u4e00\u68f5\u9ad8\u5ea6\u70ba \\(n\\) \u7684\u905e\u8ff4\u6a39\uff0c\u6bcf\u500b\u7bc0\u9ede\u4ee3\u8868\u4e00\u500b\u5b50\u554f\u984c\uff0c\u5c0d\u61c9\u4e00\u500b\u958b\u555f\u7684 <code>dfs()</code> \u51fd\u5f0f\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^n)\\) \uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> <p></p> <p> \u5716 12-15 \u00a0 \u6cb3\u5167\u5854\u554f\u984c\u7684\u905e\u8ff4\u6a39 </p> <p>Quote</p> <p>\u6cb3\u5167\u5854\u554f\u984c\u6e90\u81ea\u4e00\u500b\u53e4\u8001\u7684\u50b3\u8aaa\u3002\u5728\u53e4\u5370\u5ea6\u7684\u4e00\u500b\u5bfa\u5edf\u88e1\uff0c\u50e7\u4fb6\u5011\u6709\u4e09\u6839\u9ad8\u5927\u7684\u947d\u77f3\u67f1\u5b50\uff0c\u4ee5\u53ca \\(64\\) \u500b\u5927\u5c0f\u4e0d\u4e00\u7684\u91d1\u5713\u76e4\u3002\u50e7\u4fb6\u5011\u4e0d\u65b7\u5730\u79fb\u52d5\u5713\u76e4\uff0c\u4ed6\u5011\u76f8\u4fe1\u5728\u6700\u5f8c\u4e00\u500b\u5713\u76e4\u88ab\u6b63\u78ba\u653e\u7f6e\u7684\u90a3\u4e00\u523b\uff0c\u9019\u500b\u4e16\u754c\u5c31\u6703\u7d50\u675f\u3002</p> <p>\u7136\u800c\uff0c\u5373\u4f7f\u50e7\u4fb6\u5011\u6bcf\u79d2\u9418\u79fb\u52d5\u4e00\u6b21\uff0c\u7e3d\u5171\u9700\u8981\u5927\u7d04 \\(2^{64} \\approx 1.84\u00d710^{19}\\) \u79d2\uff0c\u5408\u7d04 \\(5850\\) \u5104\u5e74\uff0c\u9060\u9060\u8d85\u904e\u4e86\u73fe\u5728\u5c0d\u5b87\u5b99\u5e74\u9f61\u7684\u4f30\u8a08\u3002\u6240\u4ee5\uff0c\u5018\u82e5\u9019\u500b\u50b3\u8aaa\u662f\u771f\u7684\uff0c\u6211\u5011\u61c9\u8a72\u4e0d\u9700\u8981\u64d4\u5fc3\u4e16\u754c\u672b\u65e5\u7684\u5230\u4f86\u3002</p>"},{"location":"chapter_divide_and_conquer/summary/","title":"12.5 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u5206\u6cbb\u662f\u4e00\u7a2e\u5e38\u898b\u7684\u6f14\u7b97\u6cd5\u8a2d\u8a08\u7b56\u7565\uff0c\u5305\u62ec\u5206\uff08\u5283\u5206\uff09\u548c\u6cbb\uff08\u5408\u4f75\uff09\u5169\u500b\u968e\u6bb5\uff0c\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u5be6\u73fe\u3002</li> <li>\u5224\u65b7\u662f\u5426\u662f\u5206\u6cbb\u6f14\u7b97\u6cd5\u554f\u984c\u7684\u4f9d\u64da\u5305\u62ec\uff1a\u554f\u984c\u80fd\u5426\u5206\u89e3\u3001\u5b50\u554f\u984c\u662f\u5426\u7368\u7acb\u3001\u5b50\u554f\u984c\u80fd\u5426\u5408\u4f75\u3002</li> <li>\u5408\u4f75\u6392\u5e8f\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u61c9\u7528\uff0c\u5176\u905e\u8ff4\u5730\u5c07\u9663\u5217\u5283\u5206\u70ba\u7b49\u9577\u7684\u5169\u500b\u5b50\u9663\u5217\uff0c\u76f4\u5230\u53ea\u5269\u4e00\u500b\u5143\u7d20\u6642\u958b\u59cb\u9010\u5c64\u5408\u4f75\uff0c\u5f9e\u800c\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u5f15\u5165\u5206\u6cbb\u7b56\u7565\u5f80\u5f80\u53ef\u4ee5\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\u3002\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u7b56\u7565\u6e1b\u5c11\u4e86\u64cd\u4f5c\u6578\u91cf\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5f8c\u6709\u5229\u65bc\u7cfb\u7d71\u7684\u4e26\u884c\u6700\u4f73\u5316\u3002</li> <li>\u5206\u6cbb\u65e2\u53ef\u4ee5\u89e3\u6c7a\u8a31\u591a\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u4e5f\u5ee3\u6cdb\u61c9\u7528\u65bc\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u8a2d\u8a08\u4e2d\uff0c\u8655\u8655\u53ef\u898b\u5176\u8eab\u5f71\u3002</li> <li>\u76f8\u8f03\u65bc\u66b4\u529b\u641c\u5c0b\uff0c\u81ea\u9069\u61c9\u641c\u5c0b\u6548\u7387\u66f4\u9ad8\u3002\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \u7684\u641c\u5c0b\u6f14\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u5be6\u73fe\u7684\u3002</li> <li>\u4e8c\u5206\u641c\u5c0b\u662f\u5206\u6cbb\u7b56\u7565\u7684\u53e6\u4e00\u500b\u5178\u578b\u61c9\u7528\uff0c\u5b83\u4e0d\u5305\u542b\u5c07\u5b50\u554f\u984c\u7684\u89e3\u9032\u884c\u5408\u4f75\u7684\u6b65\u9a5f\u3002\u6211\u5011\u53ef\u4ee5\u900f\u904e\u905e\u8ff4\u5206\u6cbb\u5be6\u73fe\u4e8c\u5206\u641c\u5c0b\u3002</li> <li>\u5728\u69cb\u5efa\u4e8c\u5143\u6a39\u7684\u554f\u984c\u4e2d\uff0c\u69cb\u5efa\u6a39\uff08\u539f\u554f\u984c\uff09\u53ef\u4ee5\u5283\u5206\u70ba\u69cb\u5efa\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\uff08\u5b50\u554f\u984c\uff09\uff0c\u9019\u53ef\u4ee5\u900f\u904e\u5283\u5206\u524d\u5e8f\u8d70\u8a2a\u548c\u4e2d\u5e8f\u8d70\u8a2a\u7684\u7d22\u5f15\u5340\u9593\u4f86\u5be6\u73fe\u3002</li> <li>\u5728\u6cb3\u5167\u5854\u554f\u984c\u4e2d\uff0c\u4e00\u500b\u898f\u6a21\u70ba \\(n\\) \u7684\u554f\u984c\u53ef\u4ee5\u5283\u5206\u70ba\u5169\u500b\u898f\u6a21\u70ba \\(n-1\\) \u7684\u5b50\u554f\u984c\u548c\u4e00\u500b\u898f\u6a21\u70ba \\(1\\) \u7684\u5b50\u554f\u984c\u3002\u6309\u9806\u5e8f\u89e3\u6c7a\u9019\u4e09\u500b\u5b50\u554f\u984c\u5f8c\uff0c\u539f\u554f\u984c\u96a8\u4e4b\u5f97\u5230\u89e3\u6c7a\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/","title":"\u7b2c 14 \u7ae0 \u00a0 \u52d5\u614b\u898f\u5283","text":"<p>Abstract</p> <p>\u5c0f\u6eaa\u532f\u5165\u6cb3\u6d41\uff0c\u6c5f\u6cb3\u532f\u5165\u5927\u6d77\u3002</p> <p>\u52d5\u614b\u898f\u5283\u5c07\u5c0f\u554f\u984c\u7684\u89e3\u5f59\u96c6\u6210\u5927\u554f\u984c\u7684\u7b54\u6848\uff0c\u4e00\u6b65\u6b65\u5f15\u9818\u6211\u5011\u8d70\u5411\u89e3\u6c7a\u554f\u984c\u7684\u5f7c\u5cb8\u3002</p>"},{"location":"chapter_dynamic_programming/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>14.1 \u00a0 \u521d\u63a2\u52d5\u614b\u898f\u5283</li> <li>14.2 \u00a0 DP \u554f\u984c\u7279\u6027</li> <li>14.3 \u00a0 DP \u89e3\u984c\u601d\u8def</li> <li>14.4 \u00a0 0-1 \u80cc\u5305\u554f\u984c</li> <li>14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c</li> <li>14.6 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u554f\u984c</li> <li>14.7 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_dynamic_programming/dp_problem_features/","title":"14.2 \u00a0 \u52d5\u614b\u898f\u5283\u554f\u984c\u7279\u6027","text":"<p>\u5728\u4e0a\u4e00\u7bc0\u4e2d\uff0c\u6211\u5011\u5b78\u7fd2\u4e86\u52d5\u614b\u898f\u5283\u662f\u5982\u4f55\u900f\u904e\u5b50\u554f\u984c\u5206\u89e3\u4f86\u6c42\u89e3\u539f\u554f\u984c\u7684\u3002\u5be6\u969b\u4e0a\uff0c\u5b50\u554f\u984c\u5206\u89e3\u662f\u4e00\u7a2e\u901a\u7528\u7684\u6f14\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52d5\u614b\u898f\u5283\u3001\u56de\u6eaf\u4e2d\u7684\u5074\u91cd\u9ede\u4e0d\u540c\u3002</p> <ul> <li>\u5206\u6cbb\u6f14\u7b97\u6cd5\u905e\u8ff4\u5730\u5c07\u539f\u554f\u984c\u5283\u5206\u70ba\u591a\u500b\u76f8\u4e92\u7368\u7acb\u7684\u5b50\u554f\u984c\uff0c\u76f4\u81f3\u6700\u5c0f\u5b50\u554f\u984c\uff0c\u4e26\u5728\u56de\u6eaf\u4e2d\u5408\u4f75\u5b50\u554f\u984c\u7684\u89e3\uff0c\u6700\u7d42\u5f97\u5230\u539f\u554f\u984c\u7684\u89e3\u3002</li> <li>\u52d5\u614b\u898f\u5283\u4e5f\u5c0d\u554f\u984c\u9032\u884c\u905e\u8ff4\u5206\u89e3\uff0c\u4f46\u8207\u5206\u6cbb\u6f14\u7b97\u6cd5\u7684\u4e3b\u8981\u5340\u5225\u662f\uff0c\u52d5\u614b\u898f\u5283\u4e2d\u7684\u5b50\u554f\u984c\u662f\u76f8\u4e92\u4f9d\u8cf4\u7684\uff0c\u5728\u5206\u89e3\u904e\u7a0b\u4e2d\u6703\u51fa\u73fe\u8a31\u591a\u91cd\u758a\u5b50\u554f\u984c\u3002</li> <li>\u56de\u6eaf\u6f14\u7b97\u6cd5\u5728\u5617\u8a66\u548c\u56de\u9000\u4e2d\u7aae\u8209\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u4e26\u900f\u904e\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u5c0b\u5206\u652f\u3002\u539f\u554f\u984c\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u6c7a\u7b56\u6b65\u9a5f\u69cb\u6210\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u6bcf\u500b\u6c7a\u7b56\u6b65\u9a5f\u4e4b\u524d\u7684\u5b50\u5e8f\u5217\u770b\u4f5c\u4e00\u500b\u5b50\u554f\u984c\u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u52d5\u614b\u898f\u5283\u5e38\u7528\u4f86\u6c42\u89e3\u6700\u6700\u4f73\u5316\u554f\u984c\uff0c\u5b83\u5011\u4e0d\u50c5\u5305\u542b\u91cd\u758a\u5b50\u554f\u984c\uff0c\u9084\u5177\u6709\u53e6\u5916\u5169\u5927\u7279\u6027\uff1a\u6700\u512a\u5b50\u7d50\u69cb\u3001\u7121\u5f8c\u6548\u6027\u3002</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","title":"14.2.1 \u00a0 \u6700\u512a\u5b50\u7d50\u69cb","text":"<p>\u6211\u5011\u5c0d\u722c\u6a13\u68af\u554f\u984c\u7a0d\u4f5c\u6539\u52d5\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9069\u5408\u5c55\u793a\u6700\u512a\u5b50\u7d50\u69cb\u6982\u5ff5\u3002</p> <p>\u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9</p> <p>\u7d66\u5b9a\u4e00\u500b\u6a13\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u968e\u6216\u8005 \\(2\\) \u968e\uff0c\u6bcf\u4e00\u968e\u6a13\u68af\u4e0a\u90fd\u8cbc\u6709\u4e00\u500b\u975e\u8ca0\u6574\u6578\uff0c\u8868\u793a\u4f60\u5728\u8a72\u81fa\u968e\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u50f9\u3002\u7d66\u5b9a\u4e00\u500b\u975e\u8ca0\u6574\u6578\u9663\u5217 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u500b\u81fa\u968e\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u50f9\uff0c\\(cost[0]\\) \u70ba\u5730\u9762\uff08\u8d77\u59cb\u9ede\uff09\u3002\u8acb\u8a08\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u50f9\u624d\u80fd\u5230\u9054\u9802\u90e8\uff1f</p> <p>\u5982\u5716 14-6 \u6240\u793a\uff0c\u82e5\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u968e\u7684\u4ee3\u50f9\u5206\u5225\u70ba \\(1\\)\u3001\\(10\\)\u3001\\(1\\) \uff0c\u5247\u5f9e\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u968e\u7684\u6700\u5c0f\u4ee3\u50f9\u70ba \\(2\\) \u3002</p> <p></p> <p> \u5716 14-6 \u00a0 \u722c\u5230\u7b2c 3 \u968e\u7684\u6700\u5c0f\u4ee3\u50f9 </p> <p>\u8a2d \\(dp[i]\\) \u70ba\u722c\u5230\u7b2c \\(i\\) \u968e\u7d2f\u8a08\u4ed8\u51fa\u7684\u4ee3\u50f9\uff0c\u7531\u65bc\u7b2c \\(i\\) \u968e\u53ea\u53ef\u80fd\u5f9e \\(i - 1\\) \u968e\u6216 \\(i - 2\\) \u968e\u8d70\u4f86\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u65bc \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u70ba\u4e86\u5118\u53ef\u80fd\u6e1b\u5c11\u4ee3\u50f9\uff0c\u6211\u5011\u61c9\u8a72\u9078\u64c7\u5169\u8005\u4e2d\u8f03\u5c0f\u7684\u90a3\u4e00\u500b\uff1a</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>\u9019\u4fbf\u53ef\u4ee5\u5f15\u51fa\u6700\u512a\u5b50\u7d50\u69cb\u7684\u542b\u7fa9\uff1a\u539f\u554f\u984c\u7684\u6700\u512a\u89e3\u662f\u5f9e\u5b50\u554f\u984c\u7684\u6700\u512a\u89e3\u69cb\u5efa\u5f97\u4f86\u7684\u3002</p> <p>\u672c\u984c\u986f\u7136\u5177\u6709\u6700\u512a\u5b50\u7d50\u69cb\uff1a\u6211\u5011\u5f9e\u5169\u500b\u5b50\u554f\u984c\u6700\u512a\u89e3 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u4e2d\u6311\u9078\u51fa\u8f03\u512a\u7684\u90a3\u4e00\u500b\uff0c\u4e26\u7528\u5b83\u69cb\u5efa\u51fa\u539f\u554f\u984c \\(dp[i]\\) \u7684\u6700\u512a\u89e3\u3002</p> <p>\u90a3\u9ebc\uff0c\u4e0a\u4e00\u7bc0\u7684\u722c\u6a13\u68af\u984c\u76ee\u6709\u6c92\u6709\u6700\u512a\u5b50\u7d50\u69cb\u5462\uff1f\u5b83\u7684\u76ee\u6a19\u662f\u6c42\u89e3\u65b9\u6848\u6578\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u500b\u8a08\u6578\u554f\u984c\uff0c\u4f46\u5982\u679c\u63db\u4e00\u7a2e\u554f\u6cd5\uff1a\u201c\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6578\u91cf\u201d\u3002\u6211\u5011\u610f\u5916\u5730\u767c\u73fe\uff0c\u96d6\u7136\u984c\u76ee\u4fee\u6539\u524d\u5f8c\u662f\u7b49\u50f9\u7684\uff0c\u4f46\u6700\u512a\u5b50\u7d50\u69cb\u6d6e\u73fe\u51fa\u4f86\u4e86\uff1a\u7b2c \\(n\\) \u968e\u6700\u5927\u65b9\u6848\u6578\u91cf\u7b49\u65bc\u7b2c \\(n-1\\) \u968e\u548c\u7b2c \\(n-2\\) \u968e\u6700\u5927\u65b9\u6848\u6578\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8aaa\uff0c\u6700\u512a\u5b50\u7d50\u69cb\u7684\u89e3\u91cb\u65b9\u5f0f\u6bd4\u8f03\u9748\u6d3b\uff0c\u5728\u4e0d\u540c\u554f\u984c\u4e2d\u6703\u6709\u4e0d\u540c\u7684\u542b\u7fa9\u3002</p> <p>\u6839\u64da\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72c0\u614b \\(dp[1] = cost[1]\\) \u548c \\(dp[2] = cost[2]\\) \uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5f97\u5230\u52d5\u614b\u898f\u5283\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1], dp[2] = cost[1], cost[2]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3 ... n {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int *dp = calloc(n + 1, sizeof(int));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 */\nfun minCostClimbingStairsDP(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    val dp = IntArray(n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (i in 3..n) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  dp = Array.new(n + 1, 0)\n  # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1], dp[2] = cost[1], cost[2]\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283\nfn minCostClimbingStairsDP(comptime cost: []i32) i32 {\n    comptime var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (3..n + 1) |i| {\n        dp[i] = @min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 14-7 \u00a0 \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p> <p>\u672c\u984c\u4e5f\u53ef\u4ee5\u9032\u884c\u7a7a\u9593\u6700\u4f73\u5316\uff0c\u5c07\u4e00\u7dad\u58d3\u7e2e\u81f3\u96f6\u7dad\uff0c\u4f7f\u5f97\u7a7a\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    a, b := cost[1], cost[2]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in 3 ... n {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun minCostClimbingStairsDPComp(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    var a = cost[1]\n    var b = cost[2]\n    for (i in 3..n) {\n        val tmp = b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u52d5\u614b\u898f\u5283 ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  dp = Array.new(n + 1, 0)\n  # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1], dp[2] = cost[1], cost[2]\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n\n# \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\ndef min_cost_climbing_stairs_dp_comp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  a, b = cost[1], cost[2]\n  (3...(n + 1)).each { |i| a, b = b, [a, b].min + cost[i] }\n  b\nend\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u6a13\u68af\u6700\u5c0f\u4ee3\u50f9\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn minCostClimbingStairsDPComp(cost: []i32) i32 {\n    var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    var a = cost[1];\n    var b = cost[2];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (3..n + 1) |i| {\n        var tmp = b;\n        b = @min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","title":"14.2.2 \u00a0 \u7121\u5f8c\u6548\u6027","text":"<p>\u7121\u5f8c\u6548\u6027\u662f\u52d5\u614b\u898f\u5283\u80fd\u5920\u6709\u6548\u89e3\u6c7a\u554f\u984c\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5176\u5b9a\u7fa9\u70ba\uff1a\u7d66\u5b9a\u4e00\u500b\u78ba\u5b9a\u7684\u72c0\u614b\uff0c\u5b83\u7684\u672a\u4f86\u767c\u5c55\u53ea\u8207\u7576\u524d\u72c0\u614b\u6709\u95dc\uff0c\u800c\u8207\u904e\u53bb\u7d93\u6b77\u7684\u6240\u6709\u72c0\u614b\u7121\u95dc\u3002</p> <p>\u4ee5\u722c\u6a13\u68af\u554f\u984c\u70ba\u4f8b\uff0c\u7d66\u5b9a\u72c0\u614b \\(i\\) \uff0c\u5b83\u6703\u767c\u5c55\u51fa\u72c0\u614b \\(i+1\\) \u548c\u72c0\u614b \\(i+2\\) \uff0c\u5206\u5225\u5c0d\u61c9\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u9019\u5169\u7a2e\u9078\u64c7\u6642\uff0c\u6211\u5011\u7121\u9808\u8003\u616e\u72c0\u614b \\(i\\) \u4e4b\u524d\u7684\u72c0\u614b\uff0c\u5b83\u5011\u5c0d\u72c0\u614b \\(i\\) \u7684\u672a\u4f86\u6c92\u6709\u5f71\u97ff\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u5011\u7d66\u722c\u6a13\u68af\u554f\u984c\u65b0\u589e\u4e00\u500b\u7d04\u675f\uff0c\u60c5\u6cc1\u5c31\u4e0d\u4e00\u6a23\u4e86\u3002</p> <p>\u5e36\u7d04\u675f\u722c\u6a13\u68af</p> <p>\u7d66\u5b9a\u4e00\u500b\u5171\u6709 \\(n\\) \u968e\u7684\u6a13\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u968e\u6216\u8005 \\(2\\) \u968e\uff0c\u4f46\u4e0d\u80fd\u9023\u7e8c\u5169\u8f2a\u8df3 \\(1\\) \u968e\uff0c\u8acb\u554f\u6709\u591a\u5c11\u7a2e\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u6a13\u9802\uff1f</p> <p>\u5982\u5716 14-8 \u6240\u793a\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u968e\u50c5\u5269 \\(2\\) \u7a2e\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u9023\u7e8c\u4e09\u6b21\u8df3 \\(1\\) \u968e\u7684\u65b9\u6848\u4e0d\u6eff\u8db3\u7d04\u675f\u689d\u4ef6\uff0c\u56e0\u6b64\u88ab\u6368\u68c4\u3002</p> <p></p> <p> \u5716 14-8 \u00a0 \u5e36\u7d04\u675f\u722c\u5230\u7b2c 3 \u968e\u7684\u65b9\u6848\u6578\u91cf </p> <p>\u5728\u8a72\u554f\u984c\u4e2d\uff0c\u5982\u679c\u4e0a\u4e00\u8f2a\u662f\u8df3 \\(1\\) \u968e\u4e0a\u4f86\u7684\uff0c\u90a3\u9ebc\u4e0b\u4e00\u8f2a\u5c31\u5fc5\u9808\u8df3 \\(2\\) \u968e\u3002\u9019\u610f\u5473\u8457\uff0c\u4e0b\u4e00\u6b65\u9078\u64c7\u4e0d\u80fd\u7531\u7576\u524d\u72c0\u614b\uff08\u7576\u524d\u6240\u5728\u6a13\u68af\u968e\u6578\uff09\u7368\u7acb\u6c7a\u5b9a\uff0c\u9084\u548c\u524d\u4e00\u500b\u72c0\u614b\uff08\u4e0a\u4e00\u8f2a\u6240\u5728\u6a13\u68af\u968e\u6578\uff09\u6709\u95dc\u3002</p> <p>\u4e0d\u96e3\u767c\u73fe\uff0c\u6b64\u554f\u984c\u5df2\u4e0d\u6eff\u8db3\u7121\u5f8c\u6548\u6027\uff0c\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u70ba \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f2a\u8df3 \\(1\\) \u968e\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8a31\u591a\u201c\u4e0a\u4e00\u8f2a\u662f\u8df3 \\(1\\) \u968e\u4e0a\u4f86\u7684\u201d\u65b9\u6848\uff0c\u800c\u70ba\u4e86\u6eff\u8db3\u7d04\u675f\uff0c\u6211\u5011\u5c31\u4e0d\u80fd\u5c07 \\(dp[i-1]\\) \u76f4\u63a5\u8a08\u5165 \\(dp[i]\\) \u4e2d\u3002</p> <p>\u70ba\u6b64\uff0c\u6211\u5011\u9700\u8981\u64f4\u5c55\u72c0\u614b\u5b9a\u7fa9\uff1a\u72c0\u614b \\([i, j]\\) \u8868\u793a\u8655\u5728\u7b2c \\(i\\) \u968e\u4e26\u4e14\u4e0a\u4e00\u8f2a\u8df3\u4e86 \\(j\\) \u968e\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72c0\u614b\u5b9a\u7fa9\u6709\u6548\u5730\u5340\u5206\u4e86\u4e0a\u4e00\u8f2a\u8df3\u4e86 \\(1\\) \u968e\u9084\u662f \\(2\\) \u968e\uff0c\u6211\u5011\u53ef\u4ee5\u64da\u6b64\u5224\u65b7\u7576\u524d\u72c0\u614b\u662f\u5f9e\u4f55\u800c\u4f86\u7684\u3002</p> <ul> <li>\u7576\u4e0a\u4e00\u8f2a\u8df3\u4e86 \\(1\\) \u968e\u6642\uff0c\u4e0a\u4e0a\u4e00\u8f2a\u53ea\u80fd\u9078\u64c7\u8df3 \\(2\\) \u968e\uff0c\u5373 \\(dp[i, 1]\\) \u53ea\u80fd\u5f9e \\(dp[i-1, 2]\\) \u8f49\u79fb\u904e\u4f86\u3002</li> <li>\u7576\u4e0a\u4e00\u8f2a\u8df3\u4e86 \\(2\\) \u968e\u6642\uff0c\u4e0a\u4e0a\u4e00\u8f2a\u53ef\u9078\u64c7\u8df3 \\(1\\) \u968e\u6216\u8df3 \\(2\\) \u968e\uff0c\u5373 \\(dp[i, 2]\\) \u53ef\u4ee5\u5f9e \\(dp[i-2, 1]\\) \u6216 \\(dp[i-2, 2]\\) \u8f49\u79fb\u904e\u4f86\u3002</li> </ul> <p>\u5982\u5716 14-9 \u6240\u793a\uff0c\u5728\u8a72\u5b9a\u7fa9\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72c0\u614b \\([i, j]\\) \u5c0d\u61c9\u7684\u65b9\u6848\u6578\u3002\u6b64\u6642\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u70ba\uff1a</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> \u5716 14-9 \u00a0 \u8003\u616e\u7d04\u675f\u4e0b\u7684\u905e\u63a8\u95dc\u4fc2 </p> <p>\u6700\u7d42\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u5169\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"\u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[][] dp = new int[n + 1][3];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[,] dp = new int[n + 1, 3];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp := make([][3]int, n+1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3 ... n {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return 1;\n    };\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.kt<pre><code>/* \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfun climbingStairsConstraintDP(n: Int): Int {\n    if (n == 1 || n == 2) {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    val dp = Array(n + 1) { IntArray(3) }\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (i in 3..n) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.rb<pre><code>### \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 ###\ndef climbing_stairs_constraint_dp(n)\n  return 1 if n == 1 || n == 2\n\n  # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  dp = Array.new(n + 1) { Array.new(3, 0) }\n  # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1][1], dp[1][2] = 1, 0\n  dp[2][1], dp[2][2] = 0, 1\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  for i in 3...(n + 1)\n    dp[i][1] = dp[i - 1][2]\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n  end\n\n  dp[n][1] + dp[n][2]\nend\n</code></pre> climbing_stairs_constraint_dp.zig<pre><code>// \u5e36\u7d04\u675f\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283\nfn climbingStairsConstraintDP(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (3..n + 1) |i| {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u65bc\u50c5\u9700\u591a\u8003\u616e\u524d\u9762\u4e00\u500b\u72c0\u614b\uff0c\u56e0\u6b64\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u900f\u904e\u64f4\u5c55\u72c0\u614b\u5b9a\u7fa9\uff0c\u4f7f\u5f97\u554f\u984c\u91cd\u65b0\u6eff\u8db3\u7121\u5f8c\u6548\u6027\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u554f\u984c\u5177\u6709\u975e\u5e38\u56b4\u91cd\u7684\u201c\u6709\u5f8c\u6548\u6027\u201d\u3002</p> <p>\u722c\u6a13\u68af\u8207\u969c\u7919\u751f\u6210</p> <p>\u7d66\u5b9a\u4e00\u500b\u5171\u6709 \\(n\\) \u968e\u7684\u6a13\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u968e\u6216\u8005 \\(2\\) \u968e\u3002\u898f\u5b9a\u7576\u722c\u5230\u7b2c \\(i\\) \u968e\u6642\uff0c\u7cfb\u7d71\u81ea\u52d5\u6703\u5728\u7b2c \\(2i\\) \u968e\u4e0a\u653e\u4e0a\u969c\u7919\u7269\uff0c\u4e4b\u5f8c\u6240\u6709\u8f2a\u90fd\u4e0d\u5141\u8a31\u8df3\u5230\u7b2c \\(2i\\) \u968e\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u5169\u8f2a\u5206\u5225\u8df3\u5230\u4e86\u7b2c \\(2\\)\u3001\\(3\\) \u968e\u4e0a\uff0c\u5247\u4e4b\u5f8c\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4\\)\u3001\\(6\\) \u968e\u4e0a\u3002\u8acb\u554f\u6709\u591a\u5c11\u7a2e\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u6a13\u9802\uff1f</p> <p>\u5728\u9019\u500b\u554f\u984c\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8e8d\u4f9d\u8cf4\u904e\u53bb\u6240\u6709\u7684\u72c0\u614b\uff0c\u56e0\u70ba\u6bcf\u4e00\u6b21\u8df3\u8e8d\u90fd\u6703\u5728\u66f4\u9ad8\u7684\u968e\u68af\u4e0a\u8a2d\u5b9a\u969c\u7919\uff0c\u4e26\u5f71\u97ff\u672a\u4f86\u7684\u8df3\u8e8d\u3002\u5c0d\u65bc\u9019\u985e\u554f\u984c\uff0c\u52d5\u614b\u898f\u5283\u5f80\u5f80\u96e3\u4ee5\u89e3\u6c7a\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u8907\u96dc\u7684\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\uff08\u4f8b\u5982\u65c5\u884c\u5546\u554f\u984c\uff09\u4e0d\u6eff\u8db3\u7121\u5f8c\u6548\u6027\u3002\u5c0d\u65bc\u9019\u985e\u554f\u984c\uff0c\u6211\u5011\u901a\u5e38\u6703\u9078\u64c7\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u555f\u767c\u5f0f\u641c\u5c0b\u3001\u907a\u50b3\u6f14\u7b97\u6cd5\u3001\u5f37\u5316\u5b78\u7fd2\u7b49\uff0c\u5f9e\u800c\u5728\u6709\u9650\u6642\u9593\u5167\u5f97\u5230\u53ef\u7528\u7684\u5340\u57df\u6027\u6700\u512a\u89e3\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","title":"14.3 \u00a0 \u52d5\u614b\u898f\u5283\u89e3\u984c\u601d\u8def","text":"<p>\u4e0a\u5169\u7bc0\u4ecb\u7d39\u4e86\u52d5\u614b\u898f\u5283\u554f\u984c\u7684\u4e3b\u8981\u7279\u5fb5\uff0c\u63a5\u4e0b\u4f86\u6211\u5011\u4e00\u8d77\u63a2\u7a76\u5169\u500b\u66f4\u52a0\u5be6\u7528\u7684\u554f\u984c\u3002</p> <ol> <li>\u5982\u4f55\u5224\u65b7\u4e00\u500b\u554f\u984c\u662f\u4e0d\u662f\u52d5\u614b\u898f\u5283\u554f\u984c\uff1f</li> <li>\u6c42\u89e3\u52d5\u614b\u898f\u5283\u554f\u984c\u8a72\u5f9e\u4f55\u8655\u5165\u624b\uff0c\u5b8c\u6574\u6b65\u9a5f\u662f\u4ec0\u9ebc\uff1f</li> </ol>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431","title":"14.3.1 \u00a0 \u554f\u984c\u5224\u65b7","text":"<p>\u7e3d\u7684\u4f86\u8aaa\uff0c\u5982\u679c\u4e00\u500b\u554f\u984c\u5305\u542b\u91cd\u758a\u5b50\u554f\u984c\u3001\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u4e26\u6eff\u8db3\u7121\u5f8c\u6548\u6027\uff0c\u90a3\u9ebc\u5b83\u901a\u5e38\u9069\u5408\u7528\u52d5\u614b\u898f\u5283\u6c42\u89e3\u3002\u7136\u800c\uff0c\u6211\u5011\u5f88\u96e3\u5f9e\u554f\u984c\u63cf\u8ff0\u4e2d\u76f4\u63a5\u63d0\u53d6\u51fa\u9019\u4e9b\u7279\u6027\u3002\u56e0\u6b64\u6211\u5011\u901a\u5e38\u6703\u653e\u5bec\u689d\u4ef6\uff0c\u5148\u89c0\u5bdf\u554f\u984c\u662f\u5426\u9069\u5408\u4f7f\u7528\u56de\u6eaf\uff08\u7aae\u8209\uff09\u89e3\u6c7a\u3002</p> <p>\u9069\u5408\u7528\u56de\u6eaf\u89e3\u6c7a\u7684\u554f\u984c\u901a\u5e38\u6eff\u8db3\u201c\u6c7a\u7b56\u6a39\u6a21\u578b\u201d\uff0c\u9019\u7a2e\u554f\u984c\u53ef\u4ee5\u4f7f\u7528\u6a39\u5f62\u7d50\u69cb\u4f86\u63cf\u8ff0\uff0c\u5176\u4e2d\u6bcf\u4e00\u500b\u7bc0\u9ede\u4ee3\u8868\u4e00\u500b\u6c7a\u7b56\uff0c\u6bcf\u4e00\u689d\u8def\u5f91\u4ee3\u8868\u4e00\u500b\u6c7a\u7b56\u5e8f\u5217\u3002</p> <p>\u63db\u53e5\u8a71\u8aaa\uff0c\u5982\u679c\u554f\u984c\u5305\u542b\u660e\u78ba\u7684\u6c7a\u7b56\u6982\u5ff5\uff0c\u4e26\u4e14\u89e3\u662f\u900f\u904e\u4e00\u7cfb\u5217\u6c7a\u7b56\u7522\u751f\u7684\uff0c\u90a3\u9ebc\u5b83\u5c31\u6eff\u8db3\u6c7a\u7b56\u6a39\u6a21\u578b\uff0c\u901a\u5e38\u53ef\u4ee5\u4f7f\u7528\u56de\u6eaf\u4f86\u89e3\u6c7a\u3002</p> <p>\u5728\u6b64\u57fa\u790e\u4e0a\uff0c\u52d5\u614b\u898f\u5283\u554f\u984c\u9084\u6709\u4e00\u4e9b\u5224\u65b7\u7684\u201c\u52a0\u5206\u9805\u201d\u3002</p> <ul> <li>\u554f\u984c\u5305\u542b\u6700\u5927\uff08\u5c0f\uff09\u6216\u6700\u591a\uff08\u5c11\uff09\u7b49\u6700\u6700\u4f73\u5316\u63cf\u8ff0\u3002</li> <li>\u554f\u984c\u7684\u72c0\u614b\u80fd\u5920\u4f7f\u7528\u4e00\u500b\u4e32\u5217\u3001\u591a\u7dad\u77e9\u9663\u6216\u6a39\u4f86\u8868\u793a\uff0c\u4e26\u4e14\u4e00\u500b\u72c0\u614b\u8207\u5176\u5468\u570d\u7684\u72c0\u614b\u5b58\u5728\u905e\u63a8\u95dc\u4fc2\u3002</li> </ul> <p>\u76f8\u61c9\u5730\uff0c\u4e5f\u5b58\u5728\u4e00\u4e9b\u201c\u6e1b\u5206\u9805\u201d\u3002</p> <ul> <li>\u554f\u984c\u7684\u76ee\u6a19\u662f\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u89e3\u6c7a\u65b9\u6848\uff0c\u800c\u4e0d\u662f\u627e\u51fa\u6700\u512a\u89e3\u3002</li> <li>\u554f\u984c\u63cf\u8ff0\u4e2d\u6709\u660e\u986f\u7684\u6392\u5217\u7d44\u5408\u7684\u7279\u5fb5\uff0c\u9700\u8981\u8fd4\u56de\u5177\u9ad4\u7684\u591a\u500b\u65b9\u6848\u3002</li> </ul> <p>\u5982\u679c\u4e00\u500b\u554f\u984c\u6eff\u8db3\u6c7a\u7b56\u6a39\u6a21\u578b\uff0c\u4e26\u5177\u6709\u8f03\u70ba\u660e\u986f\u7684\u201c\u52a0\u5206\u9805\u201d\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5047\u8a2d\u5b83\u662f\u4e00\u500b\u52d5\u614b\u898f\u5283\u554f\u984c\uff0c\u4e26\u5728\u6c42\u89e3\u904e\u7a0b\u4e2d\u9a57\u8b49\u5b83\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","title":"14.3.2 \u00a0 \u554f\u984c\u6c42\u89e3\u6b65\u9a5f","text":"<p>\u52d5\u614b\u898f\u5283\u7684\u89e3\u984c\u6d41\u7a0b\u6703\u56e0\u554f\u984c\u7684\u6027\u8cea\u548c\u96e3\u5ea6\u800c\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u901a\u5e38\u9075\u5faa\u4ee5\u4e0b\u6b65\u9a5f\uff1a\u63cf\u8ff0\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5efa\u7acb \\(dp\\) \u8868\uff0c\u63a8\u5c0e\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff0c\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u7b49\u3002</p> <p>\u70ba\u4e86\u66f4\u5f62\u8c61\u5730\u5c55\u793a\u89e3\u984c\u6b65\u9a5f\uff0c\u6211\u5011\u4f7f\u7528\u4e00\u500b\u7d93\u5178\u554f\u984c\u201c\u6700\u5c0f\u8def\u5f91\u548c\u201d\u4f86\u8209\u4f8b\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b \\(n \\times m\\) \u7684\u4e8c\u7dad\u7db2\u683c <code>grid</code> \uff0c\u7db2\u683c\u4e2d\u7684\u6bcf\u500b\u55ae\u5143\u683c\u5305\u542b\u4e00\u500b\u975e\u8ca0\u6574\u6578\uff0c\u8868\u793a\u8a72\u55ae\u5143\u683c\u7684\u4ee3\u50f9\u3002\u6a5f\u5668\u4eba\u4ee5\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\u70ba\u8d77\u59cb\u9ede\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u8005\u5411\u53f3\u79fb\u52d5\u4e00\u6b65\uff0c\u76f4\u81f3\u5230\u9054\u53f3\u4e0b\u89d2\u55ae\u5143\u683c\u3002\u8acb\u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\u7684\u6700\u5c0f\u8def\u5f91\u548c\u3002</p> <p>\u5716 14-10 \u5c55\u793a\u4e86\u4e00\u500b\u4f8b\u5b50\uff0c\u7d66\u5b9a\u7db2\u683c\u7684\u6700\u5c0f\u8def\u5f91\u548c\u70ba \\(13\\) \u3002</p> <p></p> <p> \u5716 14-10 \u00a0 \u6700\u5c0f\u8def\u5f91\u548c\u793a\u4f8b\u8cc7\u6599 </p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f2a\u7684\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5f9e\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u672c\u984c\u7684\u6bcf\u4e00\u8f2a\u7684\u6c7a\u7b56\u5c31\u662f\u5f9e\u7576\u524d\u683c\u5b50\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u3002\u8a2d\u7576\u524d\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u70ba \\([i, j]\\) \uff0c\u5247\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u5f8c\uff0c\u7d22\u5f15\u8b8a\u70ba \\([i+1, j]\\) \u6216 \\([i, j+1]\\) \u3002\u56e0\u6b64\uff0c\u72c0\u614b\u61c9\u5305\u542b\u884c\u7d22\u5f15\u548c\u5217\u7d22\u5f15\u5169\u500b\u8b8a\u6578\uff0c\u8a18\u70ba \\([i, j]\\) \u3002</p> <p>\u72c0\u614b \\([i, j]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\u70ba\uff1a\u5f9e\u8d77\u59cb\u9ede \\([0, 0]\\) \u8d70\u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c\uff0c\u89e3\u8a18\u70ba \\(dp[i, j]\\) \u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u5011\u5c31\u5f97\u5230\u4e86\u5716 14-11 \u6240\u793a\u7684\u4e8c\u7dad \\(dp\\) \u77e9\u9663\uff0c\u5176\u5c3a\u5bf8\u8207\u8f38\u5165\u7db2\u683c \\(grid\\) \u76f8\u540c\u3002</p> <p></p> <p> \u5716 14-11 \u00a0 \u72c0\u614b\u5b9a\u7fa9\u8207 dp \u8868 </p> <p>Note</p> <p>\u52d5\u614b\u898f\u5283\u548c\u56de\u6eaf\u904e\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u70ba\u4e00\u500b\u6c7a\u7b56\u5e8f\u5217\uff0c\u800c\u72c0\u614b\u7531\u6240\u6709\u6c7a\u7b56\u8b8a\u6578\u69cb\u6210\u3002\u5b83\u61c9\u7576\u5305\u542b\u63cf\u8ff0\u89e3\u984c\u9032\u5ea6\u7684\u6240\u6709\u8b8a\u6578\uff0c\u5176\u5305\u542b\u4e86\u8db3\u5920\u7684\u8cc7\u8a0a\uff0c\u80fd\u5920\u7528\u4f86\u63a8\u5c0e\u51fa\u4e0b\u4e00\u500b\u72c0\u614b\u3002</p> <p>\u6bcf\u500b\u72c0\u614b\u90fd\u5c0d\u61c9\u4e00\u500b\u5b50\u554f\u984c\uff0c\u6211\u5011\u6703\u5b9a\u7fa9\u4e00\u500b \\(dp\\) \u8868\u4f86\u5132\u5b58\u6240\u6709\u5b50\u554f\u984c\u7684\u89e3\uff0c\u72c0\u614b\u7684\u6bcf\u500b\u7368\u7acb\u8b8a\u6578\u90fd\u662f \\(dp\\) \u8868\u7684\u4e00\u500b\u7dad\u5ea6\u3002\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\\(dp\\) \u8868\u662f\u72c0\u614b\u548c\u5b50\u554f\u984c\u7684\u89e3\u4e4b\u9593\u7684\u5c0d\u6620\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u9032\u800c\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b</p> <p>\u5c0d\u65bc\u72c0\u614b \\([i, j]\\) \uff0c\u5b83\u53ea\u80fd\u5f9e\u4e0a\u908a\u683c\u5b50 \\([i-1, j]\\) \u548c\u5de6\u908a\u683c\u5b50 \\([i, j-1]\\) \u8f49\u79fb\u800c\u4f86\u3002\u56e0\u6b64\u6700\u512a\u5b50\u7d50\u69cb\u70ba\uff1a\u5230\u9054 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c\u7531 \\([i, j-1]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c\u8207 \\([i-1, j]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c\u4e2d\u8f03\u5c0f\u7684\u90a3\u4e00\u500b\u6c7a\u5b9a\u3002</p> <p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u63a8\u51fa\u5716 14-12 \u6240\u793a\u7684\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> \u5716 14-12 \u00a0 \u6700\u512a\u5b50\u7d50\u69cb\u8207\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b </p> <p>Note</p> <p>\u6839\u64da\u5b9a\u7fa9\u597d\u7684 \\(dp\\) \u8868\uff0c\u601d\u8003\u539f\u554f\u984c\u548c\u5b50\u554f\u984c\u7684\u95dc\u4fc2\uff0c\u627e\u51fa\u900f\u904e\u5b50\u554f\u984c\u7684\u6700\u512a\u89e3\u4f86\u69cb\u9020\u539f\u554f\u984c\u7684\u6700\u512a\u89e3\u7684\u65b9\u6cd5\uff0c\u5373\u6700\u512a\u5b50\u7d50\u69cb\u3002</p> <p>\u4e00\u65e6\u6211\u5011\u627e\u5230\u4e86\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528\u5b83\u4f86\u69cb\u5efa\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u548c\u72c0\u614b\u8f49\u79fb\u9806\u5e8f</p> <p>\u5728\u672c\u984c\u4e2d\uff0c\u8655\u5728\u9996\u884c\u7684\u72c0\u614b\u53ea\u80fd\u5f9e\u5176\u5de6\u908a\u7684\u72c0\u614b\u5f97\u4f86\uff0c\u8655\u5728\u9996\u5217\u7684\u72c0\u614b\u53ea\u80fd\u5f9e\u5176\u4e0a\u908a\u7684\u72c0\u614b\u5f97\u4f86\uff0c\u56e0\u6b64\u9996\u884c \\(i = 0\\) \u548c\u9996\u5217 \\(j = 0\\) \u662f\u908a\u754c\u689d\u4ef6\u3002</p> <p>\u5982\u5716 14-13 \u6240\u793a\uff0c\u7531\u65bc\u6bcf\u500b\u683c\u5b50\u662f\u7531\u5176\u5de6\u65b9\u683c\u5b50\u548c\u4e0a\u65b9\u683c\u5b50\u8f49\u79fb\u800c\u4f86\uff0c\u56e0\u6b64\u6211\u5011\u4f7f\u7528\u8ff4\u5708\u4f86\u8d70\u8a2a\u77e9\u9663\uff0c\u5916\u8ff4\u5708\u8d70\u8a2a\u5404\u884c\uff0c\u5167\u8ff4\u5708\u8d70\u8a2a\u5404\u5217\u3002</p> <p></p> <p> \u5716 14-13 \u00a0 \u908a\u754c\u689d\u4ef6\u8207\u72c0\u614b\u8f49\u79fb\u9806\u5e8f </p> <p>Note</p> <p>\u908a\u754c\u689d\u4ef6\u5728\u52d5\u614b\u898f\u5283\u4e2d\u7528\u65bc\u521d\u59cb\u5316 \\(dp\\) \u8868\uff0c\u5728\u641c\u5c0b\u4e2d\u7528\u65bc\u526a\u679d\u3002</p> <p>\u72c0\u614b\u8f49\u79fb\u9806\u5e8f\u7684\u6838\u5fc3\u662f\u8981\u4fdd\u8b49\u5728\u8a08\u7b97\u7576\u524d\u554f\u984c\u7684\u89e3\u6642\uff0c\u6240\u6709\u5b83\u4f9d\u8cf4\u7684\u66f4\u5c0f\u5b50\u554f\u984c\u7684\u89e3\u90fd\u5df2\u7d93\u88ab\u6b63\u78ba\u5730\u8a08\u7b97\u51fa\u4f86\u3002</p> <p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u5011\u5df2\u7d93\u53ef\u4ee5\u76f4\u63a5\u5beb\u51fa\u52d5\u614b\u898f\u5283\u7a0b\u5f0f\u78bc\u3002\u7136\u800c\u5b50\u554f\u984c\u5206\u89e3\u662f\u4e00\u7a2e\u5f9e\u9802\u81f3\u5e95\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u6309\u7167\u201c\u66b4\u529b\u641c\u5c0b \\(\\rightarrow\\) \u8a18\u61b6\u5316\u641c\u5c0b \\(\\rightarrow\\) \u52d5\u614b\u898f\u5283\u201d\u7684\u9806\u5e8f\u5be6\u73fe\u66f4\u52a0\u7b26\u5408\u601d\u7dad\u7fd2\u6163\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u5c0b","text":"<p>\u5f9e\u72c0\u614b \\([i, j]\\) \u958b\u59cb\u641c\u5c0b\uff0c\u4e0d\u65b7\u5206\u89e3\u70ba\u66f4\u5c0f\u7684\u72c0\u614b \\([i-1, j]\\) \u548c \\([i, j-1]\\) \uff0c\u905e\u8ff4\u51fd\u5f0f\u5305\u62ec\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u905e\u8ff4\u53c3\u6578\uff1a\u72c0\u614b \\([i, j]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u5f9e \\([0, 0]\\) \u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f91\u548c \\(dp[i, j]\\) \u3002</li> <li>\u7d42\u6b62\u689d\u4ef6\uff1a\u7576 \\(i = 0\\) \u4e14 \\(j = 0\\) \u6642\uff0c\u8fd4\u56de\u4ee3\u50f9 \\(grid[0, 0]\\) \u3002</li> <li>\u526a\u679d\uff1a\u7576 \\(i &lt; 0\\) \u6642\u6216 \\(j &lt; 0\\) \u6642\u7d22\u5f15\u8d8a\u754c\uff0c\u6b64\u6642\u8fd4\u56de\u4ee3\u50f9 \\(+\\infty\\) \uff0c\u4ee3\u8868\u4e0d\u53ef\u884c\u3002</li> </ul> <p>\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b\"\"\"\n    # \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfunction minPathSumDFS(grid, i, j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u578b\u5225\u662f\u56fa\u5b9a\u7bc4\u570d\u7684\u6574\u6578\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u7121\u7aae\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b */\nfun minPathSumDFS(grid: Array&lt;IntArray&gt;, i: Int, j: Int): Int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    val up = minPathSumDFS(grid, i - 1, j)\n    val left = minPathSumDFS(grid, i, j - 1)\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b ###\ndef min_path_sum_dfs(grid, i, j)\n  # \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n  return grid[i][j] if i == 0 &amp;&amp; j == 0\n  # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  up = min_path_sum_dfs(grid, i - 1, j)\n  left = min_path_sum_dfs(grid, i, j - 1)\n  # \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  [left, up].min + grid[i][j]\nend\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u66b4\u529b\u641c\u5c0b\nfn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    var up = minPathSumDFS(grid, i - 1, j);\n    var left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-14 \u7d66\u51fa\u4e86\u4ee5 \\(dp[2, 1]\\) \u70ba\u6839\u7bc0\u9ede\u7684\u905e\u8ff4\u6a39\uff0c\u5176\u4e2d\u5305\u542b\u4e00\u4e9b\u91cd\u758a\u5b50\u554f\u984c\uff0c\u5176\u6578\u91cf\u6703\u96a8\u8457\u7db2\u683c <code>grid</code> \u7684\u5c3a\u5bf8\u8b8a\u5927\u800c\u6025\u5287\u589e\u591a\u3002</p> <p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u9020\u6210\u91cd\u758a\u5b50\u554f\u984c\u7684\u539f\u56e0\u70ba\uff1a\u5b58\u5728\u591a\u689d\u8def\u5f91\u53ef\u4ee5\u5f9e\u5de6\u4e0a\u89d2\u5230\u9054\u67d0\u4e00\u55ae\u5143\u683c\u3002</p> <p></p> <p> \u5716 14-14 \u00a0 \u66b4\u529b\u641c\u5c0b\u905e\u8ff4\u6a39 </p> <p>\u6bcf\u500b\u72c0\u614b\u90fd\u6709\u5411\u4e0b\u548c\u5411\u53f3\u5169\u7a2e\u9078\u64c7\uff0c\u5f9e\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u7e3d\u5171\u9700\u8981 \\(m + n - 2\\) \u6b65\uff0c\u6240\u4ee5\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^{m + n})\\) \u3002\u8acb\u6ce8\u610f\uff0c\u9019\u7a2e\u8a08\u7b97\u65b9\u5f0f\u672a\u8003\u616e\u81e8\u8fd1\u7db2\u683c\u908a\u754c\u7684\u60c5\u6cc1\uff0c\u7576\u5230\u9054\u7db2\u8def\u908a\u754c\u6642\u53ea\u5269\u4e0b\u4e00\u7a2e\u9078\u64c7\uff0c\u56e0\u6b64\u5be6\u969b\u7684\u8def\u5f91\u6578\u91cf\u6703\u5c11\u4e00\u4e9b\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b","text":"<p>\u6211\u5011\u5f15\u5165\u4e00\u500b\u548c\u7db2\u683c <code>grid</code> \u76f8\u540c\u5c3a\u5bf8\u7684\u8a18\u61b6\u4e32\u5217 <code>mem</code> \uff0c\u7528\u65bc\u8a18\u9304\u5404\u500b\u5b50\u554f\u984c\u7684\u89e3\uff0c\u4e26\u5c07\u91cd\u758a\u5b50\u554f\u984c\u9032\u884c\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\"\"\"\n    # \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u578b\u5225\u662f\u56fa\u5b9a\u7bc4\u570d\u7684\u6574\u6578\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u7121\u7aae\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfun minPathSumDFSMem(\n    grid: Array&lt;IntArray&gt;,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j]\n    }\n    // \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    val up = minPathSumDFSMem(grid, mem, i - 1, j)\n    val left = minPathSumDFSMem(grid, mem, i, j - 1)\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b ###\ndef min_path_sum_dfs_mem(grid, mem, i, j)\n  # \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n  return grid[0][0] if i == 0 &amp;&amp; j == 0\n  # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  return mem[i][j] if mem[i][j] != -1\n  # \u5de6\u908a\u548c\u4e0a\u908a\u55ae\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n  left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n  # \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n  mem[i][j] = [left, up].min + grid[i][j]\nend\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\nfn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u70ba\u5de6\u4e0a\u89d2\u55ae\u5143\u683c\uff0c\u5247\u7d42\u6b62\u641c\u5c0b\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de +\u221e \u4ee3\u50f9\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {\n        return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    }\n    // \u8a08\u7b97\u5f9e\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    var up = minPathSumDFSMem(grid, mem, i - 1, j);\n    var left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8fd4\u56de\u5f9e\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f91\u4ee3\u50f9\n    mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 14-15 \u6240\u793a\uff0c\u5728\u5f15\u5165\u8a18\u61b6\u5316\u5f8c\uff0c\u6240\u6709\u5b50\u554f\u984c\u7684\u89e3\u53ea\u9700\u8a08\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u53d6\u6c7a\u65bc\u72c0\u614b\u7e3d\u6578\uff0c\u5373\u7db2\u683c\u5c3a\u5bf8 \\(O(nm)\\) \u3002</p> <p></p> <p> \u5716 14-15 \u00a0 \u8a18\u61b6\u5316\u641c\u5c0b\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52d5\u614b\u898f\u5283","text":"<p>\u57fa\u65bc\u8fed\u4ee3\u5be6\u73fe\u52d5\u614b\u898f\u5283\u89e3\u6cd5\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in 1 ..&lt; m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for i in 1 ..&lt; n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1 ..&lt; n {\n        for j in 1 ..&lt; m {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 */\nfun minPathSumDP(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n) { IntArray(m) }\n    dp[0][0] = grid[0][0]\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (j in 1..&lt;m) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (i in 1..&lt;n) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (i in 1..&lt;n) {\n        for (j in 1..&lt;m) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283 ###\ndef min_path_sum_dp(grid)\n  n, m = grid.length, grid.first.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n) { Array.new(m, 0) }\n  dp[0][0] = grid[0][0]\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  (1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n  (1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for i in 1...n\n    for j in 1...m\n      dp[i][j] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]\n    end\n  end\n  dp[n -1][m -1]\nend\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u52d5\u614b\u898f\u5283\nfn minPathSumDP(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][m]i32{[_]i32{0} ** m} ** n;\n    dp[0][0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (1..m) |j| {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    for (1..n) |i| {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (1..n) |i| {\n        for (1..m) |j| {\n            dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-16 \u5c55\u793a\u4e86\u6700\u5c0f\u8def\u5f91\u548c\u7684\u72c0\u614b\u8f49\u79fb\u904e\u7a0b\uff0c\u5176\u8d70\u8a2a\u4e86\u6574\u500b\u7db2\u683c\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(nm)\\) \u3002</p> <p>\u9663\u5217 <code>dp</code> \u5927\u5c0f\u70ba \\(n \\times m\\) \uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(nm)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-16 \u00a0 \u6700\u5c0f\u8def\u5f91\u548c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","title":"4. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7531\u65bc\u6bcf\u500b\u683c\u5b50\u53ea\u8207\u5176\u5de6\u908a\u548c\u4e0a\u908a\u7684\u683c\u5b50\u6709\u95dc\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u53ea\u7528\u4e00\u500b\u55ae\u884c\u9663\u5217\u4f86\u5be6\u73fe \\(dp\\) \u8868\u3002</p> <p>\u8acb\u6ce8\u610f\uff0c\u56e0\u70ba\u9663\u5217 <code>dp</code> \u53ea\u80fd\u8868\u793a\u4e00\u884c\u7684\u72c0\u614b\uff0c\u6240\u4ee5\u6211\u5011\u7121\u6cd5\u63d0\u524d\u521d\u59cb\u5316\u9996\u5217\u72c0\u614b\uff0c\u800c\u662f\u5728\u8d70\u8a2a\u6bcf\u884c\u6642\u66f4\u65b0\u5b83\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * m\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in range(1, n):\n        # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(m);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, m)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: m)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in 1 ..&lt; m {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in 1 ..&lt; n {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in 1 ..&lt; m {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n  for (int i = 1; i &lt; n; i++) {\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    dp[0] = dp[0] + grid[i][0];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; m];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in 1..n {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(m, sizeof(int));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun minPathSumDPComp(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(m)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for (j in 1..&lt;m) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (i in 1..&lt;n) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (j in 1..&lt;m) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef min_path_sum_dp_comp(grid)\n  n, m = grid.length, grid.first.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(m, 0)\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  dp[0] = grid[0][0]\n  (1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n  for i in 1...n\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    dp[0] = dp[0] + grid[i][0]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n    (1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }\n  end\n  dp[m - 1]\nend\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f91\u548c\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn minPathSumDPComp(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** m;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (1..m) |j| {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (1..n) |i| {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        for (1..m) |j| {\n            dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/","title":"14.6 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u554f\u984c","text":"<p>\u7de8\u8f2f\u8ddd\u96e2\uff0c\u4e5f\u7a31 Levenshtein \u8ddd\u96e2\uff0c\u6307\u5169\u500b\u5b57\u4e32\u4e4b\u9593\u4e92\u76f8\u8f49\u63db\u7684\u6700\u5c11\u4fee\u6539\u6b21\u6578\uff0c\u901a\u5e38\u7528\u65bc\u5728\u8cc7\u8a0a\u6aa2\u7d22\u548c\u81ea\u7136\u8a9e\u8a00\u8655\u7406\u4e2d\u5ea6\u91cf\u5169\u500b\u5e8f\u5217\u7684\u76f8\u4f3c\u5ea6\u3002</p> <p>Question</p> <p>\u8f38\u5165\u5169\u500b\u5b57\u4e32 \\(s\\) \u548c \\(t\\) \uff0c\u8fd4\u56de\u5c07 \\(s\\) \u8f49\u63db\u70ba \\(t\\) \u6240\u9700\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u3002</p> <p>\u4f60\u53ef\u4ee5\u5728\u4e00\u500b\u5b57\u4e32\u4e2d\u9032\u884c\u4e09\u7a2e\u7de8\u8f2f\u64cd\u4f5c\uff1a\u63d2\u5165\u4e00\u500b\u5b57\u5143\u3001\u522a\u9664\u4e00\u500b\u5b57\u5143\u3001\u5c07\u5b57\u5143\u66ff\u63db\u70ba\u4efb\u610f\u4e00\u500b\u5b57\u5143\u3002</p> <p>\u5982\u5716 14-27 \u6240\u793a\uff0c\u5c07 <code>kitten</code> \u8f49\u63db\u70ba <code>sitting</code> \u9700\u8981\u7de8\u8f2f 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u63db\u64cd\u4f5c\u8207 1 \u6b21\u65b0\u589e\u64cd\u4f5c\uff1b\u5c07 <code>hello</code> \u8f49\u63db\u70ba <code>algo</code> \u9700\u8981 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u63db\u64cd\u4f5c\u548c 1 \u6b21\u522a\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u5716 14-27 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u7684\u793a\u4f8b\u8cc7\u6599 </p> <p>\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c\u53ef\u4ee5\u5f88\u81ea\u7136\u5730\u7528\u6c7a\u7b56\u6a39\u6a21\u578b\u4f86\u89e3\u91cb\u3002\u5b57\u4e32\u5c0d\u61c9\u6a39\u7bc0\u9ede\uff0c\u4e00\u8f2a\u6c7a\u7b56\uff08\u4e00\u6b21\u7de8\u8f2f\u64cd\u4f5c\uff09\u5c0d\u61c9\u6a39\u7684\u4e00\u689d\u908a\u3002</p> <p>\u5982\u5716 14-28 \u6240\u793a\uff0c\u5728\u4e0d\u9650\u5236\u64cd\u4f5c\u7684\u60c5\u6cc1\u4e0b\uff0c\u6bcf\u500b\u7bc0\u9ede\u90fd\u53ef\u4ee5\u6d3e\u751f\u51fa\u8a31\u591a\u689d\u908a\uff0c\u6bcf\u689d\u908a\u5c0d\u61c9\u4e00\u7a2e\u64cd\u4f5c\uff0c\u9019\u610f\u5473\u8457\u5f9e <code>hello</code> \u8f49\u63db\u5230 <code>algo</code> \u6709\u8a31\u591a\u7a2e\u53ef\u80fd\u7684\u8def\u5f91\u3002</p> <p>\u5f9e\u6c7a\u7b56\u6a39\u7684\u89d2\u5ea6\u770b\uff0c\u672c\u984c\u7684\u76ee\u6a19\u662f\u6c42\u89e3\u7bc0\u9ede <code>hello</code> \u548c\u7bc0\u9ede <code>algo</code> \u4e4b\u9593\u7684\u6700\u77ed\u8def\u5f91\u3002</p> <p></p> <p> \u5716 14-28 \u00a0 \u57fa\u65bc\u6c7a\u7b56\u6a39\u6a21\u578b\u8868\u793a\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","title":"1. \u00a0 \u52d5\u614b\u898f\u5283\u601d\u8def","text":"<p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f2a\u7684\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5f9e\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u6bcf\u4e00\u8f2a\u7684\u6c7a\u7b56\u662f\u5c0d\u5b57\u4e32 \\(s\\) \u9032\u884c\u4e00\u6b21\u7de8\u8f2f\u64cd\u4f5c\u3002</p> <p>\u6211\u5011\u5e0c\u671b\u5728\u7de8\u8f2f\u64cd\u4f5c\u7684\u904e\u7a0b\u4e2d\uff0c\u554f\u984c\u7684\u898f\u6a21\u9010\u6f38\u7e2e\u5c0f\uff0c\u9019\u6a23\u624d\u80fd\u69cb\u5efa\u5b50\u554f\u984c\u3002\u8a2d\u5b57\u4e32 \\(s\\) \u548c \\(t\\) \u7684\u9577\u5ea6\u5206\u5225\u70ba \\(n\\) \u548c \\(m\\) \uff0c\u6211\u5011\u5148\u8003\u616e\u5169\u5b57\u4e32\u5c3e\u90e8\u7684\u5b57\u5143 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u3002</p> <ul> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u76f8\u540c\uff0c\u6211\u5011\u53ef\u4ee5\u8df3\u904e\u5b83\u5011\uff0c\u76f4\u63a5\u8003\u616e \\(s[n-2]\\) \u548c \\(t[m-2]\\) \u3002</li> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u4e0d\u540c\uff0c\u6211\u5011\u9700\u8981\u5c0d \\(s\\) \u9032\u884c\u4e00\u6b21\u7de8\u8f2f\uff08\u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\uff09\uff0c\u4f7f\u5f97\u5169\u5b57\u4e32\u5c3e\u90e8\u7684\u5b57\u5143\u76f8\u540c\uff0c\u5f9e\u800c\u53ef\u4ee5\u8df3\u904e\u5b83\u5011\uff0c\u8003\u616e\u898f\u6a21\u66f4\u5c0f\u7684\u554f\u984c\u3002</li> </ul> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u6211\u5011\u5728\u5b57\u4e32 \\(s\\) \u4e2d\u9032\u884c\u7684\u6bcf\u4e00\u8f2a\u6c7a\u7b56\uff08\u7de8\u8f2f\u64cd\u4f5c\uff09\uff0c\u90fd\u6703\u4f7f\u5f97 \\(s\\) \u548c \\(t\\) \u4e2d\u5269\u9918\u7684\u5f85\u5339\u914d\u5b57\u5143\u767c\u751f\u8b8a\u5316\u3002\u56e0\u6b64\uff0c\u72c0\u614b\u70ba\u7576\u524d\u5728 \\(s\\) \u548c \\(t\\) \u4e2d\u8003\u616e\u7684\u7b2c \\(i\\) \u548c\u7b2c \\(j\\) \u500b\u5b57\u5143\uff0c\u8a18\u70ba \\([i, j]\\) \u3002</p> <p>\u72c0\u614b \\([i, j]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\uff1a\u5c07 \\(s\\) \u7684\u524d \\(i\\) \u500b\u5b57\u5143\u66f4\u6539\u70ba \\(t\\) \u7684\u524d \\(j\\) \u500b\u5b57\u5143\u6240\u9700\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u3002</p> <p>\u81f3\u6b64\uff0c\u5f97\u5230\u4e00\u500b\u5c3a\u5bf8\u70ba \\((i+1) \\times (j+1)\\) \u7684\u4e8c\u7dad \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u9032\u800c\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b</p> <p>\u8003\u616e\u5b50\u554f\u984c \\(dp[i, j]\\) \uff0c\u5176\u5c0d\u61c9\u7684\u5169\u500b\u5b57\u4e32\u7684\u5c3e\u90e8\u5b57\u5143\u70ba \\(s[i-1]\\) \u548c \\(t[j-1]\\) \uff0c\u53ef\u6839\u64da\u4e0d\u540c\u7de8\u8f2f\u64cd\u4f5c\u5206\u70ba\u5716 14-29 \u6240\u793a\u7684\u4e09\u7a2e\u60c5\u6cc1\u3002</p> <ol> <li>\u5728 \\(s[i-1]\\) \u4e4b\u5f8c\u65b0\u589e \\(t[j-1]\\) \uff0c\u5247\u5269\u9918\u5b50\u554f\u984c \\(dp[i, j-1]\\) \u3002</li> <li>\u522a\u9664 \\(s[i-1]\\) \uff0c\u5247\u5269\u9918\u5b50\u554f\u984c \\(dp[i-1, j]\\) \u3002</li> <li>\u5c07 \\(s[i-1]\\) \u66ff\u63db\u70ba \\(t[j-1]\\) \uff0c\u5247\u5269\u9918\u5b50\u554f\u984c \\(dp[i-1, j-1]\\) \u3002</li> </ol> <p></p> <p> \u5716 14-29 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u7684\u72c0\u614b\u8f49\u79fb </p> <p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u5f97\u6700\u512a\u5b50\u7d50\u69cb\uff1a\\(dp[i, j]\\) \u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u7b49\u65bc \\(dp[i, j-1]\\)\u3001\\(dp[i-1, j]\\)\u3001\\(dp[i-1, j-1]\\) \u4e09\u8005\u4e2d\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\uff0c\u518d\u52a0\u4e0a\u672c\u6b21\u7684\u7de8\u8f2f\u6b65\u6578 \\(1\\) \u3002\u5c0d\u61c9\u7684\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u70ba\uff1a</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>\u8acb\u6ce8\u610f\uff0c\u7576 \\(s[i-1]\\) \u548c \\(t[j-1]\\) \u76f8\u540c\u6642\uff0c\u7121\u9808\u7de8\u8f2f\u7576\u524d\u5b57\u5143\uff0c\u9019\u7a2e\u60c5\u6cc1\u4e0b\u7684\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u70ba\uff1a</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>\u7b2c\u4e09\u6b65\uff1a\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u548c\u72c0\u614b\u8f49\u79fb\u9806\u5e8f</p> <p>\u7576\u5169\u5b57\u4e32\u90fd\u70ba\u7a7a\u6642\uff0c\u7de8\u8f2f\u6b65\u6578\u70ba \\(0\\) \uff0c\u5373 \\(dp[0, 0] = 0\\) \u3002\u7576 \\(s\\) \u70ba\u7a7a\u4f46 \\(t\\) \u4e0d\u70ba\u7a7a\u6642\uff0c\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u7b49\u65bc \\(t\\) \u7684\u9577\u5ea6\uff0c\u5373\u9996\u884c \\(dp[0, j] = j\\) \u3002\u7576 \\(s\\) \u4e0d\u70ba\u7a7a\u4f46 \\(t\\) \u70ba\u7a7a\u6642\uff0c\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u7b49\u65bc \\(s\\) \u7684\u9577\u5ea6\uff0c\u5373\u9996\u5217 \\(dp[i, 0] = i\\) \u3002</p> <p>\u89c0\u5bdf\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff0c\u89e3 \\(dp[i, j]\\) \u4f9d\u8cf4\u5de6\u65b9\u3001\u4e0a\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u89e3\uff0c\u56e0\u6b64\u900f\u904e\u5169\u5c64\u8ff4\u5708\u6b63\u5e8f\u8d70\u8a2a\u6574\u500b \\(dp\\) \u8868\u5373\u53ef\u3002</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"\u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1 ... n {\n        dp[i][0] = i\n    }\n    for j in 1 ... m {\n        dp[0][j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1 ... n {\n        for j in 1 ... m {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1..=n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] =\n                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 */\nfun editDistanceDP(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = Array(n + 1) { IntArray(m + 1) }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (i in 1..n) {\n        dp[i][0] = i\n    }\n    for (j in 1..m) {\n        dp[0][j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (i in 1..n) {\n        for (j in 1..m) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.rb<pre><code>### \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283 ###\ndef edit_distance_dp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(n + 1) { Array.new(m + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n  (1...(n + 1)).each { |i| dp[i][0] = i }\n  (1...(m + 1)).each { |j| dp[0][j] = j }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for i in 1...(n + 1)\n    for j in 1...(m +1)\n      if s[i - 1] == t[j - 1]\n        # \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n        dp[i][j] = dp[i - 1][j - 1]\n      else\n        # \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n        dp[i][j] = [dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]].min + 1\n      end\n    end\n  end\n  dp[n][m]\nend\n</code></pre> edit_distance.zig<pre><code>// \u7de8\u8f2f\u8ddd\u96e2\uff1a\u52d5\u614b\u898f\u5283\nfn editDistanceDP(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..n + 1) |i| {\n        dp[i][0] = @intCast(i);\n    }\n    for (1..m + 1) |j| {\n        dp[0][j] = @intCast(j);\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..m + 1) |j| {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[i][j] = @min(@min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 14-30 \u6240\u793a\uff0c\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c\u7684\u72c0\u614b\u8f49\u79fb\u904e\u7a0b\u8207\u80cc\u5305\u554f\u984c\u975e\u5e38\u985e\u4f3c\uff0c\u90fd\u53ef\u4ee5\u770b\u4f5c\u586b\u5beb\u4e00\u500b\u4e8c\u7dad\u7db2\u683c\u7684\u904e\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-30 \u00a0 \u7de8\u8f2f\u8ddd\u96e2\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","title":"3. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7531\u65bc \\(dp[i,j]\\) \u662f\u7531\u4e0a\u65b9 \\(dp[i-1, j]\\)\u3001\u5de6\u65b9 \\(dp[i, j-1]\\)\u3001\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \u8f49\u79fb\u800c\u4f86\u7684\uff0c\u800c\u6b63\u5e8f\u8d70\u8a2a\u6703\u4e1f\u5931\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \uff0c\u5012\u5e8f\u8d70\u8a2a\u7121\u6cd5\u63d0\u524d\u69cb\u5efa \\(dp[i, j-1]\\) \uff0c\u56e0\u6b64\u5169\u7a2e\u8d70\u8a2a\u9806\u5e8f\u90fd\u4e0d\u53ef\u53d6\u3002</p> <p>\u70ba\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u4e00\u500b\u8b8a\u6578 <code>leftup</code> \u4f86\u66ab\u5b58\u5de6\u4e0a\u65b9\u7684\u89e3 \\(dp[i-1, j-1]\\) \uff0c\u5f9e\u800c\u53ea\u9700\u8003\u616e\u5de6\u65b9\u548c\u4e0a\u65b9\u7684\u89e3\u3002\u6b64\u6642\u7684\u60c5\u6cc1\u8207\u5b8c\u5168\u80cc\u5305\u554f\u984c\u76f8\u540c\uff0c\u53ef\u4f7f\u7528\u6b63\u5e8f\u8d70\u8a2a\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"\u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in range(1, m + 1):\n        dp[j] = j\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in range(1, n + 1):\n        # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        leftup = dp[0]  # \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] += 1\n        # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup\n            else:\n                # \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i := 1; i &lt;= n; i++ {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        leftUp := dp[0] // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftUp\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in 1 ... m {\n        dp[j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in 1 ... n {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in 1 ... m {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n  for (int i = 1; i &lt;= n; i++) {\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n    dp[0] = i;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n        dp[j] = leftup;\n      } else {\n        // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for i in 1..=n {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        let mut leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i as i32;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    int res = dp[m];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun editDistanceDPComp(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = IntArray(m + 1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (j in 1..m) {\n        dp[j] = j\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (i in 1..n) {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (j in 1..m) {\n            val temp = dp[j]\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.rb<pre><code>### \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef edit_distance_dp_comp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(m + 1, 0)\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n  (1...(m + 1)).each { |j| dp[j] = j }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n  for i in 1...(n + 1)\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n    leftup = dp.first # \u66ab\u5b58 dp[i-1, j-1]\n    dp[0] += 1\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n    for j in 1...(m + 1)\n      temp = dp[j]\n      if s[i - 1] == t[j - 1]\n        # \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n        dp[j] = leftup\n      else\n        # \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n        dp[j] = [dp[j - 1], dp[j], leftup].min + 1\n      end\n      leftup = temp # \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n    end\n  end\n  dp[m]\nend\n</code></pre> edit_distance.zig<pre><code>// \u7de8\u8f2f\u8ddd\u96e2\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn editDistanceDPComp(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_]i32{0} ** (m + 1);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\n    for (1..m + 1) |j| {\n        dp[j] = @intCast(j);\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\n    for (1..n + 1) |i| {\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0]; // \u66ab\u5b58 dp[i-1, j-1]\n        dp[0] = @intCast(i);\n        // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u5217\n        for (1..m + 1) |j| {\n            var temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u5169\u5b57\u5143\u76f8\u7b49\uff0c\u5247\u76f4\u63a5\u8df3\u904e\u6b64\u5169\u5b57\u5143\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7de8\u8f2f\u6b65\u6578 = \u63d2\u5165\u3001\u522a\u9664\u3001\u66ff\u63db\u9019\u4e09\u7a2e\u64cd\u4f5c\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578 + 1\n                dp[j] = @min(@min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u70ba\u4e0b\u4e00\u8f2a\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"14.1 \u00a0 \u521d\u63a2\u52d5\u614b\u898f\u5283","text":"<p>\u52d5\u614b\u898f\u5283\uff08dynamic programming\uff09\u662f\u4e00\u500b\u91cd\u8981\u7684\u6f14\u7b97\u6cd5\u7bc4\u5f0f\uff0c\u5b83\u5c07\u4e00\u500b\u554f\u984c\u5206\u89e3\u70ba\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u554f\u984c\uff0c\u4e26\u900f\u904e\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\u4f86\u907f\u514d\u91cd\u8907\u8a08\u7b97\uff0c\u5f9e\u800c\u5927\u5e45\u63d0\u5347\u6642\u9593\u6548\u7387\u3002</p> <p>\u5728\u672c\u7bc0\u4e2d\uff0c\u6211\u5011\u5f9e\u4e00\u500b\u7d93\u5178\u4f8b\u984c\u5165\u624b\uff0c\u5148\u7d66\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c0\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u758a\u5b50\u554f\u984c\uff0c\u518d\u9010\u6b65\u5c0e\u51fa\u66f4\u9ad8\u6548\u7684\u52d5\u614b\u898f\u5283\u89e3\u6cd5\u3002</p> <p>\u722c\u6a13\u68af</p> <p>\u7d66\u5b9a\u4e00\u500b\u5171\u6709 \\(n\\) \u968e\u7684\u6a13\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u968e\u6216\u8005 \\(2\\) \u968e\uff0c\u8acb\u554f\u6709\u591a\u5c11\u7a2e\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u6a13\u9802\uff1f</p> <p>\u5982\u5716 14-1 \u6240\u793a\uff0c\u5c0d\u65bc\u4e00\u500b \\(3\\) \u968e\u6a13\u68af\uff0c\u5171\u6709 \\(3\\) \u7a2e\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u6a13\u9802\u3002</p> <p></p> <p> \u5716 14-1 \u00a0 \u722c\u5230\u7b2c 3 \u968e\u7684\u65b9\u6848\u6578\u91cf </p> <p>\u672c\u984c\u7684\u76ee\u6a19\u662f\u6c42\u89e3\u65b9\u6848\u6578\u91cf\uff0c\u6211\u5011\u53ef\u4ee5\u8003\u616e\u900f\u904e\u56de\u6eaf\u4f86\u7aae\u8209\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u9ad4\u4f86\u8aaa\uff0c\u5c07\u722c\u6a13\u68af\u60f3\u8c61\u70ba\u4e00\u500b\u591a\u8f2a\u9078\u64c7\u7684\u904e\u7a0b\uff1a\u5f9e\u5730\u9762\u51fa\u767c\uff0c\u6bcf\u8f2a\u9078\u64c7\u4e0a \\(1\\) \u968e\u6216 \\(2\\) \u968e\uff0c\u6bcf\u7576\u5230\u9054\u6a13\u68af\u9802\u90e8\u6642\u5c31\u5c07\u65b9\u6848\u6578\u91cf\u52a0 \\(1\\) \uff0c\u7576\u8d8a\u904e\u6a13\u68af\u9802\u90e8\u6642\u5c31\u5c07\u5176\u526a\u679d\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"\u56de\u6eaf\"\"\"\n    # \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if state == n:\n        res[0] += 1\n    # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices:\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if state + choice &gt; n:\n            continue\n        # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res)\n        # \u56de\u9000\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u56de\u6eaf\"\"\"\n    choices = [1, 2]  # \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    state = 0  # \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    res = [0]  # \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* \u56de\u6eaf */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (auto &amp;choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n)\n            continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    int state = 0;                // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    vector&lt;int&gt; res = {0};        // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (Integer choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n)\n            continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    int state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* \u56de\u6eaf */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    foreach (int choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n)\n            continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        Backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    int state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    List&lt;int&gt; res = [0]; // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if state+choice &gt; n {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state+choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n int) int {\n    // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    choices := []int{1, 2}\n    // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    state := 0\n    res := make([]int, 1)\n    // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if state == n {\n        res[0] += 1\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if state + choice &gt; n {\n            continue\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    let state = 0 // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    var res: [Int] = []\n    res.append(0) // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* \u56de\u6eaf */\nfunction backtrack(choices, state, n, res) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n) continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    const state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* \u56de\u6eaf */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n) continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    const state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n  if (state == n) {\n    res[0]++;\n  }\n  // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for (int choice in choices) {\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n    if (state + choice &gt; n) continue;\n    // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n    backtrack(choices, state + choice, n, res);\n    // \u56de\u9000\n  }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n  int state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n  List&lt;int&gt; res = [];\n  res.add(0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* \u56de\u6eaf */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for &amp;choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if state + choice &gt; n {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![1, 2]; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    let state = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    let mut res = Vec::new();\n    res.push(0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* \u56de\u6eaf */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n)\n            continue;\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res, len);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    int state = 0;           // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.kt<pre><code>/* \u56de\u6eaf */\nfun backtrack(\n    choices: MutableList&lt;Int&gt;,\n    state: Int,\n    n: Int,\n    res: MutableList&lt;Int&gt;\n) {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n)\n        res[0] = res[0] + 1\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n) continue\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u6a13\u68af\uff1a\u56de\u6eaf */\nfun climbingStairsBacktrack(n: Int): Int {\n    val choices = mutableListOf(1, 2) // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    val state = 0 // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    val res = mutableListOf&lt;Int&gt;()\n    res.add(0) // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.rb<pre><code>### \u56de\u6eaf ###\ndef backtrack(choices, state, n, res)\n  # \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n  res[0] += 1 if state == n\n  # \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n  for choice in choices\n    # \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n    next if state + choice &gt; n\n\n    # \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n    backtrack(choices, state + choice, n, res)\n  end\n  # \u56de\u9000\nend\n\n### \u722c\u6a13\u68af\uff1a\u56de\u6eaf ###\ndef climbing_stairs_backtrack(n)\n  choices = [1, 2] # \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n  state = 0 # \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n  res = [0] # \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n  backtrack(choices, state, n, res)\n  res.first\nend\n</code></pre> climbing_stairs_backtrack.zig<pre><code>// \u56de\u6eaf\nfn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void {\n    // \u7576\u722c\u5230\u7b2c n \u968e\u6642\uff0c\u65b9\u6848\u6578\u91cf\u52a0 1\n    if (state == n) {\n        res.items[0] = res.items[0] + 1;\n    }\n    // \u8d70\u8a2a\u6240\u6709\u9078\u64c7\n    for (choices) |choice| {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8a31\u8d8a\u904e\u7b2c n \u968e\n        if (state + choice &gt; n) {\n            continue;\n        }\n        // \u5617\u8a66\uff1a\u505a\u51fa\u9078\u64c7\uff0c\u66f4\u65b0\u72c0\u614b\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n// \u722c\u6a13\u68af\uff1a\u56de\u6eaf\nfn climbingStairsBacktrack(n: usize) !i32 {\n    var choices = [_]i32{ 1, 2 }; // \u53ef\u9078\u64c7\u5411\u4e0a\u722c 1 \u968e\u6216 2 \u968e\n    var state: i32 = 0; // \u5f9e\u7b2c 0 \u968e\u958b\u59cb\u722c\n    var res = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer res.deinit();\n    try res.append(0); // \u4f7f\u7528 res[0] \u8a18\u9304\u65b9\u6848\u6578\u91cf\n    backtrack(&amp;choices, state, @intCast(n), res);\n    return res.items[0];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411","title":"14.1.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u5c0b","text":"<p>\u56de\u6eaf\u6f14\u7b97\u6cd5\u901a\u5e38\u4e26\u4e0d\u986f\u5f0f\u5730\u5c0d\u554f\u984c\u9032\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c07\u6c42\u89e3\u554f\u984c\u770b\u4f5c\u4e00\u7cfb\u5217\u6c7a\u7b56\u6b65\u9a5f\uff0c\u900f\u904e\u8a66\u63a2\u548c\u526a\u679d\uff0c\u641c\u5c0b\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u5617\u8a66\u5f9e\u554f\u984c\u5206\u89e3\u7684\u89d2\u5ea6\u5206\u6790\u9019\u9053\u984c\u3002\u8a2d\u722c\u5230\u7b2c \\(i\\) \u968e\u5171\u6709 \\(dp[i]\\) \u7a2e\u65b9\u6848\uff0c\u90a3\u9ebc \\(dp[i]\\) \u5c31\u662f\u539f\u554f\u984c\uff0c\u5176\u5b50\u554f\u984c\u5305\u62ec\uff1a</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>\u7531\u65bc\u6bcf\u8f2a\u53ea\u80fd\u4e0a \\(1\\) \u968e\u6216 \\(2\\) \u968e\uff0c\u56e0\u6b64\u7576\u6211\u5011\u7ad9\u5728\u7b2c \\(i\\) \u968e\u6a13\u68af\u4e0a\u6642\uff0c\u4e0a\u4e00\u8f2a\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u968e\u6216\u7b2c \\(i - 2\\) \u968e\u4e0a\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u6211\u5011\u53ea\u80fd\u5f9e\u7b2c \\(i -1\\) \u968e\u6216\u7b2c \\(i - 2\\) \u968e\u9081\u5411\u7b2c \\(i\\) \u968e\u3002</p> <p>\u7531\u6b64\u4fbf\u53ef\u5f97\u51fa\u4e00\u500b\u91cd\u8981\u63a8\u8ad6\uff1a\u722c\u5230\u7b2c \\(i - 1\\) \u968e\u7684\u65b9\u6848\u6578\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u968e\u7684\u65b9\u6848\u6578\u5c31\u7b49\u65bc\u722c\u5230\u7b2c \\(i\\) \u968e\u7684\u65b9\u6848\u6578\u3002\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>\u9019\u610f\u5473\u8457\u5728\u722c\u6a13\u68af\u554f\u984c\u4e2d\uff0c\u5404\u500b\u5b50\u554f\u984c\u4e4b\u9593\u5b58\u5728\u905e\u63a8\u95dc\u4fc2\uff0c\u539f\u554f\u984c\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u554f\u984c\u7684\u89e3\u69cb\u5efa\u5f97\u4f86\u3002\u5716 14-2 \u5c55\u793a\u4e86\u8a72\u905e\u63a8\u95dc\u4fc2\u3002</p> <p></p> <p> \u5716 14-2 \u00a0 \u65b9\u6848\u6578\u91cf\u905e\u63a8\u95dc\u4fc2 </p> <p>\u6211\u5011\u53ef\u4ee5\u6839\u64da\u905e\u63a8\u516c\u5f0f\u5f97\u5230\u66b4\u529b\u641c\u5c0b\u89e3\u6cd5\u3002\u4ee5 \\(dp[n]\\) \u70ba\u8d77\u59cb\u9ede\uff0c\u905e\u8ff4\u5730\u5c07\u4e00\u500b\u8f03\u5927\u554f\u984c\u62c6\u89e3\u70ba\u5169\u500b\u8f03\u5c0f\u554f\u984c\u7684\u548c\uff0c\u76f4\u81f3\u5230\u9054\u6700\u5c0f\u5b50\u554f\u984c \\(dp[1]\\) \u548c \\(dp[2]\\) \u6642\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373 \\(dp[1] = 1\\)\u3001\\(dp[2] = 2\\) \uff0c\u8868\u793a\u722c\u5230\u7b2c \\(1\\)\u3001\\(2\\) \u968e\u5206\u5225\u6709 \\(1\\)\u3001\\(2\\) \u7a2e\u65b9\u6848\u3002</p> <p>\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u5b83\u548c\u6a19\u6e96\u56de\u6eaf\u7a0b\u5f0f\u78bc\u90fd\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\uff0c\u4f46\u66f4\u52a0\u7c21\u6f54\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"\u641c\u5c0b\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u641c\u5c0b\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* \u641c\u5c0b */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* \u641c\u5c0b */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* \u641c\u5c0b */\nint DFS(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* \u641c\u5c0b */\nfunc dfs(i int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* \u641c\u5c0b */\nfunc dfs(i: Int) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* \u641c\u5c0b */\nfunction dfs(i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* \u641c\u5c0b */\nfunction dfs(i: number): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* \u641c\u5c0b */\nint dfs(int i) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* \u641c\u5c0b */\nfn dfs(i: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n)\n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* \u641c\u5c0b */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.kt<pre><code>/* \u641c\u5c0b */\nfun dfs(i: Int): Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2) return i\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1) + dfs(i - 2)\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u641c\u5c0b */\nfun climbingStairsDFS(n: Int): Int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.rb<pre><code>### \u641c\u5c0b ###\ndef dfs(i)\n  # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  return i if i == 1 || i == 2\n  # dp[i] = dp[i-1] + dp[i-2]\n  dfs(i - 1) + dfs(i - 2)\nend\n\n### \u722c\u6a13\u68af\uff1a\u641c\u5c0b ###\ndef climbing_stairs_dfs(n)\n  dfs(n)\nend\n</code></pre> climbing_stairs_dfs.zig<pre><code>// \u641c\u5c0b\nfn dfs(i: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n// \u722c\u6a13\u68af\uff1a\u641c\u5c0b\nfn climbingStairsDFS(comptime n: usize) i32 {\n    return dfs(n);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-3 \u5c55\u793a\u4e86\u66b4\u529b\u641c\u5c0b\u5f62\u6210\u7684\u905e\u8ff4\u6a39\u3002\u5c0d\u65bc\u554f\u984c \\(dp[n]\\) \uff0c\u5176\u905e\u8ff4\u6a39\u7684\u6df1\u5ea6\u70ba \\(n\\) \uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^n)\\) \u3002\u6307\u6578\u968e\u5c6c\u65bc\u7206\u70b8\u5f0f\u589e\u9577\uff0c\u5982\u679c\u6211\u5011\u8f38\u5165\u4e00\u500b\u6bd4\u8f03\u5927\u7684 \\(n\\) \uff0c\u5247\u6703\u9677\u5165\u6f2b\u9577\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002</p> <p></p> <p> \u5716 14-3 \u00a0 \u722c\u6a13\u68af\u5c0d\u61c9\u905e\u8ff4\u6a39 </p> <p>\u89c0\u5bdf\u5716 14-3 \uff0c\u6307\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f\u201c\u91cd\u758a\u5b50\u554f\u984c\u201d\u5c0e\u81f4\u7684\u3002\u4f8b\u5982 \\(dp[9]\\) \u88ab\u5206\u89e3\u70ba \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\\(dp[8]\\) \u88ab\u5206\u89e3\u70ba \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u5169\u8005\u90fd\u5305\u542b\u5b50\u554f\u984c \\(dp[7]\\) \u3002</p> <p>\u4ee5\u6b64\u985e\u63a8\uff0c\u5b50\u554f\u984c\u4e2d\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u758a\u5b50\u554f\u984c\uff0c\u5b50\u5b50\u5b6b\u5b6b\u7121\u7aae\u76e1\u4e5f\u3002\u7d55\u5927\u90e8\u5206\u8a08\u7b97\u8cc7\u6e90\u90fd\u6d6a\u8cbb\u5728\u9019\u4e9b\u91cd\u758a\u7684\u5b50\u554f\u984c\u4e0a\u3002</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412","title":"14.1.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b","text":"<p>\u70ba\u4e86\u63d0\u5347\u6f14\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u5011\u5e0c\u671b\u6240\u6709\u7684\u91cd\u758a\u5b50\u554f\u984c\u90fd\u53ea\u88ab\u8a08\u7b97\u4e00\u6b21\u3002\u70ba\u6b64\uff0c\u6211\u5011\u5ba3\u544a\u4e00\u500b\u9663\u5217 <code>mem</code> \u4f86\u8a18\u9304\u6bcf\u500b\u5b50\u554f\u984c\u7684\u89e3\uff0c\u4e26\u5728\u641c\u5c0b\u904e\u7a0b\u4e2d\u5c07\u91cd\u758a\u5b50\u554f\u984c\u526a\u679d\u3002</p> <ol> <li>\u7576\u9996\u6b21\u8a08\u7b97 \\(dp[i]\\) \u6642\uff0c\u6211\u5011\u5c07\u5176\u8a18\u9304\u81f3 <code>mem[i]</code> \uff0c\u4ee5\u4fbf\u4e4b\u5f8c\u4f7f\u7528\u3002</li> <li>\u7576\u518d\u6b21\u9700\u8981\u8a08\u7b97 \\(dp[i]\\) \u6642\uff0c\u6211\u5011\u4fbf\u53ef\u76f4\u63a5\u5f9e <code>mem[i]</code> \u4e2d\u7372\u53d6\u7d50\u679c\uff0c\u5f9e\u800c\u907f\u514d\u91cd\u8907\u8a08\u7b97\u8a72\u5b50\u554f\u984c\u3002</li> </ol> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"\u8a18\u61b6\u5316\u641c\u5c0b\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # \u8a18\u9304 dp[i]\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\"\"\"\n    # mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint dfs(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint DFS(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfunc dfsMem(i int, mem []int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // \u8a18\u9304 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // \u8a18\u9304 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfunction dfs(i, mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction climbingStairsDFSMem(n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfunction dfs(i: number, mem: number[]): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint dfs(int i, List&lt;int&gt; mem) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // \u8a18\u9304 dp[i]\n  mem[i] = count;\n  return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint climbingStairsDFSMem(int n) {\n  // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nint dfs(int i, int *mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.kt<pre><code>/* \u8a18\u61b6\u5316\u641c\u5c0b */\nfun dfs(i: Int, mem: IntArray): Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2) return i\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i]\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    // \u8a18\u9304 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfun climbingStairsDFSMem(n: Int): Int {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    val mem = IntArray(n + 1)\n    mem.fill(-1)\n    return dfs(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.rb<pre><code>### \u8a18\u61b6\u5316\u641c\u5c0b ###\ndef dfs(i, mem)\n  # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  return i if i == 1 || i == 2\n  # \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n  return mem[i] if mem[i] != -1\n\n  # dp[i] = dp[i-1] + dp[i-2]\n  count = dfs(i - 1, mem) + dfs(i - 2, mem)\n  # \u8a18\u9304 dp[i]\n  mem[i] = count\nend\n\n### \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b ###\ndef climbing_stairs_dfs_mem(n)\n  # mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n  mem = Array.new(n + 1, -1)\n  dfs(n, mem)\nend\n</code></pre> climbing_stairs_dfs_mem.zig<pre><code>// \u8a18\u61b6\u5316\u641c\u5c0b\nfn dfs(i: usize, mem: []i32) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // \u82e5\u5b58\u5728\u8a18\u9304 dp[i] \uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8a18\u9304 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n// \u722c\u6a13\u68af\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\nfn climbingStairsDFSMem(comptime n: usize) i32 {\n    // mem[i] \u8a18\u9304\u722c\u5230\u7b2c i \u968e\u7684\u65b9\u6848\u7e3d\u6578\uff0c-1 \u4ee3\u8868\u7121\u8a18\u9304\n    var mem = [_]i32{ -1 } ** (n + 1);\n    return dfs(n, &amp;mem);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u5716 14-4 \uff0c\u7d93\u904e\u8a18\u61b6\u5316\u8655\u7406\u5f8c\uff0c\u6240\u6709\u91cd\u758a\u5b50\u554f\u984c\u90fd\u53ea\u9700\u8a08\u7b97\u4e00\u6b21\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(n)\\) \uff0c\u9019\u662f\u4e00\u500b\u5de8\u5927\u7684\u98db\u8e8d\u3002</p> <p></p> <p> \u5716 14-4 \u00a0 \u8a18\u61b6\u5316\u641c\u5c0b\u5c0d\u61c9\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413","title":"14.1.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52d5\u614b\u898f\u5283","text":"<p>\u8a18\u61b6\u5316\u641c\u5c0b\u662f\u4e00\u7a2e\u201c\u5f9e\u9802\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u5011\u5f9e\u539f\u554f\u984c\uff08\u6839\u7bc0\u9ede\uff09\u958b\u59cb\uff0c\u905e\u8ff4\u5730\u5c07\u8f03\u5927\u5b50\u554f\u984c\u5206\u89e3\u70ba\u8f03\u5c0f\u5b50\u554f\u984c\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u554f\u984c\uff08\u8449\u7bc0\u9ede\uff09\u3002\u4e4b\u5f8c\uff0c\u900f\u904e\u56de\u6eaf\u9010\u5c64\u6536\u96c6\u5b50\u554f\u984c\u7684\u89e3\uff0c\u69cb\u5efa\u51fa\u539f\u554f\u984c\u7684\u89e3\u3002</p> <p>\u8207\u4e4b\u76f8\u53cd\uff0c\u52d5\u614b\u898f\u5283\u662f\u4e00\u7a2e\u201c\u5f9e\u5e95\u81f3\u9802\u201d\u7684\u65b9\u6cd5\uff1a\u5f9e\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\u958b\u59cb\uff0c\u8fed\u4ee3\u5730\u69cb\u5efa\u66f4\u5927\u5b50\u554f\u984c\u7684\u89e3\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u554f\u984c\u7684\u89e3\u3002</p> <p>\u7531\u65bc\u52d5\u614b\u898f\u5283\u4e0d\u5305\u542b\u56de\u6eaf\u904e\u7a0b\uff0c\u56e0\u6b64\u53ea\u9700\u4f7f\u7528\u8ff4\u5708\u8fed\u4ee3\u5be6\u73fe\uff0c\u7121\u9808\u4f7f\u7528\u905e\u8ff4\u3002\u5728\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u6211\u5011\u521d\u59cb\u5316\u4e00\u500b\u9663\u5217 <code>dp</code> \u4f86\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\uff0c\u5b83\u8d77\u5230\u4e86\u8207\u8a18\u61b6\u5316\u641c\u5c0b\u4e2d\u9663\u5217 <code>mem</code> \u76f8\u540c\u7684\u8a18\u9304\u4f5c\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1], dp[2] = 1, 2\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3 ... n {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1] = 1;\n  dp[2] = 2;\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 */\nfun climbingStairsDP(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    val dp = IntArray(n + 1)\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (i in 3..n) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283 ###\ndef climbing_stairs_dp(n)\n  return n  if n == 1 || n == 2\n\n  # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n  dp = Array.new(n + 1, 0)\n  # \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n  dp[1], dp[2] = 1, 2\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n  (3...(n + 1)).each { |i| dp[i] = dp[i - 1] + dp[i - 2] }\n\n  dp[n]\nend\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u6a13\u68af\uff1a\u52d5\u614b\u898f\u5283\nfn climbingStairsDP(comptime n: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u65bc\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72c0\u614b\uff1a\u9810\u8a2d\u6700\u5c0f\u5b50\u554f\u984c\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for (3..n + 1) |i| {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-5 \u6a21\u64ec\u4e86\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u7684\u57f7\u884c\u904e\u7a0b\u3002</p> <p></p> <p> \u5716 14-5 \u00a0 \u722c\u6a13\u68af\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p> <p>\u8207\u56de\u6eaf\u6f14\u7b97\u6cd5\u4e00\u6a23\uff0c\u52d5\u614b\u898f\u5283\u4e5f\u4f7f\u7528\u201c\u72c0\u614b\u201d\u6982\u5ff5\u4f86\u8868\u793a\u554f\u984c\u6c42\u89e3\u7684\u7279\u5b9a\u968e\u6bb5\uff0c\u6bcf\u500b\u72c0\u614b\u90fd\u5c0d\u61c9\u4e00\u500b\u5b50\u554f\u984c\u4ee5\u53ca\u76f8\u61c9\u7684\u5340\u57df\u6027\u6700\u512a\u89e3\u3002\u4f8b\u5982\uff0c\u722c\u6a13\u68af\u554f\u984c\u7684\u72c0\u614b\u5b9a\u7fa9\u70ba\u7576\u524d\u6240\u5728\u6a13\u68af\u968e\u6578 \\(i\\) \u3002</p> <p>\u6839\u64da\u4ee5\u4e0a\u5167\u5bb9\uff0c\u6211\u5011\u53ef\u4ee5\u7e3d\u7d50\u51fa\u52d5\u614b\u898f\u5283\u7684\u5e38\u7528\u8853\u8a9e\u3002</p> <ul> <li>\u5c07\u9663\u5217 <code>dp</code> \u7a31\u70ba dp \u8868\uff0c\\(dp[i]\\) \u8868\u793a\u72c0\u614b \\(i\\) \u5c0d\u61c9\u5b50\u554f\u984c\u7684\u89e3\u3002</li> <li>\u5c07\u6700\u5c0f\u5b50\u554f\u984c\u5c0d\u61c9\u7684\u72c0\u614b\uff08\u7b2c \\(1\\) \u968e\u548c\u7b2c \\(2\\) \u968e\u6a13\u68af\uff09\u7a31\u70ba\u521d\u59cb\u72c0\u614b\u3002</li> <li>\u5c07\u905e\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u7a31\u70ba\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414","title":"14.1.4 \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7d30\u5fc3\u7684\u8b80\u8005\u53ef\u80fd\u767c\u73fe\u4e86\uff0c\u7531\u65bc \\(dp[i]\\) \u53ea\u8207 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u95dc\uff0c\u56e0\u6b64\u6211\u5011\u7121\u9808\u4f7f\u7528\u4e00\u500b\u9663\u5217 <code>dp</code> \u4f86\u5132\u5b58\u6240\u6709\u5b50\u554f\u984c\u7684\u89e3\uff0c\u800c\u53ea\u9700\u5169\u500b\u8b8a\u6578\u6efe\u52d5\u524d\u9032\u5373\u53ef\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"\u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5f9e\u8f03\u5c0f\u5b50\u554f\u984c\u9010\u6b65\u6c42\u89e3\u8f03\u5927\u5b50\u554f\u984c\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in 3 ... n {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun climbingStairsDPComp(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    var a = 1\n    var b = 2\n    for (i in 3..n) {\n        val temp = b\n        b += a\n        a = temp\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef climbing_stairs_dp_comp(n)\n  return n if n == 1 || n == 2\n\n  a, b = 1, 2\n  (3...(n + 1)).each { a, b = b, a + b }\n\n  b\nend\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u6a13\u68af\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn climbingStairsDPComp(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    var a: i32 = 1;\n    var b: i32 = 2;\n    for (3..n + 1) |_| {\n        var tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\uff0c\u7531\u65bc\u7701\u53bb\u4e86\u9663\u5217 <code>dp</code> \u4f54\u7528\u7684\u7a7a\u9593\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</p> <p>\u5728\u52d5\u614b\u898f\u5283\u554f\u984c\u4e2d\uff0c\u7576\u524d\u72c0\u614b\u5f80\u5f80\u50c5\u8207\u524d\u9762\u6709\u9650\u500b\u72c0\u614b\u6709\u95dc\uff0c\u9019\u6642\u6211\u5011\u53ef\u4ee5\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72c0\u614b\uff0c\u900f\u904e\u201c\u964d\u7dad\u201d\u4f86\u7bc0\u7701\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002\u9019\u7a2e\u7a7a\u9593\u6700\u4f73\u5316\u6280\u5de7\u88ab\u7a31\u70ba\u201c\u6efe\u52d5\u8b8a\u6578\u201d\u6216\u201c\u6efe\u52d5\u9663\u5217\u201d\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"14.4 \u00a0 0-1 \u80cc\u5305\u554f\u984c","text":"<p>\u80cc\u5305\u554f\u984c\u662f\u4e00\u500b\u975e\u5e38\u597d\u7684\u52d5\u614b\u898f\u5283\u5165\u9580\u984c\u76ee\uff0c\u662f\u52d5\u614b\u898f\u5283\u4e2d\u6700\u5e38\u898b\u7684\u554f\u984c\u5f62\u5f0f\u3002\u5176\u5177\u6709\u5f88\u591a\u8b8a\u7a2e\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u554f\u984c\u3001\u5b8c\u5168\u80cc\u5305\u554f\u984c\u3001\u591a\u91cd\u80cc\u5305\u554f\u984c\u7b49\u3002</p> <p>\u5728\u672c\u7bc0\u4e2d\uff0c\u6211\u5011\u5148\u4f86\u6c42\u89e3\u6700\u5e38\u898b\u7684 0-1 \u80cc\u5305\u554f\u984c\u3002</p> <p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u500b\u7269\u54c1\uff0c\u7b2c \\(i\\) \u500b\u7269\u54c1\u7684\u91cd\u91cf\u70ba \\(wgt[i-1]\\)\u3001\u50f9\u503c\u70ba \\(val[i-1]\\) \uff0c\u548c\u4e00\u500b\u5bb9\u91cf\u70ba \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u500b\u7269\u54c1\u53ea\u80fd\u9078\u64c7\u4e00\u6b21\uff0c\u554f\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u50f9\u503c\u3002</p> <p>\u89c0\u5bdf\u5716 14-17 \uff0c\u7531\u65bc\u7269\u54c1\u7de8\u865f \\(i\\) \u5f9e \\(1\\) \u958b\u59cb\u8a08\u6578\uff0c\u9663\u5217\u7d22\u5f15\u5f9e \\(0\\) \u958b\u59cb\u8a08\u6578\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5c0d\u61c9\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u50f9\u503c \\(val[i-1]\\) \u3002</p> <p></p> <p> \u5716 14-17 \u00a0 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u8cc7\u6599 </p> <p>\u6211\u5011\u53ef\u4ee5\u5c07 0-1 \u80cc\u5305\u554f\u984c\u770b\u4f5c\u4e00\u500b\u7531 \\(n\\) \u8f2a\u6c7a\u7b56\u7d44\u6210\u7684\u904e\u7a0b\uff0c\u5c0d\u65bc\u6bcf\u500b\u7269\u9ad4\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u5169\u7a2e\u6c7a\u7b56\uff0c\u56e0\u6b64\u8a72\u554f\u984c\u6eff\u8db3\u6c7a\u7b56\u6a39\u6a21\u578b\u3002</p> <p>\u8a72\u554f\u984c\u7684\u76ee\u6a19\u662f\u6c42\u89e3\u201c\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u50f9\u503c\u201d\uff0c\u56e0\u6b64\u8f03\u5927\u6a5f\u7387\u662f\u4e00\u500b\u52d5\u614b\u898f\u5283\u554f\u984c\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f2a\u7684\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5f9e\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u5c0d\u65bc\u6bcf\u500b\u7269\u54c1\u4f86\u8aaa\uff0c\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u8b8a\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u6e1b\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\u72c0\u614b\u5b9a\u7fa9\uff1a\u7576\u524d\u7269\u54c1\u7de8\u865f \\(i\\) \u548c\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u8a18\u70ba \\([i, c]\\) \u3002</p> <p>\u72c0\u614b \\([i, c]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\u70ba\uff1a\u524d \\(i\\) \u500b\u7269\u54c1\u5728\u5bb9\u91cf\u70ba \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u50f9\u503c\uff0c\u8a18\u70ba \\(dp[i, c]\\) \u3002</p> <p>\u5f85\u6c42\u89e3\u7684\u662f \\(dp[n, cap]\\) \uff0c\u56e0\u6b64\u9700\u8981\u4e00\u500b\u5c3a\u5bf8\u70ba \\((n+1) \\times (cap+1)\\) \u7684\u4e8c\u7dad \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u9032\u800c\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b</p> <p>\u7576\u6211\u5011\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u6c7a\u7b56\u5f8c\uff0c\u5269\u9918\u7684\u662f\u524d \\(i-1\\) \u500b\u7269\u54c1\u6c7a\u7b56\u7684\u5b50\u554f\u984c\uff0c\u53ef\u5206\u70ba\u4ee5\u4e0b\u5169\u7a2e\u60c5\u6cc1\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u8b8a\uff0c\u72c0\u614b\u8b8a\u5316\u70ba \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u6e1b\u5c11 \\(wgt[i-1]\\) \uff0c\u50f9\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72c0\u614b\u8b8a\u5316\u70ba \\([i-1, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4e0a\u8ff0\u5206\u6790\u5411\u6211\u5011\u63ed\u793a\u4e86\u672c\u984c\u7684\u6700\u512a\u5b50\u7d50\u69cb\uff1a\u6700\u5927\u50f9\u503c \\(dp[i, c]\\) \u7b49\u65bc\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\u3002\u7531\u6b64\u53ef\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u7576\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u9918\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u548c\u72c0\u614b\u8f49\u79fb\u9806\u5e8f</p> <p>\u7576\u7121\u7269\u54c1\u6216\u80cc\u5305\u5bb9\u91cf\u70ba \\(0\\) \u6642\u6700\u5927\u50f9\u503c\u70ba \\(0\\) \uff0c\u5373\u9996\u5217 \\(dp[i, 0]\\) \u548c\u9996\u884c \\(dp[0, c]\\) \u90fd\u7b49\u65bc \\(0\\) \u3002</p> <p>\u7576\u524d\u72c0\u614b \\([i, c]\\) \u5f9e\u4e0a\u65b9\u7684\u72c0\u614b \\([i-1, c]\\) \u548c\u5de6\u4e0a\u65b9\u7684\u72c0\u614b \\([i-1, c-wgt[i-1]]\\) \u8f49\u79fb\u800c\u4f86\uff0c\u56e0\u6b64\u900f\u904e\u5169\u5c64\u8ff4\u5708\u6b63\u5e8f\u8d70\u8a2a\u6574\u500b \\(dp\\) \u8868\u5373\u53ef\u3002</p> <p>\u6839\u64da\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u5011\u63a5\u4e0b\u4f86\u6309\u9806\u5e8f\u5be6\u73fe\u66b4\u529b\u641c\u5c0b\u3001\u8a18\u61b6\u5316\u641c\u5c0b\u3001\u52d5\u614b\u898f\u5283\u89e3\u6cd5\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u5c0b","text":"<p>\u641c\u5c0b\u7a0b\u5f0f\u78bc\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u905e\u8ff4\u53c3\u6578\uff1a\u72c0\u614b \\([i, c]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u5b50\u554f\u984c\u7684\u89e3 \\(dp[i, c]\\) \u3002</li> <li>\u7d42\u6b62\u689d\u4ef6\uff1a\u7576\u7269\u54c1\u7de8\u865f\u8d8a\u754c \\(i = 0\\) \u6216\u80cc\u5305\u5269\u9918\u5bb9\u91cf\u70ba \\(0\\) \u6642\uff0c\u7d42\u6b62\u905e\u8ff4\u4e26\u8fd4\u56de\u50f9\u503c \\(0\\) \u3002</li> <li>\u526a\u679d\uff1a\u82e5\u7576\u524d\u7269\u54c1\u91cd\u91cf\u8d85\u51fa\u80cc\u5305\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b\"\"\"\n    # \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfunction knapsackDFS(wgt, val, i, c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return myMax(no, yes);\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b */\nfun knapsackDFS(\n    wgt: IntArray,\n    _val: IntArray,\n    i: Int,\n    c: Int\n): Int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, _val, i - 1, c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    val no = knapsackDFS(wgt, _val, i - 1, c)\n    val yes = knapsackDFS(wgt, _val, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return max(no, yes)\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b ###\ndef knapsack_dfs(wgt, val, i, c)\n  # \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n  return 0 if i == 0 || c == 0\n  # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n  return knapsack_dfs(wgt, val, i - 1, c) if wgt[i - 1] &gt; c\n  # \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n  no = knapsack_dfs(wgt, val, i - 1, c)\n  yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n  [no, yes].max\nend\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u5c0b\nfn knapsackDFS(wgt: []i32, val: []i32, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    var no = knapsackDFS(wgt, val, i - 1, c);\n    var yes = knapsackDFS(wgt, val, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    return @max(no, yes);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 14-18 \u6240\u793a\uff0c\u7531\u65bc\u6bcf\u500b\u7269\u54c1\u90fd\u6703\u7522\u751f\u4e0d\u9078\u548c\u9078\u5169\u689d\u641c\u5c0b\u5206\u652f\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(2^n)\\) \u3002</p> <p>\u89c0\u5bdf\u905e\u8ff4\u6a39\uff0c\u5bb9\u6613\u767c\u73fe\u5176\u4e2d\u5b58\u5728\u91cd\u758a\u5b50\u554f\u984c\uff0c\u4f8b\u5982 \\(dp[1, 10]\\) \u7b49\u3002\u800c\u7576\u7269\u54c1\u8f03\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f03\u5927\uff0c\u5c24\u5176\u662f\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f03\u591a\u6642\uff0c\u91cd\u758a\u5b50\u554f\u984c\u7684\u6578\u91cf\u5c07\u6703\u5927\u5e45\u589e\u591a\u3002</p> <p></p> <p> \u5716 14-18 \u00a0 0-1 \u80cc\u5305\u554f\u984c\u7684\u66b4\u529b\u641c\u5c0b\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8a18\u61b6\u5316\u641c\u5c0b","text":"<p>\u70ba\u4e86\u4fdd\u8b49\u91cd\u758a\u5b50\u554f\u984c\u53ea\u88ab\u8a08\u7b97\u4e00\u6b21\uff0c\u6211\u5011\u85c9\u52a9\u8a18\u61b6\u4e32\u5217 <code>mem</code> \u4f86\u8a18\u9304\u5b50\u554f\u984c\u7684\u89e3\uff0c\u5176\u4e2d <code>mem[i][c]</code> \u5c0d\u61c9 \\(dp[i, c]\\) \u3002</p> <p>\u5f15\u5165\u8a18\u61b6\u5316\u4e4b\u5f8c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53d6\u6c7a\u65bc\u5b50\u554f\u984c\u6578\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\"\"\"\n    # \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b */\nfun knapsackDFSMem(\n    wgt: IntArray,\n    _val: IntArray,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    c: Int\n): Int {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    val no = knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    val yes = knapsackDFSMem(wgt, _val, mem, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b ###\ndef knapsack_dfs_mem(wgt, val, mem, i, c)\n  # \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n  return 0 if i == 0 || c == 0\n  # \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  return mem[i][c] if mem[i][c] != -1\n  # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n  return knapsack_dfs_mem(wgt, val, mem, i - 1, c) if wgt[i - 1] &gt; c\n  # \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n  no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n  yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n  mem[i][c] = [no, yes].max\nend\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u8a18\u61b6\u5316\u641c\u5c0b\nfn knapsackDFSMem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9078\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u5247\u8fd4\u56de\u50f9\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8a18\u9304\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u53ea\u80fd\u9078\u64c7\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8a08\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u50f9\u503c\n    var no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    var yes = knapsackDFSMem(wgt, val, mem, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8a18\u9304\u4e26\u8fd4\u56de\u5169\u7a2e\u65b9\u6848\u4e2d\u50f9\u503c\u66f4\u5927\u7684\u90a3\u4e00\u500b\n    mem[i][c] = @max(no, yes);\n    return mem[i][c];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-19 \u5c55\u793a\u4e86\u5728\u8a18\u61b6\u5316\u641c\u5c0b\u4e2d\u88ab\u526a\u6389\u7684\u641c\u5c0b\u5206\u652f\u3002</p> <p></p> <p> \u5716 14-19 \u00a0 0-1 \u80cc\u5305\u554f\u984c\u7684\u8a18\u61b6\u5316\u641c\u5c0b\u905e\u8ff4\u6a39 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52d5\u614b\u898f\u5283","text":"<p>\u52d5\u614b\u898f\u5283\u5be6\u8cea\u4e0a\u5c31\u662f\u5728\u72c0\u614b\u8f49\u79fb\u4e2d\u586b\u5145 \\(dp\\) \u8868\u7684\u904e\u7a0b\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = std::cmp::max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],\n                );\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfun knapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 ###\ndef knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = [dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283\nfn knapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i - 1][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 14-20 \u6240\u793a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u90fd\u7531\u9663\u5217 <code>dp</code> \u5927\u5c0f\u6c7a\u5b9a\uff0c\u5373 \\(O(n \\times cap)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-20 \u00a0 0-1 \u80cc\u5305\u554f\u984c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#4","title":"4. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7531\u65bc\u6bcf\u500b\u72c0\u614b\u90fd\u53ea\u8207\u5176\u4e0a\u4e00\u884c\u7684\u72c0\u614b\u6709\u95dc\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u5169\u500b\u9663\u5217\u6efe\u52d5\u524d\u9032\uff0c\u5c07\u7a7a\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \u3002</p> <p>\u9032\u4e00\u6b65\u601d\u8003\uff0c\u6211\u5011\u80fd\u5426\u50c5\u7528\u4e00\u500b\u9663\u5217\u5be6\u73fe\u7a7a\u9593\u6700\u4f73\u5316\u5462\uff1f\u89c0\u5bdf\u53ef\u77e5\uff0c\u6bcf\u500b\u72c0\u614b\u90fd\u662f\u7531\u6b63\u4e0a\u65b9\u6216\u5de6\u4e0a\u65b9\u7684\u683c\u5b50\u8f49\u79fb\u904e\u4f86\u7684\u3002\u5047\u8a2d\u53ea\u6709\u4e00\u500b\u9663\u5217\uff0c\u7576\u958b\u59cb\u8d70\u8a2a\u7b2c \\(i\\) \u884c\u6642\uff0c\u8a72\u9663\u5217\u5132\u5b58\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72c0\u614b\u3002</p> <ul> <li>\u5982\u679c\u63a1\u53d6\u6b63\u5e8f\u8d70\u8a2a\uff0c\u90a3\u9ebc\u8d70\u8a2a\u5230 \\(dp[i, j]\\) \u6642\uff0c\u5de6\u4e0a\u65b9 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) \u503c\u53ef\u80fd\u5df2\u7d93\u88ab\u8986\u84cb\uff0c\u6b64\u6642\u5c31\u7121\u6cd5\u5f97\u5230\u6b63\u78ba\u7684\u72c0\u614b\u8f49\u79fb\u7d50\u679c\u3002</li> <li>\u5982\u679c\u63a1\u53d6\u5012\u5e8f\u8d70\u8a2a\uff0c\u5247\u4e0d\u6703\u767c\u751f\u8986\u84cb\u554f\u984c\uff0c\u72c0\u614b\u8f49\u79fb\u53ef\u4ee5\u6b63\u78ba\u9032\u884c\u3002</li> </ul> <p>\u5716 14-21 \u5c55\u793a\u4e86\u5728\u55ae\u500b\u9663\u5217\u4e0b\u5f9e\u7b2c \\(i = 1\\) \u884c\u8f49\u63db\u81f3\u7b2c \\(i = 2\\) \u884c\u7684\u904e\u7a0b\u3002\u8acb\u601d\u8003\u6b63\u5e8f\u8d70\u8a2a\u548c\u5012\u5e8f\u8d70\u8a2a\u7684\u5340\u5225\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-21 \u00a0 0-1 \u80cc\u5305\u7684\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p> <p>\u5728\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e2d\uff0c\u6211\u5011\u50c5\u9700\u5c07\u9663\u5217 <code>dp</code> \u7684\u7b2c\u4e00\u7dad \\(i\\) \u76f4\u63a5\u522a\u9664\uff0c\u4e26\u4e14\u628a\u5167\u8ff4\u5708\u66f4\u6539\u70ba\u5012\u5e8f\u8d70\u8a2a\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        # \u5012\u5e8f\u8d70\u8a2a\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for c in (1 ... cap).reversed() {\n            if wgt[i - 1] &lt;= c {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5012\u5e8f\u8d70\u8a2a\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun knapsackDPComp(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(cap + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        // \u5012\u5e8f\u8d70\u8a2a\n        for (c in cap downTo 1) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(cap + 1, 0)\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    # \u5012\u5e8f\u8d70\u8a2a\n    for c in cap.downto(1)\n      if wgt[i - 1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[c] = dp[c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn knapsackDPComp(wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        // \u5012\u5e8f\u8d70\u8a2a\n        var c = cap;\n        while (c &gt; 0) : (c -= 1) {\n            if (wgt[i - 1] &lt; c) {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/summary/","title":"14.7 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u52d5\u614b\u898f\u5283\u5c0d\u554f\u984c\u9032\u884c\u5206\u89e3\uff0c\u4e26\u900f\u904e\u5132\u5b58\u5b50\u554f\u984c\u7684\u89e3\u4f86\u898f\u907f\u91cd\u8907\u8a08\u7b97\uff0c\u63d0\u9ad8\u8a08\u7b97\u6548\u7387\u3002</li> <li>\u4e0d\u8003\u616e\u6642\u9593\u7684\u524d\u63d0\u4e0b\uff0c\u6240\u6709\u52d5\u614b\u898f\u5283\u554f\u984c\u90fd\u53ef\u4ee5\u7528\u56de\u6eaf\uff08\u66b4\u529b\u641c\u5c0b\uff09\u9032\u884c\u6c42\u89e3\uff0c\u4f46\u905e\u8ff4\u6a39\u4e2d\u5b58\u5728\u5927\u91cf\u7684\u91cd\u758a\u5b50\u554f\u984c\uff0c\u6548\u7387\u6975\u4f4e\u3002\u900f\u904e\u5f15\u5165\u8a18\u61b6\u5316\u4e32\u5217\uff0c\u53ef\u4ee5\u5132\u5b58\u6240\u6709\u8a08\u7b97\u904e\u7684\u5b50\u554f\u984c\u7684\u89e3\uff0c\u5f9e\u800c\u4fdd\u8b49\u91cd\u758a\u5b50\u554f\u984c\u53ea\u88ab\u8a08\u7b97\u4e00\u6b21\u3002</li> <li>\u8a18\u61b6\u5316\u641c\u5c0b\u662f\u4e00\u7a2e\u5f9e\u9802\u81f3\u5e95\u7684\u905e\u8ff4\u5f0f\u89e3\u6cd5\uff0c\u800c\u8207\u4e4b\u5c0d\u61c9\u7684\u52d5\u614b\u898f\u5283\u662f\u4e00\u7a2e\u5f9e\u5e95\u81f3\u9802\u7684\u905e\u63a8\u5f0f\u89e3\u6cd5\uff0c\u5176\u5982\u540c\u201c\u586b\u5beb\u8868\u683c\u201d\u4e00\u6a23\u3002\u7531\u65bc\u7576\u524d\u72c0\u614b\u50c5\u4f9d\u8cf4\u67d0\u4e9b\u5340\u57df\u6027\u72c0\u614b\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u6d88\u9664 \\(dp\\) \u8868\u7684\u4e00\u500b\u7dad\u5ea6\uff0c\u5f9e\u800c\u964d\u4f4e\u7a7a\u9593\u8907\u96dc\u5ea6\u3002</li> <li>\u5b50\u554f\u984c\u5206\u89e3\u662f\u4e00\u7a2e\u901a\u7528\u7684\u6f14\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52d5\u614b\u898f\u5283\u3001\u56de\u6eaf\u4e2d\u5177\u6709\u4e0d\u540c\u7684\u6027\u8cea\u3002</li> <li>\u52d5\u614b\u898f\u5283\u554f\u984c\u6709\u4e09\u5927\u7279\u6027\uff1a\u91cd\u758a\u5b50\u554f\u984c\u3001\u6700\u512a\u5b50\u7d50\u69cb\u3001\u7121\u5f8c\u6548\u6027\u3002</li> <li>\u5982\u679c\u539f\u554f\u984c\u7684\u6700\u512a\u89e3\u53ef\u4ee5\u5f9e\u5b50\u554f\u984c\u7684\u6700\u512a\u89e3\u69cb\u5efa\u5f97\u4f86\uff0c\u5247\u5b83\u5c31\u5177\u6709\u6700\u512a\u5b50\u7d50\u69cb\u3002</li> <li>\u7121\u5f8c\u6548\u6027\u6307\u5c0d\u65bc\u4e00\u500b\u72c0\u614b\uff0c\u5176\u672a\u4f86\u767c\u5c55\u53ea\u8207\u8a72\u72c0\u614b\u6709\u95dc\uff0c\u800c\u8207\u904e\u53bb\u7d93\u6b77\u7684\u6240\u6709\u72c0\u614b\u7121\u95dc\u3002\u8a31\u591a\u7d44\u5408\u6700\u4f73\u5316\u554f\u984c\u4e0d\u5177\u6709\u7121\u5f8c\u6548\u6027\uff0c\u7121\u6cd5\u4f7f\u7528\u52d5\u614b\u898f\u5283\u5feb\u901f\u6c42\u89e3\u3002</li> </ul> <p>\u80cc\u5305\u554f\u984c</p> <ul> <li>\u80cc\u5305\u554f\u984c\u662f\u6700\u5178\u578b\u7684\u52d5\u614b\u898f\u5283\u554f\u984c\u4e4b\u4e00\uff0c\u5177\u6709 0-1 \u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305\u7b49\u8b8a\u7a2e\u3002</li> <li>0-1 \u80cc\u5305\u7684\u72c0\u614b\u5b9a\u7fa9\u70ba\u524d \\(i\\) \u500b\u7269\u54c1\u5728\u5bb9\u91cf\u70ba \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u50f9\u503c\u3002\u6839\u64da\u4e0d\u653e\u5165\u80cc\u5305\u548c\u653e\u5165\u80cc\u5305\u5169\u7a2e\u6c7a\u7b56\uff0c\u53ef\u5f97\u5230\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u4e26\u69cb\u5efa\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u3002\u5728\u7a7a\u9593\u6700\u4f73\u5316\u4e2d\uff0c\u7531\u65bc\u6bcf\u500b\u72c0\u614b\u4f9d\u8cf4\u6b63\u4e0a\u65b9\u548c\u5de6\u4e0a\u65b9\u7684\u72c0\u614b\uff0c\u56e0\u6b64\u9700\u8981\u5012\u5e8f\u8d70\u8a2a\u4e32\u5217\uff0c\u907f\u514d\u5de6\u4e0a\u65b9\u72c0\u614b\u88ab\u8986\u84cb\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u6bcf\u7a2e\u7269\u54c1\u7684\u9078\u53d6\u6578\u91cf\u7121\u9650\u5236\uff0c\u56e0\u6b64\u9078\u64c7\u653e\u5165\u7269\u54c1\u7684\u72c0\u614b\u8f49\u79fb\u8207 0-1 \u80cc\u5305\u554f\u984c\u4e0d\u540c\u3002\u7531\u65bc\u72c0\u614b\u4f9d\u8cf4\u6b63\u4e0a\u65b9\u548c\u6b63\u5de6\u65b9\u7684\u72c0\u614b\uff0c\u56e0\u6b64\u5728\u7a7a\u9593\u6700\u4f73\u5316\u4e2d\u61c9\u7576\u6b63\u5e8f\u8d70\u8a2a\u3002</li> <li>\u96f6\u9322\u514c\u63db\u554f\u984c\u662f\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u4e00\u500b\u8b8a\u7a2e\u3002\u5b83\u5f9e\u6c42\u201c\u6700\u5927\u201d\u50f9\u503c\u8b8a\u70ba\u6c42\u201c\u6700\u5c0f\u201d\u786c\u5e63\u6578\u91cf\uff0c\u56e0\u6b64\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\max()\\) \u61c9\u6539\u70ba \\(\\min()\\) \u3002\u5f9e\u8ffd\u6c42\u201c\u4e0d\u8d85\u904e\u201d\u80cc\u5305\u5bb9\u91cf\u5230\u8ffd\u6c42\u201c\u6070\u597d\u201d\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\uff0c\u56e0\u6b64\u4f7f\u7528 \\(amt + 1\\) \u4f86\u8868\u793a\u201c\u7121\u6cd5\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u201d\u7684\u7121\u6548\u89e3\u3002</li> <li>\u96f6\u9322\u514c\u63db\u554f\u984c II \u5f9e\u6c42\u201c\u6700\u5c11\u786c\u5e63\u6578\u91cf\u201d\u6539\u70ba\u6c42\u201c\u786c\u5e63\u7d44\u5408\u6578\u91cf\u201d\uff0c\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u76f8\u61c9\u5730\u5f9e \\(\\min()\\) \u6539\u70ba\u6c42\u548c\u904b\u7b97\u5b50\u3002</li> </ul> <p>\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c</p> <ul> <li>\u7de8\u8f2f\u8ddd\u96e2\uff08Levenshtein \u8ddd\u96e2\uff09\u7528\u65bc\u8861\u91cf\u5169\u500b\u5b57\u4e32\u4e4b\u9593\u7684\u76f8\u4f3c\u5ea6\uff0c\u5176\u5b9a\u7fa9\u70ba\u5f9e\u4e00\u500b\u5b57\u4e32\u5230\u53e6\u4e00\u500b\u5b57\u4e32\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\uff0c\u7de8\u8f2f\u64cd\u4f5c\u5305\u62ec\u65b0\u589e\u3001\u522a\u9664\u3001\u66ff\u63db\u3002</li> <li>\u7de8\u8f2f\u8ddd\u96e2\u554f\u984c\u7684\u72c0\u614b\u5b9a\u7fa9\u70ba\u5c07 \\(s\\) \u7684\u524d \\(i\\) \u500b\u5b57\u5143\u66f4\u6539\u70ba \\(t\\) \u7684\u524d \\(j\\) \u500b\u5b57\u5143\u6240\u9700\u7684\u6700\u5c11\u7de8\u8f2f\u6b65\u6578\u3002\u7576 \\(s[i] \\ne t[j]\\) \u6642\uff0c\u5177\u6709\u4e09\u7a2e\u6c7a\u7b56\uff1a\u65b0\u589e\u3001\u522a\u9664\u3001\u66ff\u63db\uff0c\u5b83\u5011\u90fd\u6709\u76f8\u61c9\u7684\u5269\u9918\u5b50\u554f\u984c\u3002\u64da\u6b64\u4fbf\u53ef\u4ee5\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\u8207\u69cb\u5efa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u3002\u800c\u7576 \\(s[i] = t[j]\\) \u6642\uff0c\u7121\u9808\u7de8\u8f2f\u7576\u524d\u5b57\u5143\u3002</li> <li>\u5728\u7de8\u8f2f\u8ddd\u96e2\u4e2d\uff0c\u72c0\u614b\u4f9d\u8cf4\u5176\u6b63\u4e0a\u65b9\u3001\u6b63\u5de6\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u72c0\u614b\uff0c\u56e0\u6b64\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u6b63\u5e8f\u6216\u5012\u5e8f\u8d70\u8a2a\u90fd\u7121\u6cd5\u6b63\u78ba\u5730\u9032\u884c\u72c0\u614b\u8f49\u79fb\u3002\u70ba\u6b64\uff0c\u6211\u5011\u5229\u7528\u4e00\u500b\u8b8a\u6578\u66ab\u5b58\u5de6\u4e0a\u65b9\u72c0\u614b\uff0c\u5f9e\u800c\u8f49\u5316\u5230\u8207\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7b49\u50f9\u7684\u60c5\u6cc1\uff0c\u53ef\u4ee5\u5728\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u9032\u884c\u6b63\u5e8f\u8d70\u8a2a\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","title":"14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c","text":"<p>\u5728\u672c\u7bc0\u4e2d\uff0c\u6211\u5011\u5148\u6c42\u89e3\u53e6\u4e00\u500b\u5e38\u898b\u7684\u80cc\u5305\u554f\u984c\uff1a\u5b8c\u5168\u80cc\u5305\uff0c\u518d\u77ad\u89e3\u5b83\u7684\u4e00\u7a2e\u7279\u4f8b\uff1a\u96f6\u9322\u514c\u63db\u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","title":"14.5.1 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u500b\u7269\u54c1\uff0c\u7b2c \\(i\\) \u500b\u7269\u54c1\u7684\u91cd\u91cf\u70ba \\(wgt[i-1]\\)\u3001\u50f9\u503c\u70ba \\(val[i-1]\\) \uff0c\u548c\u4e00\u500b\u5bb9\u91cf\u70ba \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u500b\u7269\u54c1\u53ef\u4ee5\u91cd\u8907\u9078\u53d6\uff0c\u554f\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u50f9\u503c\u3002\u793a\u4f8b\u5982\u5716 14-22 \u6240\u793a\u3002</p> <p></p> <p> \u5716 14-22 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u793a\u4f8b\u8cc7\u6599 </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","title":"1. \u00a0 \u52d5\u614b\u898f\u5283\u601d\u8def","text":"<p>\u5b8c\u5168\u80cc\u5305\u554f\u984c\u548c 0-1 \u80cc\u5305\u554f\u984c\u975e\u5e38\u76f8\u4f3c\uff0c\u5340\u5225\u50c5\u5728\u65bc\u4e0d\u9650\u5236\u7269\u54c1\u7684\u9078\u64c7\u6b21\u6578\u3002</p> <ul> <li>\u5728 0-1 \u80cc\u5305\u554f\u984c\u4e2d\uff0c\u6bcf\u7a2e\u7269\u54c1\u53ea\u6709\u4e00\u500b\uff0c\u56e0\u6b64\u5c07\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u5f8c\uff0c\u53ea\u80fd\u5f9e\u524d \\(i-1\\) \u500b\u7269\u54c1\u4e2d\u9078\u64c7\u3002</li> <li>\u5728\u5b8c\u5168\u80cc\u5305\u554f\u984c\u4e2d\uff0c\u6bcf\u7a2e\u7269\u54c1\u7684\u6578\u91cf\u662f\u7121\u9650\u7684\uff0c\u56e0\u6b64\u5c07\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u5f8c\uff0c\u4ecd\u53ef\u4ee5\u5f9e\u524d \\(i\\) \u500b\u7269\u54c1\u4e2d\u9078\u64c7\u3002</li> </ul> <p>\u5728\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u898f\u5b9a\u4e0b\uff0c\u72c0\u614b \\([i, c]\\) \u7684\u8b8a\u5316\u5206\u70ba\u5169\u7a2e\u60c5\u6cc1\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u8207 0-1 \u80cc\u5305\u554f\u984c\u76f8\u540c\uff0c\u8f49\u79fb\u81f3 \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u8207 0-1 \u80cc\u5305\u554f\u984c\u4e0d\u540c\uff0c\u8f49\u79fb\u81f3 \\([i, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u5f9e\u800c\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u8b8a\u70ba\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5c0d\u6bd4\u5169\u9053\u984c\u76ee\u7684\u7a0b\u5f0f\u78bc\uff0c\u72c0\u614b\u8f49\u79fb\u4e2d\u6709\u4e00\u8655\u5f9e \\(i-1\\) \u8b8a\u70ba \\(i\\) \uff0c\u5176\u9918\u5b8c\u5168\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 */\nfun unboundedKnapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283\nfn unboundedKnapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","title":"3. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7531\u65bc\u7576\u524d\u72c0\u614b\u662f\u5f9e\u5de6\u908a\u548c\u4e0a\u908a\u7684\u72c0\u614b\u8f49\u79fb\u800c\u4f86\u7684\uff0c\u56e0\u6b64\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u61c9\u8a72\u5c0d \\(dp\\) \u8868\u4e2d\u7684\u6bcf\u4e00\u884c\u9032\u884c\u6b63\u5e8f\u8d70\u8a2a\u3002</p> <p>\u9019\u500b\u8d70\u8a2a\u9806\u5e8f\u8207 0-1 \u80cc\u5305\u6b63\u597d\u76f8\u53cd\u3002\u8acb\u85c9\u52a9\u5716 14-23 \u4f86\u7406\u89e3\u5169\u8005\u7684\u5340\u5225\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-23 \u00a0 \u5b8c\u5168\u80cc\u5305\u554f\u984c\u5728\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p> <p>\u7a0b\u5f0f\u78bc\u5be6\u73fe\u6bd4\u8f03\u7c21\u55ae\uff0c\u50c5\u9700\u5c07\u9663\u5217 <code>dp</code> \u7684\u7b2c\u4e00\u7dad\u522a\u9664\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u8d70\u8a2a\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[c] = dp[c];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun unboundedKnapsackDPComp(\n    wgt: IntArray,\n    _val: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(cap + 1)\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### \u5b8c\u5168\u80cc\u5305\uff1a\u52d5\u614b\u898f\u5283 ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n\n### \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ##3\ndef unbounded_knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(cap + 1, 0)\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    # \u6b63\u5e8f\u8d70\u8a2a\n    for c in 1...(cap + 1)\n      if wgt[i -1] &gt; c\n        # \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n        dp[c] = dp[c]\n      else\n        # \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn unboundedKnapsackDPComp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u904e\u80cc\u5305\u5bb9\u91cf\uff0c\u5247\u4e0d\u9078\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u7269\u54c1 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","title":"14.5.2 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c","text":"<p>\u80cc\u5305\u554f\u984c\u662f\u4e00\u5927\u985e\u52d5\u614b\u898f\u5283\u554f\u984c\u7684\u4ee3\u8868\uff0c\u5176\u64c1\u6709\u5f88\u591a\u8b8a\u7a2e\uff0c\u4f8b\u5982\u96f6\u9322\u514c\u63db\u554f\u984c\u3002</p> <p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u7a2e\u786c\u5e63\uff0c\u7b2c \\(i\\) \u7a2e\u786c\u5e63\u7684\u9762\u503c\u70ba \\(coins[i - 1]\\) \uff0c\u76ee\u6a19\u91d1\u984d\u70ba \\(amt\\) \uff0c\u6bcf\u7a2e\u786c\u5e63\u53ef\u4ee5\u91cd\u8907\u9078\u53d6\uff0c\u554f\u80fd\u5920\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u7684\u6700\u5c11\u786c\u5e63\u6578\u91cf\u3002\u5982\u679c\u7121\u6cd5\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u5716 14-24 \u6240\u793a\u3002</p> <p></p> <p> \u5716 14-24 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c\u7684\u793a\u4f8b\u8cc7\u6599 </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","title":"1. \u00a0 \u52d5\u614b\u898f\u5283\u601d\u8def","text":"<p>\u96f6\u9322\u514c\u63db\u53ef\u4ee5\u770b\u4f5c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u4e00\u7a2e\u7279\u6b8a\u60c5\u6cc1\uff0c\u5169\u8005\u5177\u6709\u4ee5\u4e0b\u95dc\u806f\u8207\u4e0d\u540c\u9ede\u3002</p> <ul> <li>\u5169\u9053\u984c\u53ef\u4ee5\u76f8\u4e92\u8f49\u63db\uff0c\u201c\u7269\u54c1\u201d\u5c0d\u61c9\u201c\u786c\u5e63\u201d\u3001\u201c\u7269\u54c1\u91cd\u91cf\u201d\u5c0d\u61c9\u201c\u786c\u5e63\u9762\u503c\u201d\u3001\u201c\u80cc\u5305\u5bb9\u91cf\u201d\u5c0d\u61c9\u201c\u76ee\u6a19\u91d1\u984d\u201d\u3002</li> <li>\u6700\u4f73\u5316\u76ee\u6a19\u76f8\u53cd\uff0c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u662f\u8981\u6700\u5927\u5316\u7269\u54c1\u50f9\u503c\uff0c\u96f6\u9322\u514c\u63db\u554f\u984c\u662f\u8981\u6700\u5c0f\u5316\u786c\u5e63\u6578\u91cf\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u554f\u984c\u662f\u6c42\u201c\u4e0d\u8d85\u904e\u201d\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u89e3\uff0c\u96f6\u9322\u514c\u63db\u662f\u6c42\u201c\u6070\u597d\u201d\u6e4a\u5230\u76ee\u6a19\u91d1\u984d\u7684\u89e3\u3002</li> </ul> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f2a\u7684\u6c7a\u7b56\uff0c\u5b9a\u7fa9\u72c0\u614b\uff0c\u5f9e\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u72c0\u614b \\([i, a]\\) \u5c0d\u61c9\u7684\u5b50\u554f\u984c\u70ba\uff1a\u524d \\(i\\) \u7a2e\u786c\u5e63\u80fd\u5920\u6e4a\u51fa\u91d1\u984d \\(a\\) \u7684\u6700\u5c11\u786c\u5e63\u6578\u91cf\uff0c\u8a18\u70ba \\(dp[i, a]\\) \u3002</p> <p>\u4e8c\u7dad \\(dp\\) \u8868\u7684\u5c3a\u5bf8\u70ba \\((n+1) \\times (amt+1)\\) \u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u512a\u5b50\u7d50\u69cb\uff0c\u9032\u800c\u63a8\u5c0e\u51fa\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b</p> <p>\u672c\u984c\u8207\u5b8c\u5168\u80cc\u5305\u554f\u984c\u7684\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u5b58\u5728\u4ee5\u4e0b\u5169\u9ede\u5dee\u7570\u3002</p> <ul> <li>\u672c\u984c\u8981\u6c42\u6700\u5c0f\u503c\uff0c\u56e0\u6b64\u9700\u5c07\u904b\u7b97\u5b50 \\(\\max()\\) \u66f4\u6539\u70ba \\(\\min()\\) \u3002</li> <li>\u6700\u4f73\u5316\u4e3b\u9ad4\u662f\u786c\u5e63\u6578\u91cf\u800c\u975e\u5546\u54c1\u50f9\u503c\uff0c\u56e0\u6b64\u5728\u9078\u4e2d\u786c\u5e63\u6642\u57f7\u884c \\(+1\\) \u5373\u53ef\u3002</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>\u7b2c\u4e09\u6b65\uff1a\u78ba\u5b9a\u908a\u754c\u689d\u4ef6\u548c\u72c0\u614b\u8f49\u79fb\u9806\u5e8f</p> <p>\u7576\u76ee\u6a19\u91d1\u984d\u70ba \\(0\\) \u6642\uff0c\u6e4a\u51fa\u5b83\u7684\u6700\u5c11\u786c\u5e63\u6578\u91cf\u70ba \\(0\\) \uff0c\u5373\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u7b49\u65bc \\(0\\) \u3002</p> <p>\u7576\u7121\u786c\u5e63\u6642\uff0c\u7121\u6cd5\u6e4a\u51fa\u4efb\u610f \\(&gt; 0\\) \u7684\u76ee\u6a19\u91d1\u984d\uff0c\u5373\u662f\u7121\u6548\u89e3\u3002\u70ba\u4f7f\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\min()\\) \u51fd\u5f0f\u80fd\u5920\u8b58\u5225\u4e26\u904e\u6ffe\u7121\u6548\u89e3\uff0c\u6211\u5011\u8003\u616e\u4f7f\u7528 \\(+ \\infty\\) \u4f86\u8868\u793a\u5b83\u5011\uff0c\u5373\u4ee4\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u65bc \\(+ \\infty\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5927\u591a\u6578\u7a0b\u5f0f\u8a9e\u8a00\u4e26\u672a\u63d0\u4f9b \\(+ \\infty\\) \u8b8a\u6578\uff0c\u53ea\u80fd\u4f7f\u7528\u6574\u578b <code>int</code> \u7684\u6700\u5927\u503c\u4f86\u4ee3\u66ff\u3002\u800c\u9019\u53c8\u6703\u5c0e\u81f4\u5927\u6578\u8d8a\u754c\uff1a\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(+ 1\\) \u64cd\u4f5c\u53ef\u80fd\u767c\u751f\u6ea2\u4f4d\u3002</p> <p>\u70ba\u6b64\uff0c\u6211\u5011\u63a1\u7528\u6578\u5b57 \\(amt + 1\\) \u4f86\u8868\u793a\u7121\u6548\u89e3\uff0c\u56e0\u70ba\u6e4a\u51fa \\(amt\\) \u7684\u786c\u5e63\u6578\u91cf\u6700\u591a\u70ba \\(amt\\) \u3002\u6700\u5f8c\u8fd4\u56de\u524d\uff0c\u5224\u65b7 \\(dp[n, amt]\\) \u662f\u5426\u7b49\u65bc \\(amt + 1\\) \uff0c\u82e5\u662f\u5247\u8fd4\u56de \\(-1\\) \uff0c\u4ee3\u8868\u7121\u6cd5\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1 ... amt {\n        dp[0][a] = MAX\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1..=amt {\n        dp[0][a] = max;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfun coinChangeDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (a in 1..amt) {\n        dp[0][a] = MAX\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[n][amt] != MAX) dp[n][amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 ###\ndef coin_change_dp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n  (1...(amt + 1)).each { |a| dp[0][a] = _MAX }\n  # \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[i][a] = dp[i - 1][a]\n      else\n        # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n        dp[i][a] = [dp[i - 1][a], dp[i][a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[n][amt] != _MAX ? dp[n][amt] : -1\nend\n</code></pre> coin_change.zig<pre><code>// \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283\nfn coinChangeDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..amt + 1) |a| {\n        dp[0][a] = max;\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = @min(dp[i - 1][a], dp[i][a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[n][amt] != max) {\n        return @intCast(dp[n][amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5716 14-25 \u5c55\u793a\u4e86\u96f6\u9322\u514c\u63db\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b\uff0c\u548c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u975e\u5e38\u76f8\u4f3c\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 14-25 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c\u7684\u52d5\u614b\u898f\u5283\u904e\u7a0b </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","title":"3. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u96f6\u9322\u514c\u63db\u7684\u7a7a\u9593\u6700\u4f73\u5316\u7684\u8655\u7406\u65b9\u5f0f\u548c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u8d70\u8a2a\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u6b63\u5e8f\u8d70\u8a2a\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = malloc((amt + 1) * sizeof(int));\n    for (int j = 1; j &lt;= amt; j++) {\n        dp[j] = MAX;\n    } \n    dp[0] = 0;\n\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun coinChangeDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(amt + 1)\n    dp.fill(MAX)\n    dp[0] = 0\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[amt] != MAX) dp[amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef coin_change_dp_comp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(amt + 1, _MAX)\n  dp[0] = 0\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    # \u6b63\u5e8f\u8d70\u8a2a\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[a] = dp[a]\n      else\n        # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n        dp[a] = [dp[a], dp[a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[amt] != _MAX ? dp[amt] : -1\nend\n</code></pre> coin_change.zig<pre><code>// \u96f6\u9322\u514c\u63db\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn coinChangeDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    @memset(&amp;dp, max);\n    dp[0] = 0;\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = @min(dp[a], dp[a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[amt] != max) {\n        return @intCast(dp[amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","title":"14.5.3 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c II","text":"<p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u7a2e\u786c\u5e63\uff0c\u7b2c \\(i\\) \u7a2e\u786c\u5e63\u7684\u9762\u503c\u70ba \\(coins[i - 1]\\) \uff0c\u76ee\u6a19\u91d1\u984d\u70ba \\(amt\\) \uff0c\u6bcf\u7a2e\u786c\u5e63\u53ef\u4ee5\u91cd\u8907\u9078\u53d6\uff0c\u554f\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u7684\u786c\u5e63\u7d44\u5408\u6578\u91cf\u3002\u793a\u4f8b\u5982\u5716 14-26 \u6240\u793a\u3002</p> <p></p> <p> \u5716 14-26 \u00a0 \u96f6\u9322\u514c\u63db\u554f\u984c II \u7684\u793a\u4f8b\u8cc7\u6599 </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","title":"1. \u00a0 \u52d5\u614b\u898f\u5283\u601d\u8def","text":"<p>\u76f8\u6bd4\u65bc\u4e0a\u4e00\u984c\uff0c\u672c\u984c\u76ee\u6a19\u662f\u6c42\u7d44\u5408\u6578\u91cf\uff0c\u56e0\u6b64\u5b50\u554f\u984c\u8b8a\u70ba\uff1a\u524d \\(i\\) \u7a2e\u786c\u5e63\u80fd\u5920\u6e4a\u51fa\u91d1\u984d \\(a\\) \u7684\u7d44\u5408\u6578\u91cf\u3002\u800c \\(dp\\) \u8868\u4ecd\u7136\u662f\u5c3a\u5bf8\u70ba \\((n+1) \\times (amt + 1)\\) \u7684\u4e8c\u7dad\u77e9\u9663\u3002</p> <p>\u7576\u524d\u72c0\u614b\u7684\u7d44\u5408\u6578\u91cf\u7b49\u65bc\u4e0d\u9078\u7576\u524d\u786c\u5e63\u8207\u9078\u7576\u524d\u786c\u5e63\u9019\u5169\u7a2e\u6c7a\u7b56\u7684\u7d44\u5408\u6578\u91cf\u4e4b\u548c\u3002\u72c0\u614b\u8f49\u79fb\u65b9\u7a0b\u70ba\uff1a</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>\u7576\u76ee\u6a19\u91d1\u984d\u70ba \\(0\\) \u6642\uff0c\u7121\u9808\u9078\u64c7\u4efb\u4f55\u786c\u5e63\u5373\u53ef\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\uff0c\u56e0\u6b64\u61c9\u5c07\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u521d\u59cb\u5316\u70ba \\(1\\) \u3002\u7576\u7121\u786c\u5e63\u6642\uff0c\u7121\u6cd5\u6e4a\u51fa\u4efb\u4f55 \\(&gt;0\\) \u7684\u76ee\u6a19\u91d1\u984d\uff0c\u56e0\u6b64\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u65bc \\(0\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u521d\u59cb\u5316\u9996\u5217\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // \u72c0\u614b\u8f49\u79fb\uff1a\u5176\u9918\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0 ... n {\n        dp[i][0] = 1\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u521d\u59cb\u5316\u9996\u5217\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0..=n {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 */\nfun coinChangeIIDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (i in 0..n) {\n        dp[i][0] = 1\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283 ###\ndef coin_change_ii_dp(coins, amt)\n  n = coins.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # \u521d\u59cb\u5316\u9996\u5217\n  (0...(n + 1)).each { |i| dp[i][0] = 1 }\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[i][a] = dp[i - 1][a]\n      else\n        # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n      end\n    end\n  end\n  dp[n][amt]\nend\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u9322\u514c\u63db II\uff1a\u52d5\u614b\u898f\u5283\nfn coinChangeIIDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (0..n + 1) |i| {\n        dp[i][0] = 1;\n    }\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","title":"3. \u00a0 \u7a7a\u9593\u6700\u4f73\u5316","text":"<p>\u7a7a\u9593\u6700\u4f73\u5316\u8655\u7406\u65b9\u5f0f\u76f8\u540c\uff0c\u522a\u9664\u786c\u5e63\u7dad\u5ea6\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # \u72c0\u614b\u8f49\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u8d70\u8a2a\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // \u72c0\u614b\u8f49\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u6b63\u5e8f\u8d70\u8a2a\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // \u72c0\u614b\u8f49\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 */\nfun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(amt + 1)\n    dp[0] = 1\n    // \u72c0\u614b\u8f49\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283 ###\ndef coin_change_ii_dp_comp(coins, amt)\n  n = coins.length\n  # \u521d\u59cb\u5316 dp \u8868\n  dp = Array.new(amt + 1, 0)\n  dp[0] = 1\n  # \u72c0\u614b\u8f49\u79fb\n  for i in 1...(n + 1)\n    # \u6b63\u5e8f\u8d70\u8a2a\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n        dp[a] = dp[a]\n      else\n        # \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u4e4b\u548c\n        dp[a] = dp[a] + dp[a - coins[i - 1]]\n      end\n    end\n  end\n  dp[amt]\nend\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u9322\u514c\u63db II\uff1a\u7a7a\u9593\u6700\u4f73\u5316\u5f8c\u7684\u52d5\u614b\u898f\u5283\nfn coinChangeIIDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    dp[0] = 1;\n    // \u72c0\u614b\u8f49\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u904e\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u4e0d\u9078\u786c\u5e63 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9078\u548c\u9078\u786c\u5e63 i \u9019\u5169\u7a2e\u65b9\u6848\u7684\u8f03\u5c0f\u503c\n                dp[a] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_graph/","title":"\u7b2c 9 \u7ae0 \u00a0 \u5716","text":"<p>Abstract</p> <p>\u5728\u751f\u547d\u65c5\u9014\u4e2d\uff0c\u6211\u5011\u5c31\u50cf\u662f\u4e00\u500b\u500b\u7bc0\u9ede\uff0c\u88ab\u7121\u6578\u770b\u4e0d\u898b\u7684\u908a\u76f8\u9023\u3002</p> <p>\u6bcf\u4e00\u6b21\u7684\u76f8\u8b58\u8207\u76f8\u96e2\uff0c\u90fd\u5728\u9019\u5f35\u5de8\u5927\u7684\u7db2\u8def\u5716\u4e2d\u7559\u4e0b\u7368\u7279\u7684\u5370\u8a18\u3002</p>"},{"location":"chapter_graph/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>9.1 \u00a0 \u5716</li> <li>9.2 \u00a0 \u5716\u57fa\u790e\u64cd\u4f5c</li> <li>9.3 \u00a0 \u5716\u7684\u8d70\u8a2a</li> <li>9.4 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_graph/graph/","title":"9.1 \u00a0 \u5716","text":"<p>\u5716\uff08graph\uff09\u662f\u4e00\u7a2e\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u7531\u9802\u9ede\uff08vertex\uff09\u548c\u908a\uff08edge\uff09\u7d44\u6210\u3002\u6211\u5011\u53ef\u4ee5\u5c07\u5716 \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u70ba\u4e00\u7d44\u9802\u9ede \\(V\\) \u548c\u4e00\u7d44\u908a \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u500b\u5305\u542b 5 \u500b\u9802\u9ede\u548c 7 \u689d\u908a\u7684\u5716\u3002</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>\u5982\u679c\u5c07\u9802\u9ede\u770b\u4f5c\u7bc0\u9ede\uff0c\u5c07\u908a\u770b\u4f5c\u9023\u7dda\u5404\u500b\u7bc0\u9ede\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5c07\u5716\u770b\u4f5c\u4e00\u7a2e\u5f9e\u93c8\u7d50\u4e32\u5217\u62d3\u5c55\u800c\u4f86\u7684\u8cc7\u6599\u7d50\u69cb\u3002\u5982\u5716 9-1 \u6240\u793a\uff0c\u76f8\u8f03\u65bc\u7dda\u6027\u95dc\u4fc2\uff08\u93c8\u7d50\u4e32\u5217\uff09\u548c\u5206\u6cbb\u95dc\u4fc2\uff08\u6a39\uff09\uff0c\u7db2\u8def\u95dc\u4fc2\uff08\u5716\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u56e0\u800c\u66f4\u70ba\u8907\u96dc\u3002</p> <p></p> <p> \u5716 9-1 \u00a0 \u93c8\u7d50\u4e32\u5217\u3001\u6a39\u3001\u5716\u4e4b\u9593\u7684\u95dc\u4fc2 </p>"},{"location":"chapter_graph/graph/#911","title":"9.1.1 \u00a0 \u5716\u7684\u5e38\u898b\u578b\u5225\u8207\u8853\u8a9e","text":"<p>\u6839\u64da\u908a\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u70ba\u7121\u5411\u5716\uff08undirected graph\uff09\u548c\u6709\u5411\u5716\uff08directed graph\uff09\uff0c\u5982\u5716 9-2 \u6240\u793a\u3002</p> <ul> <li>\u5728\u7121\u5411\u5716\u4e2d\uff0c\u908a\u8868\u793a\u5169\u9802\u9ede\u4e4b\u9593\u7684\u201c\u96d9\u5411\u201d\u9023\u7dda\u95dc\u4fc2\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u95dc\u4fc2\u201d\u3002</li> <li>\u5728\u6709\u5411\u5716\u4e2d\uff0c\u908a\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u5169\u500b\u65b9\u5411\u7684\u908a\u662f\u76f8\u4e92\u7368\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u95dc\u6ce8\u201d\u8207\u201c\u88ab\u95dc\u6ce8\u201d\u95dc\u4fc2\u3002</li> </ul> <p></p> <p> \u5716 9-2 \u00a0 \u6709\u5411\u5716\u8207\u7121\u5411\u5716 </p> <p>\u6839\u64da\u6240\u6709\u9802\u9ede\u662f\u5426\u9023\u901a\uff0c\u53ef\u5206\u70ba\u9023\u901a\u5716\uff08connected graph\uff09\u548c\u975e\u9023\u901a\u5716\uff08disconnected graph\uff09\uff0c\u5982\u5716 9-3 \u6240\u793a\u3002</p> <ul> <li>\u5c0d\u65bc\u9023\u901a\u5716\uff0c\u5f9e\u67d0\u500b\u9802\u9ede\u51fa\u767c\uff0c\u53ef\u4ee5\u5230\u9054\u5176\u9918\u4efb\u610f\u9802\u9ede\u3002</li> <li>\u5c0d\u65bc\u975e\u9023\u901a\u5716\uff0c\u5f9e\u67d0\u500b\u9802\u9ede\u51fa\u767c\uff0c\u81f3\u5c11\u6709\u4e00\u500b\u9802\u9ede\u7121\u6cd5\u5230\u9054\u3002</li> </ul> <p></p> <p> \u5716 9-3 \u00a0 \u9023\u901a\u5716\u8207\u975e\u9023\u901a\u5716 </p> <p>\u6211\u5011\u9084\u53ef\u4ee5\u70ba\u908a\u65b0\u589e\u201c\u6b0a\u91cd\u201d\u8b8a\u6578\uff0c\u5f9e\u800c\u5f97\u5230\u5982\u5716 9-4 \u6240\u793a\u7684\u6709\u6b0a\u5716\uff08weighted graph\uff09\u3002\u4f8b\u5982\u5728\u300a\u738b\u8005\u69ae\u8000\u300b\u7b49\u624b\u904a\u4e2d\uff0c\u7cfb\u7d71\u6703\u6839\u64da\u5171\u540c\u904a\u6232\u6642\u9593\u4f86\u8a08\u7b97\u73a9\u5bb6\u4e4b\u9593\u7684\u201c\u89aa\u5bc6\u5ea6\u201d\uff0c\u9019\u7a2e\u89aa\u5bc6\u5ea6\u7db2\u8def\u5c31\u53ef\u4ee5\u7528\u6709\u6b0a\u5716\u4f86\u8868\u793a\u3002</p> <p></p> <p> \u5716 9-4 \u00a0 \u6709\u6b0a\u5716\u8207\u7121\u6b0a\u5716 </p> <p>\u5716\u8cc7\u6599\u7d50\u69cb\u5305\u542b\u4ee5\u4e0b\u5e38\u7528\u8853\u8a9e\u3002</p> <ul> <li>\u9130\u63a5\uff08adjacency\uff09\uff1a\u7576\u5169\u9802\u9ede\u4e4b\u9593\u5b58\u5728\u908a\u76f8\u9023\u6642\uff0c\u7a31\u9019\u5169\u9802\u9ede\u201c\u9130\u63a5\u201d\u3002\u5728\u5716 9-4 \u4e2d\uff0c\u9802\u9ede 1 \u7684\u9130\u63a5\u9802\u9ede\u70ba\u9802\u9ede 2\u30013\u30015\u3002</li> <li>\u8def\u5f91\uff08path\uff09\uff1a\u5f9e\u9802\u9ede A \u5230\u9802\u9ede B \u7d93\u904e\u7684\u908a\u69cb\u6210\u7684\u5e8f\u5217\u88ab\u7a31\u70ba\u5f9e A \u5230 B \u7684\u201c\u8def\u5f91\u201d\u3002\u5728\u5716 9-4 \u4e2d\uff0c\u908a\u5e8f\u5217 1-5-2-4 \u662f\u9802\u9ede 1 \u5230\u9802\u9ede 4 \u7684\u4e00\u689d\u8def\u5f91\u3002</li> <li>\u5ea6\uff08degree\uff09\uff1a\u4e00\u500b\u9802\u9ede\u64c1\u6709\u7684\u908a\u6578\u3002\u5c0d\u65bc\u6709\u5411\u5716\uff0c\u5165\u5ea6\uff08in-degree\uff09\u8868\u793a\u6709\u591a\u5c11\u689d\u908a\u6307\u5411\u8a72\u9802\u9ede\uff0c\u51fa\u5ea6\uff08out-degree\uff09\u8868\u793a\u6709\u591a\u5c11\u689d\u908a\u5f9e\u8a72\u9802\u9ede\u6307\u51fa\u3002</li> </ul>"},{"location":"chapter_graph/graph/#912","title":"9.1.2 \u00a0 \u5716\u7684\u8868\u793a","text":"<p>\u5716\u7684\u5e38\u7528\u8868\u793a\u65b9\u5f0f\u5305\u62ec\u201c\u9130\u63a5\u77e9\u9663\u201d\u548c\u201c\u9130\u63a5\u8868\u201d\u3002\u4ee5\u4e0b\u4f7f\u7528\u7121\u5411\u5716\u9032\u884c\u8209\u4f8b\u3002</p>"},{"location":"chapter_graph/graph/#1","title":"1. \u00a0 \u9130\u63a5\u77e9\u9663","text":"<p>\u8a2d\u5716\u7684\u9802\u9ede\u6578\u91cf\u70ba \\(n\\) \uff0c\u9130\u63a5\u77e9\u9663\uff08adjacency matrix\uff09\u4f7f\u7528\u4e00\u500b \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9663\u4f86\u8868\u793a\u5716\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u500b\u9802\u9ede\uff0c\u77e9\u9663\u5143\u7d20\u4ee3\u8868\u908a\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u5169\u500b\u9802\u9ede\u4e4b\u9593\u662f\u5426\u5b58\u5728\u908a\u3002</p> <p>\u5982\u5716 9-5 \u6240\u793a\uff0c\u8a2d\u9130\u63a5\u77e9\u9663\u70ba \\(M\\)\u3001\u9802\u9ede\u4e32\u5217\u70ba \\(V\\) \uff0c\u90a3\u9ebc\u77e9\u9663\u5143\u7d20 \\(M[i, j] = 1\\) \u8868\u793a\u9802\u9ede \\(V[i]\\) \u5230\u9802\u9ede \\(V[j]\\) \u4e4b\u9593\u5b58\u5728\u908a\uff0c\u53cd\u4e4b \\(M[i, j] = 0\\) \u8868\u793a\u5169\u9802\u9ede\u4e4b\u9593\u7121\u908a\u3002</p> <p></p> <p> \u5716 9-5 \u00a0 \u5716\u7684\u9130\u63a5\u77e9\u9663\u8868\u793a </p> <p>\u9130\u63a5\u77e9\u9663\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u5728\u7c21\u55ae\u5716\u4e2d\uff0c\u9802\u9ede\u4e0d\u80fd\u8207\u81ea\u8eab\u76f8\u9023\uff0c\u6b64\u6642\u9130\u63a5\u77e9\u9663\u4e3b\u5c0d\u89d2\u7dda\u5143\u7d20\u6c92\u6709\u610f\u7fa9\u3002</li> <li>\u5c0d\u65bc\u7121\u5411\u5716\uff0c\u5169\u500b\u65b9\u5411\u7684\u908a\u7b49\u50f9\uff0c\u6b64\u6642\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\u3002</li> <li>\u5c07\u9130\u63a5\u77e9\u9663\u7684\u5143\u7d20\u5f9e \\(1\\) \u548c \\(0\\) \u66ff\u63db\u70ba\u6b0a\u91cd\uff0c\u5247\u53ef\u8868\u793a\u6709\u6b0a\u5716\u3002</li> </ul> <p>\u4f7f\u7528\u9130\u63a5\u77e9\u9663\u8868\u793a\u5716\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u8a2a\u554f\u77e9\u9663\u5143\u7d20\u4ee5\u7372\u53d6\u908a\uff0c\u56e0\u6b64\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9663\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u8a18\u61b6\u9ad4\u4f54\u7528\u8f03\u591a\u3002</p>"},{"location":"chapter_graph/graph/#2","title":"2. \u00a0 \u9130\u63a5\u8868","text":"<p>\u9130\u63a5\u8868\uff08adjacency list\uff09\u4f7f\u7528 \\(n\\) \u500b\u93c8\u7d50\u4e32\u5217\u4f86\u8868\u793a\u5716\uff0c\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u8868\u793a\u9802\u9ede\u3002\u7b2c \\(i\\) \u500b\u93c8\u7d50\u4e32\u5217\u5c0d\u61c9\u9802\u9ede \\(i\\) \uff0c\u5176\u4e2d\u5132\u5b58\u4e86\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\uff08\u8207\u8a72\u9802\u9ede\u76f8\u9023\u7684\u9802\u9ede\uff09\u3002\u5716 9-6 \u5c55\u793a\u4e86\u4e00\u500b\u4f7f\u7528\u9130\u63a5\u8868\u5132\u5b58\u7684\u5716\u7684\u793a\u4f8b\u3002</p> <p></p> <p> \u5716 9-6 \u00a0 \u5716\u7684\u9130\u63a5\u8868\u8868\u793a </p> <p>\u9130\u63a5\u8868\u50c5\u5132\u5b58\u5be6\u969b\u5b58\u5728\u7684\u908a\uff0c\u800c\u908a\u7684\u7e3d\u6578\u901a\u5e38\u9060\u5c0f\u65bc \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u7bc0\u7701\u7a7a\u9593\u3002\u7136\u800c\uff0c\u5728\u9130\u63a5\u8868\u4e2d\u9700\u8981\u900f\u904e\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4f86\u67e5\u8a62\u908a\uff0c\u56e0\u6b64\u5176\u6642\u9593\u6548\u7387\u4e0d\u5982\u9130\u63a5\u77e9\u9663\u3002</p> <p>\u89c0\u5bdf\u5716 9-6 \uff0c\u9130\u63a5\u8868\u7d50\u69cb\u8207\u96dc\u6e4a\u8868\u4e2d\u7684\u201c\u93c8\u5f0f\u4f4d\u5740\u201d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u5011\u4e5f\u53ef\u4ee5\u63a1\u7528\u985e\u4f3c\u7684\u65b9\u6cd5\u4f86\u6700\u4f73\u5316\u6548\u7387\u3002\u6bd4\u5982\u7576\u93c8\u7d50\u4e32\u5217\u8f03\u9577\u6642\uff0c\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba AVL \u6a39\u6216\u7d05\u9ed1\u6a39\uff0c\u5f9e\u800c\u5c07\u6642\u9593\u6548\u7387\u5f9e \\(O(n)\\) \u6700\u4f73\u5316\u81f3 \\(O(\\log n)\\) \uff1b\u9084\u53ef\u4ee5\u628a\u93c8\u7d50\u4e32\u5217\u8f49\u63db\u70ba\u96dc\u6e4a\u8868\uff0c\u5f9e\u800c\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u964d\u81f3 \\(O(1)\\) \u3002</p>"},{"location":"chapter_graph/graph/#913","title":"9.1.3 \u00a0 \u5716\u7684\u5e38\u898b\u61c9\u7528","text":"<p>\u5982\u8868 9-1 \u6240\u793a\uff0c\u8a31\u591a\u73fe\u5be6\u7cfb\u7d71\u53ef\u4ee5\u7528\u5716\u4f86\u5efa\u6a21\uff0c\u76f8\u61c9\u7684\u554f\u984c\u4e5f\u53ef\u4ee5\u7d04\u5316\u70ba\u5716\u8a08\u7b97\u554f\u984c\u3002</p> <p> \u8868 9-1 \u00a0 \u73fe\u5be6\u751f\u6d3b\u4e2d\u5e38\u898b\u7684\u5716 </p> \u9802\u9ede \u908a \u5716\u8a08\u7b97\u554f\u984c \u793e\u4ea4\u7db2\u8def \u4f7f\u7528\u8005 \u597d\u53cb\u95dc\u4fc2 \u6f5b\u5728\u597d\u53cb\u63a8\u85a6 \u5730\u9435\u7dda\u8def \u7ad9\u9ede \u7ad9\u9ede\u9593\u7684\u9023\u901a\u6027 \u6700\u77ed\u8def\u7dda\u63a8\u85a6 \u592a\u967d\u7cfb \u661f\u9ad4 \u661f\u9ad4\u9593\u7684\u842c\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8ecc\u9053\u8a08\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 \u5716\u7684\u57fa\u790e\u64cd\u4f5c","text":"<p>\u5716\u7684\u57fa\u790e\u64cd\u4f5c\u53ef\u5206\u70ba\u5c0d\u201c\u908a\u201d\u7684\u64cd\u4f5c\u548c\u5c0d\u201c\u9802\u9ede\u201d\u7684\u64cd\u4f5c\u3002\u5728\u201c\u9130\u63a5\u77e9\u9663\u201d\u548c\u201c\u9130\u63a5\u8868\u201d\u5169\u7a2e\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5be6\u73fe\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002</p>"},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1 \u00a0 \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u7684\u5be6\u73fe","text":"<p>\u7d66\u5b9a\u4e00\u500b\u9802\u9ede\u6578\u91cf\u70ba \\(n\\) \u7684\u7121\u5411\u5716\uff0c\u5247\u5404\u7a2e\u64cd\u4f5c\u7684\u5be6\u73fe\u65b9\u5f0f\u5982\u5716 9-7 \u6240\u793a\u3002</p> <ul> <li>\u65b0\u589e\u6216\u522a\u9664\u908a\uff1a\u76f4\u63a5\u5728\u9130\u63a5\u77e9\u9663\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u908a\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002\u800c\u7531\u65bc\u662f\u7121\u5411\u5716\uff0c\u56e0\u6b64\u9700\u8981\u540c\u6642\u66f4\u65b0\u5169\u500b\u65b9\u5411\u7684\u908a\u3002</li> <li>\u65b0\u589e\u9802\u9ede\uff1a\u5728\u9130\u63a5\u77e9\u9663\u7684\u5c3e\u90e8\u65b0\u589e\u4e00\u884c\u4e00\u5217\uff0c\u4e26\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002</li> <li>\u522a\u9664\u9802\u9ede\uff1a\u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u4e00\u884c\u4e00\u5217\u3002\u7576\u522a\u9664\u9996\u884c\u9996\u5217\u6642\u9054\u5230\u6700\u5dee\u60c5\u6cc1\uff0c\u9700\u8981\u5c07 \\((n-1)^2\\) \u500b\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52d5\u201d\uff0c\u5f9e\u800c\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u50b3\u5165 \\(n\\) \u500b\u9802\u9ede\uff0c\u521d\u59cb\u5316\u9577\u5ea6\u70ba \\(n\\) \u7684\u9802\u9ede\u4e32\u5217 <code>vertices</code> \uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u9130\u63a5\u77e9\u9663 <code>adjMat</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002</li> </ul> \u521d\u59cb\u5316\u9130\u63a5\u77e9\u9663\u65b0\u589e\u908a\u522a\u9664\u908a\u65b0\u589e\u9802\u9ede\u522a\u9664\u9802\u9ede <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 9-7 \u00a0 \u9130\u63a5\u77e9\u9663\u7684\u521d\u59cb\u5316\u3001\u589e\u522a\u908a\u3001\u589e\u522a\u9802\u9ede </p> <p>\u4ee5\u4e0b\u662f\u57fa\u65bc\u9130\u63a5\u77e9\u9663\u8868\u793a\u5716\u7684\u5be6\u73fe\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"\u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        # \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n        self.vertices: list[int] = []\n        # \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n        self.adj_mat: list[list[int]] = []\n        # \u65b0\u589e\u9802\u9ede\n        for val in vertices:\n            self.add_vertex(val)\n        # \u65b0\u589e\u908a\n        # \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u9802\u9ede\u6578\u91cf\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"\u65b0\u589e\u9802\u9ede\"\"\"\n        n = self.size()\n        # \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        self.vertices.append(val)\n        # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"\u522a\u9664\u9802\u9ede\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        self.vertices.pop(index)\n        # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.pop(index)\n        # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"\u65b0\u589e\u908a\"\"\"\n        # \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"\u522a\u9664\u908a\"\"\"\n        # \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"\u5217\u5370\u9130\u63a5\u77e9\u9663\"\"\"\n        print(\"\u9802\u9ede\u4e32\u5217 =\", self.vertices)\n        print(\"\u9130\u63a5\u77e9\u9663 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // \u65b0\u589e\u9802\u9ede\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    void addVertex(int val) {\n        int n = size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.push_back(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"\u9802\u9ede\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.erase(vertices.begin() + index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.erase(adjMat.begin() + index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9802\u9ede\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9802\u9ede\u4e0d\u5b58\u5728\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    void print() {\n        cout &lt;&lt; \"\u9802\u9ede\u4e32\u5217 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"\u9130\u63a5\u77e9\u9663 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // \u65b0\u589e\u9802\u9ede\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public void addVertex(int val) {\n        int n = size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.add(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.remove(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    public void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    public void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    public void print() {\n        System.out.print(\"\u9802\u9ede\u4e32\u5217 = \");\n        System.out.println(vertices);\n        System.out.println(\"\u9130\u63a5\u77e9\u9663 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    List&lt;List&lt;int&gt;&gt; adjMat; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u65b0\u589e\u9802\u9ede\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public void AddVertex(int val) {\n        int n = Size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.Add(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.RemoveAt(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.RemoveAt(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    public void AddEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    public void RemoveEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    public void Print() {\n        Console.Write(\"\u9802\u9ede\u4e32\u5217 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"\u9130\u63a5\u77e9\u9663 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\ntype graphAdjMat struct {\n    // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    vertices []int\n    // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    adjMat [][]int\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // \u65b0\u589e\u9802\u9ede\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // \u521d\u59cb\u5316\u5716\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // \u65b0\u589e\u908a\n    // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* \u65b0\u589e\u9802\u9ede */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n    g.vertices = append(g.vertices, val)\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* \u522a\u9664\u9802\u9ede */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* \u65b0\u589e\u908a */\n// \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* \u522a\u9664\u908a */\n// \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t\u9802\u9ede\u4e32\u5217 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t\u9130\u63a5\u77e9\u9663 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    private var vertices: [Int] // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    private var adjMat: [[Int]] // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // \u65b0\u589e\u9802\u9ede\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    func addVertex(val: Int) {\n        let n = size()\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.append(val)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.remove(at: index)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(at: index)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    func addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    func removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    func print() {\n        Swift.print(\"\u9802\u9ede\u4e32\u5217 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"\u9130\u63a5\u77e9\u9663 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    vertices; // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    adjMat; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u65b0\u589e\u9802\u9ede\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    addVertex(val) {\n        const n = this.size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        this.vertices.splice(index, 1);\n\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    print() {\n        console.log('\u9802\u9ede\u4e32\u5217 = ', this.vertices);\n        console.log('\u9130\u63a5\u77e9\u9663 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n    vertices: number[]; // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    adjMat: number[][]; // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u65b0\u589e\u9802\u9ede\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        this.vertices.splice(index, 1);\n\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    print(): void {\n        console.log('\u9802\u9ede\u4e32\u5217 = ', this.vertices);\n        console.log('\u9130\u63a5\u77e9\u9663 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // \u9802\u9ede\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //\u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n  /* \u5efa\u69cb\u5b50 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // \u65b0\u589e\u9802\u9ede\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // \u65b0\u589e\u908a\n    // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n  int size() {\n    return vertices.length;\n  }\n\n  /* \u65b0\u589e\u9802\u9ede */\n  void addVertex(int val) {\n    int n = size();\n    // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n    vertices.add(val);\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* \u522a\u9664\u9802\u9ede */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n    vertices.removeAt(index);\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n    adjMat.removeAt(index);\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* \u65b0\u589e\u908a */\n  // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n  void addEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* \u522a\u9664\u908a */\n  // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n  void removeEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n  void printAdjMat() {\n    print(\"\u9802\u9ede\u4e32\u5217 = $vertices\");\n    print(\"\u9130\u63a5\u77e9\u9663 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u578b\u5225 */\npub struct GraphAdjMat {\n    // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    pub vertices: Vec&lt;i32&gt;,\n    // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* \u5efa\u69cb\u5b50 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // \u65b0\u589e\u9802\u9ede\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        self.vertices.push(val);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        self.adj_mat.push(vec![0; n]);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for row in &amp;mut self.adj_mat {\n            row.push(0);\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"index error\")\n        }\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        self.vertices.remove(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.remove(index);\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in &amp;mut self.adj_mat {\n            row.remove(index);\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    pub fn print(&amp;self) {\n        println!(\"\u9802\u9ede\u4e32\u5217 = {:?}\", self.vertices);\n        println!(\"\u9130\u63a5\u77e9\u9663 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u7d50\u69cb\u9ad4 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* \u5efa\u69cb\u5b50 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* \u65b0\u589e\u9802\u9ede */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"\u5716\u7684\u9802\u9ede\u6578\u91cf\u5df2\u9054\u6700\u5927\u503c\\n\");\n        return;\n    }\n    // \u65b0\u589e\u7b2c n \u500b\u9802\u9ede\uff0c\u4e26\u5c07\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* \u522a\u9664\u9802\u9ede */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"\u9802\u9ede\u7d22\u5f15\u8d8a\u754c\\n\");\n        return;\n    }\n    // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* \u65b0\u589e\u908a */\n// \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u908a\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* \u522a\u9664\u908a */\n// \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u908a\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"\u9802\u9ede\u4e32\u5217 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"\u9130\u63a5\u77e9\u9663 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices = mutableListOf&lt;Int&gt;() // \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    val adjMat = mutableListOf&lt;MutableList&lt;Int&gt;&gt;() // \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        // \u65b0\u589e\u9802\u9ede\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // \u65b0\u589e\u908a\n        // \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    fun addVertex(_val: Int) {\n        val n = size()\n        // \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n        vertices.add(_val)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n        val newRow = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size())\n            throw IndexOutOfBoundsException()\n        // \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n        vertices.removeAt(index)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.removeAt(index)\n        // \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* \u65b0\u589e\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    fun addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        // \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u522a\u9664\u908a */\n    // \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    fun removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u77e9\u9663 */\n    fun print() {\n        print(\"\u9802\u9ede\u4e32\u5217 = \")\n        println(vertices)\n        println(\"\u9130\u63a5\u77e9\u9663 =\")\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>### \u57fa\u65bc\u9130\u63a5\u77e9\u9663\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 ###\nclass GraphAdjMat\n  def initialize(vertices, edges)\n    ### \u5efa\u69cb\u5b50 ###\n    # \u9802\u9ede\u4e32\u5217\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9802\u9ede\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    @vertices = []\n    # \u9130\u63a5\u77e9\u9663\uff0c\u884c\u5217\u7d22\u5f15\u5c0d\u61c9\u201c\u9802\u9ede\u7d22\u5f15\u201d\n    @adj_mat = []\n    # \u65b0\u589e\u9802\u9ede\n    vertices.each { |val| add_vertex(val) }\n    # \u65b0\u589e\u908a\n    # \u8acb\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9802\u9ede\u7d22\u5f15\uff0c\u5373\u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    edges.each { |e| add_edge(e[0], e[1]) }\n  end\n\n  ### \u7372\u53d6\u9802\u9ede\u6578\u91cf ###\n  def size\n    @vertices.length\n  end\n\n  ### \u65b0\u589e\u9802\u9ede ###\n  def add_vertex(val)\n    n = size\n    # \u5411\u9802\u9ede\u4e32\u5217\u4e2d\u65b0\u589e\u65b0\u9802\u9ede\u7684\u503c\n    @vertices &lt;&lt; val\n    # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u884c\n    new_row = Array.new(n, 0)\n    @adj_mat &lt;&lt; new_row\n    # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u65b0\u589e\u4e00\u5217\n    @adj_mat.each { |row| row &lt;&lt; 0 }\n  end\n\n  ### \u522a\u9664\u9802\u9ede ###\n  def remove_vertex(index)\n    raise IndexError if index &gt;= size\n\n    # \u5728\u9802\u9ede\u4e32\u5217\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9802\u9ede\n    @vertices.delete_at(index)\n    # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u884c\n    @adj_mat.delete_at(index)\n    # \u5728\u9130\u63a5\u77e9\u9663\u4e2d\u522a\u9664\u7d22\u5f15 index \u7684\u5217\n    @adj_mat.each { |row| row.delete_at(index) }\n  end\n\n  ### \u65b0\u589e\u908a ###\n  def add_edge(i, j)\n    # \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    # \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    # \u5728\u7121\u5411\u5716\u4e2d\uff0c\u9130\u63a5\u77e9\u9663\u95dc\u65bc\u4e3b\u5c0d\u89d2\u7dda\u5c0d\u7a31\uff0c\u5373\u6eff\u8db3 (i, j) == (j, i)\n    @adj_mat[i][j] = 1\n    @adj_mat[j][i] = 1\n  end\n\n  ### \u522a\u9664\u908a ###\n  def remove_edge(i, j)\n    # \u53c3\u6578 i, j \u5c0d\u61c9 vertices \u5143\u7d20\u7d22\u5f15\n    # \u7d22\u5f15\u8d8a\u754c\u8207\u76f8\u7b49\u8655\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    @adj_mat[i][j] = 0\n    @adj_mat[j][i] = 0\n  end\n\n  ### \u5217\u5370\u9130\u63a5\u77e9\u9663 ###\n  def __print__\n    puts \"\u9802\u9ede\u4e32\u5217 = #{@vertices}\"\n    puts '\u9130\u63a5\u77e9\u9663 ='\n    print_matrix(@adj_mat)\n  end\nend\n</code></pre> graph_adjacency_matrix.zig<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2 \u00a0 \u57fa\u65bc\u9130\u63a5\u8868\u7684\u5be6\u73fe","text":"<p>\u8a2d\u7121\u5411\u5716\u7684\u9802\u9ede\u7e3d\u6578\u70ba \\(n\\)\u3001\u908a\u7e3d\u6578\u70ba \\(m\\) \uff0c\u5247\u53ef\u6839\u64da\u5716 9-8 \u6240\u793a\u7684\u65b9\u6cd5\u5be6\u73fe\u5404\u7a2e\u64cd\u4f5c\u3002</p> <ul> <li>\u65b0\u589e\u908a\uff1a\u5728\u9802\u9ede\u5c0d\u61c9\u93c8\u7d50\u4e32\u5217\u7684\u672b\u5c3e\u65b0\u589e\u908a\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002\u56e0\u70ba\u662f\u7121\u5411\u5716\uff0c\u6240\u4ee5\u9700\u8981\u540c\u6642\u65b0\u589e\u5169\u500b\u65b9\u5411\u7684\u908a\u3002</li> <li>\u522a\u9664\u908a\uff1a\u5728\u9802\u9ede\u5c0d\u61c9\u93c8\u7d50\u4e32\u5217\u4e2d\u67e5\u8a62\u4e26\u522a\u9664\u6307\u5b9a\u908a\uff0c\u4f7f\u7528 \\(O(m)\\) \u6642\u9593\u3002\u5728\u7121\u5411\u5716\u4e2d\uff0c\u9700\u8981\u540c\u6642\u522a\u9664\u5169\u500b\u65b9\u5411\u7684\u908a\u3002</li> <li>\u65b0\u589e\u9802\u9ede\uff1a\u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u93c8\u7d50\u4e32\u5217\uff0c\u4e26\u5c07\u65b0\u589e\u9802\u9ede\u4f5c\u70ba\u93c8\u7d50\u4e32\u5217\u982d\u7bc0\u9ede\uff0c\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002</li> <li>\u522a\u9664\u9802\u9ede\uff1a\u9700\u8d70\u8a2a\u6574\u500b\u9130\u63a5\u8868\uff0c\u522a\u9664\u5305\u542b\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u908a\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u6642\u9593\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u5728\u9130\u63a5\u8868\u4e2d\u5efa\u7acb \\(n\\) \u500b\u9802\u9ede\u548c \\(2m\\) \u689d\u908a\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u6642\u9593\u3002</li> </ul> \u521d\u59cb\u5316\u9130\u63a5\u8868\u65b0\u589e\u908a\u522a\u9664\u908a\u65b0\u589e\u9802\u9ede\u522a\u9664\u9802\u9ede <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 9-8 \u00a0 \u9130\u63a5\u8868\u7684\u521d\u59cb\u5316\u3001\u589e\u522a\u908a\u3001\u589e\u522a\u9802\u9ede </p> <p>\u4ee5\u4e0b\u662f\u9130\u63a5\u8868\u7684\u7a0b\u5f0f\u78bc\u5be6\u73fe\u3002\u5c0d\u6bd4\u5716 9-8 \uff0c\u5be6\u969b\u7a0b\u5f0f\u78bc\u6709\u4ee5\u4e0b\u4e0d\u540c\u3002</p> <ul> <li>\u70ba\u4e86\u65b9\u4fbf\u65b0\u589e\u8207\u522a\u9664\u9802\u9ede\uff0c\u4ee5\u53ca\u7c21\u5316\u7a0b\u5f0f\u78bc\uff0c\u6211\u5011\u4f7f\u7528\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\u4f86\u4ee3\u66ff\u93c8\u7d50\u4e32\u5217\u3002</li> <li>\u4f7f\u7528\u96dc\u6e4a\u8868\u4f86\u5132\u5b58\u9130\u63a5\u8868\uff0c<code>key</code> \u70ba\u9802\u9ede\u4f8b\u9805\uff0c<code>value</code> \u70ba\u8a72\u9802\u9ede\u7684\u9130\u63a5\u9802\u9ede\u4e32\u5217\uff08\u93c8\u7d50\u4e32\u5217\uff09\u3002</li> </ul> <p>\u53e6\u5916\uff0c\u6211\u5011\u5728\u9130\u63a5\u8868\u4e2d\u4f7f\u7528 <code>Vertex</code> \u985e\u5225\u4f86\u8868\u793a\u9802\u9ede\uff0c\u9019\u6a23\u505a\u7684\u539f\u56e0\u662f\uff1a\u5982\u679c\u8207\u9130\u63a5\u77e9\u9663\u4e00\u6a23\uff0c\u7528\u4e32\u5217\u7d22\u5f15\u4f86\u5340\u5206\u4e0d\u540c\u9802\u9ede\uff0c\u90a3\u9ebc\u5047\u8a2d\u8981\u522a\u9664\u7d22\u5f15\u70ba \\(i\\) \u7684\u9802\u9ede\uff0c\u5247\u9700\u8d70\u8a2a\u6574\u500b\u9130\u63a5\u8868\uff0c\u5c07\u6240\u6709\u5927\u65bc \\(i\\) \u7684\u7d22\u5f15\u5168\u90e8\u6e1b \\(1\\) \uff0c\u6548\u7387\u5f88\u4f4e\u3002\u800c\u5982\u679c\u6bcf\u500b\u9802\u9ede\u90fd\u662f\u552f\u4e00\u7684 <code>Vertex</code> \u4f8b\u9805\uff0c\u522a\u9664\u67d0\u4e00\u9802\u9ede\u4e4b\u5f8c\u5c31\u7121\u9808\u6539\u52d5\u5176\u4ed6\u9802\u9ede\u4e86\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"\u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        # \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u9802\u9ede\u6578\u91cf\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u65b0\u589e\u908a\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u65b0\u589e\u908a vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u522a\u9664\u908a\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u522a\u9664\u908a vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"\u65b0\u589e\u9802\u9ede\"\"\"\n        if vet in self.adj_list:\n            return\n        # \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"\u522a\u9664\u9802\u9ede\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        self.adj_list.pop(vet)\n        # \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"\u5217\u5370\u9130\u63a5\u8868\"\"\"\n        print(\"\u9130\u63a5\u8868 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n  public:\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* \u5728 vector \u4e2d\u522a\u9664\u6307\u5b9a\u7bc0\u9ede */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* \u5efa\u69cb\u5b50 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    int size() {\n        return adjList.size();\n    }\n\n    /* \u65b0\u589e\u908a */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9802\u9ede\");\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9802\u9ede\");\n        // \u522a\u9664\u908a vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9802\u9ede\");\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.erase(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    void print() {\n        cout &lt;&lt; \"\u9130\u63a5\u8868 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u5efa\u69cb\u5b50 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* \u65b0\u589e\u908a */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u522a\u9664\u908a vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.remove(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    public void print() {\n        System.out.println(\"\u9130\u63a5\u8868 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u5efa\u69cb\u5b50 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* \u65b0\u589e\u908a */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u522a\u9664\u908a vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList.Add(vet, []);\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.Remove(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    public void Print() {\n        Console.WriteLine(\"\u9130\u63a5\u8868 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\ntype graphAdjList struct {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    adjList map[Vertex][]Vertex\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* \u65b0\u589e\u908a */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u65b0\u589e\u908a vet1 - vet2, \u65b0\u589e\u533f\u540d struct{},\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u522a\u9664\u908a */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u522a\u9664\u908a vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u65b0\u589e\u9802\u9ede */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u522a\u9664\u9802\u9ede */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n    delete(g.adjList, vet)\n    // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* \u5217\u5370\u9130\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"\u9130\u63a5\u8868 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* \u5efa\u69cb\u5b50 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* \u65b0\u589e\u908a */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c3\u6578\u932f\u8aa4\")\n        }\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* \u522a\u9664\u908a */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c3\u6578\u932f\u8aa4\")\n        }\n        // \u522a\u9664\u908a vet1 - vet2\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList[vet] = []\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"\u53c3\u6578\u932f\u8aa4\")\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.removeValue(forKey: vet)\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    public func print() {\n        Swift.print(\"\u9130\u63a5\u8868 =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    adjList;\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* \u65b0\u589e\u908a */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u65b0\u589e\u908a vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u522a\u9664\u908a vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        this.adjList.set(vet, []);\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        this.adjList.delete(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    print() {\n        console.log('\u9130\u63a5\u8868 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* \u65b0\u589e\u908a */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u65b0\u589e\u908a vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u522a\u9664\u908a vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        this.adjList.set(vet, []);\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        this.adjList.delete(vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    print(): void {\n        console.log('\u9130\u63a5\u8868 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList {\n  // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* \u5efa\u69cb\u5b50 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n  int size() {\n    return adjList.length;\n  }\n\n  /* \u65b0\u589e\u908a */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u65b0\u589e\u908a vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* \u522a\u9664\u908a */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u522a\u9664\u908a vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* \u65b0\u589e\u9802\u9ede */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n    adjList[vet] = [];\n  }\n\n  /* \u522a\u9664\u9802\u9ede */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n    adjList.remove(vet);\n    // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* \u5217\u5370\u9130\u63a5\u8868 */\n  void printAdjList() {\n    print(\"\u9130\u63a5\u8868 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u578b\u5225 */\npub struct GraphAdjList {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;,\n}\n\nimpl GraphAdjList {\n    /* \u5efa\u69cb\u5b50 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* \u65b0\u589e\u908a */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u65b0\u589e\u908a vet1 - vet2\n        self.adj_list.get_mut(&amp;vet1).unwrap().push(vet2);\n        self.adj_list.get_mut(&amp;vet2).unwrap().push(vet1);\n    }\n\n    /* \u522a\u9664\u908a */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u522a\u9664\u908a vet1 - vet2\n        self.adj_list\n            .get_mut(&amp;vet1)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet2);\n        self.adj_list\n            .get_mut(&amp;vet2)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet1);\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet) {\n            panic!(\"value error\");\n        }\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        self.adj_list.remove(&amp;vet);\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    pub fn print(&amp;self) {\n        println!(\"\u9130\u63a5\u8868 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* \u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // \u9802\u9ede\n    struct AdjListNode *next; // \u5f8c\u7e7c\u7bc0\u9ede\n} AdjListNode;\n\n/* \u67e5\u8a62\u9802\u9ede\u5c0d\u61c9\u7684\u7bc0\u9ede */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u65b0\u589e\u908a\u8f14\u52a9\u51fd\u5f0f */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // \u982d\u63d2\u6cd5\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* \u522a\u9664\u908a\u8f14\u52a9\u51fd\u5f0f */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // \u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u641c\u7d22 vet \u5c0d\u61c9\u7bc0\u9ede\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // \u5c07 vet \u5c0d\u61c9\u7bc0\u9ede\u5f9e\u93c8\u7d50\u4e32\u5217\u4e2d\u522a\u9664\n    pre-&gt;next = cur-&gt;next;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(cur);\n}\n\n/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // \u7bc0\u9ede\u9663\u5217\n    int size;                     // \u7bc0\u9ede\u6578\u91cf\n} GraphAdjList;\n\n/* \u5efa\u69cb\u5b50 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* \u67e5\u8a62\u9802\u9ede\u5c0d\u61c9\u7684\u7bc0\u9ede */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u65b0\u589e\u908a */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // \u65b0\u589e\u908a vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* \u522a\u9664\u908a */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // \u522a\u9664\u908a vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* \u65b0\u589e\u9802\u9ede */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* \u522a\u9664\u9802\u9ede */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // \u5c07\u8a72\u9802\u9ede\u4e4b\u5f8c\u7684\u9802\u9ede\u5411\u524d\u79fb\u52d5\uff0c\u4ee5\u586b\u88dc\u7a7a\u7f3a\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    val adjList = HashMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt;()\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        // \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n        for (edge in edges) {\n            addVertex(edge[0]!!)\n            addVertex(edge[1]!!)\n            addEdge(edge[0]!!, edge[1]!!)\n        }\n    }\n\n    /* \u7372\u53d6\u9802\u9ede\u6578\u91cf */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* \u65b0\u589e\u908a */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u65b0\u589e\u908a vet1 - vet2\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1)\n    }\n\n    /* \u522a\u9664\u908a */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u522a\u9664\u908a vet1 - vet2\n        adjList[vet1]?.remove(vet2)\n        adjList[vet2]?.remove(vet1)\n    }\n\n    /* \u65b0\u589e\u9802\u9ede */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n        adjList[vet] = mutableListOf()\n    }\n\n    /* \u522a\u9664\u9802\u9ede */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n        adjList.remove(vet)\n        // \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* \u5217\u5370\u9130\u63a5\u8868 */\n    fun print() {\n        println(\"\u9130\u63a5\u8868 =\")\n        for (pair in adjList.entries) {\n            val tmp = mutableListOf&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex._val)\n            }\n            println(\"${pair.key._val}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>### \u57fa\u65bc\u9130\u63a5\u8868\u5be6\u73fe\u7684\u7121\u5411\u5716\u985e\u5225 ###\nclass GraphAdjList\n  attr_reader :adj_list\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(edges)\n    # \u9130\u63a5\u8868\uff0ckey\uff1a\u9802\u9ede\uff0cvalue\uff1a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    @adj_list = {}\n    # \u65b0\u589e\u6240\u6709\u9802\u9ede\u548c\u908a\n    for edge in edges\n      add_vertex(edge[0])\n      add_vertex(edge[1])\n      add_edge(edge[0], edge[1])\n    end\n  end\n\n  ### \u7372\u53d6\u9802\u9ede\u6578\u91cf ###\n  def size\n    @adj_list.length\n  end\n\n  ### \u65b0\u589e\u908a ###\n  def add_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    @adj_list[vet1] &lt;&lt; vet2\n    @adj_list[vet2] &lt;&lt; vet1\n  end\n\n  ### \u522a\u9664\u908a ###\n  def remove_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    # \u522a\u9664\u908a vet1 - vet2\n    @adj_list[vet1].delete(vet2)\n    @adj_list[vet2].delete(vet1)\n  end\n\n  ### \u65b0\u589e\u9802\u9ede ###\n  def add_vertex(vet)\n    return if @adj_list.include?(vet)\n\n    # \u5728\u9130\u63a5\u8868\u4e2d\u65b0\u589e\u4e00\u500b\u65b0\u93c8\u7d50\u4e32\u5217\n    @adj_list[vet] = []\n  end\n\n  ### \u522a\u9664\u9802\u9ede ###\n  def remove_vertex(vet)\n    raise ArgumentError unless @adj_list.include?(vet)\n\n    # \u5728\u9130\u63a5\u8868\u4e2d\u522a\u9664\u9802\u9ede vet \u5c0d\u61c9\u7684\u93c8\u7d50\u4e32\u5217\n    @adj_list.delete(vet)\n    # \u8d70\u8a2a\u5176\u4ed6\u9802\u9ede\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u522a\u9664\u6240\u6709\u5305\u542b vet \u7684\u908a\n    for vertex in @adj_list\n      @adj_list[vertex.first].delete(vet) if @adj_list[vertex.first].include?(vet)\n    end\n  end\n\n  ### \u5217\u5370\u9130\u63a5\u8868 ###\n  def __print__\n    puts '\u9130\u63a5\u8868 ='\n    for vertex in @adj_list\n      tmp = @adj_list[vertex.first].map { |v| v.val }\n      puts \"#{vertex.first.val}: #{tmp},\"\n    end\n  end\nend\n</code></pre> graph_adjacency_list.zig<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3 \u00a0 \u6548\u7387\u5c0d\u6bd4","text":"<p>\u8a2d\u5716\u4e2d\u5171\u6709 \\(n\\) \u500b\u9802\u9ede\u548c \\(m\\) \u689d\u908a\uff0c\u8868 9-2 \u5c0d\u6bd4\u4e86\u9130\u63a5\u77e9\u9663\u548c\u9130\u63a5\u8868\u7684\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u3002</p> <p> \u8868 9-2 \u00a0 \u9130\u63a5\u77e9\u9663\u8207\u9130\u63a5\u8868\u5c0d\u6bd4 </p> \u9130\u63a5\u77e9\u9663 \u9130\u63a5\u8868\uff08\u93c8\u7d50\u4e32\u5217\uff09 \u9130\u63a5\u8868\uff08\u96dc\u6e4a\u8868\uff09 \u5224\u65b7\u662f\u5426\u9130\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u65b0\u589e\u908a \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u522a\u9664\u908a \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u65b0\u589e\u9802\u9ede \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u522a\u9664\u9802\u9ede \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u8a18\u61b6\u9ad4\u7a7a\u9593\u4f54\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>\u89c0\u5bdf\u8868 9-2 \uff0c\u4f3c\u4e4e\u9130\u63a5\u8868\uff08\u96dc\u6e4a\u8868\uff09\u7684\u6642\u9593\u6548\u7387\u8207\u7a7a\u9593\u6548\u7387\u6700\u512a\u3002\u4f46\u5be6\u969b\u4e0a\uff0c\u5728\u9130\u63a5\u77e9\u9663\u4e2d\u64cd\u4f5c\u908a\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u4e00\u6b21\u9663\u5217\u8a2a\u554f\u6216\u8ce6\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7d9c\u5408\u4f86\u770b\uff0c\u9130\u63a5\u77e9\u9663\u9ad4\u73fe\u4e86\u201c\u4ee5\u7a7a\u9593\u63db\u6642\u9593\u201d\u7684\u539f\u5247\uff0c\u800c\u9130\u63a5\u8868\u9ad4\u73fe\u4e86\u201c\u4ee5\u6642\u9593\u63db\u7a7a\u9593\u201d\u7684\u539f\u5247\u3002</p>"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 \u5716\u7684\u8d70\u8a2a","text":"<p>\u6a39\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5c0d\u591a\u201d\u7684\u95dc\u4fc2\uff0c\u800c\u5716\u5247\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5c0d\u591a\u201d\u95dc\u4fc2\u3002\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u6a39\u770b\u4f5c\u5716\u7684\u4e00\u7a2e\u7279\u4f8b\u3002\u986f\u7136\uff0c\u6a39\u7684\u8d70\u8a2a\u64cd\u4f5c\u4e5f\u662f\u5716\u7684\u8d70\u8a2a\u64cd\u4f5c\u7684\u4e00\u7a2e\u7279\u4f8b\u3002</p> <p>\u5716\u548c\u6a39\u90fd\u9700\u8981\u61c9\u7528\u641c\u7d22\u6f14\u7b97\u6cd5\u4f86\u5be6\u73fe\u8d70\u8a2a\u64cd\u4f5c\u3002\u5716\u7684\u8d70\u8a2a\u65b9\u5f0f\u4e5f\u53ef\u5206\u70ba\u5169\u7a2e\uff1a\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u548c\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u3002</p>"},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1 \u00a0 \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a","text":"<p>\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u662f\u4e00\u7a2e\u7531\u8fd1\u53ca\u9060\u7684\u8d70\u8a2a\u65b9\u5f0f\uff0c\u5f9e\u67d0\u500b\u7bc0\u9ede\u51fa\u767c\uff0c\u59cb\u7d42\u512a\u5148\u8a2a\u554f\u8ddd\u96e2\u6700\u8fd1\u7684\u9802\u9ede\uff0c\u4e26\u4e00\u5c64\u5c64\u5411\u5916\u64f4\u5f35\u3002\u5982\u5716 9-9 \u6240\u793a\uff0c\u5f9e\u5de6\u4e0a\u89d2\u9802\u9ede\u51fa\u767c\uff0c\u9996\u5148\u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\uff0c\u7136\u5f8c\u8d70\u8a2a\u4e0b\u4e00\u500b\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9802\u9ede\u8a2a\u554f\u5b8c\u7562\u3002</p> <p></p> <p> \u5716 9-9 \u00a0 \u5716\u7684\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a </p>"},{"location":"chapter_graph/graph_traversal/#1","title":"1. \u00a0 \u6f14\u7b97\u6cd5\u5be6\u73fe","text":"<p>BFS \u901a\u5e38\u85c9\u52a9\u4f47\u5217\u4f86\u5be6\u73fe\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\u3002\u4f47\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8cea\uff0c\u9019\u8207 BFS \u7684\u201c\u7531\u8fd1\u53ca\u9060\u201d\u7684\u601d\u60f3\u7570\u66f2\u540c\u5de5\u3002</p> <ol> <li>\u5c07\u8d70\u8a2a\u8d77\u59cb\u9802\u9ede <code>startVet</code> \u52a0\u5165\u5217\u5217\uff0c\u4e26\u958b\u555f\u8ff4\u5708\u3002</li> <li>\u5728\u8ff4\u5708\u7684\u6bcf\u8f2a\u8fed\u4ee3\u4e2d\uff0c\u5f48\u51fa\u4f47\u5217\u9996\u9802\u9ede\u4e26\u8a18\u9304\u8a2a\u554f\uff0c\u7136\u5f8c\u5c07\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\u52a0\u5165\u5230\u4f47\u5217\u5c3e\u90e8\u3002</li> <li>\u8ff4\u5708\u6b65\u9a5f <code>2.</code> \uff0c\u76f4\u5230\u6240\u6709\u9802\u9ede\u88ab\u8a2a\u554f\u5b8c\u7562\u5f8c\u7d50\u675f\u3002</li> </ol> <p>\u70ba\u4e86\u9632\u6b62\u91cd\u8907\u8d70\u8a2a\u9802\u9ede\uff0c\u6211\u5011\u9700\u8981\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u96c6\u5408 <code>visited</code> \u4f86\u8a18\u9304\u54ea\u4e9b\u7bc0\u9ede\u5df2\u88ab\u8a2a\u554f\u3002</p> <p>Tip</p> <p>\u96dc\u6e4a\u96c6\u5408\u53ef\u4ee5\u770b\u4f5c\u4e00\u500b\u53ea\u5132\u5b58 <code>key</code> \u800c\u4e0d\u5132\u5b58 <code>value</code> \u7684\u96dc\u6e4a\u8868\uff0c\u5b83\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u8907\u96dc\u5ea6\u4e0b\u9032\u884c <code>key</code> \u7684\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u3002\u6839\u64da <code>key</code> \u7684\u552f\u4e00\u6027\uff0c\u96dc\u6e4a\u96c6\u5408\u901a\u5e38\u7528\u65bc\u8cc7\u6599\u53bb\u91cd\u7b49\u5834\u666f\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\"\"\"\n    # \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    # \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res = []\n    # \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    visited = set[Vertex]([start_vet])\n    # \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    que = deque[Vertex]([start_vet])\n    # \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while len(que) &gt; 0:\n        vet = que.popleft()  # \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.append(vet)  # \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        # \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            que.append(adj_vet)  # \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.add(adj_vet)  # \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    # \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.push_back(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            que.push(adjVet);        // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.emplace(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.add(vet);            // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            que.offer(adjVet);   // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.add(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.Add(vet);               // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            que.Enqueue(adjVet);   // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.Add(adjVet);   // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS, \u4f7f\u7528\u5207\u7247\u6a21\u64ec\u4f47\u5217\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    for len(queue) &gt; 0 {\n        // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        vet := queue[0]\n        queue = queue[1:]\n        // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        res = append(res, vet)\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    var que: [Vertex] = [startVet]\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while !que.isEmpty {\n        let vet = que.removeFirst() // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.append(vet) // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            que.append(adjVet) // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.insert(adjVet) // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction graphBFS(graph, startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    const res = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    const visited = new Set();\n    visited.add(startVet);\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    const que = [startVet];\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (que.length) {\n        const vet = que.shift(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            que.push(adjVet); // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.add(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    const que = [startVet];\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (que.length) {\n        const vet = que.shift(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            que.push(adjVet); // \u53ea\u5165\u5217\u672a\u8a2a\u554f\n            visited.add(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n    res.add(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n      }\n      que.add(adjVet); // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n      visited.add(adjVet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    }\n  }\n  // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    let mut res = vec![];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while !que.is_empty() {\n        let vet = que.pop_front().unwrap(); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n                }\n                que.push_back(adj_vet); // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n                visited.insert(adj_vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n            }\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* \u7bc0\u9ede\u4f47\u5217\u7d50\u69cb\u9ad4 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* \u5efa\u69cb\u5b50 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* \u5165\u5217\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* \u51fa\u5217\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* \u6aa2\u67e5\u9802\u9ede\u662f\u5426\u5df2\u88ab\u8a2a\u554f */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // \u8d70\u8a2a\u67e5\u8a62\u7bc0\u9ede\uff0c\u4f7f\u7528 O(n) \u6642\u9593\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res[(*resSize)++] = vet;      // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n                visited[(*visitedSize)++] = node-&gt;vertex; // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n            }\n            node = node-&gt;next;\n        }\n    }\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): MutableList&lt;Vertex?&gt; {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    val visited = HashSet&lt;Vertex&gt;()\n    visited.add(startVet)\n    // \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n    val que = LinkedList&lt;Vertex&gt;()\n    que.offer(startVet)\n    // \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n    while (!que.isEmpty()) {\n        val vet = que.poll() // \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n        res.add(vet)         // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n        // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet))\n                continue        // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            que.offer(adjVet)   // \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n            visited.add(adjVet) // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n        }\n    }\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>### \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a ###\ndef graph_bfs(graph, start_vet)\n  # \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  # \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  res = []\n  # \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n  visited = Set.new([start_vet])\n  # \u4f47\u5217\u7528\u65bc\u5be6\u73fe BFS\n  que = [start_vet]\n  # \u4ee5\u9802\u9ede vet \u70ba\u8d77\u9ede\uff0c\u8ff4\u5708\u76f4\u81f3\u8a2a\u554f\u5b8c\u6240\u6709\u9802\u9ede\n  while que.length &gt; 0\n    vet = que.shift # \u4f47\u5217\u9996\u9802\u9ede\u51fa\u968a\n    res &lt;&lt; vet # \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    # \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for adj_vet in graph.adj_list[vet]\n      next if visited.include?(adj_vet) # \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n      que &lt;&lt; adj_vet # \u53ea\u5165\u5217\u672a\u8a2a\u554f\u7684\u9802\u9ede\n      visited.add(adj_vet) # \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    end\n  end\n  # \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  res\nend\n</code></pre> graph_bfs.zig<pre><code>[class]{}-[func]{graphBFS}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u7a0b\u5f0f\u78bc\u76f8\u5c0d\u62bd\u8c61\uff0c\u5efa\u8b70\u5c0d\u7167\u5716 9-10 \u4f86\u52a0\u6df1\u7406\u89e3\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 9-10 \u00a0 \u5716\u7684\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u6b65\u9a5f </p> <p>\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0d\u552f\u4e00\u3002\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u9060\u201d\u7684\u9806\u5e8f\u8d70\u8a2a\uff0c\u800c\u591a\u500b\u76f8\u540c\u8ddd\u96e2\u7684\u9802\u9ede\u7684\u8d70\u8a2a\u9806\u5e8f\u5141\u8a31\u88ab\u4efb\u610f\u6253\u4e82\u3002\u4ee5\u5716 9-10 \u70ba\u4f8b\uff0c\u9802\u9ede \\(1\\)\u3001\\(3\\) \u7684\u8a2a\u554f\u9806\u5e8f\u53ef\u4ee5\u4ea4\u63db\uff0c\u9802\u9ede \\(2\\)\u3001\\(4\\)\u3001\\(6\\) \u7684\u8a2a\u554f\u9806\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u63db\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2","title":"2. \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<p>\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u6240\u6709\u9802\u9ede\u90fd\u6703\u5165\u5217\u4e26\u51fa\u968a\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u6642\u9593\uff1b\u5728\u8d70\u8a2a\u9130\u63a5\u9802\u9ede\u7684\u904e\u7a0b\u4e2d\uff0c\u7531\u65bc\u662f\u7121\u5411\u5716\uff0c\u56e0\u6b64\u6240\u6709\u908a\u90fd\u6703\u88ab\u8a2a\u554f \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u6642\u9593\uff1b\u7e3d\u9ad4\u4f7f\u7528 \\(O(|V| + |E|)\\) \u6642\u9593\u3002</p> <p>\u7a7a\u9593\u8907\u96dc\u5ea6\uff1a\u4e32\u5217 <code>res</code> \uff0c\u96dc\u6e4a\u96c6\u5408 <code>visited</code> \uff0c\u4f47\u5217 <code>que</code> \u4e2d\u7684\u9802\u9ede\u6578\u91cf\u6700\u591a\u70ba \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u9593\u3002</p>"},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2 \u00a0 \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a","text":"<p>\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u662f\u4e00\u7a2e\u512a\u5148\u8d70\u5230\u5e95\u3001\u7121\u8def\u53ef\u8d70\u518d\u56de\u982d\u7684\u8d70\u8a2a\u65b9\u5f0f\u3002\u5982\u5716 9-11 \u6240\u793a\uff0c\u5f9e\u5de6\u4e0a\u89d2\u9802\u9ede\u51fa\u767c\uff0c\u8a2a\u554f\u7576\u524d\u9802\u9ede\u7684\u67d0\u500b\u9130\u63a5\u9802\u9ede\uff0c\u76f4\u5230\u8d70\u5230\u76e1\u982d\u6642\u8fd4\u56de\uff0c\u518d\u7e7c\u7e8c\u8d70\u5230\u76e1\u982d\u4e26\u8fd4\u56de\uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9802\u9ede\u8d70\u8a2a\u5b8c\u6210\u3002</p> <p></p> <p> \u5716 9-11 \u00a0 \u5716\u7684\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a </p>"},{"location":"chapter_graph/graph_traversal/#1_1","title":"1. \u00a0 \u6f14\u7b97\u6cd5\u5be6\u73fe","text":"<p>\u9019\u7a2e\u201c\u8d70\u5230\u76e1\u982d\u518d\u8fd4\u56de\u201d\u7684\u6f14\u7b97\u6cd5\u7bc4\u5f0f\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u4f86\u5be6\u73fe\u3002\u8207\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u985e\u4f3c\uff0c\u5728\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u4e2d\uff0c\u6211\u5011\u4e5f\u9700\u8981\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u96c6\u5408 <code>visited</code> \u4f86\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\uff0c\u4ee5\u907f\u514d\u91cd\u8907\u8a2a\u554f\u9802\u9ede\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f\"\"\"\n    res.append(vet)  # \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet)  # \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    # \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        # \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\"\"\"\n    # \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    # \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res = []\n    # \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.emplace(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.Add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede                             \n        }\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append \u64cd\u4f5c\u6703\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u9808\u8b93\u539f\u5f15\u7528\u91cd\u65b0\u8ce6\u503c\u70ba\u65b0slice\u7684\u5f15\u7528\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // \u8fd4\u56de\u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.insert(vet) // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        }\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        }\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction graphDFS(graph, startVet) {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    const res = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        }\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n  visited.add(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n  // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n    }\n    // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.insert(vet); // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n                         // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n            }\n            // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    let mut res = vec![];\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* \u6aa2\u67e5\u9802\u9ede\u662f\u5426\u5df2\u88ab\u8a2a\u554f */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // \u8d70\u8a2a\u67e5\u8a62\u7bc0\u9ede\uff0c\u4f7f\u7528 O(n) \u6642\u9593\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    res[(*resSize)++] = vet;\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet)     // \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n    visited.add(vet) // \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n    // \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet))\n            continue  // \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n        // \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n// \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\nfun graphDFS(graph: GraphAdjList, startVet: Vertex?): MutableList&lt;Vertex?&gt; {\n    // \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n    val visited = HashSet&lt;Vertex?&gt;()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>### \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u8f14\u52a9\u51fd\u5f0f ###\ndef dfs(graph, visited, res, vet)\n  res &lt;&lt; vet # \u8a18\u9304\u8a2a\u554f\u9802\u9ede\n  visited.add(vet) # \u6a19\u8a18\u8a72\u9802\u9ede\u5df2\u88ab\u8a2a\u554f\n  # \u8d70\u8a2a\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  for adj_vet in graph.adj_list[vet]\n    next if visited.include?(adj_vet) # \u8df3\u904e\u5df2\u88ab\u8a2a\u554f\u7684\u9802\u9ede\n    # \u905e\u8ff4\u8a2a\u554f\u9130\u63a5\u9802\u9ede\n    dfs(graph, visited, res, adj_vet)\n  end\nend\n\n### \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a ###\ndef graph_dfs(graph, start_vet)\n  # \u4f7f\u7528\u9130\u63a5\u8868\u4f86\u8868\u793a\u5716\uff0c\u4ee5\u4fbf\u7372\u53d6\u6307\u5b9a\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\n  # \u9802\u9ede\u8d70\u8a2a\u5e8f\u5217\n  res = []\n  # \u96dc\u6e4a\u96c6\u5408\uff0c\u7528\u65bc\u8a18\u9304\u5df2\u88ab\u8a2a\u554f\u904e\u7684\u9802\u9ede\n  visited = Set.new\n  dfs(graph, visited, res, start_vet)\n  res\nend\n</code></pre> graph_dfs.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u7684\u6f14\u7b97\u6cd5\u6d41\u7a0b\u5982\u5716 9-12 \u6240\u793a\u3002</p> <ul> <li>\u76f4\u865b\u7dda\u4ee3\u8868\u5411\u4e0b\u905e\u63a8\uff0c\u8868\u793a\u958b\u555f\u4e86\u4e00\u500b\u65b0\u7684\u905e\u8ff4\u65b9\u6cd5\u4f86\u8a2a\u554f\u65b0\u9802\u9ede\u3002</li> <li>\u66f2\u865b\u7dda\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u905e\u8ff4\u65b9\u6cd5\u5df2\u7d93\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u958b\u555f\u6b64\u65b9\u6cd5\u7684\u4f4d\u7f6e\u3002</li> </ul> <p>\u70ba\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8b70\u5c07\u5716 9-12 \u8207\u7a0b\u5f0f\u78bc\u7d50\u5408\u8d77\u4f86\uff0c\u5728\u8166\u4e2d\u6a21\u64ec\uff08\u6216\u8005\u7528\u7b46\u756b\u4e0b\u4f86\uff09\u6574\u500b DFS \u904e\u7a0b\uff0c\u5305\u62ec\u6bcf\u500b\u905e\u8ff4\u65b9\u6cd5\u4f55\u6642\u958b\u555f\u3001\u4f55\u6642\u8fd4\u56de\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 9-12 \u00a0 \u5716\u7684\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u6b65\u9a5f </p> <p>\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u8207\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u985e\u4f3c\uff0c\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u5e8f\u5217\u7684\u9806\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7d66\u5b9a\u67d0\u9802\u9ede\uff0c\u5148\u5f80\u54ea\u500b\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u9130\u63a5\u9802\u9ede\u7684\u9806\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e82\uff0c\u90fd\u662f\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u3002</p> <p>\u4ee5\u6a39\u7684\u8d70\u8a2a\u70ba\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u5225\u5c0d\u61c9\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a\uff0c\u5b83\u5011\u5c55\u793a\u4e86\u4e09\u7a2e\u8d70\u8a2a\u512a\u5148\u9806\u5e8f\uff0c\u7136\u800c\u9019\u4e09\u8005\u90fd\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2_1","title":"2. \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<p>\u6642\u9593\u8907\u96dc\u5ea6\uff1a\u6240\u6709\u9802\u9ede\u90fd\u6703\u88ab\u8a2a\u554f \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u6642\u9593\uff1b\u6240\u6709\u908a\u90fd\u6703\u88ab\u8a2a\u554f \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u6642\u9593\uff1b\u7e3d\u9ad4\u4f7f\u7528 \\(O(|V| + |E|)\\) \u6642\u9593\u3002</p> <p>\u7a7a\u9593\u8907\u96dc\u5ea6\uff1a\u4e32\u5217 <code>res</code> \uff0c\u96dc\u6e4a\u96c6\u5408 <code>visited</code> \u9802\u9ede\u6578\u91cf\u6700\u591a\u70ba \\(|V|\\) \uff0c\u905e\u8ff4\u6df1\u5ea6\u6700\u5927\u70ba \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u9593\u3002</p>"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_graph/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u5716\u7531\u9802\u9ede\u548c\u908a\u7d44\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u70ba\u4e00\u7d44\u9802\u9ede\u548c\u4e00\u7d44\u908a\u69cb\u6210\u7684\u96c6\u5408\u3002</li> <li>\u76f8\u8f03\u65bc\u7dda\u6027\u95dc\u4fc2\uff08\u93c8\u7d50\u4e32\u5217\uff09\u548c\u5206\u6cbb\u95dc\u4fc2\uff08\u6a39\uff09\uff0c\u7db2\u8def\u95dc\u4fc2\uff08\u5716\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u70ba\u8907\u96dc\u3002</li> <li>\u6709\u5411\u5716\u7684\u908a\u5177\u6709\u65b9\u5411\u6027\uff0c\u9023\u901a\u5716\u4e2d\u7684\u4efb\u610f\u9802\u9ede\u5747\u53ef\u9054\uff0c\u6709\u6b0a\u5716\u7684\u6bcf\u689d\u908a\u90fd\u5305\u542b\u6b0a\u91cd\u8b8a\u6578\u3002</li> <li>\u9130\u63a5\u77e9\u9663\u5229\u7528\u77e9\u9663\u4f86\u8868\u793a\u5716\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u500b\u9802\u9ede\uff0c\u77e9\u9663\u5143\u7d20\u4ee3\u8868\u908a\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u5169\u500b\u9802\u9ede\u4e4b\u9593\u6709\u908a\u6216\u7121\u908a\u3002\u9130\u63a5\u77e9\u9663\u5728\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u9593\u4f54\u7528\u8f03\u591a\u3002</li> <li>\u9130\u63a5\u8868\u4f7f\u7528\u591a\u500b\u93c8\u7d50\u4e32\u5217\u4f86\u8868\u793a\u5716\uff0c\u7b2c \\(i\\) \u500b\u93c8\u7d50\u4e32\u5217\u5c0d\u61c9\u9802\u9ede \\(i\\) \uff0c\u5176\u4e2d\u5132\u5b58\u4e86\u8a72\u9802\u9ede\u7684\u6240\u6709\u9130\u63a5\u9802\u9ede\u3002\u9130\u63a5\u8868\u76f8\u5c0d\u65bc\u9130\u63a5\u77e9\u9663\u66f4\u52a0\u7bc0\u7701\u7a7a\u9593\uff0c\u4f46\u7531\u65bc\u9700\u8981\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4f86\u67e5\u8a62\u908a\uff0c\u56e0\u6b64\u6642\u9593\u6548\u7387\u8f03\u4f4e\u3002</li> <li>\u7576\u9130\u63a5\u8868\u4e2d\u7684\u93c8\u7d50\u4e32\u5217\u904e\u9577\u6642\uff0c\u53ef\u4ee5\u5c07\u5176\u8f49\u63db\u70ba\u7d05\u9ed1\u6a39\u6216\u96dc\u6e4a\u8868\uff0c\u5f9e\u800c\u63d0\u5347\u67e5\u8a62\u6548\u7387\u3002</li> <li>\u5f9e\u6f14\u7b97\u6cd5\u601d\u60f3\u7684\u89d2\u5ea6\u5206\u6790\uff0c\u9130\u63a5\u77e9\u9663\u9ad4\u73fe\u4e86\u201c\u4ee5\u7a7a\u9593\u63db\u6642\u9593\u201d\uff0c\u9130\u63a5\u8868\u9ad4\u73fe\u4e86\u201c\u4ee5\u6642\u9593\u63db\u7a7a\u9593\u201d\u3002</li> <li>\u5716\u53ef\u7528\u65bc\u5efa\u6a21\u5404\u985e\u73fe\u5be6\u7cfb\u7d71\uff0c\u5982\u793e\u4ea4\u7db2\u8def\u3001\u5730\u9435\u7dda\u8def\u7b49\u3002</li> <li>\u6a39\u662f\u5716\u7684\u4e00\u7a2e\u7279\u4f8b\uff0c\u6a39\u7684\u8d70\u8a2a\u4e5f\u662f\u5716\u7684\u8d70\u8a2a\u7684\u4e00\u7a2e\u7279\u4f8b\u3002</li> <li>\u5716\u7684\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u662f\u4e00\u7a2e\u7531\u8fd1\u53ca\u9060\u3001\u5c64\u5c64\u64f4\u5f35\u7684\u641c\u5c0b\u65b9\u5f0f\uff0c\u901a\u5e38\u85c9\u52a9\u4f47\u5217\u5be6\u73fe\u3002</li> <li>\u5716\u7684\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u662f\u4e00\u7a2e\u512a\u5148\u8d70\u5230\u5e95\u3001\u7121\u8def\u53ef\u8d70\u6642\u518d\u56de\u6eaf\u7684\u641c\u5c0b\u65b9\u5f0f\uff0c\u5e38\u57fa\u65bc\u905e\u8ff4\u4f86\u5be6\u73fe\u3002</li> </ul>"},{"location":"chapter_graph/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u8def\u5f91\u7684\u5b9a\u7fa9\u662f\u9802\u9ede\u5e8f\u5217\u9084\u662f\u908a\u5e8f\u5217\uff1f</p> <p>\u7dad\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8a9e\u8a00\u7248\u672c\u7684\u5b9a\u7fa9\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f91\u662f\u4e00\u500b\u908a\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f91\u662f\u4e00\u500b\u9802\u9ede\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p> <p>\u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f91\u88ab\u8996\u70ba\u4e00\u500b\u908a\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u500b\u9802\u9ede\u5e8f\u5217\u3002\u9019\u662f\u56e0\u70ba\u5169\u500b\u9802\u9ede\u4e4b\u9593\u53ef\u80fd\u5b58\u5728\u591a\u689d\u908a\u9023\u7dda\uff0c\u6b64\u6642\u6bcf\u689d\u908a\u90fd\u5c0d\u61c9\u4e00\u689d\u8def\u5f91\u3002</p> <p>Q\uff1a\u975e\u9023\u901a\u5716\u4e2d\u662f\u5426\u6703\u6709\u7121\u6cd5\u8d70\u8a2a\u5230\u7684\u9ede\uff1f</p> <p>\u5728\u975e\u9023\u901a\u5716\u4e2d\uff0c\u5f9e\u67d0\u500b\u9802\u9ede\u51fa\u767c\uff0c\u81f3\u5c11\u6709\u4e00\u500b\u9802\u9ede\u7121\u6cd5\u5230\u9054\u3002\u8d70\u8a2a\u975e\u9023\u901a\u5716\u9700\u8981\u8a2d\u5b9a\u591a\u500b\u8d77\u9ede\uff0c\u4ee5\u8d70\u8a2a\u5230\u5716\u7684\u6240\u6709\u9023\u901a\u5206\u91cf\u3002</p> <p>Q\uff1a\u5728\u9130\u63a5\u8868\u4e2d\uff0c\u201c\u8207\u8a72\u9802\u9ede\u76f8\u9023\u7684\u6240\u6709\u9802\u9ede\u201d\u7684\u9802\u9ede\u9806\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f</p> <p>\u53ef\u4ee5\u662f\u4efb\u610f\u9806\u5e8f\u3002\u4f46\u5728\u5be6\u969b\u61c9\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u6309\u7167\u6307\u5b9a\u898f\u5247\u4f86\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9802\u9ede\u65b0\u589e\u7684\u6b21\u5e8f\uff0c\u6216\u8005\u6309\u7167\u9802\u9ede\u503c\u5927\u5c0f\u7684\u9806\u5e8f\u7b49\uff0c\u9019\u6a23\u6709\u52a9\u65bc\u5feb\u901f\u67e5\u8a62\u201c\u5e36\u6709\u67d0\u7a2e\u6975\u503c\u201d\u7684\u9802\u9ede\u3002</p>"},{"location":"chapter_greedy/","title":"\u7b2c 15 \u7ae0 \u00a0 \u8caa\u5a6a","text":"<p>Abstract</p> <p>\u5411\u65e5\u8475\u671d\u8457\u592a\u967d\u8f49\u52d5\uff0c\u6642\u523b\u8ffd\u6c42\u81ea\u8eab\u6210\u9577\u7684\u6700\u5927\u53ef\u80fd\u3002</p> <p>\u8caa\u5a6a\u7b56\u7565\u5728\u4e00\u8f2a\u8f2a\u7684\u7c21\u55ae\u9078\u64c7\u4e2d\uff0c\u9010\u6b65\u5c0e\u5411\u6700\u4f73\u7b54\u6848\u3002</p>"},{"location":"chapter_greedy/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>15.1 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5</li> <li>15.2 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c</li> <li>15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u554f\u984c</li> <li>15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u7a4d\u554f\u984c</li> <li>15.5 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_greedy/fractional_knapsack_problem/","title":"15.2 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u500b\u7269\u54c1\uff0c\u7b2c \\(i\\) \u500b\u7269\u54c1\u7684\u91cd\u91cf\u70ba \\(wgt[i-1]\\)\u3001\u50f9\u503c\u70ba \\(val[i-1]\\) \uff0c\u548c\u4e00\u500b\u5bb9\u91cf\u70ba \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u500b\u7269\u54c1\u53ea\u80fd\u9078\u64c7\u4e00\u6b21\uff0c\u4f46\u53ef\u4ee5\u9078\u64c7\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u50f9\u503c\u6839\u64da\u9078\u64c7\u7684\u91cd\u91cf\u6bd4\u4f8b\u8a08\u7b97\uff0c\u554f\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u50f9\u503c\u3002\u793a\u4f8b\u5982\u5716 15-3 \u6240\u793a\u3002</p> <p></p> <p> \u5716 15-3 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c\u7684\u793a\u4f8b\u8cc7\u6599 </p> <p>\u5206\u6578\u80cc\u5305\u554f\u984c\u548c 0-1 \u80cc\u5305\u554f\u984c\u6574\u9ad4\u4e0a\u975e\u5e38\u76f8\u4f3c\uff0c\u72c0\u614b\u5305\u542b\u7576\u524d\u7269\u54c1 \\(i\\) \u548c\u5bb9\u91cf \\(c\\) \uff0c\u76ee\u6a19\u662f\u6c42\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u6700\u5927\u50f9\u503c\u3002</p> <p>\u4e0d\u540c\u9ede\u5728\u65bc\uff0c\u672c\u984c\u5141\u8a31\u53ea\u9078\u64c7\u7269\u54c1\u7684\u4e00\u90e8\u5206\u3002\u5982\u5716 15-4 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u5c0d\u7269\u54c1\u4efb\u610f\u5730\u9032\u884c\u5207\u5206\uff0c\u4e26\u6309\u7167\u91cd\u91cf\u6bd4\u4f8b\u4f86\u8a08\u7b97\u76f8\u61c9\u50f9\u503c\u3002</p> <ol> <li>\u5c0d\u65bc\u7269\u54c1 \\(i\\) \uff0c\u5b83\u5728\u55ae\u4f4d\u91cd\u91cf\u4e0b\u7684\u50f9\u503c\u70ba \\(val[i-1] / wgt[i-1]\\) \uff0c\u7c21\u7a31\u55ae\u4f4d\u50f9\u503c\u3002</li> <li>\u5047\u8a2d\u653e\u5165\u4e00\u90e8\u5206\u7269\u54c1 \\(i\\) \uff0c\u91cd\u91cf\u70ba \\(w\\) \uff0c\u5247\u80cc\u5305\u589e\u52a0\u7684\u50f9\u503c\u70ba \\(w \\times val[i-1] / wgt[i-1]\\) \u3002</li> </ol> <p></p> <p> \u5716 15-4 \u00a0 \u7269\u54c1\u5728\u55ae\u4f4d\u91cd\u91cf\u4e0b\u7684\u50f9\u503c </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#1","title":"1. \u00a0 \u8caa\u5a6a\u7b56\u7565\u78ba\u5b9a","text":"<p>\u6700\u5927\u5316\u80cc\u5305\u5167\u7269\u54c1\u7e3d\u50f9\u503c\uff0c\u672c\u8cea\u4e0a\u662f\u6700\u5927\u5316\u55ae\u4f4d\u91cd\u91cf\u4e0b\u7684\u7269\u54c1\u50f9\u503c\u3002\u7531\u6b64\u4fbf\u53ef\u63a8\u7406\u51fa\u5716 15-5 \u6240\u793a\u7684\u8caa\u5a6a\u7b56\u7565\u3002</p> <ol> <li>\u5c07\u7269\u54c1\u6309\u7167\u55ae\u4f4d\u50f9\u503c\u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\u3002</li> <li>\u8d70\u8a2a\u6240\u6709\u7269\u54c1\uff0c\u6bcf\u8f2a\u8caa\u5a6a\u5730\u9078\u64c7\u55ae\u4f4d\u50f9\u503c\u6700\u9ad8\u7684\u7269\u54c1\u3002</li> <li>\u82e5\u5269\u9918\u80cc\u5305\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u4f7f\u7528\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u586b\u6eff\u80cc\u5305\u3002</li> </ol> <p></p> <p> \u5716 15-5 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c\u7684\u8caa\u5a6a\u7b56\u7565 </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u6211\u5011\u5efa\u7acb\u4e86\u4e00\u500b\u7269\u54c1\u985e\u5225 <code>Item</code> \uff0c\u4ee5\u4fbf\u5c07\u7269\u54c1\u6309\u7167\u55ae\u4f4d\u50f9\u503c\u9032\u884c\u6392\u5e8f\u3002\u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u7576\u80cc\u5305\u5df2\u6eff\u6642\u8df3\u51fa\u4e26\u8fd4\u56de\u89e3\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig fractional_knapsack.py<pre><code>class Item:\n    \"\"\"\u7269\u54c1\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # \u7269\u54c1\u91cd\u91cf\n        self.v = v  # \u7269\u54c1\u50f9\u503c\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a\"\"\"\n    # \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v\n            cap -= item.w\n        else:\n            # \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (item.v / item.w) * cap\n            # \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* \u7269\u54c1 */\nclass Item {\n  public:\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u50f9\u503c\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* \u7269\u54c1 */\nclass Item {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u50f9\u503c\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (double) item.v / item.w * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* \u7269\u54c1 */\nclass Item(int w, int v) {\n    public int w = w; // \u7269\u54c1\u91cd\u91cf\n    public int v = v; // \u7269\u54c1\u50f9\u503c\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* \u7269\u54c1 */\ntype Item struct {\n    w int // \u7269\u54c1\u91cd\u91cf\n    v int // \u7269\u54c1\u50f9\u503c\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* \u7269\u54c1 */\nclass Item {\n    var w: Int // \u7269\u54c1\u91cd\u91cf\n    var v: Int // \u7269\u54c1\u50f9\u503c\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) }\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* \u7269\u54c1 */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // \u7269\u54c1\u91cd\u91cf\n        this.v = v; // \u7269\u54c1\u50f9\u503c\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* \u7269\u54c1 */\nclass Item {\n    w: number; // \u7269\u54c1\u91cd\u91cf\n    v: number; // \u7269\u54c1\u50f9\u503c\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* \u7269\u54c1 */\nclass Item {\n  int w; // \u7269\u54c1\u91cd\u91cf\n  int v; // \u7269\u54c1\u50f9\u503c\n\n  Item(this.w, this.v);\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n      res += item.v / item.w * cap;\n      // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* \u7269\u54c1 */\nstruct Item {\n    w: i32, // \u7269\u54c1\u91cd\u91cf\n    v: i32, // \u7269\u54c1\u50f9\u503c\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* \u7269\u54c1 */\ntypedef struct {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u50f9\u503c\n} Item;\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.kt<pre><code>/* \u7269\u54c1 */\nclass Item(\n    val w: Int, // \u7269\u54c1\n    val v: Int  // \u7269\u54c1\u50f9\u503c\n)\n\n/* \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a */\nfun fractionalKnapsack(wgt: IntArray, _val: IntArray, c: Int): Double {\n    // \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\u3001\u50f9\u503c\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], _val[i])\n    }\n    // \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n    items.sortBy { item: Item? -&gt; -(item!!.v.toDouble() / item.w) }\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n            res += item.v\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n            res += item.v.toDouble() / item.w * cap\n            // \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.rb<pre><code>### \u7269\u54c1 ###\nclass Item\n  attr_accessor :w # \u7269\u54c1\u91cd\u91cf\n  attr_accessor :v # \u7269\u54c1\u50f9\u503c\n\n  def initialize(w, v)\n    @w = w\n    @v = v\n  end\nend\n\n### \u5206\u6578\u80cc\u5305\uff1a\u8caa\u5a6a ###\ndef fractional_knapsack(wgt, val, cap)\n  # \u5efa\u7acb\u7269\u54c1\u4e32\u5217\uff0c\u5305\u542b\u5169\u500b\u5c6c\u6027\uff1a\u91cd\u91cf\uff0c\u50f9\u503c\n  items = wgt.each_with_index.map { |w, i| Item.new(w, val[i]) }\n  # \u6309\u7167\u55ae\u4f4d\u50f9\u503c item.v / item.w \u5f9e\u9ad8\u5230\u4f4e\u9032\u884c\u6392\u5e8f\n  items.sort! { |a, b| (b.v.to_f / b.w) &lt;=&gt; (a.v.to_f / a.w) }\n  # \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\n  res = 0\n  for item in items\n    if item.w &lt;= cap\n      # \u82e5\u5269\u9918\u5bb9\u91cf\u5145\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u6574\u500b\u88dd\u9032\u80cc\u5305\n      res += item.v\n      cap -= item.w\n    else\n      # \u82e5\u5269\u9918\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5247\u5c07\u7576\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88dd\u9032\u80cc\u5305\n      res += (item.v.to_f / item.w) * cap\n      # \u5df2\u7121\u5269\u9918\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u8ff4\u5708\n      break\n    end\n  end\n  res\nend\n</code></pre> fractional_knapsack.zig<pre><code>[class]{Item}-[func]{}\n\n[class]{}-[func]{fractionalKnapsack}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u9664\u6392\u5e8f\u4e4b\u5916\uff0c\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u9700\u8981\u8d70\u8a2a\u6574\u500b\u7269\u54c1\u4e32\u5217\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u70ba\u7269\u54c1\u6578\u91cf\u3002</p> <p>\u7531\u65bc\u521d\u59cb\u5316\u4e86\u4e00\u500b <code>Item</code> \u7269\u4ef6\u4e32\u5217\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#3","title":"3. \u00a0 \u6b63\u78ba\u6027\u8b49\u660e","text":"<p>\u63a1\u7528\u53cd\u8b49\u6cd5\u3002\u5047\u8a2d\u7269\u54c1 \\(x\\) \u662f\u55ae\u4f4d\u50f9\u503c\u6700\u9ad8\u7684\u7269\u54c1\uff0c\u4f7f\u7528\u67d0\u6f14\u7b97\u6cd5\u6c42\u5f97\u6700\u5927\u50f9\u503c\u70ba <code>res</code> \uff0c\u4f46\u8a72\u89e3\u4e2d\u4e0d\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u73fe\u5728\u5f9e\u80cc\u5305\u4e2d\u62ff\u51fa\u55ae\u4f4d\u91cd\u91cf\u7684\u4efb\u610f\u7269\u54c1\uff0c\u4e26\u66ff\u63db\u70ba\u55ae\u4f4d\u91cd\u91cf\u7684\u7269\u54c1 \\(x\\) \u3002\u7531\u65bc\u7269\u54c1 \\(x\\) \u7684\u55ae\u4f4d\u50f9\u503c\u6700\u9ad8\uff0c\u56e0\u6b64\u66ff\u63db\u5f8c\u7684\u7e3d\u50f9\u503c\u4e00\u5b9a\u5927\u65bc <code>res</code> \u3002\u9019\u8207 <code>res</code> \u662f\u6700\u512a\u89e3\u77db\u76fe\uff0c\u8aaa\u660e\u6700\u512a\u89e3\u4e2d\u5fc5\u9808\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u5c0d\u65bc\u8a72\u89e3\u4e2d\u7684\u5176\u4ed6\u7269\u54c1\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u69cb\u5efa\u51fa\u4e0a\u8ff0\u77db\u76fe\u3002\u7e3d\u800c\u8a00\u4e4b\uff0c\u55ae\u4f4d\u50f9\u503c\u66f4\u5927\u7684\u7269\u54c1\u7e3d\u662f\u66f4\u512a\u9078\u64c7\uff0c\u9019\u8aaa\u660e\u8caa\u5a6a\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p> <p>\u5982\u5716 15-6 \u6240\u793a\uff0c\u5982\u679c\u5c07\u7269\u54c1\u91cd\u91cf\u548c\u7269\u54c1\u55ae\u4f4d\u50f9\u503c\u5206\u5225\u770b\u4f5c\u4e00\u5f35\u4e8c\u7dad\u5716\u8868\u7684\u6a6b\u8ef8\u548c\u7e31\u8ef8\uff0c\u5247\u5206\u6578\u80cc\u5305\u554f\u984c\u53ef\u8f49\u5316\u70ba\u201c\u6c42\u5728\u6709\u9650\u6a6b\u8ef8\u5340\u9593\u4e0b\u570d\u6210\u7684\u6700\u5927\u9762\u7a4d\u201d\u3002\u9019\u500b\u985e\u6bd4\u53ef\u4ee5\u5e6b\u52a9\u6211\u5011\u5f9e\u5e7e\u4f55\u89d2\u5ea6\u7406\u89e3\u8caa\u5a6a\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</p> <p></p> <p> \u5716 15-6 \u00a0 \u5206\u6578\u80cc\u5305\u554f\u984c\u7684\u5e7e\u4f55\u8868\u793a </p>"},{"location":"chapter_greedy/greedy_algorithm/","title":"15.1 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5","text":"<p>\u8caa\u5a6a\u6f14\u7b97\u6cd5\uff08greedy algorithm\uff09\u662f\u4e00\u7a2e\u5e38\u898b\u7684\u89e3\u6c7a\u6700\u4f73\u5316\u554f\u984c\u7684\u6f14\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u554f\u984c\u7684\u6bcf\u500b\u6c7a\u7b56\u968e\u6bb5\uff0c\u90fd\u9078\u64c7\u7576\u524d\u770b\u8d77\u4f86\u6700\u512a\u7684\u9078\u64c7\uff0c\u5373\u8caa\u5a6a\u5730\u505a\u51fa\u5340\u57df\u6027\u6700\u512a\u7684\u6c7a\u7b56\uff0c\u4ee5\u671f\u7372\u5f97\u5168\u57df\u6027\u6700\u512a\u89e3\u3002\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7c21\u6f54\u4e14\u9ad8\u6548\uff0c\u5728\u8a31\u591a\u5be6\u969b\u554f\u984c\u4e2d\u6709\u8457\u5ee3\u6cdb\u7684\u61c9\u7528\u3002</p> <p>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u548c\u52d5\u614b\u898f\u5283\u90fd\u5e38\u7528\u65bc\u89e3\u6c7a\u6700\u4f73\u5316\u554f\u984c\u3002\u5b83\u5011\u4e4b\u9593\u5b58\u5728\u4e00\u4e9b\u76f8\u4f3c\u4e4b\u8655\uff0c\u6bd4\u5982\u90fd\u4f9d\u8cf4\u6700\u512a\u5b50\u7d50\u69cb\u6027\u8cea\uff0c\u4f46\u5de5\u4f5c\u539f\u7406\u4e0d\u540c\u3002</p> <ul> <li>\u52d5\u614b\u898f\u5283\u6703\u6839\u64da\u4e4b\u524d\u968e\u6bb5\u7684\u6240\u6709\u6c7a\u7b56\u4f86\u8003\u616e\u7576\u524d\u6c7a\u7b56\uff0c\u4e26\u4f7f\u7528\u904e\u53bb\u5b50\u554f\u984c\u7684\u89e3\u4f86\u69cb\u5efa\u7576\u524d\u5b50\u554f\u984c\u7684\u89e3\u3002</li> <li>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u4e0d\u6703\u8003\u616e\u904e\u53bb\u7684\u6c7a\u7b56\uff0c\u800c\u662f\u4e00\u8def\u5411\u524d\u5730\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u4e0d\u65b7\u7e2e\u5c0f\u554f\u984c\u7bc4\u570d\uff0c\u76f4\u81f3\u554f\u984c\u88ab\u89e3\u6c7a\u3002</li> </ul> <p>\u6211\u5011\u5148\u900f\u904e\u4f8b\u984c\u201c\u96f6\u9322\u514c\u63db\u201d\u77ad\u89e3\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002\u9019\u9053\u984c\u5df2\u7d93\u5728\u201c\u5b8c\u5168\u80cc\u5305\u554f\u984c\u201d\u7ae0\u7bc0\u4e2d\u4ecb\u7d39\u904e\uff0c\u76f8\u4fe1\u4f60\u5c0d\u5b83\u4e26\u4e0d\u964c\u751f\u3002</p> <p>Question</p> <p>\u7d66\u5b9a \\(n\\) \u7a2e\u786c\u5e63\uff0c\u7b2c \\(i\\) \u7a2e\u786c\u5e63\u7684\u9762\u503c\u70ba \\(coins[i - 1]\\) \uff0c\u76ee\u6a19\u91d1\u984d\u70ba \\(amt\\) \uff0c\u6bcf\u7a2e\u786c\u5e63\u53ef\u4ee5\u91cd\u8907\u9078\u53d6\uff0c\u554f\u80fd\u5920\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u7684\u6700\u5c11\u786c\u5e63\u6578\u91cf\u3002\u5982\u679c\u7121\u6cd5\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\uff0c\u5247\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u672c\u984c\u63a1\u53d6\u7684\u8caa\u5a6a\u7b56\u7565\u5982\u5716 15-1 \u6240\u793a\u3002\u7d66\u5b9a\u76ee\u6a19\u91d1\u984d\uff0c\u6211\u5011\u8caa\u5a6a\u5730\u9078\u64c7\u4e0d\u5927\u65bc\u4e14\u6700\u63a5\u8fd1\u5b83\u7684\u786c\u5e63\uff0c\u4e0d\u65b7\u8ff4\u5708\u8a72\u6b65\u9a5f\uff0c\u76f4\u81f3\u6e4a\u51fa\u76ee\u6a19\u91d1\u984d\u70ba\u6b62\u3002</p> <p></p> <p> \u5716 15-1 \u00a0 \u96f6\u9322\u514c\u63db\u7684\u8caa\u5a6a\u7b56\u7565 </p> <p>\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a\"\"\"\n    # \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    i = len(coins) - 1\n    count = 0\n    # \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while amt &gt; 0:\n        # \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # \u9078\u64c7 coins[i]\n        amt -= coins[i]\n        count += 1\n    # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    int i = coins.size() - 1;\n    int count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint coinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    int i = coins.length - 1;\n    int count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    int i = coins.Length - 1;\n    int count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    i := len(coins) - 1\n    count := 0\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    for amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i]\n        count += 1\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfunction coinChangeGreedy(coins, amt) {\n    // \u5047\u8a2d coins \u9663\u5217\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // \u5047\u8a2d coins \u9663\u5217\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n  int i = coins.length - 1;\n  int count = 0;\n  // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n  while (amt &gt; 0) {\n    // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // \u9078\u64c7 coins[i]\n    amt -= coins[i];\n    count++;\n  }\n  // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count += 1;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    int i = size - 1;\n    int count = 0;\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9078\u64c7 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.kt<pre><code>/* \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a */\nfun coinChangeGreedy(coins: IntArray, amt: Int): Int {\n    // \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n    var am = amt\n    var i = coins.size - 1\n    var count = 0\n    // \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n    while (am &gt; 0) {\n        // \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; am) {\n            i--\n        }\n        // \u9078\u64c7 coins[i]\n        am -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5247\u8fd4\u56de -1\n    return if (am == 0) count else -1\n}\n</code></pre> coin_change_greedy.rb<pre><code>### \u96f6\u9322\u514c\u63db\uff1a\u8caa\u5a6a ###\ndef coin_change_greedy(coins, amt)\n  # \u5047\u8a2d coins \u4e32\u5217\u6709\u5e8f\n  i = coins.length - 1\n  count = 0\n  # \u8ff4\u5708\u9032\u884c\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u5230\u7121\u5269\u9918\u91d1\u984d\n  while amt &gt; 0\n    # \u627e\u5230\u5c0f\u65bc\u4e14\u6700\u63a5\u8fd1\u5269\u9918\u91d1\u984d\u7684\u786c\u5e63\n    while i &gt; 0 &amp;&amp; coins[i] &gt; amt\n      i -= 1\n    end\n    # \u9078\u64c7 coins[i]\n    amt -= coins[i]\n    count += 1\n  end\n  # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c \u5247\u8fd4\u56de -1\n  amt == 0 ? count : -1\nend\n</code></pre> coin_change_greedy.zig<pre><code>[class]{}-[func]{coinChangeGreedy}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u4f60\u53ef\u80fd\u6703\u4e0d\u7531\u5730\u767c\u51fa\u611f\u5606\uff1aSo clean \uff01\u8caa\u5a6a\u6f14\u7b97\u6cd5\u50c5\u7528\u7d04\u5341\u884c\u7a0b\u5f0f\u78bc\u5c31\u89e3\u6c7a\u4e86\u96f6\u9322\u514c\u63db\u554f\u984c\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1511","title":"15.1.1 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u4e0d\u50c5\u64cd\u4f5c\u76f4\u63a5\u3001\u5be6\u73fe\u7c21\u55ae\uff0c\u800c\u4e14\u901a\u5e38\u6548\u7387\u4e5f\u5f88\u9ad8\u3002\u5728\u4ee5\u4e0a\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u8a18\u786c\u5e63\u6700\u5c0f\u9762\u503c\u70ba \\(\\min(coins)\\) \uff0c\u5247\u8caa\u5a6a\u9078\u64c7\u6700\u591a\u8ff4\u5708 \\(amt / \\min(coins)\\) \u6b21\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(amt / \\min(coins))\\) \u3002\u9019\u6bd4\u52d5\u614b\u898f\u5283\u89e3\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n \\times amt)\\) \u5c0f\u4e86\u4e00\u500b\u6578\u91cf\u7d1a\u3002</p> <p>\u7136\u800c\uff0c\u5c0d\u65bc\u67d0\u4e9b\u786c\u5e63\u9762\u503c\u7d44\u5408\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u4e26\u4e0d\u80fd\u627e\u5230\u6700\u512a\u89e3\u3002\u5716 15-2 \u7d66\u51fa\u4e86\u5169\u500b\u793a\u4f8b\u3002</p> <ul> <li>\u6b63\u4f8b \\(coins = [1, 5, 10, 20, 50, 100]\\)\uff1a\u5728\u8a72\u786c\u5e63\u7d44\u5408\u4e0b\uff0c\u7d66\u5b9a\u4efb\u610f \\(amt\\) \uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u90fd\u53ef\u4ee5\u627e\u5230\u6700\u512a\u89e3\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 20, 50]\\)\uff1a\u5047\u8a2d \\(amt = 60\\) \uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 10\\) \u7684\u514c\u63db\u7d44\u5408\uff0c\u5171\u8a08 \\(11\\) \u679a\u786c\u5e63\uff0c\u4f46\u52d5\u614b\u898f\u5283\u53ef\u4ee5\u627e\u5230\u6700\u512a\u89e3 \\(20 + 20 + 20\\) \uff0c\u50c5\u9700 \\(3\\) \u679a\u786c\u5e63\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 49, 50]\\)\uff1a\u5047\u8a2d \\(amt = 98\\) \uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 48\\) \u7684\u514c\u63db\u7d44\u5408\uff0c\u5171\u8a08 \\(49\\) \u679a\u786c\u5e63\uff0c\u4f46\u52d5\u614b\u898f\u5283\u53ef\u4ee5\u627e\u5230\u6700\u512a\u89e3 \\(49 + 49\\) \uff0c\u50c5\u9700 \\(2\\) \u679a\u786c\u5e63\u3002</li> </ul> <p></p> <p> \u5716 15-2 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u7121\u6cd5\u627e\u51fa\u6700\u512a\u89e3\u7684\u793a\u4f8b </p> <p>\u4e5f\u5c31\u662f\u8aaa\uff0c\u5c0d\u65bc\u96f6\u9322\u514c\u63db\u554f\u984c\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7121\u6cd5\u4fdd\u8b49\u627e\u5230\u5168\u57df\u6027\u6700\u512a\u89e3\uff0c\u4e26\u4e14\u6709\u53ef\u80fd\u627e\u5230\u975e\u5e38\u5dee\u7684\u89e3\u3002\u5b83\u66f4\u9069\u5408\u7528\u52d5\u614b\u898f\u5283\u89e3\u6c7a\u3002</p> <p>\u4e00\u822c\u60c5\u6cc1\u4e0b\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u9069\u7528\u60c5\u6cc1\u5206\u4ee5\u4e0b\u5169\u7a2e\u3002</p> <ol> <li>\u53ef\u4ee5\u4fdd\u8b49\u627e\u5230\u6700\u512a\u89e3\uff1a\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\u5f80\u5f80\u662f\u6700\u512a\u9078\u64c7\uff0c\u56e0\u70ba\u5b83\u5f80\u5f80\u6bd4\u56de\u6eaf\u3001\u52d5\u614b\u898f\u5283\u66f4\u9ad8\u6548\u3002</li> <li>\u53ef\u4ee5\u627e\u5230\u8fd1\u4f3c\u6700\u512a\u89e3\uff1a\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\u4e5f\u662f\u53ef\u7528\u7684\u3002\u5c0d\u65bc\u5f88\u591a\u8907\u96dc\u554f\u984c\u4f86\u8aaa\uff0c\u5c0b\u627e\u5168\u57df\u6027\u6700\u512a\u89e3\u975e\u5e38\u56f0\u96e3\uff0c\u80fd\u4ee5\u8f03\u9ad8\u6548\u7387\u627e\u5230\u6b21\u512a\u89e3\u4e5f\u662f\u975e\u5e38\u4e0d\u932f\u7684\u3002</li> </ol>"},{"location":"chapter_greedy/greedy_algorithm/#1512","title":"15.1.2 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u7279\u6027","text":"<p>\u90a3\u9ebc\u554f\u984c\u4f86\u4e86\uff0c\u4ec0\u9ebc\u6a23\u7684\u554f\u984c\u9069\u5408\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6c42\u89e3\u5462\uff1f\u6216\u8005\u8aaa\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5728\u4ec0\u9ebc\u60c5\u6cc1\u4e0b\u53ef\u4ee5\u4fdd\u8b49\u627e\u5230\u6700\u512a\u89e3\uff1f</p> <p>\u76f8\u8f03\u65bc\u52d5\u614b\u898f\u5283\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u4f7f\u7528\u689d\u4ef6\u66f4\u52a0\u82db\u523b\uff0c\u5176\u4e3b\u8981\u95dc\u6ce8\u554f\u984c\u7684\u5169\u500b\u6027\u8cea\u3002</p> <ul> <li>\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\uff1a\u53ea\u6709\u7576\u5c40\u90e8\u6700\u512a\u9078\u64c7\u59cb\u7d42\u53ef\u4ee5\u5c0e\u81f4\u5168\u57df\u6027\u6700\u512a\u89e3\u6642\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u624d\u80fd\u4fdd\u8b49\u5f97\u5230\u6700\u512a\u89e3\u3002</li> <li>\u6700\u512a\u5b50\u7d50\u69cb\uff1a\u539f\u554f\u984c\u7684\u6700\u512a\u89e3\u5305\u542b\u5b50\u554f\u984c\u7684\u6700\u512a\u89e3\u3002</li> </ul> <p>\u6700\u512a\u5b50\u7d50\u69cb\u5df2\u7d93\u5728\u201c\u52d5\u614b\u898f\u5283\u201d\u7ae0\u7bc0\u4e2d\u4ecb\u7d39\u904e\uff0c\u9019\u88e1\u4e0d\u518d\u8d05\u8ff0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u554f\u984c\u7684\u6700\u512a\u5b50\u7d50\u69cb\u4e26\u4e0d\u660e\u986f\uff0c\u4f46\u4ecd\u7136\u53ef\u4f7f\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u89e3\u6c7a\u3002</p> <p>\u6211\u5011\u4e3b\u8981\u63a2\u7a76\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u7684\u5224\u65b7\u65b9\u6cd5\u3002\u96d6\u7136\u5b83\u7684\u63cf\u8ff0\u770b\u4e0a\u53bb\u6bd4\u8f03\u7c21\u55ae\uff0c\u4f46\u5be6\u969b\u4e0a\u5c0d\u65bc\u8a31\u591a\u554f\u984c\uff0c\u8b49\u660e\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u4e26\u975e\u6613\u4e8b\u3002</p> <p>\u4f8b\u5982\u96f6\u9322\u514c\u63db\u554f\u984c\uff0c\u6211\u5011\u96d6\u7136\u80fd\u5920\u5bb9\u6613\u5730\u8209\u51fa\u53cd\u4f8b\uff0c\u5c0d\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u9032\u884c\u8b49\u507d\uff0c\u4f46\u8b49\u5be6\u7684\u96e3\u5ea6\u8f03\u5927\u3002\u5982\u679c\u554f\uff1a\u6eff\u8db3\u4ec0\u9ebc\u689d\u4ef6\u7684\u786c\u5e63\u7d44\u5408\u53ef\u4ee5\u4f7f\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6c42\u89e3\uff1f\u6211\u5011\u5f80\u5f80\u53ea\u80fd\u6191\u85c9\u76f4\u89ba\u6216\u8209\u4f8b\u5b50\u4f86\u7d66\u51fa\u4e00\u500b\u6a21\u7a1c\u5169\u53ef\u7684\u7b54\u6848\uff0c\u800c\u96e3\u4ee5\u7d66\u51fa\u56b4\u8b39\u7684\u6578\u5b78\u8b49\u660e\u3002</p> <p>Quote</p> <p>\u6709\u4e00\u7bc7\u8ad6\u6587\u7d66\u51fa\u4e86\u4e00\u500b \\(O(n^3)\\) \u6642\u9593\u8907\u96dc\u5ea6\u7684\u6f14\u7b97\u6cd5\uff0c\u7528\u65bc\u5224\u65b7\u4e00\u500b\u786c\u5e63\u7d44\u5408\u80fd\u5426\u4f7f\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u627e\u51fa\u4efb\u610f\u91d1\u984d\u7684\u6700\u512a\u89e3\u3002</p> <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>"},{"location":"chapter_greedy/greedy_algorithm/#1513","title":"15.1.3 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u89e3\u984c\u6b65\u9a5f","text":"<p>\u8caa\u5a6a\u554f\u984c\u7684\u89e3\u6c7a\u6d41\u7a0b\u5927\u9ad4\u53ef\u5206\u70ba\u4ee5\u4e0b\u4e09\u6b65\u3002</p> <ol> <li>\u554f\u984c\u5206\u6790\uff1a\u68b3\u7406\u8207\u7406\u89e3\u554f\u984c\u7279\u6027\uff0c\u5305\u62ec\u72c0\u614b\u5b9a\u7fa9\u3001\u6700\u4f73\u5316\u76ee\u6a19\u548c\u7d04\u675f\u689d\u4ef6\u7b49\u3002\u9019\u4e00\u6b65\u5728\u56de\u6eaf\u548c\u52d5\u614b\u898f\u5283\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</li> <li>\u78ba\u5b9a\u8caa\u5a6a\u7b56\u7565\uff1a\u78ba\u5b9a\u5982\u4f55\u5728\u6bcf\u4e00\u6b65\u4e2d\u505a\u51fa\u8caa\u5a6a\u9078\u64c7\u3002\u9019\u500b\u7b56\u7565\u80fd\u5920\u5728\u6bcf\u4e00\u6b65\u6e1b\u5c0f\u554f\u984c\u7684\u898f\u6a21\uff0c\u4e26\u6700\u7d42\u89e3\u6c7a\u6574\u500b\u554f\u984c\u3002</li> <li>\u6b63\u78ba\u6027\u8b49\u660e\uff1a\u901a\u5e38\u9700\u8981\u8b49\u660e\u554f\u984c\u5177\u6709\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u548c\u6700\u512a\u5b50\u7d50\u69cb\u3002\u9019\u500b\u6b65\u9a5f\u53ef\u80fd\u9700\u8981\u7528\u5230\u6578\u5b78\u8b49\u660e\uff0c\u4f8b\u5982\u6b78\u7d0d\u6cd5\u6216\u53cd\u8b49\u6cd5\u7b49\u3002</li> </ol> <p>\u78ba\u5b9a\u8caa\u5a6a\u7b56\u7565\u662f\u6c42\u89e3\u554f\u984c\u7684\u6838\u5fc3\u6b65\u9a5f\uff0c\u4f46\u5be6\u65bd\u8d77\u4f86\u53ef\u80fd\u4e26\u4e0d\u5bb9\u6613\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e0d\u540c\u554f\u984c\u7684\u8caa\u5a6a\u7b56\u7565\u7684\u5dee\u7570\u8f03\u5927\u3002\u5c0d\u65bc\u8a31\u591a\u554f\u984c\u4f86\u8aaa\uff0c\u8caa\u5a6a\u7b56\u7565\u6bd4\u8f03\u6dfa\u986f\uff0c\u6211\u5011\u900f\u904e\u4e00\u4e9b\u5927\u6982\u7684\u601d\u8003\u8207\u5617\u8a66\u5c31\u80fd\u5f97\u51fa\u3002\u800c\u5c0d\u65bc\u4e00\u4e9b\u8907\u96dc\u554f\u984c\uff0c\u8caa\u5a6a\u7b56\u7565\u53ef\u80fd\u975e\u5e38\u96b1\u853d\uff0c\u9019\u7a2e\u60c5\u6cc1\u5c31\u975e\u5e38\u8003\u9a57\u500b\u4eba\u7684\u89e3\u984c\u7d93\u9a57\u8207\u6f14\u7b97\u6cd5\u80fd\u529b\u4e86\u3002</li> <li>\u67d0\u4e9b\u8caa\u5a6a\u7b56\u7565\u5177\u6709\u8f03\u5f37\u7684\u8ff7\u60d1\u6027\u3002\u7576\u6211\u5011\u6eff\u61f7\u4fe1\u5fc3\u8a2d\u8a08\u597d\u8caa\u5a6a\u7b56\u7565\uff0c\u5beb\u51fa\u89e3\u984c\u7a0b\u5f0f\u78bc\u4e26\u63d0\u4ea4\u57f7\u884c\uff0c\u5f88\u53ef\u80fd\u767c\u73fe\u90e8\u5206\u6e2c\u8a66\u6a23\u4f8b\u7121\u6cd5\u900f\u904e\u3002\u9019\u662f\u56e0\u70ba\u8a2d\u8a08\u7684\u8caa\u5a6a\u7b56\u7565\u53ea\u662f\u201c\u90e8\u5206\u6b63\u78ba\u201d\u7684\uff0c\u4e0a\u6587\u4ecb\u7d39\u7684\u96f6\u9322\u514c\u63db\u5c31\u662f\u4e00\u500b\u5178\u578b\u6848\u4f8b\u3002</li> </ul> <p>\u70ba\u4e86\u4fdd\u8b49\u6b63\u78ba\u6027\uff0c\u6211\u5011\u61c9\u8a72\u5c0d\u8caa\u5a6a\u7b56\u7565\u9032\u884c\u56b4\u8b39\u7684\u6578\u5b78\u8b49\u660e\uff0c\u901a\u5e38\u9700\u8981\u7528\u5230\u53cd\u8b49\u6cd5\u6216\u6578\u5b78\u6b78\u7d0d\u6cd5\u3002</p> <p>\u7136\u800c\uff0c\u6b63\u78ba\u6027\u8b49\u660e\u4e5f\u5f88\u53ef\u80fd\u4e0d\u662f\u4e00\u4ef6\u6613\u4e8b\u3002\u5982\u82e5\u6c92\u6709\u982d\u7dd2\uff0c\u6211\u5011\u901a\u5e38\u6703\u9078\u64c7\u9762\u5411\u6e2c\u8a66\u7528\u4f8b\u9032\u884c\u7a0b\u5f0f\u78bc\u9664\u932f\uff0c\u4e00\u6b65\u6b65\u4fee\u6539\u8207\u9a57\u8b49\u8caa\u5a6a\u7b56\u7565\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1514","title":"15.1.4 \u00a0 \u8caa\u5a6a\u6f14\u7b97\u6cd5\u5178\u578b\u4f8b\u984c","text":"<p>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5e38\u5e38\u61c9\u7528\u5728\u6eff\u8db3\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u548c\u6700\u512a\u5b50\u7d50\u69cb\u7684\u6700\u4f73\u5316\u554f\u984c\u4e2d\uff0c\u4ee5\u4e0b\u5217\u8209\u4e86\u4e00\u4e9b\u5178\u578b\u7684\u8caa\u5a6a\u6f14\u7b97\u6cd5\u554f\u984c\u3002</p> <ul> <li>\u786c\u5e63\u627e\u96f6\u554f\u984c\uff1a\u5728\u67d0\u4e9b\u786c\u5e63\u7d44\u5408\u4e0b\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7e3d\u662f\u53ef\u4ee5\u5f97\u5230\u6700\u512a\u89e3\u3002</li> <li>\u5340\u9593\u6392\u7a0b\u554f\u984c\uff1a\u5047\u8a2d\u4f60\u6709\u4e00\u4e9b\u4efb\u52d9\uff0c\u6bcf\u500b\u4efb\u52d9\u5728\u4e00\u6bb5\u6642\u9593\u5167\u9032\u884c\uff0c\u4f60\u7684\u76ee\u6a19\u662f\u5b8c\u6210\u5118\u53ef\u80fd\u591a\u7684\u4efb\u52d9\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9078\u64c7\u7d50\u675f\u6642\u9593\u6700\u65e9\u7684\u4efb\u52d9\uff0c\u90a3\u9ebc\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u512a\u89e3\u3002</li> <li>\u5206\u6578\u80cc\u5305\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u7d44\u7269\u54c1\u548c\u4e00\u500b\u8f09\u91cd\u91cf\uff0c\u4f60\u7684\u76ee\u6a19\u662f\u9078\u64c7\u4e00\u7d44\u7269\u54c1\uff0c\u4f7f\u5f97\u7e3d\u91cd\u91cf\u4e0d\u8d85\u904e\u8f09\u91cd\u91cf\uff0c\u4e14\u7e3d\u50f9\u503c\u6700\u5927\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9078\u64c7\u50f9\u6548\u6bd4\u6700\u9ad8\uff08\u50f9\u503c / \u91cd\u91cf\uff09\u7684\u7269\u54c1\uff0c\u90a3\u9ebc\u8caa\u5a6a\u6f14\u7b97\u6cd5\u5728\u4e00\u4e9b\u60c5\u6cc1\u4e0b\u53ef\u4ee5\u5f97\u5230\u6700\u512a\u89e3\u3002</li> <li>\u80a1\u7968\u8cb7\u8ce3\u554f\u984c\uff1a\u7d66\u5b9a\u4e00\u7d44\u80a1\u7968\u7684\u6b77\u53f2\u50f9\u683c\uff0c\u4f60\u53ef\u4ee5\u9032\u884c\u591a\u6b21\u8cb7\u8ce3\uff0c\u4f46\u5982\u679c\u4f60\u5df2\u7d93\u6301\u6709\u80a1\u7968\uff0c\u90a3\u9ebc\u5728\u8ce3\u51fa\u4e4b\u524d\u4e0d\u80fd\u518d\u8cb7\uff0c\u76ee\u6a19\u662f\u7372\u53d6\u6700\u5927\u5229\u6f64\u3002</li> <li>\u970d\u592b\u66fc\u7de8\u78bc\uff1a\u970d\u592b\u66fc\u7de8\u78bc\u662f\u4e00\u7a2e\u7528\u65bc\u7121\u640d\u8cc7\u6599\u58d3\u7e2e\u7684\u8caa\u5a6a\u6f14\u7b97\u6cd5\u3002\u900f\u904e\u69cb\u5efa\u970d\u592b\u66fc\u6a39\uff0c\u6bcf\u6b21\u9078\u64c7\u51fa\u73fe\u983b\u7387\u6700\u4f4e\u7684\u5169\u500b\u7bc0\u9ede\u5408\u4f75\uff0c\u6700\u5f8c\u5f97\u5230\u7684\u970d\u592b\u66fc\u6a39\u7684\u5e36\u6b0a\u8def\u5f91\u9577\u5ea6\uff08\u7de8\u78bc\u9577\u5ea6\uff09\u6700\u5c0f\u3002</li> <li>Dijkstra \u6f14\u7b97\u6cd5\uff1a\u5b83\u662f\u4e00\u7a2e\u89e3\u6c7a\u7d66\u5b9a\u6e90\u9802\u9ede\u5230\u5176\u9918\u5404\u9802\u9ede\u7684\u6700\u77ed\u8def\u5f91\u554f\u984c\u7684\u8caa\u5a6a\u6f14\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_greedy/max_capacity_problem/","title":"15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u554f\u984c","text":"<p>Question</p> <p>\u8f38\u5165\u4e00\u500b\u9663\u5217 \\(ht\\) \uff0c\u5176\u4e2d\u7684\u6bcf\u500b\u5143\u7d20\u4ee3\u8868\u4e00\u500b\u5782\u76f4\u9694\u677f\u7684\u9ad8\u5ea6\u3002\u9663\u5217\u4e2d\u7684\u4efb\u610f\u5169\u500b\u9694\u677f\uff0c\u4ee5\u53ca\u5b83\u5011\u4e4b\u9593\u7684\u7a7a\u9593\u53ef\u4ee5\u7d44\u6210\u4e00\u500b\u5bb9\u5668\u3002</p> <p>\u5bb9\u5668\u7684\u5bb9\u91cf\u7b49\u65bc\u9ad8\u5ea6\u548c\u5bec\u5ea6\u7684\u4e58\u7a4d\uff08\u9762\u7a4d\uff09\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u8f03\u77ed\u7684\u9694\u677f\u6c7a\u5b9a\uff0c\u5bec\u5ea6\u662f\u5169\u500b\u9694\u677f\u7684\u9663\u5217\u7d22\u5f15\u4e4b\u5dee\u3002</p> <p>\u8acb\u5728\u9663\u5217\u4e2d\u9078\u64c7\u5169\u500b\u9694\u677f\uff0c\u4f7f\u5f97\u7d44\u6210\u7684\u5bb9\u5668\u7684\u5bb9\u91cf\u6700\u5927\uff0c\u8fd4\u56de\u6700\u5927\u5bb9\u91cf\u3002\u793a\u4f8b\u5982\u5716 15-7 \u6240\u793a\u3002</p> <p></p> <p> \u5716 15-7 \u00a0 \u6700\u5927\u5bb9\u91cf\u554f\u984c\u7684\u793a\u4f8b\u8cc7\u6599 </p> <p>\u5bb9\u5668\u7531\u4efb\u610f\u5169\u500b\u9694\u677f\u570d\u6210\uff0c\u56e0\u6b64\u672c\u984c\u7684\u72c0\u614b\u70ba\u5169\u500b\u9694\u677f\u7684\u7d22\u5f15\uff0c\u8a18\u70ba \\([i, j]\\) \u3002</p> <p>\u6839\u64da\u984c\u610f\uff0c\u5bb9\u91cf\u7b49\u65bc\u9ad8\u5ea6\u4e58\u4ee5\u5bec\u5ea6\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u77ed\u677f\u6c7a\u5b9a\uff0c\u5bec\u5ea6\u662f\u5169\u9694\u677f\u7684\u9663\u5217\u7d22\u5f15\u4e4b\u5dee\u3002\u8a2d\u5bb9\u91cf\u70ba \\(cap[i, j]\\) \uff0c\u5247\u53ef\u5f97\u8a08\u7b97\u516c\u5f0f\uff1a</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>\u8a2d\u9663\u5217\u9577\u5ea6\u70ba \\(n\\) \uff0c\u5169\u500b\u9694\u677f\u7684\u7d44\u5408\u6578\u91cf\uff08\u72c0\u614b\u7e3d\u6578\uff09\u70ba \\(C_n^2 = \\frac{n(n - 1)}{2}\\) \u500b\u3002\u6700\u76f4\u63a5\u5730\uff0c\u6211\u5011\u53ef\u4ee5\u7aae\u8209\u6240\u6709\u72c0\u614b\uff0c\u5f9e\u800c\u6c42\u5f97\u6700\u5927\u5bb9\u91cf\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_greedy/max_capacity_problem/#1","title":"1. \u00a0 \u8caa\u5a6a\u7b56\u7565\u78ba\u5b9a","text":"<p>\u9019\u9053\u984c\u9084\u6709\u66f4\u9ad8\u6548\u7387\u7684\u89e3\u6cd5\u3002\u5982\u5716 15-8 \u6240\u793a\uff0c\u73fe\u9078\u53d6\u4e00\u500b\u72c0\u614b \\([i, j]\\) \uff0c\u5176\u6eff\u8db3\u7d22\u5f15 \\(i &lt; j\\) \u4e14\u9ad8\u5ea6 \\(ht[i] &lt; ht[j]\\) \uff0c\u5373 \\(i\\) \u70ba\u77ed\u677f\u3001\\(j\\) \u70ba\u9577\u677f\u3002</p> <p></p> <p> \u5716 15-8 \u00a0 \u521d\u59cb\u72c0\u614b </p> <p>\u5982\u5716 15-9 \u6240\u793a\uff0c\u82e5\u6b64\u6642\u5c07\u9577\u677f \\(j\\) \u5411\u77ed\u677f \\(i\\) \u9760\u8fd1\uff0c\u5247\u5bb9\u91cf\u4e00\u5b9a\u8b8a\u5c0f\u3002</p> <p>\u9019\u662f\u56e0\u70ba\u5728\u79fb\u52d5\u9577\u677f \\(j\\) \u5f8c\uff0c\u5bec\u5ea6 \\(j-i\\) \u80af\u5b9a\u8b8a\u5c0f\uff1b\u800c\u9ad8\u5ea6\u7531\u77ed\u677f\u6c7a\u5b9a\uff0c\u56e0\u6b64\u9ad8\u5ea6\u53ea\u53ef\u80fd\u4e0d\u8b8a\uff08 \\(i\\) \u4ecd\u70ba\u77ed\u677f\uff09\u6216\u8b8a\u5c0f\uff08\u79fb\u52d5\u5f8c\u7684 \\(j\\) \u6210\u70ba\u77ed\u677f\uff09\u3002</p> <p></p> <p> \u5716 15-9 \u00a0 \u5411\u5167\u79fb\u52d5\u9577\u677f\u5f8c\u7684\u72c0\u614b </p> <p>\u53cd\u5411\u601d\u8003\uff0c\u6211\u5011\u53ea\u6709\u5411\u5167\u6536\u7e2e\u77ed\u677f \\(i\\) \uff0c\u624d\u6709\u53ef\u80fd\u4f7f\u5bb9\u91cf\u8b8a\u5927\u3002\u56e0\u70ba\u96d6\u7136\u5bec\u5ea6\u4e00\u5b9a\u8b8a\u5c0f\uff0c\u4f46\u9ad8\u5ea6\u53ef\u80fd\u6703\u8b8a\u5927\uff08\u79fb\u52d5\u5f8c\u7684\u77ed\u677f \\(i\\) \u53ef\u80fd\u6703\u8b8a\u9577\uff09\u3002\u4f8b\u5982\u5728\u5716 15-10 \u4e2d\uff0c\u79fb\u52d5\u77ed\u677f\u5f8c\u9762\u7a4d\u8b8a\u5927\u3002</p> <p></p> <p> \u5716 15-10 \u00a0 \u5411\u5167\u79fb\u52d5\u77ed\u677f\u5f8c\u7684\u72c0\u614b </p> <p>\u7531\u6b64\u4fbf\u53ef\u63a8\u51fa\u672c\u984c\u7684\u8caa\u5a6a\u7b56\u7565\uff1a\u521d\u59cb\u5316\u5169\u6307\u6a19\uff0c\u4f7f\u5176\u5206\u5217\u5bb9\u5668\u5169\u7aef\uff0c\u6bcf\u8f2a\u5411\u5167\u6536\u7e2e\u77ed\u677f\u5c0d\u61c9\u7684\u6307\u6a19\uff0c\u76f4\u81f3\u5169\u6307\u6a19\u76f8\u9047\u3002</p> <p>\u5716 15-11 \u5c55\u793a\u4e86\u8caa\u5a6a\u7b56\u7565\u7684\u57f7\u884c\u904e\u7a0b\u3002</p> <ol> <li>\u521d\u59cb\u72c0\u614b\u4e0b\uff0c\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u5206\u5217\u9663\u5217\u5169\u7aef\u3002</li> <li>\u8a08\u7b97\u7576\u524d\u72c0\u614b\u7684\u5bb9\u91cf \\(cap[i, j]\\) \uff0c\u4e26\u66f4\u65b0\u6700\u5927\u5bb9\u91cf\u3002</li> <li>\u6bd4\u8f03\u677f \\(i\\) \u548c\u677f \\(j\\) \u7684\u9ad8\u5ea6\uff0c\u4e26\u5c07\u77ed\u677f\u5411\u5167\u79fb\u52d5\u4e00\u683c\u3002</li> <li>\u8ff4\u5708\u57f7\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u76f4\u81f3 \\(i\\) \u548c \\(j\\) \u76f8\u9047\u6642\u7d50\u675f\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 15-11 \u00a0 \u6700\u5927\u5bb9\u91cf\u554f\u984c\u7684\u8caa\u5a6a\u904e\u7a0b </p>"},{"location":"chapter_greedy/max_capacity_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u7a0b\u5f0f\u78bc\u8ff4\u5708\u6700\u591a \\(n\\) \u8f2a\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002</p> <p>\u8b8a\u6578 \\(i\\)\u3001\\(j\\)\u3001\\(res\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"\u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a\"\"\"\n    # \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    i, j = 0, len(ht) - 1\n    # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    res = 0\n    # \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while i &lt; j:\n        # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    int i = 0, j = ht.size() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    int res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint maxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    int i = 0, j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    int res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint MaxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    int i = 0, j = ht.Length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    int res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfunc maxCapacity(ht []int) int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    i, j := 0, len(ht)-1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    res := 0\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    for i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    var i = ht.startIndex, j = ht.endIndex - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    var res = 0\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfunction maxCapacity(ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    let res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfunction maxCapacity(ht: number[]): number {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    let res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint maxCapacity(List&lt;int&gt; ht) {\n  // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n  int i = 0, j = ht.length - 1;\n  // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n  int res = 0;\n  // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n  while (i &lt; j) {\n    // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    let mut res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nint maxCapacity(int ht[], int htLength) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    int i = 0;\n    int j = htLength - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    int res = 0;\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.kt<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a */\nfun maxCapacity(ht: IntArray): Int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n    var i = 0\n    var j = ht.size - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n    var res = 0\n    // \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        val cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // \u5411\u5167\u79fb\u52d5\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.rb<pre><code>### \u6700\u5927\u5bb9\u91cf\uff1a\u8caa\u5a6a ###\ndef max_capacity(ht)\n  # \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u9663\u5217\u5169\u7aef\n  i, j = 0, ht.length - 1\n  # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u70ba 0\n  res = 0\n\n  # \u8ff4\u5708\u8caa\u5a6a\u9078\u64c7\uff0c\u76f4\u81f3\u5169\u677f\u76f8\u9047\n  while i &lt; j\n    # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n    cap = [ht[i], ht[j]].min * (j - i)\n    res = [res, cap].max\n    # \u5411\u5167\u79fb\u52d5\u77ed\u677f\n    if ht[i] &lt; ht[j]\n      i += 1\n    else\n      j -= 1\n    end\n  end\n\n  res\nend\n</code></pre> max_capacity.zig<pre><code>[class]{}-[func]{maxCapacity}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_greedy/max_capacity_problem/#3","title":"3. \u00a0 \u6b63\u78ba\u6027\u8b49\u660e","text":"<p>\u4e4b\u6240\u4ee5\u8caa\u5a6a\u6bd4\u7aae\u8209\u66f4\u5feb\uff0c\u662f\u56e0\u70ba\u6bcf\u8f2a\u7684\u8caa\u5a6a\u9078\u64c7\u90fd\u6703\u201c\u8df3\u904e\u201d\u4e00\u4e9b\u72c0\u614b\u3002</p> <p>\u6bd4\u5982\u5728\u72c0\u614b \\(cap[i, j]\\) \u4e0b\uff0c\\(i\\) \u70ba\u77ed\u677f\u3001\\(j\\) \u70ba\u9577\u677f\u3002\u82e5\u8caa\u5a6a\u5730\u5c07\u77ed\u677f \\(i\\) \u5411\u5167\u79fb\u52d5\u4e00\u683c\uff0c\u6703\u5c0e\u81f4\u5716 15-12 \u6240\u793a\u7684\u72c0\u614b\u88ab\u201c\u8df3\u904e\u201d\u3002\u9019\u610f\u5473\u8457\u4e4b\u5f8c\u7121\u6cd5\u9a57\u8b49\u9019\u4e9b\u72c0\u614b\u7684\u5bb9\u91cf\u5927\u5c0f\u3002</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> \u5716 15-12 \u00a0 \u79fb\u52d5\u77ed\u677f\u5c0e\u81f4\u88ab\u8df3\u904e\u7684\u72c0\u614b </p> <p>\u89c0\u5bdf\u767c\u73fe\uff0c\u9019\u4e9b\u88ab\u8df3\u904e\u7684\u72c0\u614b\u5be6\u969b\u4e0a\u5c31\u662f\u5c07\u9577\u677f \\(j\\) \u5411\u5167\u79fb\u52d5\u7684\u6240\u6709\u72c0\u614b\u3002\u524d\u9762\u6211\u5011\u5df2\u7d93\u8b49\u660e\u5167\u79fb\u9577\u677f\u4e00\u5b9a\u6703\u5c0e\u81f4\u5bb9\u91cf\u8b8a\u5c0f\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u88ab\u8df3\u904e\u7684\u72c0\u614b\u90fd\u4e0d\u53ef\u80fd\u662f\u6700\u512a\u89e3\uff0c\u8df3\u904e\u5b83\u5011\u4e0d\u6703\u5c0e\u81f4\u932f\u904e\u6700\u512a\u89e3\u3002</p> <p>\u4ee5\u4e0a\u5206\u6790\u8aaa\u660e\uff0c\u79fb\u52d5\u77ed\u677f\u7684\u64cd\u4f5c\u662f\u201c\u5b89\u5168\u201d\u7684\uff0c\u8caa\u5a6a\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/","title":"15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u7a4d\u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u6b63\u6574\u6578 \\(n\\) \uff0c\u5c07\u5176\u5207\u5206\u70ba\u81f3\u5c11\u5169\u500b\u6b63\u6574\u6578\u7684\u548c\uff0c\u6c42\u5207\u5206\u5f8c\u6240\u6709\u6574\u6578\u7684\u4e58\u7a4d\u6700\u5927\u662f\u591a\u5c11\uff0c\u5982\u5716 15-13 \u6240\u793a\u3002</p> <p></p> <p> \u5716 15-13 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u7a4d\u7684\u554f\u984c\u5b9a\u7fa9 </p> <p>\u5047\u8a2d\u6211\u5011\u5c07 \\(n\\) \u5207\u5206\u70ba \\(m\\) \u500b\u6574\u6578\u56e0\u5b50\uff0c\u5176\u4e2d\u7b2c \\(i\\) \u500b\u56e0\u5b50\u8a18\u70ba \\(n_i\\) \uff0c\u5373</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>\u672c\u984c\u7684\u76ee\u6a19\u662f\u6c42\u5f97\u6240\u6709\u6574\u6578\u56e0\u5b50\u7684\u6700\u5927\u4e58\u7a4d\uff0c\u5373</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>\u6211\u5011\u9700\u8981\u601d\u8003\u7684\u662f\uff1a\u5207\u5206\u6578\u91cf \\(m\\) \u61c9\u8a72\u591a\u5927\uff0c\u6bcf\u500b \\(n_i\\) \u61c9\u8a72\u662f\u591a\u5c11\uff1f</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#1","title":"1. \u00a0 \u8caa\u5a6a\u7b56\u7565\u78ba\u5b9a","text":"<p>\u6839\u64da\u7d93\u9a57\uff0c\u5169\u500b\u6574\u6578\u7684\u4e58\u7a4d\u5f80\u5f80\u6bd4\u5b83\u5011\u7684\u52a0\u548c\u66f4\u5927\u3002\u5047\u8a2d\u5f9e \\(n\\) \u4e2d\u5206\u51fa\u4e00\u500b\u56e0\u5b50 \\(2\\) \uff0c\u5247\u5b83\u5011\u7684\u4e58\u7a4d\u70ba \\(2(n-2)\\) \u3002\u6211\u5011\u5c07\u8a72\u4e58\u7a4d\u8207 \\(n\\) \u4f5c\u6bd4\u8f03\uff1a</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>\u5982\u5716 15-14 \u6240\u793a\uff0c\u7576 \\(n \\geq 4\\) \u6642\uff0c\u5207\u5206\u51fa\u4e00\u500b \\(2\\) \u5f8c\u4e58\u7a4d\u6703\u8b8a\u5927\uff0c\u9019\u8aaa\u660e\u5927\u65bc\u7b49\u65bc \\(4\\) \u7684\u6574\u6578\u90fd\u61c9\u8a72\u88ab\u5207\u5206\u3002</p> <p>\u8caa\u5a6a\u7b56\u7565\u4e00\uff1a\u5982\u679c\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b \\(\\geq 4\\) \u7684\u56e0\u5b50\uff0c\u90a3\u9ebc\u5b83\u5c31\u61c9\u8a72\u88ab\u7e7c\u7e8c\u5207\u5206\u3002\u6700\u7d42\u7684\u5207\u5206\u65b9\u6848\u53ea\u61c9\u51fa\u73fe \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9019\u4e09\u7a2e\u56e0\u5b50\u3002</p> <p></p> <p> \u5716 15-14 \u00a0 \u5207\u5206\u5c0e\u81f4\u4e58\u7a4d\u8b8a\u5927 </p> <p>\u63a5\u4e0b\u4f86\u601d\u8003\u54ea\u500b\u56e0\u5b50\u662f\u6700\u512a\u7684\u3002\u5728 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9019\u4e09\u500b\u56e0\u5b50\u4e2d\uff0c\u986f\u7136 \\(1\\) \u662f\u6700\u5dee\u7684\uff0c\u56e0\u70ba \\(1 \\times (n-1) &lt; n\\) \u6046\u6210\u7acb\uff0c\u5373\u5207\u5206\u51fa \\(1\\) \u53cd\u800c\u6703\u5c0e\u81f4\u4e58\u7a4d\u6e1b\u5c0f\u3002</p> <p>\u5982\u5716 15-15 \u6240\u793a\uff0c\u7576 \\(n = 6\\) \u6642\uff0c\u6709 \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\) \u3002\u9019\u610f\u5473\u8457\u5207\u5206\u51fa \\(3\\) \u6bd4\u5207\u5206\u51fa \\(2\\) \u66f4\u512a\u3002</p> <p>\u8caa\u5a6a\u7b56\u7565\u4e8c\uff1a\u5728\u5207\u5206\u65b9\u6848\u4e2d\uff0c\u6700\u591a\u96bb\u61c9\u5b58\u5728\u5169\u500b \\(2\\) \u3002\u56e0\u70ba\u4e09\u500b \\(2\\) \u7e3d\u662f\u53ef\u4ee5\u66ff\u63db\u70ba\u5169\u500b \\(3\\) \uff0c\u5f9e\u800c\u7372\u5f97\u66f4\u5927\u7684\u4e58\u7a4d\u3002</p> <p></p> <p> \u5716 15-15 \u00a0 \u6700\u512a\u5207\u5206\u56e0\u5b50 </p> <p>\u7d9c\u4e0a\u6240\u8ff0\uff0c\u53ef\u63a8\u7406\u51fa\u4ee5\u4e0b\u8caa\u5a6a\u7b56\u7565\u3002</p> <ol> <li>\u8f38\u5165\u6574\u6578 \\(n\\) \uff0c\u5f9e\u5176\u4e0d\u65b7\u5730\u5207\u5206\u51fa\u56e0\u5b50 \\(3\\) \uff0c\u76f4\u81f3\u9918\u6578\u70ba \\(0\\)\u3001\\(1\\)\u3001\\(2\\) \u3002</li> <li>\u7576\u9918\u6578\u70ba \\(0\\) \u6642\uff0c\u4ee3\u8868 \\(n\\) \u662f \\(3\\) \u7684\u500d\u6578\uff0c\u56e0\u6b64\u4e0d\u505a\u4efb\u4f55\u8655\u7406\u3002</li> <li>\u7576\u9918\u6578\u70ba \\(2\\) \u6642\uff0c\u4e0d\u7e7c\u7e8c\u5283\u5206\uff0c\u4fdd\u7559\u3002</li> <li>\u7576\u9918\u6578\u70ba \\(1\\) \u6642\uff0c\u7531\u65bc \\(2 \\times 2 &gt; 1 \\times 3\\) \uff0c\u56e0\u6b64\u61c9\u5c07\u6700\u5f8c\u4e00\u500b \\(3\\) \u66ff\u63db\u70ba \\(2\\) \u3002</li> </ol>"},{"location":"chapter_greedy/max_product_cutting_problem/#2","title":"2. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5982\u5716 15-16 \u6240\u793a\uff0c\u6211\u5011\u7121\u9808\u900f\u904e\u8ff4\u5708\u4f86\u5207\u5206\u6574\u6578\uff0c\u800c\u53ef\u4ee5\u5229\u7528\u5411\u4e0b\u6574\u9664\u904b\u7b97\u5f97\u5230 \\(3\\) \u7684\u500b\u6578 \\(a\\) \uff0c\u7528\u53d6\u6a21\u904b\u7b97\u5f97\u5230\u9918\u6578 \\(b\\) \uff0c\u6b64\u6642\u6709\uff1a</p> \\[ n = 3 a + b \\] <p>\u8acb\u6ce8\u610f\uff0c\u5c0d\u65bc \\(n \\leq 3\\) \u7684\u908a\u754c\u60c5\u6cc1\uff0c\u5fc5\u9808\u62c6\u5206\u51fa\u4e00\u500b \\(1\\) \uff0c\u4e58\u7a4d\u70ba \\(1 \\times (n - 1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"\u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a\"\"\"\n    # \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    a, b = n // 3, n % 3\n    if b == 1:\n        # \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return int(math.pow(3, a)) * 2\n    # \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint maxProductCutting(int n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return (int)pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint maxProductCutting(int n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return (int) Math.pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint MaxProductCutting(int n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfunc maxProductCutting(n int) int {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return pow(3, a) * 2\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfunction maxProductCutting(n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfunction maxProductCutting(n: number): number {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint maxProductCutting(int n) {\n  // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return (pow(3, a) * 2).toInt();\n  }\n  // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        3_i32.pow(a as u32) * 2\n    } else {\n        // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nint maxProductCutting(int n) {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return pow(3, a) * 2;\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.kt<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a */\nfun maxProductCutting(n: Int): Int {\n    // \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1)\n    }\n    // \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n    val a = n / 3\n    val b = n % 3\n    if (b == 1) {\n        // \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n        return 3.0.pow((a - 1)).toInt() * 2 * 2\n    }\n    if (b == 2) {\n        // \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n        return 3.0.pow(a).toInt() * 2 * 2\n    }\n    // \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n    return 3.0.pow(a).toInt()\n}\n</code></pre> max_product_cutting.rb<pre><code>### \u6700\u5927\u5207\u5206\u4e58\u7a4d\uff1a\u8caa\u5a6a ###\ndef max_product_cutting(n)\n  # \u7576 n &lt;= 3 \u6642\uff0c\u5fc5\u9808\u5207\u5206\u51fa\u4e00\u500b 1\n  return 1 * (n - 1) if n &lt;= 3\n  # \u8caa\u5a6a\u5730\u5207\u5206\u51fa 3 \uff0ca \u70ba 3 \u7684\u500b\u6578\uff0cb \u70ba\u9918\u6578\n  a, b = n / 3, n % 3\n  # \u7576\u9918\u6578\u70ba 1 \u6642\uff0c\u5c07\u4e00\u5c0d 1 * 3 \u8f49\u5316\u70ba 2 * 2\n  return (3.pow(a - 1) * 2 * 2).to_i if b == 1\n  # \u7576\u9918\u6578\u70ba 2 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n  return (3.pow(a) * 2).to_i if b == 2\n  # \u7576\u9918\u6578\u70ba 0 \u6642\uff0c\u4e0d\u505a\u8655\u7406\n  3.pow(a).to_i\nend\n</code></pre> max_product_cutting.zig<pre><code>[class]{}-[func]{maxProductCutting}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p></p> <p> \u5716 15-16 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u7a4d\u7684\u8a08\u7b97\u65b9\u6cd5 </p> <p>\u6642\u9593\u8907\u96dc\u5ea6\u53d6\u6c7a\u65bc\u7a0b\u5f0f\u8a9e\u8a00\u7684\u51aa\u904b\u7b97\u7684\u5be6\u73fe\u65b9\u6cd5\u3002\u4ee5 Python \u70ba\u4f8b\uff0c\u5e38\u7528\u7684\u51aa\u8a08\u7b97\u51fd\u5f0f\u6709\u4e09\u7a2e\u3002</p> <ul> <li>\u904b\u7b97\u5b50 <code>**</code> \u548c\u51fd\u5f0f <code>pow()</code> \u7684\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(\\log\u2061 a)\\) \u3002</li> <li>\u51fd\u5f0f <code>math.pow()</code> \u5167\u90e8\u547c\u53eb C \u8a9e\u8a00\u5eab\u7684 <code>pow()</code> \u51fd\u5f0f\uff0c\u5176\u57f7\u884c\u6d6e\u9ede\u53d6\u51aa\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</li> </ul> <p>\u8b8a\u6578 \\(a\\) \u548c \\(b\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#3","title":"3. \u00a0 \u6b63\u78ba\u6027\u8b49\u660e","text":"<p>\u4f7f\u7528\u53cd\u8b49\u6cd5\uff0c\u53ea\u5206\u6790 \\(n \\geq 3\\) \u7684\u60c5\u6cc1\u3002</p> <ol> <li>\u6240\u6709\u56e0\u5b50 \\(\\leq 3\\) \uff1a\u5047\u8a2d\u6700\u512a\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728 \\(\\geq 4\\) \u7684\u56e0\u5b50 \\(x\\) \uff0c\u90a3\u9ebc\u4e00\u5b9a\u53ef\u4ee5\u5c07\u5176\u7e7c\u7e8c\u5283\u5206\u70ba \\(2(x-2)\\) \uff0c\u5f9e\u800c\u7372\u5f97\u66f4\u5927\u7684\u4e58\u7a4d\u3002\u9019\u8207\u5047\u8a2d\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u4e0d\u5305\u542b \\(1\\) \uff1a\u5047\u8a2d\u6700\u512a\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728\u4e00\u500b\u56e0\u5b50 \\(1\\) \uff0c\u90a3\u9ebc\u5b83\u4e00\u5b9a\u53ef\u4ee5\u5408\u4f75\u5165\u53e6\u5916\u4e00\u500b\u56e0\u5b50\u4e2d\uff0c\u4ee5\u7372\u5f97\u66f4\u5927\u7684\u4e58\u7a4d\u3002\u9019\u8207\u5047\u8a2d\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u6700\u591a\u5305\u542b\u5169\u500b \\(2\\) \uff1a\u5047\u8a2d\u6700\u512a\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b\u4e09\u500b \\(2\\) \uff0c\u90a3\u9ebc\u4e00\u5b9a\u53ef\u4ee5\u66ff\u63db\u70ba\u5169\u500b \\(3\\) \uff0c\u4e58\u7a4d\u66f4\u5927\u3002\u9019\u8207\u5047\u8a2d\u77db\u76fe\u3002</li> </ol>"},{"location":"chapter_greedy/summary/","title":"15.5 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u901a\u5e38\u7528\u65bc\u89e3\u6c7a\u6700\u6700\u4f73\u5316\u554f\u984c\uff0c\u5176\u539f\u7406\u662f\u5728\u6bcf\u500b\u6c7a\u7b56\u968e\u6bb5\u90fd\u505a\u51fa\u5340\u57df\u6027\u6700\u512a\u7684\u6c7a\u7b56\uff0c\u4ee5\u671f\u7372\u5f97\u5168\u57df\u6027\u6700\u512a\u89e3\u3002</li> <li>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6703\u8fed\u4ee3\u5730\u505a\u51fa\u4e00\u500b\u53c8\u4e00\u500b\u7684\u8caa\u5a6a\u9078\u64c7\uff0c\u6bcf\u8f2a\u90fd\u5c07\u554f\u984c\u8f49\u5316\u6210\u4e00\u500b\u898f\u6a21\u66f4\u5c0f\u7684\u5b50\u554f\u984c\uff0c\u76f4\u5230\u554f\u984c\u88ab\u89e3\u6c7a\u3002</li> <li>\u8caa\u5a6a\u6f14\u7b97\u6cd5\u4e0d\u50c5\u5be6\u73fe\u7c21\u55ae\uff0c\u9084\u5177\u6709\u5f88\u9ad8\u7684\u89e3\u984c\u6548\u7387\u3002\u76f8\u6bd4\u65bc\u52d5\u614b\u898f\u5283\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u66f4\u4f4e\u3002</li> <li>\u5728\u96f6\u9322\u514c\u63db\u554f\u984c\u4e2d\uff0c\u5c0d\u65bc\u67d0\u4e9b\u786c\u5e63\u7d44\u5408\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8b49\u627e\u5230\u6700\u512a\u89e3\uff1b\u5c0d\u65bc\u53e6\u5916\u4e00\u4e9b\u786c\u5e63\u7d44\u5408\u5247\u4e0d\u7136\uff0c\u8caa\u5a6a\u6f14\u7b97\u6cd5\u53ef\u80fd\u627e\u5230\u5f88\u5dee\u7684\u89e3\u3002</li> <li>\u9069\u5408\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6c42\u89e3\u7684\u554f\u984c\u5177\u6709\u5169\u5927\u6027\u8cea\uff1a\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u548c\u6700\u512a\u5b50\u7d50\u69cb\u3002\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u4ee3\u8868\u8caa\u5a6a\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</li> <li>\u5c0d\u65bc\u67d0\u4e9b\u8907\u96dc\u554f\u984c\uff0c\u8caa\u5a6a\u9078\u64c7\u6027\u8cea\u7684\u8b49\u660e\u4e26\u4e0d\u7c21\u55ae\u3002\u76f8\u5c0d\u4f86\u8aaa\uff0c\u8b49\u507d\u66f4\u52a0\u5bb9\u6613\uff0c\u4f8b\u5982\u96f6\u9322\u514c\u63db\u554f\u984c\u3002</li> <li>\u6c42\u89e3\u8caa\u5a6a\u554f\u984c\u4e3b\u8981\u5206\u70ba\u4e09\u6b65\uff1a\u554f\u984c\u5206\u6790\u3001\u78ba\u5b9a\u8caa\u5a6a\u7b56\u7565\u3001\u6b63\u78ba\u6027\u8b49\u660e\u3002\u5176\u4e2d\uff0c\u78ba\u5b9a\u8caa\u5a6a\u7b56\u7565\u662f\u6838\u5fc3\u6b65\u9a5f\uff0c\u6b63\u78ba\u6027\u8b49\u660e\u5f80\u5f80\u662f\u96e3\u9ede\u3002</li> <li>\u5206\u6578\u80cc\u5305\u554f\u984c\u5728 0-1 \u80cc\u5305\u7684\u57fa\u790e\u4e0a\uff0c\u5141\u8a31\u9078\u64c7\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u8caa\u5a6a\u6f14\u7b97\u6cd5\u6c42\u89e3\u3002\u8caa\u5a6a\u7b56\u7565\u7684\u6b63\u78ba\u6027\u53ef\u4ee5\u4f7f\u7528\u53cd\u8b49\u6cd5\u4f86\u8b49\u660e\u3002</li> <li>\u6700\u5927\u5bb9\u91cf\u554f\u984c\u53ef\u4f7f\u7528\u7aae\u8209\u6cd5\u6c42\u89e3\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002\u900f\u904e\u8a2d\u8a08\u8caa\u5a6a\u7b56\u7565\uff0c\u6bcf\u8f2a\u5411\u5167\u79fb\u52d5\u77ed\u677f\uff0c\u53ef\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u5728\u6700\u5927\u5207\u5206\u4e58\u7a4d\u554f\u984c\u4e2d\uff0c\u6211\u5011\u5148\u5f8c\u63a8\u7406\u51fa\u5169\u500b\u8caa\u5a6a\u7b56\u7565\uff1a\\(\\geq 4\\) \u7684\u6574\u6578\u90fd\u61c9\u8a72\u7e7c\u7e8c\u5207\u5206\uff0c\u6700\u512a\u5207\u5206\u56e0\u5b50\u70ba \\(3\\) \u3002\u7a0b\u5f0f\u78bc\u4e2d\u5305\u542b\u51aa\u904b\u7b97\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53d6\u6c7a\u65bc\u51aa\u904b\u7b97\u5be6\u73fe\u65b9\u6cd5\uff0c\u901a\u5e38\u70ba \\(O(1)\\) \u6216 \\(O(\\log n)\\) \u3002</li> </ul>"},{"location":"chapter_hashing/","title":"\u7b2c 6 \u7ae0 \u00a0 \u96dc\u6e4a\u8868","text":"<p>Abstract</p> <p>\u5728\u8a08\u7b97\u6a5f\u4e16\u754c\u4e2d\uff0c\u96dc\u6e4a\u8868\u5982\u540c\u4e00\u4f4d\u8070\u6167\u7684\u5716\u66f8\u7ba1\u7406\u54e1\u3002</p> <p>\u4ed6\u77e5\u9053\u5982\u4f55\u8a08\u7b97\u7d22\u66f8\u865f\uff0c\u5f9e\u800c\u53ef\u4ee5\u5feb\u901f\u627e\u5230\u76ee\u6a19\u5716\u66f8\u3002</p>"},{"location":"chapter_hashing/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>6.1 \u00a0 \u96dc\u6e4a\u8868</li> <li>6.2 \u00a0 \u96dc\u6e4a\u885d\u7a81</li> <li>6.3 \u00a0 \u96dc\u6e4a\u6f14\u7b97\u6cd5</li> <li>6.4 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_hashing/hash_algorithm/","title":"6.3 \u00a0 \u96dc\u6e4a\u6f14\u7b97\u6cd5","text":"<p>\u524d\u5169\u7bc0\u4ecb\u7d39\u4e86\u96dc\u6e4a\u8868\u7684\u5de5\u4f5c\u539f\u7406\u548c\u96dc\u6e4a\u885d\u7a81\u7684\u8655\u7406\u65b9\u6cd5\u3002\u7136\u800c\u7121\u8ad6\u662f\u958b\u653e\u5b9a\u5740\u9084\u662f\u93c8\u5f0f\u4f4d\u5740\uff0c\u5b83\u5011\u53ea\u80fd\u4fdd\u8b49\u96dc\u6e4a\u8868\u53ef\u4ee5\u5728\u767c\u751f\u885d\u7a81\u6642\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u7121\u6cd5\u6e1b\u5c11\u96dc\u6e4a\u885d\u7a81\u7684\u767c\u751f\u3002</p> <p>\u5982\u679c\u96dc\u6e4a\u885d\u7a81\u904e\u65bc\u983b\u7e41\uff0c\u96dc\u6e4a\u8868\u7684\u6548\u80fd\u5247\u6703\u6025\u5287\u52a3\u5316\u3002\u5982\u5716 6-8 \u6240\u793a\uff0c\u5c0d\u65bc\u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868\uff0c\u7406\u60f3\u60c5\u6cc1\u4e0b\u9375\u503c\u5c0d\u5747\u52fb\u5206\u4f48\u5728\u5404\u500b\u6876\u4e2d\uff0c\u9054\u5230\u6700\u4f73\u67e5\u8a62\u6548\u7387\uff1b\u6700\u5dee\u60c5\u6cc1\u4e0b\u6240\u6709\u9375\u503c\u5c0d\u90fd\u5132\u5b58\u5230\u540c\u4e00\u500b\u6876\u4e2d\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</p> <p></p> <p> \u5716 6-8 \u00a0 \u96dc\u6e4a\u885d\u7a81\u7684\u6700\u4f73\u60c5\u6cc1\u8207\u6700\u5dee\u60c5\u6cc1 </p> <p>\u9375\u503c\u5c0d\u7684\u5206\u4f48\u60c5\u6cc1\u7531\u96dc\u6e4a\u51fd\u5f0f\u6c7a\u5b9a\u3002\u56de\u61b6\u96dc\u6e4a\u51fd\u5f0f\u7684\u8a08\u7b97\u6b65\u9a5f\uff0c\u5148\u8a08\u7b97\u96dc\u6e4a\u503c\uff0c\u518d\u5c0d\u9663\u5217\u9577\u5ea6\u53d6\u6a21\uff1a</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u89c0\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u7576\u96dc\u6e4a\u8868\u5bb9\u91cf <code>capacity</code> \u56fa\u5b9a\u6642\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5 <code>hash()</code> \u6c7a\u5b9a\u4e86\u8f38\u51fa\u503c\uff0c\u9032\u800c\u6c7a\u5b9a\u4e86\u9375\u503c\u5c0d\u5728\u96dc\u6e4a\u8868\u4e2d\u7684\u5206\u4f48\u60c5\u6cc1\u3002</p> <p>\u9019\u610f\u5473\u8457\uff0c\u70ba\u4e86\u964d\u4f4e\u96dc\u6e4a\u885d\u7a81\u7684\u767c\u751f\u6a5f\u7387\uff0c\u6211\u5011\u61c9\u7576\u5c07\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u96dc\u6e4a\u6f14\u7b97\u6cd5 <code>hash()</code> \u7684\u8a2d\u8a08\u4e0a\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1 \u00a0 \u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u76ee\u6a19","text":"<p>\u70ba\u4e86\u5be6\u73fe\u201c\u65e2\u5feb\u53c8\u7a69\u201d\u7684\u96dc\u6e4a\u8868\u8cc7\u6599\u7d50\u69cb\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u61c9\u5177\u5099\u4ee5\u4e0b\u7279\u9ede\u3002</p> <ul> <li>\u78ba\u5b9a\u6027\uff1a\u5c0d\u65bc\u76f8\u540c\u7684\u8f38\u5165\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u61c9\u59cb\u7d42\u7522\u751f\u76f8\u540c\u7684\u8f38\u51fa\u3002\u9019\u6a23\u624d\u80fd\u78ba\u4fdd\u96dc\u6e4a\u8868\u662f\u53ef\u9760\u7684\u3002</li> <li>\u6548\u7387\u9ad8\uff1a\u8a08\u7b97\u96dc\u6e4a\u503c\u7684\u904e\u7a0b\u61c9\u8a72\u8db3\u5920\u5feb\u3002\u8a08\u7b97\u958b\u92b7\u8d8a\u5c0f\uff0c\u96dc\u6e4a\u8868\u7684\u5be6\u7528\u6027\u8d8a\u9ad8\u3002</li> <li>\u5747\u52fb\u5206\u4f48\uff1a\u96dc\u6e4a\u6f14\u7b97\u6cd5\u61c9\u4f7f\u5f97\u9375\u503c\u5c0d\u5747\u52fb\u5206\u4f48\u5728\u96dc\u6e4a\u8868\u4e2d\u3002\u5206\u4f48\u8d8a\u5747\u52fb\uff0c\u96dc\u6e4a\u885d\u7a81\u7684\u6a5f\u7387\u5c31\u8d8a\u4f4e\u3002</li> </ul> <p>\u5be6\u969b\u4e0a\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u65bc\u5be6\u73fe\u96dc\u6e4a\u8868\uff0c\u9084\u5ee3\u6cdb\u61c9\u7528\u65bc\u5176\u4ed6\u9818\u57df\u4e2d\u3002</p> <ul> <li>\u5bc6\u78bc\u5132\u5b58\uff1a\u70ba\u4e86\u4fdd\u8b77\u4f7f\u7528\u8005\u5bc6\u78bc\u7684\u5b89\u5168\uff0c\u7cfb\u7d71\u901a\u5e38\u4e0d\u6703\u76f4\u63a5\u5132\u5b58\u4f7f\u7528\u8005\u7684\u660e\u6587\u5bc6\u78bc\uff0c\u800c\u662f\u5132\u5b58\u5bc6\u78bc\u7684\u96dc\u6e4a\u503c\u3002\u7576\u7528\u6236\u8f38\u5165\u5bc6\u78bc\u6642\uff0c\u7cfb\u7d71\u6703\u5c0d\u8f38\u5165\u7684\u5bc6\u78bc\u8a08\u7b97\u96dc\u6e4a\u503c\uff0c\u7136\u5f8c\u8207\u5132\u5b58\u7684\u96dc\u6e4a\u503c\u9032\u884c\u6bd4\u8f03\u3002\u5982\u679c\u5169\u8005\u5339\u914d\uff0c\u90a3\u9ebc\u5bc6\u78bc\u5c31\u88ab\u8996\u70ba\u6b63\u78ba\u3002</li> <li>\u8cc7\u6599\u5b8c\u6574\u6027\u6aa2\u67e5\uff1a\u8cc7\u6599\u50b3\u9001\u65b9\u53ef\u4ee5\u8a08\u7b97\u8cc7\u6599\u7684\u96dc\u6e4a\u503c\u4e26\u5c07\u5176\u4e00\u540c\u50b3\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8a08\u7b97\u63a5\u6536\u5230\u7684\u8cc7\u6599\u7684\u96dc\u6e4a\u503c\uff0c\u4e26\u8207\u63a5\u6536\u5230\u7684\u96dc\u6e4a\u503c\u9032\u884c\u6bd4\u8f03\u3002\u5982\u679c\u5169\u8005\u5339\u914d\uff0c\u90a3\u9ebc\u8cc7\u6599\u5c31\u88ab\u8996\u70ba\u5b8c\u6574\u3002</li> </ul> <p>\u5c0d\u65bc\u5bc6\u78bc\u5b78\u7684\u76f8\u95dc\u61c9\u7528\uff0c\u70ba\u4e86\u9632\u6b62\u5f9e\u96dc\u6e4a\u503c\u63a8\u5c0e\u51fa\u539f\u59cb\u5bc6\u78bc\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u9700\u8981\u5177\u5099\u66f4\u9ad8\u7b49\u7d1a\u7684\u5b89\u5168\u7279\u6027\u3002</p> <ul> <li>\u55ae\u5411\u6027\uff1a\u7121\u6cd5\u900f\u904e\u96dc\u6e4a\u503c\u53cd\u63a8\u51fa\u95dc\u65bc\u8f38\u5165\u8cc7\u6599\u7684\u4efb\u4f55\u8cc7\u8a0a\u3002</li> <li>\u6297\u78b0\u649e\u6027\uff1a\u61c9\u7576\u6975\u96e3\u627e\u5230\u5169\u500b\u4e0d\u540c\u7684\u8f38\u5165\uff0c\u4f7f\u5f97\u5b83\u5011\u7684\u96dc\u6e4a\u503c\u76f8\u540c\u3002</li> <li>\u96ea\u5d29\u6548\u61c9\uff1a\u8f38\u5165\u7684\u5fae\u5c0f\u8b8a\u5316\u61c9\u7576\u5c0e\u81f4\u8f38\u51fa\u7684\u986f\u8457\u4e14\u4e0d\u53ef\u9810\u6e2c\u7684\u8b8a\u5316\u3002</li> </ul> <p>\u8acb\u6ce8\u610f\uff0c\u201c\u5747\u52fb\u5206\u4f48\u201d\u8207\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u5169\u500b\u7368\u7acb\u7684\u6982\u5ff5\uff0c\u6eff\u8db3\u5747\u52fb\u5206\u4f48\u4e0d\u4e00\u5b9a\u6eff\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u96a8\u6a5f\u8f38\u5165 <code>key</code> \u4e0b\uff0c\u96dc\u6e4a\u51fd\u5f0f <code>key % 100</code> \u53ef\u4ee5\u7522\u751f\u5747\u52fb\u5206\u4f48\u7684\u8f38\u51fa\u3002\u7136\u800c\u8a72\u96dc\u6e4a\u6f14\u7b97\u6cd5\u904e\u65bc\u7c21\u55ae\uff0c\u6240\u6709\u5f8c\u5169\u4f4d\u76f8\u7b49\u7684 <code>key</code> \u7684\u8f38\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u5f9e\u96dc\u6e4a\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684 <code>key</code> \uff0c\u5f9e\u800c\u7834\u89e3\u5bc6\u78bc\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2 \u00a0 \u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u8a2d\u8a08","text":"<p>\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u8a2d\u8a08\u662f\u4e00\u500b\u9700\u8981\u8003\u616e\u8a31\u591a\u56e0\u7d20\u7684\u8907\u96dc\u554f\u984c\u3002\u7136\u800c\u5c0d\u65bc\u67d0\u4e9b\u8981\u6c42\u4e0d\u9ad8\u7684\u5834\u666f\uff0c\u6211\u5011\u4e5f\u80fd\u8a2d\u8a08\u4e00\u4e9b\u7c21\u55ae\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\u3002</p> <ul> <li>\u52a0\u6cd5\u96dc\u6e4a\uff1a\u5c0d\u8f38\u5165\u7684\u6bcf\u500b\u5b57\u5143\u7684 ASCII \u78bc\u9032\u884c\u76f8\u52a0\uff0c\u5c07\u5f97\u5230\u7684\u7e3d\u548c\u4f5c\u70ba\u96dc\u6e4a\u503c\u3002</li> <li>\u4e58\u6cd5\u96dc\u6e4a\uff1a\u5229\u7528\u4e58\u6cd5\u7684\u4e0d\u76f8\u95dc\u6027\uff0c\u6bcf\u8f2a\u4e58\u4ee5\u4e00\u500b\u5e38\u6578\uff0c\u5c07\u5404\u500b\u5b57\u5143\u7684 ASCII \u78bc\u7d2f\u7a4d\u5230\u96dc\u6e4a\u503c\u4e2d\u3002</li> <li>\u4e92\u65a5\u6216\u96dc\u6e4a\uff1a\u5c07\u8f38\u5165\u8cc7\u6599\u7684\u6bcf\u500b\u5143\u7d20\u900f\u904e\u4e92\u65a5\u6216\u64cd\u4f5c\u7d2f\u7a4d\u5230\u4e00\u500b\u96dc\u6e4a\u503c\u4e2d\u3002</li> <li>\u65cb\u8f49\u96dc\u6e4a\uff1a\u5c07\u6bcf\u500b\u5b57\u5143\u7684 ASCII \u78bc\u7d2f\u7a4d\u5230\u4e00\u500b\u96dc\u6e4a\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u7a4d\u4e4b\u524d\u90fd\u6703\u5c0d\u96dc\u6e4a\u503c\u9032\u884c\u65cb\u8f49\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"\u52a0\u6cd5\u96dc\u6e4a\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"\u4e58\u6cd5\u96dc\u6e4a\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"\u4e92\u65a5\u6216\u96dc\u6e4a\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"\u65cb\u8f49\u96dc\u6e4a\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* \u52a0\u6cd5\u96dc\u6e4a */\nfun addHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u4e58\u6cd5\u96dc\u6e4a */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u4e92\u65a5\u6216\u96dc\u6e4a */\nfun xorHash(key: String): Int {\n    var hash = 0\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* \u65cb\u8f49\u96dc\u6e4a */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>### \u52a0\u6cd5\u96dc\u6e4a ###\ndef add_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash += c.ord }\n\n  hash % modulus\nend\n\n### \u4e58\u6cd5\u96dc\u6e4a ###\ndef mul_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = 31 * hash + c.ord }\n\n  hash % modulus\nend\n\n### \u4e92\u65a5\u6216\u96dc\u6e4a ###\ndef xor_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash ^= c.ord }\n\n  hash % modulus\nend\n\n### \u65cb\u8f49\u96dc\u6e4a ###\ndef rot_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.ord }\n\n  hash % modulus\nend\n</code></pre> simple_hash.zig<pre><code>[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u89c0\u5bdf\u767c\u73fe\uff0c\u6bcf\u7a2e\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u6700\u5f8c\u4e00\u6b65\u90fd\u662f\u5c0d\u5927\u8cea\u6578 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u78ba\u4fdd\u96dc\u6e4a\u503c\u5728\u5408\u9069\u7684\u7bc4\u570d\u5167\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u70ba\u4ec0\u9ebc\u8981\u5f37\u8abf\u5c0d\u8cea\u6578\u53d6\u6a21\uff0c\u6216\u8005\u8aaa\u5c0d\u5408\u6578\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u9ebc\uff1f\u9019\u662f\u4e00\u500b\u6709\u8da3\u7684\u554f\u984c\u3002</p> <p>\u5148\u4e1f\u64f2\u7d50\u8ad6\uff1a\u4f7f\u7528\u5927\u8cea\u6578\u4f5c\u70ba\u6a21\u6578\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8b49\u96dc\u6e4a\u503c\u7684\u5747\u52fb\u5206\u4f48\u3002\u56e0\u70ba\u8cea\u6578\u4e0d\u8207\u5176\u4ed6\u6578\u5b57\u5b58\u5728\u516c\u7d04\u6578\uff0c\u53ef\u4ee5\u6e1b\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u7522\u751f\u7684\u9031\u671f\u6027\u6a21\u5f0f\uff0c\u5f9e\u800c\u907f\u514d\u96dc\u6e4a\u885d\u7a81\u3002</p> <p>\u8209\u500b\u4f8b\u5b50\uff0c\u5047\u8a2d\u6211\u5011\u9078\u64c7\u5408\u6578 \\(9\\) \u4f5c\u70ba\u6a21\u6578\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\uff0c\u90a3\u9ebc\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684 <code>key</code> \u90fd\u6703\u88ab\u5c0d\u6620\u5230 \\(0\\)\u3001\\(3\\)\u3001\\(6\\) \u9019\u4e09\u500b\u96dc\u6e4a\u503c\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>\u5982\u679c\u8f38\u5165 <code>key</code> \u6070\u597d\u6eff\u8db3\u9019\u7a2e\u7b49\u5dee\u6578\u5217\u7684\u8cc7\u6599\u5206\u4f48\uff0c\u90a3\u9ebc\u96dc\u6e4a\u503c\u5c31\u6703\u51fa\u73fe\u805a\u5806\u7a4d\uff0c\u5f9e\u800c\u52a0\u91cd\u96dc\u6e4a\u885d\u7a81\u3002\u73fe\u5728\uff0c\u5047\u8a2d\u5c07 <code>modulus</code> \u66ff\u63db\u70ba\u8cea\u6578 \\(13\\) \uff0c\u7531\u65bc <code>key</code> \u548c <code>modulus</code> \u4e4b\u9593\u4e0d\u5b58\u5728\u516c\u7d04\u6578\uff0c\u56e0\u6b64\u8f38\u51fa\u7684\u96dc\u6e4a\u503c\u7684\u5747\u52fb\u6027\u6703\u660e\u986f\u63d0\u5347\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u5982\u679c\u80fd\u5920\u4fdd\u8b49 <code>key</code> \u662f\u96a8\u6a5f\u5747\u52fb\u5206\u4f48\u7684\uff0c\u90a3\u9ebc\u9078\u64c7\u8cea\u6578\u6216\u8005\u5408\u6578\u4f5c\u70ba\u6a21\u6578\u90fd\u53ef\u4ee5\uff0c\u5b83\u5011\u90fd\u80fd\u8f38\u51fa\u5747\u52fb\u5206\u4f48\u7684\u96dc\u6e4a\u503c\u3002\u800c\u7576 <code>key</code> \u7684\u5206\u4f48\u5b58\u5728\u67d0\u7a2e\u9031\u671f\u6027\u6642\uff0c\u5c0d\u5408\u6578\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73fe\u805a\u96c6\u73fe\u8c61\u3002</p> <p>\u7e3d\u800c\u8a00\u4e4b\uff0c\u6211\u5011\u901a\u5e38\u9078\u53d6\u8cea\u6578\u4f5c\u70ba\u6a21\u6578\uff0c\u4e26\u4e14\u9019\u500b\u8cea\u6578\u6700\u597d\u8db3\u5920\u5927\uff0c\u4ee5\u5118\u53ef\u80fd\u6d88\u9664\u9031\u671f\u6027\u6a21\u5f0f\uff0c\u63d0\u5347\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u7a69\u5065\u6027\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3 \u00a0 \u5e38\u898b\u96dc\u6e4a\u6f14\u7b97\u6cd5","text":"<p>\u4e0d\u96e3\u767c\u73fe\uff0c\u4ee5\u4e0a\u4ecb\u7d39\u7684\u7c21\u55ae\u96dc\u6e4a\u6f14\u7b97\u6cd5\u90fd\u6bd4\u8f03\u201c\u8106\u5f31\u201d\uff0c\u9060\u9060\u6c92\u6709\u9054\u5230\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u8a2d\u8a08\u76ee\u6a19\u3002\u4f8b\u5982\uff0c\u7531\u65bc\u52a0\u6cd5\u548c\u4e92\u65a5\u6216\u6eff\u8db3\u4ea4\u63db\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u96dc\u6e4a\u548c\u4e92\u65a5\u6216\u96dc\u6e4a\u7121\u6cd5\u5340\u5206\u5167\u5bb9\u76f8\u540c\u4f46\u9806\u5e8f\u4e0d\u540c\u7684\u5b57\u4e32\uff0c\u9019\u53ef\u80fd\u6703\u52a0\u5287\u96dc\u6e4a\u885d\u7a81\uff0c\u4e26\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u554f\u984c\u3002</p> <p>\u5728\u5be6\u969b\u4e2d\uff0c\u6211\u5011\u901a\u5e38\u6703\u7528\u4e00\u4e9b\u6a19\u6e96\u96dc\u6e4a\u6f14\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002\u5b83\u5011\u53ef\u4ee5\u5c07\u4efb\u610f\u9577\u5ea6\u7684\u8f38\u5165\u8cc7\u6599\u5c0d\u6620\u5230\u6046\u5b9a\u9577\u5ea6\u7684\u96dc\u6e4a\u503c\u3002</p> <p>\u8fd1\u4e00\u500b\u4e16\u7d00\u4ee5\u4f86\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u8655\u5728\u4e0d\u65b7\u5347\u7d1a\u8207\u6700\u4f73\u5316\u7684\u904e\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u54e1\u52aa\u529b\u63d0\u5347\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u6548\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u54e1\u548c\u99ed\u5ba2\u5247\u81f4\u529b\u65bc\u5c0b\u627e\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u554f\u984c\u3002\u8868 6-2 \u5c55\u793a\u4e86\u5728\u5be6\u969b\u61c9\u7528\u4e2d\u5e38\u898b\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\u3002</p> <ul> <li>MD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u64ca\uff0c\u56e0\u6b64\u5b83\u5011\u88ab\u5404\u985e\u5b89\u5168\u61c9\u7528\u68c4\u7528\u3002</li> <li>SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73fe\u6210\u529f\u7684\u653b\u64ca\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u7528\u5728\u5404\u985e\u5b89\u5168\u61c9\u7528\u8207\u5354\u8b70\u4e2d\u3002</li> <li>SHA-3 \u76f8\u8f03 SHA-2 \u7684\u5be6\u73fe\u958b\u92b7\u66f4\u4f4e\u3001\u8a08\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u84cb\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002</li> </ul> <p> \u8868 6-2 \u00a0 \u5e38\u898b\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5 </p> MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u6642\u9593 1992 1995 2002 2008 \u8f38\u51fa\u9577\u5ea6 128 bit 160 bit 256/512 bit 224/256/384/512 bit \u96dc\u6e4a\u885d\u7a81 \u8f03\u591a \u8f03\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7d1a \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u64ca \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u64ca \u9ad8 \u9ad8 \u61c9\u7528 \u5df2\u88ab\u68c4\u7528\uff0c\u4ecd\u7528\u65bc\u8cc7\u6599\u5b8c\u6574\u6027\u6aa2\u67e5 \u5df2\u88ab\u68c4\u7528 \u52a0\u5bc6\u8ca8\u5e63\u4ea4\u6613\u9a57\u8b49\u3001\u6578\u5b57\u7c3d\u540d\u7b49 \u53ef\u7528\u65bc\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4 \u00a0 \u8cc7\u6599\u7d50\u69cb\u7684\u96dc\u6e4a\u503c","text":"<p>\u6211\u5011\u77e5\u9053\uff0c\u96dc\u6e4a\u8868\u7684 <code>key</code> \u53ef\u4ee5\u662f\u6574\u6578\u3001\u5c0f\u6578\u6216\u5b57\u4e32\u7b49\u8cc7\u6599\u578b\u5225\u3002\u7a0b\u5f0f\u8a9e\u8a00\u901a\u5e38\u6703\u70ba\u9019\u4e9b\u8cc7\u6599\u578b\u5225\u63d0\u4f9b\u5167\u5efa\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\uff0c\u7528\u65bc\u8a08\u7b97\u96dc\u6e4a\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u70ba\u4f8b\uff0c\u6211\u5011\u53ef\u4ee5\u547c\u53eb <code>hash()</code> \u51fd\u5f0f\u4f86\u8a08\u7b97\u5404\u7a2e\u8cc7\u6599\u578b\u5225\u7684\u96dc\u6e4a\u503c\u3002</p> <ul> <li>\u6574\u6578\u548c\u5e03\u6797\u91cf\u7684\u96dc\u6e4a\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002</li> <li>\u6d6e\u9ede\u6578\u548c\u5b57\u4e32\u7684\u96dc\u6e4a\u503c\u8a08\u7b97\u8f03\u70ba\u8907\u96dc\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u8acb\u81ea\u884c\u5b78\u7fd2\u3002</li> <li>\u5143\u7d44\u7684\u96dc\u6e4a\u503c\u662f\u5c0d\u5176\u4e2d\u6bcf\u4e00\u500b\u5143\u7d20\u9032\u884c\u96dc\u6e4a\uff0c\u7136\u5f8c\u5c07\u9019\u4e9b\u96dc\u6e4a\u503c\u7d44\u5408\u8d77\u4f86\uff0c\u5f97\u5230\u55ae\u4e00\u7684\u96dc\u6e4a\u503c\u3002</li> <li>\u7269\u4ef6\u7684\u96dc\u6e4a\u503c\u57fa\u65bc\u5176\u8a18\u61b6\u9ad4\u4f4d\u5740\u751f\u6210\u3002\u900f\u904e\u91cd\u5beb\u7269\u4ef6\u7684\u96dc\u6e4a\u65b9\u6cd5\uff0c\u53ef\u5be6\u73fe\u57fa\u65bc\u5167\u5bb9\u751f\u6210\u96dc\u6e4a\u503c\u3002</li> </ul> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u4e0d\u540c\u7a0b\u5f0f\u8a9e\u8a00\u7684\u5167\u5efa\u96dc\u6e4a\u503c\u8a08\u7b97\u51fd\u5f0f\u7684\u5b9a\u7fa9\u548c\u65b9\u6cd5\u4e0d\u540c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3\n\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u6797\u91cf True \u7684\u96dc\u6e4a\u503c\u70ba 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba 326484311674566659\n\nstr = \"Hello \u6f14\u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7d44 (12836, '\u5c0f\u54c8') \u7684\u96dc\u6e4a\u503c\u70ba 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u7bc0\u9ede\u7269\u4ef6 &lt;ListNode object at 0x1058fd810&gt; \u7684\u96dc\u6e4a\u503c\u70ba 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// \u5e03\u6797\u91cf 1 \u7684\u96dc\u6e4a\u503c\u70ba 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba 4614256650576692846\n\nstring str = \"Hello \u6f14\u7b97\u6cd5\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba 15466937326284535026\n\n// \u5728 C++ \u4e2d\uff0c\u5167\u5efa std:hash() \u50c5\u63d0\u4f9b\u57fa\u672c\u8cc7\u6599\u578b\u5225\u7684\u96dc\u6e4a\u503c\u8a08\u7b97\n// \u9663\u5217\u3001\u7269\u4ef6\u7684\u96dc\u6e4a\u503c\u8a08\u7b97\u9700\u8981\u81ea\u884c\u5be6\u73fe\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -1340954729\n\nString str = \"Hello \u6f14\u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u9663\u5217 [12836, \u5c0f\u54c8] \u7684\u96dc\u6e4a\u503c\u70ba 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u7bc0\u9ede\u7269\u4ef6 utils.ListNode@7dc5e7b4 \u7684\u96dc\u6e4a\u503c\u70ba 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -1340954729;\n\nstring str = \"Hello \u6f14\u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// \u9663\u5217 [12836, \u5c0f\u54c8] \u7684\u96dc\u6e4a\u503c\u70ba 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// \u7bc0\u9ede\u7269\u4ef6 0 \u7684\u96dc\u6e4a\u503c\u70ba 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go \u672a\u63d0\u4f9b\u5167\u5efa hash code \u51fd\u5f0f\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -2465384235396674631\n\nlet str = \"Hello \u6f14\u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u9663\u5217 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u96dc\u6e4a\u503c\u70ba -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u7bc0\u9ede\u7269\u4ef6 utils.ListNode \u7684\u96dc\u6e4a\u503c\u70ba -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5167\u5efa hash code \u51fd\u5f0f\n</code></pre> built_in_hash.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5167\u5efa hash code \u51fd\u5f0f\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u6797\u503c true \u7684\u96dc\u6e4a\u503c\u70ba 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba 2570631074981783\n\nString str = \"Hello \u6f14\u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u9663\u5217 [12836, \u5c0f\u54c8] \u7684\u96dc\u6e4a\u503c\u70ba 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u7bc0\u9ede\u7269\u4ef6 Instance of 'ListNode' \u7684\u96dc\u6e4a\u503c\u70ba 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba 2566941990314602357\n\nlet str = \"Hello \u6f14\u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// \u5143\u7d44 (12836, \"\u5c0f\u54c8\") \u7684\u96dc\u6e4a\u503c\u70ba 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// \u7bc0\u9ede\u7269\u4ef6 RefCell { value: ListNode { val: 42, next: None } } \u7684\u96dc\u6e4a\u503c\u70ba15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa hash code \u51fd\u5f0f\n</code></pre> built_in_hash.kt<pre><code>val num = 3\nval hashNum = num.hashCode()\n// \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba 3\n\nval bol = true\nval hashBol = bol.hashCode()\n// \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba 1231\n\nval dec = 3.14159\nval hashDec = dec.hashCode()\n// \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -1340954729\n\nval str = \"Hello \u6f14\u7b97\u6cd5\"\nval hashStr = str.hashCode()\n// \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -727081396\n\nval arr = arrayOf&lt;Any&gt;(12836, \"\u5c0f\u54c8\")\nval hashTup = arr.hashCode()\n// \u9663\u5217 [12836, \u5c0f\u54c8] \u7684\u96dc\u6e4a\u503c\u70ba 189568618\n\nval obj = ListNode(0)\nval hashObj = obj.hashCode()\n// \u7bc0\u9ede\u7269\u4ef6 utils.ListNode@1d81eb93 \u7684\u96dc\u6e4a\u503c\u70ba 495053715\n</code></pre> built_in_hash.rb<pre><code>num = 3\nhash_num = num.hash\n# \u6574\u6578 3 \u7684\u96dc\u6e4a\u503c\u70ba -4385856518450339636\n\nbol = true\nhash_bol = bol.hash\n# \u5e03\u6797\u91cf true \u7684\u96dc\u6e4a\u503c\u70ba -1617938112149317027\n\ndec = 3.14159\nhash_dec = dec.hash\n# \u5c0f\u6578 3.14159 \u7684\u96dc\u6e4a\u503c\u70ba -1479186995943067893\n\nstr = \"Hello \u6f14\u7b97\u6cd5\"\nhash_str = str.hash\n# \u5b57\u4e32\u201cHello \u6f14\u7b97\u6cd5\u201d\u7684\u96dc\u6e4a\u503c\u70ba -4075943250025831763\n\ntup = [12836, '\u5c0f\u54c8']\nhash_tup = tup.hash\n# \u5143\u7d44 (12836, '\u5c0f\u54c8') \u7684\u96dc\u6e4a\u503c\u70ba 1999544809202288822\n\nobj = ListNode.new(0)\nhash_obj = obj.hash\n# \u7bc0\u9ede\u7269\u4ef6 #&lt;ListNode:0x000078133140ab70&gt; \u7684\u96dc\u6e4a\u503c\u70ba 4302940560806366381\n</code></pre> built_in_hash.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5728\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u8b8a\u7269\u4ef6\u624d\u53ef\u4f5c\u70ba\u96dc\u6e4a\u8868\u7684 <code>key</code> \u3002\u5047\u5982\u6211\u5011\u5c07\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\u4f5c\u70ba <code>key</code> \uff0c\u7576\u4e32\u5217\u7684\u5167\u5bb9\u767c\u751f\u8b8a\u5316\u6642\uff0c\u5b83\u7684\u96dc\u6e4a\u503c\u4e5f\u96a8\u4e4b\u6539\u8b8a\uff0c\u6211\u5011\u5c31\u7121\u6cd5\u5728\u96dc\u6e4a\u8868\u4e2d\u67e5\u8a62\u5230\u539f\u5148\u7684 <code>value</code> \u4e86\u3002</p> <p>\u96d6\u7136\u81ea\u5b9a\u7fa9\u7269\u4ef6\uff08\u6bd4\u5982\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\uff09\u7684\u6210\u54e1\u8b8a\u6578\u662f\u53ef\u8b8a\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u96dc\u6e4a\u7684\u3002\u9019\u662f\u56e0\u70ba\u7269\u4ef6\u7684\u96dc\u6e4a\u503c\u901a\u5e38\u662f\u57fa\u65bc\u8a18\u61b6\u9ad4\u4f4d\u5740\u751f\u6210\u7684\uff0c\u5373\u4f7f\u7269\u4ef6\u7684\u5167\u5bb9\u767c\u751f\u4e86\u8b8a\u5316\uff0c\u4f46\u5b83\u7684\u8a18\u61b6\u9ad4\u4f4d\u5740\u4e0d\u8b8a\uff0c\u96dc\u6e4a\u503c\u4ecd\u7136\u662f\u4e0d\u8b8a\u7684\u3002</p> <p>\u7d30\u5fc3\u7684\u4f60\u53ef\u80fd\u767c\u73fe\u5728\u4e0d\u540c\u63a7\u5236\u6aaf\u4e2d\u57f7\u884c\u7a0b\u5f0f\u6642\uff0c\u8f38\u51fa\u7684\u96dc\u6e4a\u503c\u662f\u4e0d\u540c\u7684\u3002\u9019\u662f\u56e0\u70ba Python \u76f4\u8b6f\u5668\u5728\u6bcf\u6b21\u555f\u52d5\u6642\uff0c\u90fd\u6703\u70ba\u5b57\u4e32\u96dc\u6e4a\u51fd\u5f0f\u52a0\u5165\u4e00\u500b\u96a8\u6a5f\u7684\u9e7d\uff08salt\uff09\u503c\u3002\u9019\u7a2e\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u64ca\uff0c\u63d0\u5347\u96dc\u6e4a\u6f14\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002</p>"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 \u96dc\u6e4a\u885d\u7a81","text":"<p>\u4e0a\u4e00\u7bc0\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u6cc1\u4e0b\u96dc\u6e4a\u51fd\u5f0f\u7684\u8f38\u5165\u7a7a\u9593\u9060\u5927\u65bc\u8f38\u51fa\u7a7a\u9593\uff0c\u56e0\u6b64\u7406\u8ad6\u4e0a\u96dc\u6e4a\u885d\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u6bd4\u5982\uff0c\u8f38\u5165\u7a7a\u9593\u70ba\u5168\u9ad4\u6574\u6578\uff0c\u8f38\u51fa\u7a7a\u9593\u70ba\u9663\u5217\u5bb9\u91cf\u5927\u5c0f\uff0c\u5247\u5fc5\u7136\u6709\u591a\u500b\u6574\u6578\u5c0d\u6620\u81f3\u540c\u4e00\u6876\u7d22\u5f15\u3002</p> <p>\u96dc\u6e4a\u885d\u7a81\u6703\u5c0e\u81f4\u67e5\u8a62\u7d50\u679c\u932f\u8aa4\uff0c\u56b4\u91cd\u5f71\u97ff\u96dc\u6e4a\u8868\u7684\u53ef\u7528\u6027\u3002\u70ba\u4e86\u89e3\u6c7a\u8a72\u554f\u984c\uff0c\u6bcf\u7576\u9047\u5230\u96dc\u6e4a\u885d\u7a81\u6642\uff0c\u6211\u5011\u5c31\u9032\u884c\u96dc\u6e4a\u8868\u64f4\u5bb9\uff0c\u76f4\u81f3\u885d\u7a81\u6d88\u5931\u70ba\u6b62\u3002\u6b64\u65b9\u6cd5\u7c21\u55ae\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u70ba\u96dc\u6e4a\u8868\u64f4\u5bb9\u9700\u8981\u9032\u884c\u5927\u91cf\u7684\u8cc7\u6599\u642c\u904b\u8207\u96dc\u6e4a\u503c\u8a08\u7b97\u3002\u70ba\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u5011\u53ef\u4ee5\u63a1\u7528\u4ee5\u4e0b\u7b56\u7565\u3002</p> <ol> <li>\u6539\u826f\u96dc\u6e4a\u8868\u8cc7\u6599\u7d50\u69cb\uff0c\u4f7f\u5f97\u96dc\u6e4a\u8868\u53ef\u4ee5\u5728\u51fa\u73fe\u96dc\u6e4a\u885d\u7a81\u6642\u6b63\u5e38\u5de5\u4f5c\u3002</li> <li>\u50c5\u5728\u5fc5\u8981\u6642\uff0c\u5373\u7576\u96dc\u6e4a\u885d\u7a81\u6bd4\u8f03\u56b4\u91cd\u6642\uff0c\u624d\u57f7\u884c\u64f4\u5bb9\u64cd\u4f5c\u3002</li> </ol> <p>\u96dc\u6e4a\u8868\u7684\u7d50\u69cb\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u201c\u93c8\u5f0f\u4f4d\u5740\u201d\u548c\u201c\u958b\u653e\u5b9a\u5740\u201d\u3002</p>"},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1 \u00a0 \u93c8\u5f0f\u4f4d\u5740","text":"<p>\u5728\u539f\u59cb\u96dc\u6e4a\u8868\u4e2d\uff0c\u6bcf\u500b\u6876\u50c5\u80fd\u5132\u5b58\u4e00\u500b\u9375\u503c\u5c0d\u3002\u93c8\u5f0f\u4f4d\u5740\uff08separate chaining\uff09\u5c07\u55ae\u500b\u5143\u7d20\u8f49\u63db\u70ba\u93c8\u7d50\u4e32\u5217\uff0c\u5c07\u9375\u503c\u5c0d\u4f5c\u70ba\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\uff0c\u5c07\u6240\u6709\u767c\u751f\u885d\u7a81\u7684\u9375\u503c\u5c0d\u90fd\u5132\u5b58\u5728\u540c\u4e00\u93c8\u7d50\u4e32\u5217\u4e2d\u3002\u5716 6-5 \u5c55\u793a\u4e86\u4e00\u500b\u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868\u7684\u4f8b\u5b50\u3002</p> <p></p> <p> \u5716 6-5 \u00a0 \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 </p> <p>\u57fa\u65bc\u93c8\u5f0f\u4f4d\u5740\u5be6\u73fe\u7684\u96dc\u6e4a\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u767c\u751f\u4e86\u4ee5\u4e0b\u8b8a\u5316\u3002</p> <ul> <li>\u67e5\u8a62\u5143\u7d20\uff1a\u8f38\u5165 <code>key</code> \uff0c\u7d93\u904e\u96dc\u6e4a\u51fd\u5f0f\u5f97\u5230\u6876\u7d22\u5f15\uff0c\u5373\u53ef\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u982d\u7bc0\u9ede\uff0c\u7136\u5f8c\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4e26\u5c0d\u6bd4 <code>key</code> \u4ee5\u67e5\u8a62\u76ee\u6a19\u9375\u503c\u5c0d\u3002</li> <li>\u65b0\u589e\u5143\u7d20\uff1a\u9996\u5148\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u982d\u7bc0\u9ede\uff0c\u7136\u5f8c\u5c07\u7bc0\u9ede\uff08\u9375\u503c\u5c0d\uff09\u65b0\u589e\u5230\u93c8\u7d50\u4e32\u5217\u4e2d\u3002</li> <li>\u522a\u9664\u5143\u7d20\uff1a\u6839\u64da\u96dc\u6e4a\u51fd\u5f0f\u7684\u7d50\u679c\u8a2a\u554f\u93c8\u7d50\u4e32\u5217\u982d\u90e8\uff0c\u63a5\u8457\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4ee5\u67e5\u8a62\u76ee\u6a19\u7bc0\u9ede\u4e26\u5c07\u5176\u522a\u9664\u3002</li> </ul> <p>\u93c8\u5f0f\u4f4d\u5740\u5b58\u5728\u4ee5\u4e0b\u4fb7\u9650\u6027\u3002</p> <ul> <li>\u4f54\u7528\u7a7a\u9593\u589e\u5927\uff1a\u93c8\u7d50\u4e32\u5217\u5305\u542b\u7bc0\u9ede\u6307\u6a19\uff0c\u5b83\u76f8\u6bd4\u9663\u5217\u66f4\u52a0\u8017\u8cbb\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> <li>\u67e5\u8a62\u6548\u7387\u964d\u4f4e\uff1a\u56e0\u70ba\u9700\u8981\u7dda\u6027\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u4f86\u67e5\u8a62\u5c0d\u61c9\u5143\u7d20\u3002</li> </ul> <p>\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u7d66\u51fa\u4e86\u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868\u7684\u7c21\u55ae\u5be6\u73fe\uff0c\u9700\u8981\u6ce8\u610f\u5169\u9ede\u3002</p> <ul> <li>\u4f7f\u7528\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\u4ee3\u66ff\u93c8\u7d50\u4e32\u5217\uff0c\u5f9e\u800c\u7c21\u5316\u7a0b\u5f0f\u78bc\u3002\u5728\u9019\u7a2e\u8a2d\u5b9a\u4e0b\uff0c\u96dc\u6e4a\u8868\uff08\u9663\u5217\uff09\u5305\u542b\u591a\u500b\u6876\uff0c\u6bcf\u500b\u6876\u90fd\u662f\u4e00\u500b\u4e32\u5217\u3002</li> <li>\u4ee5\u4e0b\u5be6\u73fe\u5305\u542b\u96dc\u6e4a\u8868\u64f4\u5bb9\u65b9\u6cd5\u3002\u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e \\(\\frac{2}{3}\\) \u6642\uff0c\u6211\u5011\u5c07\u96dc\u6e4a\u8868\u64f4\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"\u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self.size = 0  # \u9375\u503c\u5c0d\u6578\u91cf\n        self.capacity = 4  # \u96dc\u6e4a\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n        self.extend_ratio = 2  # \u64f4\u5bb9\u500d\u6578\n        self.buckets = [[] for _ in range(self.capacity)]  # \u6876\u9663\u5217\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u96dc\u6e4a\u51fd\u5f0f\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8ca0\u8f09\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"\u67e5\u8a62\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u65b0\u589e\u64cd\u4f5c\"\"\"\n        # \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u522a\u9664\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"\u64f4\u5bb9\u96dc\u6e4a\u8868\"\"\"\n        # \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        buckets = self.buckets\n        # \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u5217\u5370\u96dc\u6e4a\u8868\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n  private:\n    int size;                       // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity;                   // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres;               // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio;                // \u64f4\u5bb9\u500d\u6578\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // \u6876\u9663\u5217\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* \u6790\u69cb\u65b9\u6cd5 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // \u91cb\u653e\u8a18\u61b6\u9ad4\n                delete pair;\n            }\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n        return \"\";\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // \u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n                delete tmp;                       // \u91cb\u653e\u8a18\u61b6\u9ad4\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // \u91cb\u653e\u8a18\u61b6\u9ad4\n                delete pair;\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    int size; // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio; // \u64f4\u5bb9\u500d\u6578\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    void put(int key, String val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    int size; // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio; // \u64f4\u5bb9\u500d\u6578\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void Extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\ntype hashMapChaining struct {\n    size        int      // \u9375\u503c\u5c0d\u6578\u91cf\n    capacity    int      // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    loadThres   float64  // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    extendRatio int      // \u64f4\u5bb9\u500d\u6578\n    buckets     [][]pair // \u6876\u9663\u5217\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* \u8ca0\u8f09\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n    return \"\"\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // \u5207\u7247\u522a\u9664\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\nfunc (m *hashMapChaining) extend() {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    var size: Int // \u9375\u503c\u5c0d\u6578\u91cf\n    var capacity: Int // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    var extendRatio: Int // \u64f4\u5bb9\u500d\u6578\n    var buckets: [[Pair]] // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de nil\n        return nil\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    func extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    #size; // \u9375\u503c\u5c0d\u6578\u91cf\n    #capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    #loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    #extendRatio; // \u64f4\u5bb9\u500d\u6578\n    #buckets; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    put(key, val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    #extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    #size: number; // \u9375\u503c\u5c0d\u6578\u91cf\n    #capacity: number; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    #loadThres: number; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    #extendRatio: number; // \u64f4\u5bb9\u500d\u6578\n    #buckets: Pair[][]; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    #extend(): void {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n  late int size; // \u9375\u503c\u5c0d\u6578\u91cf\n  late int capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n  late double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n  late int extendRatio; // \u64f4\u5bb9\u500d\u6578\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u9663\u5217\n\n  /* \u5efa\u69cb\u5b50 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* \u96dc\u6e4a\u51fd\u5f0f */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* \u8ca0\u8f09\u56e0\u5b50 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* \u67e5\u8a62\u64cd\u4f5c */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u65b0\u589e\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* \u522a\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n  void extend() {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u5217\u5370\u96dc\u6e4a\u8868 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nstruct HashMapChaining {\n    size: i32,\n    capacity: i32,\n    load_thres: f32,\n    extend_ratio: i32,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* \u5efa\u69cb\u5b50 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity as usize\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for i in 0..bucket.len() {\n            if bucket[i].key == key {\n                let pair = bucket.remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de None\n        None\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    fn extend(&amp;mut self) {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        let buckets_tmp = std::mem::replace(&amp;mut self.buckets, vec![]);\n\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val;\n                return;\n            }\n        }\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        let pair = Pair { key, val };\n        bucket.push(pair);\n        self.size += 1;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;self.buckets[index];\n\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for pair in bucket {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de None\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\ntypedef struct {\n    int size;         // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity;     // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio;  // \u64f4\u5bb9\u500d\u6578\n    Node **buckets;   // \u6876\u9663\u5217\n} HashMapChaining;\n\n/* \u5efa\u69cb\u5b50 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8ca0\u8f09\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // \u91cb\u653e\u8a18\u61b6\u9ad4\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // \u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // \u91cb\u653e\u8a18\u61b6\u9ad4\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* \u93c8\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 */\nclass HashMapChaining {\n    var size: Int // \u9375\u503c\u5c0d\u6578\u91cf\n    var capacity: Int // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    val loadThres: Double // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    val extendRatio: Int // \u64f4\u5bb9\u500d\u6578\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // \u6876\u9663\u5217\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        for (pair in bucket) {\n            if (pair.key == key) return pair._val\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5247\u8fd4\u56de null\n        return null\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fun put(key: Int, _val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair._val = _val\n                return\n            }\n        }\n        // \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n        val pair = Pair(key, _val)\n        bucket.add(pair)\n        size++\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    fun extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio\n        // mutablelist \u7121\u56fa\u5b9a\u5927\u5c0f\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair._val\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>### \u9375\u5f0f\u4f4d\u5740\u96dc\u6e4a\u8868 ###\nclass HashMapChaining\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @size = 0 # \u9375\u503c\u5c0d\u6578\u91cf\n    @capacity = 4 # \u96dc\u6e4a\u8868\u5bb9\u91cf\n    @load_thres = 2.0 / 3.0 # \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    @extend_ratio = 2 # \u64f4\u5bb9\u500d\u6578\n    @buckets = Array.new(@capacity) { [] } # \u6876\u9663\u5217\n  end\n\n  ### \u96dc\u6e4a\u51fd\u5f0f ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### \u8ca0\u8f09\u56e0\u5b50 ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### \u67e5\u8a62\u64cd\u4f5c ###\n  def get(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # \u8d70\u8a2a\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    for pair in bucket\n      return pair.val if pair.key == key\n    end\n    # \u82e5\u672a\u627e\u5230 key , \u5247\u8fd4\u56de nil\n    nil\n  end\n\n  ### \u65b0\u589e\u64cd\u4f5c ###\n  def put(key, val)\n    # \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    extend if load_factor &gt; @load_thres\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # \u8d70\u8a2a\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5247\u66f4\u65b0\u5c0d\u61c9 val \u4e26\u8fd4\u56de\n    for pair in bucket\n      if pair.key == key\n        pair.val = val\n        return\n      end\n    end\n    # \u82e5\u7121\u8a72 key \uff0c\u5247\u5c07\u9375\u503c\u5c0d\u65b0\u589e\u81f3\u5c3e\u90e8\n    pair = Pair.new(key, val)\n    bucket &lt;&lt; pair\n    @size += 1\n  end\n\n  ### \u522a\u9664\u64cd\u4f5c ###\n  def remove(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # \u8d70\u8a2a\u6876\uff0c\u5f9e\u4e2d\u522a\u9664\u9375\u503c\u5c0d\n    for pair in bucket\n      if pair.key == key\n        bucket.delete(pair)\n        @size -= 1\n        break\n      end\n    end\n  end\n\n  ### \u64f4\u5bb9\u96dc\u6e4a\u8868 ###\n  def extend\n    # \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    buckets = @buckets\n    # \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity) { [] }\n    @size = 0\n    # \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for bucket in buckets\n      for pair in bucket\n        put(pair.key, pair.val)\n      end\n    end\n  end\n\n  ### \u5217\u5370\u96dc\u6e4a\u8868 ###\n  def print\n    for bucket in @buckets\n      res = []\n      for pair in bucket\n        res &lt;&lt; \"#{pair.key} -&gt; #{pair.val}\"\n      end\n      pp res\n    end\n  end\nend\n</code></pre> hash_map_chaining.zig<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7576\u93c8\u7d50\u4e32\u5217\u5f88\u9577\u6642\uff0c\u67e5\u8a62\u6548\u7387 \\(O(n)\\) \u5f88\u5dee\u3002\u6b64\u6642\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u63db\u70ba\u201cAVL \u6a39\u201d\u6216\u201c\u7d05\u9ed1\u6a39\u201d\uff0c\u5f9e\u800c\u5c07\u67e5\u8a62\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2 \u00a0 \u958b\u653e\u5b9a\u5740","text":"<p>\u958b\u653e\u5b9a\u5740\uff08open addressing\uff09\u4e0d\u5f15\u5165\u984d\u5916\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u800c\u662f\u900f\u904e\u201c\u591a\u6b21\u63a2\u6e2c\u201d\u4f86\u8655\u7406\u96dc\u6e4a\u885d\u7a81\uff0c\u63a2\u6e2c\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7dda\u6027\u63a2\u67e5\u3001\u5e73\u65b9\u63a2\u6e2c\u548c\u591a\u6b21\u96dc\u6e4a\u7b49\u3002</p> <p>\u4e0b\u9762\u4ee5\u7dda\u6027\u63a2\u67e5\u70ba\u4f8b\uff0c\u4ecb\u7d39\u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868\u7684\u5de5\u4f5c\u6a5f\u5236\u3002</p>"},{"location":"chapter_hashing/hash_collision/#1","title":"1. \u00a0 \u7dda\u6027\u63a2\u67e5","text":"<p>\u7dda\u6027\u63a2\u67e5\u63a1\u7528\u56fa\u5b9a\u6b65\u9577\u7684\u7dda\u6027\u641c\u5c0b\u4f86\u9032\u884c\u63a2\u6e2c\uff0c\u5176\u64cd\u4f5c\u65b9\u6cd5\u8207\u666e\u901a\u96dc\u6e4a\u8868\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u82e5\u767c\u73fe\u6876\u5167\u5df2\u6709\u5143\u7d20\uff0c\u5247\u5f9e\u885d\u7a81\u4f4d\u7f6e\u5411\u5f8c\u7dda\u6027\u8d70\u8a2a\uff08\u6b65\u9577\u901a\u5e38\u70ba \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u6876\uff0c\u5c07\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002</li> <li>\u67e5\u8a62\u5143\u7d20\uff1a\u82e5\u767c\u73fe\u96dc\u6e4a\u885d\u7a81\uff0c\u5247\u4f7f\u7528\u76f8\u540c\u6b65\u9577\u5411\u5f8c\u9032\u884c\u7dda\u6027\u8d70\u8a2a\uff0c\u76f4\u5230\u627e\u5230\u5c0d\u61c9\u5143\u7d20\uff0c\u8fd4\u56de <code>value</code> \u5373\u53ef\uff1b\u5982\u679c\u9047\u5230\u7a7a\u6876\uff0c\u8aaa\u660e\u76ee\u6a19\u5143\u7d20\u4e0d\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u8fd4\u56de <code>None</code> \u3002</li> </ul> <p>\u5716 6-6 \u5c55\u793a\u4e86\u958b\u653e\u5b9a\u5740\uff08\u7dda\u6027\u63a2\u67e5\uff09\u96dc\u6e4a\u8868\u7684\u9375\u503c\u5c0d\u5206\u4f48\u3002\u6839\u64da\u6b64\u96dc\u6e4a\u51fd\u5f0f\uff0c\u6700\u5f8c\u5169\u4f4d\u76f8\u540c\u7684 <code>key</code> \u90fd\u6703\u88ab\u5c0d\u6620\u5230\u76f8\u540c\u7684\u6876\u3002\u800c\u900f\u904e\u7dda\u6027\u63a2\u67e5\uff0c\u5b83\u5011\u88ab\u4f9d\u6b21\u5132\u5b58\u5728\u8a72\u6876\u4ee5\u53ca\u4e4b\u4e0b\u7684\u6876\u4e2d\u3002</p> <p></p> <p> \u5716 6-6 \u00a0 \u958b\u653e\u5b9a\u5740\uff08\u7dda\u6027\u63a2\u67e5\uff09\u96dc\u6e4a\u8868\u7684\u9375\u503c\u5c0d\u5206\u4f48 </p> <p>\u7136\u800c\uff0c\u7dda\u6027\u63a2\u67e5\u5bb9\u6613\u7522\u751f\u201c\u805a\u96c6\u73fe\u8c61\u201d\u3002\u5177\u9ad4\u4f86\u8aaa\uff0c\u9663\u5217\u4e2d\u9023\u7e8c\u88ab\u4f54\u7528\u7684\u4f4d\u7f6e\u8d8a\u9577\uff0c\u9019\u4e9b\u9023\u7e8c\u4f4d\u7f6e\u767c\u751f\u96dc\u6e4a\u885d\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u5f9e\u800c\u9032\u4e00\u6b65\u4fc3\u4f7f\u8a72\u4f4d\u7f6e\u7684\u805a\u5806\u7a4d\u751f\u9577\uff0c\u5f62\u6210\u60e1\u6027\u8ff4\u5708\uff0c\u6700\u7d42\u5c0e\u81f4\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u52a3\u5316\u3002</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6211\u5011\u4e0d\u80fd\u5728\u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868\u4e2d\u76f4\u63a5\u522a\u9664\u5143\u7d20\u3002\u9019\u662f\u56e0\u70ba\u522a\u9664\u5143\u7d20\u6703\u5728\u9663\u5217\u5167\u7522\u751f\u4e00\u500b\u7a7a\u6876 <code>None</code> \uff0c\u800c\u7576\u67e5\u8a62\u5143\u7d20\u6642\uff0c\u7dda\u6027\u63a2\u67e5\u5230\u8a72\u7a7a\u6876\u5c31\u6703\u8fd4\u56de\uff0c\u56e0\u6b64\u5728\u8a72\u7a7a\u6876\u4e4b\u4e0b\u7684\u5143\u7d20\u90fd\u7121\u6cd5\u518d\u88ab\u8a2a\u554f\u5230\uff0c\u7a0b\u5f0f\u53ef\u80fd\u8aa4\u5224\u9019\u4e9b\u5143\u7d20\u4e0d\u5b58\u5728\uff0c\u5982\u5716 6-7 \u6240\u793a\u3002</p> <p></p> <p> \u5716 6-7 \u00a0 \u5728\u958b\u653e\u5b9a\u5740\u4e2d\u522a\u9664\u5143\u7d20\u5c0e\u81f4\u7684\u67e5\u8a62\u554f\u984c </p> <p>\u70ba\u4e86\u89e3\u6c7a\u8a72\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u63a1\u7528\u61f6\u522a\u9664\uff08lazy deletion\uff09\u6a5f\u5236\uff1a\u5b83\u4e0d\u76f4\u63a5\u5f9e\u96dc\u6e4a\u8868\u4e2d\u79fb\u9664\u5143\u7d20\uff0c\u800c\u662f\u5229\u7528\u4e00\u500b\u5e38\u6578 <code>TOMBSTONE</code> \u4f86\u6a19\u8a18\u9019\u500b\u6876\u3002\u5728\u8a72\u6a5f\u5236\u4e0b\uff0c<code>None</code> \u548c <code>TOMBSTONE</code> \u90fd\u4ee3\u8868\u7a7a\u6876\uff0c\u90fd\u53ef\u4ee5\u653e\u7f6e\u9375\u503c\u5c0d\u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u7dda\u6027\u63a2\u67e5\u5230 <code>TOMBSTONE</code> \u6642\u61c9\u8a72\u7e7c\u7e8c\u8d70\u8a2a\uff0c\u56e0\u70ba\u5176\u4e4b\u4e0b\u53ef\u80fd\u9084\u5b58\u5728\u9375\u503c\u5c0d\u3002</p> <p>\u7136\u800c\uff0c\u61f6\u522a\u9664\u53ef\u80fd\u6703\u52a0\u901f\u96dc\u6e4a\u8868\u7684\u6548\u80fd\u9000\u5316\u3002\u9019\u662f\u56e0\u70ba\u6bcf\u6b21\u522a\u9664\u64cd\u4f5c\u90fd\u6703\u7522\u751f\u4e00\u500b\u522a\u9664\u6a19\u8a18\uff0c\u96a8\u8457 <code>TOMBSTONE</code> \u7684\u589e\u52a0\uff0c\u641c\u5c0b\u6642\u9593\u4e5f\u6703\u589e\u52a0\uff0c\u56e0\u70ba\u7dda\u6027\u63a2\u67e5\u53ef\u80fd\u9700\u8981\u8df3\u904e\u591a\u500b <code>TOMBSTONE</code> \u624d\u80fd\u627e\u5230\u76ee\u6a19\u5143\u7d20\u3002</p> <p>\u70ba\u6b64\uff0c\u8003\u616e\u7dda\u4e0a\u6027\u63a2\u67e5\u4e2d\u8a18\u9304\u9047\u5230\u7684\u9996\u500b <code>TOMBSTONE</code> \u7684\u7d22\u5f15\uff0c\u4e26\u5c07\u641c\u5c0b\u5230\u7684\u76ee\u6a19\u5143\u7d20\u8207\u8a72 <code>TOMBSTONE</code> \u4ea4\u63db\u4f4d\u7f6e\u3002\u9019\u6a23\u505a\u7684\u597d\u8655\u662f\u7576\u6bcf\u6b21\u67e5\u8a62\u6216\u65b0\u589e\u5143\u7d20\u6642\uff0c\u5143\u7d20\u6703\u88ab\u79fb\u52d5\u81f3\u8ddd\u96e2\u7406\u60f3\u4f4d\u7f6e\uff08\u63a2\u6e2c\u8d77\u59cb\u9ede\uff09\u66f4\u8fd1\u7684\u6876\uff0c\u5f9e\u800c\u6700\u4f73\u5316\u67e5\u8a62\u6548\u7387\u3002</p> <p>\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e86\u4e00\u500b\u5305\u542b\u61f6\u522a\u9664\u7684\u958b\u653e\u5b9a\u5740\uff08\u7dda\u6027\u63a2\u67e5\uff09\u96dc\u6e4a\u8868\u3002\u70ba\u4e86\u66f4\u52a0\u5145\u5206\u5730\u4f7f\u7528\u96dc\u6e4a\u8868\u7684\u7a7a\u9593\uff0c\u6211\u5011\u5c07\u96dc\u6e4a\u8868\u770b\u4f5c\u4e00\u500b\u201c\u74b0\u5f62\u9663\u5217\u201d\uff0c\u7576\u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u56de\u5230\u982d\u90e8\u7e7c\u7e8c\u8d70\u8a2a\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"\u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self.size = 0  # \u9375\u503c\u5c0d\u6578\u91cf\n        self.capacity = 4  # \u96dc\u6e4a\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n        self.extend_ratio = 2  # \u64f4\u5bb9\u500d\u6578\n        self.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u9663\u5217\n        self.TOMBSTONE = Pair(-1, \"-1\")  # \u522a\u9664\u6a19\u8a18\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u96dc\u6e4a\u51fd\u5f0f\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8ca0\u8f09\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"\u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while self.buckets[index] is not None:\n            # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].key == key:\n                # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                return index  # \u8fd4\u56de\u6876\u7d22\u5f15\n            # \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % self.capacity\n        # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8a62\u64cd\u4f5c\"\"\"\n        # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u65b0\u589e\u64cd\u4f5c\"\"\"\n        # \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u522a\u9664\u64cd\u4f5c\"\"\"\n        # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"\u64f4\u5bb9\u96dc\u6e4a\u8868\"\"\"\n        # \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        buckets_tmp = self.buckets\n        # \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u5217\u5370\u96dc\u6e4a\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity = 4;                     // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    const double loadThres = 2.0 / 3.0;     // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    const int extendRatio = 2;            // \u64f4\u5bb9\u500d\u6578\n    vector&lt;Pair *&gt; buckets;               // \u6876\u9663\u5217\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u522a\u9664\u6a19\u8a18\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* \u6790\u69cb\u65b9\u6cd5 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (buckets[index] != nullptr) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (buckets[index]-&gt;key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    string get(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n        return \"\";\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    private int size; // \u9375\u503c\u5c0d\u6578\u91cf\n    private int capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    private final double loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    private final int extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n    private Pair[] buckets; // \u6876\u9663\u5217\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public String get(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void put(int key, String val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    private void extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    int size; // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n    Pair[] buckets; // \u6876\u9663\u5217\n    Pair TOMBSTONE = new(-1, \"-1\"); // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public string? Get(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    void Extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\ntype hashMapOpenAddressing struct {\n    size        int     // \u9375\u503c\u5c0d\u6578\u91cf\n    capacity    int     // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    loadThres   float64 // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    extendRatio int     // \u64f4\u5bb9\u500d\u6578\n    buckets     []*pair // \u6876\u9663\u5217\n    TOMBSTONE   *pair   // \u522a\u9664\u6a19\u8a18\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // \u6839\u64da\u9375\u8a08\u7b97\u96dc\u6e4a\u503c\n}\n\n/* \u8ca0\u8f09\u56e0\u5b50 */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // \u8a08\u7b97\u7576\u524d\u8ca0\u8f09\u56e0\u5b50\n}\n\n/* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // \u7372\u53d6\u521d\u59cb\u7d22\u5f15\n    firstTombstone := -1     // \u8a18\u9304\u9047\u5230\u7684\u7b2c\u4e00\u500bTOMBSTONE\u7684\u4f4d\u7f6e\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n            }\n            return index // \u8fd4\u56de\u627e\u5230\u7684\u7d22\u5f15\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\u7684\u4f4d\u7f6e\n        }\n        index = (index + 1) % h.capacity // \u7dda\u6027\u63a2\u67e5\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    }\n    return \"\" // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de \"\"\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    }\n    index := h.findBucket(key) // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        h.size++\n    } else {\n        h.buckets[index].val = val // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val\n    }\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        h.size--\n    }\n}\n\n/* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    h.capacity *= h.extendRatio           // \u66f4\u65b0\u5bb9\u91cf\n    h.buckets = make([]*pair, h.capacity) // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    h.size = 0                            // \u91cd\u7f6e\u5927\u5c0f\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    var size: Int // \u9375\u503c\u5c0d\u6578\u91cf\n    var capacity: Int // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    var extendRatio: Int // \u64f4\u5bb9\u500d\u6578\n    var buckets: [Pair?] // \u6876\u9663\u5217\n    var TOMBSTONE: Pair // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while buckets[index] != nil {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if buckets[index]!.key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return nil\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    func extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    #size; // \u9375\u503c\u5c0d\u6578\u91cf\n    #capacity; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    #loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    #extendRatio; // \u64f4\u5bb9\u500d\u6578\n    #buckets; // \u6876\u9663\u5217\n    #TOMBSTONE; // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.#size = 0; // \u9375\u503c\u5c0d\u6578\u91cf\n        this.#capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n        this.#loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n        this.#extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n        this.#buckets = Array(this.#capacity).fill(null); // \u6876\u9663\u5217\n        this.#TOMBSTONE = new Pair(-1, '-1'); // \u522a\u9664\u6a19\u8a18\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (this.#buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (this.#buckets[index].key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % this.#capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    put(key, val) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    remove(key) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    #extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    private size: number; // \u9375\u503c\u5c0d\u6578\u91cf\n    private capacity: number; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    private loadThres: number; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    private extendRatio: number; // \u64f4\u5bb9\u500d\u6578\n    private buckets: Array&lt;Pair | null&gt;; // \u6876\u9663\u5217\n    private TOMBSTONE: Pair; // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor() {\n        this.size = 0; // \u9375\u503c\u5c0d\u6578\u91cf\n        this.capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n        this.loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n        this.extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n        this.buckets = Array(this.capacity).fill(null); // \u6876\u9663\u5217\n        this.TOMBSTONE = new Pair(-1, '-1'); // \u522a\u9664\u6a19\u8a18\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (this.buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (this.buckets[index]!.key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % this.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key: number): string | null {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    private extend(): void {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        const bucketsTmp = this.buckets;\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n  late int _size; // \u9375\u503c\u5c0d\u6578\u91cf\n  int _capacity = 4; // \u96dc\u6e4a\u8868\u5bb9\u91cf\n  double _loadThres = 2.0 / 3.0; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n  int _extendRatio = 2; // \u64f4\u5bb9\u500d\u6578\n  late List&lt;Pair?&gt; _buckets; // \u6876\u9663\u5217\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u522a\u9664\u6a19\u8a18\n\n  /* \u5efa\u69cb\u5b50 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* \u96dc\u6e4a\u51fd\u5f0f */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* \u8ca0\u8f09\u56e0\u5b50 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n    while (_buckets[index] != null) {\n      // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n      if (_buckets[index]!.key == key) {\n        // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n        }\n        return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n      }\n      // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n      index = (index + 1) % _capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* \u67e5\u8a62\u64cd\u4f5c */\n  String? get(int key) {\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u65b0\u589e\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* \u522a\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n  void extend() {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u5217\u5370\u96dc\u6e4a\u8868 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nstruct HashMapOpenAddressing {\n    size: usize,                // \u9375\u503c\u5c0d\u6578\u91cf\n    capacity: usize,            // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    load_thres: f64,            // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    extend_ratio: usize,        // \u64f4\u5bb9\u500d\u6578\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // \u6876\u9663\u5217\n    TOMBSTONE: Option&lt;Pair&gt;,    // \u522a\u9664\u6a19\u8a18\n}\n\nimpl HashMapOpenAddressing {\n    /* \u5efa\u69cb\u5b50 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while self.buckets[index].is_some() {\n            // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u5efa\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % self.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        None\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    fn extend(&amp;mut self) {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        let buckets_tmp = self.buckets.clone();\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\ntypedef struct {\n    int size;         // \u9375\u503c\u5c0d\u6578\u91cf\n    int capacity;     // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    double loadThres; // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    int extendRatio;  // \u64f4\u5bb9\u500d\u6578\n    Pair **buckets;   // \u6876\u9663\u5217\n    Pair *TOMBSTONE;  // \u522a\u9664\u6a19\u8a18\n} HashMapOpenAddressing;\n\n/* \u5efa\u69cb\u5b50 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap-&gt;TOMBSTONE);\n    free(hashMap);\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8ca0\u8f09\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n            }\n            return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n        }\n        // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u7a7a\u5b57\u4e32\n    return \"\";\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;size = 0;\n    // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 */\nclass HashMapOpenAddressing {\n    private var size: Int               // \u9375\u503c\u5c0d\u6578\u91cf\n    private var capacity: Int           // \u96dc\u6e4a\u8868\u5bb9\u91cf\n    private val loadThres: Double       // \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    private val extendRatio: Int        // \u64f4\u5bb9\u500d\u6578\n    private var buckets: Array&lt;Pair?&gt;   // \u6876\u9663\u5217\n    private val TOMBSTONE: Pair         // \u522a\u9664\u6a19\u8a18\n\n    /* \u5efa\u69cb\u5b50 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = arrayOfNulls(capacity)\n        TOMBSTONE = Pair(-1, \"-1\")\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8ca0\u8f09\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n            if (buckets[index]?.key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?._val\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de null\n        return null\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fun put(key: Int, _val: String) {\n        // \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u4e26\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!._val = _val\n            return\n        }\n        // \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n        buckets[index] = Pair(key, _val)\n        size++\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        // \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* \u64f4\u5bb9\u96dc\u6e4a\u8868 */\n    fun extend() {\n        // \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair._val}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>### \u958b\u653e\u5b9a\u5740\u96dc\u6e4a\u8868 ###\nclass HashMapOpenAddressing\n  TOMBSTONE = Pair.new(-1, '-1') # \u522a\u9664\u6a19\u8a18\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @size = 0 # \u9375\u503c\u5c0d\u6578\u91cf\n    @capacity = 4 # \u96dc\u6e4a\u8868\u5bb9\u91cf\n    @load_thres = 2.0 / 3.0 # \u89f8\u767c\u64f4\u5bb9\u7684\u8ca0\u8f09\u56e0\u5b50\u95be\u503c\n    @extend_ratio = 2 # \u64f4\u5bb9\u500d\u6578\n    @buckets = Array.new(@capacity) # \u6876\u9663\u5217\n  end\n\n  ### \u96dc\u6e4a\u51fd\u5f0f ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### \u8ca0\u8f09\u56e0\u5b50 ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15 ###\n  def find_bucket(key)\n    index = hash_func(key)\n    first_tombstone = -1\n    # \u7dda\u6027\u63a2\u67e5\uff0c\u7576\u9047\u5230\u7a7a\u6876\u6642\u8df3\u51fa\n    while !@buckets[index].nil?\n      # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n      if @buckets[index].key == key\n        # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u522a\u9664\u6a19\u8a18\uff0c\u5247\u5c07\u9375\u503c\u5c0d\u79fb\u52d5\u81f3\u8a72\u7d22\u5f15\u8655\n        if first_tombstone != -1\n          @buckets[first_tombstone] = @buckets[index]\n          @buckets[index] = TOMBSTONE\n          return first_tombstone # \u8fd4\u56de\u79fb\u52d5\u5f8c\u7684\u6876\u7d22\u5f15\n        end\n        return index # \u8fd4\u56de\u6876\u7d22\u5f15\n      end\n      # \u8a18\u9304\u9047\u5230\u7684\u9996\u500b\u522a\u9664\u6a19\u8a18\n      first_tombstone = index if first_tombstone == -1 &amp;&amp; @buckets[index] == TOMBSTONE\n      # \u8a08\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u904e\u5c3e\u90e8\u5247\u8fd4\u56de\u982d\u90e8\n      index = (index + 1) % @capacity\n    end\n    # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de\u65b0\u589e\u9ede\u7684\u7d22\u5f15\n    first_tombstone == -1 ? index : first_tombstone\n  end\n\n  ### \u67e5\u8a62\u64cd\u4f5c ###\n  def get(key)\n    # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    index = find_bucket(key)\n    # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8fd4\u56de\u5c0d\u61c9 val\n    return @buckets[index].val unless [nil, TOMBSTONE].include?(@buckets[index])\n    # \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u8fd4\u56de nil\n    nil\n  end\n\n  ### \u65b0\u589e\u64cd\u4f5c ###\n  def put(key, val)\n    # \u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e\u95be\u503c\u6642\uff0c\u57f7\u884c\u64f4\u5bb9\n    extend if load_factor &gt; @load_thres\n    # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    index = find_bucket(key)\n    # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u8986\u84cb val \u958b\u8fd4\u56de\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index].val = val\n      return\n    end\n    # \u82e5\u9375\u503c\u5c0d\u4e0d\u5b58\u5728\uff0c\u5247\u65b0\u589e\u8a72\u9375\u503c\u5c0d\n    @buckets[index] = Pair.new(key, val)\n    @size += 1\n  end\n\n  ### \u522a\u9664\u64cd\u4f5c ###\n  def remove(key)\n    # \u641c\u5c0b key \u5c0d\u61c9\u7684\u6876\u7d22\u5f15\n    index = find_bucket(key)\n    # \u82e5\u627e\u5230\u9375\u503c\u5c0d\uff0c\u5247\u7528\u522a\u9664\u6a19\u8a18\u8986\u84cb\u5b83\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index] = TOMBSTONE\n      @size -= 1\n    end\n  end\n\n  ### \u64f4\u5bb9\u96dc\u6e4a\u8868 ###\n  def extend\n    # \u66ab\u5b58\u539f\u96dc\u6e4a\u8868\n    buckets_tmp = @buckets\n    # \u521d\u59cb\u5316\u64f4\u5bb9\u5f8c\u7684\u65b0\u96dc\u6e4a\u8868\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity)\n    @size = 0\n    # \u5c07\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u642c\u904b\u81f3\u65b0\u96dc\u6e4a\u8868\n    for pair in buckets_tmp\n      put(pair.key, pair.val) unless [nil, TOMBSTONE].include?(pair)\n    end\n  end\n\n  ### \u5217\u5370\u96dc\u6e4a\u8868 ###\n  def print\n    for pair in @buckets\n      if pair.nil?\n        puts \"Nil\"\n      elsif pair == TOMBSTONE\n        puts \"TOMBSTONE\"\n      else\n        puts \"#{pair.key} -&gt; #{pair.val}\"\n      end\n    end\n  end\nend\n</code></pre> hash_map_open_addressing.zig<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre>"},{"location":"chapter_hashing/hash_collision/#2","title":"2. \u00a0 \u5e73\u65b9\u63a2\u6e2c","text":"<p>\u5e73\u65b9\u63a2\u6e2c\u8207\u7dda\u6027\u63a2\u67e5\u985e\u4f3c\uff0c\u90fd\u662f\u958b\u653e\u5b9a\u5740\u7684\u5e38\u898b\u7b56\u7565\u4e4b\u4e00\u3002\u7576\u767c\u751f\u885d\u7a81\u6642\uff0c\u5e73\u65b9\u63a2\u6e2c\u4e0d\u662f\u7c21\u55ae\u5730\u8df3\u904e\u4e00\u500b\u56fa\u5b9a\u7684\u6b65\u6578\uff0c\u800c\u662f\u8df3\u904e\u201c\u63a2\u6e2c\u6b21\u6578\u7684\u5e73\u65b9\u201d\u7684\u6b65\u6578\uff0c\u5373 \\(1, 4, 9, \\dots\\) \u6b65\u3002</p> <p>\u5e73\u65b9\u63a2\u6e2c\u4e3b\u8981\u5177\u6709\u4ee5\u4e0b\u512a\u52e2\u3002</p> <ul> <li>\u5e73\u65b9\u63a2\u6e2c\u900f\u904e\u8df3\u904e\u63a2\u6e2c\u6b21\u6578\u5e73\u65b9\u7684\u8ddd\u96e2\uff0c\u8a66\u5716\u7de9\u89e3\u7dda\u6027\u63a2\u67e5\u7684\u805a\u96c6\u6548\u61c9\u3002</li> <li>\u5e73\u65b9\u63a2\u6e2c\u6703\u8df3\u904e\u66f4\u5927\u7684\u8ddd\u96e2\u4f86\u5c0b\u627e\u7a7a\u4f4d\u7f6e\uff0c\u6709\u52a9\u65bc\u8cc7\u6599\u5206\u4f48\u5f97\u66f4\u52a0\u5747\u52fb\u3002</li> </ul> <p>\u7136\u800c\uff0c\u5e73\u65b9\u63a2\u6e2c\u4e26\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002</p> <ul> <li>\u4ecd\u7136\u5b58\u5728\u805a\u96c6\u73fe\u8c61\uff0c\u5373\u67d0\u4e9b\u4f4d\u7f6e\u6bd4\u5176\u4ed6\u4f4d\u7f6e\u66f4\u5bb9\u6613\u88ab\u4f54\u7528\u3002</li> <li>\u7531\u65bc\u5e73\u65b9\u7684\u589e\u9577\uff0c\u5e73\u65b9\u63a2\u6e2c\u53ef\u80fd\u4e0d\u6703\u63a2\u6e2c\u6574\u500b\u96dc\u6e4a\u8868\uff0c\u9019\u610f\u5473\u8457\u5373\u4f7f\u96dc\u6e4a\u8868\u4e2d\u6709\u7a7a\u6876\uff0c\u5e73\u65b9\u63a2\u6e2c\u4e5f\u53ef\u80fd\u7121\u6cd5\u8a2a\u554f\u5230\u5b83\u3002</li> </ul>"},{"location":"chapter_hashing/hash_collision/#3","title":"3. \u00a0 \u591a\u6b21\u96dc\u6e4a","text":"<p>\u9867\u540d\u601d\u7fa9\uff0c\u591a\u6b21\u96dc\u6e4a\u65b9\u6cd5\u4f7f\u7528\u591a\u500b\u96dc\u6e4a\u51fd\u5f0f \\(f_1(x)\\)\u3001\\(f_2(x)\\)\u3001\\(f_3(x)\\)\u3001\\(\\dots\\) \u9032\u884c\u63a2\u6e2c\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u82e5\u96dc\u6e4a\u51fd\u5f0f \\(f_1(x)\\) \u51fa\u73fe\u885d\u7a81\uff0c\u5247\u5617\u8a66 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u985e\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u4f4d\u5f8c\u63d2\u5165\u5143\u7d20\u3002</li> <li>\u67e5\u8a62\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u96dc\u6e4a\u51fd\u5f0f\u9806\u5e8f\u4e0b\u9032\u884c\u67e5\u8a62\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6a19\u5143\u7d20\u6642\u8fd4\u56de\uff1b\u82e5\u9047\u5230\u7a7a\u4f4d\u6216\u5df2\u5617\u8a66\u6240\u6709\u96dc\u6e4a\u51fd\u5f0f\uff0c\u8aaa\u660e\u96dc\u6e4a\u8868\u4e2d\u4e0d\u5b58\u5728\u8a72\u5143\u7d20\uff0c\u5247\u8fd4\u56de <code>None</code> \u3002</li> </ul> <p>\u8207\u7dda\u6027\u63a2\u67e5\u76f8\u6bd4\uff0c\u591a\u6b21\u96dc\u6e4a\u65b9\u6cd5\u4e0d\u6613\u7522\u751f\u805a\u96c6\uff0c\u4f46\u591a\u500b\u96dc\u6e4a\u51fd\u5f0f\u6703\u5e36\u4f86\u984d\u5916\u7684\u8a08\u7b97\u91cf\u3002</p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u958b\u653e\u5b9a\u5740\uff08\u7dda\u6027\u63a2\u67e5\u3001\u5e73\u65b9\u63a2\u6e2c\u548c\u591a\u6b21\u96dc\u6e4a\uff09\u96dc\u6e4a\u8868\u90fd\u5b58\u5728\u201c\u4e0d\u80fd\u76f4\u63a5\u522a\u9664\u5143\u7d20\u201d\u7684\u554f\u984c\u3002</p>"},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3 \u00a0 \u7a0b\u5f0f\u8a9e\u8a00\u7684\u9078\u64c7","text":"<p>\u5404\u7a2e\u7a0b\u5f0f\u8a9e\u8a00\u63a1\u53d6\u4e86\u4e0d\u540c\u7684\u96dc\u6e4a\u8868\u5be6\u73fe\u7b56\u7565\uff0c\u4e0b\u9762\u8209\u5e7e\u500b\u4f8b\u5b50\u3002</p> <ul> <li>Python \u63a1\u7528\u958b\u653e\u5b9a\u5740\u3002\u5b57\u5178 <code>dict</code> \u4f7f\u7528\u507d\u96a8\u6a5f\u6578\u9032\u884c\u63a2\u6e2c\u3002</li> <li>Java \u63a1\u7528\u93c8\u5f0f\u4f4d\u5740\u3002\u81ea JDK 1.8 \u4ee5\u4f86\uff0c\u7576 <code>HashMap</code> \u5167\u9663\u5217\u9577\u5ea6\u9054\u5230 64 \u4e14\u93c8\u7d50\u4e32\u5217\u9577\u5ea6\u9054\u5230 8 \u6642\uff0c\u93c8\u7d50\u4e32\u5217\u6703\u8f49\u63db\u70ba\u7d05\u9ed1\u6a39\u4ee5\u63d0\u5347\u67e5\u8a62\u6548\u80fd\u3002</li> <li>Go \u63a1\u7528\u93c8\u5f0f\u4f4d\u5740\u3002Go \u898f\u5b9a\u6bcf\u500b\u6876\u6700\u591a\u5132\u5b58 8 \u500b\u9375\u503c\u5c0d\uff0c\u8d85\u51fa\u5bb9\u91cf\u5247\u9023\u7dda\u4e00\u500b\u6ea2\u4f4d\u6876\uff1b\u7576\u6ea2\u4f4d\u6876\u904e\u591a\u6642\uff0c\u6703\u57f7\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u64f4\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u78ba\u4fdd\u6548\u80fd\u3002</li> </ul>"},{"location":"chapter_hashing/hash_map/","title":"6.1 \u00a0 \u96dc\u6e4a\u8868","text":"<p>\u96dc\u6e4a\u8868\uff08hash table\uff09\uff0c\u53c8\u7a31\u6563\u5217\u8868\uff0c\u5b83\u900f\u904e\u5efa\u7acb\u9375 <code>key</code> \u8207\u503c <code>value</code> \u4e4b\u9593\u7684\u5c0d\u6620\uff0c\u5be6\u73fe\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8a62\u3002\u5177\u9ad4\u800c\u8a00\uff0c\u6211\u5011\u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u4e00\u500b\u9375 <code>key</code> \uff0c\u5247\u53ef\u4ee5\u5728 \\(O(1)\\) \u6642\u9593\u5167\u7372\u53d6\u5c0d\u61c9\u7684\u503c <code>value</code> \u3002</p> <p>\u5982\u5716 6-1 \u6240\u793a\uff0c\u7d66\u5b9a \\(n\\) \u500b\u5b78\u751f\uff0c\u6bcf\u500b\u5b78\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b78\u865f\u201d\u5169\u9805\u8cc7\u6599\u3002\u5047\u5982\u6211\u5011\u5e0c\u671b\u5be6\u73fe\u201c\u8f38\u5165\u4e00\u500b\u5b78\u865f\uff0c\u8fd4\u56de\u5c0d\u61c9\u7684\u59d3\u540d\u201d\u7684\u67e5\u8a62\u529f\u80fd\uff0c\u5247\u53ef\u4ee5\u63a1\u7528\u5716 6-1 \u6240\u793a\u7684\u96dc\u6e4a\u8868\u4f86\u5be6\u73fe\u3002</p> <p></p> <p> \u5716 6-1 \u00a0 \u96dc\u6e4a\u8868\u7684\u62bd\u8c61\u8868\u793a </p> <p>\u9664\u96dc\u6e4a\u8868\u5916\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u4e5f\u53ef\u4ee5\u5be6\u73fe\u67e5\u8a62\u529f\u80fd\uff0c\u5b83\u5011\u7684\u6548\u7387\u5c0d\u6bd4\u5982\u8868 6-1 \u6240\u793a\u3002</p> <ul> <li>\u65b0\u589e\u5143\u7d20\uff1a\u50c5\u9700\u5c07\u5143\u7d20\u65b0\u589e\u81f3\u9663\u5217\uff08\u93c8\u7d50\u4e32\u5217\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u6642\u9593\u3002</li> <li>\u67e5\u8a62\u5143\u7d20\uff1a\u7531\u65bc\u9663\u5217\uff08\u93c8\u7d50\u4e32\u5217\uff09\u662f\u4e82\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u8d70\u8a2a\u5176\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002</li> <li>\u522a\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8a62\u5230\u5143\u7d20\uff0c\u518d\u5f9e\u9663\u5217\uff08\u93c8\u7d50\u4e32\u5217\uff09\u4e2d\u522a\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002</li> </ul> <p> \u8868 6-1 \u00a0 \u5143\u7d20\u67e5\u8a62\u6548\u7387\u5c0d\u6bd4 </p> \u9663\u5217 \u93c8\u7d50\u4e32\u5217 \u96dc\u6e4a\u8868 \u67e5\u8a62\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u65b0\u589e\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u522a\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>\u89c0\u5bdf\u767c\u73fe\uff0c\u5728\u96dc\u6e4a\u8868\u4e2d\u9032\u884c\u589e\u522a\u67e5\u6539\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1 \u00a0 \u96dc\u6e4a\u8868\u5e38\u7528\u64cd\u4f5c","text":"<p>\u96dc\u6e4a\u8868\u7684\u5e38\u898b\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8a62\u64cd\u4f5c\u3001\u65b0\u589e\u9375\u503c\u5c0d\u548c\u522a\u9664\u9375\u503c\u5c0d\u7b49\uff0c\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map.py<pre><code># \u521d\u59cb\u5316\u96dc\u6e4a\u8868\nhmap: dict = {}\n\n# \u65b0\u589e\u64cd\u4f5c\n# \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u56c9\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9d28\"\n\n# \u67e5\u8a62\u64cd\u4f5c\n# \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nname: str = hmap[15937]\n\n# \u522a\u9664\u64cd\u4f5c\n# \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nunordered_map&lt;int, string&gt; map;\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u56c9\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9d28\";\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");\nmap.put(15937, \"\u5c0f\u56c9\");\nmap.put(16750, \"\u5c0f\u7b97\");\nmap.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9d28\");\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nDictionary&lt;int, string&gt; map = new() {\n    /* \u65b0\u589e\u64cd\u4f5c */\n    // \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\n    { 12836, \"\u5c0f\u54c8\" },\n    { 15937, \"\u5c0f\u56c9\" },\n    { 16750, \"\u5c0f\u7b97\" },\n    { 13276, \"\u5c0f\u6cd5\" },\n    { 10583, \"\u5c0f\u9d28\" }\n};\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nhmap := make(map[int]string)\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u56c9\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9d28\"\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nname := hmap[15937]\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nvar map: [Int: String] = [:]\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u56c9\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9d28\"\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nconst map = new Map();\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u56c9');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9d28');\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nconst map = new Map&lt;number, string&gt;();\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u56c9');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9d28');\nconsole.info('\\n\u65b0\u589e\u5b8c\u6210\u5f8c\uff0c\u96dc\u6e4a\u8868\u70ba\\nKey -&gt; Value');\nconsole.info(map);\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f38\u5165\u5b78\u865f 15937 \uff0c\u67e5\u8a62\u5230\u59d3\u540d ' + name);\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.delete(10583);\nconsole.info('\\n\u522a\u9664 10583 \u5f8c\uff0c\u96dc\u6e4a\u8868\u70ba\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nMap&lt;int, String&gt; map = {};\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u56c9\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9d28\";\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap.insert(12836, \"\u5c0f\u54c8\".to_string());\nmap.insert(15937, \"\u5c0f\u56c9\".to_string());\nmap.insert(16750, \"\u5c0f\u7b97\".to_string());\nmap.insert(13279, \"\u5c0f\u6cd5\".to_string());\nmap.insert(10583, \"\u5c0f\u9d28\".to_string());\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u96dc\u6e4a\u8868\n</code></pre> hash_map.kt<pre><code>/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nval map = HashMap&lt;Int,String&gt;()\n\n/* \u65b0\u589e\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u56c9\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9d28\"\n\n/* \u67e5\u8a62\u64cd\u4f5c */\n// \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nval name = map[15937]\n\n/* \u522a\u9664\u64cd\u4f5c */\n// \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nmap.remove(10583)\n</code></pre> hash_map.rb<pre><code># \u521d\u59cb\u5316\u96dc\u6e4a\u8868\nhmap = {}\n\n# \u65b0\u589e\u64cd\u4f5c\n# \u5728\u96dc\u6e4a\u8868\u4e2d\u65b0\u589e\u9375\u503c\u5c0d (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u56c9\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9d28\"\n\n# \u67e5\u8a62\u64cd\u4f5c\n# \u5411\u96dc\u6e4a\u8868\u4e2d\u8f38\u5165\u9375 key \uff0c\u5f97\u5230\u503c value\nname = hmap[15937]\n\n# \u522a\u9664\u64cd\u4f5c\n# \u5728\u96dc\u6e4a\u8868\u4e2d\u522a\u9664\u9375\u503c\u5c0d (key, value)\nhmap.delete(10583)\n</code></pre> hash_map.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u96dc\u6e4a\u8868\u6709\u4e09\u7a2e\u5e38\u7528\u7684\u8d70\u8a2a\u65b9\u5f0f\uff1a\u8d70\u8a2a\u9375\u503c\u5c0d\u3001\u8d70\u8a2a\u9375\u548c\u8d70\u8a2a\u503c\u3002\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map.py<pre><code># \u8d70\u8a2a\u96dc\u6e4a\u8868\n# \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# \u55ae\u7368\u8d70\u8a2a\u9375 key\nfor key in hmap.keys():\n    print(key)\n# \u55ae\u7368\u8d70\u8a2a\u503c value\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u5668\u8d70\u8a2a key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 key\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c value\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 key\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c value\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 key\nfor key := range hmap {\n    fmt.Println(key)\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c value\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 Key\nfor key in map.keys {\n    print(key)\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c Value\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\nconsole.info('\\n\u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u55ae\u7368\u8d70\u8a2a\u9375 Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u55ae\u7368\u8d70\u8a2a\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\nconsole.info('\\n\u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u55ae\u7368\u8d70\u8a2a\u9375 Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u55ae\u7368\u8d70\u8a2a\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// \u55ae\u7368\u8d70\u8a2a\u9375 Key\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// \u55ae\u7368\u8d70\u8a2a\u503c Value\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// \u55ae\u7368\u8d70\u8a2a\u9375 Key\nfor key in map.keys() {\n    println!(\"{key}\");\n}\n\n// \u55ae\u7368\u8d70\u8a2a\u503c Value\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u96dc\u6e4a\u8868\n</code></pre> hash_map.kt<pre><code>/* \u8d70\u8a2a\u96dc\u6e4a\u8868 */\n// \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nfor ((key, value) in map) {\n    println(\"$key -&gt; $value\")\n}\n// \u55ae\u7368\u8d70\u8a2a\u9375 key\nfor (key in map.keys) {\n    println(key)\n}\n// \u55ae\u7368\u8d70\u8a2a\u503c value\nfor (_val in map.values) {\n    println(_val)\n}\n</code></pre> hash_map.rb<pre><code># \u8d70\u8a2a\u96dc\u6e4a\u8868\n# \u8d70\u8a2a\u9375\u503c\u5c0d key-&gt;value\nhmap.entries.each { |key, value| puts \"#{key} -&gt; #{value}\" }\n\n# \u55ae\u7368\u8d70\u8a2a\u9375 key\nhmap.keys.each { |key| puts key }\n\n# \u55ae\u7368\u8d70\u8a2a\u503c value\nhmap.values.each { |val| puts val }\n</code></pre> hash_map.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2 \u00a0 \u96dc\u6e4a\u8868\u7c21\u55ae\u5be6\u73fe","text":"<p>\u6211\u5011\u5148\u8003\u616e\u6700\u7c21\u55ae\u7684\u60c5\u6cc1\uff0c\u50c5\u7528\u4e00\u500b\u9663\u5217\u4f86\u5be6\u73fe\u96dc\u6e4a\u8868\u3002\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u6211\u5011\u5c07\u9663\u5217\u4e2d\u7684\u6bcf\u500b\u7a7a\u4f4d\u7a31\u70ba\u6876\uff08bucket\uff09\uff0c\u6bcf\u500b\u6876\u53ef\u5132\u5b58\u4e00\u500b\u9375\u503c\u5c0d\u3002\u56e0\u6b64\uff0c\u67e5\u8a62\u64cd\u4f5c\u5c31\u662f\u627e\u5230 <code>key</code> \u5c0d\u61c9\u7684\u6876\uff0c\u4e26\u5728\u6876\u4e2d\u7372\u53d6 <code>value</code> \u3002</p> <p>\u90a3\u9ebc\uff0c\u5982\u4f55\u57fa\u65bc <code>key</code> \u5b9a\u4f4d\u5c0d\u61c9\u7684\u6876\u5462\uff1f\u9019\u662f\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\uff08hash function\uff09\u5be6\u73fe\u7684\u3002\u96dc\u6e4a\u51fd\u5f0f\u7684\u4f5c\u7528\u662f\u5c07\u4e00\u500b\u8f03\u5927\u7684\u8f38\u5165\u7a7a\u9593\u5c0d\u6620\u5230\u4e00\u500b\u8f03\u5c0f\u7684\u8f38\u51fa\u7a7a\u9593\u3002\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u8f38\u5165\u7a7a\u9593\u662f\u6240\u6709 <code>key</code> \uff0c\u8f38\u51fa\u7a7a\u9593\u662f\u6240\u6709\u6876\uff08\u9663\u5217\u7d22\u5f15\uff09\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u8f38\u5165\u4e00\u500b <code>key</code> \uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u5f97\u5230\u8a72 <code>key</code> \u5c0d\u61c9\u7684\u9375\u503c\u5c0d\u5728\u9663\u5217\u4e2d\u7684\u5132\u5b58\u4f4d\u7f6e\u3002</p> <p>\u8f38\u5165\u4e00\u500b <code>key</code> \uff0c\u96dc\u6e4a\u51fd\u5f0f\u7684\u8a08\u7b97\u904e\u7a0b\u5206\u70ba\u4ee5\u4e0b\u5169\u6b65\u3002</p> <ol> <li>\u900f\u904e\u67d0\u7a2e\u96dc\u6e4a\u6f14\u7b97\u6cd5 <code>hash()</code> \u8a08\u7b97\u5f97\u5230\u96dc\u6e4a\u503c\u3002</li> <li>\u5c07\u96dc\u6e4a\u503c\u5c0d\u6876\u6578\u91cf\uff08\u9663\u5217\u9577\u5ea6\uff09<code>capacity</code> \u53d6\u6a21\uff0c\u5f9e\u800c\u7372\u53d6\u8a72 <code>key</code> \u5c0d\u61c9\u7684\u9663\u5217\u7d22\u5f15 <code>index</code> \u3002</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u96a8\u5f8c\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u5229\u7528 <code>index</code> \u5728\u96dc\u6e4a\u8868\u4e2d\u8a2a\u554f\u5c0d\u61c9\u7684\u6876\uff0c\u5f9e\u800c\u7372\u53d6 <code>value</code> \u3002</p> <p>\u8a2d\u9663\u5217\u9577\u5ea6 <code>capacity = 100</code>\u3001\u96dc\u6e4a\u6f14\u7b97\u6cd5 <code>hash(key) = key</code> \uff0c\u6613\u5f97\u96dc\u6e4a\u51fd\u5f0f\u70ba <code>key % 100</code> \u3002\u5716 6-2 \u4ee5 <code>key</code> \u5b78\u865f\u548c <code>value</code> \u59d3\u540d\u70ba\u4f8b\uff0c\u5c55\u793a\u4e86\u96dc\u6e4a\u51fd\u5f0f\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p></p> <p> \u5716 6-2 \u00a0 \u96dc\u6e4a\u51fd\u5f0f\u5de5\u4f5c\u539f\u7406 </p> <p>\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e86\u4e00\u500b\u7c21\u55ae\u96dc\u6e4a\u8868\u3002\u5176\u4e2d\uff0c\u6211\u5011\u5c07 <code>key</code> \u548c <code>value</code> \u5c01\u88dd\u6210\u4e00\u500b\u985e\u5225 <code>Pair</code> \uff0c\u4ee5\u8868\u793a\u9375\u503c\u5c0d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_hash_map.py<pre><code>class Pair:\n    \"\"\"\u9375\u503c\u5c0d\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        # \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u96dc\u6e4a\u51fd\u5f0f\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8a62\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"\u65b0\u589e\u64cd\u4f5c\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"\u522a\u9664\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        # \u7f6e\u70ba None \uff0c\u4ee3\u8868\u522a\u9664\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"\u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"\u7372\u53d6\u6240\u6709\u9375\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"\u7372\u53d6\u6240\u6709\u503c\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"\u5217\u5370\u96dc\u6e4a\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* \u9375\u503c\u5c0d */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\u4e26\u7f6e\u70ba nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* \u9375\u503c\u5c0d */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        buckets.set(index, null);\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* \u9375\u503c\u5c0d int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        buckets[index] = null;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* \u9375\u503c\u5c0d */\ntype pair struct {\n    key int\n    val string\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u96dc\u6e4a\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n    // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* \u96dc\u6e4a\u51fd\u5f0f */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* \u67e5\u8a62\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // \u7f6e\u70ba nil \uff0c\u4ee3\u8868\u522a\u9664\n    a.buckets[index] = nil\n}\n\n/* \u7372\u53d6\u6240\u6709\u9375\u5c0d */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* \u7372\u53d6\u6240\u6709\u9375 */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* \u7372\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* \u9375\u503c\u5c0d */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // \u7f6e\u70ba nil \uff0c\u4ee3\u8868\u522a\u9664\n        buckets[index] = nil\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* \u9375\u503c\u5c0d Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        this.#buckets[index] = null;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* \u9375\u503c\u5c0d Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        this.buckets[index] = null;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* \u9375\u503c\u5c0d */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n    _buckets = List.filled(100, null);\n  }\n\n  /* \u96dc\u6e4a\u51fd\u5f0f */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* \u67e5\u8a62\u64cd\u4f5c */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* \u65b0\u589e\u64cd\u4f5c */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* \u522a\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* \u7372\u53d6\u6240\u6709\u9375 */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* \u7372\u53d6\u6240\u6709\u503c */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* \u5217\u5370\u96dc\u6e4a\u8868 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* \u9375\u503c\u5c0d */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // \u7f6e\u70ba None \uff0c\u4ee3\u8868\u522a\u9664\n        self.buckets[index] = None;\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* \u9375\u503c\u5c0d int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\ntypedef struct {\n    Pair *buckets[MAX_SIZE];\n} ArrayHashMap;\n\n/* \u5efa\u69cb\u5b50 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    for (int i=0; i &lt; MAX_SIZE; i++) {\n        hmap-&gt;buckets[i] = NULL;\n    }\n    return hmap;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* \u65b0\u589e\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* \u522a\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7d71\u8a08\u6709\u6548\u9375\u503c\u5c0d\u6578\u91cf */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* \u7372\u53d6\u6240\u6709\u9375 */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7d71\u8a08\u6709\u6548\u9375\u503c\u5c0d\u6578\u91cf */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* \u7372\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7d71\u8a08\u6709\u6548\u9375\u503c\u5c0d\u6578\u91cf */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* \u5217\u5370\u96dc\u6e4a\u8868 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* \u9375\u503c\u5c0d */\nclass Pair(\n    var key: Int,\n    var _val: String\n)\n\n/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 */\nclass ArrayHashMap {\n    // \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    /* \u96dc\u6e4a\u51fd\u5f0f */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8a62\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair._val\n    }\n\n    /* \u65b0\u589e\u64cd\u4f5c */\n    fun put(key: Int, _val: String) {\n        val pair = Pair(key, _val)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u522a\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n        buckets[index] = null\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = mutableListOf&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u9375 */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = mutableListOf&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u7372\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = mutableListOf&lt;String&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                valueSet.add(pair._val)\n        }\n        return valueSet\n    }\n\n    /* \u5217\u5370\u96dc\u6e4a\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val _val = kv._val\n            println(\"$key -&gt; $_val\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>### \u9375\u503c\u5c0d ###\nclass Pair\n  attr_accessor :key, :val\n\n  def initialize(key, val)\n    @key = key\n    @val = val\n  end\nend\n\n### \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868 ###\nclass ArrayHashMap\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    # \u521d\u59cb\u5316\u9663\u5217\uff0c\u5305\u542b 100 \u500b\u6876\n    @buckets = Array.new(100)\n  end\n\n  ### \u96dc\u6e4a\u51fd\u5f0f ###\n  def hash_func(key)\n    index = key % 100\n  end\n\n  ### \u67e5\u8a62\u64cd\u4f5c ###\n  def get(key)\n    index = hash_func(key)\n    pair = @buckets[index]\n\n    return if pair.nil?\n    pair.val\n  end\n\n  ### \u65b0\u589e\u64cd\u4f5c ###\n  def put(key, val)\n    pair = Pair.new(key, val)\n    index = hash_func(key)\n    @buckets[index] = pair\n  end\n\n  ### \u522a\u9664\u64cd\u4f5c ###\n  def remove(key)\n    index = hash_func(key)\n    # \u7f6e\u70ba nil \uff0c\u4ee3\u8868\u522a\u9664\n    @buckets[index] = nil\n  end\n\n  ### \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d ###\n  def entry_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair unless pair.nil? }\n    result\n  end\n\n  ### \u7372\u53d6\u6240\u6709\u9375 ###\n  def key_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.key unless pair.nil? }\n    result\n  end\n\n  ### \u7372\u53d6\u6240\u6709\u503c ###\n  def value_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.val unless pair.nil? }\n    result\n  end\n\n  ### \u5217\u5370\u96dc\u6e4a\u8868 ###\n  def print\n    @buckets.each { |pair| puts \"#{pair.key} -&gt; #{pair.val}\" unless pair.nil? }\n  end\nend\n</code></pre> array_hash_map.zig<pre><code>// \u9375\u503c\u5c0d\nconst Pair = struct {\n    key: usize = undefined,\n    val: []const u8 = undefined,\n\n   pub fn init(key: usize, val: []const u8) Pair {\n        return Pair {\n            .key = key,\n            .val = val,\n        };\n    }\n};\n\n// \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u96dc\u6e4a\u8868\nfn ArrayHashMap(comptime T: type) type {\n    return struct {\n        bucket: ?std.ArrayList(?T) = null,\n        mem_allocator: std.mem.Allocator = undefined,\n\n        const Self = @This();\n\n        // \u5efa\u69cb\u5b50\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            self.mem_allocator = allocator;\n            // \u521d\u59cb\u5316\u4e00\u500b\u9577\u5ea6\u70ba 100 \u7684\u6876\uff08\u9663\u5217\uff09\n            self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n            var i: i32 = 0;\n            while (i &lt; 100) : (i += 1) {\n                try self.bucket.?.append(null);\n            }\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\n        pub fn deinit(self: *Self) void {\n            if (self.bucket != null) self.bucket.?.deinit();\n        }\n\n        // \u96dc\u6e4a\u51fd\u5f0f\n        fn hashFunc(key: usize) usize {\n            var index = key % 100;\n            return index;\n        }\n\n        // \u67e5\u8a62\u64cd\u4f5c\n        pub fn get(self: *Self, key: usize) []const u8 {\n            var index = hashFunc(key);\n            var pair = self.bucket.?.items[index];\n            return pair.?.val;\n        }\n\n        // \u65b0\u589e\u64cd\u4f5c\n        pub fn put(self: *Self, key: usize, val: []const u8) !void {\n            var pair = Pair.init(key, val);\n            var index = hashFunc(key);\n            self.bucket.?.items[index] = pair;\n        }\n\n        // \u522a\u9664\u64cd\u4f5c\n        pub fn remove(self: *Self, key: usize) !void {\n            var index = hashFunc(key);\n            // \u7f6e\u70ba null \uff0c\u4ee3\u8868\u522a\u9664\n            self.bucket.?.items[index] = null;\n        }       \n\n        // \u7372\u53d6\u6240\u6709\u9375\u503c\u5c0d\n        pub fn pairSet(self: *Self) !std.ArrayList(T) {\n            var entry_set = std.ArrayList(T).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try entry_set.append(item.?);\n            }\n            return entry_set;\n        }  \n\n        // \u7372\u53d6\u6240\u6709\u9375\n        pub fn keySet(self: *Self) !std.ArrayList(usize) {\n            var key_set = std.ArrayList(usize).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try key_set.append(item.?.key);\n            }\n            return key_set;\n        }  \n\n        // \u7372\u53d6\u6240\u6709\u503c\n        pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n            var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try value_set.append(item.?.val);\n            }\n            return value_set;\n        }\n\n        // \u5217\u5370\u96dc\u6e4a\u8868\n        pub fn print(self: *Self) !void {\n            var entry_set = try self.pairSet();\n            defer entry_set.deinit();\n            for (entry_set.items) |item| {\n                std.debug.print(\"{} -&gt; {s}\\n\", .{item.key, item.val});\n            }\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3 \u00a0 \u96dc\u6e4a\u885d\u7a81\u8207\u64f4\u5bb9","text":"<p>\u5f9e\u672c\u8cea\u4e0a\u770b\uff0c\u96dc\u6e4a\u51fd\u5f0f\u7684\u4f5c\u7528\u662f\u5c07\u6240\u6709 <code>key</code> \u69cb\u6210\u7684\u8f38\u5165\u7a7a\u9593\u5c0d\u6620\u5230\u9663\u5217\u6240\u6709\u7d22\u5f15\u69cb\u6210\u7684\u8f38\u51fa\u7a7a\u9593\uff0c\u800c\u8f38\u5165\u7a7a\u9593\u5f80\u5f80\u9060\u5927\u65bc\u8f38\u51fa\u7a7a\u9593\u3002\u56e0\u6b64\uff0c\u7406\u8ad6\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u500b\u8f38\u5165\u5c0d\u61c9\u76f8\u540c\u8f38\u51fa\u201d\u7684\u60c5\u6cc1\u3002</p> <p>\u5c0d\u65bc\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u96dc\u6e4a\u51fd\u5f0f\uff0c\u7576\u8f38\u5165\u7684 <code>key</code> \u5f8c\u5169\u4f4d\u76f8\u540c\u6642\uff0c\u96dc\u6e4a\u51fd\u5f0f\u7684\u8f38\u51fa\u7d50\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8a62\u5b78\u865f\u70ba 12836 \u548c 20336 \u7684\u5169\u500b\u5b78\u751f\u6642\uff0c\u6211\u5011\u5f97\u5230\uff1a</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>\u5982\u5716 6-3 \u6240\u793a\uff0c\u5169\u500b\u5b78\u865f\u6307\u5411\u4e86\u540c\u4e00\u500b\u59d3\u540d\uff0c\u9019\u986f\u7136\u662f\u4e0d\u5c0d\u7684\u3002\u6211\u5011\u5c07\u9019\u7a2e\u591a\u500b\u8f38\u5165\u5c0d\u61c9\u540c\u4e00\u8f38\u51fa\u7684\u60c5\u6cc1\u7a31\u70ba\u96dc\u6e4a\u885d\u7a81\uff08hash collision\uff09\u3002</p> <p></p> <p> \u5716 6-3 \u00a0 \u96dc\u6e4a\u885d\u7a81\u793a\u4f8b </p> <p>\u5bb9\u6613\u60f3\u5230\uff0c\u96dc\u6e4a\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u500b <code>key</code> \u88ab\u5206\u914d\u5230\u540c\u4e00\u500b\u6876\u4e2d\u7684\u6a5f\u7387\u5c31\u8d8a\u4f4e\uff0c\u885d\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u64f4\u5bb9\u96dc\u6e4a\u8868\u4f86\u6e1b\u5c11\u96dc\u6e4a\u885d\u7a81\u3002</p> <p>\u5982\u5716 6-4 \u6240\u793a\uff0c\u64f4\u5bb9\u524d\u9375\u503c\u5c0d <code>(136, A)</code> \u548c <code>(236, D)</code> \u767c\u751f\u885d\u7a81\uff0c\u64f4\u5bb9\u5f8c\u885d\u7a81\u6d88\u5931\u3002</p> <p></p> <p> \u5716 6-4 \u00a0 \u96dc\u6e4a\u8868\u64f4\u5bb9 </p> <p>\u985e\u4f3c\u65bc\u9663\u5217\u64f4\u5bb9\uff0c\u96dc\u6e4a\u8868\u64f4\u5bb9\u9700\u5c07\u6240\u6709\u9375\u503c\u5c0d\u5f9e\u539f\u96dc\u6e4a\u8868\u9077\u79fb\u81f3\u65b0\u96dc\u6e4a\u8868\uff0c\u975e\u5e38\u8017\u6642\uff1b\u4e26\u4e14\u7531\u65bc\u96dc\u6e4a\u8868\u5bb9\u91cf <code>capacity</code> \u6539\u8b8a\uff0c\u6211\u5011\u9700\u8981\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u4f86\u91cd\u65b0\u8a08\u7b97\u6240\u6709\u9375\u503c\u5c0d\u7684\u5132\u5b58\u4f4d\u7f6e\uff0c\u9019\u9032\u4e00\u6b65\u589e\u52a0\u4e86\u64f4\u5bb9\u904e\u7a0b\u7684\u8a08\u7b97\u958b\u92b7\u3002\u70ba\u6b64\uff0c\u7a0b\u5f0f\u8a9e\u8a00\u901a\u5e38\u6703\u9810\u7559\u8db3\u5920\u5927\u7684\u96dc\u6e4a\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u983b\u7e41\u64f4\u5bb9\u3002</p> <p>\u8ca0\u8f09\u56e0\u5b50\uff08load factor\uff09\u662f\u96dc\u6e4a\u8868\u7684\u4e00\u500b\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u7fa9\u70ba\u96dc\u6e4a\u8868\u7684\u5143\u7d20\u6578\u91cf\u9664\u4ee5\u6876\u6578\u91cf\uff0c\u7528\u65bc\u8861\u91cf\u96dc\u6e4a\u885d\u7a81\u7684\u56b4\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u4f5c\u70ba\u96dc\u6e4a\u8868\u64f4\u5bb9\u7684\u89f8\u767c\u689d\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u7576\u8ca0\u8f09\u56e0\u5b50\u8d85\u904e \\(0.75\\) \u6642\uff0c\u7cfb\u7d71\u6703\u5c07\u96dc\u6e4a\u8868\u64f4\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</p>"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_hashing/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u8f38\u5165 <code>key</code> \uff0c\u96dc\u6e4a\u8868\u80fd\u5920\u5728 \\(O(1)\\) \u6642\u9593\u5167\u67e5\u8a62\u5230 <code>value</code> \uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002</li> <li>\u5e38\u898b\u7684\u96dc\u6e4a\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8a62\u3001\u65b0\u589e\u9375\u503c\u5c0d\u3001\u522a\u9664\u9375\u503c\u5c0d\u548c\u8d70\u8a2a\u96dc\u6e4a\u8868\u7b49\u3002</li> <li>\u96dc\u6e4a\u51fd\u5f0f\u5c07 <code>key</code> \u5c0d\u6620\u70ba\u9663\u5217\u7d22\u5f15\uff0c\u5f9e\u800c\u8a2a\u554f\u5c0d\u61c9\u6876\u4e26\u7372\u53d6 <code>value</code> \u3002</li> <li>\u5169\u500b\u4e0d\u540c\u7684 <code>key</code> \u53ef\u80fd\u5728\u7d93\u904e\u96dc\u6e4a\u51fd\u5f0f\u5f8c\u5f97\u5230\u76f8\u540c\u7684\u9663\u5217\u7d22\u5f15\uff0c\u5c0e\u81f4\u67e5\u8a62\u7d50\u679c\u51fa\u932f\uff0c\u9019\u7a2e\u73fe\u8c61\u88ab\u7a31\u70ba\u96dc\u6e4a\u885d\u7a81\u3002</li> <li>\u96dc\u6e4a\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u96dc\u6e4a\u885d\u7a81\u7684\u6a5f\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u900f\u904e\u64f4\u5bb9\u96dc\u6e4a\u8868\u4f86\u7de9\u89e3\u96dc\u6e4a\u885d\u7a81\u3002\u8207\u9663\u5217\u64f4\u5bb9\u985e\u4f3c\uff0c\u96dc\u6e4a\u8868\u64f4\u5bb9\u64cd\u4f5c\u7684\u958b\u92b7\u5f88\u5927\u3002</li> <li>\u8ca0\u8f09\u56e0\u5b50\u5b9a\u7fa9\u70ba\u96dc\u6e4a\u8868\u4e2d\u5143\u7d20\u6578\u91cf\u9664\u4ee5\u6876\u6578\u91cf\uff0c\u53cd\u6620\u4e86\u96dc\u6e4a\u885d\u7a81\u7684\u56b4\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89f8\u767c\u96dc\u6e4a\u8868\u64f4\u5bb9\u7684\u689d\u4ef6\u3002</li> <li>\u93c8\u5f0f\u4f4d\u5740\u900f\u904e\u5c07\u55ae\u500b\u5143\u7d20\u8f49\u5316\u70ba\u93c8\u7d50\u4e32\u5217\uff0c\u5c07\u6240\u6709\u885d\u7a81\u5143\u7d20\u5132\u5b58\u5728\u540c\u4e00\u500b\u93c8\u7d50\u4e32\u5217\u4e2d\u3002\u7136\u800c\uff0c\u93c8\u7d50\u4e32\u5217\u904e\u9577\u6703\u964d\u4f4e\u67e5\u8a62\u6548\u7387\uff0c\u53ef\u4ee5\u900f\u904e\u9032\u4e00\u6b65\u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u63db\u70ba\u7d05\u9ed1\u6a39\u4f86\u63d0\u9ad8\u6548\u7387\u3002</li> <li>\u958b\u653e\u5b9a\u5740\u900f\u904e\u591a\u6b21\u63a2\u6e2c\u4f86\u8655\u7406\u96dc\u6e4a\u885d\u7a81\u3002\u7dda\u6027\u63a2\u67e5\u4f7f\u7528\u56fa\u5b9a\u6b65\u9577\uff0c\u7f3a\u9ede\u662f\u4e0d\u80fd\u522a\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u7522\u751f\u805a\u96c6\u3002\u591a\u6b21\u96dc\u6e4a\u4f7f\u7528\u591a\u500b\u96dc\u6e4a\u51fd\u5f0f\u9032\u884c\u63a2\u6e2c\uff0c\u76f8\u8f03\u7dda\u6027\u63a2\u67e5\u66f4\u4e0d\u6613\u7522\u751f\u805a\u96c6\uff0c\u4f46\u591a\u500b\u96dc\u6e4a\u51fd\u5f0f\u589e\u52a0\u4e86\u8a08\u7b97\u91cf\u3002</li> <li>\u4e0d\u540c\u7a0b\u5f0f\u8a9e\u8a00\u63a1\u53d6\u4e86\u4e0d\u540c\u7684\u96dc\u6e4a\u8868\u5be6\u73fe\u3002\u4f8b\u5982\uff0cJava \u7684 <code>HashMap</code> \u4f7f\u7528\u93c8\u5f0f\u4f4d\u5740\uff0c\u800c Python \u7684 <code>Dict</code> \u63a1\u7528\u958b\u653e\u5b9a\u5740\u3002</li> <li>\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u6211\u5011\u5e0c\u671b\u96dc\u6e4a\u6f14\u7b97\u6cd5\u5177\u6709\u78ba\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u52fb\u5206\u4f48\u7684\u7279\u9ede\u3002\u5728\u5bc6\u78bc\u5b78\u4e2d\uff0c\u96dc\u6e4a\u6f14\u7b97\u6cd5\u9084\u61c9\u8a72\u5177\u5099\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u61c9\u3002</li> <li>\u96dc\u6e4a\u6f14\u7b97\u6cd5\u901a\u5e38\u63a1\u7528\u5927\u8cea\u6578\u4f5c\u70ba\u6a21\u6578\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8b49\u96dc\u6e4a\u503c\u5747\u52fb\u5206\u4f48\uff0c\u6e1b\u5c11\u96dc\u6e4a\u885d\u7a81\u3002</li> <li>\u5e38\u898b\u7684\u96dc\u6e4a\u6f14\u7b97\u6cd5\u5305\u62ec MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002MD5 \u5e38\u7528\u65bc\u6821\u9a57\u6a94\u6848\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u65bc\u5b89\u5168\u61c9\u7528\u8207\u5354\u8b70\u3002</li> <li>\u7a0b\u5f0f\u8a9e\u8a00\u901a\u5e38\u6703\u70ba\u8cc7\u6599\u578b\u5225\u63d0\u4f9b\u5167\u5efa\u96dc\u6e4a\u6f14\u7b97\u6cd5\uff0c\u7528\u65bc\u8a08\u7b97\u96dc\u6e4a\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u8b8a\u7269\u4ef6\u662f\u53ef\u96dc\u6e4a\u7684\u3002</li> </ul>"},{"location":"chapter_hashing/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u96dc\u6e4a\u8868\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5728\u4ec0\u9ebc\u60c5\u6cc1\u4e0b\u662f \\(O(n)\\) \uff1f</p> <p>\u7576\u96dc\u6e4a\u885d\u7a81\u6bd4\u8f03\u56b4\u91cd\u6642\uff0c\u96dc\u6e4a\u8868\u7684\u6642\u9593\u8907\u96dc\u5ea6\u6703\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u7576\u96dc\u6e4a\u51fd\u5f0f\u8a2d\u8a08\u5f97\u6bd4\u8f03\u597d\u3001\u5bb9\u91cf\u8a2d\u5b9a\u6bd4\u8f03\u5408\u7406\u3001\u885d\u7a81\u6bd4\u8f03\u5e73\u5747\u6642\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u5011\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u5167\u5efa\u7684\u96dc\u6e4a\u8868\u6642\uff0c\u901a\u5e38\u8a8d\u70ba\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(1)\\) \u3002</p> <p>Q\uff1a\u70ba\u4ec0\u9ebc\u4e0d\u4f7f\u7528\u96dc\u6e4a\u51fd\u5f0f \\(f(x) = x\\) \u5462\uff1f\u9019\u6a23\u5c31\u4e0d\u6703\u6709\u885d\u7a81\u4e86\u3002</p> <p>\u5728 \\(f(x) = x\\) \u96dc\u6e4a\u51fd\u5f0f\u4e0b\uff0c\u6bcf\u500b\u5143\u7d20\u5c0d\u61c9\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u9019\u8207\u9663\u5217\u7b49\u50f9\u3002\u7136\u800c\uff0c\u8f38\u5165\u7a7a\u9593\u901a\u5e38\u9060\u5927\u65bc\u8f38\u51fa\u7a7a\u9593\uff08\u9663\u5217\u9577\u5ea6\uff09\uff0c\u56e0\u6b64\u96dc\u6e4a\u51fd\u5f0f\u7684\u6700\u5f8c\u4e00\u6b65\u5f80\u5f80\u662f\u5c0d\u9663\u5217\u9577\u5ea6\u53d6\u6a21\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u96dc\u6e4a\u8868\u7684\u76ee\u6a19\u662f\u5c07\u4e00\u500b\u8f03\u5927\u7684\u72c0\u614b\u7a7a\u9593\u5c0d\u6620\u5230\u4e00\u500b\u8f03\u5c0f\u7684\u7a7a\u9593\uff0c\u4e26\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8a62\u6548\u7387\u3002</p> <p>Q\uff1a\u96dc\u6e4a\u8868\u5e95\u5c64\u5be6\u73fe\u662f\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u4e8c\u5143\u6a39\uff0c\u4f46\u70ba\u4ec0\u9ebc\u6548\u7387\u53ef\u4ee5\u6bd4\u5b83\u5011\u66f4\u9ad8\u5462\uff1f</p> <p>\u9996\u5148\uff0c\u96dc\u6e4a\u8868\u7684\u6642\u9593\u6548\u7387\u8b8a\u9ad8\uff0c\u4f46\u7a7a\u9593\u6548\u7387\u8b8a\u4f4e\u4e86\u3002\u96dc\u6e4a\u8868\u6709\u76f8\u7576\u4e00\u90e8\u5206\u8a18\u61b6\u9ad4\u672a\u4f7f\u7528\u3002</p> <p>\u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u5834\u666f\u4e0b\u6642\u9593\u6548\u7387\u8b8a\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u500b\u529f\u80fd\u80fd\u5920\u5728\u76f8\u540c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e0b\u4f7f\u7528\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\uff0c\u90a3\u9ebc\u901a\u5e38\u6bd4\u96dc\u6e4a\u8868\u66f4\u5feb\u3002\u9019\u662f\u56e0\u70ba\u96dc\u6e4a\u51fd\u5f0f\u8a08\u7b97\u9700\u8981\u958b\u92b7\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u7684\u5e38\u6578\u9805\u66f4\u5927\u3002</p> <p>\u6700\u5f8c\uff0c\u96dc\u6e4a\u8868\u7684\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u80fd\u767c\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u93c8\u5f0f\u4f4d\u5740\u4e2d\uff0c\u6211\u5011\u63a1\u53d6\u5728\u93c8\u7d50\u4e32\u5217\u6216\u7d05\u9ed1\u6a39\u4e2d\u57f7\u884c\u67e5\u8a62\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u6642\u9593\u7684\u98a8\u96aa\u3002</p> <p>Q\uff1a\u591a\u6b21\u96dc\u6e4a\u6709\u4e0d\u80fd\u76f4\u63a5\u522a\u9664\u5143\u7d20\u7684\u7f3a\u9677\u55ce\uff1f\u6a19\u8a18\u70ba\u5df2\u522a\u9664\u7684\u7a7a\u9593\u9084\u80fd\u518d\u6b21\u4f7f\u7528\u55ce\uff1f</p> <p>\u591a\u6b21\u96dc\u6e4a\u662f\u958b\u653e\u5b9a\u5740\u7684\u4e00\u7a2e\uff0c\u958b\u653e\u5b9a\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u522a\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u900f\u904e\u6a19\u8a18\u522a\u9664\u3002\u6a19\u8a18\u70ba\u5df2\u522a\u9664\u7684\u7a7a\u9593\u53ef\u4ee5\u518d\u6b21\u4f7f\u7528\u3002\u7576\u5c07\u65b0\u5143\u7d20\u63d2\u5165\u96dc\u6e4a\u8868\uff0c\u4e26\u4e14\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u627e\u5230\u6a19\u8a18\u70ba\u5df2\u522a\u9664\u7684\u4f4d\u7f6e\u6642\uff0c\u8a72\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u5143\u7d20\u4f7f\u7528\u3002\u9019\u6a23\u505a\u65e2\u80fd\u4fdd\u6301\u96dc\u6e4a\u8868\u7684\u63a2\u6e2c\u5e8f\u5217\u4e0d\u8b8a\uff0c\u53c8\u80fd\u4fdd\u8b49\u96dc\u6e4a\u8868\u7684\u7a7a\u9593\u4f7f\u7528\u7387\u3002</p> <p>Q\uff1a\u70ba\u4ec0\u9ebc\u7dda\u4e0a\u6027\u63a2\u67e5\u4e2d\uff0c\u67e5\u8a62\u5143\u7d20\u7684\u6642\u5019\u6703\u51fa\u73fe\u96dc\u6e4a\u885d\u7a81\u5462\uff1f</p> <p>\u67e5\u8a62\u7684\u6642\u5019\u900f\u904e\u96dc\u6e4a\u51fd\u5f0f\u627e\u5230\u5c0d\u61c9\u7684\u6876\u548c\u9375\u503c\u5c0d\uff0c\u767c\u73fe <code>key</code> \u4e0d\u5339\u914d\uff0c\u9019\u5c31\u4ee3\u8868\u6709\u96dc\u6e4a\u885d\u7a81\u3002\u56e0\u6b64\uff0c\u7dda\u6027\u63a2\u67e5\u6cd5\u6703\u6839\u64da\u9810\u5148\u8a2d\u5b9a\u7684\u6b65\u9577\u4f9d\u6b21\u5411\u4e0b\u67e5\u8a62\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u78ba\u7684\u9375\u503c\u5c0d\u6216\u7121\u6cd5\u627e\u5230\u8df3\u51fa\u70ba\u6b62\u3002</p> <p>Q\uff1a\u70ba\u4ec0\u9ebc\u96dc\u6e4a\u8868\u64f4\u5bb9\u80fd\u5920\u7de9\u89e3\u96dc\u6e4a\u885d\u7a81\uff1f</p> <p>\u96dc\u6e4a\u51fd\u5f0f\u7684\u6700\u5f8c\u4e00\u6b65\u5f80\u5f80\u662f\u5c0d\u9663\u5217\u9577\u5ea6 \\(n\\) \u53d6\u6a21\uff08\u53d6\u9918\uff09\uff0c\u8b93\u8f38\u51fa\u503c\u843d\u5728\u9663\u5217\u7d22\u5f15\u7bc4\u570d\u5167\uff1b\u5728\u64f4\u5bb9\u5f8c\uff0c\u9663\u5217\u9577\u5ea6 \\(n\\) \u767c\u751f\u8b8a\u5316\uff0c\u800c <code>key</code> \u5c0d\u61c9\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u767c\u751f\u8b8a\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u500b\u6876\u7684\u591a\u500b <code>key</code> \uff0c\u5728\u64f4\u5bb9\u5f8c\u53ef\u80fd\u6703\u88ab\u5206\u914d\u5230\u591a\u500b\u6876\u4e2d\uff0c\u5f9e\u800c\u5be6\u73fe\u96dc\u6e4a\u885d\u7a81\u7684\u7de9\u89e3\u3002</p>"},{"location":"chapter_heap/","title":"\u7b2c 8 \u7ae0 \u00a0 \u5806\u7a4d","text":"<p>Abstract</p> <p>\u5806\u7a4d\u5c31\u50cf\u662f\u5c71\u5dbd\u5cf0\u5dd2\uff0c\u5c64\u758a\u8d77\u4f0f\u3001\u5f62\u614b\u5404\u7570\u3002</p> <p>\u5ea7\u5ea7\u5c71\u5cf0\u9ad8\u4f4e\u932f\u843d\uff0c\u800c\u6700\u9ad8\u7684\u5c71\u5cf0\u7e3d\u662f\u6700\u5148\u6620\u5165\u773c\u7c3e\u3002</p>"},{"location":"chapter_heap/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>8.1 \u00a0 \u5806\u7a4d</li> <li>8.2 \u00a0 \u5efa\u5806\u7a4d\u64cd\u4f5c</li> <li>8.3 \u00a0 Top-k \u554f\u984c</li> <li>8.4 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_heap/build_heap/","title":"8.2 \u00a0 \u5efa\u5806\u7a4d\u64cd\u4f5c","text":"<p>\u5728\u67d0\u4e9b\u60c5\u6cc1\u4e0b\uff0c\u6211\u5011\u5e0c\u671b\u4f7f\u7528\u4e00\u500b\u4e32\u5217\u7684\u6240\u6709\u5143\u7d20\u4f86\u69cb\u5efa\u4e00\u500b\u5806\u7a4d\uff0c\u9019\u500b\u904e\u7a0b\u88ab\u7a31\u70ba\u201c\u5efa\u5806\u7a4d\u64cd\u4f5c\u201d\u3002</p>"},{"location":"chapter_heap/build_heap/#821","title":"8.2.1 \u00a0 \u85c9\u52a9\u5165\u5806\u7a4d\u64cd\u4f5c\u5be6\u73fe","text":"<p>\u6211\u5011\u9996\u5148\u5efa\u7acb\u4e00\u500b\u7a7a\u5806\u7a4d\uff0c\u7136\u5f8c\u8d70\u8a2a\u4e32\u5217\uff0c\u4f9d\u6b21\u5c0d\u6bcf\u500b\u5143\u7d20\u57f7\u884c\u201c\u5165\u5806\u7a4d\u64cd\u4f5c\u201d\uff0c\u5373\u5148\u5c07\u5143\u7d20\u65b0\u589e\u81f3\u5806\u7a4d\u7684\u5c3e\u90e8\uff0c\u518d\u5c0d\u8a72\u5143\u7d20\u57f7\u884c\u201c\u5f9e\u5e95\u81f3\u9802\u201d\u5806\u7a4d\u5316\u3002</p> <p>\u6bcf\u7576\u4e00\u500b\u5143\u7d20\u5165\u5806\u7a4d\uff0c\u5806\u7a4d\u7684\u9577\u5ea6\u5c31\u52a0\u4e00\u3002\u7531\u65bc\u7bc0\u9ede\u662f\u5f9e\u9802\u5230\u5e95\u4f9d\u6b21\u88ab\u65b0\u589e\u9032\u4e8c\u5143\u6a39\u7684\uff0c\u56e0\u6b64\u5806\u7a4d\u662f\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u69cb\u5efa\u7684\u3002</p> <p>\u8a2d\u5143\u7d20\u6578\u91cf\u70ba \\(n\\) \uff0c\u6bcf\u500b\u5143\u7d20\u7684\u5165\u5806\u7a4d\u64cd\u4f5c\u4f7f\u7528 \\(O(\\log{n})\\) \u6642\u9593\uff0c\u56e0\u6b64\u8a72\u5efa\u5806\u7a4d\u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</p>"},{"location":"chapter_heap/build_heap/#822","title":"8.2.2 \u00a0 \u900f\u904e\u8d70\u8a2a\u5806\u7a4d\u5316\u5be6\u73fe","text":"<p>\u5be6\u969b\u4e0a\uff0c\u6211\u5011\u53ef\u4ee5\u5be6\u73fe\u4e00\u7a2e\u66f4\u70ba\u9ad8\u6548\u7684\u5efa\u5806\u7a4d\u65b9\u6cd5\uff0c\u5171\u5206\u70ba\u5169\u6b65\u3002</p> <ol> <li>\u5c07\u4e32\u5217\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u5730\u65b0\u589e\u5230\u5806\u7a4d\u4e2d\uff0c\u6b64\u6642\u5806\u7a4d\u7684\u6027\u8cea\u5c1a\u672a\u5f97\u5230\u6eff\u8db3\u3002</li> <li>\u5012\u5e8f\u8d70\u8a2a\u5806\u7a4d\uff08\u5c64\u5e8f\u8d70\u8a2a\u7684\u5012\u5e8f\uff09\uff0c\u4f9d\u6b21\u5c0d\u6bcf\u500b\u975e\u8449\u7bc0\u9ede\u57f7\u884c\u201c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\u201d\u3002</li> </ol> <p>\u6bcf\u7576\u5806\u7a4d\u5316\u4e00\u500b\u7bc0\u9ede\u5f8c\uff0c\u4ee5\u8a72\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\u7684\u5b50\u6a39\u5c31\u5f62\u6210\u4e00\u500b\u5408\u6cd5\u7684\u5b50\u5806\u7a4d\u3002\u800c\u7531\u65bc\u662f\u5012\u5e8f\u8d70\u8a2a\uff0c\u56e0\u6b64\u5806\u7a4d\u662f\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u69cb\u5efa\u7684\u3002</p> <p>\u4e4b\u6240\u4ee5\u9078\u64c7\u5012\u5e8f\u8d70\u8a2a\uff0c\u662f\u56e0\u70ba\u9019\u6a23\u80fd\u5920\u4fdd\u8b49\u7576\u524d\u7bc0\u9ede\u4e4b\u4e0b\u7684\u5b50\u6a39\u5df2\u7d93\u662f\u5408\u6cd5\u7684\u5b50\u5806\u7a4d\uff0c\u9019\u6a23\u5806\u7a4d\u5316\u7576\u524d\u7bc0\u9ede\u624d\u662f\u6709\u6548\u7684\u3002</p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u7531\u65bc\u8449\u7bc0\u9ede\u6c92\u6709\u5b50\u7bc0\u9ede\uff0c\u56e0\u6b64\u5b83\u5011\u5929\u7136\u5c31\u662f\u5408\u6cd5\u7684\u5b50\u5806\u7a4d\uff0c\u7121\u9808\u5806\u7a4d\u5316\u3002\u5982\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\uff0c\u6700\u5f8c\u4e00\u500b\u975e\u8449\u7bc0\u9ede\u662f\u6700\u5f8c\u4e00\u500b\u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\uff0c\u6211\u5011\u5f9e\u5b83\u958b\u59cb\u5012\u5e8f\u8d70\u8a2a\u4e26\u57f7\u884c\u5806\u7a4d\u5316\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"\u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d\"\"\"\n    # \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    self.max_heap = nums\n    # \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nMaxHeap(vector&lt;int&gt; nums) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    maxHeap = nums;\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    maxHeap = new List&lt;int&gt;(nums);\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u5207\u7247\u5efa\u5806\u7a4d */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\ninit(nums: [Int]) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    maxHeap = nums\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u5efa\u7acb\u7a7a\u5806\u7a4d\u6216\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nconstructor(nums) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u5efa\u7acb\u7a7a\u5806\u7a4d\u6216\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nconstructor(nums?: number[]) {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nMaxHeap(List&lt;int&gt; nums) {\n  // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n  _maxHeap = nums;\n  // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    let mut heap = MaxHeap { max_heap: nums };\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u5207\u7247\u5efa\u5806\u7a4d */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // \u6240\u6709\u5143\u7d20\u5165\u5806\u7a4d\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* \u5927\u9802\u5806\u7a4d */\nclass MaxHeap(nums: MutableList&lt;Int&gt;?) {\n    // \u4f7f\u7528\u4e32\u5217\u800c\u975e\u9663\u5217\uff0c\u9019\u6a23\u7121\u9808\u8003\u616e\u64f4\u5bb9\u554f\u984c\n    private val maxHeap = mutableListOf&lt;Int&gt;()\n\n    /* \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d */\n    init {\n        // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n        maxHeap.addAll(nums!!)\n        // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u4ea4\u63db\u5143\u7d20 */\n    private fun swap(i: Int, j: Int) {\n        val temp = maxHeap[i]\n        maxHeap[i] = maxHeap[j]\n        maxHeap[j] = temp\n    }\n\n    /* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        /* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\n        return size() == 0\n    }\n\n    /* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* \u5143\u7d20\u5165\u5806\u7a4d */\n    fun push(_val: Int) {\n        // \u65b0\u589e\u7bc0\u9ede\n        maxHeap.add(_val)\n        // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n        siftUp(size() - 1)\n    }\n\n    /* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\n    private fun siftUp(it: Int) {\n        // Kotlin\u7684\u51fd\u5f0f\u53c3\u6578\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5efa\u7acb\u81e8\u6642\u8b8a\u6578\n        var i = it\n        while (true) {\n            // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n            val p = parent(i)\n            // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // \u4ea4\u63db\u5169\u7bc0\u9ede\n            swap(i, p)\n            // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n            i = p\n        }\n    }\n\n    /* \u5143\u7d20\u51fa\u5806\u7a4d */\n    fun pop(): Int {\n        // \u5224\u7a7a\u8655\u7406\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        swap(0, size() - 1)\n        // \u522a\u9664\u7bc0\u9ede\n        val _val = maxHeap.removeAt(size() - 1)\n        // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n        siftDown(0)\n        // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n        return _val\n    }\n\n    /* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\n    private fun siftDown(it: Int) {\n        // Kotlin\u7684\u51fd\u5f0f\u53c3\u6578\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5efa\u7acb\u81e8\u6642\u8b8a\u6578\n        var i = it\n        while (true) {\n            // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n            if (ma == i) break\n            // \u4ea4\u63db\u5169\u7bc0\u9ede\n            swap(i, ma)\n            // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n            i = ma\n        }\n    }\n\n    /* \u5217\u5370\u5806\u7a4d\uff08\u4e8c\u5143\u6a39\uff09 */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>### \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d ###\ndef initialize(nums)\n  # \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n  @max_heap = nums\n  # \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n  parent(size - 1).downto(0) do |i|\n    sift_down(i)\n  end\nend\n</code></pre> my_heap.zig<pre><code>// \u5efa\u69cb\u5b50\uff0c\u6839\u64da\u8f38\u5165\u4e32\u5217\u5efa\u5806\u7a4d\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n    if (self.max_heap != null) return;\n    self.max_heap = std.ArrayList(T).init(allocator);\n    // \u5c07\u4e32\u5217\u5143\u7d20\u539f\u5c01\u4e0d\u52d5\u65b0\u589e\u9032\u5806\u7a4d\n    try self.max_heap.?.appendSlice(nums);\n    // \u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    var i: usize = parent(self.size() - 1) + 1;\n    while (i &gt; 0) : (i -= 1) {\n        try self.siftDown(i - 1);\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/build_heap/#823","title":"8.2.3 \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<p>\u4e0b\u9762\uff0c\u6211\u5011\u4f86\u5617\u8a66\u63a8\u7b97\u7b2c\u4e8c\u7a2e\u5efa\u5806\u7a4d\u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <ul> <li>\u5047\u8a2d\u5b8c\u5168\u4e8c\u5143\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\(n\\) \uff0c\u5247\u8449\u7bc0\u9ede\u6578\u91cf\u70ba \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u70ba\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u9700\u8981\u5806\u7a4d\u5316\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\((n - 1) / 2\\) \u3002</li> <li>\u5728\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\u7684\u904e\u7a0b\u4e2d\uff0c\u6bcf\u500b\u7bc0\u9ede\u6700\u591a\u5806\u7a4d\u5316\u5230\u8449\u7bc0\u9ede\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6578\u70ba\u4e8c\u5143\u6a39\u9ad8\u5ea6 \\(\\log n\\) \u3002</li> </ul> <p>\u5c07\u4e0a\u8ff0\u5169\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u7a4d\u904e\u7a0b\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002\u4f46\u9019\u500b\u4f30\u7b97\u7d50\u679c\u4e26\u4e0d\u6e96\u78ba\uff0c\u56e0\u70ba\u6211\u5011\u6c92\u6709\u8003\u616e\u5230\u4e8c\u5143\u6a39\u5e95\u5c64\u7bc0\u9ede\u6578\u91cf\u9060\u591a\u65bc\u9802\u5c64\u7bc0\u9ede\u7684\u6027\u8cea\u3002</p> <p>\u63a5\u4e0b\u4f86\u6211\u5011\u4f86\u9032\u884c\u66f4\u70ba\u6e96\u78ba\u7684\u8a08\u7b97\u3002\u70ba\u4e86\u964d\u4f4e\u8a08\u7b97\u96e3\u5ea6\uff0c\u5047\u8a2d\u7d66\u5b9a\u4e00\u500b\u7bc0\u9ede\u6578\u91cf\u70ba \\(n\\) \u3001\u9ad8\u5ea6\u70ba \\(h\\) \u7684\u201c\u5b8c\u7f8e\u4e8c\u5143\u6a39\u201d\uff0c\u8a72\u5047\u8a2d\u4e0d\u6703\u5f71\u97ff\u8a08\u7b97\u7d50\u679c\u7684\u6b63\u78ba\u6027\u3002</p> <p></p> <p> \u5716 8-5 \u00a0 \u5b8c\u7f8e\u4e8c\u5143\u6a39\u7684\u5404\u5c64\u7bc0\u9ede\u6578\u91cf </p> <p>\u5982\u5716 8-5 \u6240\u793a\uff0c\u7bc0\u9ede\u201c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\u201d\u7684\u6700\u5927\u8fed\u4ee3\u6b21\u6578\u7b49\u65bc\u8a72\u7bc0\u9ede\u5230\u8449\u7bc0\u9ede\u7684\u8ddd\u96e2\uff0c\u800c\u8a72\u8ddd\u96e2\u6b63\u662f\u201c\u7bc0\u9ede\u9ad8\u5ea6\u201d\u3002\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u5c0d\u5404\u5c64\u7684\u201c\u7bc0\u9ede\u6578\u91cf \\(\\times\\) \u7bc0\u9ede\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5f97\u5230\u6240\u6709\u7bc0\u9ede\u7684\u5806\u7a4d\u5316\u8fed\u4ee3\u6b21\u6578\u7684\u7e3d\u548c\u3002</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>\u5316\u7c21\u4e0a\u5f0f\u9700\u8981\u85c9\u52a9\u4e2d\u5b78\u7684\u6578\u5217\u77e5\u8b58\uff0c\u5148\u5c07 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>\u4f7f\u7528\u932f\u4f4d\u76f8\u6e1b\u6cd5\uff0c\u7528\u4e0b\u5f0f \\(2 T(h)\\) \u6e1b\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97\uff1a</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>\u89c0\u5bdf\u4e0a\u5f0f\uff0c\u767c\u73fe \\(T(h)\\) \u662f\u4e00\u500b\u7b49\u6bd4\u6578\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u6642\u9593\u8907\u96dc\u5ea6\u70ba\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>\u9032\u4e00\u6b65\uff0c\u9ad8\u5ea6\u70ba \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u5143\u6a39\u7684\u7bc0\u9ede\u6578\u91cf\u70ba \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u8907\u96dc\u5ea6\u70ba \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 \u5806\u7a4d","text":"<p>\u5806\u7a4d\uff08heap\uff09\u662f\u4e00\u7a2e\u6eff\u8db3\u7279\u5b9a\u689d\u4ef6\u7684\u5b8c\u5168\u4e8c\u5143\u6a39\uff0c\u4e3b\u8981\u53ef\u5206\u70ba\u5169\u7a2e\u578b\u5225\uff0c\u5982\u5716 8-1 \u6240\u793a\u3002</p> <ul> <li>\u5c0f\u9802\u5806\u7a4d\uff08min heap\uff09\uff1a\u4efb\u610f\u7bc0\u9ede\u7684\u503c \\(\\leq\\) \u5176\u5b50\u7bc0\u9ede\u7684\u503c\u3002</li> <li>\u5927\u9802\u5806\u7a4d\uff08max heap\uff09\uff1a\u4efb\u610f\u7bc0\u9ede\u7684\u503c \\(\\geq\\) \u5176\u5b50\u7bc0\u9ede\u7684\u503c\u3002</li> </ul> <p></p> <p> \u5716 8-1 \u00a0 \u5c0f\u9802\u5806\u7a4d\u8207\u5927\u9802\u5806\u7a4d </p> <p>\u5806\u7a4d\u4f5c\u70ba\u5b8c\u5168\u4e8c\u5143\u6a39\u7684\u4e00\u500b\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u6700\u5e95\u5c64\u7bc0\u9ede\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c64\u7684\u7bc0\u9ede\u90fd\u88ab\u586b\u6eff\u3002</li> <li>\u6211\u5011\u5c07\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede\u7a31\u70ba\u201c\u5806\u7a4d\u9802\u201d\uff0c\u5c07\u5e95\u5c64\u6700\u9760\u53f3\u7684\u7bc0\u9ede\u7a31\u70ba\u201c\u5806\u7a4d\u5e95\u201d\u3002</li> <li>\u5c0d\u65bc\u5927\u9802\u5806\u7a4d\uff08\u5c0f\u9802\u5806\u7a4d\uff09\uff0c\u5806\u7a4d\u9802\u5143\u7d20\uff08\u6839\u7bc0\u9ede\uff09\u7684\u503c\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002</li> </ul>"},{"location":"chapter_heap/heap/#811","title":"8.1.1 \u00a0 \u5806\u7a4d\u7684\u5e38\u7528\u64cd\u4f5c","text":"<p>\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u63d0\u4f9b\u7684\u662f\u512a\u5148\u4f47\u5217\uff08priority queue\uff09\uff0c\u9019\u662f\u4e00\u7a2e\u62bd\u8c61\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u5b9a\u7fa9\u70ba\u5177\u6709\u512a\u5148\u9806\u5e8f\u6392\u5e8f\u7684\u4f47\u5217\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u5806\u7a4d\u901a\u5e38\u7528\u65bc\u5be6\u73fe\u512a\u5148\u4f47\u5217\uff0c\u5927\u9802\u5806\u7a4d\u76f8\u7576\u65bc\u5143\u7d20\u6309\u5f9e\u5927\u5230\u5c0f\u7684\u9806\u5e8f\u51fa\u5217\u7684\u512a\u5148\u4f47\u5217\u3002\u5f9e\u4f7f\u7528\u89d2\u5ea6\u4f86\u770b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u201c\u512a\u5148\u4f47\u5217\u201d\u548c\u201c\u5806\u7a4d\u201d\u770b\u4f5c\u7b49\u50f9\u7684\u8cc7\u6599\u7d50\u69cb\u3002\u56e0\u6b64\uff0c\u672c\u66f8\u5c0d\u5169\u8005\u4e0d\u505a\u7279\u5225\u5340\u5206\uff0c\u7d71\u4e00\u7a31\u4f5c\u201c\u5806\u7a4d\u201d\u3002</p> <p>\u5806\u7a4d\u7684\u5e38\u7528\u64cd\u4f5c\u898b\u8868 8-1 \uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u64da\u7a0b\u5f0f\u8a9e\u8a00\u4f86\u78ba\u5b9a\u3002</p> <p> \u8868 8-1 \u00a0 \u5806\u7a4d\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u6642\u9593\u8907\u96dc\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u5806\u7a4d \\(O(\\log n)\\) <code>pop()</code> \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d \\(O(\\log n)\\) <code>peek()</code> \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20\uff08\u5c0d\u65bc\u5927 / \u5c0f\u9802\u5806\u7a4d\u5206\u5225\u70ba\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) <code>size()</code> \u7372\u53d6\u5806\u7a4d\u7684\u5143\u7d20\u6578\u91cf \\(O(1)\\) <code>isEmpty()</code> \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a \\(O(1)\\) <p>\u5728\u5be6\u969b\u61c9\u7528\u4e2d\uff0c\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u63d0\u4f9b\u7684\u5806\u7a4d\u985e\u5225\uff08\u6216\u512a\u5148\u4f47\u5217\u985e\u5225\uff09\u3002</p> <p>\u985e\u4f3c\u65bc\u6392\u5e8f\u6f14\u7b97\u6cd5\u4e2d\u7684\u201c\u5f9e\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u5f9e\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u8a2d\u5b9a\u4e00\u500b <code>flag</code> \u6216\u4fee\u6539 <code>Comparator</code> \u5be6\u73fe\u201c\u5c0f\u9802\u5806\u7a4d\u201d\u8207\u201c\u5927\u9802\u5806\u7a4d\u201d\u4e4b\u9593\u7684\u8f49\u63db\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap.py<pre><code># \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u7d44\u9810\u8a2d\u5be6\u73fe\u5c0f\u9802\u5806\u7a4d\n# \u8003\u616e\u5c07\u201c\u5143\u7d20\u53d6\u8ca0\u201d\u5f8c\u518d\u5165\u5806\u7a4d\uff0c\u9019\u6a23\u5c31\u53ef\u4ee5\u5c07\u5927\u5c0f\u95dc\u4fc2\u985b\u5012\uff0c\u5f9e\u800c\u5be6\u73fe\u5927\u9802\u5806\u7a4d\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u6642\u5c0d\u61c9\u5c0f\u9802\u5806\u7a4d\uff0cflag = -1 \u6642\u5c0d\u61c9\u5927\u9802\u5806\u7a4d\n\n# \u5143\u7d20\u5165\u5806\u7a4d\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\n# \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u7372\u53d6\u5806\u7a4d\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\uff08\u4f7f\u7528 lambda \u8868\u793a\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\uff08\u4f7f\u7528 lambda \u8868\u793a\u5f0f\u4fee\u6539 Comparer \u5373\u53ef\uff09\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y.CompareTo(x)));\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go \u8a9e\u8a00\u4e2d\u53ef\u4ee5\u900f\u904e\u5be6\u73fe heap.Interface \u4f86\u69cb\u5efa\u6574\u6578\u5927\u9802\u5806\u7a4d\n// \u5be6\u73fe heap.Interface \u9700\u8981\u540c\u6642\u5be6\u73fe sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5be6\u73fe\u63a8\u5165\u5143\u7d20\u5230\u5806\u7a4d\nfunc (h *intHeap) Push(x any) {\n    // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u70ba\u53c3\u6578\n    // \u56e0\u70ba\u5b83\u5011\u4e0d\u50c5\u6703\u5c0d\u5207\u7247\u7684\u5167\u5bb9\u9032\u884c\u8abf\u6574\uff0c\u9084\u6703\u4fee\u6539\u5207\u7247\u7684\u9577\u5ea6\u3002\n    *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5be6\u73fe\u5f48\u51fa\u5806\u7a4d\u9802\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n    // \u5f85\u51fa\u5806\u7a4d\u5143\u7d20\u5b58\u653e\u5728\u6700\u5f8c\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n    // \u5982\u679c\u5be6\u73fe\u5c0f\u9802\u5806\u7a4d\uff0c\u5247\u9700\u8981\u8abf\u6574\u70ba\u5c0f\u65bc\u865f\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* \u521d\u59cb\u5316\u5806\u7a4d */\n    // \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* \u5143\u7d20\u5165\u5806\u7a4d */\n    // \u547c\u53eb heap.Interface \u7684\u65b9\u6cd5\uff0c\u4f86\u65b0\u589e\u5143\u7d20\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\n    top := maxHeap.Top()\n    fmt.Printf(\"\u5806\u7a4d\u9802\u5143\u7d20\u70ba %d\\n\", top)\n\n    /* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n    // \u547c\u53eb heap.Interface \u7684\u65b9\u6cd5\uff0c\u4f86\u79fb\u9664\u5143\u7d20\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\n    size := len(*maxHeap)\n    fmt.Printf(\"\u5806\u7a4d\u5143\u7d20\u6578\u91cf\u70ba %d\\n\", size)\n\n    /* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"\u5806\u7a4d\u662f\u5426\u70ba\u7a7a %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// Swift \u7684 Heap \u578b\u5225\u540c\u6642\u652f\u6301\u6700\u5927\u5806\u7a4d\u548c\u6700\u5c0f\u5806\u7a4d\uff0c\u4e14\u9700\u8981\u5f15\u5165 swift-collections\nvar heap = Heap&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nvar peek = heap.max()!\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nlet size = heap.count\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nlet isEmpty = heap.isEmpty\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.dart<pre><code>// Dart \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.kt<pre><code>/* \u521d\u59cb\u5316\u5806\u7a4d */\n// \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// \u521d\u59cb\u5316\u5927\u9802\u5806\u7a4d\uff08\u4f7f\u7528 lambda \u8868\u793a\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* \u5143\u7d20\u5165\u5806\u7a4d */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* \u7372\u53d6\u5806\u7a4d\u9802\u5143\u7d20 */\nvar peek = maxHeap.peek() // 5\n\n/* \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d */\n// \u51fa\u5806\u7a4d\u5143\u7d20\u6703\u5f62\u6210\u4e00\u500b\u5f9e\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* \u7372\u53d6\u5806\u7a4d\u5927\u5c0f */\nval size = maxHeap.size\n\n/* \u5224\u65b7\u5806\u7a4d\u662f\u5426\u70ba\u7a7a */\nval isEmpty = maxHeap.isEmpty()\n\n/* \u8f38\u5165\u4e32\u5217\u4e26\u5efa\u5806\u7a4d */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code># Ruby \u672a\u63d0\u4f9b\u5167\u5efa Heap \u985e\u5225\n</code></pre> heap.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/heap/#812","title":"8.1.2 \u00a0 \u5806\u7a4d\u7684\u5be6\u73fe","text":"<p>\u4e0b\u6587\u5be6\u73fe\u7684\u662f\u5927\u9802\u5806\u7a4d\u3002\u82e5\u8981\u5c07\u5176\u8f49\u63db\u70ba\u5c0f\u9802\u5806\u7a4d\uff0c\u53ea\u9700\u5c07\u6240\u6709\u5927\u5c0f\u908f\u8f2f\u5224\u65b7\u9032\u884c\u9006\u8f49\uff08\u4f8b\u5982\uff0c\u5c07 \\(\\geq\\) \u66ff\u63db\u70ba \\(\\leq\\) \uff09\u3002\u611f\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u81ea\u884c\u5be6\u73fe\u3002</p>"},{"location":"chapter_heap/heap/#1","title":"1. \u00a0 \u5806\u7a4d\u7684\u5132\u5b58\u8207\u8868\u793a","text":"<p>\u201c\u4e8c\u5143\u6a39\u201d\u7ae0\u7bc0\u8b1b\u904e\uff0c\u5b8c\u5168\u4e8c\u5143\u6a39\u975e\u5e38\u9069\u5408\u7528\u9663\u5217\u4f86\u8868\u793a\u3002\u7531\u65bc\u5806\u7a4d\u6b63\u662f\u4e00\u7a2e\u5b8c\u5168\u4e8c\u5143\u6a39\uff0c\u56e0\u6b64\u6211\u5011\u5c07\u63a1\u7528\u9663\u5217\u4f86\u5132\u5b58\u5806\u7a4d\u3002</p> <p>\u7576\u4f7f\u7528\u9663\u5217\u8868\u793a\u4e8c\u5143\u6a39\u6642\uff0c\u5143\u7d20\u4ee3\u8868\u7bc0\u9ede\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u7bc0\u9ede\u5728\u4e8c\u5143\u6a39\u4e2d\u7684\u4f4d\u7f6e\u3002\u7bc0\u9ede\u6307\u6a19\u900f\u904e\u7d22\u5f15\u5c0d\u6620\u516c\u5f0f\u4f86\u5be6\u73fe\u3002</p> <p>\u5982\u5716 8-2 \u6240\u793a\uff0c\u7d66\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\u70ba \\(2i + 1\\) \uff0c\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\u70ba \\(2i + 2\\) \uff0c\u7236\u7bc0\u9ede\u7684\u7d22\u5f15\u70ba \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u7576\u7d22\u5f15\u8d8a\u754c\u6642\uff0c\u8868\u793a\u7a7a\u7bc0\u9ede\u6216\u7bc0\u9ede\u4e0d\u5b58\u5728\u3002</p> <p></p> <p> \u5716 8-2 \u00a0 \u5806\u7a4d\u7684\u8868\u793a\u8207\u5132\u5b58 </p> <p>\u6211\u5011\u53ef\u4ee5\u5c07\u7d22\u5f15\u5c0d\u6620\u516c\u5f0f\u5c01\u88dd\u6210\u51fd\u5f0f\uff0c\u65b9\u4fbf\u5f8c\u7e8c\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"\u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"\u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"\u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n    return (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n</code></pre> my_heap.cpp<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.java<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.cs<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint Parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.go<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n    // \u5411\u4e0b\u6574\u9664\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.js<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.ts<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.dart<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rs<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.c<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u53d6\u6574\n}\n</code></pre> my_heap.kt<pre><code>/* \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rb<pre><code>### \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 ###\ndef left(i)\n  2 * i + 1\nend\n\n### \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 ###\ndef right(i)\n  2 * i + 2\nend\n\n### \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 ###\ndef parent(i)\n  (i - 1) / 2     # \u5411\u4e0b\u6574\u9664\nend\n</code></pre> my_heap.zig<pre><code>// \u7372\u53d6\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\nfn left(i: usize) usize {\n    return 2 * i + 1;\n}\n\n// \u7372\u53d6\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\nfn right(i: usize) usize {\n    return 2 * i + 2;\n}\n\n// \u7372\u53d6\u7236\u7bc0\u9ede\u7684\u7d22\u5f15\nfn parent(i: usize) usize {\n    // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n    return @divFloor(i - 1, 2);\n}\n</code></pre>"},{"location":"chapter_heap/heap/#2","title":"2. \u00a0 \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20","text":"<p>\u5806\u7a4d\u9802\u5143\u7d20\u5373\u70ba\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede\uff0c\u4e5f\u5c31\u662f\u4e32\u5217\u7684\u9996\u500b\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"\u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>### \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 ###\ndef peek\n  @max_heap[0]\nend\n</code></pre> my_heap.zig<pre><code>// \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20\nfn peek(self: *Self) T {\n    return self.max_heap.?.items[0];\n}  \n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/heap/#3","title":"3. \u00a0 \u5143\u7d20\u5165\u5806\u7a4d","text":"<p>\u7d66\u5b9a\u5143\u7d20 <code>val</code> \uff0c\u6211\u5011\u9996\u5148\u5c07\u5176\u65b0\u589e\u5230\u5806\u7a4d\u5e95\u3002\u65b0\u589e\u4e4b\u5f8c\uff0c\u7531\u65bc <code>val</code> \u53ef\u80fd\u5927\u65bc\u5806\u7a4d\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7a4d\u7684\u6210\u7acb\u689d\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u58de\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u5fa9\u5f9e\u63d2\u5165\u7bc0\u9ede\u5230\u6839\u7bc0\u9ede\u7684\u8def\u5f91\u4e0a\u7684\u5404\u500b\u7bc0\u9ede\uff0c\u9019\u500b\u64cd\u4f5c\u88ab\u7a31\u70ba\u5806\u7a4d\u5316\uff08heapify\uff09\u3002</p> <p>\u8003\u616e\u5f9e\u5165\u5806\u7a4d\u7bc0\u9ede\u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u57f7\u884c\u5806\u7a4d\u5316\u3002\u5982\u5716 8-3 \u6240\u793a\uff0c\u6211\u5011\u6bd4\u8f03\u63d2\u5165\u7bc0\u9ede\u8207\u5176\u7236\u7bc0\u9ede\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u7bc0\u9ede\u66f4\u5927\uff0c\u5247\u5c07\u5b83\u5011\u4ea4\u63db\u3002\u7136\u5f8c\u7e7c\u7e8c\u57f7\u884c\u6b64\u64cd\u4f5c\uff0c\u5f9e\u5e95\u81f3\u9802\u4fee\u5fa9\u5806\u7a4d\u4e2d\u7684\u5404\u500b\u7bc0\u9ede\uff0c\u76f4\u81f3\u8d8a\u904e\u6839\u7bc0\u9ede\u6216\u9047\u5230\u7121\u9808\u4ea4\u63db\u7684\u7bc0\u9ede\u6642\u7d50\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 8-3 \u00a0 \u5143\u7d20\u5165\u5806\u7a4d\u6b65\u9a5f </p> <p>\u8a2d\u7bc0\u9ede\u7e3d\u6578\u70ba \\(n\\) \uff0c\u5247\u6a39\u7684\u9ad8\u5ea6\u70ba \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u7a4d\u5316\u64cd\u4f5c\u7684\u8ff4\u5708\u8f2a\u6578\u6700\u591a\u70ba \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u7a4d\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"\u5143\u7d20\u5165\u5806\u7a4d\"\"\"\n    # \u65b0\u589e\u7bc0\u9ede\n    self.max_heap.append(val)\n    # \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\"\"\"\n    while True:\n        # \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        p = self.parent(i)\n        # \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # \u4ea4\u63db\u5169\u7bc0\u9ede\n        self.swap(i, p)\n        # \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid push(int val) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.push_back(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        int p = parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(maxHeap[i], maxHeap[p]);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid push(int val) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.add(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        int p = parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid Push(int val) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.Add(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    SiftUp(Size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid SiftUp(int i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        int p = Parent(i);\n        // \u82e5\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\uff0c\u5247\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        Swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfunc (h *maxHeap) push(val any) {\n    // \u65b0\u589e\u7bc0\u9ede\n    h.data = append(h.data, val)\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    h.siftUp(len(h.data) - 1)\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        p := h.parent(i)\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        h.swap(i, p)\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfunc push(val: Int) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.append(val)\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(i: size() - 1)\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        let p = parent(i: i)\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i: i, j: p)\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\npush(val) {\n    // \u65b0\u589e\u7bc0\u9ede\n    this.#maxHeap.push(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    this.#siftUp(this.size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\n#siftUp(i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        const p = this.#parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        this.#swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\npush(val: number): void {\n    // \u65b0\u589e\u7bc0\u9ede\n    this.maxHeap.push(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    this.siftUp(this.size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nsiftUp(i: number): void {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        const p = this.parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        this.swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid push(int val) {\n  // \u65b0\u589e\u7bc0\u9ede\n  _maxHeap.add(val);\n  // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n  siftUp(size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid siftUp(int i) {\n  while (true) {\n    // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n    int p = _parent(i);\n    // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // \u4ea4\u63db\u5169\u7bc0\u9ede\n    _swap(i, p);\n    // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfn push(&amp;mut self, val: i32) {\n    // \u65b0\u589e\u7bc0\u9ede\n    self.max_heap.push(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    self.sift_up(self.size() - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // \u7bc0\u9ede i \u5df2\u7d93\u662f\u5806\u7a4d\u9802\u7bc0\u9ede\u4e86\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if i == 0 {\n            break;\n        }\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        let p = Self::parent(i);\n        // \u7576\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        self.swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid push(MaxHeap *maxHeap, int val) {\n    // \u9810\u8a2d\u60c5\u6cc1\u4e0b\uff0c\u4e0d\u61c9\u8a72\u65b0\u589e\u9019\u9ebc\u591a\u7bc0\u9ede\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"heap is full!\");\n        return;\n    }\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        int p = parent(maxHeap, i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(maxHeap, i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfun push(_val: Int) {\n    // \u65b0\u589e\u7bc0\u9ede\n    maxHeap.add(_val)\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    siftUp(size() - 1)\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 */\nfun siftUp(it: Int) {\n    // Kotlin\u7684\u51fd\u5f0f\u53c3\u6578\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5efa\u7acb\u81e8\u6642\u8b8a\u6578\n    var i = it\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        val p = parent(i)\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i, p)\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>### \u5143\u7d20\u5165\u5806\u7a4d ###\ndef push(val)\n  # \u65b0\u589e\u7bc0\u9ede\n  @max_heap &lt;&lt; val\n  # \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n  sift_up(size - 1)\nend\n\n### \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316 ###\ndef sift_up(i)\n  loop do\n    # \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n    p = parent(i)\n    # \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n    break if p &lt; 0 || @max_heap[i] &lt;= @max_heap[p]\n    # \u4ea4\u63db\u5169\u7bc0\u9ede\n    swap(i, p)\n    # \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n    i = p\n  end\nend\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u5165\u5806\u7a4d\nfn push(self: *Self, val: T) !void {\n    // \u65b0\u589e\u7bc0\u9ede\n    try self.max_heap.?.append(val);\n    // \u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\n    try self.siftUp(self.size() - 1);\n}  \n\n// \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u7372\u53d6\u7bc0\u9ede i \u7684\u7236\u7bc0\u9ede\n        var p = parent(i);\n        // \u7576\u201c\u8d8a\u904e\u6839\u7bc0\u9ede\u201d\u6216\u201c\u7bc0\u9ede\u7121\u9808\u4fee\u5fa9\u201d\u6642\uff0c\u7d50\u675f\u5806\u7a4d\u5316\n        if (p &lt; 0 or self.max_heap.?.items[i] &lt;= self.max_heap.?.items[p]) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        try self.swap(i, p);\n        // \u8ff4\u5708\u5411\u4e0a\u5806\u7a4d\u5316\n        i = p;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/heap/#4","title":"4. \u00a0 \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d","text":"<p>\u5806\u7a4d\u9802\u5143\u7d20\u662f\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede\uff0c\u5373\u4e32\u5217\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u5011\u76f4\u63a5\u5f9e\u4e32\u5217\u4e2d\u522a\u9664\u9996\u5143\u7d20\uff0c\u90a3\u9ebc\u4e8c\u5143\u6a39\u4e2d\u6240\u6709\u7bc0\u9ede\u7684\u7d22\u5f15\u90fd\u6703\u767c\u751f\u8b8a\u5316\uff0c\u9019\u5c07\u4f7f\u5f97\u5f8c\u7e8c\u4f7f\u7528\u5806\u7a4d\u5316\u9032\u884c\u4fee\u5fa9\u8b8a\u5f97\u56f0\u96e3\u3002\u70ba\u4e86\u5118\u91cf\u6e1b\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u8b8a\u52d5\uff0c\u6211\u5011\u63a1\u7528\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9a5f\u3002</p> <ol> <li>\u4ea4\u63db\u5806\u7a4d\u9802\u5143\u7d20\u8207\u5806\u7a4d\u5e95\u5143\u7d20\uff08\u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff09\u3002</li> <li>\u4ea4\u63db\u5b8c\u6210\u5f8c\uff0c\u5c07\u5806\u7a4d\u5e95\u5f9e\u4e32\u5217\u4e2d\u522a\u9664\uff08\u6ce8\u610f\uff0c\u7531\u65bc\u5df2\u7d93\u4ea4\u63db\uff0c\u56e0\u6b64\u5be6\u969b\u4e0a\u522a\u9664\u7684\u662f\u539f\u4f86\u7684\u5806\u7a4d\u9802\u5143\u7d20\uff09\u3002</li> <li>\u5f9e\u6839\u7bc0\u9ede\u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u57f7\u884c\u5806\u7a4d\u5316\u3002</li> </ol> <p>\u5982\u5716 8-4 \u6240\u793a\uff0c\u201c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\u201d\u7684\u64cd\u4f5c\u65b9\u5411\u8207\u201c\u5f9e\u5e95\u81f3\u9802\u5806\u7a4d\u5316\u201d\u76f8\u53cd\uff0c\u6211\u5011\u5c07\u6839\u7bc0\u9ede\u7684\u503c\u8207\u5176\u5169\u500b\u5b50\u7bc0\u9ede\u7684\u503c\u9032\u884c\u6bd4\u8f03\uff0c\u5c07\u6700\u5927\u7684\u5b50\u7bc0\u9ede\u8207\u6839\u7bc0\u9ede\u4ea4\u63db\u3002\u7136\u5f8c\u8ff4\u5708\u57f7\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u904e\u8449\u7bc0\u9ede\u6216\u9047\u5230\u7121\u9808\u4ea4\u63db\u7684\u7bc0\u9ede\u6642\u7d50\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 8-4 \u00a0 \u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u6b65\u9a5f </p> <p>\u8207\u5143\u7d20\u5165\u5806\u7a4d\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e5f\u70ba \\(O(\\log n)\\) \u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"\u5143\u7d20\u51fa\u5806\u7a4d\"\"\"\n    # \u5224\u7a7a\u8655\u7406\n    if self.is_empty():\n        raise IndexError(\"\u5806\u7a4d\u70ba\u7a7a\")\n    # \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1)\n    # \u522a\u9664\u7bc0\u9ede\n    val = self.max_heap.pop()\n    # \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    self.sift_down(0)\n    # \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\"\"\"\n    while True:\n        # \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u63db\u5169\u7bc0\u9ede\n        self.swap(i, ma)\n        # \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nvoid pop() {\n    // \u5224\u7a7a\u8655\u7406\n    if (isEmpty()) {\n        throw out_of_range(\"\u5806\u7a4d\u70ba\u7a7a\");\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // \u522a\u9664\u7bc0\u9ede\n    maxHeap.pop_back();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(0);\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint pop() {\n    // \u5224\u7a7a\u8655\u7406\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    int val = maxHeap.remove(size() - 1);\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint Pop() {\n    // \u5224\u7a7a\u8655\u7406\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    Swap(0, Size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    SiftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid SiftDown(int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u201c\u7bc0\u9ede i \u6700\u5927\u201d\u6216\u201c\u8d8a\u904e\u8449\u7bc0\u9ede\u201d\uff0c\u5247\u7d50\u675f\u5806\u7a4d\u5316\n        if (ma == i) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        Swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfunc (h *maxHeap) pop() any {\n    // \u5224\u7a7a\u8655\u7406\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    h.swap(0, h.size()-1)\n    // \u522a\u9664\u7bc0\u9ede\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    h.siftDown(0)\n\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if max == i {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        h.swap(i, max)\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfunc pop() -&gt; Int {\n    // \u5224\u7a7a\u8655\u7406\n    if isEmpty() {\n        fatalError(\"\u5806\u7a4d\u70ba\u7a7a\")\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(i: 0, j: size() - 1)\n    // \u522a\u9664\u7bc0\u9ede\n    let val = maxHeap.remove(at: size() - 1)\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(i: 0)\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i: i, j: ma)\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\npop() {\n    // \u5224\u7a7a\u8655\u7406\n    if (this.isEmpty()) throw new Error('\u5806\u7a4d\u70ba\u7a7a');\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    this.#swap(0, this.size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    const val = this.#maxHeap.pop();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    this.#siftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\n#siftDown(i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        this.#swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\npop(): number {\n    // \u5224\u7a7a\u8655\u7406\n    if (this.isEmpty()) throw new RangeError('Heap is empty.');\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    this.swap(0, this.size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    const val = this.maxHeap.pop();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    this.siftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nsiftDown(i: number): void {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        this.swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint pop() {\n  // \u5224\u7a7a\u8655\u7406\n  if (isEmpty()) throw Exception('\u5806\u7a4d\u70ba\u7a7a');\n  // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n  _swap(0, size() - 1);\n  // \u522a\u9664\u7bc0\u9ede\n  int val = _maxHeap.removeLast();\n  // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n  siftDown(0);\n  // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n  return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int i) {\n  while (true) {\n    // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u63db\u5169\u7bc0\u9ede\n    _swap(i, ma);\n    // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfn pop(&amp;mut self) -&gt; i32 {\n    // \u5224\u7a7a\u8655\u7406\n    if self.is_empty() {\n        panic!(\"index out of bounds\");\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    let val = self.max_heap.pop().unwrap();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    self.sift_down(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    val\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        self.swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint pop(MaxHeap *maxHeap) {\n    // \u5224\u7a7a\u8655\u7406\n    if (isEmpty(maxHeap)) {\n        printf(\"heap is empty!\");\n        return INT_MAX;\n    }\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(maxHeap, 0);\n\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (max == i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(maxHeap, i, max);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfun pop(): Int {\n    // \u5224\u7a7a\u8655\u7406\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1)\n    // \u522a\u9664\u7bc0\u9ede\n    val _val = maxHeap.removeAt(size() - 1)\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    siftDown(0)\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return _val\n}\n\n/* \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfun siftDown(it: Int) {\n    // Kotlin\u7684\u51fd\u5f0f\u53c3\u6578\u4e0d\u53ef\u8b8a\uff0c\u56e0\u6b64\u5efa\u7acb\u81e8\u6642\u8b8a\u6578\n    var i = it\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(i, ma)\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>### \u5143\u7d20\u51fa\u5806\u7a4d ###\ndef pop\n  # \u5224\u7a7a\u8655\u7406\n  raise IndexError, \"\u5806\u7a4d\u70ba\u7a7a\" if is_empty?\n  # \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n  swap(0, size - 1)\n  # \u522a\u9664\u7bc0\u9ede\n  val = @max_heap.pop\n  # \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n  sift_down(0)\n  # \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n  val\nend\n\n### \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 ###\ndef sift_down(i)\n  loop do\n    # \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n    l, r, ma = left(i), right(i), i\n    ma = l if l &lt; size &amp;&amp; @max_heap[l] &gt; @max_heap[ma]\n    ma = r if r &lt; size &amp;&amp; @max_heap[r] &gt; @max_heap[ma]\n\n    # \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n    break if ma == i\n\n    # \u4ea4\u63db\u5169\u7bc0\u9ede\n    swap(i, ma)\n    # \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n    i = ma\n  end\nend\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u51fa\u5806\u7a4d\nfn pop(self: *Self) !T {\n    // \u5224\u65b7\u8655\u7406\n    if (self.isEmpty()) unreachable;\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    try self.swap(0, self.size() - 1);\n    // \u522a\u9664\u7bc0\u9ede\n    var val = self.max_heap.?.pop();\n    // \u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\n    try self.siftDown(0);\n    // \u8fd4\u56de\u5806\u7a4d\u9802\u5143\u7d20\n    return val;\n} \n\n// \u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        var l = left(i);\n        var r = right(i);\n        var ma = i;\n        if (l &lt; self.size() and self.max_heap.?.items[l] &gt; self.max_heap.?.items[ma]) ma = l;\n        if (r &lt; self.size() and self.max_heap.?.items[r] &gt; self.max_heap.?.items[ma]) ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        try self.swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_heap/heap/#813","title":"8.1.3 \u00a0 \u5806\u7a4d\u7684\u5e38\u898b\u61c9\u7528","text":"<ul> <li>\u512a\u5148\u4f47\u5217\uff1a\u5806\u7a4d\u901a\u5e38\u4f5c\u70ba\u5be6\u73fe\u512a\u5148\u4f47\u5217\u7684\u9996\u9078\u8cc7\u6599\u7d50\u69cb\uff0c\u5176\u5165\u5217\u548c\u51fa\u5217\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(\\log n)\\) \uff0c\u800c\u5efa\u5806\u7a4d\u64cd\u4f5c\u70ba \\(O(n)\\) \uff0c\u9019\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002</li> <li>\u5806\u7a4d\u6392\u5e8f\uff1a\u7d66\u5b9a\u4e00\u7d44\u8cc7\u6599\uff0c\u6211\u5011\u53ef\u4ee5\u7528\u5b83\u5011\u5efa\u7acb\u4e00\u500b\u5806\u7a4d\uff0c\u7136\u5f8c\u4e0d\u65b7\u5730\u57f7\u884c\u5143\u7d20\u51fa\u5806\u7a4d\u64cd\u4f5c\uff0c\u5f9e\u800c\u5f97\u5230\u6709\u5e8f\u8cc7\u6599\u3002\u7136\u800c\uff0c\u6211\u5011\u901a\u5e38\u6703\u4f7f\u7528\u4e00\u7a2e\u66f4\u512a\u96c5\u7684\u65b9\u5f0f\u5be6\u73fe\u5806\u7a4d\u6392\u5e8f\uff0c\u8a73\u898b\u201c\u5806\u7a4d\u6392\u5e8f\u201d\u7ae0\u7bc0\u3002</li> <li>\u7372\u53d6\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\uff1a\u9019\u662f\u4e00\u500b\u7d93\u5178\u7684\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u540c\u6642\u4e5f\u662f\u4e00\u7a2e\u5178\u578b\u61c9\u7528\uff0c\u4f8b\u5982\u9078\u64c7\u71b1\u5ea6\u524d 10 \u7684\u65b0\u805e\u4f5c\u70ba\u5fae\u535a\u71b1\u641c\uff0c\u9078\u53d6\u92b7\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002</li> </ul>"},{"location":"chapter_heap/summary/","title":"8.4 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_heap/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u5806\u7a4d\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u5143\u6a39\uff0c\u6839\u64da\u6210\u7acb\u689d\u4ef6\u53ef\u5206\u70ba\u5927\u9802\u5806\u7a4d\u548c\u5c0f\u9802\u5806\u7a4d\u3002\u5927\uff08\u5c0f\uff09\u9802\u5806\u7a4d\u7684\u5806\u7a4d\u9802\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002</li> <li>\u512a\u5148\u4f47\u5217\u7684\u5b9a\u7fa9\u662f\u5177\u6709\u51fa\u5217\u512a\u5148\u9806\u5e8f\u7684\u4f47\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u7a4d\u4f86\u5be6\u73fe\u3002</li> <li>\u5806\u7a4d\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5c0d\u61c9\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806\u7a4d \\(O(\\log n)\\)\u3001\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d \\(O(\\log n)\\) \u548c\u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 \\(O(1)\\) \u7b49\u3002</li> <li>\u5b8c\u5168\u4e8c\u5143\u6a39\u975e\u5e38\u9069\u5408\u7528\u9663\u5217\u8868\u793a\uff0c\u56e0\u6b64\u6211\u5011\u901a\u5e38\u4f7f\u7528\u9663\u5217\u4f86\u5132\u5b58\u5806\u7a4d\u3002</li> <li>\u5806\u7a4d\u5316\u64cd\u4f5c\u7528\u65bc\u7dad\u8b77\u5806\u7a4d\u7684\u6027\u8cea\uff0c\u5728\u5165\u5806\u7a4d\u548c\u51fa\u5806\u7a4d\u64cd\u4f5c\u4e2d\u90fd\u6703\u7528\u5230\u3002</li> <li>\u8f38\u5165 \\(n\\) \u500b\u5143\u7d20\u4e26\u5efa\u5806\u7a4d\u7684\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u6700\u4f73\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</li> <li>Top-k \u662f\u4e00\u500b\u7d93\u5178\u6f14\u7b97\u6cd5\u554f\u984c\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u7a4d\u8cc7\u6599\u7d50\u69cb\u9ad8\u6548\u89e3\u6c7a\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log k)\\) \u3002</li> </ul>"},{"location":"chapter_heap/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u8cc7\u6599\u7d50\u69cb\u7684\u201c\u5806\u7a4d\u201d\u8207\u8a18\u61b6\u9ad4\u7ba1\u7406\u7684\u201c\u5806\u7a4d\u201d\u662f\u540c\u4e00\u500b\u6982\u5ff5\u55ce\uff1f</p> <p>\u5169\u8005\u4e0d\u662f\u540c\u4e00\u500b\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u201c\u5806\u7a4d\u201d\u3002\u8a08\u7b97\u6a5f\u7cfb\u7d71\u8a18\u61b6\u9ad4\u4e2d\u7684\u5806\u7a4d\u662f\u52d5\u614b\u8a18\u61b6\u9ad4\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5f0f\u5728\u57f7\u884c\u6642\u53ef\u4ee5\u4f7f\u7528\u5b83\u4f86\u5132\u5b58\u8cc7\u6599\u3002\u7a0b\u5f0f\u53ef\u4ee5\u8acb\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u7a4d\u8a18\u61b6\u9ad4\uff0c\u7528\u65bc\u5132\u5b58\u5982\u7269\u4ef6\u548c\u9663\u5217\u7b49\u8907\u96dc\u7d50\u69cb\u3002\u7576\u9019\u4e9b\u8cc7\u6599\u4e0d\u518d\u9700\u8981\u6642\uff0c\u7a0b\u5f0f\u9700\u8981\u91cb\u653e\u9019\u4e9b\u8a18\u61b6\u9ad4\uff0c\u4ee5\u9632\u6b62\u8a18\u61b6\u9ad4\u6d41\u5931\u3002\u76f8\u8f03\u65bc\u5806\u758a\u8a18\u61b6\u9ad4\uff0c\u5806\u7a4d\u8a18\u61b6\u9ad4\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8b39\u614e\uff0c\u4f7f\u7528\u4e0d\u7576\u53ef\u80fd\u6703\u5c0e\u81f4\u8a18\u61b6\u9ad4\u6d41\u5931\u548c\u91ce\u6307\u6a19\u7b49\u554f\u984c\u3002</p>"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-k \u554f\u984c","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u7121\u5e8f\u9663\u5217 <code>nums</code> \uff0c\u8acb\u8fd4\u56de\u9663\u5217\u4e2d\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\u3002</p> <p>\u5c0d\u65bc\u8a72\u554f\u984c\uff0c\u6211\u5011\u5148\u4ecb\u7d39\u5169\u7a2e\u601d\u8def\u6bd4\u8f03\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7d39\u6548\u7387\u66f4\u9ad8\u7684\u5806\u7a4d\u89e3\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#831","title":"8.3.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u8d70\u8a2a\u9078\u64c7","text":"<p>\u6211\u5011\u53ef\u4ee5\u9032\u884c\u5716 8-6 \u6240\u793a\u7684 \\(k\\) \u8f2a\u8d70\u8a2a\uff0c\u5206\u5225\u5728\u6bcf\u8f2a\u4e2d\u63d0\u53d6\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(\\dots\\)\u3001\\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(nk)\\) \u3002</p> <p>\u6b64\u65b9\u6cd5\u53ea\u9069\u7528\u65bc \\(k \\ll n\\) \u7684\u60c5\u6cc1\uff0c\u56e0\u70ba\u7576 \\(k\\) \u8207 \\(n\\) \u6bd4\u8f03\u63a5\u8fd1\u6642\uff0c\u5176\u6642\u9593\u8907\u96dc\u5ea6\u8da8\u5411\u65bc \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u6642\u3002</p> <p></p> <p> \u5716 8-6 \u00a0 \u8d70\u8a2a\u5c0b\u627e\u6700\u5927\u7684 k \u500b\u5143\u7d20 </p> <p>Tip</p> <p>\u7576 \\(k = n\\) \u6642\uff0c\u6211\u5011\u53ef\u4ee5\u5f97\u5230\u5b8c\u6574\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6b64\u6642\u7b49\u50f9\u65bc\u201c\u9078\u64c7\u6392\u5e8f\u201d\u6f14\u7b97\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#832","title":"8.3.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"<p>\u5982\u5716 8-7 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u5148\u5c0d\u9663\u5217 <code>nums</code> \u9032\u884c\u6392\u5e8f\uff0c\u518d\u8fd4\u56de\u6700\u53f3\u908a\u7684 \\(k\\) \u500b\u5143\u7d20\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</p> <p>\u986f\u7136\uff0c\u8a72\u65b9\u6cd5\u201c\u8d85\u984d\u201d\u5b8c\u6210\u4efb\u52d9\u4e86\uff0c\u56e0\u70ba\u6211\u5011\u53ea\u9700\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002</p> <p></p> <p> \u5716 8-7 \u00a0 \u6392\u5e8f\u5c0b\u627e\u6700\u5927\u7684 k \u500b\u5143\u7d20 </p>"},{"location":"chapter_heap/top_k/#833","title":"8.3.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806\u7a4d","text":"<p>\u6211\u5011\u53ef\u4ee5\u57fa\u65bc\u5806\u7a4d\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u6c7a Top-k \u554f\u984c\uff0c\u6d41\u7a0b\u5982\u5716 8-8 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4e00\u500b\u5c0f\u9802\u5806\u7a4d\uff0c\u5176\u5806\u7a4d\u9802\u5143\u7d20\u6700\u5c0f\u3002</li> <li>\u5148\u5c07\u9663\u5217\u7684\u524d \\(k\\) \u500b\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u7a4d\u3002</li> <li>\u5f9e\u7b2c \\(k + 1\\) \u500b\u5143\u7d20\u958b\u59cb\uff0c\u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\uff0c\u4e26\u5c07\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\u3002</li> <li>\u8d70\u8a2a\u5b8c\u6210\u5f8c\uff0c\u5806\u7a4d\u4e2d\u5132\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 8-8 \u00a0 \u57fa\u65bc\u5806\u7a4d\u5c0b\u627e\u6700\u5927\u7684 k \u500b\u5143\u7d20 </p> <p>\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"\u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20\"\"\"\n    # \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    heap = []\n    # \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for i in range(k, len(nums)):\n        # \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (int i = k; i &lt; nums.length; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for i := k; i &lt; len(nums); i++ {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u500b\u5c0f\u9802\u5806\u7a4d\uff0c\u4e26\u5c07\u524d k \u500b\u5143\u7d20\u5efa\u5806\u7a4d\n    var heap = Heap(nums.prefix(k))\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for i in nums.indices.dropFirst(k) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfunction pushMinHeap(maxHeap, val) {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfunction popMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u7a4d\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    // \u8acb\u6ce8\u610f\uff1a\u6211\u5011\u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u7528\u5927\u9802\u5806\u7a4d\u4f86\u6a21\u64ec\u5c0f\u9802\u5806\u7a4d\n    const maxHeap = new MaxHeap([]);\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u7a4d\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806\u7a4d */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u7a4d\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    // \u8acb\u6ce8\u610f\uff1a\u6211\u5011\u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u7528\u5927\u9802\u5806\u7a4d\u4f86\u6a21\u64ec\u5c0f\u9802\u5806\u7a4d\n    const maxHeap = new MaxHeap([]);\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u7a4d\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\uff0c\u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n  for (int i = k; i &lt; nums.length; i++) {\n    // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap \u662f\u5927\u9802\u5806\u7a4d\uff0c\u4f7f\u7528 Reverse \u5c07\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u5be6\u73fe\u5c0f\u9802\u5806\u7a4d\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for &amp;num in nums.iter().skip(k) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* \u5143\u7d20\u5165\u5806\u7a4d */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // \u5143\u7d20\u53d6\u53cd\n    push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806\u7a4d */\nint popMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -pop(maxHeap);\n}\n\n/* \u8a2a\u554f\u5806\u7a4d\u9802\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u7a4d\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u4e26\u5b58\u5165 res \u9663\u5217\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* \u53d6\u51fa\u5806\u7a4d\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u4e26\u5b58\u5165 res \u9663\u5217\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20\u7684\u51fd\u5f0f\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    // \u8acb\u6ce8\u610f\uff1a\u6211\u5011\u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u7528\u5927\u9802\u5806\u7a4d\u4f86\u6a21\u64ec\u5c0f\u9802\u5806\u7a4d\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n    val heap = PriorityQueue&lt;Int&gt;()\n    // \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n    for (i in k..&lt;nums.size) {\n        // \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>### \u57fa\u65bc\u5806\u7a4d\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5927\u7684 k \u500b\u5143\u7d20 ###\ndef top_k_heap(nums, k)\n  # \u521d\u59cb\u5316\u5c0f\u9802\u5806\u7a4d\n  # \u8acb\u6ce8\u610f\uff1a\u6211\u5011\u5c07\u5806\u7a4d\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u5f9e\u800c\u7528\u5927\u9802\u5806\u7a4d\u4f86\u6a21\u64ec\u5c0f\u9802\u5806\u7a4d\n  max_heap = MaxHeap.new([])\n\n  # \u5c07\u9663\u5217\u7684\u524d k \u500b\u5143\u7d20\u5165\u5806\u7a4d\n  for i in 0...k\n    push_min_heap(max_heap, nums[i])\n  end\n\n  # \u5f9e\u7b2c k+1 \u500b\u5143\u7d20\u958b\u59cb\uff0c\u4fdd\u6301\u5806\u7a4d\u7684\u9577\u5ea6\u70ba k\n  for i in k...nums.length\n    # \u82e5\u7576\u524d\u5143\u7d20\u5927\u65bc\u5806\u7a4d\u9802\u5143\u7d20\uff0c\u5247\u5c07\u5806\u7a4d\u9802\u5143\u7d20\u51fa\u5806\u7a4d\u3001\u7576\u524d\u5143\u7d20\u5165\u5806\u7a4d\n    if nums[i] &gt; peek_min_heap(max_heap)\n      pop_min_heap(max_heap)\n      push_min_heap(max_heap, nums[i])\n    end\n  end\n\n  get_min_heap(max_heap)\nend\n</code></pre> top_k.zig<pre><code>[class]{}-[func]{topKHeap}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u7e3d\u5171\u57f7\u884c\u4e86 \\(n\\) \u8f2a\u5165\u5806\u7a4d\u548c\u51fa\u5806\u7a4d\uff0c\u5806\u7a4d\u7684\u6700\u5927\u9577\u5ea6\u70ba \\(k\\) \uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log k)\\) \u3002\u8a72\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u7576 \\(k\\) \u8f03\u5c0f\u6642\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u8da8\u5411 \\(O(n)\\) \uff1b\u7576 \\(k\\) \u8f03\u5927\u6642\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u4e0d\u6703\u8d85\u904e \\(O(n \\log n)\\) \u3002</p> <p>\u53e6\u5916\uff0c\u8a72\u65b9\u6cd5\u9069\u7528\u65bc\u52d5\u614b\u8cc7\u6599\u6d41\u7684\u4f7f\u7528\u5834\u666f\u3002\u5728\u4e0d\u65b7\u52a0\u5165\u8cc7\u6599\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u6301\u7e8c\u7dad\u8b77\u5806\u7a4d\u5167\u7684\u5143\u7d20\uff0c\u5f9e\u800c\u5be6\u73fe\u6700\u5927\u7684 \\(k\\) \u500b\u5143\u7d20\u7684\u52d5\u614b\u66f4\u65b0\u3002</p>"},{"location":"chapter_hello_algo/","title":"\u5e8f","text":"<p>\u5e7e\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u201c\u528d\u6307 Offer\u201d\u7cfb\u5217\u984c\u89e3\uff0c\u53d7\u5230\u4e86\u8a31\u591a\u8b80\u8005\u7684\u9f13\u52f5\u548c\u652f\u6301\u3002\u5728\u8207\u8b80\u8005\u4ea4\u6d41\u671f\u9593\uff0c\u6211\u6700\u5e38\u88ab\u554f\u7684\u4e00\u500b\u554f\u984c\u662f\u201c\u5982\u4f55\u5165\u9580\u6f14\u7b97\u6cd5\u201d\u3002\u9010\u6f38\u5730\uff0c\u6211\u5c0d\u9019\u500b\u554f\u984c\u7522\u751f\u4e86\u6fc3\u539a\u7684\u8208\u8da3\u3002</p> <p>\u5169\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u984c\u4f3c\u4e4e\u662f\u6700\u53d7\u6b61\u8fce\u7684\u65b9\u6cd5\uff0c\u7c21\u55ae\u3001\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\u5237\u984c\u5c31\u5982\u540c\u73a9\u201c\u6383\u96f7\u201d\u904a\u6232\uff0c\u81ea\u5b78\u80fd\u529b\u5f37\u7684\u4eba\u80fd\u5920\u9806\u5229\u5c07\u5730\u96f7\u9010\u500b\u6392\u6389\uff0c\u800c\u57fa\u790e\u4e0d\u8db3\u7684\u4eba\u5f88\u53ef\u80fd\u88ab\u70b8\u5f97\u6eff\u982d\u662f\u5305\uff0c\u4e26\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7e2e\u3002\u901a\u8b80\u6559\u6750\u4e5f\u662f\u4e00\u7a2e\u5e38\u898b\u505a\u6cd5\uff0c\u4f46\u5c0d\u65bc\u9762\u5411\u6c42\u8077\u7684\u4eba\u4f86\u8aaa\uff0c\u7562\u696d\u8ad6\u6587\u3001\u6295\u905e\u7c21\u6b77\u3001\u6e96\u5099\u7b46\u8a66\u548c\u9762\u8a66\u5df2\u7d93\u6d88\u8017\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u5543\u539a\u91cd\u7684\u66f8\u5f80\u5f80\u8b8a\u6210\u4e86\u4e00\u9805\u8271\u9245\u7684\u6311\u6230\u3002</p> <p>\u5982\u679c\u4f60\u4e5f\u9762\u81e8\u985e\u4f3c\u7684\u56f0\u64fe\uff0c\u90a3\u9ebc\u5f88\u5e78\u904b\u9019\u672c\u66f8\u201c\u627e\u201d\u5230\u4e86\u4f60\u3002\u672c\u66f8\u662f\u6211\u5c0d\u9019\u500b\u554f\u984c\u7d66\u51fa\u7684\u7b54\u6848\uff0c\u5373\u4f7f\u4e0d\u662f\u6700\u512a\u89e3\uff0c\u4e5f\u81f3\u5c11\u662f\u4e00\u6b21\u7a4d\u6975\u7684\u5617\u8a66\u3002\u672c\u66f8\u96d6\u7136\u4e0d\u8db3\u4ee5\u8b93\u4f60\u76f4\u63a5\u62ff\u5230 Offer\uff0c\u4f46\u6703\u5f15\u5c0e\u4f60\u63a2\u7d22\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u201c\u77e5\u8b58\u5730\u5716\u201d\uff0c\u5e36\u4f60\u77ad\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72c0\u3001\u5927\u5c0f\u548c\u5206\u4f48\u4f4d\u7f6e\uff0c\u8b93\u4f60\u638c\u63e1\u5404\u7a2e\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u9019\u4e9b\u672c\u9818\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u81ea\u5982\u5730\u5237\u984c\u548c\u95b1\u8b80\u6587\u737b\uff0c\u9010\u6b65\u69cb\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8b58\u9ad4\u7cfb\u3002</p> <p>\u6211\u6df1\u6df1\u8d0a\u540c\u8cbb\u66fc\u6559\u6388\u6240\u8a00\uff1a\u201cKnowledge isn't free. You have to pay attention.\u201d\u5f9e\u9019\u500b\u610f\u7fa9\u4e0a\u770b\uff0c\u9019\u672c\u66f8\u4e26\u975e\u5b8c\u5168\u201c\u514d\u8cbb\u201d\u3002\u70ba\u4e86\u4e0d\u8f9c\u8ca0\u4f60\u70ba\u672c\u66f8\u6240\u4ed8\u51fa\u7684\u5bf6\u8cb4\u201c\u6ce8\u610f\u529b\u201d\uff0c\u6211\u6703\u7aed\u76e1\u6240\u80fd\uff0c\u6295\u5165\u6700\u5927\u7684\u201c\u6ce8\u610f\u529b\u201d\u4f86\u5b8c\u6210\u672c\u66f8\u7684\u5275\u4f5c\u3002</p> <p>\u672c\u4eba\u81ea\u77e5\u5b78\u758f\u624d\u6dfa\uff0c\u66f8\u4e2d\u5167\u5bb9\u96d6\u7136\u5df2\u7d93\u904e\u4e00\u6bb5\u6642\u9593\u7684\u6253\u78e8\uff0c\u4f46\u4e00\u5b9a\u4ecd\u6709\u8a31\u591a\u932f\u8aa4\uff0c\u61c7\u8acb\u5404\u4f4d\u8001\u5e2b\u548c\u540c\u5b78\u6279\u8a55\u6307\u6b63\u3002</p> <p></p> Hello\uff0c\u6f14\u7b97\u6cd5\uff01 <p>\u8a08\u7b97\u6a5f\u7684\u51fa\u73fe\u7d66\u4e16\u754c\u5e36\u4f86\u4e86\u5de8\u5927\u8b8a\u9769\uff0c\u5b83\u6191\u85c9\u9ad8\u901f\u7684\u8a08\u7b97\u80fd\u529b\u548c\u51fa\u8272\u7684\u53ef\u7a0b\u5f0f\u8a2d\u8a08\u6027\uff0c\u6210\u70ba\u4e86\u57f7\u884c\u6f14\u7b97\u6cd5\u8207\u8655\u7406\u8cc7\u6599\u7684\u7406\u60f3\u5a92\u4ecb\u3002\u7121\u8ad6\u662f\u96fb\u5b50\u904a\u6232\u7684\u903c\u771f\u756b\u9762\u3001\u81ea\u52d5\u99d5\u99db\u7684\u667a\u6167\u6c7a\u7b56\uff0c\u9084\u662f AlphaGo \u7684\u7cbe\u5f69\u68cb\u5c40\u3001ChatGPT \u7684\u81ea\u7136\u4e92\u52d5\uff0c\u9019\u4e9b\u61c9\u7528\u90fd\u662f\u6f14\u7b97\u6cd5\u5728\u8a08\u7b97\u6a5f\u4e0a\u7684\u7cbe\u5999\u6f14\u7e79\u3002</p> <p>\u4e8b\u5be6\u4e0a\uff0c\u5728\u8a08\u7b97\u6a5f\u554f\u4e16\u4e4b\u524d\uff0c\u6f14\u7b97\u6cd5\u548c\u8cc7\u6599\u7d50\u69cb\u5c31\u5df2\u7d93\u5b58\u5728\u65bc\u4e16\u754c\u7684\u5404\u500b\u89d2\u843d\u3002\u65e9\u671f\u7684\u6f14\u7b97\u6cd5\u76f8\u5c0d\u7c21\u55ae\uff0c\u4f8b\u5982\u53e4\u4ee3\u7684\u8a08\u6578\u65b9\u6cd5\u548c\u5de5\u5177\u88fd\u4f5c\u6b65\u9a5f\u7b49\u3002\u96a8\u8457\u6587\u660e\u7684\u9032\u6b65\uff0c\u6f14\u7b97\u6cd5\u9010\u6f38\u8b8a\u5f97\u66f4\u52a0\u7cbe\u7d30\u548c\u8907\u96dc\u3002\u5f9e\u5de7\u596a\u5929\u5de5\u7684\u5320\u4eba\u6280\u85dd\u3001\u5230\u89e3\u653e\u751f\u7522\u529b\u7684\u5de5\u696d\u7522\u54c1\u3001\u518d\u5230\u5b87\u5b99\u57f7\u884c\u7684\u79d1\u5b78\u898f\u5f8b\uff0c\u5e7e\u4e4e\u6bcf\u4e00\u4ef6\u5e73\u51e1\u6216\u4ee4\u4eba\u9a5a\u6b4e\u7684\u4e8b\u7269\u80cc\u5f8c\uff0c\u90fd\u96b1\u85cf\u8457\u7cbe\u5999\u7684\u6f14\u7b97\u6cd5\u601d\u60f3\u3002</p> <p>\u540c\u6a23\uff0c\u8cc7\u6599\u7d50\u69cb\u7121\u8655\u4e0d\u5728\uff1a\u5927\u5230\u793e\u6703\u7db2\u8def\uff0c\u5c0f\u5230\u5730\u9435\u7dda\u8def\uff0c\u8a31\u591a\u7cfb\u7d71\u90fd\u53ef\u4ee5\u5efa\u6a21\u70ba\u201c\u5716\u201d\uff1b\u5927\u5230\u4e00\u500b\u570b\u5bb6\uff0c\u5c0f\u5230\u4e00\u500b\u5bb6\u5ead\uff0c\u793e\u6703\u7684\u4e3b\u8981\u7d44\u7e54\u5f62\u5f0f\u5448\u73fe\u51fa\u201c\u6a39\u201d\u7684\u7279\u5fb5\uff1b\u51ac\u5929\u7684\u8863\u670d\u5c31\u50cf\u201c\u5806\u758a\u201d\uff0c\u6700\u5148\u7a7f\u4e0a\u7684\u6700\u5f8c\u624d\u80fd\u812b\u4e0b\uff1b\u7fbd\u6bdb\u7403\u7b52\u5247\u5982\u540c\u201c\u4f47\u5217\u201d\uff0c\u4e00\u7aef\u653e\u5165\u3001\u53e6\u4e00\u7aef\u53d6\u51fa\uff1b\u5b57\u5178\u5c31\u50cf\u4e00\u500b\u201c\u96dc\u6e4a\u8868\u201d\uff0c\u80fd\u5920\u5feb\u901f\u67e5\u8a62\u76ee\u6a19\u8a5e\u689d\u3002</p> <p>\u672c\u66f8\u65e8\u5728\u900f\u904e\u6e05\u6670\u6613\u61c2\u7684\u52d5\u756b\u5716\u89e3\u548c\u53ef\u57f7\u884c\u7684\u7a0b\u5f0f\u78bc\u793a\u4f8b\uff0c\u4f7f\u8b80\u8005\u7406\u89e3\u6f14\u7b97\u6cd5\u548c\u8cc7\u6599\u7d50\u69cb\u7684\u6838\u5fc3\u6982\u5ff5\uff0c\u4e26\u80fd\u5920\u900f\u904e\u7a0b\u5f0f\u8a2d\u8a08\u4f86\u5be6\u73fe\u5b83\u5011\u3002\u5728\u6b64\u57fa\u790e\u4e0a\uff0c\u672c\u66f8\u81f4\u529b\u65bc\u63ed\u793a\u6f14\u7b97\u6cd5\u5728\u8907\u96dc\u4e16\u754c\u4e2d\u7684\u751f\u52d5\u9ad4\u73fe\uff0c\u5c55\u73fe\u6f14\u7b97\u6cd5\u4e4b\u7f8e\u3002\u5e0c\u671b\u672c\u66f8\u80fd\u5920\u5e6b\u52a9\u5230\u4f60\uff01</p>"},{"location":"chapter_introduction/","title":"\u7b2c 1 \u7ae0 \u00a0 \u521d\u8b58\u6f14\u7b97\u6cd5","text":"<p>Abstract</p> <p>\u4e00\u4f4d\u5c11\u5973\u7fe9\u7fe9\u8d77\u821e\uff0c\u8207\u8cc7\u6599\u4ea4\u7e54\u5728\u4e00\u8d77\uff0c\u88d9\u896c\u4e0a\u98c4\u63da\u8457\u6f14\u7b97\u6cd5\u7684\u65cb\u5f8b\u3002</p> <p>\u5979\u9080\u8acb\u4f60\u5171\u821e\uff0c\u8acb\u7dca\u8ddf\u5979\u7684\u6b65\u4f10\uff0c\u8e0f\u5165\u5145\u6eff\u908f\u8f2f\u8207\u7f8e\u611f\u7684\u6f14\u7b97\u6cd5\u4e16\u754c\u3002</p>"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>1.1 \u00a0 \u6f14\u7b97\u6cd5\u7121\u8655\u4e0d\u5728</li> <li>1.2 \u00a0 \u6f14\u7b97\u6cd5\u662f\u4ec0\u9ebc</li> <li>1.3 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1 \u00a0 \u6f14\u7b97\u6cd5\u7121\u8655\u4e0d\u5728","text":"<p>\u7576\u6211\u5011\u807d\u5230\u201c\u6f14\u7b97\u6cd5\u201d\u9019\u500b\u8a5e\u6642\uff0c\u5f88\u81ea\u7136\u5730\u6703\u60f3\u5230\u6578\u5b78\u3002\u7136\u800c\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u6f14\u7b97\u6cd5\u4e26\u4e0d\u6d89\u53ca\u8907\u96dc\u6578\u5b78\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8cf4\u57fa\u672c\u908f\u8f2f\uff0c\u9019\u4e9b\u908f\u8f2f\u5728\u6211\u5011\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u8655\u8655\u53ef\u898b\u3002</p> <p>\u5728\u6b63\u5f0f\u63a2\u8a0e\u6f14\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u500b\u6709\u8da3\u7684\u4e8b\u5be6\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7d93\u5728\u4e0d\u77e5\u4e0d\u89ba\u4e2d\u5b78\u6703\u4e86\u8a31\u591a\u6f14\u7b97\u6cd5\uff0c\u4e26\u7fd2\u6163\u5c07\u5b83\u5011\u61c9\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\u6211\u5c07\u8209\u5e7e\u500b\u5177\u9ad4\u7684\u4f8b\u5b50\u4f86\u8b49\u5be6\u9019\u4e00\u9ede\u3002</p> <p>\u4f8b\u4e00\uff1a\u67e5\u5b57\u5178\u3002\u5728\u5b57\u5178\u88e1\uff0c\u6bcf\u500b\u6f22\u5b57\u90fd\u5c0d\u61c9\u4e00\u500b\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u5b57\u6bcd\u9806\u5e8f\u6392\u5217\u7684\u3002\u5047\u8a2d\u6211\u5011\u9700\u8981\u67e5\u8a62\u4e00\u500b\u62fc\u97f3\u9996\u5b57\u6bcd\u70ba \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u6703\u6309\u7167\u5716 1-1 \u6240\u793a\u7684\u65b9\u5f0f\u5be6\u73fe\u3002</p> <ol> <li>\u7ffb\u958b\u5b57\u5178\u7d04\u4e00\u534a\u7684\u9801\u6578\uff0c\u6aa2\u8996\u8a72\u9801\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u9ebc\uff0c\u5047\u8a2d\u9996\u5b57\u6bcd\u70ba \\(m\\) \u3002</li> <li>\u7531\u65bc\u5728\u62fc\u97f3\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u65bc \\(m\\) \u4e4b\u5f8c\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u8a62\u7bc4\u570d\u7e2e\u5c0f\u5230\u5f8c\u534a\u90e8\u5206\u3002</li> <li>\u4e0d\u65b7\u91cd\u8907\u6b65\u9a5f <code>1.</code> \u548c\u6b65\u9a5f <code>2.</code> \uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u70ba \\(r\\) \u7684\u9801\u78bc\u70ba\u6b62\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 1-1 \u00a0 \u67e5\u5b57\u5178\u6b65\u9a5f </p> <p>\u67e5\u5b57\u5178\u9019\u500b\u5c0f\u5b78\u751f\u5fc5\u5099\u6280\u80fd\uff0c\u5be6\u969b\u4e0a\u5c31\u662f\u8457\u540d\u7684\u201c\u4e8c\u5206\u641c\u5c0b\u201d\u6f14\u7b97\u6cd5\u3002\u5f9e\u8cc7\u6599\u7d50\u69cb\u7684\u89d2\u5ea6\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u5b57\u5178\u8996\u70ba\u4e00\u500b\u5df2\u6392\u5e8f\u7684\u201c\u9663\u5217\u201d\uff1b\u5f9e\u6f14\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u201c\u4e8c\u5206\u641c\u5c0b\u201d\u3002</p> <p>\u4f8b\u4e8c\uff1a\u6574\u7406\u64b2\u514b\u3002\u6211\u5011\u5728\u6253\u724c\u6642\uff0c\u6bcf\u5c40\u90fd\u9700\u8981\u6574\u7406\u624b\u4e2d\u7684\u64b2\u514b\u724c\uff0c\u4f7f\u5176\u5f9e\u5c0f\u5230\u5927\u6392\u5217\uff0c\u5be6\u73fe\u6d41\u7a0b\u5982\u5716 1-2 \u6240\u793a\u3002</p> <ol> <li>\u5c07\u64b2\u514b\u724c\u5283\u5206\u70ba\u201c\u6709\u5e8f\u201d\u548c\u201c\u7121\u5e8f\u201d\u5169\u90e8\u5206\uff0c\u4e26\u5047\u8a2d\u521d\u59cb\u72c0\u614b\u4e0b\u6700\u5de6 1 \u5f35\u64b2\u514b\u724c\u5df2\u7d93\u6709\u5e8f\u3002</li> <li>\u5728\u7121\u5e8f\u90e8\u5206\u62bd\u51fa\u4e00\u5f35\u64b2\u514b\u724c\uff0c\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\u7684\u6b63\u78ba\u4f4d\u7f6e\uff1b\u5b8c\u6210\u5f8c\u6700\u5de6 2 \u5f35\u64b2\u514b\u5df2\u7d93\u6709\u5e8f\u3002</li> <li>\u4e0d\u65b7\u8ff4\u5708\u6b65\u9a5f <code>2.</code> \uff0c\u6bcf\u4e00\u8f2a\u5c07\u4e00\u5f35\u64b2\u514b\u724c\u5f9e\u7121\u5e8f\u90e8\u5206\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\uff0c\u76f4\u81f3\u6240\u6709\u64b2\u514b\u724c\u90fd\u6709\u5e8f\u3002</li> </ol> <p></p> <p> \u5716 1-2 \u00a0 \u64b2\u514b\u6392\u5e8f\u6b65\u9a5f </p> <p>\u4e0a\u8ff0\u6574\u7406\u64b2\u514b\u724c\u7684\u65b9\u6cd5\u672c\u8cea\u4e0a\u662f\u201c\u63d2\u5165\u6392\u5e8f\u201d\u6f14\u7b97\u6cd5\uff0c\u5b83\u5728\u8655\u7406\u5c0f\u578b\u8cc7\u6599\u96c6\u6642\u975e\u5e38\u9ad8\u6548\u3002\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\u7684\u6392\u5e8f\u5eab\u51fd\u5f0f\u4e2d\u90fd\u6709\u63d2\u5165\u6392\u5e8f\u7684\u8eab\u5f71\u3002</p> <p>\u4f8b\u4e09\uff1a\u8ca8\u5e63\u627e\u96f6\u3002\u5047\u8a2d\u6211\u5011\u5728\u8d85\u5e02\u8cfc\u8cb7\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7d66\u4e86\u6536\u9280\u54e1 \\(100\\) \u5143\uff0c\u5247\u6536\u9280\u54e1\u9700\u8981\u627e\u6211\u5011 \\(31\\) \u5143\u3002\u4ed6\u6703\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u5982\u5716 1-3 \u6240\u793a\u7684\u601d\u8003\u3002</p> <ol> <li>\u53ef\u9078\u9805\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8ca8\u5e63\uff0c\u5305\u62ec \\(1\\) \u5143\u3001\\(5\\) \u5143\u3001\\(10\\) \u5143\u3001\\(20\\) \u5143\u3002</li> <li>\u5f9e\u53ef\u9078\u9805\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u9918 \\(31 - 20 = 11\\) \u5143\u3002</li> <li>\u5f9e\u5269\u9918\u53ef\u9078\u9805\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u9918 \\(11 - 10 = 1\\) \u5143\u3002</li> <li>\u5f9e\u5269\u9918\u53ef\u9078\u9805\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u9918 \\(1 - 1 = 0\\) \u5143\u3002</li> <li>\u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u70ba \\(20 + 10 + 1 = 31\\) \u5143\u3002</li> </ol> <p></p> <p> \u5716 1-3 \u00a0 \u8ca8\u5e63\u627e\u96f6\u904e\u7a0b </p> <p>\u5728\u4ee5\u4e0a\u6b65\u9a5f\u4e2d\uff0c\u6211\u5011\u6bcf\u4e00\u6b65\u90fd\u63a1\u53d6\u7576\u524d\u770b\u4f86\u6700\u597d\u7684\u9078\u64c7\uff08\u5118\u53ef\u80fd\u7528\u5927\u9762\u984d\u7684\u8ca8\u5e63\uff09\uff0c\u6700\u7d42\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u5f9e\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u9019\u7a2e\u65b9\u6cd5\u672c\u8cea\u4e0a\u662f\u201c\u8caa\u5a6a\u201d\u6f14\u7b97\u6cd5\u3002</p> <p>\u5c0f\u5230\u70f9\u98ea\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u969b\u822a\u884c\uff0c\u5e7e\u4e4e\u6240\u6709\u554f\u984c\u7684\u89e3\u6c7a\u90fd\u96e2\u4e0d\u958b\u6f14\u7b97\u6cd5\u3002\u8a08\u7b97\u6a5f\u7684\u51fa\u73fe\u4f7f\u5f97\u6211\u5011\u80fd\u5920\u900f\u904e\u7a0b\u5f0f\u8a2d\u8a08\u5c07\u8cc7\u6599\u7d50\u69cb\u5132\u5b58\u5728\u8a18\u61b6\u9ad4\u4e2d\uff0c\u540c\u6642\u7de8\u5beb\u7a0b\u5f0f\u78bc\u547c\u53eb CPU \u548c GPU \u57f7\u884c\u6f14\u7b97\u6cd5\u3002\u9019\u6a23\u4e00\u4f86\uff0c\u6211\u5011\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u554f\u984c\u8f49\u79fb\u5230\u8a08\u7b97\u6a5f\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u6c7a\u5404\u7a2e\u8907\u96dc\u554f\u984c\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u5c0d\u8cc7\u6599\u7d50\u69cb\u3001\u6f14\u7b97\u6cd5\u3001\u9663\u5217\u548c\u4e8c\u5206\u641c\u5c0b\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u8acb\u7e7c\u7e8c\u5f80\u4e0b\u95b1\u8b80\uff0c\u672c\u66f8\u5c07\u5f15\u5c0e\u4f60\u9081\u5165\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u77e5\u8b58\u6bbf\u5802\u3002</p>"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u6f14\u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u7121\u8655\u4e0d\u5728\uff0c\u4e26\u4e0d\u662f\u9059\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8b58\u3002\u5be6\u969b\u4e0a\uff0c\u6211\u5011\u5df2\u7d93\u5728\u4e0d\u77e5\u4e0d\u89ba\u4e2d\u5b78\u6703\u4e86\u8a31\u591a\u6f14\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u6c7a\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u554f\u984c\u3002</li> <li>\u67e5\u5b57\u5178\u7684\u539f\u7406\u8207\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\u9ad4\u73fe\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u6f14\u7b97\u6cd5\u601d\u60f3\u3002</li> <li>\u6574\u7406\u64b2\u514b\u7684\u904e\u7a0b\u8207\u63d2\u5165\u6392\u5e8f\u6f14\u7b97\u6cd5\u975e\u5e38\u985e\u4f3c\u3002\u63d2\u5165\u6392\u5e8f\u6f14\u7b97\u6cd5\u9069\u5408\u6392\u5e8f\u5c0f\u578b\u8cc7\u6599\u96c6\u3002</li> <li>\u8ca8\u5e63\u627e\u96f6\u7684\u6b65\u9a5f\u672c\u8cea\u4e0a\u662f\u8caa\u5a6a\u6f14\u7b97\u6cd5\uff0c\u6bcf\u4e00\u6b65\u90fd\u63a1\u53d6\u7576\u524d\u770b\u4f86\u6700\u597d\u7684\u9078\u64c7\u3002</li> <li>\u6f14\u7b97\u6cd5\u662f\u5728\u6709\u9650\u6642\u9593\u5167\u89e3\u6c7a\u7279\u5b9a\u554f\u984c\u7684\u4e00\u7d44\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9a5f\uff0c\u800c\u8cc7\u6599\u7d50\u69cb\u662f\u8a08\u7b97\u6a5f\u4e2d\u7d44\u7e54\u548c\u5132\u5b58\u8cc7\u6599\u7684\u65b9\u5f0f\u3002</li> <li>\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7dca\u5bc6\u76f8\u9023\u3002\u8cc7\u6599\u7d50\u69cb\u662f\u6f14\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u6f14\u7b97\u6cd5\u70ba\u8cc7\u6599\u7d50\u69cb\u6ce8\u5165\u751f\u547d\u529b\u3002</li> <li>\u6211\u5011\u53ef\u4ee5\u5c07\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u985e\u6bd4\u70ba\u62fc\u88dd\u7a4d\u6728\uff0c\u7a4d\u6728\u4ee3\u8868\u8cc7\u6599\uff0c\u7a4d\u6728\u7684\u5f62\u72c0\u548c\u9023\u7dda\u65b9\u5f0f\u7b49\u4ee3\u8868\u8cc7\u6599\u7d50\u69cb\uff0c\u62fc\u88dd\u7a4d\u6728\u7684\u6b65\u9a5f\u5247\u5c0d\u61c9\u6f14\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_introduction/summary/#1-q-a","title":"1. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u4f5c\u70ba\u4e00\u540d\u7a0b\u5f0f\u8a2d\u8a08\u5e2b\uff0c\u6211\u5728\u65e5\u5e38\u5de5\u4f5c\u4e2d\u5f9e\u672a\u7528\u6f14\u7b97\u6cd5\u89e3\u6c7a\u904e\u554f\u984c\uff0c\u5e38\u7528\u6f14\u7b97\u6cd5\u90fd\u88ab\u7a0b\u5f0f\u8a9e\u8a00\u5c01\u88dd\u597d\u4e86\uff0c\u76f4\u63a5\u7528\u5c31\u53ef\u4ee5\u4e86\uff1b\u9019\u662f\u5426\u610f\u5473\u8457\u6211\u5011\u5de5\u4f5c\u4e2d\u7684\u554f\u984c\u9084\u6c92\u6709\u5230\u9054\u9700\u8981\u6f14\u7b97\u6cd5\u7684\u7a0b\u5ea6\uff1f</p> <p>\u5982\u679c\u628a\u5177\u9ad4\u7684\u5de5\u4f5c\u6280\u80fd\u6bd4\u4f5c\u662f\u6b66\u529f\u7684\u201c\u62db\u5f0f\u201d\u7684\u8a71\uff0c\u90a3\u9ebc\u57fa\u790e\u79d1\u76ee\u61c9\u8a72\u66f4\u50cf\u662f\u201c\u5167\u529f\u201d\u3002</p> <p>\u6211\u8a8d\u70ba\u5b78\u6f14\u7b97\u6cd5\uff08\u4ee5\u53ca\u5176\u4ed6\u57fa\u790e\u79d1\u76ee\uff09\u7684\u610f\u7fa9\u4e0d\u662f\u5728\u65bc\u5728\u5de5\u4f5c\u4e2d\u5f9e\u96f6\u5be6\u73fe\u5b83\uff0c\u800c\u662f\u57fa\u65bc\u5b78\u5230\u7684\u77e5\u8b58\uff0c\u5728\u89e3\u6c7a\u554f\u984c\u6642\u80fd\u5920\u4f5c\u51fa\u5c08\u696d\u7684\u53cd\u61c9\u548c\u5224\u65b7\uff0c\u5f9e\u800c\u63d0\u5347\u5de5\u4f5c\u7684\u6574\u9ad4\u8cea\u91cf\u3002\u8209\u4e00\u500b\u7c21\u55ae\u4f8b\u5b50\uff0c\u6bcf\u7a2e\u7a0b\u5f0f\u8a9e\u8a00\u90fd\u5167\u5efa\u4e86\u6392\u5e8f\u51fd\u5f0f\uff1a</p> <ul> <li>\u5982\u679c\u6211\u5011\u6c92\u6709\u5b78\u904e\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\uff0c\u90a3\u9ebc\u7d66\u5b9a\u4efb\u4f55\u8cc7\u6599\uff0c\u6211\u5011\u53ef\u80fd\u90fd\u585e\u7d66\u9019\u500b\u6392\u5e8f\u51fd\u5f0f\u53bb\u505a\u4e86\u3002\u57f7\u884c\u9806\u66a2\u3001\u6548\u80fd\u4e0d\u932f\uff0c\u770b\u4e0a\u53bb\u4e26\u6c92\u6709\u4ec0\u9ebc\u554f\u984c\u3002</li> <li>\u4f46\u5982\u679c\u5b78\u904e\u6f14\u7b97\u6cd5\uff0c\u6211\u5011\u5c31\u6703\u77e5\u9053\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(n \\log n)\\) \uff1b\u800c\u5982\u679c\u7d66\u5b9a\u7684\u8cc7\u6599\u662f\u56fa\u5b9a\u4f4d\u6578\u7684\u6574\u6578\uff08\u4f8b\u5982\u5b78\u865f\uff09\uff0c\u90a3\u9ebc\u6211\u5011\u5c31\u53ef\u4ee5\u7528\u6548\u7387\u66f4\u9ad8\u7684\u201c\u57fa\u6578\u6392\u5e8f\u201d\u4f86\u505a\uff0c\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u964d\u70ba \\(O(nk)\\) \uff0c\u5176\u4e2d \\(k\\) \u70ba\u4f4d\u6578\u3002\u7576\u8cc7\u6599\u9ad4\u91cf\u5f88\u5927\u6642\uff0c\u7bc0\u7701\u51fa\u4f86\u7684\u57f7\u884c\u6642\u9593\u5c31\u80fd\u5275\u9020\u8f03\u5927\u50f9\u503c\uff08\u6210\u672c\u964d\u4f4e\u3001\u9ad4\u9a57\u8b8a\u597d\u7b49\uff09\u3002</li> </ul> <p>\u5728\u5de5\u7a0b\u9818\u57df\u4e2d\uff0c\u5927\u91cf\u554f\u984c\u662f\u96e3\u4ee5\u9054\u5230\u6700\u512a\u89e3\u7684\uff0c\u8a31\u591a\u554f\u984c\u53ea\u662f\u88ab\u201c\u5dee\u4e0d\u591a\u201d\u5730\u89e3\u6c7a\u4e86\u3002\u554f\u984c\u7684\u96e3\u6613\u7a0b\u5ea6\u4e00\u65b9\u9762\u53d6\u6c7a\u65bc\u554f\u984c\u672c\u8eab\u7684\u6027\u8cea\uff0c\u53e6\u4e00\u65b9\u9762\u4e5f\u53d6\u6c7a\u65bc\u89c0\u6e2c\u554f\u984c\u7684\u4eba\u7684\u77e5\u8b58\u5132\u5099\u3002\u4eba\u7684\u77e5\u8b58\u8d8a\u5b8c\u5099\u3001\u7d93\u9a57\u8d8a\u591a\uff0c\u5206\u6790\u554f\u984c\u5c31\u6703\u8d8a\u6df1\u5165\uff0c\u554f\u984c\u5c31\u80fd\u88ab\u89e3\u6c7a\u5f97\u66f4\u512a\u96c5\u3002</p>"},{"location":"chapter_introduction/what_is_dsa/","title":"1.2 \u00a0 \u6f14\u7b97\u6cd5\u662f\u4ec0\u9ebc","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1 \u00a0 \u6f14\u7b97\u6cd5\u5b9a\u7fa9","text":"<p>\u6f14\u7b97\u6cd5\uff08algorithm\uff09\u662f\u5728\u6709\u9650\u6642\u9593\u5167\u89e3\u6c7a\u7279\u5b9a\u554f\u984c\u7684\u4e00\u7d44\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9a5f\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u554f\u984c\u662f\u660e\u78ba\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f38\u5165\u548c\u8f38\u51fa\u5b9a\u7fa9\u3002</li> <li>\u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u5920\u5728\u6709\u9650\u6b65\u9a5f\u3001\u6642\u9593\u548c\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e0b\u5b8c\u6210\u3002</li> <li>\u5404\u6b65\u9a5f\u90fd\u6709\u78ba\u5b9a\u7684\u542b\u7fa9\uff0c\u5728\u76f8\u540c\u7684\u8f38\u5165\u548c\u57f7\u884c\u689d\u4ef6\u4e0b\uff0c\u8f38\u51fa\u59cb\u7d42\u76f8\u540c\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2 \u00a0 \u8cc7\u6599\u7d50\u69cb\u5b9a\u7fa9","text":"<p>\u8cc7\u6599\u7d50\u69cb\uff08data structure\uff09\u662f\u7d44\u7e54\u548c\u5132\u5b58\u8cc7\u6599\u7684\u65b9\u5f0f\uff0c\u6db5\u84cb\u8cc7\u6599\u5167\u5bb9\u3001\u8cc7\u6599\u4e4b\u9593\u95dc\u4fc2\u548c\u8cc7\u6599\u64cd\u4f5c\u65b9\u6cd5\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u8a2d\u8a08\u76ee\u6a19\u3002</p> <ul> <li>\u7a7a\u9593\u4f54\u7528\u5118\u91cf\u5c11\uff0c\u4ee5\u7bc0\u7701\u8a08\u7b97\u6a5f\u8a18\u61b6\u9ad4\u3002</li> <li>\u8cc7\u6599\u64cd\u4f5c\u5118\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u84cb\u8cc7\u6599\u8a2a\u554f\u3001\u65b0\u589e\u3001\u522a\u9664\u3001\u66f4\u65b0\u7b49\u3002</li> <li>\u63d0\u4f9b\u7c21\u6f54\u7684\u8cc7\u6599\u8868\u793a\u548c\u908f\u8f2f\u8cc7\u8a0a\uff0c\u4ee5\u4fbf\u6f14\u7b97\u6cd5\u9ad8\u6548\u57f7\u884c\u3002</li> </ul> <p>\u8cc7\u6599\u7d50\u69cb\u8a2d\u8a08\u662f\u4e00\u500b\u5145\u6eff\u6b0a\u8861\u7684\u904e\u7a0b\u3002\u5982\u679c\u60f3\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u63d0\u5347\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u5354\u3002\u4e0b\u9762\u8209\u5169\u500b\u4f8b\u5b50\u3002</p> <ul> <li>\u93c8\u7d50\u4e32\u5217\u76f8\u8f03\u65bc\u9663\u5217\uff0c\u5728\u8cc7\u6599\u65b0\u589e\u548c\u522a\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u72a7\u7272\u4e86\u8cc7\u6599\u8a2a\u554f\u901f\u5ea6\u3002</li> <li>\u5716\u76f8\u8f03\u65bc\u93c8\u7d50\u4e32\u5217\uff0c\u63d0\u4f9b\u4e86\u66f4\u8c50\u5bcc\u7684\u908f\u8f2f\u8cc7\u8a0a\uff0c\u4f46\u9700\u8981\u4f54\u7528\u66f4\u5927\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3 \u00a0 \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u95dc\u4fc2","text":"<p>\u5982\u5716 1-4 \u6240\u793a\uff0c\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u9ad8\u5ea6\u76f8\u95dc\u3001\u7dca\u5bc6\u7d50\u5408\uff0c\u5177\u9ad4\u8868\u73fe\u5728\u4ee5\u4e0b\u4e09\u500b\u65b9\u9762\u3002</p> <ul> <li>\u8cc7\u6599\u7d50\u69cb\u662f\u6f14\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u8cc7\u6599\u7d50\u69cb\u70ba\u6f14\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7d50\u69cb\u5316\u5132\u5b58\u7684\u8cc7\u6599\uff0c\u4ee5\u53ca\u64cd\u4f5c\u8cc7\u6599\u7684\u65b9\u6cd5\u3002</li> <li>\u6f14\u7b97\u6cd5\u70ba\u8cc7\u6599\u7d50\u69cb\u6ce8\u5165\u751f\u547d\u529b\u3002\u8cc7\u6599\u7d50\u69cb\u672c\u8eab\u50c5\u5132\u5b58\u8cc7\u6599\u8cc7\u8a0a\uff0c\u7d50\u5408\u6f14\u7b97\u6cd5\u624d\u80fd\u89e3\u6c7a\u7279\u5b9a\u554f\u984c\u3002</li> <li>\u6f14\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u65bc\u4e0d\u540c\u7684\u8cc7\u6599\u7d50\u69cb\u5be6\u73fe\uff0c\u4f46\u57f7\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\uff0c\u9078\u64c7\u5408\u9069\u7684\u8cc7\u6599\u7d50\u69cb\u662f\u95dc\u9375\u3002</li> </ul> <p></p> <p> \u5716 1-4 \u00a0 \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u95dc\u4fc2 </p> <p>\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7336\u5982\u5716 1-5 \u6240\u793a\u7684\u62fc\u88dd\u7a4d\u6728\u3002\u4e00\u5957\u7a4d\u6728\uff0c\u9664\u4e86\u5305\u542b\u8a31\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u9084\u9644\u6709\u8a73\u7d30\u7684\u7d44\u88dd\u8aaa\u660e\u66f8\u3002\u6211\u5011\u6309\u7167\u8aaa\u660e\u66f8\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7d44\u88dd\u51fa\u7cbe\u7f8e\u7684\u7a4d\u6728\u6a21\u578b\u3002</p> <p></p> <p> \u5716 1-5 \u00a0 \u62fc\u88dd\u7a4d\u6728 </p> <p>\u5169\u8005\u7684\u8a73\u7d30\u5c0d\u61c9\u95dc\u4fc2\u5982\u8868 1-1 \u6240\u793a\u3002</p> <p> \u8868 1-1 \u00a0 \u5c07\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u985e\u6bd4\u70ba\u62fc\u88dd\u7a4d\u6728 </p> \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5 \u62fc\u88dd\u7a4d\u6728 \u8f38\u5165\u8cc7\u6599 \u672a\u62fc\u88dd\u7684\u7a4d\u6728 \u8cc7\u6599\u7d50\u69cb \u7a4d\u6728\u7d44\u7e54\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72c0\u3001\u5927\u5c0f\u3001\u9023\u7dda\u65b9\u5f0f\u7b49 \u6f14\u7b97\u6cd5 \u628a\u7a4d\u6728\u62fc\u6210\u76ee\u6a19\u5f62\u614b\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9a5f \u8f38\u51fa\u8cc7\u6599 \u7a4d\u6728\u6a21\u578b <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u662f\u7368\u7acb\u65bc\u7a0b\u5f0f\u8a9e\u8a00\u7684\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u66f8\u5f97\u4ee5\u63d0\u4f9b\u57fa\u65bc\u591a\u7a2e\u7a0b\u5f0f\u8a9e\u8a00\u7684\u5be6\u73fe\u3002</p> <p>\u7d04\u5b9a\u4fd7\u6210\u7684\u7c21\u7a31</p> <p>\u5728\u5be6\u969b\u8a0e\u8ad6\u6642\uff0c\u6211\u5011\u901a\u5e38\u6703\u5c07\u201c\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u201d\u7c21\u7a31\u70ba\u201c\u6f14\u7b97\u6cd5\u201d\u3002\u6bd4\u5982\u773e\u6240\u5468\u77e5\u7684 LeetCode \u6f14\u7b97\u6cd5\u984c\u76ee\uff0c\u5be6\u969b\u4e0a\u540c\u6642\u8003\u67e5\u8cc7\u6599\u7d50\u69cb\u548c\u6f14\u7b97\u6cd5\u5169\u65b9\u9762\u7684\u77e5\u8b58\u3002</p>"},{"location":"chapter_preface/","title":"\u7b2c 0 \u7ae0 \u00a0 \u524d\u8a00","text":"<p>Abstract</p> <p>\u6f14\u7b97\u6cd5\u7336\u5982\u7f8e\u5999\u7684\u4ea4\u97ff\u6a02\uff0c\u6bcf\u4e00\u884c\u7a0b\u5f0f\u78bc\u90fd\u50cf\u97fb\u5f8b\u822c\u6d41\u6dcc\u3002</p> <p>\u9858\u9019\u672c\u66f8\u5728\u4f60\u7684\u8166\u6d77\u4e2d\u8f15\u8f15\u97ff\u8d77\uff0c\u7559\u4e0b\u7368\u7279\u800c\u6df1\u523b\u7684\u65cb\u5f8b\u3002</p>"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>0.1 \u00a0 \u95dc\u65bc\u672c\u66f8</li> <li>0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u66f8</li> <li>0.3 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_preface/about_the_book/","title":"0.1 \u00a0 \u95dc\u65bc\u672c\u66f8","text":"<p>\u672c\u5c08\u6848\u65e8\u5728\u5efa\u7acb\u4e00\u672c\u958b\u6e90\u3001\u514d\u8cbb\u3001\u5c0d\u65b0\u624b\u53cb\u597d\u7684\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u5165\u9580\u6559\u7a0b\u3002</p> <ul> <li>\u5168\u66f8\u63a1\u7528\u52d5\u756b\u5716\u89e3\uff0c\u5167\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b78\u7fd2\u66f2\u7dda\u5e73\u6ed1\uff0c\u5f15\u5c0e\u521d\u5b78\u8005\u63a2\u7d22\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u77e5\u8b58\u5730\u5716\u3002</li> <li>\u6e90\u7a0b\u5f0f\u78bc\u53ef\u4e00\u9375\u57f7\u884c\uff0c\u5e6b\u52a9\u8b80\u8005\u5728\u7df4\u7fd2\u4e2d\u63d0\u5347\u7a0b\u5f0f\u8a2d\u8a08\u6280\u80fd\uff0c\u77ad\u89e3\u6f14\u7b97\u6cd5\u5de5\u4f5c\u539f\u7406\u548c\u8cc7\u6599\u7d50\u69cb\u5e95\u5c64\u5be6\u73fe\u3002</li> <li>\u63d0\u5021\u8b80\u8005\u4e92\u52a9\u5b78\u7fd2\uff0c\u6b61\u8fce\u5927\u5bb6\u5728\u8a55\u8ad6\u5340\u63d0\u51fa\u554f\u984c\u8207\u5206\u4eab\u898b\u89e3\uff0c\u5728\u4ea4\u6d41\u8a0e\u8ad6\u4e2d\u5171\u540c\u9032\u6b65\u3002</li> </ul>"},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1 \u00a0 \u8b80\u8005\u7269\u4ef6","text":"<p>\u82e5\u4f60\u662f\u6f14\u7b97\u6cd5\u521d\u5b78\u8005\uff0c\u5f9e\u672a\u63a5\u89f8\u904e\u6f14\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7d93\u6709\u4e00\u4e9b\u5237\u984c\u7d93\u9a57\uff0c\u5c0d\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8a8d\u8b58\uff0c\u5728\u6703\u8207\u4e0d\u6703\u4e4b\u9593\u53cd\u8986\u6a6b\u8df3\uff0c\u90a3\u9ebc\u672c\u66f8\u6b63\u662f\u70ba\u4f60\u91cf\u8eab\u5b9a\u88fd\u7684\uff01</p> <p>\u5982\u679c\u4f60\u5df2\u7d93\u7a4d\u7d2f\u4e00\u5b9a\u7684\u5237\u984c\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u984c\u578b\uff0c\u90a3\u9ebc\u672c\u66f8\u53ef\u52a9\u4f60\u56de\u9867\u8207\u68b3\u7406\u6f14\u7b97\u6cd5\u77e5\u8b58\u9ad4\u7cfb\uff0c\u5009\u5eab\u6e90\u7a0b\u5f0f\u78bc\u53ef\u4ee5\u7576\u4f5c\u201c\u5237\u984c\u5de5\u5177\u5eab\u201d\u6216\u201c\u6f14\u7b97\u6cd5\u5b57\u5178\u201d\u4f86\u4f7f\u7528\u3002</p> <p>\u82e5\u4f60\u662f\u6f14\u7b97\u6cd5\u201c\u5927\u795e\u201d\uff0c\u6211\u5011\u671f\u5f85\u6536\u5230\u4f60\u7684\u5bf6\u8cb4\u5efa\u8b70\uff0c\u6216\u8005\u4e00\u8d77\u53c3\u8207\u5275\u4f5c\u3002</p> <p>\u524d\u7f6e\u689d\u4ef6</p> <p>\u4f60\u9700\u8981\u81f3\u5c11\u5177\u5099\u4efb\u4e00\u8a9e\u8a00\u7684\u7a0b\u5f0f\u8a2d\u8a08\u57fa\u790e\uff0c\u80fd\u5920\u95b1\u8b80\u548c\u7de8\u5beb\u7c21\u55ae\u7a0b\u5f0f\u78bc\u3002</p>"},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2 \u00a0 \u5167\u5bb9\u7d50\u69cb","text":"<p>\u672c\u66f8\u7684\u4e3b\u8981\u5167\u5bb9\u5982\u5716 0-1 \u6240\u793a\u3002</p> <ul> <li>\u8907\u96dc\u5ea6\u5206\u6790\uff1a\u8cc7\u6599\u7d50\u69cb\u548c\u6f14\u7b97\u6cd5\u7684\u8a55\u50f9\u7dad\u5ea6\u8207\u65b9\u6cd5\u3002\u6642\u9593\u8907\u96dc\u5ea6\u548c\u7a7a\u9593\u8907\u96dc\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u898b\u578b\u5225\u3001\u793a\u4f8b\u7b49\u3002</li> <li>\u8cc7\u6599\u7d50\u69cb\uff1a\u57fa\u672c\u8cc7\u6599\u578b\u5225\u548c\u8cc7\u6599\u7d50\u69cb\u7684\u5206\u985e\u65b9\u6cd5\u3002\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u5806\u758a\u3001\u4f47\u5217\u3001\u96dc\u6e4a\u8868\u3001\u6a39\u3001\u5806\u7a4d\u3001\u5716\u7b49\u8cc7\u6599\u7d50\u69cb\u7684\u5b9a\u7fa9\u3001\u512a\u7f3a\u9ede\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u898b\u578b\u5225\u3001\u5178\u578b\u61c9\u7528\u3001\u5be6\u73fe\u65b9\u6cd5\u7b49\u3002</li> <li>\u6f14\u7b97\u6cd5\uff1a\u641c\u5c0b\u3001\u6392\u5e8f\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52d5\u614b\u898f\u5283\u3001\u8caa\u5a6a\u7b49\u6f14\u7b97\u6cd5\u7684\u5b9a\u7fa9\u3001\u512a\u7f3a\u9ede\u3001\u6548\u7387\u3001\u61c9\u7528\u5834\u666f\u3001\u89e3\u984c\u6b65\u9a5f\u548c\u793a\u4f8b\u554f\u984c\u7b49\u3002</li> </ul> <p></p> <p> \u5716 0-1 \u00a0 \u672c\u66f8\u4e3b\u8981\u5167\u5bb9 </p>"},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3 \u00a0 \u81f4\u8b1d","text":"<p>\u672c\u66f8\u5728\u958b\u6e90\u793e\u7fa4\u773e\u591a\u8ca2\u737b\u8005\u7684\u5171\u540c\u52aa\u529b\u4e0b\u4e0d\u65b7\u5b8c\u5584\u3002\u611f\u8b1d\u6bcf\u4e00\u4f4d\u6295\u5165\u6642\u9593\u8207\u7cbe\u529b\u7684\u64b0\u7a3f\u4eba\uff0c\u4ed6\u5011\u662f\uff08\u6309\u7167 GitHub \u81ea\u52d5\u751f\u6210\u7684\u9806\u5e8f\uff09\uff1akrahets\u3001coderonion\u3001Gonglja\u3001nuomi1\u3001Reanon\u3001justin-tse\u3001hpstory\u3001danielsss\u3001curtishd\u3001night-cruise\u3001S-N-O-R-L-A-X\u3001msk397\u3001gvenusleo\u3001khoaxuantu\u3001RiverTwilight\u3001rongyi\u3001gyt95\u3001zhuoqinyue\u3001K3v123\u3001Zuoxun\u3001mingXta\u3001hello-ikun\u3001FangYuan33\u3001GN-Yu\u3001yuelinxin\u3001longsizhuo\u3001Cathay-Chen\u3001guowei-gong\u3001xBLACKICEx\u3001IsChristina\u3001JoseHung\u3001qualifier1024\u3001QiLOL\u3001pengchzn\u3001Guanngxu\u3001L-Super\u3001WSL0809\u3001Slone123c\u3001lhxsm\u3001yuan0221\u3001what-is-me\u3001theNefelibatas\u3001longranger2\u3001cy-by-side\u3001xiongsp\u3001JeffersonHuang\u3001Transmigration-zhou\u3001magentaqin\u3001Wonderdch\u3001malone6\u3001xiaomiusa87\u3001gaofer\u3001bluebean-cloud\u3001a16su\u3001Shyam-Chen\u3001nanlei\u3001hongyun-robot\u3001Phoenix0415\u3001MolDuM\u3001Nigh\u3001he-weilai\u3001junminhong\u3001mgisr\u3001iron-irax\u3001yd-j\u3001XiaChuerwu\u3001XC-Zero\u3001seven1240\u3001SamJin98\u3001wodray\u3001reeswell\u3001NI-SW\u3001Horbin-Magician\u3001Enlightenus\u3001xjr7670\u3001YangXuanyi\u3001DullSword\u3001boloboloda\u3001iStig\u3001qq909244296\u3001jiaxianhua\u3001wenjianmin\u3001keshida\u3001kilikilikid\u3001lclc6\u3001lwbaptx\u3001liuxjerry\u3001lucaswangdev\u3001lyl625760\u3001hts0000\u3001gledfish\u3001fbigm\u3001echo1937\u3001szu17dmy\u3001dshlstarr\u3001Yucao-cy\u3001coderlef\u3001czruby\u3001bongbongbakudan\u3001beintentional\u3001ZongYangL\u3001ZhongYuuu\u3001luluxia\u3001xb534\u3001bitsmi\u3001ElaBosak233\u3001baagod\u3001zhouLion\u3001yishangzhang\u3001yi427\u3001yabo083\u3001weibk\u3001wangwang105\u3001th1nk3r-ing\u3001tao363\u30014yDX3906\u3001syd168\u3001steventimes\u3001sslmj2020\u3001smilelsb\u3001siqyka\u3001selear\u3001sdshaoda\u3001Xi-Row\u3001popozhu\u3001nuquist19\u3001noobcodemaker\u3001XiaoK29\u3001chadyi\u3001ZhongGuanbin\u3001shanghai-Jerry\u3001JackYang-hellobobo\u3001Javesun99\u3001lipusheng\u3001BlindTerran\u3001ShiMaRing\u3001FreddieLi\u3001FloranceYeh\u3001iFleey\u3001fanchenggang\u3001gltianwen\u3001goerll\u3001Dr-XYZ\u3001nedchu\u3001curly210102\u3001CuB3y0nd\u3001KraHsu\u3001CarrotDLaw\u3001youshaoXG\u3001bubble9um\u3001fanenr\u3001eagleanurag\u3001LifeGoesOnionOnionOnion\u300152coder\u3001foursevenlove\u3001KorsChen\u3001hezhizhen\u3001linzeyan\u3001ZJKung\u3001GaochaoZhu\u3001hopkings2008\u3001yang-le\u3001Evilrabbit520\u3001Turing-1024-Lee\u3001thomasq0\u3001Suremotoo\u3001Allen-Scai\u3001Risuntsy\u3001Richard-Zhang1019\u3001qingpeng9802\u3001primexiao\u3001nidhoggfgg\u30011ch0\u3001MwumLi\u3001martinx\u3001ZnYang2018\u3001hugtyftg\u3001logan-qiu\u3001psychelzh\u3001Keynman\u3001KeiichiKasai\u30010130w \u3002</p> <p>\u672c\u66f8\u7684\u7a0b\u5f0f\u78bc\u5be9\u95b1\u5de5\u4f5c\u7531 coderonion\u3001curtishd\u3001Gonglja\u3001gvenusleo\u3001hpstory\u3001justin-tse\u3001khoaxuantu\u3001krahets\u3001night-cruise\u3001nuomi1\u3001Reanon \u548c rongyi \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u9806\u5e8f\u6392\u5217\uff09\u3002\u611f\u8b1d\u4ed6\u5011\u4ed8\u51fa\u7684\u6642\u9593\u8207\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u5011\u78ba\u4fdd\u4e86\u5404\u8a9e\u8a00\u7a0b\u5f0f\u78bc\u7684\u898f\u7bc4\u8207\u7d71\u4e00\u3002</p> <p>\u5728\u672c\u66f8\u7684\u5275\u4f5c\u904e\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8a31\u591a\u4eba\u7684\u5e6b\u52a9\u3002</p> <ul> <li>\u611f\u8b1d\u6211\u5728\u516c\u53f8\u7684\u5c0e\u5e2b\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u66a2\u8ac7\u4e2d\u4f60\u9f13\u52f5\u6211\u201c\u5feb\u884c\u52d5\u8d77\u4f86\u201d\uff0c\u5805\u5b9a\u4e86\u6211\u5beb\u9019\u672c\u66f8\u7684\u6c7a\u5fc3\uff1b</li> <li>\u611f\u8b1d\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u70ba\u672c\u66f8\u7684\u9996\u4f4d\u8b80\u8005\uff0c\u5f9e\u6f14\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8a31\u591a\u5bf6\u8cb4\u5efa\u8b70\uff0c\u4f7f\u5f97\u672c\u66f8\u66f4\u9069\u5408\u65b0\u624b\u95b1\u8b80\uff1b</li> <li>\u611f\u8b1d\u9a30\u5bf6\u3001\u7426\u5bf6\u3001\u98db\u5bf6\u70ba\u672c\u66f8\u8d77\u4e86\u4e00\u500b\u5bcc\u6709\u5275\u610f\u7684\u540d\u5b57\uff0c\u559a\u8d77\u5927\u5bb6\u5beb\u4e0b\u7b2c\u4e00\u884c\u7a0b\u5f0f\u78bc\u201cHello World!\u201d\u7684\u7f8e\u597d\u56de\u61b6\uff1b</li> <li>\u611f\u8b1d\u6821\u9293\u5728\u667a\u6167\u8ca1\u7522\u6b0a\u65b9\u9762\u63d0\u4f9b\u7684\u5c08\u696d\u5e6b\u52a9\uff0c\u9019\u5c0d\u672c\u958b\u6e90\u66f8\u7684\u5b8c\u5584\u8d77\u5230\u4e86\u91cd\u8981\u4f5c\u7528\uff1b</li> <li>\u611f\u8b1d\u8607\u6f7c\u70ba\u672c\u66f8\u8a2d\u8a08\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c logo \uff0c\u4e26\u5728\u6211\u7684\u5f37\u8feb\u75c7\u7684\u9a45\u4f7f\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\uff1b</li> <li>\u611f\u8b1d @squidfunk \u63d0\u4f9b\u7684\u6392\u7248\u5efa\u8b70\uff0c\u4ee5\u53ca\u4ed6\u958b\u767c\u7684\u958b\u6e90\u6587\u4ef6\u4e3b\u984c Material-for-MkDocs \u3002</li> </ul> <p>\u5728\u5beb\u4f5c\u904e\u7a0b\u4e2d\uff0c\u6211\u95b1\u8b80\u4e86\u8a31\u591a\u95dc\u65bc\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u9019\u4e9b\u4f5c\u54c1\u70ba\u672c\u66f8\u63d0\u4f9b\u4e86\u512a\u79c0\u7684\u7bc4\u672c\uff0c\u78ba\u4fdd\u4e86\u672c\u66f8\u5167\u5bb9\u7684\u6e96\u78ba\u6027\u8207\u54c1\u8cea\u3002\u5728\u6b64\u611f\u8b1d\u6240\u6709\u8001\u5e2b\u548c\u524d\u8f29\u7684\u5091\u51fa\u8ca2\u737b\uff01</p> <p>\u672c\u66f8\u5021\u5c0e\u624b\u8166\u4e26\u7528\u7684\u5b78\u7fd2\u65b9\u5f0f\uff0c\u5728\u9019\u4e00\u9ede\u4e0a\u6211\u6df1\u53d7\u300a\u52d5\u624b\u5b78\u6df1\u5ea6\u5b78\u7fd2\u300b\u7684\u555f\u767c\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8b80\u8005\u5f37\u70c8\u63a8\u85a6\u9019\u672c\u512a\u79c0\u7684\u8457\u4f5c\u3002</p> <p>\u8877\u5fc3\u611f\u8b1d\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u5011\u4e00\u76f4\u4ee5\u4f86\u7684\u652f\u6301\u8207\u9f13\u52f5\uff0c\u8b93\u6211\u6709\u6a5f\u6703\u505a\u9019\u4ef6\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002</p>"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u66f8","text":"<p>Tip</p> <p>\u70ba\u4e86\u7372\u5f97\u6700\u4f73\u7684\u95b1\u8b80\u9ad4\u9a57\uff0c\u5efa\u8b70\u4f60\u901a\u8b80\u672c\u7bc0\u5167\u5bb9\u3002</p>"},{"location":"chapter_preface/suggestions/#021","title":"0.2.1 \u00a0 \u884c\u6587\u98a8\u683c\u7d04\u5b9a","text":"<ul> <li>\u6a19\u984c\u5f8c\u6a19\u8a3b <code>*</code> \u7684\u662f\u9078\u8b80\u7ae0\u7bc0\uff0c\u5167\u5bb9\u76f8\u5c0d\u56f0\u96e3\u3002\u5982\u679c\u4f60\u7684\u6642\u9593\u6709\u9650\uff0c\u53ef\u4ee5\u5148\u8df3\u904e\u3002</li> <li>\u5c08\u696d\u8853\u8a9e\u6703\u4f7f\u7528\u9ed1\u9ad4\uff08\u7d19\u8cea\u7248\u548c PDF \u7248\uff09\u6216\u65b0\u589e\u4e0b\u5283\u7dda\uff08\u7db2\u9801\u7248\uff09\uff0c\u4f8b\u5982\u9663\u5217\uff08array\uff09\u3002\u5efa\u8b70\u8a18\u4f4f\u5b83\u5011\uff0c\u4ee5\u4fbf\u95b1\u8b80\u6587\u737b\u3002</li> <li>\u91cd\u9ede\u5167\u5bb9\u548c\u7e3d\u7d50\u6027\u8a9e\u53e5\u6703 \u52a0\u7c97\uff0c\u9019\u985e\u6587\u5b57\u503c\u5f97\u7279\u5225\u95dc\u6ce8\u3002</li> <li>\u6709\u7279\u6307\u542b\u7fa9\u7684\u8a5e\u53e5\u6703\u4f7f\u7528\u201c\u5f15\u865f\u201d\u6a19\u8a3b\uff0c\u4ee5\u907f\u514d\u6b67\u7fa9\u3002</li> <li>\u7576\u6d89\u53ca\u7a0b\u5f0f\u8a9e\u8a00\u4e4b\u9593\u4e0d\u4e00\u81f4\u7684\u540d\u8a5e\u6642\uff0c\u672c\u66f8\u5747\u4ee5 Python \u70ba\u6e96\uff0c\u4f8b\u5982\u4f7f\u7528 <code>None</code> \u4f86\u8868\u793a\u201c\u7a7a\u201d\u3002</li> <li>\u672c\u66f8\u90e8\u5206\u653e\u68c4\u4e86\u7a0b\u5f0f\u8a9e\u8a00\u7684\u8a3b\u91cb\u898f\u7bc4\uff0c\u4ee5\u63db\u53d6\u66f4\u52a0\u7dca\u6e4a\u7684\u5167\u5bb9\u6392\u7248\u3002\u8a3b\u91cb\u4e3b\u8981\u5206\u70ba\u4e09\u7a2e\u985e\u578b\uff1a\u6a19\u984c\u8a3b\u91cb\u3001\u5167\u5bb9\u8a3b\u91cb\u3001\u591a\u884c\u8a3b\u91cb\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>\"\"\"\u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49\"\"\"\n\n# \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n\"\"\"\n\u591a\u884c\n\u8a3b\u91cb\n\"\"\"\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>/* \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 */\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n/**\n * \u591a\u884c\n * \u8a3b\u91cb\n */\n</code></pre> <pre><code>### \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49 ###\n\n# \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n# \u591a\u884c\n# \u8a3b\u91cb\n</code></pre> <pre><code>// \u6a19\u984c\u8a3b\u91cb\uff0c\u7528\u65bc\u6a19\u8a3b\u51fd\u5f0f\u3001\u985e\u5225\u3001\u6e2c\u8a66\u6a23\u4f8b\u7b49\n\n// \u5167\u5bb9\u8a3b\u91cb\uff0c\u7528\u65bc\u8a73\u89e3\u7a0b\u5f0f\u78bc\n\n// \u591a\u884c\n// \u8a3b\u91cb\n</code></pre>"},{"location":"chapter_preface/suggestions/#022","title":"0.2.2 \u00a0 \u5728\u52d5\u756b\u5716\u89e3\u4e2d\u9ad8\u6548\u5b78\u7fd2","text":"<p>\u76f8\u8f03\u65bc\u6587\u5b57\uff0c\u5f71\u7247\u548c\u5716\u7247\u5177\u6709\u66f4\u9ad8\u7684\u8cc7\u8a0a\u5bc6\u5ea6\u548c\u7d50\u69cb\u5316\u7a0b\u5ea6\uff0c\u66f4\u6613\u65bc\u7406\u89e3\u3002\u5728\u672c\u66f8\u4e2d\uff0c\u91cd\u9ede\u548c\u96e3\u9ede\u77e5\u8b58\u5c07\u4e3b\u8981\u900f\u904e\u52d5\u756b\u4ee5\u5716\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5247\u4f5c\u70ba\u89e3\u91cb\u8207\u88dc\u5145\u3002</p> <p>\u5982\u679c\u4f60\u5728\u95b1\u8b80\u672c\u66f8\u6642\uff0c\u767c\u73fe\u67d0\u6bb5\u5167\u5bb9\u63d0\u4f9b\u77ad\u5982\u5716 0-2 \u6240\u793a\u7684\u52d5\u756b\u5716\u89e3\uff0c\u8acb\u4ee5\u5716\u70ba\u4e3b\u3001\u4ee5\u6587\u5b57\u70ba\u8f14\uff0c\u7d9c\u5408\u5169\u8005\u4f86\u7406\u89e3\u5167\u5bb9\u3002</p> <p></p> <p> \u5716 0-2 \u00a0 \u52d5\u756b\u5716\u89e3\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#023","title":"0.2.3 \u00a0 \u5728\u7a0b\u5f0f\u78bc\u5be6\u8e10\u4e2d\u52a0\u6df1\u7406\u89e3","text":"<p>\u672c\u66f8\u7684\u914d\u5957\u7a0b\u5f0f\u78bc\u8a17\u7ba1\u5728 GitHub \u5009\u5eab\u3002\u5982\u5716 0-3 \u6240\u793a\uff0c\u6e90\u7a0b\u5f0f\u78bc\u9644\u6709\u6e2c\u8a66\u6a23\u4f8b\uff0c\u53ef\u4e00\u9375\u57f7\u884c\u3002</p> <p>\u5982\u679c\u6642\u9593\u5141\u8a31\uff0c\u5efa\u8b70\u4f60\u53c3\u7167\u7a0b\u5f0f\u78bc\u81ea\u884c\u6572\u4e00\u904d\u3002\u5982\u679c\u5b78\u7fd2\u6642\u9593\u6709\u9650\uff0c\u8acb\u81f3\u5c11\u901a\u8b80\u4e26\u57f7\u884c\u6240\u6709\u7a0b\u5f0f\u78bc\u3002</p> <p>\u8207\u95b1\u8b80\u7a0b\u5f0f\u78bc\u76f8\u6bd4\uff0c\u7de8\u5beb\u7a0b\u5f0f\u78bc\u7684\u904e\u7a0b\u5f80\u5f80\u80fd\u5e36\u4f86\u66f4\u591a\u6536\u7a6b\u3002\u52d5\u624b\u5b78\uff0c\u624d\u662f\u771f\u7684\u5b78\u3002</p> <p></p> <p> \u5716 0-3 \u00a0 \u57f7\u884c\u7a0b\u5f0f\u78bc\u793a\u4f8b </p> <p>\u57f7\u884c\u7a0b\u5f0f\u78bc\u7684\u524d\u7f6e\u5de5\u4f5c\u4e3b\u8981\u5206\u70ba\u4e09\u6b65\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u5b89\u88dd\u672c\u5730\u7a0b\u5f0f\u8a2d\u8a08\u74b0\u5883\u3002\u8acb\u53c3\u7167\u9644\u9304\u6240\u793a\u7684\u6559\u7a0b\u9032\u884c\u5b89\u88dd\uff0c\u5982\u679c\u5df2\u5b89\u88dd\uff0c\u5247\u53ef\u8df3\u904e\u6b64\u6b65\u9a5f\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u514b\u9686\u6216\u4e0b\u8f09\u7a0b\u5f0f\u78bc\u5009\u5eab\u3002\u524d\u5f80 GitHub \u5009\u5eab\u3002\u5982\u679c\u5df2\u7d93\u5b89\u88dd Git \uff0c\u53ef\u4ee5\u900f\u904e\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u5009\u5eab\uff1a</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>\u7576\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u5728\u5716 0-4 \u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u9ede\u9078\u201cDownload ZIP\u201d\u6309\u9215\u76f4\u63a5\u4e0b\u8f09\u7a0b\u5f0f\u78bc\u58d3\u7e2e\u5305\uff0c\u7136\u5f8c\u5728\u672c\u5730\u89e3\u58d3\u5373\u53ef\u3002</p> <p></p> <p> \u5716 0-4 \u00a0 \u514b\u9686\u5009\u5eab\u8207\u4e0b\u8f09\u7a0b\u5f0f\u78bc </p> <p>\u7b2c\u4e09\u6b65\uff1a\u57f7\u884c\u6e90\u7a0b\u5f0f\u78bc\u3002\u5982\u5716 0-5 \u6240\u793a\uff0c\u5c0d\u65bc\u9802\u90e8\u6a19\u6709\u6a94\u6848\u540d\u7a31\u7684\u7a0b\u5f0f\u78bc\u584a\uff0c\u6211\u5011\u53ef\u4ee5\u5728\u5009\u5eab\u7684 <code>codes</code> \u6a94\u6848\u593e\u5167\u627e\u5230\u5c0d\u61c9\u7684\u6e90\u7a0b\u5f0f\u78bc\u6a94\u6848\u3002\u6e90\u7a0b\u5f0f\u78bc\u6a94\u6848\u53ef\u4e00\u9375\u57f7\u884c\uff0c\u5c07\u5e6b\u52a9\u4f60\u7bc0\u7701\u4e0d\u5fc5\u8981\u7684\u9664\u932f\u6642\u9593\uff0c\u8b93\u4f60\u80fd\u5920\u5c08\u6ce8\u65bc\u5b78\u7fd2\u5167\u5bb9\u3002</p> <p></p> <p> \u5716 0-5 \u00a0 \u7a0b\u5f0f\u78bc\u584a\u8207\u5c0d\u61c9\u7684\u6e90\u7a0b\u5f0f\u78bc\u6a94\u6848 </p> <p>\u9664\u4e86\u672c\u5730\u57f7\u884c\u7a0b\u5f0f\u78bc\uff0c\u7db2\u9801\u7248\u9084\u652f\u6301 Python \u7a0b\u5f0f\u78bc\u7684\u8996\u89ba\u5316\u57f7\u884c\uff08\u57fa\u65bc pythontutor \u5be6\u73fe\uff09\u3002\u5982\u5716 0-6 \u6240\u793a\uff0c\u4f60\u53ef\u4ee5\u9ede\u9078\u7a0b\u5f0f\u78bc\u584a\u4e0b\u65b9\u7684\u201c\u8996\u89ba\u5316\u57f7\u884c\u201d\u4f86\u5c55\u958b\u6aa2\u8996\uff0c\u89c0\u5bdf\u6f14\u7b97\u6cd5\u7a0b\u5f0f\u78bc\u7684\u57f7\u884c\u904e\u7a0b\uff1b\u4e5f\u53ef\u4ee5\u9ede\u9078\u201c\u5168\u5c4f\u89c0\u770b\u201d\uff0c\u4ee5\u7372\u5f97\u66f4\u597d\u7684\u95b1\u89bd\u9ad4\u9a57\u3002</p> <p></p> <p> \u5716 0-6 \u00a0 Python \u7a0b\u5f0f\u78bc\u7684\u8996\u89ba\u5316\u57f7\u884c </p>"},{"location":"chapter_preface/suggestions/#024","title":"0.2.4 \u00a0 \u5728\u63d0\u554f\u8a0e\u8ad6\u4e2d\u5171\u540c\u6210\u9577","text":"<p>\u5728\u95b1\u8b80\u672c\u66f8\u6642\uff0c\u8acb\u4e0d\u8981\u8f15\u6613\u8df3\u904e\u90a3\u4e9b\u6c92\u5b78\u660e\u767d\u7684\u77e5\u8b58\u9ede\u3002\u6b61\u8fce\u5728\u8a55\u8ad6\u5340\u63d0\u51fa\u4f60\u7684\u554f\u984c\uff0c\u6211\u548c\u5c0f\u5925\u4f34\u5011\u5c07\u7aed\u8aa0\u70ba\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u6cc1\u4e0b\u53ef\u5728\u5169\u5929\u5167\u56de\u8986\u3002</p> <p>\u5982\u5716 0-7 \u6240\u793a\uff0c\u7db2\u9801\u7248\u6bcf\u500b\u7ae0\u7bc0\u7684\u5e95\u90e8\u90fd\u914d\u6709\u8a55\u8ad6\u5340\u3002\u5e0c\u671b\u4f60\u80fd\u591a\u95dc\u6ce8\u8a55\u8ad6\u5340\u7684\u5167\u5bb9\u3002\u4e00\u65b9\u9762\uff0c\u4f60\u53ef\u4ee5\u77ad\u89e3\u5927\u5bb6\u9047\u5230\u7684\u554f\u984c\uff0c\u5f9e\u800c\u67e5\u6f0f\u88dc\u7f3a\uff0c\u6fc0\u767c\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u671f\u5f85\u4f60\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u5925\u4f34\u7684\u554f\u984c\uff0c\u5206\u4eab\u4f60\u7684\u898b\u89e3\uff0c\u5e6b\u52a9\u4ed6\u4eba\u9032\u6b65\u3002</p> <p></p> <p> \u5716 0-7 \u00a0 \u8a55\u8ad6\u5340\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#025","title":"0.2.5 \u00a0 \u6f14\u7b97\u6cd5\u5b78\u7fd2\u8def\u7dda","text":"<p>\u5f9e\u7e3d\u9ad4\u4e0a\u770b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5b78\u7fd2\u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u7684\u904e\u7a0b\u5283\u5206\u70ba\u4e09\u500b\u968e\u6bb5\u3002</p> <ol> <li>\u968e\u6bb5\u4e00\uff1a\u6f14\u7b97\u6cd5\u5165\u9580\u3002\u6211\u5011\u9700\u8981\u719f\u6089\u5404\u7a2e\u8cc7\u6599\u7d50\u69cb\u7684\u7279\u9ede\u548c\u7528\u6cd5\uff0c\u5b78\u7fd2\u4e0d\u540c\u6f14\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u7684\u5167\u5bb9\u3002</li> <li>\u968e\u6bb5\u4e8c\uff1a\u5237\u6f14\u7b97\u6cd5\u984c\u3002\u5efa\u8b70\u5f9e\u71b1\u9580\u984c\u76ee\u958b\u5237\uff0c\u5148\u7a4d\u7d2f\u81f3\u5c11 100 \u9053\u984c\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u6f14\u7b97\u6cd5\u554f\u984c\u3002\u521d\u6b21\u5237\u984c\u6642\uff0c\u201c\u77e5\u8b58\u907a\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u500b\u6311\u6230\uff0c\u4f46\u8acb\u653e\u5fc3\uff0c\u9019\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u5011\u53ef\u4ee5\u6309\u7167\u201c\u827e\u8cd3\u6d69\u65af\u907a\u5fd8\u66f2\u7dda\u201d\u4f86\u8907\u7fd2\u984c\u76ee\uff0c\u901a\u5e38\u5728\u9032\u884c 3\uff5e5 \u8f2a\u7684\u91cd\u8907\u5f8c\uff0c\u5c31\u80fd\u5c07\u5176\u7262\u8a18\u5728\u5fc3\u3002\u63a8\u85a6\u7684\u984c\u55ae\u548c\u5237\u984c\u8a08\u5283\u8acb\u898b\u6b64 GitHub \u5009\u5eab\u3002</li> <li>\u968e\u6bb5\u4e09\uff1a\u642d\u5efa\u77e5\u8b58\u9ad4\u7cfb\u3002\u5728\u5b78\u7fd2\u65b9\u9762\uff0c\u6211\u5011\u53ef\u4ee5\u95b1\u8b80\u6f14\u7b97\u6cd5\u5c08\u6b04\u6587\u7ae0\u3001\u89e3\u984c\u6846\u67b6\u548c\u6f14\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65b7\u8c50\u5bcc\u77e5\u8b58\u9ad4\u7cfb\u3002\u5728\u5237\u984c\u65b9\u9762\uff0c\u53ef\u4ee5\u5617\u8a66\u63a1\u7528\u9032\u968e\u5237\u984c\u7b56\u7565\uff0c\u5982\u6309\u5c08\u984c\u5206\u985e\u3001\u4e00\u984c\u591a\u89e3\u3001\u4e00\u89e3\u591a\u984c\u7b49\uff0c\u76f8\u95dc\u7684\u5237\u984c\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u500b\u793e\u7fa4\u627e\u5230\u3002</li> </ol> <p>\u5982\u5716 0-8 \u6240\u793a\uff0c\u672c\u66f8\u5167\u5bb9\u4e3b\u8981\u6db5\u84cb\u201c\u968e\u6bb5\u4e00\u201d\uff0c\u65e8\u5728\u5e6b\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u958b\u968e\u6bb5\u4e8c\u548c\u968e\u6bb5\u4e09\u7684\u5b78\u7fd2\u3002</p> <p></p> <p> \u5716 0-8 \u00a0 \u6f14\u7b97\u6cd5\u5b78\u7fd2\u8def\u7dda </p>"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u672c\u66f8\u7684\u4e3b\u8981\u53d7\u773e\u662f\u6f14\u7b97\u6cd5\u521d\u5b78\u8005\u3002\u5982\u679c\u4f60\u5df2\u6709\u4e00\u5b9a\u57fa\u790e\uff0c\u672c\u66f8\u80fd\u5e6b\u52a9\u4f60\u7cfb\u7d71\u56de\u9867\u6f14\u7b97\u6cd5\u77e5\u8b58\uff0c\u66f8\u4e2d\u6e90\u7a0b\u5f0f\u78bc\u4e5f\u53ef\u4f5c\u70ba\u201c\u5237\u984c\u5de5\u5177\u5eab\u201d\u4f7f\u7528\u3002</li> <li>\u66f8\u4e2d\u5167\u5bb9\u4e3b\u8981\u5305\u62ec\u8907\u96dc\u5ea6\u5206\u6790\u3001\u8cc7\u6599\u7d50\u69cb\u548c\u6f14\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u84cb\u4e86\u8a72\u9818\u57df\u7684\u5927\u90e8\u5206\u4e3b\u984c\u3002</li> <li>\u5c0d\u65bc\u6f14\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b78\u968e\u6bb5\u95b1\u8b80\u4e00\u672c\u5165\u9580\u66f8\u81f3\u95dc\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8a31\u591a\u5f4e\u8def\u3002</li> <li>\u66f8\u4e2d\u7684\u52d5\u756b\u5716\u89e3\u901a\u5e38\u7528\u65bc\u4ecb\u7d39\u91cd\u9ede\u548c\u96e3\u9ede\u77e5\u8b58\u3002\u95b1\u8b80\u672c\u66f8\u6642\uff0c\u61c9\u7d66\u4e88\u9019\u4e9b\u5167\u5bb9\u66f4\u591a\u95dc\u6ce8\u3002</li> <li>\u5be6\u8e10\u4e43\u5b78\u7fd2\u7a0b\u5f0f\u8a2d\u8a08\u4e4b\u6700\u4f73\u9014\u5f91\u3002\u5f37\u70c8\u5efa\u8b70\u57f7\u884c\u6e90\u7a0b\u5f0f\u78bc\u4e26\u89aa\u81ea\u6572\u7a0b\u5f0f\u78bc\u3002</li> <li>\u672c\u66f8\u7db2\u9801\u7248\u7684\u6bcf\u500b\u7ae0\u7bc0\u90fd\u8a2d\u6709\u8a55\u8ad6\u5340\uff0c\u6b61\u8fce\u96a8\u6642\u5206\u4eab\u4f60\u7684\u7591\u60d1\u8207\u898b\u89e3\u3002</li> </ul>"},{"location":"chapter_reference/","title":"\u53c3\u8003\u6587\u737b","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p> <p>[4] \u56b4\u851a\u654f. \u8cc7\u6599\u7d50\u69cb\uff08C \u8a9e\u8a00\u7248\uff09.</p> <p>[5] \u9127\u4fca\u8f1d. \u8cc7\u6599\u7d50\u69cb\uff08C++ \u8a9e\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.</p> <p>[6] \u99ac\u514b \u827e\u502b \u7dad\u65af\u8457\uff0c\u9673\u8d8a\u8b6f. \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u5206\u6790\uff1aJava\u8a9e\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.</p> <p>[7] \u7a0b\u5091. \u5927\u8a71\u8cc7\u6599\u7d50\u69cb.</p> <p>[8] \u738b\u722d. \u8cc7\u6599\u7d50\u69cb\u8207\u6f14\u7b97\u6cd5\u4e4b\u7f8e.</p> <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p> <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>"},{"location":"chapter_searching/","title":"\u7b2c 10 \u7ae0 \u00a0 \u641c\u5c0b","text":"<p>Abstract</p> <p>\u641c\u5c0b\u662f\u4e00\u5834\u672a\u77e5\u7684\u5192\u96aa\uff0c\u6211\u5011\u6216\u8a31\u9700\u8981\u8d70\u904d\u795e\u79d8\u7a7a\u9593\u7684\u6bcf\u500b\u89d2\u843d\uff0c\u53c8\u6216\u8a31\u53ef\u4ee5\u5feb\u901f\u9396\u5b9a\u76ee\u6a19\u3002</p> <p>\u5728\u9019\u5834\u5c0b\u8993\u4e4b\u65c5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u63a2\u7d22\u90fd\u53ef\u80fd\u5f97\u5230\u4e00\u500b\u672a\u66fe\u6599\u60f3\u7684\u7b54\u6848\u3002</p>"},{"location":"chapter_searching/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>10.1 \u00a0 \u4e8c\u5206\u641c\u5c0b</li> <li>10.2 \u00a0 \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede</li> <li>10.3 \u00a0 \u4e8c\u5206\u641c\u5c0b\u908a\u754c</li> <li>10.4 \u00a0 \u96dc\u6e4a\u6700\u4f73\u5316\u7b56\u7565</li> <li>10.5 \u00a0 \u91cd\u8b58\u641c\u5c0b\u6f14\u7b97\u6cd5</li> <li>10.6 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_searching/binary_search/","title":"10.1 \u00a0 \u4e8c\u5206\u641c\u5c0b","text":"<p>\u4e8c\u5206\u641c\u5c0b\uff08binary search\uff09\u662f\u4e00\u7a2e\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u9ad8\u6548\u641c\u5c0b\u6f14\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u8cc7\u6599\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f2a\u7e2e\u5c0f\u4e00\u534a\u641c\u5c0b\u7bc4\u570d\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6a19\u5143\u7d20\u6216\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u70ba\u6b62\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217 <code>nums</code> \uff0c\u5143\u7d20\u6309\u5f9e\u5c0f\u5230\u5927\u7684\u9806\u5e8f\u6392\u5217\u4e14\u4e0d\u91cd\u8907\u3002\u8acb\u67e5\u8a62\u4e26\u8fd4\u56de\u5143\u7d20 <code>target</code> \u5728\u8a72\u9663\u5217\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u9663\u5217\u4e0d\u5305\u542b\u8a72\u5143\u7d20\uff0c\u5247\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u5716 10-1 \u6240\u793a\u3002</p> <p></p> <p> \u5716 10-1 \u00a0 \u4e8c\u5206\u641c\u5c0b\u793a\u4f8b\u8cc7\u6599 </p> <p>\u5982\u5716 10-2 \u6240\u793a\uff0c\u6211\u5011\u5148\u521d\u59cb\u5316\u6307\u6a19 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u5c0b\u5340\u9593 \\([0, n - 1]\\) \u3002\u8acb\u6ce8\u610f\uff0c\u4e2d\u62ec\u865f\u8868\u793a\u9589\u5340\u9593\uff0c\u5176\u5305\u542b\u908a\u754c\u503c\u672c\u8eab\u3002</p> <p>\u63a5\u4e0b\u4f86\uff0c\u8ff4\u5708\u57f7\u884c\u4ee5\u4e0b\u5169\u6b65\u3002</p> <ol> <li>\u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\: \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002</li> <li>\u5224\u65b7 <code>nums[m]</code> \u548c <code>target</code> \u7684\u5927\u5c0f\u95dc\u4fc2\uff0c\u5206\u70ba\u4ee5\u4e0b\u4e09\u7a2e\u60c5\u6cc1\u3002<ol> <li>\u7576 <code>nums[m] &lt; target</code> \u6642\uff0c\u8aaa\u660e <code>target</code> \u5728\u5340\u9593 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u57f7\u884c \\(i = m + 1\\) \u3002</li> <li>\u7576 <code>nums[m] &gt; target</code> \u6642\uff0c\u8aaa\u660e <code>target</code> \u5728\u5340\u9593 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u57f7\u884c \\(j = m - 1\\) \u3002</li> <li>\u7576 <code>nums[m] = target</code> \u6642\uff0c\u8aaa\u660e\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \u3002</li> </ol> </li> </ol> <p>\u82e5\u9663\u5217\u4e0d\u5305\u542b\u76ee\u6a19\u5143\u7d20\uff0c\u641c\u5c0b\u5340\u9593\u6700\u7d42\u6703\u7e2e\u5c0f\u70ba\u7a7a\u3002\u6b64\u6642\u8fd4\u56de \\(-1\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 10-2 \u00a0 \u4e8c\u5206\u641c\u5c0b\u6d41\u7a0b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u65bc \\(i\\) \u548c \\(j\\) \u90fd\u662f <code>int</code> \u578b\u5225\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u6703\u8d85\u51fa <code>int</code> \u578b\u5225\u7684\u53d6\u503c\u7bc4\u570d\u3002\u70ba\u4e86\u907f\u514d\u5927\u6578\u8d8a\u754c\uff0c\u6211\u5011\u901a\u5e38\u63a1\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u4f86\u8a08\u7b97\u4e2d\u9ede\u3002</p> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09\"\"\"\n    # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j = 0, len(nums) - 1\n    # \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while i &lt;= j:\n        # \u7406\u8ad6\u4e0a Python \u7684\u6578\u5b57\u53ef\u4ee5\u7121\u9650\u5927\uff08\u53d6\u6c7a\u65bc\u8a18\u61b6\u9ad4\u5927\u5c0f\uff09\uff0c\u7121\u9808\u8003\u616e\u5927\u6578\u8d8a\u754c\u554f\u984c\n        m = (i + j) // 2  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.size() - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint binarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.length - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint BinarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.Length - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else                       // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfunc binarySearch(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j := 0, len(nums)-1\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    for i &lt;= j {\n        m := i + (j-i)/2      // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfunction binarySearch(nums, target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else return m; // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n  int i = 0, j = nums.length - 1;\n  // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n      j = m - 1;\n    } else {\n      // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nint binarySearch(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = len - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = 0\n    var j = nums.size - 1\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        else  // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09 ###\ndef binary_search(nums, target)\n  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n  i, j = 0, nums.length - 1\n\n  # \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n  while i &lt;= j\n    # \u7406\u8ad6\u4e0a Ruby \u7684\u6578\u5b57\u53ef\u4ee5\u7121\u9650\u5927\uff08\u53d6\u6c7a\u65bc\u8a18\u61b6\u9ad4\u5927\u5c0f\uff09\uff0c\u7121\u9808\u8003\u616e\u5927\u6578\u8d8a\u754c\u554f\u984c\n    m = (i + j) / 2   # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n\n    if nums[m] &lt; target\n      i = m + 1 # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    elsif nums[m] &gt; target\n      j = m - 1 # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    else\n      return m  # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    end\n  end\n\n  -1  # \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\nend\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u641c\u5c0b\uff08\u96d9\u9589\u5340\u9593\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1] \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i: usize = 0;\n    var j: usize = nums.items.len - 1;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i &gt; j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {                                // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \uff1a\u5728\u4e8c\u5206\u8ff4\u5708\u4e2d\uff0c\u5340\u9593\u6bcf\u8f2a\u7e2e\u5c0f\u4e00\u534a\uff0c\u56e0\u6b64\u8ff4\u5708\u6b21\u6578\u70ba \\(\\log_2 n\\) \u3002</p> <p>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \uff1a\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7a7a\u9593\u3002</p>"},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1 \u00a0 \u5340\u9593\u8868\u793a\u65b9\u6cd5","text":"<p>\u9664\u4e86\u4e0a\u8ff0\u96d9\u9589\u5340\u9593\u5916\uff0c\u5e38\u898b\u7684\u5340\u9593\u8868\u793a\u9084\u6709\u201c\u5de6\u9589\u53f3\u958b\u201d\u5340\u9593\uff0c\u5b9a\u7fa9\u70ba \\([0, n)\\) \uff0c\u5373\u5de6\u908a\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u908a\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8a72\u8868\u793a\u4e0b\uff0c\u5340\u9593 \\([i, j)\\) \u5728 \\(i = j\\) \u6642\u70ba\u7a7a\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u57fa\u65bc\u8a72\u8868\u793a\u5be6\u73fe\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09\"\"\"\n    # \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j = 0, len(nums)\n    # \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while i &lt; j:\n        m = (i + j) // 2  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n        elif nums[m] &gt; target:\n            j = m  # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.size();\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.length;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint BinarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.Length;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        else                       // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j := 0, len(nums)\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    for i &lt; j {\n        m := i + (j-i)/2      // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = nums.startIndex\n    var j = nums.endIndex\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfunction binarySearchLCRO(nums, target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        else return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n  int i = 0, j = nums.length;\n  // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n      j = m;\n    } else {\n      // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = len;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 */\nfun binarySearchLCRO(nums: IntArray, target: Int): Int {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = 0\n    var j = nums.size\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt; j) {\n        val m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m\n        else  // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09 ###\ndef binary_search_lcro(nums, target)\n  # \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n  i, j = 0, nums.length\n\n  # \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n  while i &lt; j\n    # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n    elsif nums[m] &gt; target\n      j = m - 1 # \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n    else\n      return m  # \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    end\n  end\n\n  -1  # \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\nend\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u641c\u5c0b\uff08\u5de6\u9589\u53f3\u958b\u5340\u9593\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u5de6\u9589\u53f3\u958b\u5340\u9593 [0, n) \uff0c\u5373 i, j \u5206\u5225\u6307\u5411\u9663\u5217\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i: usize = 0;\n    var j: usize = nums.items.len;\n    // \u8ff4\u5708\uff0c\u7576\u641c\u5c0b\u5340\u9593\u70ba\u7a7a\u6642\u8df3\u51fa\uff08\u7576 i = j \u6642\u70ba\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u6cc1\u8aaa\u660e target \u5728\u5340\u9593 [i, m) \u4e2d\n            j = m;\n        } else {                                // \u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6a19\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u5982\u5716 10-3 \u6240\u793a\uff0c\u5728\u5169\u7a2e\u5340\u9593\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u8ff4\u5708\u689d\u4ef6\u548c\u7e2e\u5c0f\u5340\u9593\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002</p> <p>\u7531\u65bc\u201c\u96d9\u9589\u5340\u9593\u201d\u8868\u793a\u4e2d\u7684\u5de6\u53f3\u908a\u754c\u90fd\u88ab\u5b9a\u7fa9\u70ba\u9589\u5340\u9593\uff0c\u56e0\u6b64\u900f\u904e\u6307\u6a19 \\(i\\) \u548c\u6307\u6a19 \\(j\\) \u7e2e\u5c0f\u5340\u9593\u7684\u64cd\u4f5c\u4e5f\u662f\u5c0d\u7a31\u7684\u3002\u9019\u6a23\u66f4\u4e0d\u5bb9\u6613\u51fa\u932f\uff0c\u56e0\u6b64\u4e00\u822c\u5efa\u8b70\u63a1\u7528\u201c\u96d9\u9589\u5340\u9593\u201d\u7684\u5beb\u6cd5\u3002</p> <p></p> <p> \u5716 10-3 \u00a0 \u5169\u7a2e\u5340\u9593\u5b9a\u7fa9 </p>"},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2 \u00a0 \u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u4e8c\u5206\u641c\u5c0b\u5728\u6642\u9593\u548c\u7a7a\u9593\u65b9\u9762\u90fd\u6709\u8f03\u597d\u7684\u6548\u80fd\u3002</p> <ul> <li>\u4e8c\u5206\u641c\u5c0b\u7684\u6642\u9593\u6548\u7387\u9ad8\u3002\u5728\u5927\u8cc7\u6599\u91cf\u4e0b\uff0c\u5c0d\u6578\u968e\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5177\u6709\u986f\u8457\u512a\u52e2\u3002\u4f8b\u5982\uff0c\u7576\u8cc7\u6599\u5927\u5c0f \\(n = 2^{20}\\) \u6642\uff0c\u7dda\u6027\u67e5\u8a62\u9700\u8981 \\(2^{20} = 1048576\\) \u8f2a\u8ff4\u5708\uff0c\u800c\u4e8c\u5206\u641c\u5c0b\u50c5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f2a\u8ff4\u5708\u3002</li> <li>\u4e8c\u5206\u641c\u5c0b\u7121\u9808\u984d\u5916\u7a7a\u9593\u3002\u76f8\u8f03\u65bc\u9700\u8981\u85c9\u52a9\u984d\u5916\u7a7a\u9593\u7684\u641c\u5c0b\u6f14\u7b97\u6cd5\uff08\u4f8b\u5982\u96dc\u6e4a\u67e5\u8a62\uff09\uff0c\u4e8c\u5206\u641c\u5c0b\u66f4\u52a0\u7bc0\u7701\u7a7a\u9593\u3002</li> </ul> <p>\u7136\u800c\uff0c\u4e8c\u5206\u641c\u5c0b\u4e26\u975e\u9069\u7528\u65bc\u6240\u6709\u60c5\u6cc1\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e8c\u5206\u641c\u5c0b\u50c5\u9069\u7528\u65bc\u6709\u5e8f\u8cc7\u6599\u3002\u82e5\u8f38\u5165\u8cc7\u6599\u7121\u5e8f\uff0c\u70ba\u4e86\u4f7f\u7528\u4e8c\u5206\u641c\u5c0b\u800c\u5c08\u9580\u9032\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u511f\u5931\u3002\u56e0\u70ba\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u70ba \\(O(n \\log n)\\) \uff0c\u6bd4\u7dda\u6027\u67e5\u8a62\u548c\u4e8c\u5206\u641c\u5c0b\u90fd\u66f4\u9ad8\u3002\u5c0d\u65bc\u983b\u7e41\u63d2\u5165\u5143\u7d20\u7684\u5834\u666f\uff0c\u70ba\u4fdd\u6301\u9663\u5217\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c07\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8cb4\u7684\u3002</li> <li>\u4e8c\u5206\u641c\u5c0b\u50c5\u9069\u7528\u65bc\u9663\u5217\u3002\u4e8c\u5206\u641c\u5c0b\u9700\u8981\u8df3\u8e8d\u5f0f\uff08\u975e\u9023\u7e8c\u5730\uff09\u8a2a\u554f\u5143\u7d20\uff0c\u800c\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\u57f7\u884c\u8df3\u8e8d\u5f0f\u8a2a\u554f\u7684\u6548\u7387\u8f03\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9069\u5408\u61c9\u7528\u5728\u93c8\u7d50\u4e32\u5217\u6216\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u5c0f\u8cc7\u6599\u91cf\u4e0b\uff0c\u7dda\u6027\u67e5\u8a62\u6548\u80fd\u66f4\u4f73\u3002\u7dda\u4e0a\u6027\u67e5\u8a62\u4e2d\uff0c\u6bcf\u8f2a\u53ea\u9700 1 \u6b21\u5224\u65b7\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u641c\u5c0b\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65b7\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u6e1b\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u500b\u55ae\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u7576\u8cc7\u6599\u91cf \\(n\\) \u8f03\u5c0f\u6642\uff0c\u7dda\u6027\u67e5\u8a62\u53cd\u800c\u6bd4\u4e8c\u5206\u641c\u5c0b\u66f4\u5feb\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_edge/","title":"10.3 \u00a0 \u4e8c\u5206\u641c\u5c0b\u908a\u754c","text":""},{"location":"chapter_searching/binary_search_edge/#1031","title":"10.3.1 \u00a0 \u67e5\u8a62\u5de6\u908a\u754c","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u6709\u5e8f\u9663\u5217 <code>nums</code> \uff0c\u5176\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\u3002\u8acb\u8fd4\u56de\u9663\u5217\u4e2d\u6700\u5de6\u4e00\u500b\u5143\u7d20 <code>target</code> \u7684\u7d22\u5f15\u3002\u82e5\u9663\u5217\u4e2d\u4e0d\u5305\u542b\u8a72\u5143\u7d20\uff0c\u5247\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u56de\u61b6\u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\u7684\u65b9\u6cd5\uff0c\u641c\u5c0b\u5b8c\u6210\u5f8c \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u500b <code>target</code> \uff0c\u56e0\u6b64\u67e5\u8a62\u63d2\u5165\u9ede\u672c\u8cea\u4e0a\u662f\u5728\u67e5\u8a62\u6700\u5de6\u4e00\u500b <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u8003\u616e\u900f\u904e\u67e5\u8a62\u63d2\u5165\u9ede\u7684\u51fd\u5f0f\u5be6\u73fe\u67e5\u8a62\u5de6\u908a\u754c\u3002\u8acb\u6ce8\u610f\uff0c\u9663\u5217\u4e2d\u53ef\u80fd\u4e0d\u5305\u542b <code>target</code> \uff0c\u9019\u7a2e\u60c5\u6cc1\u53ef\u80fd\u5c0e\u81f4\u4ee5\u4e0b\u5169\u7a2e\u7d50\u679c\u3002</p> <ul> <li>\u63d2\u5165\u9ede\u7684\u7d22\u5f15 \\(i\\) \u8d8a\u754c\u3002</li> <li>\u5143\u7d20 <code>nums[i]</code> \u8207 <code>target</code> \u4e0d\u76f8\u7b49\u3002</li> </ul> <p>\u7576\u9047\u5230\u4ee5\u4e0a\u5169\u7a2e\u60c5\u6cc1\u6642\uff0c\u76f4\u63a5\u8fd4\u56de \\(-1\\) \u5373\u53ef\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target\"\"\"\n    # \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    i = binary_search_insertion(nums, target)\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    int i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    i := binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.endIndex || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfunction binarySearchLeftEdge(nums, target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n  int i = binarySearchInsertion(nums, target);\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    let i = binary_search_insertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    int i = binarySearchInsertion(nums, numSize, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target */\nfun binarySearchLeftEdge(nums: IntArray, target: Int): Int {\n    // \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n    val i = binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size || nums[i] != target) {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\u6700\u5de6\u4e00\u500b target ###\ndef binary_search_left_edge(nums, target)\n  # \u7b49\u50f9\u65bc\u67e5\u8a62 target \u7684\u63d2\u5165\u9ede\n  i = binary_search_insertion(nums, target)\n\n  # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  return -1 if i == nums.length || nums[i] != target\n\n  i # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nend\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchLeftEdge}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_edge/#1032","title":"10.3.2 \u00a0 \u67e5\u8a62\u53f3\u908a\u754c","text":"<p>\u90a3\u9ebc\u5982\u4f55\u67e5\u8a62\u6700\u53f3\u4e00\u500b <code>target</code> \u5462\uff1f\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\u662f\u4fee\u6539\u7a0b\u5f0f\u78bc\uff0c\u66ff\u63db\u5728 <code>nums[m] == target</code> \u60c5\u6cc1\u4e0b\u7684\u6307\u6a19\u6536\u7e2e\u64cd\u4f5c\u3002\u7a0b\u5f0f\u78bc\u5728\u6b64\u7701\u7565\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u81ea\u884c\u5be6\u73fe\u3002</p> <p>\u4e0b\u9762\u6211\u5011\u4ecb\u7d39\u5169\u7a2e\u66f4\u52a0\u53d6\u5de7\u7684\u65b9\u6cd5\u3002</p>"},{"location":"chapter_searching/binary_search_edge/#1","title":"1. \u00a0 \u8907\u7528\u67e5\u8a62\u5de6\u908a\u754c","text":"<p>\u5be6\u969b\u4e0a\uff0c\u6211\u5011\u53ef\u4ee5\u5229\u7528\u67e5\u8a62\u6700\u5de6\u5143\u7d20\u7684\u51fd\u5f0f\u4f86\u67e5\u8a62\u6700\u53f3\u5143\u7d20\uff0c\u5177\u9ad4\u65b9\u6cd5\u70ba\uff1a\u5c07\u67e5\u8a62\u6700\u53f3\u4e00\u500b <code>target</code> \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b <code>target + 1</code>\u3002</p> <p>\u5982\u5716 10-7 \u6240\u793a\uff0c\u67e5\u8a62\u5b8c\u6210\u5f8c\uff0c\u6307\u6a19 \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u500b <code>target + 1</code>\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u800c \\(j\\) \u6307\u5411\u6700\u53f3\u4e00\u500b <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de \\(j\\) \u5373\u53ef\u3002</p> <p></p> <p> \u5716 10-7 \u00a0 \u5c07\u67e5\u8a62\u53f3\u908a\u754c\u8f49\u5316\u70ba\u67e5\u8a62\u5de6\u908a\u754c </p> <p>\u8acb\u6ce8\u610f\uff0c\u8fd4\u56de\u7684\u63d2\u5165\u9ede\u662f \\(i\\) \uff0c\u56e0\u6b64\u9700\u8981\u5c07\u5176\u6e1b \\(1\\) \uff0c\u5f9e\u800c\u7372\u5f97 \\(j\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target\"\"\"\n    # \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    i = binary_search_insertion(nums, target + 1)\n    # j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    j = i - 1\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 or nums[j] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    int i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint binarySearchRightEdge(int[] nums, int target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    i := binarySearchInsertion(nums, target+1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    j := i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    let j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfunction binarySearchRightEdge(nums, target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n  int i = binarySearchInsertion(nums, target + 1);\n  // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n  int j = i - 1;\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    let i = binary_search_insertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    let j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target */\nfun binarySearchRightEdge(nums: IntArray, target: Int): Int {\n    // \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n    val i = binarySearchInsertion(nums, target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n    val j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\u6700\u53f3\u4e00\u500b target ###\ndef binary_search_right_edge(nums, target)\n  # \u8f49\u5316\u70ba\u67e5\u8a62\u6700\u5de6\u4e00\u500b target + 1\n  i = binary_search_insertion(nums, target + 1)\n\n  # j \u6307\u5411\u6700\u53f3\u4e00\u500b target \uff0ci \u6307\u5411\u9996\u500b\u5927\u65bc target \u7684\u5143\u7d20\n  j = i - 1\n\n  # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  return -1 if j == -1 || nums[j] != target\n\n  j # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nend\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchRightEdge}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_edge/#2","title":"2. \u00a0 \u8f49\u5316\u70ba\u67e5\u8a62\u5143\u7d20","text":"<p>\u6211\u5011\u77e5\u9053\uff0c\u7576\u9663\u5217\u4e0d\u5305\u542b <code>target</code> \u6642\uff0c\u6700\u7d42 \\(i\\) \u548c \\(j\\) \u6703\u5206\u5225\u6307\u5411\u9996\u500b\u5927\u65bc\u3001\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\u3002</p> <p>\u56e0\u6b64\uff0c\u5982\u5716 10-8 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u69cb\u9020\u4e00\u500b\u9663\u5217\u4e2d\u4e0d\u5b58\u5728\u7684\u5143\u7d20\uff0c\u7528\u65bc\u67e5\u8a62\u5de6\u53f3\u908a\u754c\u3002</p> <ul> <li>\u67e5\u8a62\u6700\u5de6\u4e00\u500b <code>target</code> \uff1a\u53ef\u4ee5\u8f49\u5316\u70ba\u67e5\u8a62 <code>target - 0.5</code> \uff0c\u4e26\u8fd4\u56de\u6307\u6a19 \\(i\\) \u3002</li> <li>\u67e5\u8a62\u6700\u53f3\u4e00\u500b <code>target</code> \uff1a\u53ef\u4ee5\u8f49\u5316\u70ba\u67e5\u8a62 <code>target + 0.5</code> \uff0c\u4e26\u8fd4\u56de\u6307\u6a19 \\(j\\) \u3002</li> </ul> <p></p> <p> \u5716 10-8 \u00a0 \u5c07\u67e5\u8a62\u908a\u754c\u8f49\u5316\u70ba\u67e5\u8a62\u5143\u7d20 </p> <p>\u7a0b\u5f0f\u78bc\u5728\u6b64\u7701\u7565\uff0c\u4ee5\u4e0b\u5169\u9ede\u503c\u5f97\u6ce8\u610f\u3002</p> <ul> <li>\u7d66\u5b9a\u9663\u5217\u4e0d\u5305\u542b\u5c0f\u6578\uff0c\u9019\u610f\u5473\u8457\u6211\u5011\u7121\u9808\u95dc\u5fc3\u5982\u4f55\u8655\u7406\u76f8\u7b49\u7684\u60c5\u6cc1\u3002</li> <li>\u56e0\u70ba\u8a72\u65b9\u6cd5\u5f15\u5165\u4e86\u5c0f\u6578\uff0c\u6240\u4ee5\u9700\u8981\u5c07\u51fd\u5f0f\u4e2d\u7684\u8b8a\u6578 <code>target</code> \u6539\u70ba\u6d6e\u9ede\u6578\u578b\u5225\uff08Python \u7121\u9808\u6539\u52d5\uff09\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_insertion/","title":"10.2 \u00a0 \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede","text":"<p>\u4e8c\u5206\u641c\u5c0b\u4e0d\u50c5\u53ef\u7528\u65bc\u641c\u5c0b\u76ee\u6a19\u5143\u7d20\uff0c\u9084\u53ef\u7528\u65bc\u89e3\u6c7a\u8a31\u591a\u8b8a\u7a2e\u554f\u984c\uff0c\u6bd4\u5982\u641c\u5c0b\u76ee\u6a19\u5143\u7d20\u7684\u63d2\u5165\u4f4d\u7f6e\u3002</p>"},{"location":"chapter_searching/binary_search_insertion/#1021","title":"10.2.1 \u00a0 \u7121\u91cd\u8907\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u6709\u5e8f\u9663\u5217 <code>nums</code> \u548c\u4e00\u500b\u5143\u7d20 <code>target</code> \uff0c\u9663\u5217\u4e0d\u5b58\u5728\u91cd\u8907\u5143\u7d20\u3002\u73fe\u5c07 <code>target</code> \u63d2\u5165\u9663\u5217 <code>nums</code> \u4e2d\uff0c\u4e26\u4fdd\u6301\u5176\u6709\u5e8f\u6027\u3002\u82e5\u9663\u5217\u4e2d\u5df2\u5b58\u5728\u5143\u7d20 <code>target</code> \uff0c\u5247\u63d2\u5165\u5230\u5176\u5de6\u65b9\u3002\u8acb\u8fd4\u56de\u63d2\u5165\u5f8c <code>target</code> \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15\u3002\u793a\u4f8b\u5982\u5716 10-4 \u6240\u793a\u3002</p> <p></p> <p> \u5716 10-4 \u00a0 \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\u793a\u4f8b\u8cc7\u6599 </p> <p>\u5982\u679c\u60f3\u8907\u7528\u4e0a\u4e00\u7bc0\u7684\u4e8c\u5206\u641c\u5c0b\u7a0b\u5f0f\u78bc\uff0c\u5247\u9700\u8981\u56de\u7b54\u4ee5\u4e0b\u5169\u500b\u554f\u984c\u3002</p> <p>\u554f\u984c\u4e00\uff1a\u7576\u9663\u5217\u4e2d\u5305\u542b <code>target</code> \u6642\uff0c\u63d2\u5165\u9ede\u7684\u7d22\u5f15\u662f\u5426\u662f\u8a72\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u984c\u76ee\u8981\u6c42\u5c07 <code>target</code> \u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u5de6\u908a\uff0c\u9019\u610f\u5473\u8457\u65b0\u63d2\u5165\u7684 <code>target</code> \u66ff\u63db\u4e86\u539f\u4f86 <code>target</code> \u7684\u4f4d\u7f6e\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u7576\u9663\u5217\u5305\u542b <code>target</code> \u6642\uff0c\u63d2\u5165\u9ede\u7684\u7d22\u5f15\u5c31\u662f\u8a72 <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u554f\u984c\u4e8c\uff1a\u7576\u9663\u5217\u4e2d\u4e0d\u5b58\u5728 <code>target</code> \u6642\uff0c\u63d2\u5165\u9ede\u662f\u54ea\u500b\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u9032\u4e00\u6b65\u601d\u8003\u4e8c\u5206\u641c\u5c0b\u904e\u7a0b\uff1a\u7576 <code>nums[m] &lt; target</code> \u6642 \\(i\\) \u79fb\u52d5\uff0c\u9019\u610f\u5473\u8457\u6307\u6a19 \\(i\\) \u5728\u5411\u5927\u65bc\u7b49\u65bc <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002\u540c\u7406\uff0c\u6307\u6a19 \\(j\\) \u59cb\u7d42\u5728\u5411\u5c0f\u65bc\u7b49\u65bc <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</p> <p>\u56e0\u6b64\u4e8c\u5206\u7d50\u675f\u6642\u4e00\u5b9a\u6709\uff1a\\(i\\) \u6307\u5411\u9996\u500b\u5927\u65bc <code>target</code> \u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u9996\u500b\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\u3002\u6613\u5f97\u7576\u9663\u5217\u4e0d\u5305\u542b <code>target</code> \u6642\uff0c\u63d2\u5165\u7d22\u5f15\u70ba \\(i\\) \u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n            return m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    } else {\n      return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n    }\n  }\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfn binary_search_insertion_simple(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 */\nfun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u7121\u91cd\u8907\u5143\u7d20\uff09 ###\ndef binary_search_insertion_simple(nums, target)\n  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    elsif nums[m] &gt; target\n      j = m - 1 # target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    else\n      return m  # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede m\n    end\n  end\n\n  i # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u9ede i\nend\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertionSimple}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_insertion/#1022","title":"10.2.2 \u00a0 \u5b58\u5728\u91cd\u8907\u5143\u7d20\u7684\u60c5\u6cc1","text":"<p>Question</p> <p>\u5728\u4e0a\u4e00\u984c\u7684\u57fa\u790e\u4e0a\uff0c\u898f\u5b9a\u9663\u5217\u53ef\u80fd\u5305\u542b\u91cd\u8907\u5143\u7d20\uff0c\u5176\u9918\u4e0d\u8b8a\u3002</p> <p>\u5047\u8a2d\u9663\u5217\u4e2d\u5b58\u5728\u591a\u500b <code>target</code> \uff0c\u5247\u666e\u901a\u4e8c\u5206\u641c\u5c0b\u53ea\u80fd\u8fd4\u56de\u5176\u4e2d\u4e00\u500b <code>target</code> \u7684\u7d22\u5f15\uff0c\u800c\u7121\u6cd5\u78ba\u5b9a\u8a72\u5143\u7d20\u7684\u5de6\u908a\u548c\u53f3\u908a\u9084\u6709\u591a\u5c11 <code>target</code>\u3002</p> <p>\u984c\u76ee\u8981\u6c42\u5c07\u76ee\u6a19\u5143\u7d20\u63d2\u5165\u5230\u6700\u5de6\u908a\uff0c\u6240\u4ee5\u6211\u5011\u9700\u8981\u67e5\u8a62\u9663\u5217\u4e2d\u6700\u5de6\u4e00\u500b <code>target</code> \u7684\u7d22\u5f15\u3002\u521d\u6b65\u8003\u616e\u900f\u904e\u5716 10-5 \u6240\u793a\u7684\u6b65\u9a5f\u5be6\u73fe\u3002</p> <ol> <li>\u57f7\u884c\u4e8c\u5206\u641c\u5c0b\uff0c\u5f97\u5230\u4efb\u610f\u4e00\u500b <code>target</code> \u7684\u7d22\u5f15\uff0c\u8a18\u70ba \\(k\\) \u3002</li> <li>\u5f9e\u7d22\u5f15 \\(k\\) \u958b\u59cb\uff0c\u5411\u5de6\u9032\u884c\u7dda\u6027\u8d70\u8a2a\uff0c\u7576\u627e\u5230\u6700\u5de6\u908a\u7684 <code>target</code> \u6642\u8fd4\u56de\u3002</li> </ol> <p></p> <p> \u5716 10-5 \u00a0 \u7dda\u6027\u67e5\u8a62\u91cd\u8907\u5143\u7d20\u7684\u63d2\u5165\u9ede </p> <p>\u6b64\u65b9\u6cd5\u96d6\u7136\u53ef\u7528\uff0c\u4f46\u5176\u5305\u542b\u7dda\u6027\u67e5\u8a62\uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002\u7576\u9663\u5217\u4e2d\u5b58\u5728\u5f88\u591a\u91cd\u8907\u7684 <code>target</code> \u6642\uff0c\u8a72\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\u3002</p> <p>\u73fe\u8003\u616e\u62d3\u5c55\u4e8c\u5206\u641c\u5c0b\u7a0b\u5f0f\u78bc\u3002\u5982\u5716 10-6 \u6240\u793a\uff0c\u6574\u9ad4\u6d41\u7a0b\u4fdd\u6301\u4e0d\u8b8a\uff0c\u6bcf\u8f2a\u5148\u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65b7 <code>target</code> \u548c <code>nums[m]</code> \u7684\u5927\u5c0f\u95dc\u4fc2\uff0c\u5206\u70ba\u4ee5\u4e0b\u5e7e\u7a2e\u60c5\u6cc1\u3002</p> <ul> <li>\u7576 <code>nums[m] &lt; target</code> \u6216 <code>nums[m] &gt; target</code> \u6642\uff0c\u8aaa\u660e\u9084\u6c92\u6709\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u63a1\u7528\u666e\u901a\u4e8c\u5206\u641c\u5c0b\u7684\u7e2e\u5c0f\u5340\u9593\u64cd\u4f5c\uff0c\u5f9e\u800c\u4f7f\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u5411 <code>target</code> \u9760\u8fd1\u3002</li> <li>\u7576 <code>nums[m] == target</code> \u6642\uff0c\u8aaa\u660e\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\u5728\u5340\u9593 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u63a1\u7528 \\(j = m - 1\\) \u4f86\u7e2e\u5c0f\u5340\u9593\uff0c\u5f9e\u800c\u4f7f\u6307\u6a19 \\(j\\) \u5411\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</li> </ul> <p>\u8ff4\u5708\u5b8c\u6210\u5f8c\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u908a\u7684 <code>target</code> \uff0c\\(j\\) \u6307\u5411\u9996\u500b\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\uff0c\u56e0\u6b64\u7d22\u5f15 \\(i\\) \u5c31\u662f\u63d2\u5165\u9ede\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 10-6 \u00a0 \u4e8c\u5206\u641c\u5c0b\u91cd\u8907\u5143\u7d20\u7684\u63d2\u5165\u9ede\u7684\u6b65\u9a5f </p> <p>\u89c0\u5bdf\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\uff0c\u5224\u65b7\u5206\u652f <code>nums[m] &gt; target</code> \u548c <code>nums[m] == target</code> \u7684\u64cd\u4f5c\u76f8\u540c\uff0c\u56e0\u6b64\u5169\u8005\u53ef\u4ee5\u5408\u4f75\u3002</p> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u6211\u5011\u4ecd\u7136\u53ef\u4ee5\u5c07\u5224\u65b7\u689d\u4ef6\u4fdd\u6301\u5c55\u958b\uff0c\u56e0\u70ba\u5176\u908f\u8f2f\u66f4\u52a0\u6e05\u6670\u3001\u53ef\u8b80\u6027\u66f4\u597d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        else:\n            j = m - 1  # \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n    # \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n            j = m - 1\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    } else {\n      j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n    }\n  }\n  // \u8fd4\u56de\u63d2\u5165\u9ede i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 */\nfun binarySearchInsertion(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target \u5728\u5340\u9593 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target \u5728\u5340\u9593 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u9ede i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### \u4e8c\u5206\u641c\u5c0b\u63d2\u5165\u9ede\uff08\u5b58\u5728\u91cd\u8907\u5143\u7d20\uff09 ###\ndef binary_search_insertion(nums, target)\n  # \u521d\u59cb\u5316\u96d9\u9589\u5340\u9593 [0, n-1]\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # \u8a08\u7b97\u4e2d\u9ede\u7d22\u5f15 m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target \u5728\u5340\u9593 [m+1, j] \u4e2d\n    elsif nums[m] &gt; target\n      j = m - 1 # target \u5728\u5340\u9593 [i, m-1] \u4e2d\n    else\n      j = m - 1 # \u9996\u500b\u5c0f\u65bc target \u7684\u5143\u7d20\u5728\u5340\u9593 [i, m-1] \u4e2d\n    end\n  end\n\n  i # \u8fd4\u56de\u63d2\u5165\u9ede i\nend\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertion}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>Tip</p> <p>\u672c\u7bc0\u7684\u7a0b\u5f0f\u78bc\u90fd\u662f\u201c\u96d9\u9589\u5340\u9593\u201d\u5beb\u6cd5\u3002\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u81ea\u884c\u5be6\u73fe\u201c\u5de6\u9589\u53f3\u958b\u201d\u5beb\u6cd5\u3002</p> <p>\u7e3d\u7684\u4f86\u770b\uff0c\u4e8c\u5206\u641c\u5c0b\u7121\u975e\u5c31\u662f\u7d66\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u5206\u5225\u8a2d\u5b9a\u641c\u5c0b\u76ee\u6a19\uff0c\u76ee\u6a19\u53ef\u80fd\u662f\u4e00\u500b\u5177\u9ad4\u7684\u5143\u7d20\uff08\u4f8b\u5982 <code>target</code> \uff09\uff0c\u4e5f\u53ef\u80fd\u662f\u4e00\u500b\u5143\u7d20\u7bc4\u570d\uff08\u4f8b\u5982\u5c0f\u65bc <code>target</code> \u7684\u5143\u7d20\uff09\u3002</p> <p>\u5728\u4e0d\u65b7\u7684\u8ff4\u5708\u4e8c\u5206\u4e2d\uff0c\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u90fd\u9010\u6f38\u903c\u8fd1\u9810\u5148\u8a2d\u5b9a\u7684\u76ee\u6a19\u3002\u6700\u7d42\uff0c\u5b83\u5011\u6216\u662f\u6210\u529f\u627e\u5230\u7b54\u6848\uff0c\u6216\u662f\u8d8a\u904e\u908a\u754c\u5f8c\u505c\u6b62\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.4 \u00a0 \u96dc\u6e4a\u6700\u4f73\u5316\u7b56\u7565","text":"<p>\u5728\u6f14\u7b97\u6cd5\u984c\u4e2d\uff0c\u6211\u5011\u5e38\u900f\u904e\u5c07\u7dda\u6027\u67e5\u8a62\u66ff\u63db\u70ba\u96dc\u6e4a\u67e5\u8a62\u4f86\u964d\u4f4e\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u3002\u6211\u5011\u85c9\u52a9\u4e00\u500b\u6f14\u7b97\u6cd5\u984c\u4f86\u52a0\u6df1\u7406\u89e3\u3002</p> <p>Question</p> <p>\u7d66\u5b9a\u4e00\u500b\u6574\u6578\u9663\u5217 <code>nums</code> \u548c\u4e00\u500b\u76ee\u6a19\u5143\u7d20 <code>target</code> \uff0c\u8acb\u5728\u9663\u5217\u4e2d\u641c\u7d22\u201c\u548c\u201d\u70ba <code>target</code> \u7684\u5169\u500b\u5143\u7d20\uff0c\u4e26\u8fd4\u56de\u5b83\u5011\u7684\u9663\u5217\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u500b\u89e3\u5373\u53ef\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1041","title":"10.4.1 \u00a0 \u7dda\u6027\u67e5\u8a62\uff1a\u4ee5\u6642\u9593\u63db\u7a7a\u9593","text":"<p>\u8003\u616e\u76f4\u63a5\u8d70\u8a2a\u6240\u6709\u53ef\u80fd\u7684\u7d44\u5408\u3002\u5982\u5716 10-9 \u6240\u793a\uff0c\u6211\u5011\u958b\u555f\u4e00\u500b\u5169\u5c64\u8ff4\u5708\uff0c\u5728\u6bcf\u8f2a\u4e2d\u5224\u65b7\u5169\u500b\u6574\u6578\u7684\u548c\u662f\u5426\u70ba <code>target</code> \uff0c\u82e5\u662f\uff0c\u5247\u8fd4\u56de\u5b83\u5011\u7684\u7d22\u5f15\u3002</p> <p></p> <p> \u5716 10-9 \u00a0 \u7dda\u6027\u67e5\u8a62\u6c42\u89e3\u5169\u6578\u4e4b\u548c </p> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209\"\"\"\n    # \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; j &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u5217\u8209 */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 */\nfun twoSumBruteForce(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    for (i in 0..&lt;size - 1) {\n        for (j in i + 1..&lt;size) {\n            if (nums[i] + nums[j] == target) return intArrayOf(i, j)\n        }\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209 ###\ndef two_sum_brute_force(nums, target)\n  # \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n  for i in 0...(nums.length - 1)\n    for j in (i + 1)...nums.length\n      return [i, j] if nums[i] + nums[j] == target\n    end\n  end\n\n  []\nend\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u5217\u8209\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\n    var size: usize = nums.len;\n    var i: usize = 0;\n    // \u5169\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n^2)\n    while (i &lt; size - 1) : (i += 1) {\n        var j = i + 1;\n        while (j &lt; size) : (j += 1) {\n            if (nums[i] + nums[j] == target) {\n                return [_]i32{@intCast(i), @intCast(j)};\n            }\n        }\n    }\n    return null;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6b64\u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \uff0c\u5728\u5927\u8cc7\u6599\u91cf\u4e0b\u975e\u5e38\u8017\u6642\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1042","title":"10.4.2 \u00a0 \u96dc\u6e4a\u67e5\u8a62\uff1a\u4ee5\u7a7a\u9593\u63db\u6642\u9593","text":"<p>\u8003\u616e\u85c9\u52a9\u4e00\u500b\u96dc\u6e4a\u8868\uff0c\u9375\u503c\u5c0d\u5206\u5225\u70ba\u9663\u5217\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u8ff4\u5708\u8d70\u8a2a\u9663\u5217\uff0c\u6bcf\u8f2a\u57f7\u884c\u5716 10-10 \u6240\u793a\u7684\u6b65\u9a5f\u3002</p> <ol> <li>\u5224\u65b7\u6578\u5b57 <code>target - nums[i]</code> \u662f\u5426\u5728\u96dc\u6e4a\u8868\u4e2d\uff0c\u82e5\u662f\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\u9019\u5169\u500b\u5143\u7d20\u7684\u7d22\u5f15\u3002</li> <li>\u5c07\u9375\u503c\u5c0d <code>nums[i]</code> \u548c\u7d22\u5f15 <code>i</code> \u65b0\u589e\u9032\u96dc\u6e4a\u8868\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> \u5716 10-10 \u00a0 \u8f14\u52a9\u96dc\u6e4a\u8868\u6c42\u89e3\u5169\u6578\u4e4b\u548c </p> <p>\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff0c\u50c5\u9700\u55ae\u5c64\u8ff4\u5708\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868\"\"\"\n    # \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    dic = {}\n    # \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    unordered_map&lt;int, int&gt; dic;\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    Dictionary&lt;int, int&gt; dic = [];\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    hashTable := map[int]int{}\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    var dic: [Int: Int] = [:]\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfunction twoSumHashTable(nums, target) {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    let m = {};\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    let m: Map&lt;number, number&gt; = new Map();\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a \u8f14\u52a9\u96dc\u6e4a\u8868 */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n  Map&lt;int, int&gt; dic = HashMap();\n  // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    let mut dic = HashMap::new();\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32),\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u96dc\u6e4a\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u65bc uthash.h \u5be6\u73fe\n} HashTable;\n\n/* \u96dc\u6e4a\u8868\u67e5\u8a62 */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* \u96dc\u6e4a\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(HashTable **h, int key, int val) {\n    HashTable *t = find(*h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(*h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(&amp;hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 */\nfun twoSumHashTable(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    val dic = HashMap&lt;Int, Int&gt;()\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    for (i in 0..&lt;size) {\n        if (dic.containsKey(target - nums[i])) {\n            return intArrayOf(dic[target - nums[i]]!!, i)\n        }\n        dic[nums[i]] = i\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868 ###\ndef two_sum_hash_table(nums, target)\n  # \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n  dic = {}\n  # \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n  for i in 0...nums.length\n    return [dic[target - nums[i]], i] if dic.has_key?(target - nums[i])\n\n    dic[nums[i]] = i\n  end\n\n  []\nend\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e8c\uff1a\u8f14\u52a9\u96dc\u6e4a\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\n    var size: usize = nums.len;\n    // \u8f14\u52a9\u96dc\u6e4a\u8868\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    var dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\n    defer dic.deinit();\n    var i: usize = 0;\n    // \u55ae\u5c64\u8ff4\u5708\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\n    while (i &lt; size) : (i += 1) {\n        if (dic.contains(target - nums[i])) {\n            return [_]i32{dic.get(target - nums[i]).?, @intCast(i)};\n        }\n        try dic.put(nums[i], @intCast(i));\n    }\n    return null;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u6b64\u65b9\u6cd5\u900f\u904e\u96dc\u6e4a\u67e5\u8a62\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u57f7\u884c\u6548\u7387\u3002</p> <p>\u7531\u65bc\u9700\u8981\u7dad\u8b77\u4e00\u500b\u984d\u5916\u7684\u96dc\u6e4a\u8868\uff0c\u56e0\u6b64\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \u3002\u5118\u7ba1\u5982\u6b64\uff0c\u8a72\u65b9\u6cd5\u7684\u6574\u9ad4\u6642\u7a7a\u6548\u7387\u66f4\u70ba\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u984c\u7684\u6700\u512a\u89e3\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.5 \u00a0 \u91cd\u8b58\u641c\u5c0b\u6f14\u7b97\u6cd5","text":"<p>\u641c\u5c0b\u6f14\u7b97\u6cd5\uff08searching algorithm\uff09\u7528\u65bc\u5728\u8cc7\u6599\u7d50\u69cb\uff08\u4f8b\u5982\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u6a39\u6216\u5716\uff09\u4e2d\u641c\u7d22\u4e00\u500b\u6216\u4e00\u7d44\u6eff\u8db3\u7279\u5b9a\u689d\u4ef6\u7684\u5143\u7d20\u3002</p> <p>\u641c\u5c0b\u6f14\u7b97\u6cd5\u53ef\u6839\u64da\u5be6\u73fe\u601d\u8def\u5206\u70ba\u4ee5\u4e0b\u5169\u985e\u3002</p> <ul> <li>\u900f\u904e\u8d70\u8a2a\u8cc7\u6599\u7d50\u69cb\u4f86\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\uff0c\u4f8b\u5982\u9663\u5217\u3001\u93c8\u7d50\u4e32\u5217\u3001\u6a39\u548c\u5716\u7684\u8d70\u8a2a\u7b49\u3002</li> <li>\u5229\u7528\u8cc7\u6599\u7d44\u7e54\u7d50\u69cb\u6216\u8cc7\u6599\u5305\u542b\u7684\u5148\u9a57\u8cc7\u8a0a\uff0c\u5be6\u73fe\u9ad8\u6548\u5143\u7d20\u67e5\u8a62\uff0c\u4f8b\u5982\u4e8c\u5206\u641c\u5c0b\u3001\u96dc\u6e4a\u67e5\u8a62\u548c\u4e8c\u5143\u641c\u5c0b\u6a39\u67e5\u8a62\u7b49\u3002</li> </ul> <p>\u4e0d\u96e3\u767c\u73fe\uff0c\u9019\u4e9b\u77e5\u8b58\u9ede\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u7bc0\u4e2d\u4ecb\u7d39\u904e\uff0c\u56e0\u6b64\u641c\u5c0b\u6f14\u7b97\u6cd5\u5c0d\u65bc\u6211\u5011\u4f86\u8aaa\u4e26\u4e0d\u964c\u751f\u3002\u5728\u672c\u7bc0\u4e2d\uff0c\u6211\u5011\u5c07\u5f9e\u66f4\u52a0\u7cfb\u7d71\u7684\u8996\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5be9\u8996\u641c\u5c0b\u6f14\u7b97\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1051","title":"10.5.1 \u00a0 \u66b4\u529b\u641c\u5c0b","text":"<p>\u66b4\u529b\u641c\u5c0b\u900f\u904e\u8d70\u8a2a\u8cc7\u6599\u7d50\u69cb\u7684\u6bcf\u500b\u5143\u7d20\u4f86\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\u3002</p> <ul> <li>\u201c\u7dda\u6027\u641c\u5c0b\u201d\u9069\u7528\u65bc\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u7b49\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u3002\u5b83\u5f9e\u8cc7\u6599\u7d50\u69cb\u7684\u4e00\u7aef\u958b\u59cb\uff0c\u9010\u500b\u8a2a\u554f\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6a19\u5143\u7d20\u6216\u5230\u9054\u53e6\u4e00\u7aef\u4ecd\u6c92\u6709\u627e\u5230\u76ee\u6a19\u5143\u7d20\u70ba\u6b62\u3002</li> <li>\u201c\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\u201d\u548c\u201c\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u201d\u662f\u5716\u548c\u6a39\u7684\u5169\u7a2e\u8d70\u8a2a\u7b56\u7565\u3002\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\u5f9e\u521d\u59cb\u7bc0\u9ede\u958b\u59cb\u9010\u5c64\u641c\u5c0b\uff0c\u7531\u8fd1\u53ca\u9060\u5730\u8a2a\u554f\u5404\u500b\u7bc0\u9ede\u3002\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u5f9e\u521d\u59cb\u7bc0\u9ede\u958b\u59cb\uff0c\u6cbf\u8457\u4e00\u689d\u8def\u5f91\u8d70\u5230\u982d\uff0c\u518d\u56de\u6eaf\u4e26\u5617\u8a66\u5176\u4ed6\u8def\u5f91\uff0c\u76f4\u5230\u8d70\u8a2a\u5b8c\u6574\u500b\u8cc7\u6599\u7d50\u69cb\u3002</li> </ul> <p>\u66b4\u529b\u641c\u5c0b\u7684\u512a\u9ede\u662f\u7c21\u55ae\u4e14\u901a\u7528\u6027\u597d\uff0c\u7121\u9808\u5c0d\u8cc7\u6599\u505a\u9810\u8655\u7406\u548c\u85c9\u52a9\u984d\u5916\u7684\u8cc7\u6599\u7d50\u69cb\u3002</p> <p>\u7136\u800c\uff0c\u6b64\u985e\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u70ba\u5143\u7d20\u6578\u91cf\uff0c\u56e0\u6b64\u5728\u8cc7\u6599\u91cf\u8f03\u5927\u7684\u60c5\u6cc1\u4e0b\u6548\u80fd\u8f03\u5dee\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1052","title":"10.5.2 \u00a0 \u81ea\u9069\u61c9\u641c\u5c0b","text":"<p>\u81ea\u9069\u61c9\u641c\u5c0b\u5229\u7528\u8cc7\u6599\u7684\u7279\u6709\u5c6c\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u4f86\u6700\u4f73\u5316\u641c\u5c0b\u904e\u7a0b\uff0c\u5f9e\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\u3002</p> <ul> <li>\u201c\u4e8c\u5206\u641c\u5c0b\u201d\u5229\u7528\u8cc7\u6599\u7684\u6709\u5e8f\u6027\u5be6\u73fe\u9ad8\u6548\u67e5\u8a62\uff0c\u50c5\u9069\u7528\u65bc\u9663\u5217\u3002</li> <li>\u201c\u96dc\u6e4a\u67e5\u8a62\u201d\u5229\u7528\u96dc\u6e4a\u8868\u5c07\u641c\u5c0b\u8cc7\u6599\u548c\u76ee\u6a19\u8cc7\u6599\u5efa\u7acb\u70ba\u9375\u503c\u5c0d\u5c0d\u6620\uff0c\u5f9e\u800c\u5be6\u73fe\u67e5\u8a62\u64cd\u4f5c\u3002</li> <li>\u201c\u6a39\u67e5\u8a62\u201d\u5728\u7279\u5b9a\u7684\u6a39\u7d50\u69cb\uff08\u4f8b\u5982\u4e8c\u5143\u641c\u5c0b\u6a39\uff09\u4e2d\uff0c\u57fa\u65bc\u6bd4\u8f03\u7bc0\u9ede\u503c\u4f86\u5feb\u901f\u6392\u9664\u7bc0\u9ede\uff0c\u5f9e\u800c\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\u3002</li> </ul> <p>\u6b64\u985e\u6f14\u7b97\u6cd5\u7684\u512a\u9ede\u662f\u6548\u7387\u9ad8\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</p> <p>\u7136\u800c\uff0c\u4f7f\u7528\u9019\u4e9b\u6f14\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5c0d\u8cc7\u6599\u9032\u884c\u9810\u8655\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u641c\u5c0b\u9700\u8981\u9810\u5148\u5c0d\u9663\u5217\u9032\u884c\u6392\u5e8f\uff0c\u96dc\u6e4a\u67e5\u8a62\u548c\u6a39\u67e5\u8a62\u90fd\u9700\u8981\u85c9\u52a9\u984d\u5916\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u7dad\u8b77\u9019\u4e9b\u8cc7\u6599\u7d50\u69cb\u4e5f\u9700\u8981\u984d\u5916\u7684\u6642\u9593\u548c\u7a7a\u9593\u958b\u92b7\u3002</p> <p>Tip</p> <p>\u81ea\u9069\u61c9\u641c\u5c0b\u6f14\u7b97\u6cd5\u5e38\u88ab\u7a31\u70ba\u67e5\u8a62\u6f14\u7b97\u6cd5\uff0c\u4e3b\u8981\u7528\u65bc\u5728\u7279\u5b9a\u8cc7\u6599\u7d50\u69cb\u4e2d\u5feb\u901f\u6aa2\u7d22\u76ee\u6a19\u5143\u7d20\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1053","title":"10.5.3 \u00a0 \u641c\u5c0b\u65b9\u6cd5\u9078\u53d6","text":"<p>\u7d66\u5b9a\u5927\u5c0f\u70ba \\(n\\) \u7684\u4e00\u7d44\u8cc7\u6599\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u7dda\u6027\u641c\u5c0b\u3001\u4e8c\u5206\u641c\u5c0b\u3001\u6a39\u67e5\u8a62\u3001\u96dc\u6e4a\u67e5\u8a62\u7b49\u591a\u7a2e\u65b9\u6cd5\u5f9e\u4e2d\u641c\u7d22\u76ee\u6a19\u5143\u7d20\u3002\u5404\u500b\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u5716 10-11 \u6240\u793a\u3002</p> <p></p> <p> \u5716 10-11 \u00a0 \u591a\u7a2e\u641c\u5c0b\u7b56\u7565 </p> <p>\u4e0a\u8ff0\u5e7e\u7a2e\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u8207\u7279\u6027\u5982\u8868 10-1 \u6240\u793a\u3002</p> <p> \u8868 10-1 \u00a0 \u67e5\u8a62\u6f14\u7b97\u6cd5\u6548\u7387\u5c0d\u6bd4 </p> \u7dda\u6027\u641c\u5c0b \u4e8c\u5206\u641c\u5c0b \u6a39\u67e5\u8a62 \u96dc\u6e4a\u67e5\u8a62 \u67e5\u8a62\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u522a\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u984d\u5916\u7a7a\u9593 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u8cc7\u6599\u9810\u8655\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6a39 \\(O(n \\log n)\\) \u5efa\u96dc\u6e4a\u8868 \\(O(n)\\) \u8cc7\u6599\u662f\u5426\u6709\u5e8f \u7121\u5e8f \u6709\u5e8f \u6709\u5e8f \u7121\u5e8f <p>\u641c\u5c0b\u6f14\u7b97\u6cd5\u7684\u9078\u64c7\u9084\u53d6\u6c7a\u65bc\u8cc7\u6599\u9ad4\u91cf\u3001\u641c\u5c0b\u6548\u80fd\u8981\u6c42\u3001\u8cc7\u6599\u67e5\u8a62\u8207\u66f4\u65b0\u983b\u7387\u7b49\u3002</p> <p>\u7dda\u6027\u641c\u5c0b</p> <ul> <li>\u901a\u7528\u6027\u8f03\u597d\uff0c\u7121\u9808\u4efb\u4f55\u8cc7\u6599\u9810\u8655\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u5011\u50c5\u9700\u67e5\u8a62\u4e00\u6b21\u8cc7\u6599\uff0c\u90a3\u9ebc\u5176\u4ed6\u4e09\u7a2e\u65b9\u6cd5\u7684\u8cc7\u6599\u9810\u8655\u7406\u7684\u6642\u9593\u6bd4\u7dda\u6027\u641c\u5c0b\u7684\u6642\u9593\u9084\u8981\u66f4\u9577\u3002</li> <li>\u9069\u7528\u65bc\u9ad4\u91cf\u8f03\u5c0f\u7684\u8cc7\u6599\uff0c\u6b64\u60c5\u6cc1\u4e0b\u6642\u9593\u8907\u96dc\u5ea6\u5c0d\u6548\u7387\u5f71\u97ff\u8f03\u5c0f\u3002</li> <li>\u9069\u7528\u65bc\u8cc7\u6599\u66f4\u65b0\u983b\u7387\u8f03\u9ad8\u7684\u5834\u666f\uff0c\u56e0\u70ba\u8a72\u65b9\u6cd5\u4e0d\u9700\u8981\u5c0d\u8cc7\u6599\u9032\u884c\u4efb\u4f55\u984d\u5916\u7dad\u8b77\u3002</li> </ul> <p>\u4e8c\u5206\u641c\u5c0b</p> <ul> <li>\u9069\u7528\u65bc\u5927\u8cc7\u6599\u91cf\u7684\u60c5\u6cc1\uff0c\u6548\u7387\u8868\u73fe\u7a69\u5b9a\uff0c\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \u3002</li> <li>\u8cc7\u6599\u91cf\u4e0d\u80fd\u904e\u5927\uff0c\u56e0\u70ba\u5132\u5b58\u9663\u5217\u9700\u8981\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u3002</li> <li>\u4e0d\u9069\u7528\u65bc\u9ad8\u983b\u589e\u522a\u8cc7\u6599\u7684\u5834\u666f\uff0c\u56e0\u70ba\u7dad\u8b77\u6709\u5e8f\u9663\u5217\u7684\u958b\u92b7\u8f03\u5927\u3002</li> </ul> <p>\u96dc\u6e4a\u67e5\u8a62</p> <ul> <li>\u9069\u5408\u5c0d\u67e5\u8a62\u6548\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u5834\u666f\uff0c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\) \u3002</li> <li>\u4e0d\u9069\u5408\u9700\u8981\u6709\u5e8f\u8cc7\u6599\u6216\u7bc4\u570d\u67e5\u8a62\u7684\u5834\u666f\uff0c\u56e0\u70ba\u96dc\u6e4a\u8868\u7121\u6cd5\u7dad\u8b77\u8cc7\u6599\u7684\u6709\u5e8f\u6027\u3002</li> <li>\u5c0d\u96dc\u6e4a\u51fd\u5f0f\u548c\u96dc\u6e4a\u885d\u7a81\u8655\u7406\u7b56\u7565\u7684\u4f9d\u8cf4\u6027\u8f03\u9ad8\uff0c\u5177\u6709\u8f03\u5927\u7684\u6548\u80fd\u52a3\u5316\u98a8\u96aa\u3002</li> <li>\u4e0d\u9069\u5408\u8cc7\u6599\u91cf\u904e\u5927\u7684\u60c5\u6cc1\uff0c\u56e0\u70ba\u96dc\u6e4a\u8868\u9700\u8981\u984d\u5916\u7a7a\u9593\u4f86\u6700\u5927\u7a0b\u5ea6\u5730\u6e1b\u5c11\u885d\u7a81\uff0c\u5f9e\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8a62\u6548\u80fd\u3002</li> </ul> <p>\u6a39\u67e5\u8a62</p> <ul> <li>\u9069\u7528\u65bc\u6d77\u91cf\u8cc7\u6599\uff0c\u56e0\u70ba\u6a39\u7bc0\u9ede\u5728\u8a18\u61b6\u9ad4\u4e2d\u662f\u5206\u6563\u5132\u5b58\u7684\u3002</li> <li>\u9069\u5408\u9700\u8981\u7dad\u8b77\u6709\u5e8f\u8cc7\u6599\u6216\u7bc4\u570d\u67e5\u8a62\u7684\u5834\u666f\u3002</li> <li>\u5728\u6301\u7e8c\u589e\u522a\u7bc0\u9ede\u7684\u904e\u7a0b\u4e2d\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u53ef\u80fd\u7522\u751f\u50be\u659c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u82e5\u4f7f\u7528 AVL \u6a39\u6216\u7d05\u9ed1\u6a39\uff0c\u5247\u5404\u9805\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a69\u5b9a\u57f7\u884c\uff0c\u4f46\u7dad\u8b77\u6a39\u5e73\u8861\u7684\u64cd\u4f5c\u6703\u589e\u52a0\u984d\u5916\u7684\u958b\u92b7\u3002</li> </ul>"},{"location":"chapter_searching/summary/","title":"10.6 \u00a0 \u5c0f\u7d50","text":"<ul> <li>\u4e8c\u5206\u641c\u5c0b\u4f9d\u8cf4\u8cc7\u6599\u7684\u6709\u5e8f\u6027\uff0c\u900f\u904e\u8ff4\u5708\u9010\u6b65\u7e2e\u6e1b\u4e00\u534a\u641c\u5c0b\u5340\u9593\u4f86\u9032\u884c\u67e5\u8a62\u3002\u5b83\u8981\u6c42\u8f38\u5165\u8cc7\u6599\u6709\u5e8f\uff0c\u4e14\u50c5\u9069\u7528\u65bc\u9663\u5217\u6216\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u66b4\u529b\u641c\u5c0b\u900f\u904e\u8d70\u8a2a\u8cc7\u6599\u7d50\u69cb\u4f86\u5b9a\u4f4d\u8cc7\u6599\u3002\u7dda\u6027\u641c\u5c0b\u9069\u7528\u65bc\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\uff0c\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\u548c\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u9069\u7528\u65bc\u5716\u548c\u6a39\u3002\u6b64\u985e\u6f14\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u7121\u9808\u5c0d\u8cc7\u6599\u9032\u884c\u9810\u8655\u7406\uff0c\u4f46\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\) \u8f03\u9ad8\u3002</li> <li>\u96dc\u6e4a\u67e5\u8a62\u3001\u6a39\u67e5\u8a62\u548c\u4e8c\u5206\u641c\u5c0b\u5c6c\u65bc\u9ad8\u6548\u641c\u5c0b\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u8cc7\u6599\u7d50\u69cb\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6a19\u5143\u7d20\u3002\u6b64\u985e\u6f14\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u85c9\u52a9\u984d\u5916\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u5be6\u969b\u4e2d\uff0c\u6211\u5011\u9700\u8981\u5c0d\u8cc7\u6599\u9ad4\u91cf\u3001\u641c\u5c0b\u6548\u80fd\u8981\u6c42\u3001\u8cc7\u6599\u67e5\u8a62\u548c\u66f4\u65b0\u983b\u7387\u7b49\u56e0\u7d20\u9032\u884c\u5177\u9ad4\u5206\u6790\uff0c\u5f9e\u800c\u9078\u64c7\u5408\u9069\u7684\u641c\u5c0b\u65b9\u6cd5\u3002</li> <li>\u7dda\u6027\u641c\u5c0b\u9069\u7528\u65bc\u5c0f\u578b\u6216\u983b\u7e41\u66f4\u65b0\u7684\u8cc7\u6599\uff1b\u4e8c\u5206\u641c\u5c0b\u9069\u7528\u65bc\u5927\u578b\u3001\u6392\u5e8f\u7684\u8cc7\u6599\uff1b\u96dc\u6e4a\u67e5\u8a62\u9069\u7528\u65bc\u5c0d\u67e5\u8a62\u6548\u7387\u8981\u6c42\u8f03\u9ad8\u4e14\u7121\u9808\u7bc4\u570d\u67e5\u8a62\u7684\u8cc7\u6599\uff1b\u6a39\u67e5\u8a62\u9069\u7528\u65bc\u9700\u8981\u7dad\u8b77\u9806\u5e8f\u548c\u652f\u6301\u7bc4\u570d\u67e5\u8a62\u7684\u5927\u578b\u52d5\u614b\u8cc7\u6599\u3002</li> <li>\u7528\u96dc\u6e4a\u67e5\u8a62\u66ff\u63db\u7dda\u6027\u67e5\u8a62\u662f\u4e00\u7a2e\u5e38\u7528\u7684\u6700\u4f73\u5316\u57f7\u884c\u6642\u9593\u7684\u7b56\u7565\uff0c\u53ef\u5c07\u6642\u9593\u8907\u96dc\u5ea6\u5f9e \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</li> </ul>"},{"location":"chapter_sorting/","title":"\u7b2c 11 \u7ae0 \u00a0 \u6392\u5e8f","text":"<p>Abstract</p> <p>\u6392\u5e8f\u7336\u5982\u4e00\u628a\u5c07\u6df7\u4e82\u8b8a\u70ba\u79e9\u5e8f\u7684\u9b54\u6cd5\u9470\u5319\uff0c\u4f7f\u6211\u5011\u80fd\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u7406\u89e3\u8207\u8655\u7406\u8cc7\u6599\u3002</p> <p>\u7121\u8ad6\u662f\u7c21\u55ae\u7684\u5347\u5e8f\uff0c\u9084\u662f\u8907\u96dc\u7684\u5206\u985e\u6392\u5217\uff0c\u6392\u5e8f\u90fd\u5411\u6211\u5011\u5c55\u793a\u4e86\u8cc7\u6599\u7684\u548c\u8ae7\u7f8e\u611f\u3002</p>"},{"location":"chapter_sorting/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>11.1 \u00a0 \u6392\u5e8f\u6f14\u7b97\u6cd5</li> <li>11.2 \u00a0 \u9078\u64c7\u6392\u5e8f</li> <li>11.3 \u00a0 \u6ce1\u6cab\u6392\u5e8f</li> <li>11.4 \u00a0 \u63d2\u5165\u6392\u5e8f</li> <li>11.5 \u00a0 \u5feb\u901f\u6392\u5e8f</li> <li>11.6 \u00a0 \u5408\u4f75\u6392\u5e8f</li> <li>11.7 \u00a0 \u5806\u7a4d\u6392\u5e8f</li> <li>11.8 \u00a0 \u6876\u6392\u5e8f</li> <li>11.9 \u00a0 \u8a08\u6578\u6392\u5e8f</li> <li>11.10 \u00a0 \u57fa\u6578\u6392\u5e8f</li> <li>11.11 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_sorting/bubble_sort/","title":"11.3 \u00a0 \u6ce1\u6cab\u6392\u5e8f","text":"<p>\u6ce1\u6cab\u6392\u5e8f\uff08bubble sort\uff09\u900f\u904e\u9023\u7e8c\u5730\u6bd4\u8f03\u8207\u4ea4\u63db\u76f8\u9130\u5143\u7d20\u5be6\u73fe\u6392\u5e8f\u3002\u9019\u500b\u904e\u7a0b\u5c31\u50cf\u6c23\u6ce1\u5f9e\u5e95\u90e8\u5347\u5230\u9802\u90e8\u4e00\u6a23\uff0c\u56e0\u6b64\u5f97\u540d\u6ce1\u6cab\u6392\u5e8f\u3002</p> <p>\u5982\u5716 11-4 \u6240\u793a\uff0c\u5192\u6ce1\u904e\u7a0b\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u63db\u64cd\u4f5c\u4f86\u6a21\u64ec\uff1a\u5f9e\u9663\u5217\u6700\u5de6\u7aef\u958b\u59cb\u5411\u53f3\u8d70\u8a2a\uff0c\u4f9d\u6b21\u6bd4\u8f03\u76f8\u9130\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 &gt; \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u63db\u4e8c\u8005\u3002\u8d70\u8a2a\u5b8c\u6210\u5f8c\uff0c\u6700\u5927\u7684\u5143\u7d20\u6703\u88ab\u79fb\u52d5\u5230\u9663\u5217\u7684\u6700\u53f3\u7aef\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-4 \u00a0 \u5229\u7528\u5143\u7d20\u4ea4\u63db\u64cd\u4f5c\u6a21\u64ec\u5192\u6ce1 </p>"},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8a2d\u9663\u5217\u7684\u9577\u5ea6\u70ba \\(n\\) \uff0c\u6ce1\u6cab\u6392\u5e8f\u7684\u6b65\u9a5f\u5982\u5716 11-5 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5c0d \\(n\\) \u500b\u5143\u7d20\u57f7\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c07\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u6b63\u78ba\u4f4d\u7f6e\u3002</li> <li>\u63a5\u4e0b\u4f86\uff0c\u5c0d\u5269\u9918 \\(n - 1\\) \u500b\u5143\u7d20\u57f7\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c07\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u6b63\u78ba\u4f4d\u7f6e\u3002</li> <li>\u4ee5\u6b64\u985e\u63a8\uff0c\u7d93\u904e \\(n - 1\\) \u8f2a\u201c\u5192\u6ce1\u201d\u5f8c\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u63db\u81f3\u6b63\u78ba\u4f4d\u7f6e\u3002</li> <li>\u50c5\u5269\u7684\u4e00\u500b\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u7121\u9808\u6392\u5e8f\uff0c\u56e0\u6b64\u9663\u5217\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> <p></p> <p> \u5716 11-5 \u00a0 \u6ce1\u6cab\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"\u6ce1\u6cab\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in range(n - 1, 0, -1):\n        # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                // \u9019\u88e1\u4f7f\u7528\u4e86 std::swap() \u51fd\u5f0f\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid BubbleSort(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums.swapAt(j, j + 1)\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfunction bubbleSort(nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums.swap(j, j + 1);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* \u6ce1\u6cab\u6392\u5e8f */\nfun bubbleSort(nums: IntArray) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### \u6ce1\u6cab\u6392\u5e8f ###\ndef bubble_sort(nums)\n  n = nums.length\n  # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for i in (n - 1).downto(1)\n    # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n      end\n    end\n  end\nend\n</code></pre> bubble_sort.zig<pre><code>// \u6ce1\u6cab\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2 \u00a0 \u6548\u7387\u6700\u4f73\u5316","text":"<p>\u6211\u5011\u767c\u73fe\uff0c\u5982\u679c\u67d0\u8f2a\u201c\u5192\u6ce1\u201d\u4e2d\u6c92\u6709\u57f7\u884c\u4efb\u4f55\u4ea4\u63db\u64cd\u4f5c\uff0c\u8aaa\u660e\u9663\u5217\u5df2\u7d93\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7d50\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u500b\u6a19\u8a8c\u4f4d <code>flag</code> \u4f86\u76e3\u6e2c\u9019\u7a2e\u60c5\u6cc1\uff0c\u4e00\u65e6\u51fa\u73fe\u5c31\u7acb\u5373\u8fd4\u56de\u3002</p> <p>\u7d93\u904e\u6700\u4f73\u5316\uff0c\u6ce1\u6cab\u6392\u5e8f\u7684\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u548c\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u4ecd\u70ba \\(O(n^2)\\) \uff1b\u4f46\u7576\u8f38\u5165\u9663\u5217\u5b8c\u5168\u6709\u5e8f\u6642\uff0c\u53ef\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"\u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09\"\"\"\n    n = len(nums)\n    # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in range(n - 1, 0, -1):\n        flag = False  # \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n        if not flag:\n            break  # \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n</code></pre> bubble_sort.cpp<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                // \u9019\u88e1\u4f7f\u7528\u4e86 std::swap() \u51fd\u5f0f\n                swap(nums[j], nums[j + 1]);\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nvoid BubbleSortWithFlag(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag) break;     // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if flag == false { // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        var flag = false // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums.swapAt(j, j + 1)\n                flag = true // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if !flag { // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n    // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n      }\n    }\n    if (!flag) break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09 */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                nums.swap(j, j + 1);\n                flag = true; // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if !flag {\n            break; // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n        };\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        bool flag = false;\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09 */\nfun bubbleSortWithFlag(nums: IntArray) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        var flag = false // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                flag = true // \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n            }\n        }\n        if (!flag) break // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09###\ndef bubble_sort_with_flag(nums)\n  n = nums.length\n  # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n  for i in (n - 1).downto(1)\n    flag = false # \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n\n    # \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n        flag = true # \u8a18\u9304\u4ea4\u63db\u5143\u7d20\n      end\n    end\n\n    break unless flag # \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n  end\nend\n</code></pre> bubble_sort.zig<pre><code>// \u6ce1\u6cab\u6392\u5e8f\uff08\u6a19\u8a8c\u6700\u4f73\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var flag = false;   // \u521d\u59cb\u5316\u6a19\u8a8c\u4f4d\n        var j: usize = 0;\n        // \u5167\u8ff4\u5708\uff1a\u5c07\u672a\u6392\u5e8f\u5340\u9593 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u63db\u81f3\u8a72\u5340\u9593\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u63db nums[j] \u8207 nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true;\n            }\n        }\n        if (!flag) break;   // \u6b64\u8f2a\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u63db\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\)\u3001\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5404\u8f2a\u201c\u5192\u6ce1\u201d\u8d70\u8a2a\u7684\u9663\u5217\u9577\u5ea6\u4f9d\u6b21\u70ba \\(n - 1\\)\u3001\\(n - 2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u7e3d\u548c\u70ba \\((n - 1) n / 2\\) \u3002\u5728\u5f15\u5165 <code>flag</code> \u6700\u4f73\u5316\u5f8c\uff0c\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054\u5230 \\(O(n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u7531\u65bc\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u63db\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/","title":"11.8 \u00a0 \u6876\u6392\u5e8f","text":"<p>\u524d\u8ff0\u5e7e\u7a2e\u6392\u5e8f\u6f14\u7b97\u6cd5\u90fd\u5c6c\u65bc\u201c\u57fa\u65bc\u6bd4\u8f03\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\u201d\uff0c\u5b83\u5011\u900f\u904e\u6bd4\u8f03\u5143\u7d20\u9593\u7684\u5927\u5c0f\u4f86\u5be6\u73fe\u6392\u5e8f\u3002\u6b64\u985e\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u7121\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5c07\u63a2\u8a0e\u5e7e\u7a2e\u201c\u975e\u6bd4\u8f03\u6392\u5e8f\u6f14\u7b97\u6cd5\u201d\uff0c\u5b83\u5011\u7684\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u9054\u5230\u7dda\u6027\u968e\u3002</p> <p>\u6876\u6392\u5e8f\uff08bucket sort\uff09\u662f\u5206\u6cbb\u7b56\u7565\u7684\u4e00\u500b\u5178\u578b\u61c9\u7528\u3002\u5b83\u900f\u904e\u8a2d\u5b9a\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u9806\u5e8f\u7684\u6876\uff0c\u6bcf\u500b\u6876\u5c0d\u61c9\u4e00\u500b\u6578\u64da\u7bc4\u570d\uff0c\u5c07\u8cc7\u6599\u5e73\u5747\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\uff1b\u7136\u5f8c\uff0c\u5728\u6bcf\u500b\u6876\u5167\u90e8\u5206\u5225\u57f7\u884c\u6392\u5e8f\uff1b\u6700\u7d42\u6309\u7167\u6876\u7684\u9806\u5e8f\u5c07\u6240\u6709\u8cc7\u6599\u5408\u4f75\u3002</p>"},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8003\u616e\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217\uff0c\u5176\u5143\u7d20\u662f\u7bc4\u570d \\([0, 1)\\) \u5167\u7684\u6d6e\u9ede\u6578\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u5716 11-13 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316 \\(k\\) \u500b\u6876\uff0c\u5c07 \\(n\\) \u500b\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u500b\u6876\u4e2d\u3002</li> <li>\u5c0d\u6bcf\u500b\u6876\u5206\u5225\u57f7\u884c\u6392\u5e8f\uff08\u9019\u88e1\u63a1\u7528\u7a0b\u5f0f\u8a9e\u8a00\u7684\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff09\u3002</li> <li>\u6309\u7167\u6876\u5f9e\u5c0f\u5230\u5927\u7684\u9806\u5e8f\u5408\u4f75\u7d50\u679c\u3002</li> </ol> <p></p> <p> \u5716 11-13 \u00a0 \u6876\u6392\u5e8f\u6f14\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"\u6876\u6392\u5e8f\"\"\"\n    # \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for num in nums:\n        # \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        i = int(num * k)\n        # \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].append(num)\n    # 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for bucket in buckets:\n        # \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort()\n    # 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        int i = num * k;\n        // \u5c07 num \u65b0\u589e\u9032\u6876 bucket_idx\n        buckets[i].push_back(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        int i = (int) (num * k);\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets.get(i).add(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (List&lt;Float&gt; bucket : buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        Collections.sort(bucket);\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* \u6876\u6392\u5e8f */\nvoid BucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    foreach (float num in nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        int i = (int)(num * k);\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].Add(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.Sort();\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for _, num := range nums {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        i := int(num * float64(k))\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for i := 0; i &lt; k; i++ {\n        // \u4f7f\u7528\u5167\u5efa\u5207\u7247\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        sort.Float64s(buckets[i])\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for num in nums {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        let i = Int(num * Double(k))\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].append(num)\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for i in buckets.indices {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        buckets[i].sort()\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n  for (double _num in nums) {\n    // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 _num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n    int i = (_num * k).toInt();\n    // \u5c07 _num \u65b0\u589e\u9032\u6876 bucket_idx\n    buckets[i].add(_num);\n  }\n  // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* \u6876\u6392\u5e8f */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for &amp;num in nums.iter() {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        let i = (num * k as f64) as usize;\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for bucket in &amp;mut buckets {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    let mut i = 0;\n    for bucket in buckets.iter() {\n        for &amp;num in bucket.iter() {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int n) {\n    int k = n / 2;                                 // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\n    int *sizes = malloc(k * sizeof(int));          // \u8a18\u9304\u6bcf\u500b\u6876\u7684\u5927\u5c0f\n    float **buckets = malloc(k * sizeof(float *)); // \u52d5\u614b\u9663\u5217\u7684\u9663\u5217\uff08\u6876\uff09\n    // \u70ba\u6bcf\u500b\u6876\u9810\u5206\u914d\u8db3\u5920\u7684\u7a7a\u9593\n    for (int i = 0; i &lt; k; ++i) {\n        buckets[i] = (float *)malloc(n * sizeof(float));\n        sizes[i] = 0;\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (int i = 0; i &lt; n; ++i) {\n        int idx = (int)(nums[i] * k);\n        buckets[idx][sizes[idx]++] = nums[i];\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (int i = 0; i &lt; k; ++i) {\n        qsort(buckets[i], sizes[i], sizeof(float), compare);\n    }\n    // 3. \u5408\u4f75\u6392\u5e8f\u5f8c\u7684\u6876\n    int idx = 0;\n    for (int i = 0; i &lt; k; ++i) {\n        for (int j = 0; j &lt; sizes[i]; ++j) {\n            nums[idx++] = buckets[i][j];\n        }\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        free(buckets[i]);\n    }\n}\n</code></pre> bucket_sort.kt<pre><code>/* \u6876\u6392\u5e8f */\nfun bucketSort(nums: FloatArray) {\n    // \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n    val k = nums.size / 2\n    val buckets = mutableListOf&lt;MutableList&lt;Float&gt;&gt;()\n    for (i in 0..&lt;k) {\n        buckets.add(mutableListOf())\n    }\n    // 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n    for (num in nums) {\n        // \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n        val i = (num * k).toInt()\n        // \u5c07 num \u65b0\u589e\u9032\u6876 i\n        buckets[i].add(num)\n    }\n    // 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n    for (bucket in buckets) {\n        // \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n        bucket.sort()\n    }\n    // 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n    var i = 0\n    for (bucket in buckets) {\n        for (num in bucket) {\n            nums[i++] = num\n        }\n    }\n}\n</code></pre> bucket_sort.rb<pre><code>### \u6876\u6392\u5e8f ###\ndef bucket_sort(nums)\n  # \u521d\u59cb\u5316 k = n/2 \u500b\u6876\uff0c\u9810\u671f\u5411\u6bcf\u500b\u6876\u5206\u914d 2 \u500b\u5143\u7d20\n  k = nums.length / 2\n  buckets = Array.new(k) { [] }\n\n  # 1. \u5c07\u9663\u5217\u5143\u7d20\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\n  nums.each do |num|\n    # \u8f38\u5165\u8cc7\u6599\u7bc4\u570d\u70ba [0, 1)\uff0c\u4f7f\u7528 num * k \u5c0d\u6620\u5230\u7d22\u5f15\u7bc4\u570d [0, k-1]\n    i = (num * k).to_i\n    # \u5c07 num \u65b0\u589e\u9032\u6876 i\n    buckets[i] &lt;&lt; num\n  end\n\n  # 2. \u5c0d\u5404\u500b\u6876\u57f7\u884c\u6392\u5e8f\n  buckets.each do |bucket|\n    # \u4f7f\u7528\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u66ff\u63db\u6210\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\n    bucket.sort!\n  end\n\n  # 3. \u8d70\u8a2a\u6876\u5408\u4f75\u7d50\u679c\n  i = 0\n  buckets.each do |bucket|\n    bucket.each do |num|\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> bucket_sort.zig<pre><code>[class]{}-[func]{bucketSort}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<p>\u6876\u6392\u5e8f\u9069\u7528\u65bc\u8655\u7406\u9ad4\u91cf\u5f88\u5927\u7684\u8cc7\u6599\u3002\u4f8b\u5982\uff0c\u8f38\u5165\u8cc7\u6599\u5305\u542b 100 \u842c\u500b\u5143\u7d20\uff0c\u7531\u65bc\u7a7a\u9593\u9650\u5236\uff0c\u7cfb\u7d71\u8a18\u61b6\u9ad4\u7121\u6cd5\u4e00\u6b21\u6027\u8f09\u5165\u6240\u6709\u8cc7\u6599\u3002\u6b64\u6642\uff0c\u53ef\u4ee5\u5c07\u8cc7\u6599\u5206\u6210 1000 \u500b\u6876\uff0c\u7136\u5f8c\u5206\u5225\u5c0d\u6bcf\u500b\u6876\u9032\u884c\u6392\u5e8f\uff0c\u6700\u5f8c\u5c07\u7d50\u679c\u5408\u4f75\u3002</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + k)\\) \uff1a\u5047\u8a2d\u5143\u7d20\u5728\u5404\u500b\u6876\u5167\u5e73\u5747\u5206\u4f48\uff0c\u90a3\u9ebc\u6bcf\u500b\u6876\u5167\u7684\u5143\u7d20\u6578\u91cf\u70ba \\(\\frac{n}{k}\\) \u3002\u5047\u8a2d\u6392\u5e8f\u55ae\u500b\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u6642\u9593\uff0c\u5247\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u6642\u9593\u3002\u7576\u6876\u6578\u91cf \\(k\\) \u6bd4\u8f03\u5927\u6642\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u5247\u8da8\u5411\u65bc \\(O(n)\\) \u3002\u5408\u4f75\u7d50\u679c\u6642\u9700\u8981\u8d70\u8a2a\u6240\u6709\u6876\u548c\u5143\u7d20\uff0c\u82b1\u8cbb \\(O(n + k)\\) \u6642\u9593\u3002\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u6240\u6709\u8cc7\u6599\u88ab\u5206\u914d\u5230\u4e00\u500b\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8a72\u6876\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + k)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9700\u8981\u85c9\u52a9 \\(k\\) \u500b\u6876\u548c\u7e3d\u5171 \\(n\\) \u500b\u5143\u7d20\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u6876\u6392\u5e8f\u662f\u5426\u7a69\u5b9a\u53d6\u6c7a\u65bc\u6392\u5e8f\u6876\u5167\u5143\u7d20\u7684\u6f14\u7b97\u6cd5\u662f\u5426\u7a69\u5b9a\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/#1183","title":"11.8.3 \u00a0 \u5982\u4f55\u5be6\u73fe\u5e73\u5747\u5206\u914d","text":"<p>\u6876\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u7406\u8ad6\u4e0a\u53ef\u4ee5\u9054\u5230 \\(O(n)\\) \uff0c\u95dc\u9375\u5728\u65bc\u5c07\u5143\u7d20\u5747\u52fb\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\uff0c\u56e0\u70ba\u5be6\u969b\u8cc7\u6599\u5f80\u5f80\u4e0d\u662f\u5747\u52fb\u5206\u4f48\u7684\u3002\u4f8b\u5982\uff0c\u6211\u5011\u60f3\u8981\u5c07\u6dd8\u5bf6\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u50f9\u683c\u7bc4\u570d\u5e73\u5747\u5206\u914d\u5230 10 \u500b\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u50f9\u683c\u5206\u4f48\u4e0d\u5747\uff0c\u4f4e\u65bc 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u65bc 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c07\u50f9\u683c\u5340\u9593\u5e73\u5747\u5283\u5206\u70ba 10 \u500b\uff0c\u5404\u500b\u6876\u4e2d\u7684\u5546\u54c1\u6578\u91cf\u5dee\u8ddd\u6703\u975e\u5e38\u5927\u3002</p> <p>\u70ba\u5be6\u73fe\u5e73\u5747\u5206\u914d\uff0c\u6211\u5011\u53ef\u4ee5\u5148\u8a2d\u5b9a\u4e00\u689d\u5927\u81f4\u7684\u5206\u754c\u7dda\uff0c\u5c07\u8cc7\u6599\u7c97\u7565\u5730\u5206\u5230 3 \u500b\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u7562\u5f8c\uff0c\u518d\u5c07\u5546\u54c1\u8f03\u591a\u7684\u6876\u7e7c\u7e8c\u5283\u5206\u70ba 3 \u500b\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6578\u91cf\u5927\u81f4\u76f8\u7b49\u3002</p> <p>\u5982\u5716 11-14 \u6240\u793a\uff0c\u9019\u7a2e\u65b9\u6cd5\u672c\u8cea\u4e0a\u662f\u5efa\u7acb\u4e00\u68f5\u905e\u8ff4\u6a39\uff0c\u76ee\u6a19\u662f\u8b93\u8449\u7bc0\u9ede\u7684\u503c\u5118\u53ef\u80fd\u5e73\u5747\u3002\u7576\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f2a\u5c07\u8cc7\u6599\u5283\u5206\u70ba 3 \u500b\u6876\uff0c\u5177\u9ad4\u5283\u5206\u65b9\u5f0f\u53ef\u6839\u64da\u8cc7\u6599\u7279\u9ede\u9748\u6d3b\u9078\u64c7\u3002</p> <p></p> <p> \u5716 11-14 \u00a0 \u905e\u8ff4\u5283\u5206\u6876 </p> <p>\u5982\u679c\u6211\u5011\u63d0\u524d\u77e5\u9053\u5546\u54c1\u50f9\u683c\u7684\u6a5f\u7387\u5206\u4f48\uff0c\u5247\u53ef\u4ee5\u6839\u64da\u8cc7\u6599\u6a5f\u7387\u5206\u4f48\u8a2d\u5b9a\u6bcf\u500b\u6876\u7684\u50f9\u683c\u5206\u754c\u7dda\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8cc7\u6599\u5206\u4f48\u4e26\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7d71\u8a08\uff0c\u4e5f\u53ef\u4ee5\u6839\u64da\u8cc7\u6599\u7279\u9ede\u63a1\u7528\u67d0\u7a2e\u6a5f\u7387\u6a21\u578b\u9032\u884c\u8fd1\u4f3c\u3002</p> <p>\u5982\u5716 11-15 \u6240\u793a\uff0c\u6211\u5011\u5047\u8a2d\u5546\u54c1\u50f9\u683c\u670d\u5f9e\u6b63\u614b\u5206\u4f48\uff0c\u9019\u6a23\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8a2d\u5b9a\u50f9\u683c\u5340\u9593\uff0c\u5f9e\u800c\u5c07\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u500b\u6876\u4e2d\u3002</p> <p></p> <p> \u5716 11-15 \u00a0 \u6839\u64da\u6a5f\u7387\u5206\u4f48\u5283\u5206\u6876 </p>"},{"location":"chapter_sorting/counting_sort/","title":"11.9 \u00a0 \u8a08\u6578\u6392\u5e8f","text":"<p>\u8a08\u6578\u6392\u5e8f\uff08counting sort\uff09\u900f\u904e\u7d71\u8a08\u5143\u7d20\u6578\u91cf\u4f86\u5be6\u73fe\u6392\u5e8f\uff0c\u901a\u5e38\u61c9\u7528\u65bc\u6574\u6578\u9663\u5217\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1 \u00a0 \u7c21\u55ae\u5be6\u73fe","text":"<p>\u5148\u4f86\u770b\u4e00\u500b\u7c21\u55ae\u7684\u4f8b\u5b50\u3002\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217 <code>nums</code> \uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8ca0\u6574\u6578\u201d\uff0c\u8a08\u6578\u6392\u5e8f\u7684\u6574\u9ad4\u6d41\u7a0b\u5982\u5716 11-16 \u6240\u793a\u3002</p> <ol> <li>\u8d70\u8a2a\u9663\u5217\uff0c\u627e\u51fa\u5176\u4e2d\u7684\u6700\u5927\u6578\u5b57\uff0c\u8a18\u70ba \\(m\\) \uff0c\u7136\u5f8c\u5efa\u7acb\u4e00\u500b\u9577\u5ea6\u70ba \\(m + 1\\) \u7684\u8f14\u52a9\u9663\u5217 <code>counter</code> \u3002</li> <li>\u85c9\u52a9 <code>counter</code> \u7d71\u8a08 <code>nums</code> \u4e2d\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\uff0c\u5176\u4e2d <code>counter[num]</code> \u5c0d\u61c9\u6578\u5b57 <code>num</code> \u7684\u51fa\u73fe\u6b21\u6578\u3002\u7d71\u8a08\u65b9\u6cd5\u5f88\u7c21\u55ae\uff0c\u53ea\u9700\u8d70\u8a2a <code>nums</code>\uff08\u8a2d\u7576\u524d\u6578\u5b57\u70ba <code>num</code>\uff09\uff0c\u6bcf\u8f2a\u5c07 <code>counter[num]</code> \u589e\u52a0 \\(1\\) \u5373\u53ef\u3002</li> <li>\u7531\u65bc <code>counter</code> \u7684\u5404\u500b\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u7576\u65bc\u6240\u6709\u6578\u5b57\u5df2\u7d93\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u8d70\u8a2a <code>counter</code> \uff0c\u6839\u64da\u5404\u6578\u5b57\u51fa\u73fe\u6b21\u6578\u5f9e\u5c0f\u5230\u5927\u7684\u9806\u5e8f\u586b\u5165 <code>nums</code> \u5373\u53ef\u3002</li> </ol> <p></p> <p> \u5716 11-16 \u00a0 \u8a08\u6578\u6392\u5e8f\u6d41\u7a0b </p> <p>\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"\u8a08\u6578\u6392\u5e8f\"\"\"\n    # \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\n    # 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    m = 0\n    for num in nums:\n        m = max(m, num)\n    # 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid countingSortNaive(int[] nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid CountingSortNaive(int[] nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfunc countingSortNaive(nums []int) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    var i = 0\n    for num in 0 ..&lt; m + 1 {\n        for _ in 0 ..&lt; counter[num] {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfunction countingSortNaive(nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfunction countingSortNaive(nums: number[]): void {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73fe\u6b21\u6578\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = *nums.iter().max().unwrap();\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nvoid countingSortNaive(int nums[], int size) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int *counter = calloc(m + 1, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\nfun countingSortNaive(nums: IntArray) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n    var i = 0\n    for (num in 0..&lt;m + 1) {\n        var j = 0\n        while (j &lt; counter[num]) {\n            nums[i] = num\n            j++\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### \u8a08\u6578\u6392\u5e8f ###\ndef counting_sort_naive(nums)\n  # \u7c21\u55ae\u5be6\u73fe\uff0c\u7121\u6cd5\u7528\u65bc\u6392\u5e8f\u7269\u4ef6\n  # 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n  m = 0\n  nums.each { |num| m = [m, num].max }\n  # 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. \u8d70\u8a2a counter \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u539f\u9663\u5217 nums\n  i = 0\n  for num in 0...(m + 1)\n    (0...counter[num]).each do\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSortNaive}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u8a08\u6578\u6392\u5e8f\u8207\u6876\u6392\u5e8f\u7684\u95dc\u806f</p> <p>\u5f9e\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u8a08\u6578\u6392\u5e8f\u4e2d\u7684\u8a08\u6578\u9663\u5217 <code>counter</code> \u7684\u6bcf\u500b\u7d22\u5f15\u8996\u70ba\u4e00\u500b\u6876\uff0c\u5c07\u7d71\u8a08\u6578\u91cf\u7684\u904e\u7a0b\u770b\u4f5c\u5c07\u5404\u500b\u5143\u7d20\u5206\u914d\u5230\u5c0d\u61c9\u7684\u6876\u4e2d\u3002\u672c\u8cea\u4e0a\uff0c\u8a08\u6578\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u8cc7\u6599\u4e0b\u7684\u4e00\u500b\u7279\u4f8b\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2 \u00a0 \u5b8c\u6574\u5be6\u73fe","text":"<p>\u7d30\u5fc3\u7684\u8b80\u8005\u53ef\u80fd\u767c\u73fe\u4e86\uff0c\u5982\u679c\u8f38\u5165\u8cc7\u6599\u662f\u7269\u4ef6\uff0c\u4e0a\u8ff0\u6b65\u9a5f <code>3.</code> \u5c31\u5931\u6548\u4e86\u3002\u5047\u8a2d\u8f38\u5165\u8cc7\u6599\u662f\u5546\u54c1\u7269\u4ef6\uff0c\u6211\u5011\u60f3\u6309\u7167\u5546\u54c1\u50f9\u683c\uff08\u985e\u5225\u7684\u6210\u54e1\u8b8a\u6578\uff09\u5c0d\u5546\u54c1\u9032\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u6f14\u7b97\u6cd5\u53ea\u80fd\u7d66\u51fa\u50f9\u683c\u7684\u6392\u5e8f\u7d50\u679c\u3002</p> <p>\u90a3\u9ebc\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u8cc7\u6599\u7684\u6392\u5e8f\u7d50\u679c\u5462\uff1f\u6211\u5011\u9996\u5148\u8a08\u7b97 <code>counter</code> \u7684\u201c\u524d\u7db4\u548c\u201d\u3002\u9867\u540d\u601d\u7fa9\uff0c\u7d22\u5f15 <code>i</code> \u8655\u7684\u524d\u7db4\u548c <code>prefix[i]</code> \u7b49\u65bc\u9663\u5217\u524d <code>i</code> \u500b\u5143\u7d20\u4e4b\u548c\uff1a</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>\u524d\u7db4\u548c\u5177\u6709\u660e\u78ba\u7684\u610f\u7fa9\uff0c<code>prefix[num] - 1</code> \u4ee3\u8868\u5143\u7d20 <code>num</code> \u5728\u7d50\u679c\u9663\u5217 <code>res</code> \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\u3002\u9019\u500b\u8cc7\u8a0a\u975e\u5e38\u95dc\u9375\uff0c\u56e0\u70ba\u5b83\u544a\u8a34\u6211\u5011\u5404\u500b\u5143\u7d20\u61c9\u8a72\u51fa\u73fe\u5728\u7d50\u679c\u9663\u5217\u7684\u54ea\u500b\u4f4d\u7f6e\u3002\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5012\u5e8f\u8d70\u8a2a\u539f\u9663\u5217 <code>nums</code> \u7684\u6bcf\u500b\u5143\u7d20 <code>num</code> \uff0c\u5728\u6bcf\u8f2a\u8fed\u4ee3\u4e2d\u57f7\u884c\u4ee5\u4e0b\u5169\u6b65\u3002</p> <ol> <li>\u5c07 <code>num</code> \u586b\u5165\u9663\u5217 <code>res</code> \u7684\u7d22\u5f15 <code>prefix[num] - 1</code> \u8655\u3002</li> <li>\u4ee4\u524d\u7db4\u548c <code>prefix[num]</code> \u6e1b\u5c0f \\(1\\) \uff0c\u5f9e\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e <code>num</code> \u7684\u7d22\u5f15\u3002</li> </ol> <p>\u8d70\u8a2a\u5b8c\u6210\u5f8c\uff0c\u9663\u5217 <code>res</code> \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7d50\u679c\uff0c\u6700\u5f8c\u4f7f\u7528 <code>res</code> \u8986\u84cb\u539f\u9663\u5217 <code>nums</code> \u5373\u53ef\u3002\u5716 11-17 \u5c55\u793a\u4e86\u5b8c\u6574\u7684\u8a08\u6578\u6392\u5e8f\u6d41\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-17 \u00a0 \u8a08\u6578\u6392\u5e8f\u6b65\u9a5f </p> <p>\u8a08\u6578\u6392\u5e8f\u7684\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"\u8a08\u6578\u6392\u5e8f\"\"\"\n    # \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\n    # 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    m = max(nums)\n    # 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    # \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num] -= 1  # \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    # \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--;              // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid CountingSort(int[] nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        res[counter[num]-1] = num\n        // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n        counter[num]--\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for i in 0 ..&lt; m {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let num = nums[i]\n        res[counter[num] - 1] = num // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num] -= 1 // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73fe\u6b21\u6578\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n  // \u5373 counter[_num]-1 \u662f _num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n  // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // \u5c07 _num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n    counter[_num]--; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e _num \u7684\u7d22\u5f15\n  }\n  // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    let m = *nums.iter().max().unwrap() as usize;\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    let mut counter = vec![0; m + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for i in 0..m {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num as usize] -= 1; // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    nums.copy_from_slice(&amp;res)\n}\n</code></pre> counting_sort.c<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]--;              // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    memcpy(nums, res, size * sizeof(int));\n    // 5. \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(res);\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* \u8a08\u6578\u6392\u5e8f */\n// \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\nfun countingSort(nums: IntArray) {\n    // 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n    for (i in 0..&lt;m) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u8d70\u8a2a nums \uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n    // \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n    val n = nums.size\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val num = nums[i]\n        res[counter[num] - 1] = num // \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n        counter[num]-- // \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n    for (i in 0..&lt;n) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### \u8a08\u6578\u6392\u5e8f ###\ndef counting_sort(nums)\n  # \u5b8c\u6574\u5be6\u73fe\uff0c\u53ef\u6392\u5e8f\u7269\u4ef6\uff0c\u4e26\u4e14\u662f\u7a69\u5b9a\u6392\u5e8f\n  # 1. \u7d71\u8a08\u9663\u5217\u6700\u5927\u5143\u7d20 m\n  m = nums.max\n  # 2. \u7d71\u8a08\u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73fe\u6b21\u6578\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. \u6c42 counter \u7684\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u6b21\u6578\u201d\u8f49\u63db\u70ba\u201c\u5c3e\u7d22\u5f15\u201d\n  # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u5f8c\u4e00\u6b21\u51fa\u73fe\u7684\u7d22\u5f15\n  (0...m).each { |i| counter[i + 1] += counter[i] }\n  # 4. \u5012\u5e8f\u8d70\u8a2a nums, \u5c07\u5404\u5143\u7d20\u586b\u5165\u7d50\u679c\u9663\u5217 res\n  # \u521d\u59cb\u5316\u9663\u5217 res \u7528\u65bc\u8a18\u9304\u7d50\u679c\n  n = nums.length\n  res = Array.new(n, 0)\n  (n - 1).downto(0).each do |i|\n    num = nums[i]\n    res[counter[num] - 1] = num # \u5c07 num \u653e\u7f6e\u5230\u5c0d\u61c9\u7d22\u5f15\u8655\n    counter[num] -= 1 # \u4ee4\u524d\u7db4\u548c\u81ea\u6e1b 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n  end\n  # \u4f7f\u7528\u7d50\u679c\u9663\u5217 res \u8986\u84cb\u539f\u9663\u5217 nums\n  (0...n).each { |i| nums[i] = res[i] }\nend\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSort}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + m)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f \uff1a\u6d89\u53ca\u8d70\u8a2a <code>nums</code> \u548c\u8d70\u8a2a <code>counter</code> \uff0c\u90fd\u4f7f\u7528\u7dda\u6027\u6642\u9593\u3002\u4e00\u822c\u60c5\u6cc1\u4e0b \\(n \\gg m\\) \uff0c\u6642\u9593\u8907\u96dc\u5ea6\u8da8\u65bc \\(O(n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + m)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u85c9\u52a9\u4e86\u9577\u5ea6\u5206\u5225\u70ba \\(n\\) \u548c \\(m\\) \u7684\u9663\u5217 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u7531\u65bc\u5411 <code>res</code> \u4e2d\u586b\u5145\u5143\u7d20\u7684\u9806\u5e8f\u662f\u201c\u5f9e\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u8d70\u8a2a <code>nums</code> \u53ef\u4ee5\u907f\u514d\u6539\u8b8a\u76f8\u7b49\u5143\u7d20\u4e4b\u9593\u7684\u76f8\u5c0d\u4f4d\u7f6e\uff0c\u5f9e\u800c\u5be6\u73fe\u7a69\u5b9a\u6392\u5e8f\u3002\u5be6\u969b\u4e0a\uff0c\u6b63\u5e8f\u8d70\u8a2a <code>nums</code> \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u78ba\u7684\u6392\u5e8f\u7d50\u679c\uff0c\u4f46\u7d50\u679c\u662f\u975e\u7a69\u5b9a\u7684\u3002</li> </ul>"},{"location":"chapter_sorting/counting_sort/#1194","title":"11.9.4 \u00a0 \u4fb7\u9650\u6027","text":"<p>\u770b\u5230\u9019\u88e1\uff0c\u4f60\u4e5f\u8a31\u6703\u89ba\u5f97\u8a08\u6578\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u50c5\u900f\u904e\u7d71\u8a08\u6578\u91cf\u5c31\u53ef\u4ee5\u5be6\u73fe\u9ad8\u6548\u7684\u6392\u5e8f\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8a08\u6578\u6392\u5e8f\u7684\u524d\u7f6e\u689d\u4ef6\u76f8\u5c0d\u8f03\u70ba\u56b4\u683c\u3002</p> <p>\u8a08\u6578\u6392\u5e8f\u53ea\u9069\u7528\u65bc\u975e\u8ca0\u6574\u6578\u3002\u82e5\u60f3\u5c07\u5176\u7528\u65bc\u5176\u4ed6\u578b\u5225\u7684\u8cc7\u6599\uff0c\u9700\u8981\u78ba\u4fdd\u9019\u4e9b\u8cc7\u6599\u53ef\u4ee5\u8f49\u63db\u70ba\u975e\u8ca0\u6574\u6578\uff0c\u4e26\u4e14\u5728\u8f49\u63db\u904e\u7a0b\u4e2d\u4e0d\u80fd\u6539\u8b8a\u5404\u500b\u5143\u7d20\u4e4b\u9593\u7684\u76f8\u5c0d\u5927\u5c0f\u95dc\u4fc2\u3002\u4f8b\u5982\uff0c\u5c0d\u65bc\u5305\u542b\u8ca0\u6578\u7684\u6574\u6578\u9663\u5217\uff0c\u53ef\u4ee5\u5148\u7d66\u6240\u6709\u6578\u5b57\u52a0\u4e0a\u4e00\u500b\u5e38\u6578\uff0c\u5c07\u5168\u90e8\u6578\u5b57\u8f49\u5316\u70ba\u6b63\u6578\uff0c\u6392\u5e8f\u5b8c\u6210\u5f8c\u518d\u8f49\u63db\u56de\u53bb\u3002</p> <p>\u8a08\u6578\u6392\u5e8f\u9069\u7528\u65bc\u8cc7\u6599\u91cf\u5927\u4f46\u8cc7\u6599\u7bc4\u570d\u8f03\u5c0f\u7684\u60c5\u6cc1\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5247\u6703\u4f54\u7528\u904e\u591a\u7a7a\u9593\u3002\u800c\u7576 \\(n \\ll m\\) \u6642\uff0c\u8a08\u6578\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u6642\u9593\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\u9084\u8981\u6162\u3002</p>"},{"location":"chapter_sorting/heap_sort/","title":"11.7 \u00a0 \u5806\u7a4d\u6392\u5e8f","text":"<p>Tip</p> <p>\u95b1\u8b80\u672c\u7bc0\u524d\uff0c\u8acb\u78ba\u4fdd\u5df2\u5b78\u5b8c\u201c\u5806\u7a4d\u201c\u7ae0\u7bc0\u3002</p> <p>\u5806\u7a4d\u6392\u5e8f\uff08heap sort\uff09\u662f\u4e00\u7a2e\u57fa\u65bc\u5806\u7a4d\u8cc7\u6599\u7d50\u69cb\u5be6\u73fe\u7684\u9ad8\u6548\u6392\u5e8f\u6f14\u7b97\u6cd5\u3002\u6211\u5011\u53ef\u4ee5\u5229\u7528\u5df2\u7d93\u5b78\u904e\u7684\u201c\u5efa\u5806\u7a4d\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u7a4d\u64cd\u4f5c\u201d\u5be6\u73fe\u5806\u7a4d\u6392\u5e8f\u3002</p> <ol> <li>\u8f38\u5165\u9663\u5217\u4e26\u5efa\u7acb\u5c0f\u9802\u5806\u7a4d\uff0c\u6b64\u6642\u6700\u5c0f\u5143\u7d20\u4f4d\u65bc\u5806\u7a4d\u9802\u3002</li> <li>\u4e0d\u65b7\u57f7\u884c\u51fa\u5806\u7a4d\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8a18\u9304\u51fa\u5806\u7a4d\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u5f9e\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002</li> </ol> <p>\u4ee5\u4e0a\u65b9\u6cd5\u96d6\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u85c9\u52a9\u4e00\u500b\u984d\u5916\u9663\u5217\u4f86\u5132\u5b58\u5f48\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f03\u6d6a\u8cbb\u7a7a\u9593\u3002\u5728\u5be6\u969b\u4e2d\uff0c\u6211\u5011\u901a\u5e38\u4f7f\u7528\u4e00\u7a2e\u66f4\u52a0\u512a\u96c5\u7684\u5be6\u73fe\u65b9\u5f0f\u3002</p>"},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8a2d\u9663\u5217\u7684\u9577\u5ea6\u70ba \\(n\\) \uff0c\u5806\u7a4d\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u5716 11-12 \u6240\u793a\u3002</p> <ol> <li>\u8f38\u5165\u9663\u5217\u4e26\u5efa\u7acb\u5927\u9802\u5806\u7a4d\u3002\u5b8c\u6210\u5f8c\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u65bc\u5806\u7a4d\u9802\u3002</li> <li>\u5c07\u5806\u7a4d\u9802\u5143\u7d20\uff08\u7b2c\u4e00\u500b\u5143\u7d20\uff09\u8207\u5806\u7a4d\u5e95\u5143\u7d20\uff08\u6700\u5f8c\u4e00\u500b\u5143\u7d20\uff09\u4ea4\u63db\u3002\u5b8c\u6210\u4ea4\u63db\u5f8c\uff0c\u5806\u7a4d\u7684\u9577\u5ea6\u6e1b \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6578\u91cf\u52a0 \\(1\\) \u3002</li> <li>\u5f9e\u5806\u7a4d\u9802\u5143\u7d20\u958b\u59cb\uff0c\u5f9e\u9802\u5230\u5e95\u57f7\u884c\u5806\u7a4d\u5316\u64cd\u4f5c\uff08sift down\uff09\u3002\u5b8c\u6210\u5806\u7a4d\u5316\u5f8c\uff0c\u5806\u7a4d\u7684\u6027\u8cea\u5f97\u5230\u4fee\u5fa9\u3002</li> <li>\u8ff4\u5708\u57f7\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\u3002\u8ff4\u5708 \\(n - 1\\) \u8f2a\u5f8c\uff0c\u5373\u53ef\u5b8c\u6210\u9663\u5217\u6392\u5e8f\u3002</li> </ol> <p>Tip</p> <p>\u5be6\u969b\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u7a4d\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u500b\u5f48\u51fa\u5143\u7d20\u7684\u6b65\u9a5f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-12 \u00a0 \u5806\u7a4d\u6392\u5e8f\u6b65\u9a5f </p> <p>\u5728\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e2d\uff0c\u6211\u5011\u4f7f\u7528\u4e86\u8207\u201c\u5806\u7a4d\u201d\u7ae0\u7bc0\u76f8\u540c\u7684\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 <code>sift_down()</code> \u51fd\u5f0f\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u65bc\u5806\u7a4d\u7684\u9577\u5ea6\u6703\u96a8\u8457\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u6e1b\u5c0f\uff0c\u56e0\u6b64\u6211\u5011\u9700\u8981\u7d66 <code>sift_down()</code> \u51fd\u5f0f\u65b0\u589e\u4e00\u500b\u9577\u5ea6\u53c3\u6578 \\(n\\) \uff0c\u7528\u65bc\u6307\u5b9a\u5806\u7a4d\u7684\u7576\u524d\u6709\u6548\u9577\u5ea6\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"\u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316\"\"\"\n    while True:\n        # \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u63db\u5169\u7bc0\u9ede\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"\u5806\u7a4d\u6392\u5e8f\"\"\"\n    # \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for i in range(len(nums) - 1, 0, -1):\n        # \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        nums[0], nums[i] = nums[i], nums[0]\n        # \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        swap(nums[i], nums[ma]);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        swap(nums[0], nums[i]);\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid HeapSort(int[] nums) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        nums.swapAt(i, ma)\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for i in nums.indices.dropFirst().reversed() {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        nums.swapAt(0, i)\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfunction heapSort(nums) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u63db\u5169\u7bc0\u9ede\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n    i = ma;\n  }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid heapSort(List&lt;int&gt; nums) {\n  // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        nums.swap(i, ma);\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for i in (0..nums.len() / 2).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for i in (1..nums.len()).rev() {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        nums.swap(0, i);\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (int i = n - 1; i &gt; 0; --i) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.kt<pre><code>/* \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 */\nfun siftDown(nums: IntArray, n: Int, li: Int) {\n    var i = li\n    while (true) {\n        // \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n        val l = 2 * i + 1\n        val r = 2 * i + 2\n        var ma = i\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) \n            ma = l\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) \n            ma = r\n        // \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n        if (ma == i) \n            break\n        // \u4ea4\u63db\u5169\u7bc0\u9ede\n        val temp = nums[i]\n        nums[i] = nums[ma]\n        nums[ma] = temp\n        // \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u7a4d\u6392\u5e8f */\nfun heapSort(nums: IntArray) {\n    // \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n    for (i in nums.size / 2 - 1 downTo 0) {\n        siftDown(nums, nums.size, i)\n    }\n    // \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n    for (i in nums.size - 1 downTo 1) {\n        // \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n        val temp = nums[0]\n        nums[0] = nums[i]\n        nums[i] = temp\n        // \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.rb<pre><code>### \u5806\u7a4d\u7684\u9577\u5ea6\u70ba n \uff0c\u5f9e\u7bc0\u9ede i \u958b\u59cb\uff0c\u5f9e\u9802\u81f3\u5e95\u5806\u7a4d\u5316 ###\ndef sift_down(nums, n, i)\n  while true\n    # \u5224\u65b7\u7bc0\u9ede i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7bc0\u9ede\uff0c\u8a18\u70ba ma\n    l = 2 * i + 1\n    r = 2 * i + 2\n    ma = i\n    ma = l if l &lt; n &amp;&amp; nums[l] &gt; nums[ma]\n    ma = r if r &lt; n &amp;&amp; nums[r] &gt; nums[ma]\n    # \u82e5\u7bc0\u9ede i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5247\u7121\u9808\u7e7c\u7e8c\u5806\u7a4d\u5316\uff0c\u8df3\u51fa\n    break if ma == i\n    # \u4ea4\u63db\u5169\u7bc0\u9ede\n    nums[i], nums[ma] = nums[ma], nums[i]\n    # \u8ff4\u5708\u5411\u4e0b\u5806\u7a4d\u5316\n    i = ma\n  end\nend\n\n### \u5806\u7a4d\u6392\u5e8f ###\ndef heap_sort(nums)\n  # \u5efa\u5806\u7a4d\u64cd\u4f5c\uff1a\u5806\u7a4d\u5316\u9664\u8449\u7bc0\u9ede\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\n  (nums.length / 2 - 1).downto(0) do |i|\n    sift_down(nums, nums.length, i)\n  end\n  # \u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u8ff4\u5708 n-1 \u8f2a\n  (nums.length - 1).downto(1) do |i|\n    # \u4ea4\u63db\u6839\u7bc0\u9ede\u8207\u6700\u53f3\u8449\u7bc0\u9ede\uff08\u4ea4\u63db\u9996\u5143\u7d20\u8207\u5c3e\u5143\u7d20\uff09\n    nums[0], nums[i] = nums[i], nums[0]\n    # \u4ee5\u6839\u7bc0\u9ede\u70ba\u8d77\u9ede\uff0c\u5f9e\u9802\u81f3\u5e95\u9032\u884c\u5806\u7a4d\u5316\n    sift_down(nums, i, 0)\n  end\nend\n</code></pre> heap_sort.zig<pre><code>[class]{}-[func]{siftDown}\n\n[class]{}-[func]{heapSort}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5efa\u5806\u7a4d\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002\u5f9e\u5806\u7a4d\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(\\log n)\\) \uff0c\u5171\u8ff4\u5708 \\(n - 1\\) \u8f2a\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u5e7e\u500b\u6307\u6a19\u8b8a\u6578\u4f7f\u7528 \\(O(1)\\) \u7a7a\u9593\u3002\u5143\u7d20\u4ea4\u63db\u548c\u5806\u7a4d\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u9663\u5217\u4e0a\u9032\u884c\u7684\u3002</li> <li>\u975e\u7a69\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u63db\u5806\u7a4d\u9802\u5143\u7d20\u548c\u5806\u7a4d\u5e95\u5143\u7d20\u6642\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5c0d\u4f4d\u7f6e\u53ef\u80fd\u767c\u751f\u8b8a\u5316\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/","title":"11.4 \u00a0 \u63d2\u5165\u6392\u5e8f","text":"<p>\u63d2\u5165\u6392\u5e8f\uff08insertion sort\uff09\u662f\u4e00\u7a2e\u7c21\u55ae\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u8207\u624b\u52d5\u6574\u7406\u4e00\u526f\u724c\u7684\u904e\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002</p> <p>\u5177\u9ad4\u4f86\u8aaa\uff0c\u6211\u5011\u5728\u672a\u6392\u5e8f\u5340\u9593\u9078\u64c7\u4e00\u500b\u57fa\u6e96\u5143\u7d20\uff0c\u5c07\u8a72\u5143\u7d20\u8207\u5176\u5de6\u5074\u5df2\u6392\u5e8f\u5340\u9593\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f03\u5927\u5c0f\uff0c\u4e26\u5c07\u8a72\u5143\u7d20\u63d2\u5165\u5230\u6b63\u78ba\u7684\u4f4d\u7f6e\u3002</p> <p>\u5716 11-6 \u5c55\u793a\u4e86\u9663\u5217\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u6d41\u7a0b\u3002\u8a2d\u57fa\u6e96\u5143\u7d20\u70ba <code>base</code> \uff0c\u6211\u5011\u9700\u8981\u5c07\u5f9e\u76ee\u6a19\u7d22\u5f15\u5230 <code>base</code> \u4e4b\u9593\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\uff0c\u7136\u5f8c\u5c07 <code>base</code> \u8ce6\u503c\u7d66\u76ee\u6a19\u7d22\u5f15\u3002</p> <p></p> <p> \u5716 11-6 \u00a0 \u55ae\u6b21\u63d2\u5165\u64cd\u4f5c </p>"},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u6574\u9ad4\u6d41\u7a0b\u5982\u5716 11-7 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72c0\u614b\u4e0b\uff0c\u9663\u5217\u7684\u7b2c 1 \u500b\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u9078\u53d6\u9663\u5217\u7684\u7b2c 2 \u500b\u5143\u7d20\u4f5c\u70ba <code>base</code> \uff0c\u5c07\u5176\u63d2\u5165\u5230\u6b63\u78ba\u4f4d\u7f6e\u5f8c\uff0c\u9663\u5217\u7684\u524d 2 \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9078\u53d6\u7b2c 3 \u500b\u5143\u7d20\u4f5c\u70ba <code>base</code> \uff0c\u5c07\u5176\u63d2\u5165\u5230\u6b63\u78ba\u4f4d\u7f6e\u5f8c\uff0c\u9663\u5217\u7684\u524d 3 \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u985e\u63a8\uff0c\u5728\u6700\u5f8c\u4e00\u8f2a\u4e2d\uff0c\u9078\u53d6\u6700\u5f8c\u4e00\u500b\u5143\u7d20\u4f5c\u70ba <code>base</code> \uff0c\u5c07\u5176\u63d2\u5165\u5230\u6b63\u78ba\u4f4d\u7f6e\u5f8c\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u5716 11-7 \u00a0 \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n    # \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j -= 1\n        nums[j + 1] = base  # \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n</code></pre> insertion_sort.cpp<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base;        // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid InsertionSort(int[] nums) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = bas;         // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--\n        }\n        nums[j+1] = base // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for i in nums.indices.dropFirst() {\n        let base = nums[i]\n        var j = i - 1\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j -= 1\n        }\n        nums[j + 1] = base // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n      j--;\n    }\n    nums[j + 1] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i], (i - 1) as i32);\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base; // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.kt<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfun insertionSort(nums: IntArray) {\n    //\u5916\u8ff4\u5708: \u5df2\u6392\u5e8f\u5143\u7d20\u70ba 1, 2, ..., n\n    for (i in nums.indices) {\n        val base = nums[i]\n        var j = i - 1\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j] // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n            j--\n        }\n        nums[j + 1] = base        // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.rb<pre><code>### \u63d2\u5165\u6392\u5e8f ###\ndef insertion_sort(nums)\n  n = nums.length\n  # \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n  for i in 1...n\n    base = nums[i]\n    j = i - 1\n    # \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n    while j &gt;= 0 &amp;&amp; nums[j] &gt; base\n      nums[j + 1] = nums[j] # \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n      j -= 1\n    end\n    nums[j + 1] = base # \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n  end\nend\n</code></pre> insertion_sort.zig<pre><code>// \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n    // \u5916\u8ff4\u5708\uff1a\u5df2\u6392\u5e8f\u5340\u9593\u70ba [0, i-1]\n    var i: usize = 1;\n    while (i &lt; nums.len) : (i += 1) {\n        var base = nums[i];\n        var j: usize = i;\n        // \u5167\u8ff4\u5708\uff1a\u5c07 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593 [0, i-1] \u4e2d\u7684\u6b63\u78ba\u4f4d\u7f6e\n        while (j &gt;= 1 and nums[j - 1] &gt; base) : (j -= 1) {\n            nums[j] = nums[j - 1];  // \u5c07 nums[j] \u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n        }\n        nums[j] = base;             // \u5c07 base \u8ce6\u503c\u5230\u6b63\u78ba\u4f4d\u7f6e\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\)\u3001\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u5225\u9700\u8981\u8ff4\u5708 \\(n - 1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\((n - 1) n / 2\\) \uff0c\u56e0\u6b64\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u8cc7\u6599\u6642\uff0c\u63d2\u5165\u64cd\u4f5c\u6703\u63d0\u524d\u7d42\u6b62\u3002\u7576\u8f38\u5165\u9663\u5217\u5b8c\u5168\u6709\u5e8f\u6642\uff0c\u63d2\u5165\u6392\u5e8f\u9054\u5230\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6 \\(O(n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u904e\u7a0b\u4e2d\uff0c\u6211\u5011\u6703\u5c07\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u5074\uff0c\u4e0d\u6703\u6539\u8b8a\u5b83\u5011\u7684\u9806\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/#1143","title":"11.4.3 \u00a0 \u63d2\u5165\u6392\u5e8f\u7684\u512a\u52e2","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u800c\u6211\u5011\u5373\u5c07\u5b78\u7fd2\u7684\u5feb\u901f\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002\u5118\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u66f4\u9ad8\uff0c\u4f46\u5728\u8cc7\u6599\u91cf\u8f03\u5c0f\u7684\u60c5\u6cc1\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002</p> <p>\u9019\u500b\u7d50\u8ad6\u8207\u7dda\u6027\u67e5\u8a62\u548c\u4e8c\u5206\u641c\u5c0b\u7684\u9069\u7528\u60c5\u6cc1\u7684\u7d50\u8ad6\u985e\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u9019\u985e \\(O(n \\log n)\\) \u7684\u6f14\u7b97\u6cd5\u5c6c\u65bc\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u55ae\u5143\u8a08\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u8cc7\u6599\u91cf\u8f03\u5c0f\u6642\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6578\u503c\u6bd4\u8f03\u63a5\u8fd1\uff0c\u8907\u96dc\u5ea6\u4e0d\u4f54\u4e3b\u5c0e\u5730\u4f4d\uff0c\u6bcf\u8f2a\u4e2d\u7684\u55ae\u5143\u64cd\u4f5c\u6578\u91cf\u8d77\u5230\u6c7a\u5b9a\u6027\u4f5c\u7528\u3002</p> <p>\u5be6\u969b\u4e0a\uff0c\u8a31\u591a\u7a0b\u5f0f\u8a9e\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5167\u5efa\u6392\u5e8f\u51fd\u5f0f\u63a1\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u70ba\uff1a\u5c0d\u65bc\u9577\u9663\u5217\uff0c\u63a1\u7528\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5c0d\u65bc\u77ed\u9663\u5217\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002</p> <p>\u96d6\u7136\u6ce1\u6cab\u6392\u5e8f\u3001\u9078\u64c7\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n^2)\\) \uff0c\u4f46\u5728\u5be6\u969b\u60c5\u6cc1\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u983b\u7387\u986f\u8457\u9ad8\u65bc\u6ce1\u6cab\u6392\u5e8f\u548c\u9078\u64c7\u6392\u5e8f\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u6ce1\u6cab\u6392\u5e8f\u57fa\u65bc\u5143\u7d20\u4ea4\u63db\u5be6\u73fe\uff0c\u9700\u8981\u85c9\u52a9\u4e00\u500b\u81e8\u6642\u8b8a\u6578\uff0c\u5171\u6d89\u53ca 3 \u500b\u55ae\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u65bc\u5143\u7d20\u8ce6\u503c\u5be6\u73fe\uff0c\u50c5\u9700 1 \u500b\u55ae\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u6ce1\u6cab\u6392\u5e8f\u7684\u8a08\u7b97\u958b\u92b7\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002</li> <li>\u9078\u64c7\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u6cc1\u4e0b\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(n^2)\\) \u3002\u5982\u679c\u7d66\u5b9a\u4e00\u7d44\u90e8\u5206\u6709\u5e8f\u7684\u8cc7\u6599\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9078\u64c7\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u9078\u64c7\u6392\u5e8f\u4e0d\u7a69\u5b9a\uff0c\u7121\u6cd5\u61c9\u7528\u65bc\u591a\u7d1a\u6392\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/","title":"11.6 \u00a0 \u5408\u4f75\u6392\u5e8f","text":"<p>\u5408\u4f75\u6392\u5e8f\uff08merge sort\uff09\u662f\u4e00\u7a2e\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u5305\u542b\u5716 11-10 \u6240\u793a\u7684\u201c\u5283\u5206\u201d\u548c\u201c\u5408\u4f75\u201d\u968e\u6bb5\u3002</p> <ol> <li>\u5283\u5206\u968e\u6bb5\uff1a\u900f\u904e\u905e\u8ff4\u4e0d\u65b7\u5730\u5c07\u9663\u5217\u5f9e\u4e2d\u9ede\u8655\u5206\u958b\uff0c\u5c07\u9577\u9663\u5217\u7684\u6392\u5e8f\u554f\u984c\u8f49\u63db\u70ba\u77ed\u9663\u5217\u7684\u6392\u5e8f\u554f\u984c\u3002</li> <li>\u5408\u4f75\u968e\u6bb5\uff1a\u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u5283\u5206\uff0c\u958b\u59cb\u5408\u4f75\uff0c\u6301\u7e8c\u5730\u5c07\u5de6\u53f3\u5169\u500b\u8f03\u77ed\u7684\u6709\u5e8f\u9663\u5217\u5408\u4f75\u70ba\u4e00\u500b\u8f03\u9577\u7684\u6709\u5e8f\u9663\u5217\uff0c\u76f4\u81f3\u7d50\u675f\u3002</li> </ol> <p></p> <p> \u5716 11-10 \u00a0 \u5408\u4f75\u6392\u5e8f\u7684\u5283\u5206\u8207\u5408\u4f75\u968e\u6bb5 </p>"},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5982\u5716 11-11 \u6240\u793a\uff0c\u201c\u5283\u5206\u968e\u6bb5\u201d\u5f9e\u9802\u81f3\u5e95\u905e\u8ff4\u5730\u5c07\u9663\u5217\u5f9e\u4e2d\u9ede\u5207\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\u3002</p> <ol> <li>\u8a08\u7b97\u9663\u5217\u4e2d\u9ede <code>mid</code> \uff0c\u905e\u8ff4\u5283\u5206\u5de6\u5b50\u9663\u5217\uff08\u5340\u9593 <code>[left, mid]</code> \uff09\u548c\u53f3\u5b50\u9663\u5217\uff08\u5340\u9593 <code>[mid + 1, right]</code> \uff09\u3002</li> <li>\u905e\u8ff4\u57f7\u884c\u6b65\u9a5f <code>1.</code> \uff0c\u76f4\u81f3\u5b50\u9663\u5217\u5340\u9593\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u3002</li> </ol> <p>\u201c\u5408\u4f75\u968e\u6bb5\u201d\u5f9e\u5e95\u81f3\u9802\u5730\u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u5408\u4f75\u70ba\u4e00\u500b\u6709\u5e8f\u9663\u5217\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5f9e\u9577\u5ea6\u70ba 1 \u7684\u5b50\u9663\u5217\u958b\u59cb\u5408\u4f75\uff0c\u5408\u4f75\u968e\u6bb5\u4e2d\u7684\u6bcf\u500b\u5b50\u9663\u5217\u90fd\u662f\u6709\u5e8f\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-11 \u00a0 \u5408\u4f75\u6392\u5e8f\u6b65\u9a5f </p> <p>\u89c0\u5bdf\u767c\u73fe\uff0c\u5408\u4f75\u6392\u5e8f\u8207\u4e8c\u5143\u6a39\u5f8c\u5e8f\u8d70\u8a2a\u7684\u905e\u8ff4\u9806\u5e8f\u662f\u4e00\u81f4\u7684\u3002</p> <ul> <li>\u5f8c\u5e8f\u8d70\u8a2a\uff1a\u5148\u905e\u8ff4\u5de6\u5b50\u6a39\uff0c\u518d\u905e\u8ff4\u53f3\u5b50\u6a39\uff0c\u6700\u5f8c\u8655\u7406\u6839\u7bc0\u9ede\u3002</li> <li>\u5408\u4f75\u6392\u5e8f\uff1a\u5148\u905e\u8ff4\u5de6\u5b50\u9663\u5217\uff0c\u518d\u905e\u8ff4\u53f3\u5b50\u9663\u5217\uff0c\u6700\u5f8c\u8655\u7406\u5408\u4f75\u3002</li> </ul> <p>\u5408\u4f75\u6392\u5e8f\u7684\u5be6\u73fe\u5982\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u6240\u793a\u3002\u8acb\u6ce8\u610f\uff0c<code>nums</code> \u7684\u5f85\u5408\u4f75\u5340\u9593\u70ba <code>[left, right]</code> \uff0c\u800c <code>tmp</code> \u7684\u5c0d\u61c9\u5340\u9593\u70ba <code>[0, right - left]</code> \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"\u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\"\"\"\n    # \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    # \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    tmp = [0] * (right - left + 1)\n    # \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k = left, mid + 1, 0\n    # \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"\u5408\u4f75\u6392\u5e8f\"\"\"\n    # \u7d42\u6b62\u689d\u4ef6\n    if left &gt;= right:\n        return  # \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    # \u5283\u5206\u968e\u6bb5\n    mid = (left + right) // 2 # \u8a08\u7b97\u4e2d\u9ede\n    merge_sort(nums, left, mid)  # \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    merge_sort(nums, mid + 1, right)  # \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    # \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    vector&lt;int&gt; tmp(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right)\n        return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    int mid = left + (right - left) / 2;    // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid);      // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right)\n        return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    int mid = left + (right - left) / 2; // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid MergeSort(int[] nums, int left, int right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return;       // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    int mid = left + (right - left) / 2;    // \u8a08\u7b97\u4e2d\u9ede\n    MergeSort(nums, left, mid);      // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    MergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfunc merge(nums []int, left, mid, right int) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    tmp := make([]int, right-left+1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k := left, mid+1, 0\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if left &gt;= right {\n        return\n    }\n    // \u5283\u5206\u968e\u6bb5\n    mid := left + (right - left) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left, j = mid + 1, k = 0\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n        }\n        k += 1\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if left &gt;= right { // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n        return\n    }\n    // \u5283\u5206\u968e\u6bb5\n    let mid = left + (right - left) / 2 // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums: &amp;nums, left: left, right: mid) // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfunction merge(nums, left, mid, right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    let mid = Math.floor(left + (right - left) / 2); // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    let mid = Math.floor(left + (right - left) / 2); // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n  // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n  int i = left, j = mid + 1, k = 0;\n  // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // \u7d42\u6b62\u689d\u4ef6\n  if (left &gt;= right) return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n  // \u5283\u5206\u968e\u6bb5\n  int mid = left + (right - left) ~/ 2; // \u8a08\u7b97\u4e2d\u9ede\n  mergeSort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n  mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n  // \u5408\u4f75\u968e\u6bb5\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if left &gt;= right {\n        return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    }\n\n    // \u5283\u5206\u968e\u6bb5\n    let mid = left + (right - left) / 2; // \u8a08\u7b97\u4e2d\u9ede\n    merge_sort(nums, left, mid); // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    merge_sort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nvoid merge(int *nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(tmp);\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right)\n        return; // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    int mid = left + (right - left) / 2;    // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid);      // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right); // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.kt<pre><code>/* \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 */\nfun merge(nums: IntArray, left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n    // \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp \uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n    val tmp = IntArray(right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left\n    var j = mid + 1\n    var k = 0\n    // \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++]\n        else\n            tmp[k++] = nums[j++]\n    }\n    // \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++]\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++]\n    }\n    // \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n    for (l in tmp.indices) {\n        nums[left + l] = tmp[l]\n    }\n}\n\n/* \u5408\u4f75\u6392\u5e8f */\nfun mergeSort(nums: IntArray, left: Int, right: Int) {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return  // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    val mid = left + (right - left) / 2 // \u8a08\u7b97\u4e2d\u9ede\n    mergeSort(nums, left, mid) // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    mergeSort(nums, mid + 1, right) // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.rb<pre><code>### \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217 ###\ndef merge(nums, left, mid, right)\n  # \u5de6\u5b50\u9663\u5217\u5340\u9593\u70ba [left, mid], \u53f3\u5b50\u9663\u5217\u5340\u9593\u70ba [mid+1, right]\n  # \u5efa\u7acb\u4e00\u500b\u81e8\u6642\u9663\u5217 tmp\uff0c\u7528\u65bc\u5b58\u653e\u5408\u4f75\u5f8c\u7684\u7d50\u679c\n  tmp = Array.new(right - left + 1, 0)\n  # \u521d\u59cb\u5316\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\n  i, j, k = left, mid + 1, 0\n  # \u7576\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u9084\u6709\u5143\u7d20\u6642\uff0c\u9032\u884c\u6bd4\u8f03\u4e26\u5c07\u8f03\u5c0f\u7684\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n  while i &lt;= mid &amp;&amp; j &lt;= right\n    if nums[i] &lt;= nums[j]\n      tmp[k] = nums[i]\n      i += 1\n    else\n      tmp[k] = nums[j]\n      j += 1\n    end\n    k += 1\n  end\n  # \u5c07\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u7684\u5269\u9918\u5143\u7d20\u8907\u88fd\u5230\u81e8\u6642\u9663\u5217\u4e2d\n  while i &lt;= mid\n    tmp[k] = nums[i]\n    i += 1\n    k += 1\n  end\n  while j &lt;= right\n    tmp[k] = nums[j]\n    j += 1\n    k += 1\n  end\n  # \u5c07\u81e8\u6642\u9663\u5217 tmp \u4e2d\u7684\u5143\u7d20\u8907\u88fd\u56de\u539f\u9663\u5217 nums \u7684\u5c0d\u61c9\u5340\u9593\n  (0...tmp.length).each do |k|\n    nums[left + k] = tmp[k]\n  end\nend\n\n### \u5408\u4f75\u6392\u5e8f ###\ndef merge_sort(nums, left, right)\n  # \u7d42\u6b62\u689d\u4ef6\n  # \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n  return if left &gt;= right\n  # \u5283\u5206\u968e\u6bb5\n  mid = left + (right - left) / 2 # \u8a08\u7b97\u4e2d\u9ede\n  merge_sort(nums, left, mid) # \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n  merge_sort(nums, mid + 1, right) # \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n  # \u5408\u4f75\u968e\u6bb5\n  merge(nums, left, mid, right)\nend\n</code></pre> merge_sort.zig<pre><code>// \u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\n// \u5de6\u5b50\u9663\u5217\u5340\u9593 [left, mid]\n// \u53f3\u5b50\u9663\u5217\u5340\u9593 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n    // \u521d\u59cb\u5316\u8f14\u52a9\u9663\u5217\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var tmp = try mem_allocator.alloc(i32, right + 1 - left);\n    std.mem.copy(i32, tmp, nums[left..right+1]);\n    // \u5de6\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7d50\u675f\u7d22\u5f15\n    var leftStart = left - left;\n    var leftEnd = mid - left;\n    // \u53f3\u5b50\u9663\u5217\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7d50\u675f\u7d22\u5f15\n    var rightStart = mid + 1 - left;\n    var rightEnd = right - left;\n    // i, j \u5206\u5225\u6307\u5411\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\u7684\u9996\u5143\u7d20\n    var i = leftStart;\n    var j = rightStart;\n    // \u900f\u904e\u8986\u84cb\u539f\u9663\u5217 nums \u4f86\u5408\u4f75\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\n    var k = left;\n    while (k &lt;= right) : (k += 1) {\n        // \u82e5\u201c\u5de6\u5b50\u9663\u5217\u5df2\u5168\u90e8\u5408\u4f75\u5b8c\u201d\uff0c\u5247\u9078\u53d6\u53f3\u5b50\u9663\u5217\u5143\u7d20\uff0c\u4e26\u4e14 j++\n        if (i &gt; leftEnd) {\n            nums[k] = tmp[j];\n            j += 1;\n        // \u5426\u5247\uff0c\u82e5\u201c\u53f3\u5b50\u9663\u5217\u5df2\u5168\u90e8\u5408\u4f75\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u9663\u5217\u5143\u7d20 &lt;= \u53f3\u5b50\u9663\u5217\u5143\u7d20\u201d\uff0c\u5247\u9078\u53d6\u5de6\u5b50\u9663\u5217\u5143\u7d20\uff0c\u4e26\u4e14 i++\n        } else if  (j &gt; rightEnd or tmp[i] &lt;= tmp[j]) {\n            nums[k] = tmp[i];\n            i += 1;\n        // \u5426\u5247\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u9663\u5217\u90fd\u672a\u5168\u90e8\u5408\u4f75\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u9663\u5217\u5143\u7d20 &gt; \u53f3\u5b50\u9663\u5217\u5143\u7d20\u201d\uff0c\u5247\u9078\u53d6\u53f3\u5b50\u9663\u5217\u5143\u7d20\uff0c\u4e26\u4e14 j++\n        } else {\n            nums[k] = tmp[j];\n            j += 1;\n        }\n    }\n}\n\n// \u5408\u4f75\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n    // \u7d42\u6b62\u689d\u4ef6\n    if (left &gt;= right) return;              // \u7576\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    // \u5283\u5206\u968e\u6bb5\n    var mid = left + (right - left) / 2;    // \u8a08\u7b97\u4e2d\u9ede\n    try mergeSort(nums, left, mid);         // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\n    try mergeSort(nums, mid + 1, right);    // \u905e\u8ff4\u53f3\u5b50\u9663\u5217\n    // \u5408\u4f75\u968e\u6bb5\n    try merge(nums, left, mid, right);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5283\u5206\u7522\u751f\u9ad8\u5ea6\u70ba \\(\\log n\\) \u7684\u905e\u8ff4\u6a39\uff0c\u6bcf\u5c64\u5408\u4f75\u7684\u7e3d\u64cd\u4f5c\u6578\u91cf\u70ba \\(n\\) \uff0c\u56e0\u6b64\u7e3d\u9ad4\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u905e\u8ff4\u6df1\u5ea6\u70ba \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002\u5408\u4f75\u64cd\u4f5c\u9700\u8981\u85c9\u52a9\u8f14\u52a9\u9663\u5217\u5be6\u73fe\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u4f75\u904e\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u8b8a\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/#1163","title":"11.6.3 \u00a0 \u93c8\u7d50\u4e32\u5217\u6392\u5e8f","text":"<p>\u5c0d\u65bc\u93c8\u7d50\u4e32\u5217\uff0c\u5408\u4f75\u6392\u5e8f\u76f8\u8f03\u65bc\u5176\u4ed6\u6392\u5e8f\u6f14\u7b97\u6cd5\u5177\u6709\u986f\u8457\u512a\u52e2\uff0c\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u6392\u5e8f\u4efb\u52d9\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(1)\\) \u3002</p> <ul> <li>\u5283\u5206\u968e\u6bb5\uff1a\u53ef\u4ee5\u4f7f\u7528\u201c\u8fed\u4ee3\u201d\u66ff\u4ee3\u201c\u905e\u8ff4\u201d\u4f86\u5be6\u73fe\u93c8\u7d50\u4e32\u5217\u5283\u5206\u5de5\u4f5c\uff0c\u5f9e\u800c\u7701\u53bb\u905e\u8ff4\u4f7f\u7528\u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002</li> <li>\u5408\u4f75\u968e\u6bb5\uff1a\u5728\u93c8\u7d50\u4e32\u5217\u4e2d\uff0c\u7bc0\u9ede\u589e\u522a\u64cd\u4f5c\u50c5\u9700\u6539\u8b8a\u5f15\u7528\uff08\u6307\u6a19\uff09\u5373\u53ef\u5be6\u73fe\uff0c\u56e0\u6b64\u5408\u4f75\u968e\u6bb5\uff08\u5c07\u5169\u500b\u77ed\u6709\u5e8f\u93c8\u7d50\u4e32\u5217\u5408\u4f75\u70ba\u4e00\u500b\u9577\u6709\u5e8f\u93c8\u7d50\u4e32\u5217\uff09\u7121\u9808\u5efa\u7acb\u984d\u5916\u93c8\u7d50\u4e32\u5217\u3002</li> </ul> <p>\u5177\u9ad4\u5be6\u73fe\u7d30\u7bc0\u6bd4\u8f03\u8907\u96dc\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u67e5\u95b1\u76f8\u95dc\u8cc7\u6599\u9032\u884c\u5b78\u7fd2\u3002</p>"},{"location":"chapter_sorting/quick_sort/","title":"11.5 \u00a0 \u5feb\u901f\u6392\u5e8f","text":"<p>\u5feb\u901f\u6392\u5e8f\uff08quick sort\uff09\u662f\u4e00\u7a2e\u57fa\u65bc\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u57f7\u884c\u9ad8\u6548\uff0c\u61c9\u7528\u5ee3\u6cdb\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u201c\u54e8\u5175\u5283\u5206\u201d\uff0c\u5176\u76ee\u6a19\u662f\uff1a\u9078\u64c7\u9663\u5217\u4e2d\u7684\u67d0\u500b\u5143\u7d20\u4f5c\u70ba\u201c\u57fa\u6e96\u6578\u201d\uff0c\u5c07\u6240\u6709\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u5074\uff0c\u800c\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u5074\u3002\u5177\u9ad4\u4f86\u8aaa\uff0c\u54e8\u5175\u5283\u5206\u7684\u6d41\u7a0b\u5982\u5716 11-8 \u6240\u793a\u3002</p> <ol> <li>\u9078\u53d6\u9663\u5217\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u70ba\u57fa\u6e96\u6578\uff0c\u521d\u59cb\u5316\u5169\u500b\u6307\u6a19 <code>i</code> \u548c <code>j</code> \u5206\u5225\u6307\u5411\u9663\u5217\u7684\u5169\u7aef\u3002</li> <li>\u8a2d\u5b9a\u4e00\u500b\u8ff4\u5708\uff0c\u5728\u6bcf\u8f2a\u4e2d\u4f7f\u7528 <code>i</code>\uff08<code>j</code>\uff09\u5206\u5225\u5c0b\u627e\u7b2c\u4e00\u500b\u6bd4\u57fa\u6e96\u6578\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u5f8c\u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\u3002</li> <li>\u8ff4\u5708\u57f7\u884c\u6b65\u9a5f <code>2.</code> \uff0c\u76f4\u5230 <code>i</code> \u548c <code>j</code> \u76f8\u9047\u6642\u505c\u6b62\uff0c\u6700\u5f8c\u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u500b\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-8 \u00a0 \u54e8\u5175\u5283\u5206\u6b65\u9a5f </p> <p>\u54e8\u5175\u5283\u5206\u5b8c\u6210\u5f8c\uff0c\u539f\u9663\u5217\u88ab\u5283\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u9663\u5217\u3001\u57fa\u6e96\u6578\u3001\u53f3\u5b50\u9663\u5217\uff0c\u4e14\u6eff\u8db3\u201c\u5de6\u5b50\u9663\u5217\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u6e96\u6578 \\(\\leq\\) \u53f3\u5b50\u9663\u5217\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u5011\u63a5\u4e0b\u4f86\u53ea\u9700\u5c0d\u9019\u5169\u500b\u5b50\u9663\u5217\u9032\u884c\u6392\u5e8f\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565</p> <p>\u54e8\u5175\u5283\u5206\u7684\u5be6\u8cea\u662f\u5c07\u4e00\u500b\u8f03\u9577\u9663\u5217\u7684\u6392\u5e8f\u554f\u984c\u7c21\u5316\u70ba\u5169\u500b\u8f03\u77ed\u9663\u5217\u7684\u6392\u5e8f\u554f\u984c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5283\u5206\"\"\"\n    # \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u63db\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u54e8\u5175\u5283\u5206 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums[i], nums[j]); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums[i], nums[left]);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;                   // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nint partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;             // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nint Partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;             // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u54e8\u5175\u5283\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u63db\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u54e8\u5175\u5283\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        nums.swapAt(i, j) // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    nums.swapAt(i, left) // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.js<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\npartition(nums, left, right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u63db\n        this.swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\npartition(nums: number[], left: number, right: number): number {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u63db\n        this.swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n  return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u54e8\u5175\u5283\u5206 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    nums.swap(i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    i // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nint partition(int nums[], int left, int right) {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n        swap(nums, i, j);\n    }\n    // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    swap(nums, i, left);\n    // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n    return i;\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5143\u7d20\u4ea4\u63db */\nfun swap(nums: IntArray, i: Int, j: Int) {\n    val temp = nums[i]\n    nums[i] = nums[j]\n    nums[j] = temp\n}\n\n/* \u54e8\u5175\u5283\u5206 */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j)  // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left)   // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i              // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rb<pre><code>### \u54e8\u5175\u5283\u5206 ###\ndef partition(nums, left, right)\n  # \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    end\n    # \u5143\u7d20\u4ea4\u63db\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n  nums[i], nums[left] = nums[left], nums[i]\n  i # \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\nend\n</code></pre> quick_sort.zig<pre><code>// \u5143\u7d20\u4ea4\u63db\nfn swap(nums: []i32, i: usize, j: usize) void {\n    var tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n// \u54e8\u5175\u5283\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;               // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5feb\u901f\u6392\u5e8f\u7684\u6574\u9ad4\u6d41\u7a0b\u5982\u5716 11-9 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5c0d\u539f\u9663\u5217\u57f7\u884c\u4e00\u6b21\u201c\u54e8\u5175\u5283\u5206\u201d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u3002</li> <li>\u7136\u5f8c\uff0c\u5c0d\u5de6\u5b50\u9663\u5217\u548c\u53f3\u5b50\u9663\u5217\u5206\u5225\u905e\u8ff4\u57f7\u884c\u201c\u54e8\u5175\u5283\u5206\u201d\u3002</li> <li>\u6301\u7e8c\u905e\u8ff4\uff0c\u76f4\u81f3\u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\uff0c\u5f9e\u800c\u5b8c\u6210\u6574\u500b\u9663\u5217\u7684\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u5716 11-9 \u00a0 \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n    # \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if left &gt;= right:\n        return\n    # \u54e8\u5175\u5283\u5206\n    pivot = self.partition(nums, left, right)\n    # \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5283\u5206\n    int pivot = partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5283\u5206\n    int pivot = partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5283\u5206\n    int pivot = Partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5283\u5206\n    pivot := q.partition(nums, left, right)\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5283\u5206\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5283\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5283\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n  if (left &gt;= right) return;\n  // \u54e8\u5175\u5283\u5206\n  int pivot = _partition(nums, left, right);\n  // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if left &gt;= right {\n        return;\n    }\n    // \u54e8\u5175\u5283\u5206\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int nums[], int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5283\u5206\n    int pivot = partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) return\n    // \u54e8\u5175\u5283\u5206\n    val pivot = partition(nums, left, right)\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>### \u5feb\u901f\u6392\u5e8f\u985e\u5225 ###\ndef quick_sort(nums, left, right)\n  # \u5b50\u9663\u5217\u9577\u5ea6\u4e0d\u70ba 1 \u6642\u905e\u8ff4\n  if left &lt; right\n    # \u54e8\u5175\u5283\u5206\n    pivot = partition(nums, left, right)\n    # \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quick_sort(nums, left, pivot - 1)\n    quick_sort(nums, pivot + 1, right)\n  end\n  nums\nend\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5283\u5206\n    var pivot = partition(nums, left, right);\n    // \u905e\u8ff4\u5de6\u5b50\u9663\u5217\u3001\u53f3\u5b50\u9663\u5217\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5728\u5e73\u5747\u60c5\u6cc1\u4e0b\uff0c\u54e8\u5175\u5283\u5206\u7684\u905e\u8ff4\u5c64\u6578\u70ba \\(\\log n\\) \uff0c\u6bcf\u5c64\u4e2d\u7684\u7e3d\u8ff4\u5708\u6578\u70ba \\(n\\) \uff0c\u7e3d\u9ad4\u4f7f\u7528 \\(O(n \\log n)\\) \u6642\u9593\u3002\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u6bcf\u8f2a\u54e8\u5175\u5283\u5206\u64cd\u4f5c\u90fd\u5c07\u9577\u5ea6\u70ba \\(n\\) \u7684\u9663\u5217\u5283\u5206\u70ba\u9577\u5ea6\u70ba \\(0\\) \u548c \\(n - 1\\) \u7684\u5169\u500b\u5b50\u9663\u5217\uff0c\u6b64\u6642\u905e\u8ff4\u5c64\u6578\u9054\u5230 \\(n\\) \uff0c\u6bcf\u5c64\u4e2d\u7684\u8ff4\u5708\u6578\u70ba \\(n\\) \uff0c\u7e3d\u9ad4\u4f7f\u7528 \\(O(n^2)\\) \u6642\u9593\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u5728\u8f38\u5165\u9663\u5217\u5b8c\u5168\u5012\u5e8f\u7684\u60c5\u6cc1\u4e0b\uff0c\u9054\u5230\u6700\u5dee\u905e\u8ff4\u6df1\u5ea6 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002\u6392\u5e8f\u64cd\u4f5c\u662f\u5728\u539f\u9663\u5217\u4e0a\u9032\u884c\u7684\uff0c\u672a\u85c9\u52a9\u984d\u5916\u9663\u5217\u3002</li> <li>\u975e\u7a69\u5b9a\u6392\u5e8f\uff1a\u5728\u54e8\u5175\u5283\u5206\u7684\u6700\u5f8c\u4e00\u6b65\uff0c\u57fa\u6e96\u6578\u53ef\u80fd\u6703\u88ab\u4ea4\u63db\u81f3\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u5074\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1153","title":"11.5.3 \u00a0 \u5feb\u901f\u6392\u5e8f\u70ba\u4ec0\u9ebc\u5feb","text":"<p>\u5f9e\u540d\u7a31\u4e0a\u5c31\u80fd\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u61c9\u8a72\u5177\u6709\u4e00\u5b9a\u7684\u512a\u52e2\u3002\u5118\u7ba1\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u8207\u201c\u5408\u4f75\u6392\u5e8f\u201d\u548c\u201c\u5806\u7a4d\u6392\u5e8f\u201d\u76f8\u540c\uff0c\u4f46\u901a\u5e38\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u51fa\u73fe\u6700\u5dee\u60c5\u6cc1\u7684\u6a5f\u7387\u5f88\u4f4e\uff1a\u96d6\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u6c92\u6709\u5408\u4f75\u6392\u5e8f\u7a69\u5b9a\uff0c\u4f46\u5728\u7d55\u5927\u591a\u6578\u60c5\u6cc1\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u80fd\u5728 \\(O(n \\log n)\\) \u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e0b\u57f7\u884c\u3002</li> <li>\u5feb\u53d6\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u5728\u57f7\u884c\u54e8\u5175\u5283\u5206\u64cd\u4f5c\u6642\uff0c\u7cfb\u7d71\u53ef\u5c07\u6574\u500b\u5b50\u9663\u5217\u8f09\u5165\u5230\u5feb\u53d6\uff0c\u56e0\u6b64\u8a2a\u554f\u5143\u7d20\u7684\u6548\u7387\u8f03\u9ad8\u3002\u800c\u50cf\u201c\u5806\u7a4d\u6392\u5e8f\u201d\u9019\u985e\u6f14\u7b97\u6cd5\u9700\u8981\u8df3\u8e8d\u5f0f\u8a2a\u554f\u5143\u7d20\uff0c\u5f9e\u800c\u7f3a\u4e4f\u9019\u4e00\u7279\u6027\u3002</li> <li>\u8907\u96dc\u5ea6\u7684\u5e38\u6578\u4fc2\u6578\u5c0f\uff1a\u5728\u4e0a\u8ff0\u4e09\u7a2e\u6f14\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6bd4\u8f03\u3001\u8ce6\u503c\u3001\u4ea4\u63db\u7b49\u64cd\u4f5c\u7684\u7e3d\u6578\u91cf\u6700\u5c11\u3002\u9019\u8207\u201c\u63d2\u5165\u6392\u5e8f\u201d\u6bd4\u201c\u6ce1\u6cab\u6392\u5e8f\u201d\u66f4\u5feb\u7684\u539f\u56e0\u985e\u4f3c\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1154","title":"11.5.4 \u00a0 \u57fa\u6e96\u6578\u6700\u4f73\u5316","text":"<p>\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f38\u5165\u4e0b\u7684\u6642\u9593\u6548\u7387\u53ef\u80fd\u964d\u4f4e\u3002\u8209\u4e00\u500b\u6975\u7aef\u4f8b\u5b50\uff0c\u5047\u8a2d\u8f38\u5165\u9663\u5217\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u65bc\u6211\u5011\u9078\u64c7\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u70ba\u57fa\u6e96\u6578\uff0c\u90a3\u9ebc\u5728\u54e8\u5175\u5283\u5206\u5b8c\u6210\u5f8c\uff0c\u57fa\u6e96\u6578\u88ab\u4ea4\u63db\u81f3\u9663\u5217\u6700\u53f3\u7aef\uff0c\u5c0e\u81f4\u5de6\u5b50\u9663\u5217\u9577\u5ea6\u70ba \\(n - 1\\)\u3001\u53f3\u5b50\u9663\u5217\u9577\u5ea6\u70ba \\(0\\) \u3002\u5982\u6b64\u905e\u8ff4\u4e0b\u53bb\uff0c\u6bcf\u8f2a\u54e8\u5175\u5283\u5206\u5f8c\u90fd\u6709\u4e00\u500b\u5b50\u9663\u5217\u7684\u9577\u5ea6\u70ba \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u70ba\u201c\u6ce1\u6cab\u6392\u5e8f\u201d\u7684\u8fd1\u4f3c\u5f62\u5f0f\u3002</p> <p>\u70ba\u4e86\u5118\u91cf\u907f\u514d\u9019\u7a2e\u60c5\u6cc1\u767c\u751f\uff0c\u6211\u5011\u53ef\u4ee5\u6700\u4f73\u5316\u54e8\u5175\u5283\u5206\u4e2d\u7684\u57fa\u6e96\u6578\u7684\u9078\u53d6\u7b56\u7565\u3002\u4f8b\u5982\uff0c\u6211\u5011\u53ef\u4ee5\u96a8\u6a5f\u9078\u53d6\u4e00\u500b\u5143\u7d20\u4f5c\u70ba\u57fa\u6e96\u6578\u3002\u7136\u800c\uff0c\u5982\u679c\u904b\u6c23\u4e0d\u4f73\uff0c\u6bcf\u6b21\u90fd\u9078\u5230\u4e0d\u7406\u60f3\u7684\u57fa\u6e96\u6578\uff0c\u6548\u7387\u4ecd\u7136\u4e0d\u76e1\u5982\u4eba\u610f\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7a0b\u5f0f\u8a9e\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u507d\u96a8\u6a5f\u6578\u201d\u3002\u5982\u679c\u6211\u5011\u91dd\u5c0d\u507d\u96a8\u6a5f\u6578\u5e8f\u5217\u69cb\u5efa\u4e00\u500b\u7279\u5b9a\u7684\u6e2c\u8a66\u6a23\u4f8b\uff0c\u90a3\u9ebc\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002</p> <p>\u70ba\u4e86\u9032\u4e00\u6b65\u6539\u9032\uff0c\u6211\u5011\u53ef\u4ee5\u5728\u9663\u5217\u4e2d\u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\uff08\u901a\u5e38\u70ba\u9663\u5217\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u9ede\u5143\u7d20\uff09\uff0c\u4e26\u5c07\u9019\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\u4f5c\u70ba\u57fa\u6e96\u6578\u3002\u9019\u6a23\u4e00\u4f86\uff0c\u57fa\u6e96\u6578\u201c\u65e2\u4e0d\u592a\u5c0f\u4e5f\u4e0d\u592a\u5927\u201d\u7684\u6a5f\u7387\u5c07\u5927\u5e45\u63d0\u5347\u3002\u7576\u7136\uff0c\u6211\u5011\u9084\u53ef\u4ee5\u9078\u53d6\u66f4\u591a\u5019\u9078\u5143\u7d20\uff0c\u4ee5\u9032\u4e00\u6b65\u63d0\u9ad8\u6f14\u7b97\u6cd5\u7684\u7a69\u5065\u6027\u3002\u63a1\u7528\u9019\u7a2e\u65b9\u6cd5\u5f8c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6a5f\u7387\u5927\u5927\u964d\u4f4e\u3002</p> <p>\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"\u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\"\"\"\n    l, m, r = nums[left], nums[mid], nums[right]\n    if (l &lt;= m &lt;= r) or (r &lt;= m &lt;= l):\n        return mid  # m \u5728 l \u548c r \u4e4b\u9593\n    if (m &lt;= l &lt;= r) or (r &lt;= l &lt;= m):\n        return left  # l \u5728 m \u548c r \u4e4b\u9593\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09\"\"\"\n    # \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    # \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u63db\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums[left], nums[med]);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums[i], nums[j]); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums[i], nums[left]);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;                   // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint medianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;             // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint Partition(int[] nums, int left, int right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    Swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;             // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    l, m, r := nums[left], nums[mid], nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m \u5728 l \u548c r \u4e4b\u9593\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l \u5728 m \u548c r \u4e4b\u9593\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- //\u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ //\u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        //\u5143\u7d20\u4ea4\u63db\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    //\u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    nums[i], nums[left] = nums[left], nums[i]\n    return i //\u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    let l = nums[left]\n    let m = nums[mid]\n    let r = nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m \u5728 l \u548c r \u4e4b\u9593\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l \u5728 m \u548c r \u4e4b\u9593\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    let med = medianThree(nums: nums, left: left, mid: left + (right - left) / 2, right: right)\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    nums.swapAt(left, med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nmedianThree(nums, left, mid, right) {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m \u5728 l \u548c r \u4e4b\u9593\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l \u5728 m \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        this.swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m \u5728 l \u548c r \u4e4b\u9593\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l \u5728 m \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        this.swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  int l = nums[left], m = nums[mid], r = nums[right];\n  if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n    return mid; // m \u5728 l \u548c r \u4e4b\u9593\n  if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n    return left; // l \u5728 m \u548c r \u4e4b\u9593\n  return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n  _swap(nums, left, med);\n  // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n  return i; // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    let (l, m, r) = (nums[left], nums[mid], nums[right]);\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    }\n    right\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    nums.swap(left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    nums.swap(i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    i // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nint medianThree(int nums[], int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nint partitionMedian(int nums[], int left, int right) {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;            // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.kt<pre><code>/* \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 */\nfun medianThree(nums: IntArray, left: Int, mid: Int, right: Int): Int {\n    val l = nums[left]\n    val m = nums[mid]\n    val r = nums[right]\n    if ((m in l..r) || (m in r..l))\n        return mid  // m \u5728 l \u548c r \u4e4b\u9593\n    if ((l in m..r) || (l in r..m))\n        return left // l \u5728 m \u548c r \u4e4b\u9593\n    return right\n}\n\n/* \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09 */\nfun partitionMedian(nums: IntArray, left: Int, right: Int): Int {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    val med = medianThree(nums, left, (left + right) / 2, right)\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums, left, med)\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--                      // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++                      // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j)             // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left)              // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i                         // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rb<pre><code>### \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578 ###\ndef median_three(nums, left, mid, right)\n  # \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n  _l, _m, _r = nums[left], nums[mid], nums[right]\n  # m \u5728 l \u548c r \u4e4b\u9593\n  return mid if (_l &lt;= _m &amp;&amp; _m &lt;= _r) || (_r &lt;= _m &amp;&amp; _m &lt;= _l)\n  # l \u5728 m \u548c r \u4e4b\u9593\n  return left if (_m &lt;= _l &amp;&amp; _l &lt;= _r) || (_r &lt;= _l &amp;&amp; _l &lt;= _m)\n  return right\nend\n\n### \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09###\ndef partition(nums, left, right)\n  ### \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n  med = median_three(nums, left, (left + right) / 2, right)\n  # \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u65b7\n  nums[left], nums[med] = nums[med], nums[left]\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n    end\n    # \u5143\u7d20\u4ea4\u63db\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n  nums[i], nums[left] = nums[left], nums[i]\n  i # \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\nend\n</code></pre> quick_sort.zig<pre><code>// \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n    var l = nums[left];\n    var m = nums[mid];\n    var r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u9593\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u9593\n    return right;\n}\n\n// \u54e8\u5175\u5283\u5206\uff08\u4e09\u6578\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u9078\u53d6\u4e09\u500b\u5019\u9078\u5143\u7d20\u7684\u4e2d\u4f4d\u6578\n    var med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c07\u4e2d\u4f4d\u6578\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u70ba\u57fa\u6e96\u6578\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u5f9e\u53f3\u5411\u5de6\u627e\u9996\u500b\u5c0f\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u5f9e\u5de6\u5411\u53f3\u627e\u9996\u500b\u5927\u65bc\u57fa\u6e96\u6578\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u63db\u9019\u5169\u500b\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c07\u57fa\u6e96\u6578\u4ea4\u63db\u81f3\u5169\u5b50\u9663\u5217\u7684\u5206\u754c\u7dda\n    return i;               // \u8fd4\u56de\u57fa\u6e96\u6578\u7684\u7d22\u5f15\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5 \u00a0 \u5c3e\u905e\u8ff4\u6700\u4f73\u5316","text":"<p>\u5728\u67d0\u4e9b\u8f38\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u4f54\u7528\u7a7a\u9593\u8f03\u591a\u3002\u4ee5\u5b8c\u5168\u6709\u5e8f\u7684\u8f38\u5165\u9663\u5217\u70ba\u4f8b\uff0c\u8a2d\u905e\u8ff4\u4e2d\u7684\u5b50\u9663\u5217\u9577\u5ea6\u70ba \\(m\\) \uff0c\u6bcf\u8f2a\u54e8\u5175\u5283\u5206\u64cd\u4f5c\u90fd\u5c07\u7522\u751f\u9577\u5ea6\u70ba \\(0\\) \u7684\u5de6\u5b50\u9663\u5217\u548c\u9577\u5ea6\u70ba \\(m - 1\\) \u7684\u53f3\u5b50\u9663\u5217\uff0c\u9019\u610f\u5473\u8457\u6bcf\u4e00\u5c64\u905e\u8ff4\u547c\u53eb\u6e1b\u5c11\u7684\u554f\u984c\u898f\u6a21\u975e\u5e38\u5c0f\uff08\u53ea\u6e1b\u5c11\u4e00\u500b\u5143\u7d20\uff09\uff0c\u905e\u8ff4\u6a39\u7684\u9ad8\u5ea6\u6703\u9054\u5230 \\(n - 1\\) \uff0c\u6b64\u6642\u9700\u8981\u4f54\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u5806\u758a\u5e40\u7a7a\u9593\u3002</p> <p>\u70ba\u4e86\u9632\u6b62\u5806\u758a\u5e40\u7a7a\u9593\u7684\u7d2f\u7a4d\uff0c\u6211\u5011\u53ef\u4ee5\u5728\u6bcf\u8f2a\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u5f8c\uff0c\u6bd4\u8f03\u5169\u500b\u5b50\u9663\u5217\u7684\u9577\u5ea6\uff0c\u50c5\u5c0d\u8f03\u77ed\u7684\u5b50\u9663\u5217\u9032\u884c\u905e\u8ff4\u3002\u7531\u65bc\u8f03\u77ed\u5b50\u9663\u5217\u7684\u9577\u5ea6\u4e0d\u6703\u8d85\u904e \\(n / 2\\) \uff0c\u56e0\u6b64\u9019\u7a2e\u65b9\u6cd5\u80fd\u78ba\u4fdd\u905e\u8ff4\u6df1\u5ea6\u4e0d\u8d85\u904e \\(\\log n\\) \uff0c\u5f9e\u800c\u5c07\u6700\u5dee\u7a7a\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u81f3 \\(O(\\log n)\\) \u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09\"\"\"\n    # \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while left &lt; right:\n        # \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        pivot = self.partition(nums, left, right)\n        # \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1  # \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1  # \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1;                 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1;                 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        int pivot = Partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1;  // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            QuickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    for left &lt; right {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        pivot := q.partition(nums, left, right)\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1                 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            q.quickSort(nums, pivot+1, right) // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1                 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nquickSort(nums, left, right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n  while (left &lt; right) {\n    // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n    int pivot = _partition(nums, left, right);\n    // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n      left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n    } else {\n      quickSort(nums, pivot + 1, right); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n      right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums); // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1; // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            quickSortTailCall(nums, left, pivot - 1);\n            // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n            left = pivot + 1;\n        } else {\n            // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            quickSortTailCall(nums, pivot + 1, right);\n            // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n            right = pivot - 1;\n        }\n    }\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09 */\nfun quickSortTailCall(nums: IntArray, left: Int, right: Int) {\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\n    var l = left\n    var r = right\n    while (l &lt; r) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        val pivot = partition(nums, l, r)\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - l &lt; r - pivot) {\n            quickSort(nums, l, pivot - 1) // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            l = pivot + 1 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, r) // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            r = pivot - 1 // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.rb<pre><code>### \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09###\ndef quick_sort(nums, left, right)\n  # \u5b50\u9663\u5217\u9577\u5ea6\u4e0d\u70ba 1 \u6642\u905e\u8ff4\n  while left &lt; right\n    # \u54e8\u5175\u5283\u5206\n    pivot = partition(nums, left, right)\n    # \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n    if pivot - left &lt; right - pivot\n      quick_sort(nums, left, pivot - 1)\n      left = pivot + 1 # \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n    else\n      quick_sort(nums, pivot + 1, right)\n      right = pivot - 1 # \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n    end\n  end\nend\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\n    var left = left_;\n    var right = right_;\n    // \u5b50\u9663\u5217\u9577\u5ea6\u70ba 1 \u6642\u7d42\u6b62\u905e\u8ff4\n    while (left &lt; right) {\n        // \u54e8\u5175\u5283\u5206\u64cd\u4f5c\n        var pivot = partition(nums, left, right);\n        // \u5c0d\u5169\u500b\u5b50\u9663\u5217\u4e2d\u8f03\u77ed\u7684\u90a3\u500b\u57f7\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1);   // \u905e\u8ff4\u6392\u5e8f\u5de6\u5b50\u9663\u5217\n            left = pivot + 1;                   // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right);  // \u905e\u8ff4\u6392\u5e8f\u53f3\u5b50\u9663\u5217\n            right = pivot - 1;                  // \u5269\u9918\u672a\u6392\u5e8f\u5340\u9593\u70ba [left, pivot - 1]\n        }\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/radix_sort/","title":"11.10 \u00a0 \u57fa\u6578\u6392\u5e8f","text":"<p>\u4e0a\u4e00\u7bc0\u4ecb\u7d39\u4e86\u8a08\u6578\u6392\u5e8f\uff0c\u5b83\u9069\u7528\u65bc\u8cc7\u6599\u91cf \\(n\\) \u8f03\u5927\u4f46\u8cc7\u6599\u7bc4\u570d \\(m\\) \u8f03\u5c0f\u7684\u60c5\u6cc1\u3002\u5047\u8a2d\u6211\u5011\u9700\u8981\u5c0d \\(n = 10^6\\) \u500b\u5b78\u865f\u9032\u884c\u6392\u5e8f\uff0c\u800c\u5b78\u865f\u662f\u4e00\u500b \\(8\\) \u4f4d\u6578\u5b57\uff0c\u9019\u610f\u5473\u8457\u8cc7\u6599\u7bc4\u570d \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8a08\u6578\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u8a18\u61b6\u9ad4\u7a7a\u9593\uff0c\u800c\u57fa\u6578\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u9019\u7a2e\u60c5\u6cc1\u3002</p> <p>\u57fa\u6578\u6392\u5e8f\uff08radix sort\uff09\u7684\u6838\u5fc3\u601d\u60f3\u8207\u8a08\u6578\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u900f\u904e\u7d71\u8a08\u500b\u6578\u4f86\u5be6\u73fe\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u790e\u4e0a\uff0c\u57fa\u6578\u6392\u5e8f\u5229\u7528\u6578\u5b57\u5404\u4f4d\u4e4b\u9593\u7684\u905e\u9032\u95dc\u4fc2\uff0c\u4f9d\u6b21\u5c0d\u6bcf\u4e00\u4f4d\u9032\u884c\u6392\u5e8f\uff0c\u5f9e\u800c\u5f97\u5230\u6700\u7d42\u7684\u6392\u5e8f\u7d50\u679c\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1 \u00a0 \u6f14\u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u4ee5\u5b78\u865f\u8cc7\u6599\u70ba\u4f8b\uff0c\u5047\u8a2d\u6578\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6578\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u5716 11-18 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4f4d\u6578 \\(k = 1\\) \u3002</li> <li>\u5c0d\u5b78\u865f\u7684\u7b2c \\(k\\) \u4f4d\u57f7\u884c\u201c\u8a08\u6578\u6392\u5e8f\u201d\u3002\u5b8c\u6210\u5f8c\uff0c\u8cc7\u6599\u6703\u6839\u64da\u7b2c \\(k\\) \u4f4d\u5f9e\u5c0f\u5230\u5927\u6392\u5e8f\u3002</li> <li>\u5c07 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u5f8c\u8fd4\u56de\u6b65\u9a5f <code>2.</code> \u7e7c\u7e8c\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u5f8c\u7d50\u675f\u3002</li> </ol> <p></p> <p> \u5716 11-18 \u00a0 \u57fa\u6578\u6392\u5e8f\u6f14\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4e0b\u9762\u5256\u6790\u7a0b\u5f0f\u78bc\u5be6\u73fe\u3002\u5c0d\u65bc\u4e00\u500b \\(d\\) \u9032\u4f4d\u5236\u7684\u6578\u5b57 \\(x\\) \uff0c\u8981\u7372\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8a08\u7b97\u516c\u5f0f\uff1a</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>\u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5c0d\u6d6e\u9ede\u6578 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\: d\\) \u8868\u793a\u5c0d \\(d\\) \u53d6\u6a21\uff08\u53d6\u9918\uff09\u3002\u5c0d\u65bc\u5b78\u865f\u8cc7\u6599\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002</p> <p>\u6b64\u5916\uff0c\u6211\u5011\u9700\u8981\u5c0f\u5e45\u6539\u52d5\u8a08\u6578\u6392\u5e8f\u7a0b\u5f0f\u78bc\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u64da\u6578\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u9032\u884c\u6392\u5e8f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"\u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n    # \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"\u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n    # \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    counter = [0] * 10\n    n = len(nums)\n    # \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for i in range(n):\n        d = digit(nums[i], exp)  # \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d] += 1  # \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    # \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]  # \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1  # \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    # \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"\u57fa\u6578\u6392\u5e8f\"\"\"\n    # \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    m = max(nums)\n    # \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    exp = 1\n    while exp &lt;= m:\n        # \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # \u5373 exp = 10^(k-1)\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++;                // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    int m = *max_element(nums.begin(), nums.end());\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    int[] counter = new int[10];\n    int n = nums.length;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++;                // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.cs<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint Digit(int num, int exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++;                // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid RadixSort(int[] nums) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    counter := make([]int, 10)\n    n := len(nums)\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++             // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]    // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--        // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfunc radixSort(nums []int) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    (num / exp) % 10\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    var counter = Array(repeating: 0, count: 10)\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d] += 1 // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i] // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1 // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfunction radixSort(nums) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* \u7372\u53d6\u5143\u7d20 _num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int _num, int exp) {\n  // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n  return (_num ~/ exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n    counter[d]++; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n  }\n  // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n    res[j] = nums[i]; // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n    counter[d]--; // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n  }\n  // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid radixSort(List&lt;int&gt; nums) {\n  // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n  // dart \u4e2d int \u7684\u9577\u5ea6\u662f 64 \u4f4d\u7684\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // \u5373 exp = 10^(k-1)\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return ((num / exp) % 10) as usize;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for i in 0..n {\n        let d = digit(nums[i], exp); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d] += 1; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1; // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    nums.copy_from_slice(&amp;res);\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    let m = *nums.into_iter().max().unwrap();\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    memset(counter, 0, sizeof(int) * 10); // \u521d\u59cb\u5316\u70ba 0 \u4ee5\u652f\u6301\u5f8c\u7e8c\u8a18\u61b6\u9ad4\u91cb\u653e\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (int i = 0; i &lt; size; i++) {\n        // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        int d = digit(nums[i], exp);\n        // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n        counter[d]++;\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(res);\n    free(counter);\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    int max = INT32_MIN;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.kt<pre><code>/* \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfun digit(num: Int, exp: Int): Int {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfun countingSortDigit(nums: IntArray, exp: Int) {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    val counter = IntArray(10)\n    val n = nums.size\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (i in 0..&lt;n) {\n        val d = digit(nums[i], exp) // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d]++                // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    for (i in 1..9) {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val d = digit(nums[i], exp)\n        val j = counter[d] - 1 // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--           // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    for (i in 0..&lt;n)\n        nums[i] = res[i]\n}\n\n/* \u57fa\u6578\u6392\u5e8f */\nfun radixSort(nums: IntArray) {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    var m = Int.MIN_VALUE\n    for (num in nums) if (num &gt; m) m = num\n    var exp = 1\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    while (exp &lt;= m) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n        exp *= 10\n    }\n}\n</code></pre> radix_sort.rb<pre><code>### \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) ###\ndef digit(num, exp)\n  # \u8f49\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n  (num / exp) % 10\nend\n\n### \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09###\ndef counting_sort_digit(nums, exp)\n  # \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n  counter = Array.new(10, 0)\n  n = nums.length\n  # \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n  for i in 0...n\n    d = digit(nums[i], exp) # \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n    counter[d] += 1 # \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n  end\n  # \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n  (1...10).each { |i| counter[i] += counter[i - 1] }\n  # \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n  res = Array.new(n, 0)\n  for i in (n - 1).downto(0)\n    d = digit(nums[i], exp)\n    j = counter[d] - 1 # \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n    res[j] = nums[i] # \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n    counter[d] -= 1 # \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n  end\n  # \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n  (0...n).each { |i| nums[i] = res[i] }\nend\n\n### \u57fa\u6578\u6392\u5e8f ###\ndef radix_sort(nums)\n  # \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n  m = nums.max\n  # \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n  exp = 1\n  while exp &lt;= m\n    # \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n    # k = 1 -&gt; exp = 1\n    # k = 2 -&gt; exp = 10\n    # \u5373 exp = 10^(k-1)\n    counting_sort_digit(nums, exp)\n    exp *= 10\n  end\nend\n</code></pre> radix_sort.zig<pre><code>// \u7372\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n    // \u50b3\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u8907\u57f7\u884c\u6602\u8cb4\u7684\u6b21\u65b9\u8a08\u7b97\n    return @mod(@divFloor(num, exp), 10);\n}\n\n// \u8a08\u6578\u6392\u5e8f\uff08\u6839\u64da nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n    // \u5341\u9032\u4f4d\u5236\u7684\u4f4d\u7bc4\u570d\u70ba 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u9577\u5ea6\u70ba 10 \u7684\u6876\u9663\u5217\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    // defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var counter = try mem_allocator.alloc(usize, 10);\n    @memset(counter, 0);\n    var n = nums.len;\n    // \u7d71\u8a08 0~9 \u5404\u6578\u5b57\u7684\u51fa\u73fe\u6b21\u6578\n    for (nums) |num| {\n        var d: u32 = @bitCast(digit(num, exp)); // \u7372\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8a18\u70ba d\n        counter[d] += 1; // \u7d71\u8a08\u6578\u5b57 d \u7684\u51fa\u73fe\u6b21\u6578\n    }\n    // \u6c42\u524d\u7db4\u548c\uff0c\u5c07\u201c\u51fa\u73fe\u500b\u6578\u201d\u8f49\u63db\u70ba\u201c\u9663\u5217\u7d22\u5f15\u201d\n    var i: usize = 1;\n    while (i &lt; 10) : (i += 1) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u8d70\u8a2a\uff0c\u6839\u64da\u6876\u5167\u7d71\u8a08\u7d50\u679c\uff0c\u5c07\u5404\u5143\u7d20\u586b\u5165 res\n    var res = try mem_allocator.alloc(i32, n);\n    i = n - 1;\n    while (i &gt;= 0) : (i -= 1) {\n        var d: u32 = @bitCast(digit(nums[i], exp));\n        var j = counter[d] - 1; // \u7372\u53d6 d \u5728\u9663\u5217\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c07\u7576\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1;        // \u5c07 d \u7684\u6578\u91cf\u6e1b 1\n        if (i == 0) break;\n    }\n    // \u4f7f\u7528\u7d50\u679c\u8986\u84cb\u539f\u9663\u5217 nums\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        nums[i] = res[i];\n    }\n}\n\n// \u57fa\u6578\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n    // \u7372\u53d6\u9663\u5217\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u65bc\u5224\u65b7\u6700\u5927\u4f4d\u6578\n    var m: i32 = std.math.minInt(i32);\n    for (nums) |num| {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u5f9e\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u9806\u5e8f\u8d70\u8a2a\n    var exp: i32 = 1;\n    while (exp &lt;= m) : (exp *= 10) {\n        // \u5c0d\u9663\u5217\u5143\u7d20\u7684\u7b2c k \u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        try countingSortDigit(nums, exp);    \n    }\n} \n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u70ba\u4ec0\u9ebc\u5f9e\u6700\u4f4e\u4f4d\u958b\u59cb\u6392\u5e8f\uff1f</p> <p>\u5728\u9023\u7e8c\u7684\u6392\u5e8f\u8f2a\u6b21\u4e2d\uff0c\u5f8c\u4e00\u8f2a\u6392\u5e8f\u6703\u8986\u84cb\u524d\u4e00\u8f2a\u6392\u5e8f\u7684\u7d50\u679c\u3002\u8209\u4f8b\u4f86\u8aaa\uff0c\u5982\u679c\u7b2c\u4e00\u8f2a\u6392\u5e8f\u7d50\u679c \\(a &lt; b\\) \uff0c\u800c\u7b2c\u4e8c\u8f2a\u6392\u5e8f\u7d50\u679c \\(a &gt; b\\) \uff0c\u90a3\u9ebc\u7b2c\u4e8c\u8f2a\u7684\u7d50\u679c\u5c07\u53d6\u4ee3\u7b2c\u4e00\u8f2a\u7684\u7d50\u679c\u3002\u7531\u65bc\u6578\u5b57\u7684\u9ad8\u4f4d\u512a\u5148\u9806\u5e8f\u9ad8\u65bc\u4f4e\u4f4d\uff0c\u56e0\u6b64\u61c9\u8a72\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<p>\u76f8\u8f03\u65bc\u8a08\u6578\u6392\u5e8f\uff0c\u57fa\u6578\u6392\u5e8f\u9069\u7528\u65bc\u6578\u503c\u7bc4\u570d\u8f03\u5927\u7684\u60c5\u6cc1\uff0c\u4f46\u524d\u63d0\u662f\u8cc7\u6599\u5fc5\u9808\u53ef\u4ee5\u8868\u793a\u70ba\u56fa\u5b9a\u4f4d\u6578\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6578\u4e0d\u80fd\u904e\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u9ede\u6578\u4e0d\u9069\u5408\u4f7f\u7528\u57fa\u6578\u6392\u5e8f\uff0c\u56e0\u70ba\u5176\u4f4d\u6578 \\(k\\) \u904e\u5927\uff0c\u53ef\u80fd\u5c0e\u81f4\u6642\u9593\u8907\u96dc\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002</p> <ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(nk)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u8a2d\u8cc7\u6599\u91cf\u70ba \\(n\\)\u3001\u8cc7\u6599\u70ba \\(d\\) \u9032\u4f4d\u5236\u3001\u6700\u5927\u4f4d\u6578\u70ba \\(k\\) \uff0c\u5247\u5c0d\u67d0\u4e00\u4f4d\u57f7\u884c\u8a08\u6578\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u6642\u9593\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u6642\u9593\u3002\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5c0d\u8f03\u5c0f\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u8da8\u5411 \\(O(n)\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n + d)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u8207\u8a08\u6578\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6578\u6392\u5e8f\u9700\u8981\u85c9\u52a9\u9577\u5ea6\u70ba \\(n\\) \u548c \\(d\\) \u7684\u9663\u5217 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a69\u5b9a\u6392\u5e8f\uff1a\u7576\u8a08\u6578\u6392\u5e8f\u7a69\u5b9a\u6642\uff0c\u57fa\u6578\u6392\u5e8f\u4e5f\u7a69\u5b9a\uff1b\u7576\u8a08\u6578\u6392\u5e8f\u4e0d\u7a69\u5b9a\u6642\uff0c\u57fa\u6578\u6392\u5e8f\u7121\u6cd5\u4fdd\u8b49\u5f97\u5230\u6b63\u78ba\u7684\u6392\u5e8f\u7d50\u679c\u3002</li> </ul>"},{"location":"chapter_sorting/selection_sort/","title":"11.2 \u00a0 \u9078\u64c7\u6392\u5e8f","text":"<p>\u9078\u64c7\u6392\u5e8f\uff08selection sort\uff09\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u7c21\u55ae\uff1a\u958b\u555f\u4e00\u500b\u8ff4\u5708\uff0c\u6bcf\u8f2a\u5f9e\u672a\u6392\u5e8f\u5340\u9593\u9078\u64c7\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c07\u5176\u653e\u5230\u5df2\u6392\u5e8f\u5340\u9593\u7684\u672b\u5c3e\u3002</p> <p>\u8a2d\u9663\u5217\u7684\u9577\u5ea6\u70ba \\(n\\) \uff0c\u9078\u64c7\u6392\u5e8f\u7684\u6f14\u7b97\u6cd5\u6d41\u7a0b\u5982\u5716 11-2 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72c0\u614b\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u5340\u9593\u70ba \\([0, n-1]\\) \u3002</li> <li>\u9078\u53d6\u5340\u9593 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c07\u5176\u8207\u7d22\u5f15 \\(0\\) \u8655\u7684\u5143\u7d20\u4ea4\u63db\u3002\u5b8c\u6210\u5f8c\uff0c\u9663\u5217\u524d 1 \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9078\u53d6\u5340\u9593 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c07\u5176\u8207\u7d22\u5f15 \\(1\\) \u8655\u7684\u5143\u7d20\u4ea4\u63db\u3002\u5b8c\u6210\u5f8c\uff0c\u9663\u5217\u524d 2 \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u985e\u63a8\u3002\u7d93\u904e \\(n - 1\\) \u8f2a\u9078\u64c7\u8207\u4ea4\u63db\u5f8c\uff0c\u9663\u5217\u524d \\(n - 1\\) \u500b\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u50c5\u5269\u7684\u4e00\u500b\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u7121\u9808\u6392\u5e8f\uff0c\u56e0\u6b64\u9663\u5217\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 11-2 \u00a0 \u9078\u64c7\u6392\u5e8f\u6b65\u9a5f </p> <p>\u5728\u7a0b\u5f0f\u78bc\u4e2d\uff0c\u6211\u5011\u7528 \\(k\\) \u4f86\u8a18\u9304\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"\u9078\u64c7\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for i in range(n - 1):\n        # \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        # \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for i := 0; i &lt; n-1; i++ {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n                k = j\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for i in nums.indices.dropLast() {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n  for (int i = 0; i &lt; n - 1; i++) {\n    // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n    }\n    // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfn selection_sort(nums: &amp;mut [i32]) {\n    if nums.is_empty() {\n        return;\n    }\n    let n = nums.len();\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for i in 0..n - 1 {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        let mut k = i;\n        for j in i + 1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.kt<pre><code>/* \u9078\u64c7\u6392\u5e8f */\nfun selectionSort(nums: IntArray) {\n    val n = nums.size\n    // \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n    for (i in 0..&lt;n - 1) {\n        var k = i\n        // \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n        for (j in i + 1..&lt;n) {\n            if (nums[j] &lt; nums[k])\n                k = j // \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n        val temp = nums[i]\n        nums[i] = nums[k]\n        nums[k] = temp\n    }\n}\n</code></pre> selection_sort.rb<pre><code>### \u9078\u64c7\u6392\u5e8f ###\ndef selection_sort(nums)\n  n = nums.length\n  # \u5916\u8ff4\u5708\uff1a\u672a\u6392\u5e8f\u5340\u9593\u70ba [i, n-1]\n  for i in 0...(n - 1)\n    # \u5167\u8ff4\u5708\uff1a\u627e\u5230\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u6700\u5c0f\u5143\u7d20\n    k = i\n    for j in (i + 1)...n\n      if nums[j] &lt; nums[k]\n        k = j # \u8a18\u9304\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n      end\n    end\n    # \u5c07\u8a72\u6700\u5c0f\u5143\u7d20\u8207\u672a\u6392\u5e8f\u5340\u9593\u7684\u9996\u500b\u5143\u7d20\u4ea4\u63db\n    nums[i], nums[k] = nums[k], nums[i]\n  end\nend\n</code></pre> selection_sort.zig<pre><code>[class]{}-[func]{selectionSort}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1 \u00a0 \u6f14\u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\)\u3001\u975e\u81ea\u9069\u61c9\u6392\u5e8f\uff1a\u5916\u8ff4\u5708\u5171 \\(n - 1\\) \u8f2a\uff0c\u7b2c\u4e00\u8f2a\u7684\u672a\u6392\u5e8f\u5340\u9593\u9577\u5ea6\u70ba \\(n\\) \uff0c\u6700\u5f8c\u4e00\u8f2a\u7684\u672a\u6392\u5e8f\u5340\u9593\u9577\u5ea6\u70ba \\(2\\) \uff0c\u5373\u5404\u8f2a\u5916\u8ff4\u5708\u5206\u5225\u5305\u542b \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(3\\)\u3001\\(2\\) \u8f2a\u5167\u8ff4\u5708\uff0c\u6c42\u548c\u70ba \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u6a19 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6578\u5927\u5c0f\u7684\u984d\u5916\u7a7a\u9593\u3002</li> <li>\u975e\u7a69\u5b9a\u6392\u5e8f\uff1a\u5982\u5716 11-3 \u6240\u793a\uff0c\u5143\u7d20 <code>nums[i]</code> \u6709\u53ef\u80fd\u88ab\u4ea4\u63db\u81f3\u8207\u5176\u76f8\u7b49\u7684\u5143\u7d20\u7684\u53f3\u908a\uff0c\u5c0e\u81f4\u5169\u8005\u7684\u76f8\u5c0d\u9806\u5e8f\u767c\u751f\u6539\u8b8a\u3002</li> </ul> <p> \u5716 11-3 \u00a0 \u9078\u64c7\u6392\u5e8f\u975e\u7a69\u5b9a\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1 \u00a0 \u6392\u5e8f\u6f14\u7b97\u6cd5","text":"<p>\u6392\u5e8f\u6f14\u7b97\u6cd5\uff08sorting algorithm\uff09\u7528\u65bc\u5c0d\u4e00\u7d44\u8cc7\u6599\u6309\u7167\u7279\u5b9a\u9806\u5e8f\u9032\u884c\u6392\u5217\u3002\u6392\u5e8f\u6f14\u7b97\u6cd5\u6709\u8457\u5ee3\u6cdb\u7684\u61c9\u7528\uff0c\u56e0\u70ba\u6709\u5e8f\u8cc7\u6599\u901a\u5e38\u80fd\u5920\u88ab\u66f4\u9ad8\u6548\u5730\u67e5\u8a62\u3001\u5206\u6790\u548c\u8655\u7406\u3002</p> <p>\u5982\u5716 11-1 \u6240\u793a\uff0c\u6392\u5e8f\u6f14\u7b97\u6cd5\u4e2d\u7684\u8cc7\u6599\u578b\u5225\u53ef\u4ee5\u662f\u6574\u6578\u3001\u6d6e\u9ede\u6578\u3001\u5b57\u5143\u6216\u5b57\u4e32\u7b49\u3002\u6392\u5e8f\u7684\u5224\u65b7\u898f\u5247\u53ef\u6839\u64da\u9700\u6c42\u8a2d\u5b9a\uff0c\u5982\u6578\u5b57\u5927\u5c0f\u3001\u5b57\u5143 ASCII \u78bc\u9806\u5e8f\u6216\u81ea\u5b9a\u7fa9\u898f\u5247\u3002</p> <p></p> <p> \u5716 11-1 \u00a0 \u8cc7\u6599\u578b\u5225\u548c\u5224\u65b7\u898f\u5247\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1 \u00a0 \u8a55\u50f9\u7dad\u5ea6","text":"<p>\u57f7\u884c\u6548\u7387\uff1a\u6211\u5011\u671f\u671b\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5118\u91cf\u4f4e\uff0c\u4e14\u7e3d\u9ad4\u64cd\u4f5c\u6578\u91cf\u8f03\u5c11\uff08\u6642\u9593\u8907\u96dc\u5ea6\u4e2d\u7684\u5e38\u6578\u9805\u8b8a\u5c0f\uff09\u3002\u5c0d\u65bc\u5927\u8cc7\u6599\u91cf\u7684\u60c5\u6cc1\uff0c\u57f7\u884c\u6548\u7387\u986f\u5f97\u5c24\u70ba\u91cd\u8981\u3002</p> <p>\u5c31\u5730\u6027\uff1a\u9867\u540d\u601d\u7fa9\uff0c\u539f\u5730\u6392\u5e8f\u900f\u904e\u5728\u539f\u9663\u5217\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5be6\u73fe\u6392\u5e8f\uff0c\u7121\u9808\u85c9\u52a9\u984d\u5916\u7684\u8f14\u52a9\u9663\u5217\uff0c\u5f9e\u800c\u7bc0\u7701\u8a18\u61b6\u9ad4\u3002\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u8cc7\u6599\u642c\u904b\u64cd\u4f5c\u8f03\u5c11\uff0c\u57f7\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002</p> <p>\u7a69\u5b9a\u6027\uff1a\u7a69\u5b9a\u6392\u5e8f\u5728\u5b8c\u6210\u6392\u5e8f\u5f8c\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u9663\u5217\u4e2d\u7684\u76f8\u5c0d\u9806\u5e8f\u4e0d\u767c\u751f\u6539\u8b8a\u3002</p> <p>\u7a69\u5b9a\u6392\u5e8f\u662f\u591a\u7d1a\u6392\u5e8f\u5834\u666f\u7684\u5fc5\u8981\u689d\u4ef6\u3002\u5047\u8a2d\u6211\u5011\u6709\u4e00\u500b\u5132\u5b58\u5b78\u751f\u8cc7\u8a0a\u7684\u8868\u683c\uff0c\u7b2c 1 \u5217\u548c\u7b2c 2 \u5217\u5206\u5225\u662f\u59d3\u540d\u548c\u5e74\u9f61\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u975e\u7a69\u5b9a\u6392\u5e8f\u53ef\u80fd\u5c0e\u81f4\u8f38\u5165\u8cc7\u6599\u7684\u6709\u5e8f\u6027\u55aa\u5931\uff1a</p> <pre><code># \u8f38\u5165\u8cc7\u6599\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# \u5047\u8a2d\u4f7f\u7528\u975e\u7a69\u5b9a\u6392\u5e8f\u6f14\u7b97\u6cd5\u6309\u5e74\u9f61\u6392\u5e8f\u4e32\u5217\uff0c\n# \u7d50\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5c0d\u4f4d\u7f6e\u6539\u8b8a\uff0c\n# \u8f38\u5165\u8cc7\u6599\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8cea\u4e1f\u5931\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>\u81ea\u9069\u61c9\u6027\uff1a\u81ea\u9069\u61c9\u6392\u5e8f\u80fd\u5920\u5229\u7528\u8f38\u5165\u8cc7\u6599\u5df2\u6709\u7684\u9806\u5e8f\u8cc7\u8a0a\u4f86\u6e1b\u5c11\u8a08\u7b97\u91cf\uff0c\u9054\u5230\u66f4\u512a\u7684\u6642\u9593\u6548\u7387\u3002\u81ea\u9069\u61c9\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u901a\u5e38\u512a\u65bc\u5e73\u5747\u6642\u9593\u8907\u96dc\u5ea6\u3002</p> <p>\u662f\u5426\u57fa\u65bc\u6bd4\u8f03\uff1a\u57fa\u65bc\u6bd4\u8f03\u7684\u6392\u5e8f\u4f9d\u8cf4\u6bd4\u8f03\u904b\u7b97\u5b50\uff08\\(&lt;\\)\u3001\\(=\\)\u3001\\(&gt;\\)\uff09\u4f86\u5224\u65b7\u5143\u7d20\u7684\u76f8\u5c0d\u9806\u5e8f\uff0c\u5f9e\u800c\u6392\u5e8f\u6574\u500b\u9663\u5217\uff0c\u7406\u8ad6\u6700\u512a\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n \\log n)\\) \u3002\u800c\u975e\u6bd4\u8f03\u6392\u5e8f\u4e0d\u4f7f\u7528\u6bd4\u8f03\u904b\u7b97\u5b50\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u53ef\u9054 \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5c0d\u8f03\u5dee\u3002</p>"},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2 \u00a0 \u7406\u60f3\u6392\u5e8f\u6f14\u7b97\u6cd5","text":"<p>\u57f7\u884c\u5feb\u3001\u539f\u5730\u3001\u7a69\u5b9a\u3001\u81ea\u9069\u61c9\u3001\u901a\u7528\u6027\u597d\u3002\u986f\u7136\uff0c\u8fc4\u4eca\u70ba\u6b62\u5c1a\u672a\u767c\u73fe\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9078\u64c7\u6392\u5e8f\u6f14\u7b97\u6cd5\u6642\uff0c\u9700\u8981\u6839\u64da\u5177\u9ad4\u7684\u8cc7\u6599\u7279\u9ede\u548c\u554f\u984c\u9700\u6c42\u4f86\u6c7a\u5b9a\u3002</p> <p>\u63a5\u4e0b\u4f86\uff0c\u6211\u5011\u5c07\u5171\u540c\u5b78\u7fd2\u5404\u7a2e\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u4e26\u57fa\u65bc\u4e0a\u8ff0\u8a55\u50f9\u7dad\u5ea6\u5c0d\u5404\u500b\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u512a\u7f3a\u9ede\u9032\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_sorting/summary/","title":"11.11 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_sorting/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u6ce1\u6cab\u6392\u5e8f\u900f\u904e\u4ea4\u63db\u76f8\u9130\u5143\u7d20\u4f86\u5be6\u73fe\u6392\u5e8f\u3002\u900f\u904e\u65b0\u589e\u4e00\u500b\u6a19\u8a8c\u4f4d\u4f86\u5be6\u73fe\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u6ce1\u6cab\u6392\u5e8f\u7684\u6700\u4f73\u6642\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u5230 \\(O(n)\\) \u3002</li> <li>\u63d2\u5165\u6392\u5e8f\u6bcf\u8f2a\u5c07\u672a\u6392\u5e8f\u5340\u9593\u5167\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u5340\u9593\u7684\u6b63\u78ba\u4f4d\u7f6e\uff0c\u5f9e\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u96d6\u7136\u63d2\u5165\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \uff0c\u4f46\u7531\u65bc\u55ae\u5143\u64cd\u4f5c\u76f8\u5c0d\u8f03\u5c11\uff0c\u56e0\u6b64\u5728\u5c0f\u8cc7\u6599\u91cf\u7684\u6392\u5e8f\u4efb\u52d9\u4e2d\u975e\u5e38\u53d7\u6b61\u8fce\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\u57fa\u65bc\u54e8\u5175\u5283\u5206\u64cd\u4f5c\u5be6\u73fe\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5283\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9078\u53d6\u5230\u6700\u5dee\u7684\u57fa\u6e96\u6578\uff0c\u5c0e\u81f4\u6642\u9593\u8907\u96dc\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6578\u57fa\u6e96\u6578\u6216\u96a8\u6a5f\u57fa\u6e96\u6578\u53ef\u4ee5\u964d\u4f4e\u9019\u7a2e\u52a3\u5316\u7684\u6a5f\u7387\u3002\u5c3e\u905e\u8ff4\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u6e1b\u5c11\u905e\u8ff4\u6df1\u5ea6\uff0c\u5c07\u7a7a\u9593\u8907\u96dc\u5ea6\u6700\u4f73\u5316\u5230 \\(O(\\log n)\\) \u3002</li> <li>\u5408\u4f75\u6392\u5e8f\u5305\u62ec\u5283\u5206\u548c\u5408\u4e26\u5169\u500b\u968e\u6bb5\uff0c\u5178\u578b\u5730\u9ad4\u73fe\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5408\u4f75\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u9663\u5217\u9700\u8981\u5efa\u7acb\u8f14\u52a9\u9663\u5217\uff0c\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u93c8\u7d50\u4e32\u5217\u7684\u7a7a\u9593\u8907\u96dc\u5ea6\u53ef\u4ee5\u6700\u4f73\u5316\u81f3 \\(O(1)\\) \u3002</li> <li>\u6876\u6392\u5e8f\u5305\u542b\u4e09\u500b\u6b65\u9a5f\uff1a\u8cc7\u6599\u5206\u6876\u3001\u6876\u5167\u6392\u5e8f\u548c\u5408\u4e26\u7d50\u679c\u3002\u5b83\u540c\u6a23\u9ad4\u73fe\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9069\u7528\u65bc\u8cc7\u6599\u9ad4\u91cf\u5f88\u5927\u7684\u60c5\u6cc1\u3002\u6876\u6392\u5e8f\u7684\u95dc\u9375\u5728\u65bc\u5c0d\u8cc7\u6599\u9032\u884c\u5e73\u5747\u5206\u914d\u3002</li> <li>\u8a08\u6578\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u500b\u7279\u4f8b\uff0c\u5b83\u900f\u904e\u7d71\u8a08\u8cc7\u6599\u51fa\u73fe\u7684\u6b21\u6578\u4f86\u5be6\u73fe\u6392\u5e8f\u3002\u8a08\u6578\u6392\u5e8f\u9069\u7528\u65bc\u8cc7\u6599\u91cf\u5927\u4f46\u8cc7\u6599\u7bc4\u570d\u6709\u9650\u7684\u60c5\u6cc1\uff0c\u4e26\u4e14\u8981\u6c42\u8cc7\u6599\u80fd\u5920\u8f49\u63db\u70ba\u6b63\u6574\u6578\u3002</li> <li>\u57fa\u6578\u6392\u5e8f\u900f\u904e\u9010\u4f4d\u6392\u5e8f\u4f86\u5be6\u73fe\u8cc7\u6599\u6392\u5e8f\uff0c\u8981\u6c42\u8cc7\u6599\u80fd\u5920\u8868\u793a\u70ba\u56fa\u5b9a\u4f4d\u6578\u7684\u6578\u5b57\u3002</li> <li>\u7e3d\u7684\u4f86\u8aaa\uff0c\u6211\u5011\u5e0c\u671b\u627e\u5230\u4e00\u7a2e\u6392\u5e8f\u6f14\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a69\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u81ea\u9069\u61c9\u6027\u7b49\u512a\u9ede\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u8cc7\u6599\u7d50\u69cb\u548c\u6f14\u7b97\u6cd5\u4e00\u6a23\uff0c\u6c92\u6709\u4e00\u7a2e\u6392\u5e8f\u6f14\u7b97\u6cd5\u80fd\u5920\u540c\u6642\u6eff\u8db3\u6240\u6709\u9019\u4e9b\u689d\u4ef6\u3002\u5728\u5be6\u969b\u61c9\u7528\u4e2d\uff0c\u6211\u5011\u9700\u8981\u6839\u64da\u8cc7\u6599\u7684\u7279\u6027\u4f86\u9078\u64c7\u5408\u9069\u7684\u6392\u5e8f\u6f14\u7b97\u6cd5\u3002</li> <li>\u5716 11-19 \u5c0d\u6bd4\u4e86\u4e3b\u6d41\u6392\u5e8f\u6f14\u7b97\u6cd5\u7684\u6548\u7387\u3001\u7a69\u5b9a\u6027\u3001\u5c31\u5730\u6027\u548c\u81ea\u9069\u61c9\u6027\u7b49\u3002</li> </ul> <p> \u5716 11-19 \u00a0 \u6392\u5e8f\u6f14\u7b97\u6cd5\u5c0d\u6bd4 </p>"},{"location":"chapter_sorting/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6392\u5e8f\u6f14\u7b97\u6cd5\u7a69\u5b9a\u6027\u5728\u4ec0\u9ebc\u60c5\u6cc1\u4e0b\u662f\u5fc5\u9700\u7684\uff1f</p> <p>\u5728\u73fe\u5be6\u4e2d\uff0c\u6211\u5011\u6709\u53ef\u80fd\u57fa\u65bc\u7269\u4ef6\u7684\u67d0\u500b\u5c6c\u6027\u9032\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b78\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u5169\u500b\u5c6c\u6027\uff0c\u6211\u5011\u5e0c\u671b\u5be6\u73fe\u4e00\u500b\u591a\u7d1a\u6392\u5e8f\uff1a\u5148\u6309\u7167\u59d3\u540d\u9032\u884c\u6392\u5e8f\uff0c\u5f97\u5230 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> \uff1b\u518d\u5c0d\u8eab\u9ad8\u9032\u884c\u6392\u5e8f\u3002\u7531\u65bc\u6392\u5e8f\u6f14\u7b97\u6cd5\u4e0d\u7a69\u5b9a\uff0c\u56e0\u6b64\u53ef\u80fd\u5f97\u5230 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> \u3002</p> <p>\u53ef\u4ee5\u767c\u73fe\uff0c\u5b78\u751f D \u548c C \u7684\u4f4d\u7f6e\u767c\u751f\u4e86\u4ea4\u63db\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u58de\u4e86\uff0c\u800c\u9019\u662f\u6211\u5011\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002</p> <p>Q\uff1a\u54e8\u5175\u5283\u5206\u4e2d\u201c\u5f9e\u53f3\u5f80\u5de6\u67e5\u8a62\u201d\u8207\u201c\u5f9e\u5de6\u5f80\u53f3\u67e5\u8a62\u201d\u7684\u9806\u5e8f\u53ef\u4ee5\u4ea4\u63db\u55ce\uff1f</p> <p>\u4e0d\u884c\uff0c\u7576\u6211\u5011\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u70ba\u57fa\u6e96\u6578\u6642\uff0c\u5fc5\u9808\u5148\u201c\u5f9e\u53f3\u5f80\u5de6\u67e5\u8a62\u201d\u518d\u201c\u5f9e\u5de6\u5f80\u53f3\u67e5\u8a62\u201d\u3002\u9019\u500b\u7d50\u8ad6\u6709\u4e9b\u53cd\u76f4\u89ba\uff0c\u6211\u5011\u4f86\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002</p> <p>\u54e8\u5175\u5283\u5206 <code>partition()</code> \u7684\u6700\u5f8c\u4e00\u6b65\u662f\u4ea4\u63db <code>nums[left]</code> \u548c <code>nums[i]</code> \u3002\u5b8c\u6210\u4ea4\u63db\u5f8c\uff0c\u57fa\u6e96\u6578\u5de6\u908a\u7684\u5143\u7d20\u90fd <code>&lt;=</code> \u57fa\u6e96\u6578\uff0c\u9019\u5c31\u8981\u6c42\u6700\u5f8c\u4e00\u6b65\u4ea4\u63db\u524d <code>nums[left] &gt;= nums[i]</code> \u5fc5\u9808\u6210\u7acb\u3002\u5047\u8a2d\u6211\u5011\u5148\u201c\u5f9e\u5de6\u5f80\u53f3\u67e5\u8a62\u201d\uff0c\u90a3\u9ebc\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u6e96\u6578\u66f4\u5927\u7684\u5143\u7d20\uff0c\u5247\u6703\u5728 <code>i == j</code> \u6642\u8df3\u51fa\u8ff4\u5708\uff0c\u6b64\u6642\u53ef\u80fd <code>nums[j] == nums[i] &gt; nums[left]</code>\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u6b64\u6642\u6700\u5f8c\u4e00\u6b65\u4ea4\u63db\u64cd\u4f5c\u6703\u628a\u4e00\u500b\u6bd4\u57fa\u6e96\u6578\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u63db\u81f3\u9663\u5217\u6700\u5de6\u7aef\uff0c\u5c0e\u81f4\u54e8\u5175\u5283\u5206\u5931\u6557\u3002</p> <p>\u8209\u500b\u4f8b\u5b50\uff0c\u7d66\u5b9a\u9663\u5217 <code>[0, 0, 0, 0, 1]</code> \uff0c\u5982\u679c\u5148\u201c\u5f9e\u5de6\u5411\u53f3\u67e5\u8a62\u201d\uff0c\u54e8\u5175\u5283\u5206\u5f8c\u9663\u5217\u70ba <code>[1, 0, 0, 0, 0]</code> \uff0c\u9019\u500b\u7d50\u679c\u662f\u4e0d\u6b63\u78ba\u7684\u3002</p> <p>\u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u5011\u9078\u64c7 <code>nums[right]</code> \u70ba\u57fa\u6e96\u6578\uff0c\u90a3\u9ebc\u6b63\u597d\u53cd\u904e\u4f86\uff0c\u5fc5\u9808\u5148\u201c\u5f9e\u5de6\u5f80\u53f3\u67e5\u8a62\u201d\u3002</p> <p>Q\uff1a\u95dc\u65bc\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\uff0c\u70ba\u4ec0\u9ebc\u9078\u77ed\u7684\u9663\u5217\u80fd\u4fdd\u8b49\u905e\u8ff4\u6df1\u5ea6\u4e0d\u8d85\u904e \\(\\log n\\) \uff1f</p> <p>\u905e\u8ff4\u6df1\u5ea6\u5c31\u662f\u7576\u524d\u672a\u8fd4\u56de\u7684\u905e\u8ff4\u65b9\u6cd5\u7684\u6578\u91cf\u3002\u6bcf\u8f2a\u54e8\u5175\u5283\u5206\u6211\u5011\u5c07\u539f\u9663\u5217\u5283\u5206\u70ba\u5169\u500b\u5b50\u9663\u5217\u3002\u5728\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\u5f8c\uff0c\u5411\u4e0b\u905e\u8ff4\u7684\u5b50\u9663\u5217\u9577\u5ea6\u6700\u5927\u70ba\u539f\u9663\u5217\u9577\u5ea6\u7684\u4e00\u534a\u3002\u5047\u8a2d\u6700\u5dee\u60c5\u6cc1\uff0c\u4e00\u76f4\u70ba\u4e00\u534a\u9577\u5ea6\uff0c\u90a3\u9ebc\u6700\u7d42\u7684\u905e\u8ff4\u6df1\u5ea6\u5c31\u662f \\(\\log n\\) \u3002</p> <p>\u56de\u9867\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u5011\u6709\u53ef\u80fd\u6703\u9023\u7e8c\u5730\u905e\u8ff4\u9577\u5ea6\u8f03\u5927\u7684\u9663\u5217\uff0c\u6700\u5dee\u60c5\u6cc1\u4e0b\u70ba \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u905e\u8ff4\u6df1\u5ea6\u70ba \\(n\\) \u3002\u5c3e\u905e\u8ff4\u6700\u4f73\u5316\u53ef\u4ee5\u907f\u514d\u9019\u7a2e\u60c5\u6cc1\u51fa\u73fe\u3002</p> <p>Q\uff1a\u7576\u9663\u5217\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u6642\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f \\(O(n^2)\\) \u55ce\uff1f\u8a72\u5982\u4f55\u8655\u7406\u9019\u7a2e\u9000\u5316\u60c5\u6cc1\uff1f</p> <p>\u662f\u7684\u3002\u5c0d\u65bc\u9019\u7a2e\u60c5\u6cc1\uff0c\u53ef\u4ee5\u8003\u616e\u900f\u904e\u54e8\u5175\u5283\u5206\u5c07\u9663\u5217\u5283\u5206\u70ba\u4e09\u500b\u90e8\u5206\uff1a\u5c0f\u65bc\u3001\u7b49\u65bc\u3001\u5927\u65bc\u57fa\u6e96\u6578\u3002\u50c5\u5411\u4e0b\u905e\u8ff4\u5c0f\u65bc\u548c\u5927\u65bc\u7684\u5169\u90e8\u5206\u3002\u5728\u8a72\u65b9\u6cd5\u4e0b\uff0c\u8f38\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u9663\u5217\uff0c\u50c5\u4e00\u8f2a\u54e8\u5175\u5283\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002</p> <p>Q\uff1a\u6876\u6392\u5e8f\u7684\u6700\u5dee\u6642\u9593\u8907\u96dc\u5ea6\u70ba\u4ec0\u9ebc\u662f \\(O(n^2)\\) \uff1f</p> <p>\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u500b\u6876\u4e2d\u3002\u5982\u679c\u6211\u5011\u63a1\u7528\u4e00\u500b \\(O(n^2)\\) \u6f14\u7b97\u6cd5\u4f86\u6392\u5e8f\u9019\u4e9b\u5143\u7d20\uff0c\u5247\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_stack_and_queue/","title":"\u7b2c 5 \u7ae0 \u00a0 \u5806\u758a\u8207\u4f47\u5217","text":"<p>Abstract</p> <p>\u5806\u758a\u5982\u540c\u758a\u8c93\u8c93\uff0c\u800c\u4f47\u5217\u5c31\u50cf\u8c93\u8c93\u6392\u968a\u3002</p> <p>\u5169\u8005\u5206\u5225\u4ee3\u8868\u5148\u5165\u5f8c\u51fa\u548c\u5148\u5165\u5148\u51fa\u7684\u908f\u8f2f\u95dc\u4fc2\u3002</p>"},{"location":"chapter_stack_and_queue/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>5.1 \u00a0 \u5806\u758a</li> <li>5.2 \u00a0 \u4f47\u5217</li> <li>5.3 \u00a0 \u96d9\u5411\u4f47\u5217</li> <li>5.4 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_stack_and_queue/deque/","title":"5.3 \u00a0 \u96d9\u5411\u4f47\u5217","text":"<p>\u5728\u4f47\u5217\u4e2d\uff0c\u6211\u5011\u50c5\u80fd\u522a\u9664\u982d\u90e8\u5143\u7d20\u6216\u5728\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\u3002\u5982\u5716 5-7 \u6240\u793a\uff0c\u96d9\u5411\u4f47\u5217\uff08double-ended queue\uff09\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u9748\u6d3b\u6027\uff0c\u5141\u8a31\u5728\u982d\u90e8\u548c\u5c3e\u90e8\u57f7\u884c\u5143\u7d20\u7684\u65b0\u589e\u6216\u522a\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u5716 5-7 \u00a0 \u96d9\u5411\u4f47\u5217\u7684\u64cd\u4f5c </p>"},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1 \u00a0 \u96d9\u5411\u4f47\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u96d9\u5411\u4f47\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-3 \u6240\u793a\uff0c\u5177\u9ad4\u7684\u65b9\u6cd5\u540d\u7a31\u9700\u8981\u6839\u64da\u6240\u4f7f\u7528\u7684\u7a0b\u5f0f\u8a9e\u8a00\u4f86\u78ba\u5b9a\u3002</p> <p> \u8868 5-3 \u00a0 \u96d9\u5411\u4f47\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u6642\u9593\u8907\u96dc\u5ea6 <code>push_first()</code> \u5c07\u5143\u7d20\u65b0\u589e\u81f3\u4f47\u5217\u9996 \\(O(1)\\) <code>push_last()</code> \u5c07\u5143\u7d20\u65b0\u589e\u81f3\u4f47\u5217\u5c3e \\(O(1)\\) <code>pop_first()</code> \u522a\u9664\u4f47\u5217\u9996\u5143\u7d20 \\(O(1)\\) <code>pop_last()</code> \u522a\u9664\u4f47\u5217\u5c3e\u5143\u7d20 \\(O(1)\\) <code>peek_first()</code> \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 \\(O(1)\\) <code>peek_last()</code> \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 \\(O(1)\\) <p>\u540c\u6a23\u5730\uff0c\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u5df2\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217\u985e\u5225\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig deque.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217\ndeq: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u5217\ndeq.append(2)      # \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeq.append(5)\ndeq.append(4)\ndeq.appendleft(3)  # \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeq.appendleft(1)\n\n# \u8a2a\u554f\u5143\u7d20\nfront: int = deq[0]  # \u4f47\u5217\u9996\u5143\u7d20\nrear: int = deq[-1]  # \u4f47\u5217\u5c3e\u5143\u7d20\n\n# \u5143\u7d20\u51fa\u5217\npop_front: int = deq.popleft()  # \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\npop_rear: int = deq.pop()       # \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n# \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\nsize: int = len(deq)\n\n# \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\nis_empty: bool = len(deq) == 0\n</code></pre> deque.cpp<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\ndeque&lt;int&gt; deque;\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push_back(2);   // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.push_front(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint front = deque.front(); // \u4f47\u5217\u9996\u5143\u7d20\nint back = deque.back();   // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\ndeque.pop_front();  // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\ndeque.pop_back();   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.offerLast(2);   // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.offerFirst(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u4f47\u5217\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\nint popFirst = deque.pollFirst();  // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\nint popLast = deque.pollLast();    // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c07\u93c8\u7d50\u4e32\u5217 LinkedList \u770b\u4f5c\u96d9\u5411\u4f47\u5217\u4f86\u4f7f\u7528\nLinkedList&lt;int&gt; deque = new();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.AddLast(2);   // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.AddFirst(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u4f47\u5217\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\ndeque.RemoveFirst();  // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\ndeque.RemoveLast();   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = deque.Count;\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c07 list \u4f5c\u70ba\u96d9\u5411\u4f47\u5217\u4f7f\u7528\ndeque := list.New()\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.PushBack(2)      // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.PushFront(1)\n\n/* \u8a2a\u554f\u5143\u7d20 */\nfront := deque.Front() // \u4f47\u5217\u9996\u5143\u7d20\nrear := deque.Back()   // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\ndeque.Remove(front)    // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\ndeque.Remove(rear)     // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nsize := deque.Len()\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// Swift \u6c92\u6709\u5167\u5efa\u7684\u96d9\u5411\u4f47\u5217\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u96d9\u5411\u4f47\u5217\u4f86\u4f7f\u7528\nvar deque: [Int] = []\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.append(2) // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.insert(1, at: 0)\n\n/* \u8a2a\u554f\u5143\u7d20 */\nlet peekFirst = deque.first! // \u4f47\u5217\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u4f7f\u7528 Array \u6a21\u64ec\u6642 popFirst \u7684\u8907\u96dc\u5ea6\u70ba O(n)\nlet popFirst = deque.removeFirst() // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\nlet popLast = deque.removeLast() // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nlet size = deque.count\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// JavaScript \u6c92\u6709\u5167\u5efa\u7684\u96d9\u7aef\u4f47\u5217\uff0c\u53ea\u80fd\u628a Array \u7576\u4f5c\u96d9\u7aef\u4f47\u5217\u4f86\u4f7f\u7528\nconst deque = [];\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cunshift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nconst peekFirst = deque[0];\nconst peekLast = deque[deque.length - 1];\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cshift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\nconst popFront = deque.shift();\nconst popBack = deque.pop();\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nconst size = deque.length;\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// TypeScript \u6c92\u6709\u5167\u5efa\u7684\u96d9\u7aef\u4f47\u5217\uff0c\u53ea\u80fd\u628a Array \u7576\u4f5c\u96d9\u7aef\u4f47\u5217\u4f86\u4f7f\u7528\nconst deque: number[] = [];\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cunshift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconst peekLast: number = deque[deque.length - 1];\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cshift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\nconst popFront: number = deque.shift() as number;\nconst popBack: number = deque.pop() as number;\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nconst size: number = deque.length;\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u7fa9\u70ba\u96d9\u5411\u4f47\u5217\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.addLast(2);  // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.addFirst(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nint peekFirst = deque.first; // \u4f47\u5217\u9996\u5143\u7d20\nint peekLast = deque.last;   // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\nint popFirst = deque.removeFirst(); // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\nint popLast = deque.removeLast();   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = deque.length;\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push_back(2);  // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.push_front(1);\n\n/* \u8a2a\u554f\u5143\u7d20 */\nif let Some(front) = deque.front() { // \u4f47\u5217\u9996\u5143\u7d20\n}\nif let Some(rear) = deque.back() {   // \u4f47\u5217\u5c3e\u5143\u7d20\n}\n\n/* \u5143\u7d20\u51fa\u5217 */\nif let Some(pop_front) = deque.pop_front() { // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\n}\nif let Some(pop_rear) = deque.pop_back() {   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n}\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u96d9\u5411\u4f47\u5217\n</code></pre> deque.kt<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\nval deque = LinkedList&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.offerLast(2)  // \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\ndeque.offerLast(5)\ndeque.offerLast(4)\ndeque.offerFirst(3) // \u65b0\u589e\u81f3\u4f47\u5217\u9996\ndeque.offerFirst(1)\n\n/* \u8a2a\u554f\u5143\u7d20 */\nval peekFirst = deque.peekFirst() // \u4f47\u5217\u9996\u5143\u7d20\nval peekLast = deque.peekLast()   // \u4f47\u5217\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u5217 */\nval popFirst = deque.pollFirst() // \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217\nval popLast = deque.pollLast()   // \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nval size = deque.size\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nval isEmpty = deque.isEmpty()\n</code></pre> deque.rb<pre><code># \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217\n# Ruby \u6c92\u6709\u5167\u76f4\u7684\u96d9\u7aef\u4f47\u5217\uff0c\u53ea\u80fd\u628a Array \u7576\u4f5c\u96d9\u7aef\u4f47\u5217\u4f86\u4f7f\u7528\ndeque = []\n\n# \u5143\u7d20\u5982\u968a\ndeque &lt;&lt; 2\ndeque &lt;&lt; 5\ndeque &lt;&lt; 4\n# \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cArray#unshift \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\ndeque.unshift(3)\ndeque.unshift(1)\n\n# \u8a2a\u554f\u5143\u7d20\npeek_first = deque.first\npeek_last = deque.last\n\n# \u5143\u7d20\u51fa\u5217\n# \u8acb\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0c Array#shift \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\npop_front = deque.shift\npop_back = deque.pop\n\n# \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\nsize = deque.length\n\n# \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\nis_empty = size.zero?\n</code></pre> deque.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2 \u00a0 \u96d9\u5411\u4f47\u5217\u5be6\u73fe *","text":"<p>\u96d9\u5411\u4f47\u5217\u7684\u5be6\u73fe\u8207\u4f47\u5217\u985e\u4f3c\uff0c\u53ef\u4ee5\u9078\u64c7\u93c8\u7d50\u4e32\u5217\u6216\u9663\u5217\u4f5c\u70ba\u5e95\u5c64\u8cc7\u6599\u7d50\u69cb\u3002</p>"},{"location":"chapter_stack_and_queue/deque/#1","title":"1. \u00a0 \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7684\u5be6\u73fe","text":"<p>\u56de\u9867\u4e0a\u4e00\u7bc0\u5167\u5bb9\uff0c\u6211\u5011\u4f7f\u7528\u666e\u901a\u55ae\u5411\u93c8\u7d50\u4e32\u5217\u4f86\u5be6\u73fe\u4f47\u5217\uff0c\u56e0\u70ba\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u522a\u9664\u982d\u7bc0\u9ede\uff08\u5c0d\u61c9\u51fa\u5217\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e\u65b0\u7bc0\u9ede\uff08\u5c0d\u61c9\u5165\u5217\u64cd\u4f5c\uff09\u3002</p> <p>\u5c0d\u65bc\u96d9\u5411\u4f47\u5217\u800c\u8a00\uff0c\u982d\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u57f7\u884c\u5165\u5217\u548c\u51fa\u5217\u64cd\u4f5c\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u96d9\u5411\u4f47\u5217\u9700\u8981\u5be6\u73fe\u53e6\u4e00\u500b\u5c0d\u7a31\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u70ba\u6b64\uff0c\u6211\u5011\u63a1\u7528\u201c\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u201d\u4f5c\u70ba\u96d9\u5411\u4f47\u5217\u7684\u5e95\u5c64\u8cc7\u6599\u7d50\u69cb\u3002</p> <p>\u5982\u5716 5-8 \u6240\u793a\uff0c\u6211\u5011\u5c07\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7684\u982d\u7bc0\u9ede\u548c\u5c3e\u7bc0\u9ede\u8996\u70ba\u96d9\u5411\u4f47\u5217\u7684\u4f47\u5217\u9996\u548c\u4f47\u5217\u5c3e\uff0c\u540c\u6642\u5be6\u73fe\u5728\u5169\u7aef\u65b0\u589e\u548c\u522a\u9664\u7bc0\u9ede\u7684\u529f\u80fd\u3002</p> LinkedListDequepush_last()push_first()pop_last()pop_first() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 5-8 \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u96d9\u5411\u4f47\u5217\u7684\u5165\u5217\u51fa\u5217\u64cd\u4f5c </p> <p>\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n        self.prev: ListNode | None = None  # \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n\nclass LinkedListDeque:\n    \"\"\"\u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._front: ListNode | None = None  # \u982d\u7bc0\u9ede front\n        self._rear: ListNode | None = None  # \u5c3e\u7bc0\u9ede rear\n        self._size: int = 0  # \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"\u5165\u5217\u64cd\u4f5c\"\"\"\n        node = ListNode(num)\n        # \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if self.is_empty():\n            self._front = self._rear = node\n        # \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        elif is_front:\n            # \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # \u66f4\u65b0\u982d\u7bc0\u9ede\n        # \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        else:\n            # \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        self._size += 1  # \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n\n    def push_first(self, num: int):\n        \"\"\"\u4f47\u5217\u9996\u5165\u5217\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"\u4f47\u5217\u5c3e\u5165\u5217\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"\u51fa\u5217\u64cd\u4f5c\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        # \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if is_front:\n            val: int = self._front.val  # \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            # \u522a\u9664\u982d\u7bc0\u9ede\n            fnext: ListNode | None = self._front.next\n            if fnext != None:\n                fnext.prev = None\n                self._front.next = None\n            self._front = fnext  # \u66f4\u65b0\u982d\u7bc0\u9ede\n        # \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        else:\n            val: int = self._rear.val  # \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            # \u522a\u9664\u5c3e\u7bc0\u9ede\n            rprev: ListNode | None = self._rear.prev\n            if rprev != None:\n                rprev.next = None\n                self._rear.prev = None\n            self._rear = rprev  # \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        self._size -= 1  # \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u4f47\u5217\u9996\u51fa\u5217\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u4f47\u5217\u5c3e\u51fa\u5217\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nstruct DoublyListNode {\n    int val;              // \u7bc0\u9ede\u503c\n    DoublyListNode *next; // \u5f8c\u7e7c\u7bc0\u9ede\u6307\u6a19\n    DoublyListNode *prev; // \u524d\u9a45\u7bc0\u9ede\u6307\u6a19\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    int queSize = 0;              // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* \u6790\u69cb\u65b9\u6cd5 */\n    ~LinkedListDeque() {\n        // \u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u522a\u9664\u7bc0\u9ede\uff0c\u91cb\u653e\u8a18\u61b6\u9ad4\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"\u4f47\u5217\u70ba\u7a7a\");\n        int val;\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if (isFront) {\n            val = front-&gt;val; // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            DoublyListNode *fNext = front-&gt;next;\n            if (fNext != nullptr) {\n                fNext-&gt;prev = nullptr;\n                front-&gt;next = nullptr;\n            }\n            delete front;\n            front = fNext; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        } else {\n            val = rear-&gt;val; // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            DoublyListNode *rPrev = rear-&gt;prev;\n            if (rPrev != nullptr) {\n                rPrev-&gt;next = nullptr;\n                rear-&gt;prev = nullptr;\n            }\n            delete rear;\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n        return rear-&gt;val;\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n    int val; // \u7bc0\u9ede\u503c\n    ListNode next; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n    ListNode prev; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    private ListNode front, rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    private int queSize = 0; // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if (isFront) {\n            val = front.val; // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            ListNode fNext = front.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front.next = null;\n            }\n            front = fNext; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        } else {\n            val = rear.val; // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            ListNode rPrev = rear.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear.prev = null;\n            }\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode(int val) {\n    public int val = val;       // \u7bc0\u9ede\u503c\n    public ListNode? next = null; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n    public ListNode? prev = null; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    ListNode? front, rear; // \u982d\u7bc0\u9ede front, \u5c3e\u7bc0\u9ede rear\n    int queSize = 0;      // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front!.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede                           \n        }\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n\n        queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if (isFront) {\n            val = front?.val; // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            ListNode? fNext = front?.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front!.next = null;\n            }\n            front = fNext;   // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        else {\n            val = rear?.val;  // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            ListNode? rPrev = rear?.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear!.prev = null;\n            }\n            rear = rPrev;    // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n\n        queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\ntype linkedListDeque struct {\n    // \u4f7f\u7528\u5167\u5efa\u5305 list\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u96d9\u7aef\u4f47\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* \u4f47\u5217\u9996\u5143\u7d20\u5165\u5217 */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* \u4f47\u5217\u5c3e\u5143\u7d20\u5165\u5217 */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217 */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u4f47\u5217\u5c3e\u5143\u7d20\u51fa\u5217 */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u7372\u53d6 List \u7528\u65bc\u5217\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var next: ListNode? // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n    weak var prev: ListNode? // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? // \u982d\u7bc0\u9ede front\n    private var rear: ListNode? // \u5c3e\u7bc0\u9ede rear\n    private var _size: Int // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        else if isFront {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        _size += 1 // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        let val: Int\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if isFront {\n            val = front!.val // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            let fNext = front?.next\n            if fNext != nil {\n                fNext?.prev = nil\n                front?.next = nil\n            }\n            front = fNext // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        else {\n            val = rear!.val // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            let rPrev = rear?.prev\n            if rPrev != nil {\n                rPrev?.next = nil\n                rear?.prev = nil\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        _size -= 1 // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return val\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return rear!.val\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n    prev; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528 (\u6307\u6a19)\n    next; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528 (\u6307\u6a19)\n    val; // \u7bc0\u9ede\u503c\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    #front; // \u982d\u7bc0\u9ede front\n    #rear; // \u5c3e\u7bc0\u9ede rear\n    #queSize; // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        this.#queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        this.#queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // \u5132\u5b58\u5c3e\u7bc0\u9ede\u503c\n        // \u522a\u9664\u5c3e\u7bc0\u9ede\n        let temp = this.#rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.#rear.prev = null;\n        }\n        this.#rear = temp; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // \u5132\u5b58\u5c3e\u7bc0\u9ede\u503c\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        let temp = this.#front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.#front.next = null;\n        }\n        this.#front = temp; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5217\u5370\u96d9\u5411\u4f47\u5217 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n    prev: ListNode; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528 (\u6307\u6a19)\n    next: ListNode; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528 (\u6307\u6a19)\n    val: number; // \u7bc0\u9ede\u503c\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    private front: ListNode; // \u982d\u7bc0\u9ede front\n    private rear: ListNode; // \u5c3e\u7bc0\u9ede rear\n    private queSize: number; // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        this.queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        }\n        this.queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear.val; // \u5132\u5b58\u5c3e\u7bc0\u9ede\u503c\n        // \u522a\u9664\u5c3e\u7bc0\u9ede\n        let temp: ListNode = this.rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.rear.prev = null;\n        }\n        this.rear = temp; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        this.queSize--;\n        return value;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // \u5132\u5b58\u5c3e\u7bc0\u9ede\u503c\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        let temp: ListNode = this.front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.front.next = null;\n        }\n        this.front = temp; // \u66f4\u65b0\u982d\u7bc0\u9ede\n        this.queSize--;\n        return value;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5217\u5370\u96d9\u5411\u4f47\u5217 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode {\n  int val; // \u7bc0\u9ede\u503c\n  ListNode? next; // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n  ListNode? prev; // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u5c0d\u5217 */\nclass LinkedListDeque {\n  late ListNode? _front; // \u982d\u7bc0\u9ede _front\n  late ListNode? _rear; // \u5c3e\u7bc0\u9ede _rear\n  int _queSize = 0; // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u9577\u5ea6 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* \u5165\u5217\u64cd\u4f5c */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n      _front = _rear = node;\n    } else if (isFront) {\n      // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n      // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n    } else {\n      // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n      // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    }\n    _queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n  }\n\n  /* \u4f47\u5217\u9996\u5165\u5217 */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* \u4f47\u5217\u5c3e\u5165\u5217 */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* \u51fa\u5217\u64cd\u4f5c */\n  int? pop(bool isFront) {\n    // \u82e5\u4f47\u5217\u70ba\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    if (isFront) {\n      // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n      val = _front!.val; // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n      // \u522a\u9664\u982d\u7bc0\u9ede\n      ListNode? fNext = _front!.next;\n      if (fNext != null) {\n        fNext.prev = null;\n        _front!.next = null;\n      }\n      _front = fNext; // \u66f4\u65b0\u982d\u7bc0\u9ede\n    } else {\n      // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n      val = _rear!.val; // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n      // \u522a\u9664\u5c3e\u7bc0\u9ede\n      ListNode? rPrev = _rear!.prev;\n      if (rPrev != null) {\n        rPrev.next = null;\n        _rear!.prev = null;\n      }\n      _rear = rPrev; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    }\n    _queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    return val;\n  }\n\n  /* \u4f47\u5217\u9996\u51fa\u5217 */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // \u7bc0\u9ede\u503c\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5f8c\u7e7c\u7bc0\u9ede\u6307\u6a19\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u524d\u9a45\u7bc0\u9ede\u6307\u6a19\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u982d\u7bc0\u9ede front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u7bc0\u9ede rear\n    que_size: usize,                         // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    pub fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        if is_front {\n            match self.front.take() {\n                // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // \u66f4\u65b0\u982d\u7bc0\u9ede\n                }\n            }\n        }\n        // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        else {\n            match self.rear.take() {\n                // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n                }\n            }\n        }\n        self.que_size += 1; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    pub fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // \u82e5\u4f47\u5217\u70ba\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n        if self.is_empty() {\n            return None;\n        };\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // \u66f4\u65b0\u982d\u7bc0\u9ede\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n                Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n            })\n        }\n        // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n                Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n            })\n        }\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\ntypedef struct DoublyListNode {\n    int val;                     // \u7bc0\u9ede\u503c\n    struct DoublyListNode *next; // \u5f8c\u7e7c\u7bc0\u9ede\n    struct DoublyListNode *prev; // \u524d\u9a45\u7bc0\u9ede\n} DoublyListNode;\n\n/* \u5efa\u69cb\u5b50 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\ntypedef struct {\n    DoublyListNode *front, *rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    int queSize;                  // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n} LinkedListDeque;\n\n/* \u5efa\u69cb\u5b50 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // \u91cb\u653e\u6240\u6709\u7bc0\u9ede\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91cb\u653e deque \u7d50\u69cb\u9ad4\n    free(deque);\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* \u5165\u5217 */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n    else if (isFront) {\n        // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // \u66f4\u65b0\u982d\u7bc0\u9ede\n    }\n    // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n    else {\n        // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n}\n\n/* \u4f47\u5217\u9996\u5165\u5217 */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* \u4f47\u5217\u5c3e\u5165\u5217 */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* \u51fa\u5217 */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n    if (isFront) {\n        val = peekFirst(deque); // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        if (fNext) {\n            fNext-&gt;prev = NULL;\n            deque-&gt;front-&gt;next = NULL;\n        }\n        delDoublyListNode(deque-&gt;front);\n        deque-&gt;front = fNext; // \u66f4\u65b0\u982d\u7bc0\u9ede\n    }\n    // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n    else {\n        val = peekLast(deque); // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        if (rPrev) {\n            rPrev-&gt;next = NULL;\n            deque-&gt;rear-&gt;prev = NULL;\n        }\n        delDoublyListNode(deque-&gt;rear);\n        deque-&gt;rear = rPrev; // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    }\n    deque-&gt;queSize--; // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    return val;\n}\n\n/* \u4f47\u5217\u9996\u51fa\u5217 */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* \u4f47\u5217\u5c3e\u51fa\u5217 */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* \u5217\u5370\u4f47\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // \u8907\u88fd\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u8cc7\u6599\u5230\u9663\u5217\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede */\nclass ListNode(var _val: Int) {\n    // \u7bc0\u9ede\u503c\n    var next: ListNode? = null // \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n    var prev: ListNode? = null // \u524d\u9a45\u7bc0\u9ede\u5f15\u7528\n}\n\n/* \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? = null // \u982d\u7bc0\u9ede front\n    private var rear: ListNode? = null // \u5c3e\u7bc0\u9ede rear\n    private var queSize: Int = 0 // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u5217\u64cd\u4f5c */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n        } else if (isFront) {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u982d\u7bc0\u9ede\n            // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n        } else {\n            // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize++ // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* \u51fa\u5217\u64cd\u4f5c */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) \n            throw IndexOutOfBoundsException()\n        val _val: Int\n        // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n        if (isFront) {\n            _val = front!!._val // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            val fNext = front!!.next\n            if (fNext != null) {\n                fNext.prev = null\n                front!!.next = null\n            }\n            front = fNext // \u66f4\u65b0\u982d\u7bc0\u9ede\n            // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n        } else {\n            _val = rear!!._val // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n            // \u522a\u9664\u5c3e\u7bc0\u9ede\n            val rPrev = rear!!.prev\n            if (rPrev != null) {\n                rPrev.next = null\n                rear!!.prev = null\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n        }\n        queSize-- // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        return _val\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!._val\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>=begin\nFile: linkedlist_deque.rb\nCreated Time: 2024-04-06\nAuthor: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)\n=end\n\n### \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\nclass ListNode\n  attr_accessor :val\n  attr_accessor :next # \u5f8c\u7e7c\u7bc0\u9ede\u5f15\u7528\n  attr_accessor :prev # \u524d\u8ec0\u7bc0\u9ede\u5f15\u7528\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(val)\n    @val = val\n  end\nend\n\n### \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 ###\nclass LinkedListDeque\n  ### \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 ###\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @front = nil  # \u982d\u7bc0\u9ede front\n    @rear = nil   # \u5c3e\u7bc0\u9ede rear\n    @size = 0     # \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n  end\n\n  ### \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### \u5165\u5217\u64cd\u4f5c ###\n  def push(num, is_front)\n    node = ListNode.new(num)\n    # \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c \u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n    if is_empty?\n      @front = @rear = node\n    # \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n    elsif is_front\n      # \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n      @front.prev = node\n      node.next = @front\n      @front = node # \u66f4\u65b0\u982d\u7bc0\u9ede\n    # \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n    else\n      # \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n      @rear.next = node\n      node.prev = @rear\n      @rear = node # \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    end\n    @size += 1 # \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n  end\n\n  ### \u4f47\u5217\u9996\u5165\u5217 ###\n  def push_first(num)\n    push(num, true)\n  end\n\n  ### \u4f47\u5217\u5c3e\u5165\u5217 ###\n  def push_last(num)\n    push(num, false)\n  end\n\n  ### \u51fa\u5217\u64cd\u4f5c ###\n  def pop(is_front)\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    # \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n    if is_front\n      val = @front.val # \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n      # \u522a\u9664\u982d\u7bc0\u9ede\n      fnext = @front.next\n      unless fnext.nil?\n        fnext.prev = nil\n        @front.next = nil\n      end\n      @front = fnext # \u66f4\u65b0\u982d\u7bc0\u9ede\n    # \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n    else\n      val = @rear.val # \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n      # \u522a\u9664\u5c3e\u7bc0\u9ede\n      rprev = @rear.prev\n      unless rprev.nil?\n        rprev.next = nil\n        @rear.prev = nil\n      end\n      @rear = rprev # \u66f4\u65b0\u5c3e\u7bc0\u9ede\n    end\n    @size -= 1 # \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n\n    val\n  end\n\n  ### \u4f47\u5217\u9996\u51fa\u5217 ###\n  def pop_first\n    pop(true)\n  end\n\n  ### \u4f47\u5217\u9996\u51fa\u5217 ###\n  def pop_last\n    pop(false)\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 ###\n  def peek_first\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @front.val\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 ###\n  def peek_last\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @rear.val\n  end\n\n  ### \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 ###\n  def to_array\n    node = @front\n    res = Array.new(size, 0)\n    for i in 0...size\n      res[i] = node.val\n      node = node.next\n    end\n    res\n  end\nend\n</code></pre> linkedlist_deque.zig<pre><code>// \u96d9\u5411\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\nfn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = undefined,     // \u7bc0\u9ede\u503c\n        next: ?*Self = null,    // \u5f8c\u7e7c\u7bc0\u9ede\u6307\u6a19\n        prev: ?*Self = null,    // \u524d\u9a45\u7bc0\u9ede\u6307\u6a19\n\n        // Initialize a list node with specific value\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n\n// \u57fa\u65bc\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217\nfn LinkedListDeque(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*ListNode(T) = null,                    // \u982d\u7bc0\u9ede front\n        rear: ?*ListNode(T) = null,                     // \u5c3e\u7bc0\u9ede rear\n        que_size: usize = 0,                             // \u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u4f47\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u5165\u5217\u64cd\u4f5c\n        pub fn push(self: *Self, num: T, is_front: bool) !void {\n            var node = try self.mem_allocator.create(ListNode(T));\n            node.init(num);\n            // \u82e5\u93c8\u7d50\u4e32\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n            if (self.isEmpty()) {\n                self.front = node;\n                self.rear = node;\n            // \u4f47\u5217\u9996\u5165\u5217\u64cd\u4f5c\n            } else if (is_front) {\n                // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u982d\u90e8\n                self.front.?.prev = node;\n                node.next = self.front;\n                self.front = node;  // \u66f4\u65b0\u982d\u7bc0\u9ede\n            // \u4f47\u5217\u5c3e\u5165\u5217\u64cd\u4f5c\n            } else {\n                // \u5c07 node \u65b0\u589e\u81f3\u93c8\u7d50\u4e32\u5217\u5c3e\u90e8\n                self.rear.?.next = node;\n                node.prev = self.rear;\n                self.rear = node;   // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n            }\n            self.que_size += 1;      // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n        } \n\n        // \u4f47\u5217\u9996\u5165\u5217\n        pub fn pushFirst(self: *Self, num: T) !void {\n            try self.push(num, true);\n        } \n\n        // \u4f47\u5217\u5c3e\u5165\u5217\n        pub fn pushLast(self: *Self, num: T) !void {\n            try self.push(num, false);\n        } \n\n        // \u51fa\u5217\u64cd\u4f5c\n        pub fn pop(self: *Self, is_front: bool) T {\n            if (self.isEmpty()) @panic(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n            var val: T = undefined;\n            // \u4f47\u5217\u9996\u51fa\u5217\u64cd\u4f5c\n            if (is_front) {\n                val = self.front.?.val;     // \u66ab\u5b58\u982d\u7bc0\u9ede\u503c\n                // \u522a\u9664\u982d\u7bc0\u9ede\n                var fNext = self.front.?.next;\n                if (fNext != null) {\n                    fNext.?.prev = null;\n                    self.front.?.next = null;\n                }\n                self.front = fNext;         // \u66f4\u65b0\u982d\u7bc0\u9ede\n            // \u4f47\u5217\u5c3e\u51fa\u5217\u64cd\u4f5c\n            } else {\n                val = self.rear.?.val;      // \u66ab\u5b58\u5c3e\u7bc0\u9ede\u503c\n                // \u522a\u9664\u5c3e\u7bc0\u9ede\n                var rPrev = self.rear.?.prev;\n                if (rPrev != null) {\n                    rPrev.?.next = null;\n                    self.rear.?.prev = null;\n                }\n                self.rear = rPrev;          // \u66f4\u65b0\u5c3e\u7bc0\u9ede\n            }\n            self.que_size -= 1;              // \u66f4\u65b0\u4f47\u5217\u9577\u5ea6\n            return val;\n        } \n\n        // \u4f47\u5217\u9996\u51fa\u5217\n        pub fn popFirst(self: *Self) T {\n            return self.pop(true);\n        } \n\n        // \u4f47\u5217\u5c3e\u51fa\u5217\n        pub fn popLast(self: *Self) T {\n            return self.pop(false);\n        } \n\n        // \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\n        pub fn peekFirst(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20\n        pub fn peekLast(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n            return self.rear.?.val;\n        }\n\n        // \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#2","title":"2. \u00a0 \u57fa\u65bc\u9663\u5217\u7684\u5be6\u73fe","text":"<p>\u5982\u5716 5-9 \u6240\u793a\uff0c\u8207\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u4f47\u5217\u985e\u4f3c\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u4f7f\u7528\u74b0\u5f62\u9663\u5217\u4f86\u5be6\u73fe\u96d9\u5411\u4f47\u5217\u3002</p> ArrayDequepush_last()push_first()pop_last()pop_first() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 5-9 \u00a0 \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u96d9\u5411\u4f47\u5217\u7684\u5165\u5217\u51fa\u5217\u64cd\u4f5c </p> <p>\u5728\u4f47\u5217\u7684\u5be6\u73fe\u57fa\u790e\u4e0a\uff0c\u50c5\u9700\u589e\u52a0\u201c\u4f47\u5217\u9996\u5165\u5217\u201d\u548c\u201c\u4f47\u5217\u5c3e\u51fa\u5217\u201d\u7684\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"\u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"\u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15\"\"\"\n        # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        # \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        # \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"\u4f47\u5217\u9996\u5165\u5217\"\"\"\n        if self._size == self.capacity():\n            print(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        self._front = self.index(self._front - 1)\n        # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"\u4f47\u5217\u5c3e\u5165\u5217\"\"\"\n        if self._size == self.capacity():\n            print(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        # \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        rear = self.index(self._front + self._size)\n        # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u4f47\u5217\u9996\u51fa\u5217\"\"\"\n        num = self.peek_first()\n        # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u4f47\u5217\u5c3e\u51fa\u5217\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        # \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        # \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;        // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize;      // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    int index(int i) {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\" &lt;&lt; endl;\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\" &lt;&lt; endl;\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    int popFirst() {\n        int num = peekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    private int[] nums; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private int front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private int queSize; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    private int index(int i) {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    public int popFirst() {\n        int num = peekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    public int[] toArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    int[] nums;  // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;   // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    int Index(int i) {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = Index(front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        int rear = Index(front + queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    public int[] ToArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\ntype arrayDeque struct {\n    nums        []int // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    front       int   // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    queSize     int   // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n    queCapacity int   // \u4f47\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7d0d\u5143\u7d20\u6578\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u4f47\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u4f47\u5217\u9996\u5165\u5217 */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n        return\n    }\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n    q.front = q.index(q.front - 1)\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* \u4f47\u5217\u5c3e\u5165\u5217 */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n        return\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    rear := q.index(q.front + q.queSize)\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u4f47\u5217\u9996\u51fa\u5217 */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    if num == nil {\n        return nil\n    }\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* \u4f47\u5217\u5c3e\u51fa\u5217 */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    if num == nil {\n        return nil\n    }\n    q.queSize--\n    return num\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* \u7372\u53d6 Slice \u7528\u65bc\u5217\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n    // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    private var nums: [Int] // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private var front: Int // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private var _size: Int // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    private func index(i: Int) -&gt; Int {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        (i + capacity()) % capacity()\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = index(i: front - 1)\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num\n        _size += 1\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        let rear = index(i: front + size())\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        }\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    func toArray() -&gt; [Int] {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    #nums; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    #front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    #queSize; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    index(i) {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u96d9\u5411\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        this.#front = this.index(this.#front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u96d9\u5411\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        const rear = this.index(this.#front + this.#queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    popFirst() {\n        const num = this.peekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    toArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n    private nums: number[]; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private front: number; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private queSize: number; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    index(i: number): number {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u96d9\u5411\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        this.front = this.index(this.front - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u96d9\u5411\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        const rear: number = this.index(this.front + this.queSize);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.nums[this.front];\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    toArray(): number[] {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n  late int _front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n  late int _queSize; // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n  /* \u5efa\u69cb\u5b50 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n  int index(int i) {\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + capacity()) % capacity();\n  }\n\n  /* \u4f47\u5217\u9996\u5165\u5217 */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n    }\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe _front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n    _front = index(_front - 1);\n    // \u5c07 _num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* \u4f47\u5217\u5c3e\u5165\u5217 */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    int rear = index(_front + _queSize);\n    // \u5c07 _num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u4f47\u5217\u9996\u51fa\u5217 */\n  int popFirst() {\n    int _num = peekFirst();\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u53f3\u79fb\u52d5\u4e00\u4f4d\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\");\n    }\n    // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n  List&lt;int&gt; toArray() {\n    // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\nstruct ArrayDeque {\n    nums: Vec&lt;i32&gt;,  // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    front: usize,    // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    que_size: usize, // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n}\n\nimpl ArrayDeque {\n    /* \u5efa\u69cb\u5b50 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![0; capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    pub fn push_first(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        self.front = self.index(self.front as i32 - 1);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    pub fn push_last(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    fn pop_first(&amp;mut self) -&gt; i32 {\n        let num = self.peek_first();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    fn pop_last(&amp;mut self) -&gt; i32 {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fn peek_first(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        };\n        self.nums[self.front]\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    fn peek_last(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\")\n        };\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        let mut res = vec![0; self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;       // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e + 1\n    int queCapacity; // \u4f47\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u5efa\u69cb\u5b50 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // \u521d\u59cb\u5316\u9663\u5217\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u56de\u5230\u982d\u90e8\n    // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u4f47\u5217\u9996\u5165\u5217 */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\\r\\n\");\n        return;\n    }\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u56de\u5230\u5c3e\u90e8\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // \u5c07 num \u65b0\u589e\u5230\u4f47\u5217\u9996\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u4f47\u5217\u5c3e\u5165\u5217 */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\\r\\n\");\n        return;\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n    // \u8a2a\u554f\u7570\u5e38\uff1a\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n    // \u8a2a\u554f\u7570\u5e38\uff1a\u96d9\u5411\u4f47\u5217\u70ba\u7a7a\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* \u4f47\u5217\u9996\u51fa\u5217 */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u4f47\u5217\u5c3e\u51fa\u5217 */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\nint *toArray(ArrayDeque *deque, int *queSize) {\n    *queSize = deque-&gt;queSize;\n    int *res = (int *)calloc(deque-&gt;queSize, sizeof(int));\n    int j = deque-&gt;front;\n    for (int i = 0; i &lt; deque-&gt;queSize; i++) {\n        res[i] = deque-&gt;nums[j % deque-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_deque.kt<pre><code>/* \u5efa\u69cb\u5b50 */\nclass ArrayDeque(capacity: Int) {\n    private var nums: IntArray = IntArray(capacity) // \u7528\u65bc\u5132\u5b58\u96d9\u5411\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private var front: Int = 0 // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private var queSize: Int = 0 // \u96d9\u5411\u4f47\u5217\u9577\u5ea6\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 */\n    private fun index(i: Int): Int {\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n        // \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity()\n    }\n\n    /* \u4f47\u5217\u9996\u5165\u5217 */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1)\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n        nums[front] = num\n        queSize++\n    }\n\n    /* \u4f47\u5217\u5c3e\u5165\u5217 */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u96d9\u5411\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        val rear = index(front + queSize)\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u4f47\u5217\u9996\u51fa\u5217 */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* \u4f47\u5217\u5c3e\u51fa\u5217 */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\n    fun toArray(): IntArray {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>### \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u96d9\u5411\u4f47\u5217 ###\nclass ArrayDeque\n  ### \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u9577\u5ea6 ###\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(capacity)\n    @nums = Array.new(capacity, 0)\n    @front = 0\n    @size = 0\n  end\n\n  ### \u7372\u53d6\u96d9\u5411\u4f47\u5217\u7684\u5bb9\u91cf ###\n  def capacity\n    @nums.length\n  end\n\n  ### \u5224\u65b7\u96d9\u5411\u4f47\u5217\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### \u4f47\u5217\u9996\u5165\u5217 ###\n  def push_first(num)\n    if size == capacity\n      puts '\u96d9\u5411\u4f47\u5217\u5df2\u6eff'\n      return\n    end\n\n    # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5de6\u79fb\u52d5\u4e00\u4f4d\n    # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe front \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\u56de\u5230\u5c3e\u90e8\n    @front = index(@front - 1)\n    # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u9996\n    @nums[@front] = num\n    @size += 1\n  end\n\n  ### \u4f47\u5217\u5c3e\u5165\u5217 ###\n  def push_last(num)\n    if size == capacity\n      puts '\u96d9\u5411\u4f47\u5217\u5df2\u6eff'\n      return\n    end\n\n    # \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    rear = index(@front + size)\n    # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### \u4f47\u5217\u9996\u51fa\u5217 ###\n  def pop_first\n    num = peek_first\n    # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\n    @front = index(@front + 1)\n    @size -= 1\n    num\n  end\n\n  ### \u4f47\u5217\u5c3e\u51fa\u5217 ###\n  def pop_last\n    num = peek_last\n    @size -= 1\n    num\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 ###\n  def peek_first\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u5c3e\u5143\u7d20 ###\n  def peek_last\n    raise IndexError, '\u96d9\u5411\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    # \u8a08\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last = index(@front + size - 1)\n    @nums[last]\n  end\n\n  ### \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 ###\n  def to_array\n    # \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    res = []\n    for i in 0...size\n      res &lt;&lt; @nums[index(@front + i)]\n    end\n    res\n  end\n\n  private\n\n  ### \u8a08\u7b97\u74b0\u5f62\u9663\u5217\u7d22\u5f15 ###\n  def index(i)\n    # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe\u9663\u5217\u9996\u5c3e\u76f8\u9023\n    # \u7576 i \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\uff0c\u56de\u5230\u982d\u90e8\n    # \u7576 i \u8d8a\u904e\u9663\u5217\u982d\u90e8\u5f8c\uff0c\u56de\u5230\u5c3e\u90e8\n    (i + capacity) % capacity\n  end\nend\n</code></pre> array_deque.zig<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3 \u00a0 \u96d9\u5411\u4f47\u5217\u61c9\u7528","text":"<p>\u96d9\u5411\u4f47\u5217\u517c\u5177\u5806\u758a\u8207\u4f47\u5217\u7684\u908f\u8f2f\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5be6\u73fe\u9019\u5169\u8005\u7684\u6240\u6709\u61c9\u7528\u5834\u666f\uff0c\u540c\u6642\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002</p> <p>\u6211\u5011\u77e5\u9053\uff0c\u8edf\u9ad4\u7684\u201c\u64a4\u92b7\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u5806\u758a\u4f86\u5be6\u73fe\uff1a\u7cfb\u7d71\u5c07\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c <code>push</code> \u5230\u5806\u758a\u4e2d\uff0c\u7136\u5f8c\u900f\u904e <code>pop</code> \u5be6\u73fe\u64a4\u92b7\u3002\u7136\u800c\uff0c\u8003\u616e\u5230\u7cfb\u7d71\u8cc7\u6e90\u7684\u9650\u5236\uff0c\u8edf\u9ad4\u901a\u5e38\u6703\u9650\u5236\u64a4\u92b7\u7684\u6b65\u6578\uff08\u4f8b\u5982\u50c5\u5141\u8a31\u5132\u5b58 \\(50\\) \u6b65\uff09\u3002\u7576\u5806\u758a\u7684\u9577\u5ea6\u8d85\u904e \\(50\\) \u6642\uff0c\u8edf\u9ad4\u9700\u8981\u5728\u5806\u758a\u5e95\uff08\u4f47\u5217\u9996\uff09\u57f7\u884c\u522a\u9664\u64cd\u4f5c\u3002\u4f46\u5806\u758a\u7121\u6cd5\u5be6\u73fe\u8a72\u529f\u80fd\uff0c\u6b64\u6642\u5c31\u9700\u8981\u4f7f\u7528\u96d9\u5411\u4f47\u5217\u4f86\u66ff\u4ee3\u5806\u758a\u3002\u8acb\u6ce8\u610f\uff0c\u201c\u64a4\u92b7\u201d\u7684\u6838\u5fc3\u908f\u8f2f\u4ecd\u7136\u9075\u5faa\u5806\u758a\u7684\u5148\u5165\u5f8c\u51fa\u539f\u5247\uff0c\u53ea\u662f\u96d9\u5411\u4f47\u5217\u80fd\u5920\u66f4\u52a0\u9748\u6d3b\u5730\u5be6\u73fe\u4e00\u4e9b\u984d\u5916\u908f\u8f2f\u3002</p>"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 \u4f47\u5217","text":"<p>\u4f47\u5217\uff08queue\uff09\u662f\u4e00\u7a2e\u9075\u5faa\u5148\u5165\u5148\u51fa\u898f\u5247\u7684\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u3002\u9867\u540d\u601d\u7fa9\uff0c\u4f47\u5217\u6a21\u64ec\u4e86\u6392\u968a\u73fe\u8c61\uff0c\u5373\u65b0\u4f86\u7684\u4eba\u4e0d\u65b7\u52a0\u5165\u4f47\u5217\u5c3e\u90e8\uff0c\u800c\u4f4d\u65bc\u4f47\u5217\u982d\u90e8\u7684\u4eba\u9010\u500b\u96e2\u958b\u3002</p> <p>\u5982\u5716 5-4 \u6240\u793a\uff0c\u6211\u5011\u5c07\u4f47\u5217\u982d\u90e8\u7a31\u70ba\u201c\u4f47\u5217\u9996\u201d\uff0c\u5c3e\u90e8\u7a31\u70ba\u201c\u4f47\u5217\u5c3e\u201d\uff0c\u5c07\u628a\u5143\u7d20\u52a0\u5165\u5217\u5c3e\u7684\u64cd\u4f5c\u7a31\u70ba\u201c\u5165\u5217\u201d\uff0c\u522a\u9664\u4f47\u5217\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u7a31\u70ba\u201c\u51fa\u5217\u201d\u3002</p> <p></p> <p> \u5716 5-4 \u00a0 \u4f47\u5217\u7684\u5148\u5165\u5148\u51fa\u898f\u5247 </p>"},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1 \u00a0 \u4f47\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u4f47\u5217\u7684\u5e38\u898b\u64cd\u4f5c\u5982\u8868 5-2 \u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7a0b\u5f0f\u8a9e\u8a00\u7684\u65b9\u6cd5\u540d\u7a31\u53ef\u80fd\u6703\u6709\u6240\u4e0d\u540c\u3002\u6211\u5011\u5728\u6b64\u63a1\u7528\u8207\u5806\u758a\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002</p> <p> \u8868 5-2 \u00a0 \u4f47\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u6642\u9593\u8907\u96dc\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u5217\uff0c\u5373\u5c07\u5143\u7d20\u65b0\u589e\u81f3\u4f47\u5217\u5c3e \\(O(1)\\) <code>pop()</code> \u4f47\u5217\u9996\u5143\u7d20\u51fa\u5217 \\(O(1)\\) <code>peek()</code> \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 \\(O(1)\\) <p>\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u4e2d\u73fe\u6210\u7684\u4f47\u5217\u985e\u5225\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig queue.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u4f47\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u5011\u4e00\u822c\u5c07\u96d9\u5411\u4f47\u5217\u985e\u5225 deque \u7576\u4f5c\u4f47\u5217\u4f7f\u7528\n# \u96d6\u7136 queue.Queue() \u662f\u7d14\u6b63\u7684\u4f47\u5217\u985e\u5225\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u63a8\u85a6\nque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u5217\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\nfront: int = que[0]\n\n# \u5143\u7d20\u51fa\u5217\npop: int = que.popleft()\n\n# \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\nsize: int = len(que)\n\n# \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\nqueue&lt;int&gt; queue;\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint front = queue.front();\n\n/* \u5143\u7d20\u51fa\u5217 */\nqueue.pop();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek = queue.peek();\n\n/* \u5143\u7d20\u51fa\u5217 */\nint pop = queue.poll();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\nQueue&lt;int&gt; queue = new();\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n\n/* \u5143\u7d20\u51fa\u5217 */\nint pop = queue.Dequeue();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = queue.Count;\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c07 list \u4f5c\u70ba\u4f47\u5217\u4f86\u4f7f\u7528\nqueue := list.New()\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\npeek := queue.Front()\n\n/* \u5143\u7d20\u51fa\u5217 */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nsize := queue.Len()\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// Swift \u6c92\u6709\u5167\u5efa\u7684\u4f47\u5217\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u4f47\u5217\u4f86\u4f7f\u7528\nvar queue: [Int] = []\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nlet peek = queue.first!\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u7531\u65bc\u662f\u9663\u5217\uff0c\u56e0\u6b64 removeFirst \u7684\u8907\u96dc\u5ea6\u70ba O(n)\nlet pool = queue.removeFirst()\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nlet size = queue.count\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// JavaScript \u6c92\u6709\u5167\u5efa\u7684\u4f47\u5217\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u4f47\u5217\u4f86\u4f7f\u7528\nconst queue = [];\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u5e95\u5c64\u662f\u9663\u5217\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\nconst pop = queue.shift();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// TypeScript \u6c92\u6709\u5167\u5efa\u7684\u4f47\u5217\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u4f47\u5217\u4f86\u4f7f\u7528\nconst queue: number[] = [];\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u5217 */\n// \u5e95\u5c64\u662f\u9663\u5217\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\nconst pop = queue.shift();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u4f47\u5217\u985e\u5225 Qeque \u662f\u96d9\u5411\u4f47\u5217\uff0c\u4e5f\u53ef\u4f5c\u70ba\u4f47\u5217\u4f7f\u7528\nQueue&lt;int&gt; queue = Queue();\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek = queue.first;\n\n/* \u5143\u7d20\u51fa\u5217 */\nint pop = queue.removeFirst();\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size = queue.length;\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* \u521d\u59cb\u5316\u96d9\u5411\u4f47\u5217 */\n// \u5728 Rust \u4e2d\u4f7f\u7528\u96d9\u5411\u4f47\u5217\u4f5c\u70ba\u666e\u901a\u4f47\u5217\u4f86\u4f7f\u7528\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u5217 */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nif let Some(front) = deque.front() {\n}\n\n/* \u5143\u7d20\u51fa\u5217 */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u4f47\u5217\n</code></pre> queue.kt<pre><code>/* \u521d\u59cb\u5316\u4f47\u5217 */\nval queue = LinkedList&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5217 */\nqueue.offer(1)\nqueue.offer(3)\nqueue.offer(2)\nqueue.offer(5)\nqueue.offer(4)\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nval peek = queue.peek()\n\n/* \u5143\u7d20\u51fa\u5217 */\nval pop = queue.poll()\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nval size = queue.size\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nval isEmpty = queue.isEmpty()\n</code></pre> queue.rb<pre><code># \u521d\u59cb\u5316\u4f47\u5217\n# Ruby \u5167\u5efa\u7684\u4f47\u5217\uff08Thread::Queue) \u6c92\u6709 peek \u548c\u8d70\u8a2a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u4f47\u5217\u4f86\u4f7f\u7528\nqueue = []\n\n# \u5143\u7d20\u5165\u5217\nqueue.push(1)\nqueue.push(3)\nqueue.push(2)\nqueue.push(5)\nqueue.push(4)\n\n# \u8a2a\u554f\u4f47\u5217\u5143\u7d20\npeek = queue.first\n\n# \u5143\u7d20\u51fa\u5217\n# \u6e05\u6ce8\u610f\uff0c\u7531\u65bc\u662f\u9663\u5217\uff0cArray#shift \u65b9\u6cd5\u6642\u9593\u8907\u96dc\u5ea6\u70ba O(n)\npop = queue.shift\n\n# \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\nsize = queue.length\n\n# \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\nis_empty = queue.empty?\n</code></pre> queue.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2 \u00a0 \u4f47\u5217\u5be6\u73fe","text":"<p>\u70ba\u4e86\u5be6\u73fe\u4f47\u5217\uff0c\u6211\u5011\u9700\u8981\u4e00\u7a2e\u8cc7\u6599\u7d50\u69cb\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u65b0\u589e\u5143\u7d20\uff0c\u4e26\u5728\u53e6\u4e00\u7aef\u522a\u9664\u5143\u7d20\uff0c\u93c8\u7d50\u4e32\u5217\u548c\u9663\u5217\u90fd\u7b26\u5408\u8981\u6c42\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#1","title":"1. \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u7684\u5be6\u73fe","text":"<p>\u5982\u5716 5-5 \u6240\u793a\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u7684\u201c\u982d\u7bc0\u9ede\u201d\u548c\u201c\u5c3e\u7bc0\u9ede\u201d\u5206\u5225\u8996\u70ba\u201c\u4f47\u5217\u9996\u201d\u548c\u201c\u4f47\u5217\u5c3e\u201d\uff0c\u898f\u5b9a\u4f47\u5217\u5c3e\u50c5\u53ef\u65b0\u589e\u7bc0\u9ede\uff0c\u4f47\u5217\u9996\u50c5\u53ef\u522a\u9664\u7bc0\u9ede\u3002</p> LinkedListQueuepush()pop() <p></p> <p></p> <p></p> <p> \u5716 5-5 \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u4f47\u5217\u7684\u5165\u5217\u51fa\u5217\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u7528\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u4f47\u5217\u7684\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._front: ListNode | None = None  # \u982d\u7bc0\u9ede front\n        self._rear: ListNode | None = None  # \u5c3e\u7bc0\u9ede rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u5217\"\"\"\n        # \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        node = ListNode(num)\n        # \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u5217\"\"\"\n        num = self.peek()\n        # \u522a\u9664\u982d\u7bc0\u9ede\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u4f47\u5217\u70ba\u7a7a\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f49\u5316\u70ba\u4e32\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // \u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u522a\u9664\u7bc0\u9ede\uff0c\u91cb\u653e\u8a18\u61b6\u9ad4\n        freeMemoryLinkedList(front);\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u5217 */\n    void push(int num) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        ListNode *node = new ListNode(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    int pop() {\n        int num = peek();\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"\u4f47\u5217\u70ba\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Vector \u4e26\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    private ListNode front, rear; // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5217 */\n    public void push(int num) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        ListNode node = new ListNode(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (front == null) {\n            front = node;\n            rear = node;\n        // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    public int pop() {\n        int num = peek();\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u5217 */\n    public void Push(int num) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        ListNode node = new(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (front == null) {\n            front = node;\n            rear = node;\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    public int Pop() {\n        int num = Peek();\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\ntype linkedListQueue struct {\n    // \u4f7f\u7528\u5167\u5efa\u5305 list \u4f86\u5be6\u73fe\u4f47\u5217\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u4f47\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u5217 */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u5217 */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u7372\u53d6 List \u7528\u65bc\u5217\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    private var front: ListNode? // \u982d\u7bc0\u9ede\n    private var rear: ListNode? // \u5c3e\u7bc0\u9ede\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u5217 */\n    func push(num: Int) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        let node = ListNode(x: num)\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* \u51fa\u5217 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    #front; // \u982d\u7bc0\u9ede #front\n    #rear; // \u5c3e\u7bc0\u9ede #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u5217 */\n    push(num) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        const node = new ListNode(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    pop() {\n        const num = this.peek();\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peek() {\n        if (this.size === 0) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        return this.#front.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n    private front: ListNode | null; // \u982d\u7bc0\u9ede front\n    private rear: ListNode | null; // \u5c3e\u7bc0\u9ede rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u5217 */\n    push(num: number): void {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        const node = new ListNode(num);\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.size === 0) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        return this.front!.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue {\n  ListNode? _front; // \u982d\u7bc0\u9ede _front\n  ListNode? _rear; // \u5c3e\u7bc0\u9ede _rear\n  int _queSize = 0; // \u4f47\u5217\u9577\u5ea6\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u5217 */\n  void push(int _num) {\n    // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e _num\n    final node = ListNode(_num);\n    // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* \u51fa\u5217 */\n  int pop() {\n    final int _num = peek();\n    // \u522a\u9664\u982d\u7bc0\u9ede\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('\u4f47\u5217\u70ba\u7a7a');\n    }\n    return _front!.val;\n  }\n\n  /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u982d\u7bc0\u9ede front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u7bc0\u9ede rear\n    que_size: usize,                         // \u4f47\u5217\u7684\u9577\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u5217 */\n    pub fn push(&amp;mut self, num: T) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u5217 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* \u5efa\u69cb\u5b50 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // \u91cb\u653e\u6240\u6709\u7bc0\u9ede\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91cb\u653e queue \u7d50\u69cb\u9ad4\n    free(queue);\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* \u5165\u5217 */\nvoid push(LinkedListQueue *queue, int num) {\n    // \u5c3e\u7bc0\u9ede\u8655\u65b0\u589e node\n    ListNode *node = newListNode(num);\n    // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* \u51fa\u5217 */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* \u5217\u5370\u4f47\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // \u8907\u88fd\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u8cc7\u6599\u5230\u9663\u5217\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass LinkedListQueue(\n    // \u982d\u7bc0\u9ede front \uff0c\u5c3e\u7bc0\u9ede rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u5217 */\n    fun push(num: Int) {\n        // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n        val node = ListNode(num)\n        // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n        if (front == null) {\n            front = node\n            rear = node\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* \u51fa\u5217 */\n    fun pop(): Int {\n        val num = peek()\n        // \u522a\u9664\u982d\u7bc0\u9ede\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>### \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u982d\u73fe\u7684\u4f47\u5217 ###\nclass LinkedListQueue\n  ### \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 ###\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @front = nil  # \u982d\u7bc0\u9ede front\n    @rear = nil   # \u5c3e\u7bc0\u9ede rear\n    @size = 0\n  end\n\n  ### \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    @front.nil?\n  end\n\n  ### \u5165\u5217 ###\n  def push(num)\n    # \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n    node = ListNode.new(num)\n\n    # \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\uff0c\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n    if @front.nil?\n      @front = node\n      @rear = node\n    # \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u4ee4\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n    else\n      @rear.next = node\n      @rear = node\n    end\n\n    @size += 1\n  end\n\n  ### \u51fa\u5217 ###\n  def pop\n    num = peek\n    # \u522a\u9664\u982d\u7bc0\u9ede\n    @front = @front.next\n    @size -= 1\n    num\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 ###\n  def peek\n    raise IndexError, '\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @front.val\n  end\n\n  ### \u5c07\u93c8\u7d50\u4e32\u5217\u70ba Array \u4e26\u8fd4\u56de ###\n  def to_array\n    queue = []\n    temp = @front\n    while temp\n      queue &lt;&lt; temp.val\n      temp = temp.next\n    end\n    queue\n  end\nend\n</code></pre> linkedlist_queue.zig<pre><code>// \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u4f47\u5217\nfn LinkedListQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*inc.ListNode(T) = null,                // \u982d\u7bc0\u9ede front\n        rear: ?*inc.ListNode(T) = null,                 // \u5c3e\u7bc0\u9ede rear\n        que_size: usize = 0,                            // \u4f47\u5217\u7684\u9577\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u4f47\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u4f47\u5217\u70ba\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u5165\u5217\n        pub fn push(self: *Self, num: T) !void {\n            // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            // \u5982\u679c\u4f47\u5217\u70ba\u7a7a\uff0c\u5247\u4ee4\u982d\u3001\u5c3e\u7bc0\u9ede\u90fd\u6307\u5411\u8a72\u7bc0\u9ede\n            if (self.front == null) {\n                self.front = node;\n                self.rear = node;\n            // \u5982\u679c\u4f47\u5217\u4e0d\u70ba\u7a7a\uff0c\u5247\u5c07\u8a72\u7bc0\u9ede\u65b0\u589e\u5230\u5c3e\u7bc0\u9ede\u5f8c\n            } else {\n                self.rear.?.next = node;\n                self.rear = node;\n            }\n            self.que_size += 1;\n        } \n\n        // \u51fa\u5217\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u522a\u9664\u982d\u7bc0\u9ede\n            self.front = self.front.?.next;\n            self.que_size -= 1;\n            return num;\n        } \n\n        // \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u63db\u70ba\u9663\u5217\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#2","title":"2. \u00a0 \u57fa\u65bc\u9663\u5217\u7684\u5be6\u73fe","text":"<p>\u5728\u9663\u5217\u4e2d\u522a\u9664\u9996\u5143\u7d20\u7684\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff0c\u9019\u6703\u5c0e\u81f4\u51fa\u5217\u64cd\u4f5c\u6548\u7387\u8f03\u4f4e\u3002\u7136\u800c\uff0c\u6211\u5011\u53ef\u4ee5\u63a1\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u4f86\u907f\u514d\u9019\u500b\u554f\u984c\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u4e00\u500b\u8b8a\u6578 <code>front</code> \u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u4e26\u7dad\u8b77\u4e00\u500b\u8b8a\u6578 <code>size</code> \u7528\u65bc\u8a18\u9304\u4f47\u5217\u9577\u5ea6\u3002\u5b9a\u7fa9 <code>rear = front + size</code> \uff0c\u9019\u500b\u516c\u5f0f\u8a08\u7b97\u51fa\u7684 <code>rear</code> \u6307\u5411\u4f47\u5217\u5c3e\u5143\u7d20\u4e4b\u5f8c\u7684\u4e0b\u4e00\u500b\u4f4d\u7f6e\u3002</p> <p>\u57fa\u65bc\u6b64\u8a2d\u8a08\uff0c\u9663\u5217\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u5340\u9593\u70ba <code>[front, rear - 1]</code>\uff0c\u5404\u7a2e\u64cd\u4f5c\u7684\u5be6\u73fe\u65b9\u6cd5\u5982\u5716 5-6 \u6240\u793a\u3002</p> <ul> <li>\u5165\u5217\u64cd\u4f5c\uff1a\u5c07\u8f38\u5165\u5143\u7d20\u8ce6\u503c\u7d66 <code>rear</code> \u7d22\u5f15\u8655\uff0c\u4e26\u5c07 <code>size</code> \u589e\u52a0 1 \u3002</li> <li>\u51fa\u5217\u64cd\u4f5c\uff1a\u53ea\u9700\u5c07 <code>front</code> \u589e\u52a0 1 \uff0c\u4e26\u5c07 <code>size</code> \u6e1b\u5c11 1 \u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u5230\uff0c\u5165\u5217\u548c\u51fa\u5217\u64cd\u4f5c\u90fd\u53ea\u9700\u9032\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(1)\\) \u3002</p> ArrayQueuepush()pop() <p></p> <p></p> <p></p> <p> \u5716 5-6 \u00a0 \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u4f47\u5217\u7684\u5165\u5217\u51fa\u5217\u64cd\u4f5c </p> <p>\u4f60\u53ef\u80fd\u6703\u767c\u73fe\u4e00\u500b\u554f\u984c\uff1a\u5728\u4e0d\u65b7\u9032\u884c\u5165\u5217\u548c\u51fa\u5217\u7684\u904e\u7a0b\u4e2d\uff0c<code>front</code> \u548c <code>rear</code> \u90fd\u5728\u5411\u53f3\u79fb\u52d5\uff0c\u7576\u5b83\u5011\u5230\u9054\u9663\u5217\u5c3e\u90e8\u6642\u5c31\u7121\u6cd5\u7e7c\u7e8c\u79fb\u52d5\u4e86\u3002\u70ba\u4e86\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u9663\u5217\u8996\u70ba\u9996\u5c3e\u76f8\u63a5\u7684\u201c\u74b0\u5f62\u9663\u5217\u201d\u3002</p> <p>\u5c0d\u65bc\u74b0\u5f62\u9663\u5217\uff0c\u6211\u5011\u9700\u8981\u8b93 <code>front</code> \u6216 <code>rear</code> \u5728\u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u6642\uff0c\u76f4\u63a5\u56de\u5230\u9663\u5217\u982d\u90e8\u7e7c\u7e8c\u8d70\u8a2a\u3002\u9019\u7a2e\u9031\u671f\u6027\u898f\u5f8b\u53ef\u4ee5\u900f\u904e\u201c\u53d6\u9918\u64cd\u4f5c\u201d\u4f86\u5be6\u73fe\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"\u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._nums: list[int] = [0] * size  # \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n        self._front: int = 0  # \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n        self._size: int = 0  # \u4f47\u5217\u9577\u5ea6\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u5217\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"\u4f47\u5217\u5df2\u6eff\")\n        # \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        rear: int = (self._front + self._size) % self.capacity()\n        # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u5217\"\"\"\n        num: int = self.peek()\n        # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u4f47\u5217\u70ba\u7a7a\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u4e32\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n  private:\n    int *nums;       // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;       // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize;     // \u4f47\u5217\u9577\u5ea6\n    int queCapacity; // \u4f47\u5217\u5bb9\u91cf\n\n  public:\n    ArrayQueue(int capacity) {\n        // \u521d\u59cb\u5316\u9663\u5217\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5217 */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"\u4f47\u5217\u5df2\u6eff\" &lt;&lt; endl;\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        int rear = (front + queSize) % queCapacity;\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    int pop() {\n        int num = peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"\u4f47\u5217\u70ba\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u5c07\u9663\u5217\u8f49\u5316\u70ba Vector \u4e26\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    private int[] nums; // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private int front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private int queSize; // \u4f47\u5217\u9577\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u5217 */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        int rear = (front + queSize) % capacity();\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    public int pop() {\n        int num = peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    public int[] toArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    int[] nums;  // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;   // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize; // \u4f47\u5217\u9577\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u5217 */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        int rear = (front + queSize) % Capacity();\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    public int Pop() {\n        int num = Peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    public int[] ToArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\ntype arrayQueue struct {\n    nums        []int // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    front       int   // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    queSize     int   // \u4f47\u5217\u9577\u5ea6\n    queCapacity int   // \u4f47\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7d0d\u5143\u7d20\u6578\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u4f47\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u5165\u5217 */\nfunc (q *arrayQueue) push(num int) {\n    // \u7576 rear == queCapacity \u8868\u793a\u4f47\u5217\u5df2\u6eff\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n    rear := (q.front + q.queSize) % q.queCapacity\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u51fa\u5217 */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    if num == nil {\n        return nil\n    }\n\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u7372\u53d6 Slice \u7528\u65bc\u5217\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    private var nums: [Int] // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private var front: Int // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private var _size: Int // \u4f47\u5217\u9577\u5ea6\n\n    init(capacity: Int) {\n        // \u521d\u59cb\u5316\u9663\u5217\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u5217 */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        let rear = (front + size()) % capacity()\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u51fa\u5217 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u4f47\u5217\u70ba\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    func toArray() -&gt; [Int] {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    #nums; // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    #front = 0; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    #queSize = 0; // \u4f47\u5217\u9577\u5ea6\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5165\u5217 */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        const rear = (this.#front + this.size) % this.capacity;\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    pop() {\n        const num = this.peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peek() {\n        if (this.isEmpty()) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n    private nums: number[]; // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private front: number; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private queSize: number; // \u4f47\u5217\u9577\u5ea6\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5165\u5217 */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('\u4f47\u5217\u5df2\u6eff');\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        const rear = (this.front + this.queSize) % this.capacity;\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u51fa\u5217 */\n    pop(): number {\n        const num = this.peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('\u4f47\u5217\u70ba\u7a7a');\n        return this.nums[this.front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray(): number[] {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n  late int _front; // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n  late int _queSize; // \u4f47\u5217\u9577\u5ea6\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u5217 */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"\u4f47\u5217\u5df2\u6eff\");\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n    int rear = (_front + _queSize) % capaCity();\n    // \u5c07 _num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u51fa\u5217 */\n  int pop() {\n    int _num = peek();\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u4f47\u5217\u70ba\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8fd4\u56de Array */\n  List&lt;int&gt; toArray() {\n    // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nstruct ArrayQueue {\n    nums: Vec&lt;i32&gt;,    // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    front: i32,        // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    que_size: i32,     // \u4f47\u5217\u9577\u5ea6\n    que_capacity: i32, // \u4f47\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n    /* \u5efa\u69cb\u5b50 */\n    fn new(capacity: i32) -&gt; ArrayQueue {\n        ArrayQueue {\n            nums: vec![0; capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u5165\u5217 */\n    fn push(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u4f47\u5217\u5df2\u6eff\");\n            return;\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u5217 */\n    fn pop(&amp;mut self) -&gt; i32 {\n        let num = self.peek();\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![0; self.que_size as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    int front;       // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e + 1\n    int queCapacity; // \u4f47\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u5efa\u69cb\u5b50 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // \u521d\u59cb\u5316\u9663\u5217\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* \u5165\u5217 */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"\u4f47\u5217\u5df2\u6eff\\r\\n\");\n        return;\n    }\n    // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* \u51fa\u5217 */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* \u8fd4\u56de\u9663\u5217\u7528\u65bc\u5217\u5370 */\nint *toArray(ArrayQueue *queue, int *queSize) {\n    *queSize = queue-&gt;queSize;\n    int *res = (int *)calloc(queue-&gt;queSize, sizeof(int));\n    int j = queue-&gt;front;\n    for (int i = 0; i &lt; queue-&gt;queSize; i++) {\n        res[i] = queue-&gt;nums[j % queue-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_queue.kt<pre><code>/* \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 */\nclass ArrayQueue(capacity: Int) {\n    private val nums: IntArray = IntArray(capacity) // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    private var front: Int = 0 // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    private var queSize: Int = 0 // \u4f47\u5217\u9577\u5ea6\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u5165\u5217 */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u4f47\u5217\u5df2\u6eff\")\n            return\n        }\n        // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n        // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n        val rear = (front + queSize) % capacity()\n        // \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u51fa\u5217 */\n    fun pop(): Int {\n        val num = peek()\n        // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u9663\u5217 */\n    fun toArray(): IntArray {\n        // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>### \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217 ###\nclass ArrayQueue\n  ### \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6 ###\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(size)\n    @nums = Array.new(size, 0) # \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217\n    @front = 0 # \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n    @size = 0 # \u4f47\u5217\u9577\u5ea6\n  end\n\n  ### \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf ###\n  def capacity\n    @nums.length\n  end\n\n  ### \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### \u5165\u5217 ###\n  def push(num)\n    raise IndexError, '\u4f47\u5217\u5df2\u6eff' if size == capacity\n\n    # \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n    # \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n    rear = (@front + size) % capacity\n    # \u5c07 num \u65b0\u589e\u81f3\u4f47\u5217\u5c3e\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### \u51fa\u5217 ###\n  def pop\n    num = peek\n    # \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n    @front = (@front + 1) % capacity\n    @size -= 1\n    num\n  end\n\n  ### \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20 ###\n  def peek\n    raise IndexError, '\u4f47\u5217\u70ba\u7a7a' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### \u8fd4\u56de\u4e32\u5217\u7528\u65bc\u5217\u5370 ###\n  def to_array\n    res = Array.new(size, 0)\n    j = @front\n\n    for i in 0...size\n      res[i] = @nums[j % capacity]\n      j += 1\n    end\n\n    res\n  end\nend\n</code></pre> array_queue.zig<pre><code>// \u57fa\u65bc\u74b0\u5f62\u9663\u5217\u5be6\u73fe\u7684\u4f47\u5217\nfn ArrayQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        nums: []T = undefined,                          // \u7528\u65bc\u5132\u5b58\u4f47\u5217\u5143\u7d20\u7684\u9663\u5217     \n        cap: usize = 0,                                 // \u4f47\u5217\u5bb9\u91cf\n        front: usize = 0,                               // \u4f47\u5217\u9996\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u9996\u5143\u7d20\n        queSize: usize = 0,                             // \u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e + 1\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u9663\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.cap = cap;\n            self.nums = try self.mem_allocator.alloc(T, self.cap);\n            @memset(self.nums, @as(T, 0));\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u4f47\u5217\u7684\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.cap;\n        }\n\n        // \u7372\u53d6\u4f47\u5217\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.queSize;\n        }\n\n        // \u5224\u65b7\u4f47\u5217\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.queSize == 0;\n        }\n\n        // \u5165\u5217\n        pub fn push(self: *Self, num: T) !void {\n            if (self.size() == self.capacity()) {\n                std.debug.print(\"\u4f47\u5217\u5df2\u6eff\\n\", .{});\n                return;\n            }\n            // \u8a08\u7b97\u4f47\u5217\u5c3e\u6307\u6a19\uff0c\u6307\u5411\u4f47\u5217\u5c3e\u7d22\u5f15 + 1\n            // \u900f\u904e\u53d6\u9918\u64cd\u4f5c\u5be6\u73fe rear \u8d8a\u904e\u9663\u5217\u5c3e\u90e8\u5f8c\u56de\u5230\u982d\u90e8\n            var rear = (self.front + self.queSize) % self.capacity();\n            // \u5728\u5c3e\u7bc0\u9ede\u5f8c\u65b0\u589e num\n            self.nums[rear] = num;\n            self.queSize += 1;\n        } \n\n        // \u51fa\u5217\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u4f47\u5217\u9996\u6307\u6a19\u5411\u5f8c\u79fb\u52d5\u4e00\u4f4d\uff0c\u82e5\u8d8a\u904e\u5c3e\u90e8\uff0c\u5247\u8fd4\u56de\u5230\u9663\u5217\u982d\u90e8\n            self.front = (self.front + 1) % self.capacity();\n            self.queSize -= 1;\n            return num;\n        } \n\n        // \u8a2a\u554f\u4f47\u5217\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u4f47\u5217\u70ba\u7a7a\");\n            return self.nums[self.front];\n        } \n\n        // \u8fd4\u56de\u9663\u5217\n        pub fn toArray(self: *Self) ![]T {\n            // \u50c5\u8f49\u63db\u6709\u6548\u9577\u5ea6\u7bc4\u570d\u5167\u7684\u4e32\u5217\u5143\u7d20\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            var j: usize = self.front;\n            while (i &lt; self.size()) : ({ i += 1; j += 1; }) {\n                res[i] = self.nums[j % self.capacity()];\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u4ee5\u4e0a\u5be6\u73fe\u7684\u4f47\u5217\u4ecd\u7136\u5177\u6709\u4fb7\u9650\u6027\uff1a\u5176\u9577\u5ea6\u4e0d\u53ef\u8b8a\u3002\u7136\u800c\uff0c\u9019\u500b\u554f\u984c\u4e0d\u96e3\u89e3\u6c7a\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u9663\u5217\u66ff\u63db\u70ba\u52d5\u614b\u9663\u5217\uff0c\u5f9e\u800c\u5f15\u5165\u64f4\u5bb9\u6a5f\u5236\u3002\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u5617\u8a66\u81ea\u884c\u5be6\u73fe\u3002</p> <p>\u5169\u7a2e\u5be6\u73fe\u7684\u5c0d\u6bd4\u7d50\u8ad6\u8207\u5806\u758a\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d05\u8ff0\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3 \u00a0 \u4f47\u5217\u5178\u578b\u61c9\u7528","text":"<ul> <li>\u6dd8\u5bf6\u8a02\u55ae\u3002\u8cfc\u7269\u8005\u4e0b\u55ae\u5f8c\uff0c\u8a02\u55ae\u5c07\u52a0\u5165\u5217\u5217\u4e2d\uff0c\u7cfb\u7d71\u96a8\u5f8c\u6703\u6839\u64da\u9806\u5e8f\u8655\u7406\u4f47\u5217\u4e2d\u7684\u8a02\u55ae\u3002\u5728\u96d9\u5341\u4e00\u671f\u9593\uff0c\u77ed\u6642\u9593\u5167\u6703\u7522\u751f\u6d77\u91cf\u8a02\u55ae\uff0c\u9ad8\u4f75\u767c\u6210\u70ba\u5de5\u7a0b\u5e2b\u5011\u9700\u8981\u91cd\u9ede\u653b\u514b\u7684\u554f\u984c\u3002</li> <li>\u5404\u985e\u5f85\u8fa6\u4e8b\u9805\u3002\u4efb\u4f55\u9700\u8981\u5be6\u73fe\u201c\u5148\u4f86\u5f8c\u5230\u201d\u529f\u80fd\u7684\u5834\u666f\uff0c\u4f8b\u5982\u5370\u8868\u6a5f\u7684\u4efb\u52d9\u4f47\u5217\u3001\u9910\u5ef3\u7684\u51fa\u9910\u4f47\u5217\u7b49\uff0c\u4f47\u5217\u5728\u9019\u4e9b\u5834\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7dad\u8b77\u8655\u7406\u9806\u5e8f\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/stack/","title":"5.1 \u00a0 \u5806\u758a","text":"<p>\u5806\u758a\uff08stack\uff09\u662f\u4e00\u7a2e\u9075\u5faa\u5148\u5165\u5f8c\u51fa\u908f\u8f2f\u7684\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\u3002</p> <p>\u6211\u5011\u53ef\u4ee5\u5c07\u5806\u758a\u985e\u6bd4\u70ba\u684c\u9762\u4e0a\u7684\u4e00\u758a\u76e4\u5b50\uff0c\u5982\u679c\u60f3\u53d6\u51fa\u5e95\u90e8\u7684\u76e4\u5b50\uff0c\u5247\u9700\u8981\u5148\u5c07\u4e0a\u9762\u7684\u76e4\u5b50\u4f9d\u6b21\u79fb\u8d70\u3002\u6211\u5011\u5c07\u76e4\u5b50\u66ff\u63db\u70ba\u5404\u7a2e\u578b\u5225\u7684\u5143\u7d20\uff08\u5982\u6574\u6578\u3001\u5b57\u5143\u3001\u7269\u4ef6\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u5806\u758a\u9019\u7a2e\u8cc7\u6599\u7d50\u69cb\u3002</p> <p>\u5982\u5716 5-1 \u6240\u793a\uff0c\u6211\u5011\u628a\u5806\u7a4d\u758a\u5143\u7d20\u7684\u9802\u90e8\u7a31\u70ba\u201c\u5806\u758a\u9802\u201d\uff0c\u5e95\u90e8\u7a31\u70ba\u201c\u5806\u758a\u5e95\u201d\u3002\u5c07\u628a\u5143\u7d20\u65b0\u589e\u5230\u5806\u758a\u9802\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u5165\u5806\u758a\u201d\uff0c\u522a\u9664\u5806\u758a\u9802\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u51fa\u5806\u758a\u201d\u3002</p> <p></p> <p> \u5716 5-1 \u00a0 \u5806\u758a\u7684\u5148\u5165\u5f8c\u51fa\u898f\u5247 </p>"},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1 \u00a0 \u5806\u758a\u7684\u5e38\u7528\u64cd\u4f5c","text":"<p>\u5806\u758a\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-1 \u6240\u793a\uff0c\u5177\u9ad4\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u64da\u6240\u4f7f\u7528\u7684\u7a0b\u5f0f\u8a9e\u8a00\u4f86\u78ba\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u5011\u4ee5\u5e38\u898b\u7684 <code>push()</code>\u3001<code>pop()</code>\u3001<code>peek()</code> \u547d\u540d\u70ba\u4f8b\u3002</p> <p> \u8868 5-1 \u00a0 \u5806\u758a\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5 \u63cf\u8ff0 \u6642\u9593\u8907\u96dc\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u5806\u758a\uff08\u65b0\u589e\u81f3\u5806\u758a\u9802\uff09 \\(O(1)\\) <code>pop()</code> \u5806\u758a\u9802\u5143\u7d20\u51fa\u5806\u758a \\(O(1)\\) <code>peek()</code> \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 \\(O(1)\\) <p>\u901a\u5e38\u60c5\u6cc1\u4e0b\uff0c\u6211\u5011\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7a0b\u5f0f\u8a9e\u8a00\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8a9e\u8a00\u53ef\u80fd\u6c92\u6709\u5c08\u9580\u63d0\u4f9b\u5806\u758a\u985e\u5225\uff0c\u9019\u6642\u6211\u5011\u53ef\u4ee5\u5c07\u8a72\u8a9e\u8a00\u7684\u201c\u9663\u5217\u201d\u6216\u201c\u93c8\u7d50\u4e32\u5217\u201d\u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\uff0c\u4e26\u5728\u7a0b\u5f0f\u908f\u8f2f\u4e0a\u5ffd\u7565\u8207\u5806\u758a\u7121\u95dc\u7684\u64cd\u4f5c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig stack.py<pre><code># \u521d\u59cb\u5316\u5806\u758a\n# Python \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a list \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nstack: list[int] = []\n\n# \u5143\u7d20\u5165\u5806\u758a\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\npeek: int = stack[-1]\n\n# \u5143\u7d20\u51fa\u5806\u758a\npop: int = stack.pop()\n\n# \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\nsize: int = len(stack)\n\n# \u5224\u65b7\u662f\u5426\u70ba\u7a7a\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\nstack&lt;int&gt; stack;\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint top = stack.top();\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nstack.pop(); // \u7121\u8fd4\u56de\u503c\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek = stack.peek();\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nint pop = stack.pop();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\nStack&lt;int&gt; stack = new();\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek = stack.Peek();\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nint pop = stack.Pop();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size = stack.Count;\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// \u5728 Go \u4e2d\uff0c\u63a8\u85a6\u5c07 Slice \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nvar stack []int\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\npeek := stack[len(stack)-1]\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nsize := len(stack)\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// Swift \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nvar stack: [Int] = []\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nlet peek = stack.last!\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nlet pop = stack.removeLast()\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nlet size = stack.count\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// JavaScript \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nconst stack = [];\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nconst pop = stack.pop();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// TypeScript \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nconst stack: number[] = [];\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nconst pop = stack.pop();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// Dart \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a List \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nList&lt;int&gt; stack = [];\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek = stack.last;\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nint pop = stack.removeLast();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size = stack.length;\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\n// \u628a Vec \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nlet top = stack.last().unwrap();\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nlet pop = stack.pop().unwrap();\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nlet size = stack.len();\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C \u672a\u63d0\u4f9b\u5167\u5efa\u5806\u758a\n</code></pre> stack.kt<pre><code>/* \u521d\u59cb\u5316\u5806\u758a */\nval stack = Stack&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5806\u758a */\nstack.push(1)\nstack.push(3)\nstack.push(2)\nstack.push(5)\nstack.push(4)\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nval peek = stack.peek()\n\n/* \u5143\u7d20\u51fa\u5806\u758a */\nval pop = stack.pop()\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nval size = stack.size\n\n/* \u5224\u65b7\u662f\u5426\u70ba\u7a7a */\nval isEmpty = stack.isEmpty()\n</code></pre> stack.rb<pre><code># \u521d\u59cb\u5316\u5806\u758a\n# Ruby \u6c92\u6709\u5167\u5efa\u7684\u5806\u758a\u985e\u5225\uff0c\u53ef\u4ee5\u628a Array \u7576\u4f5c\u5806\u758a\u4f86\u4f7f\u7528\nstack = []\n\n# \u5143\u7d20\u5165\u5806\u758a\nstack &lt;&lt; 1\nstack &lt;&lt; 3\nstack &lt;&lt; 2\nstack &lt;&lt; 5\nstack &lt;&lt; 4\n\n# \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\npeek = stack.last\n\n# \u5143\u7d20\u51fa\u5806\u758a\npop = stack.pop\n\n# \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\nsize = stack.length\n\n# \u5224\u65b7\u662f\u5426\u70ba\u7a7a\nis_empty = stack.empty?\n</code></pre> stack.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2 \u00a0 \u5806\u758a\u7684\u5be6\u73fe","text":"<p>\u70ba\u4e86\u6df1\u5165\u77ad\u89e3\u5806\u758a\u7684\u57f7\u884c\u6a5f\u5236\uff0c\u6211\u5011\u4f86\u5617\u8a66\u81ea\u5df1\u5be6\u73fe\u4e00\u500b\u5806\u758a\u985e\u5225\u3002</p> <p>\u5806\u758a\u9075\u5faa\u5148\u5165\u5f8c\u51fa\u7684\u539f\u5247\uff0c\u56e0\u6b64\u6211\u5011\u53ea\u80fd\u5728\u5806\u758a\u9802\u65b0\u589e\u6216\u522a\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u9663\u5217\u548c\u93c8\u7d50\u4e32\u5217\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u65b0\u589e\u548c\u522a\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u5806\u758a\u53ef\u4ee5\u8996\u70ba\u4e00\u7a2e\u53d7\u9650\u5236\u7684\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u6211\u5011\u53ef\u4ee5\u201c\u906e\u853d\u201d\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u7684\u90e8\u5206\u7121\u95dc\u64cd\u4f5c\uff0c\u4f7f\u5176\u5c0d\u5916\u8868\u73fe\u7684\u908f\u8f2f\u7b26\u5408\u5806\u758a\u7684\u7279\u6027\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#1","title":"1. \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u7684\u5be6\u73fe","text":"<p>\u4f7f\u7528\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u5806\u758a\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u93c8\u7d50\u4e32\u5217\u7684\u982d\u7bc0\u9ede\u8996\u70ba\u5806\u758a\u9802\uff0c\u5c3e\u7bc0\u9ede\u8996\u70ba\u5806\u758a\u5e95\u3002</p> <p>\u5982\u5716 5-2 \u6240\u793a\uff0c\u5c0d\u65bc\u5165\u5806\u758a\u64cd\u4f5c\uff0c\u6211\u5011\u53ea\u9700\u5c07\u5143\u7d20\u63d2\u5165\u93c8\u7d50\u4e32\u5217\u982d\u90e8\uff0c\u9019\u7a2e\u7bc0\u9ede\u63d2\u5165\u65b9\u6cd5\u88ab\u7a31\u70ba\u201c\u982d\u63d2\u6cd5\u201d\u3002\u800c\u5c0d\u65bc\u51fa\u5806\u758a\u64cd\u4f5c\uff0c\u53ea\u9700\u5c07\u982d\u7bc0\u9ede\u5f9e\u93c8\u7d50\u4e32\u5217\u4e2d\u522a\u9664\u5373\u53ef\u3002</p> LinkedListStackpush()pop() <p></p> <p></p> <p></p> <p> \u5716 5-2 \u00a0 \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u5806\u758a\u7684\u5165\u5806\u758a\u51fa\u5806\u758a\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u5806\u758a\u7684\u793a\u4f8b\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, val: int):\n        \"\"\"\u5165\u5806\u758a\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u5806\u758a\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u5806\u758a\u70ba\u7a7a\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f49\u5316\u70ba\u4e32\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    int stkSize;        // \u5806\u758a\u7684\u9577\u5ea6\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // \u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u522a\u9664\u7bc0\u9ede\uff0c\u91cb\u653e\u8a18\u61b6\u9ad4\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u5806\u758a\u70ba\u7a7a\");\n        return stackTop-&gt;val;\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    private ListNode stackPeek; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    private int stkSize = 0; // \u5806\u758a\u7684\u9577\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    ListNode? stackPeek;  // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    int stkSize = 0;   // \u5806\u758a\u7684\u9577\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\ntype linkedListStack struct {\n    // \u4f7f\u7528\u5167\u5efa\u5305 list \u4f86\u5be6\u73fe\u5806\u758a\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u5806\u758a */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u5806\u758a */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u5806\u758a */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u7372\u53d6 List \u7528\u65bc\u5217\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    private var _peek: ListNode? // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    private var _size: Int // \u5806\u758a\u7684\u9577\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u5806\u758a */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* \u51fa\u5806\u758a */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u5806\u758a\u70ba\u7a7a\")\n        }\n        return _peek!.val\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    #stackPeek; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    #stkSize = 0; // \u5806\u758a\u7684\u9577\u5ea6\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.#stackPeek.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    private stkSize: number = 0; // \u5806\u758a\u7684\u9577\u5ea6\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* \u51fa\u5806\u758a */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.stackPeek.val;\n    }\n\n    /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u985e\u5225\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack {\n  ListNode? _stackPeek; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n  int _stkSize = 0; // \u5806\u758a\u7684\u9577\u5ea6\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* \u5165\u5806\u758a */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* \u51fa\u5806\u758a */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"\u5806\u758a\u70ba\u7a7a\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba List \u4e26\u8fd4\u56de */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    stk_size: usize,                              // \u5806\u758a\u7684\u9577\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* \u51fa\u5806\u758a */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            match old_head.borrow_mut().next.take() {\n                Some(new_head) =&gt; {\n                    self.stack_peek = Some(new_head);\n                }\n                None =&gt; {\n                    self.stack_peek = None;\n                }\n            }\n            self.stk_size -= 1;\n            Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.push(node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\ntypedef struct {\n    ListNode *top; // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    int size;      // \u5806\u758a\u7684\u9577\u5ea6\n} LinkedListStack;\n\n/* \u5efa\u69cb\u5b50 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* \u5165\u5806\u758a */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // \u66f4\u65b0\u65b0\u52a0\u7bc0\u9ede\u6307\u6a19\u57df\n    node-&gt;val = num;     // \u66f4\u65b0\u65b0\u52a0\u7bc0\u9ede\u8cc7\u6599\u57df\n    s-&gt;top = node;       // \u66f4\u65b0\u5806\u758a\u9802\n    s-&gt;size++;           // \u66f4\u65b0\u5806\u758a\u5927\u5c0f\n}\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"\u5806\u758a\u70ba\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* \u51fa\u5806\u758a */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // \u91cb\u653e\u8a18\u61b6\u9ad4\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n    private var stkSize: Int = 0 // \u5806\u758a\u7684\u9577\u5ea6\n) {\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u5806\u758a */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* \u51fa\u5806\u758a */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--\n        return num\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?._val\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?._val!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>### \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a ###\nclass LinkedListStack\n  attr_reader :size\n\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @size = 0\n  end\n\n  ### \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    @peek.nil?\n  end\n\n  ### \u5165\u5806\u758a ###\n  def push(val)\n    node = ListNode.new(val)\n    node.next = @peek\n    @peek = node\n    @size += 1\n  end\n\n  ### \u51fa\u5806\u758a ###\n  def pop\n    num = peek\n    @peek = @peek.next\n    @size -= 1\n    num\n  end\n\n  ### \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 ###\n  def peek\n    raise IndexError, '\u5806\u758a\u70ba\u7a7a' if is_empty?\n\n    @peek.val\n  end\n\n  ### \u5c07\u93c8\u7d50\u4e32\u5217\u8f49\u5316\u70ba Array \u4e26\u53cd\u56de ###\n  def to_array\n    arr = []\n    node = @peek\n    while node\n      arr &lt;&lt; node.val\n      node = node.next\n    end\n    arr.reverse\n  end\nend\n</code></pre> linkedlist_stack.zig<pre><code>// \u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a\nfn LinkedListStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack_top: ?*inc.ListNode(T) = null,             // \u5c07\u982d\u7bc0\u9ede\u4f5c\u70ba\u5806\u758a\u9802\n        stk_size: usize = 0,                             // \u5806\u758a\u7684\u9577\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,    // \u8a18\u61b6\u9ad4\u5206\u914d\u5668\n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u5806\u758a\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.stack_top = null;\n            self.stk_size = 0;\n        }\n\n        // \u6790\u69cb\u51fd\u5f0f\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stk_size;\n        }\n\n        // \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u5806\u758a\u70ba\u7a7a\");\n            return self.stack_top.?.val;\n        }  \n\n        // \u5165\u5806\u758a\n        pub fn push(self: *Self, num: T) !void {\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            node.next = self.stack_top;\n            self.stack_top = node;\n            self.stk_size += 1;\n        } \n\n        // \u51fa\u5806\u758a\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            self.stack_top = self.stack_top.?.next;\n            self.stk_size -= 1;\n            return num;\n        } \n\n        // \u5c07\u5806\u758a\u8f49\u63db\u70ba\u9663\u5217\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.stack_top;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[res.len - i - 1] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#2","title":"2. \u00a0 \u57fa\u65bc\u9663\u5217\u7684\u5be6\u73fe","text":"<p>\u4f7f\u7528\u9663\u5217\u5be6\u73fe\u5806\u758a\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u9663\u5217\u7684\u5c3e\u90e8\u4f5c\u70ba\u5806\u758a\u9802\u3002\u5982\u5716 5-3 \u6240\u793a\uff0c\u5165\u5806\u758a\u8207\u51fa\u5806\u758a\u64cd\u4f5c\u5206\u5225\u5c0d\u61c9\u5728\u9663\u5217\u5c3e\u90e8\u65b0\u589e\u5143\u7d20\u8207\u522a\u9664\u5143\u7d20\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u70ba \\(O(1)\\) \u3002</p> ArrayStackpush()pop() <p></p> <p></p> <p></p> <p> \u5716 5-3 \u00a0 \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u5806\u758a\u7684\u5165\u5806\u758a\u51fa\u5806\u758a\u64cd\u4f5c </p> <p>\u7531\u65bc\u5165\u5806\u758a\u7684\u5143\u7d20\u53ef\u80fd\u6703\u6e90\u6e90\u4e0d\u65b7\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u52d5\u614b\u9663\u5217\uff0c\u9019\u6a23\u5c31\u7121\u9808\u81ea\u884c\u8655\u7406\u9663\u5217\u64f4\u5bb9\u554f\u984c\u3002\u4ee5\u4e0b\u70ba\u793a\u4f8b\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_stack.py<pre><code>class ArrayStack:\n    \"\"\"\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\"\"\"\n\n    def __init__(self):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"\u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a\"\"\"\n        return self.size() == 0\n\n    def push(self, item: int):\n        \"\"\"\u5165\u5806\u758a\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u5806\u758a\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u5806\u758a\u70ba\u7a7a\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u5806\u758a\u70ba\u7a7a\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u4e32\u5217\u7528\u65bc\u5217\u5370\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u5806\u758a\u70ba\u7a7a\");\n        return stack.back();\n    }\n\n    /* \u8fd4\u56de Vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\n        stack = [];\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\ntype arrayStack struct {\n    data []int // \u8cc7\u6599\n}\n\n/* \u521d\u59cb\u5316\u5806\u758a */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // \u8a2d\u5b9a\u5806\u758a\u7684\u9577\u5ea6\u70ba 0\uff0c\u5bb9\u91cf\u70ba 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* \u5806\u758a\u7684\u9577\u5ea6 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* \u5806\u758a\u662f\u5426\u70ba\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* \u5165\u5806\u758a */\nfunc (s *arrayStack) push(v int) {\n    // \u5207\u7247\u6703\u81ea\u52d5\u64f4\u5bb9\n    s.data = append(s.data, v)\n}\n\n/* \u51fa\u5806\u758a */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* \u7372\u53d6\u5806\u758a\u9802\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* \u7372\u53d6 Slice \u7528\u65bc\u5217\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // \u521d\u59cb\u5316\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\n        stack = []\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* \u5165\u5806\u758a */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* \u51fa\u5806\u758a */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u5806\u758a\u70ba\u7a7a\")\n        }\n        return stack.removeLast()\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u5806\u758a\u70ba\u7a7a\")\n        }\n        return stack.last!\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    pop() {\n        if (this.isEmpty()) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.#stack.pop();\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    top() {\n        if (this.isEmpty()) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* \u5165\u5806\u758a */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.stack.pop();\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u5806\u758a\u70ba\u7a7a');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* \u5165\u5806\u758a */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* \u51fa\u5806\u758a */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"\u5806\u758a\u70ba\u7a7a\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u5806\u758a\u70ba\u7a7a\");\n    }\n    return _stack.last;\n  }\n\n  /* \u5c07\u5806\u758a\u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* \u521d\u59cb\u5316\u5806\u758a */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* \u5165\u5806\u758a */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* \u51fa\u5806\u758a */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"\u5806\u758a\u70ba\u7a7a\")\n        };\n        self.stack.last()\n    }\n\n    /* \u8fd4\u56de &amp;Vec */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* \u5efa\u69cb\u5b50 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // \u521d\u59cb\u5316\u4e00\u500b\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u64f4\u5bb9\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* \u5165\u5806\u758a */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"\u5806\u758a\u5df2\u6eff\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"\u5806\u758a\u70ba\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* \u51fa\u5806\u758a */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a */\nclass ArrayStack {\n    // \u521d\u59cb\u5316\u4e32\u5217\uff08\u52d5\u614b\u9663\u5217\uff09\n    private val stack = mutableListOf&lt;Int&gt;()\n\n    /* \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u5806\u758a */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* \u51fa\u5806\u758a */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* \u5c07 List \u8f49\u5316\u70ba Array \u4e26\u8fd4\u56de */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toTypedArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>### \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a ###\nclass ArrayStack\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize\n    @stack = []\n  end\n\n  ### \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6 ###\n  def size\n    @stack.length\n  end\n\n  ### \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a ###\n  def is_empty?\n    @stack.empty?\n  end\n\n  ### \u5165\u5806\u758a ###\n  def push(item)\n    @stack &lt;&lt; item\n  end\n\n  ### \u51fa\u5806\u758a ###\n  def pop\n    raise IndexError, '\u5806\u758a\u70ba\u7a7a' if is_empty?\n\n    @stack.pop\n  end\n\n  ### \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20 ###\n  def peek\n    raise IndexError, '\u5806\u758a\u70ba\u7a7a' if is_empty?\n\n    @stack.last\n  end\n\n  ### \u8fd4\u56de\u4e32\u5217\u7528\u65bc\u5217\u5370 ###\n  def to_array\n    @stack\n  end\nend\n</code></pre> array_stack.zig<pre><code>// \u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\nfn ArrayStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack: ?std.ArrayList(T) = null,     \n\n        // \u5efa\u69cb\u5b50\uff08\u5206\u914d\u8a18\u61b6\u9ad4+\u521d\u59cb\u5316\u5806\u758a\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n            if (self.stack == null) {\n                self.stack = std.ArrayList(T).init(allocator);\n            }\n        }\n\n        // \u6790\u69cb\u65b9\u6cd5\uff08\u91cb\u653e\u8a18\u61b6\u9ad4\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.stack == null) return;\n            self.stack.?.deinit();\n        }\n\n        // \u7372\u53d6\u5806\u758a\u7684\u9577\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stack.?.items.len;\n        }\n\n        // \u5224\u65b7\u5806\u758a\u662f\u5426\u70ba\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8a2a\u554f\u5806\u758a\u9802\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u5806\u758a\u70ba\u7a7a\");\n            return self.stack.?.items[self.size() - 1];\n        }  \n\n        // \u5165\u5806\u758a\n        pub fn push(self: *Self, num: T) !void {\n            try self.stack.?.append(num);\n        } \n\n        // \u51fa\u5806\u758a\n        pub fn pop(self: *Self) T {\n            var num = self.stack.?.pop();\n            return num;\n        } \n\n        // \u8fd4\u56de ArrayList\n        pub fn toList(self: *Self) std.ArrayList(T) {\n            return self.stack.?;\n        }\n    };\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3 \u00a0 \u5169\u7a2e\u5be6\u73fe\u5c0d\u6bd4","text":"<p>\u652f\u6301\u64cd\u4f5c</p> <p>\u5169\u7a2e\u5be6\u73fe\u90fd\u652f\u6301\u5806\u758a\u5b9a\u7fa9\u4e2d\u7684\u5404\u9805\u64cd\u4f5c\u3002\u9663\u5217\u5be6\u73fe\u984d\u5916\u652f\u6301\u96a8\u6a5f\u8a2a\u554f\uff0c\u4f46\u9019\u5df2\u8d85\u51fa\u4e86\u5806\u758a\u7684\u5b9a\u7fa9\u7bc4\u7587\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u6703\u7528\u5230\u3002</p> <p>\u6642\u9593\u6548\u7387</p> <p>\u5728\u57fa\u65bc\u9663\u5217\u7684\u5be6\u73fe\u4e2d\uff0c\u5165\u5806\u758a\u548c\u51fa\u5806\u758a\u64cd\u4f5c\u90fd\u5728\u9810\u5148\u5206\u914d\u597d\u7684\u9023\u7e8c\u8a18\u61b6\u9ad4\u4e2d\u9032\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u5feb\u53d6\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f03\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u5806\u758a\u6642\u8d85\u51fa\u9663\u5217\u5bb9\u91cf\uff0c\u6703\u89f8\u767c\u64f4\u5bb9\u6a5f\u5236\uff0c\u5c0e\u81f4\u8a72\u6b21\u5165\u5806\u758a\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u8b8a\u70ba \\(O(n)\\) \u3002</p> <p>\u5728\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u7684\u5be6\u73fe\u4e2d\uff0c\u93c8\u7d50\u4e32\u5217\u7684\u64f4\u5bb9\u975e\u5e38\u9748\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u9663\u5217\u64f4\u5bb9\u6642\u6548\u7387\u964d\u4f4e\u7684\u554f\u984c\u3002\u4f46\u662f\uff0c\u5165\u5806\u758a\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u7bc0\u9ede\u7269\u4ef6\u4e26\u4fee\u6539\u6307\u6a19\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5c0d\u8f03\u4f4e\u3002\u4e0d\u904e\uff0c\u5982\u679c\u5165\u5806\u758a\u5143\u7d20\u672c\u8eab\u5c31\u662f\u7bc0\u9ede\u7269\u4ef6\uff0c\u90a3\u9ebc\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9a5f\uff0c\u5f9e\u800c\u63d0\u9ad8\u6548\u7387\u3002</p> <p>\u7d9c\u4e0a\u6240\u8ff0\uff0c\u7576\u5165\u5806\u758a\u8207\u51fa\u5806\u758a\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u8cc7\u6599\u578b\u5225\u6642\uff0c\u4f8b\u5982 <code>int</code> \u6216 <code>double</code> \uff0c\u6211\u5011\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7d50\u8ad6\u3002</p> <ul> <li>\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\u5728\u89f8\u767c\u64f4\u5bb9\u6642\u6548\u7387\u6703\u964d\u4f4e\uff0c\u4f46\u7531\u65bc\u64f4\u5bb9\u662f\u4f4e\u983b\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u7684\u5806\u758a\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a69\u5b9a\u7684\u6548\u7387\u8868\u73fe\u3002</li> </ul> <p>\u7a7a\u9593\u6548\u7387</p> <p>\u5728\u521d\u59cb\u5316\u4e32\u5217\u6642\uff0c\u7cfb\u7d71\u6703\u70ba\u4e32\u5217\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8a72\u5bb9\u91cf\u53ef\u80fd\u8d85\u51fa\u5be6\u969b\u9700\u6c42\uff1b\u4e26\u4e14\uff0c\u64f4\u5bb9\u6a5f\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u9032\u884c\u64f4\u5bb9\u7684\uff0c\u64f4\u5bb9\u5f8c\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5be6\u969b\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u65bc\u9663\u5217\u5be6\u73fe\u7684\u5806\u758a\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u9593\u6d6a\u8cbb\u3002</p> <p>\u7136\u800c\uff0c\u7531\u65bc\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u9700\u8981\u984d\u5916\u5132\u5b58\u6307\u6a19\uff0c\u56e0\u6b64\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u4f54\u7528\u7684\u7a7a\u9593\u76f8\u5c0d\u8f03\u5927\u3002</p> <p>\u7d9c\u4e0a\uff0c\u6211\u5011\u4e0d\u80fd\u7c21\u55ae\u5730\u78ba\u5b9a\u54ea\u7a2e\u5be6\u73fe\u66f4\u52a0\u7bc0\u7701\u8a18\u61b6\u9ad4\uff0c\u9700\u8981\u91dd\u5c0d\u5177\u9ad4\u60c5\u6cc1\u9032\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4 \u00a0 \u5806\u758a\u7684\u5178\u578b\u61c9\u7528","text":"<ul> <li>\u700f\u89bd\u5668\u4e2d\u7684\u5f8c\u9000\u8207\u524d\u9032\u3001\u8edf\u9ad4\u4e2d\u7684\u64a4\u92b7\u8207\u53cd\u64a4\u92b7\u3002\u6bcf\u7576\u6211\u5011\u958b\u555f\u65b0\u7684\u7db2\u9801\uff0c\u700f\u89bd\u5668\u5c31\u6703\u5c0d\u4e0a\u4e00\u500b\u7db2\u9801\u57f7\u884c\u5165\u5806\u758a\uff0c\u9019\u6a23\u6211\u5011\u5c31\u53ef\u4ee5\u901a\u904e\u5f8c\u9000\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u500b\u7db2\u9801\u3002\u5f8c\u9000\u64cd\u4f5c\u5be6\u969b\u4e0a\u662f\u5728\u57f7\u884c\u51fa\u5806\u758a\u3002\u5982\u679c\u8981\u540c\u6642\u652f\u6301\u5f8c\u9000\u548c\u524d\u9032\uff0c\u90a3\u9ebc\u9700\u8981\u5169\u500b\u5806\u758a\u4f86\u914d\u5408\u5be6\u73fe\u3002</li> <li>\u7a0b\u5f0f\u8a18\u61b6\u9ad4\u7ba1\u7406\u3002\u6bcf\u6b21\u547c\u53eb\u51fd\u5f0f\u6642\uff0c\u7cfb\u7d71\u90fd\u6703\u5728\u5806\u758a\u9802\u65b0\u589e\u4e00\u500b\u5806\u758a\u5e40\uff0c\u7528\u65bc\u8a18\u9304\u51fd\u5f0f\u7684\u4e0a\u4e0b\u6587\u8cc7\u8a0a\u3002\u5728\u905e\u8ff4\u51fd\u5f0f\u4e2d\uff0c\u5411\u4e0b\u905e\u63a8\u968e\u6bb5\u6703\u4e0d\u65b7\u57f7\u884c\u5165\u5806\u758a\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u968e\u6bb5\u5247\u6703\u4e0d\u65b7\u57f7\u884c\u51fa\u5806\u758a\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/","title":"5.4 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_stack_and_queue/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u5806\u758a\u662f\u4e00\u7a2e\u9075\u5faa\u5148\u5165\u5f8c\u51fa\u539f\u5247\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u53ef\u900f\u904e\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u4f86\u5be6\u73fe\u3002</li> <li>\u5728\u6642\u9593\u6548\u7387\u65b9\u9762\uff0c\u5806\u758a\u7684\u9663\u5217\u5be6\u73fe\u5177\u6709\u8f03\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u64f4\u5bb9\u904e\u7a0b\u4e2d\uff0c\u55ae\u6b21\u5165\u5806\u758a\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u6703\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5806\u758a\u7684\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\u5177\u6709\u66f4\u70ba\u7a69\u5b9a\u7684\u6548\u7387\u8868\u73fe\u3002</li> <li>\u5728\u7a7a\u9593\u6548\u7387\u65b9\u9762\uff0c\u5806\u758a\u7684\u9663\u5217\u5be6\u73fe\u53ef\u80fd\u5c0e\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u9593\u6d6a\u8cbb\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u93c8\u7d50\u4e32\u5217\u7bc0\u9ede\u6240\u4f54\u7528\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u6bd4\u9663\u5217\u5143\u7d20\u66f4\u5927\u3002</li> <li>\u4f47\u5217\u662f\u4e00\u7a2e\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5247\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u540c\u6a23\u53ef\u4ee5\u900f\u904e\u9663\u5217\u6216\u93c8\u7d50\u4e32\u5217\u4f86\u5be6\u73fe\u3002\u5728\u6642\u9593\u6548\u7387\u548c\u7a7a\u9593\u6548\u7387\u7684\u5c0d\u6bd4\u4e0a\uff0c\u4f47\u5217\u7684\u7d50\u8ad6\u8207\u524d\u8ff0\u5806\u758a\u7684\u7d50\u8ad6\u76f8\u4f3c\u3002</li> <li>\u96d9\u5411\u4f47\u5217\u662f\u4e00\u7a2e\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u4f47\u5217\uff0c\u5b83\u5141\u8a31\u5728\u5169\u7aef\u9032\u884c\u5143\u7d20\u7684\u65b0\u589e\u548c\u522a\u9664\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u700f\u89bd\u5668\u7684\u524d\u9032\u5f8c\u9000\u662f\u5426\u662f\u96d9\u5411\u93c8\u7d50\u4e32\u5217\u5be6\u73fe\uff1f</p> <p>\u700f\u89bd\u5668\u7684\u524d\u9032\u5f8c\u9000\u529f\u80fd\u672c\u8cea\u4e0a\u662f\u201c\u5806\u758a\u201d\u7684\u9ad4\u73fe\u3002\u7576\u7528\u6236\u8a2a\u554f\u4e00\u500b\u65b0\u9801\u9762\u6642\uff0c\u8a72\u9801\u9762\u6703\u88ab\u65b0\u589e\u5230\u5806\u758a\u9802\uff1b\u7576\u7528\u6236\u9ede\u9078\u5f8c\u9000\u6309\u9215\u6642\uff0c\u8a72\u9801\u9762\u6703\u5f9e\u5806\u758a\u9802\u5f48\u51fa\u3002\u4f7f\u7528\u96d9\u5411\u4f47\u5217\u53ef\u4ee5\u65b9\u4fbf\u5730\u5be6\u73fe\u4e00\u4e9b\u984d\u5916\u64cd\u4f5c\uff0c\u9019\u500b\u5728\u201c\u96d9\u5411\u4f47\u5217\u201d\u7ae0\u7bc0\u6709\u63d0\u5230\u3002</p> <p>Q\uff1a\u5728\u51fa\u5806\u758a\u5f8c\uff0c\u662f\u5426\u9700\u8981\u91cb\u653e\u51fa\u5806\u758a\u7bc0\u9ede\u7684\u8a18\u61b6\u9ad4\uff1f</p> <p>\u5982\u679c\u5f8c\u7e8c\u4ecd\u9700\u8981\u4f7f\u7528\u5f48\u51fa\u7bc0\u9ede\uff0c\u5247\u4e0d\u9700\u8981\u91cb\u653e\u8a18\u61b6\u9ad4\u3002\u82e5\u4e4b\u5f8c\u4e0d\u9700\u8981\u7528\u5230\uff0c<code>Java</code> \u548c <code>Python</code> \u7b49\u8a9e\u8a00\u64c1\u6709\u81ea\u52d5\u5783\u573e\u56de\u6536\u6a5f\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52d5\u91cb\u653e\u8a18\u61b6\u9ad4\uff1b\u5728 <code>C</code> \u548c <code>C++</code> \u4e2d\u9700\u8981\u624b\u52d5\u91cb\u653e\u8a18\u61b6\u9ad4\u3002</p> <p>Q\uff1a\u96d9\u5411\u4f47\u5217\u50cf\u662f\u5169\u500b\u5806\u758a\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u9ebc\uff1f</p> <p>\u96d9\u5411\u4f47\u5217\u5c31\u50cf\u662f\u5806\u758a\u548c\u4f47\u5217\u7684\u7d44\u5408\u6216\u5169\u500b\u5806\u758a\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73fe\u7684\u662f\u5806\u758a + \u4f47\u5217\u7684\u908f\u8f2f\uff0c\u56e0\u6b64\u53ef\u4ee5\u5be6\u73fe\u5806\u758a\u8207\u4f47\u5217\u7684\u6240\u6709\u61c9\u7528\uff0c\u4e26\u4e14\u66f4\u52a0\u9748\u6d3b\u3002</p> <p>Q\uff1a\u64a4\u92b7\uff08undo\uff09\u548c\u53cd\u64a4\u92b7\uff08redo\uff09\u5177\u9ad4\u662f\u5982\u4f55\u5be6\u73fe\u7684\uff1f</p> <p>\u4f7f\u7528\u5169\u500b\u5806\u758a\uff0c\u5806\u758a <code>A</code> \u7528\u65bc\u64a4\u92b7\uff0c\u5806\u758a <code>B</code> \u7528\u65bc\u53cd\u64a4\u92b7\u3002</p> <ol> <li>\u6bcf\u7576\u4f7f\u7528\u8005\u57f7\u884c\u4e00\u500b\u64cd\u4f5c\uff0c\u5c07\u9019\u500b\u64cd\u4f5c\u58d3\u5165\u5806\u758a <code>A</code> \uff0c\u4e26\u6e05\u7a7a\u5806\u758a <code>B</code> \u3002</li> <li>\u7576\u7528\u6236\u57f7\u884c\u201c\u64a4\u92b7\u201d\u6642\uff0c\u5f9e\u5806\u758a <code>A</code> \u4e2d\u5f48\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u4e26\u5c07\u5176\u58d3\u5165\u5806\u758a <code>B</code> \u3002</li> <li>\u7576\u7528\u6236\u57f7\u884c\u201c\u53cd\u64a4\u92b7\u201d\u6642\uff0c\u5f9e\u5806\u758a <code>B</code> \u4e2d\u5f48\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u4e26\u5c07\u5176\u58d3\u5165\u5806\u758a <code>A</code> \u3002</li> </ol>"},{"location":"chapter_tree/","title":"\u7b2c 7 \u7ae0 \u00a0 \u6a39","text":"<p>Abstract</p> <p>\u53c3\u5929\u5927\u6a39\u5145\u6eff\u751f\u547d\u529b\uff0c\u6839\u6df1\u8449\u8302\uff0c\u5206\u679d\u6276\u758f\u3002</p> <p>\u5b83\u70ba\u6211\u5011\u5c55\u73fe\u4e86\u8cc7\u6599\u5206\u6cbb\u7684\u751f\u52d5\u5f62\u614b\u3002</p>"},{"location":"chapter_tree/#_1","title":"\u672c\u7ae0\u5167\u5bb9","text":"<ul> <li>7.1 \u00a0 \u4e8c\u5143\u6a39</li> <li>7.2 \u00a0 \u4e8c\u5143\u6a39\u8d70\u8a2a</li> <li>7.3 \u00a0 \u4e8c\u5143\u6a39\u9663\u5217\u8868\u793a</li> <li>7.4 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39</li> <li>7.5 \u00a0 AVL *</li> <li>7.6 \u00a0 \u5c0f\u7d50</li> </ul>"},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3 \u00a0 \u4e8c\u5143\u6a39\u9663\u5217\u8868\u793a","text":"<p>\u5728\u93c8\u7d50\u4e32\u5217\u8868\u793a\u4e0b\uff0c\u4e8c\u5143\u6a39\u7684\u5132\u5b58\u55ae\u5143\u70ba\u7bc0\u9ede <code>TreeNode</code> \uff0c\u7bc0\u9ede\u4e4b\u9593\u900f\u904e\u6307\u6a19\u76f8\u9023\u7dda\u3002\u4e0a\u4e00\u7bc0\u4ecb\u7d39\u4e86\u93c8\u7d50\u4e32\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u7684\u5404\u9805\u57fa\u672c\u64cd\u4f5c\u3002</p> <p>\u90a3\u9ebc\uff0c\u6211\u5011\u80fd\u5426\u7528\u9663\u5217\u4f86\u8868\u793a\u4e8c\u5143\u6a39\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1 \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u5143\u6a39","text":"<p>\u5148\u5206\u6790\u4e00\u500b\u7c21\u55ae\u6848\u4f8b\u3002\u7d66\u5b9a\u4e00\u68f5\u5b8c\u7f8e\u4e8c\u5143\u6a39\uff0c\u6211\u5011\u5c07\u6240\u6709\u7bc0\u9ede\u6309\u7167\u5c64\u5e8f\u8d70\u8a2a\u7684\u9806\u5e8f\u5132\u5b58\u5728\u4e00\u500b\u9663\u5217\u4e2d\uff0c\u5247\u6bcf\u500b\u7bc0\u9ede\u90fd\u5c0d\u61c9\u552f\u4e00\u7684\u9663\u5217\u7d22\u5f15\u3002</p> <p>\u6839\u64da\u5c64\u5e8f\u8d70\u8a2a\u7684\u7279\u6027\uff0c\u6211\u5011\u53ef\u4ee5\u63a8\u5c0e\u51fa\u7236\u7bc0\u9ede\u7d22\u5f15\u8207\u5b50\u7bc0\u9ede\u7d22\u5f15\u4e4b\u9593\u7684\u201c\u5c0d\u6620\u516c\u5f0f\u201d\uff1a\u82e5\u67d0\u7bc0\u9ede\u7684\u7d22\u5f15\u70ba \\(i\\) \uff0c\u5247\u8a72\u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7d22\u5f15\u70ba \\(2i + 1\\) \uff0c\u53f3\u5b50\u7bc0\u9ede\u7d22\u5f15\u70ba \\(2i + 2\\) \u3002\u5716 7-12 \u5c55\u793a\u4e86\u5404\u500b\u7bc0\u9ede\u7d22\u5f15\u4e4b\u9593\u7684\u5c0d\u6620\u95dc\u4fc2\u3002</p> <p></p> <p> \u5716 7-12 \u00a0 \u5b8c\u7f8e\u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a </p> <p>\u5c0d\u6620\u516c\u5f0f\u7684\u89d2\u8272\u76f8\u7576\u65bc\u93c8\u7d50\u4e32\u5217\u4e2d\u7684\u7bc0\u9ede\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002\u7d66\u5b9a\u9663\u5217\u4e2d\u7684\u4efb\u610f\u4e00\u500b\u7bc0\u9ede\uff0c\u6211\u5011\u90fd\u53ef\u4ee5\u900f\u904e\u5c0d\u6620\u516c\u5f0f\u4f86\u8a2a\u554f\u5b83\u7684\u5de6\uff08\u53f3\uff09\u5b50\u7bc0\u9ede\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2 \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u5143\u6a39","text":"<p>\u5b8c\u7f8e\u4e8c\u5143\u6a39\u662f\u4e00\u500b\u7279\u4f8b\uff0c\u5728\u4e8c\u5143\u6a39\u7684\u4e2d\u9593\u5c64\u901a\u5e38\u5b58\u5728\u8a31\u591a <code>None</code> \u3002\u7531\u65bc\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u4e26\u4e0d\u5305\u542b\u9019\u4e9b <code>None</code> \uff0c\u56e0\u6b64\u6211\u5011\u7121\u6cd5\u50c5\u6191\u8a72\u5e8f\u5217\u4f86\u63a8\u6e2c <code>None</code> \u7684\u6578\u91cf\u548c\u5206\u4f48\u4f4d\u7f6e\u3002\u9019\u610f\u5473\u8457\u5b58\u5728\u591a\u7a2e\u4e8c\u5143\u6a39\u7d50\u69cb\u90fd\u7b26\u5408\u8a72\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u3002</p> <p>\u5982\u5716 7-13 \u6240\u793a\uff0c\u7d66\u5b9a\u4e00\u68f5\u975e\u5b8c\u7f8e\u4e8c\u5143\u6a39\uff0c\u4e0a\u8ff0\u9663\u5217\u8868\u793a\u65b9\u6cd5\u5df2\u7d93\u5931\u6548\u3002</p> <p></p> <p> \u5716 7-13 \u00a0 \u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u5c0d\u61c9\u591a\u7a2e\u4e8c\u5143\u6a39\u53ef\u80fd\u6027 </p> <p>\u70ba\u4e86\u89e3\u6c7a\u6b64\u554f\u984c\uff0c\u6211\u5011\u53ef\u4ee5\u8003\u616e\u5728\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u4e2d\u986f\u5f0f\u5730\u5beb\u51fa\u6240\u6709 <code>None</code> \u3002\u5982\u5716 7-14 \u6240\u793a\uff0c\u9019\u6a23\u8655\u7406\u5f8c\uff0c\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u5143\u6a39\u4e86\u3002\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a\n# \u4f7f\u7528 None \u4f86\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c INT_MAX \u6a19\u8a18\u7a7a\u4f4d\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88dd\u985e\u5225 Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u4f86\u6a19\u8a18\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u578b\u5225 \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u4f86\u6a19\u8a18\u7a7a\u4f4d\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 any \u578b\u5225\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u4f86\u6a19\u8a18\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u578b\u5225 \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u4f86\u6a19\u8a18\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 null \u4f86\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 null \u4f86\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u578b\u5225 \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u4f86\u6a19\u8a18\u7a7a\u4f4d\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 None \u4f86\u6a19\u8a18\u7a7a\u4f4d\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6a19\u8a18\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u7bc0\u9ede\u503c\u4e0d\u80fd\u70ba INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a */\n// \u4f7f\u7528 null \u4f86\u8868\u793a\u7a7a\u4f4d\nval tree = arrayOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>### \u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a ###\n# \u4f7f\u7528 nil \u4f86\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>\n</code></pre> <p></p> <p> \u5716 7-14 \u00a0 \u4efb\u610f\u578b\u5225\u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a </p> <p>\u503c\u5f97\u8aaa\u660e\u7684\u662f\uff0c\u5b8c\u5168\u4e8c\u5143\u6a39\u975e\u5e38\u9069\u5408\u4f7f\u7528\u9663\u5217\u4f86\u8868\u793a\u3002\u56de\u9867\u5b8c\u5168\u4e8c\u5143\u6a39\u7684\u5b9a\u7fa9\uff0c<code>None</code> \u53ea\u51fa\u73fe\u5728\u6700\u5e95\u5c64\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u56e0\u6b64\u6240\u6709 <code>None</code> \u4e00\u5b9a\u51fa\u73fe\u5728\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u7684\u672b\u5c3e\u3002</p> <p>\u9019\u610f\u5473\u8457\u4f7f\u7528\u9663\u5217\u8868\u793a\u5b8c\u5168\u4e8c\u5143\u6a39\u6642\uff0c\u53ef\u4ee5\u7701\u7565\u5132\u5b58\u6240\u6709 <code>None</code> \uff0c\u975e\u5e38\u65b9\u4fbf\u3002\u5716 7-15 \u7d66\u51fa\u4e86\u4e00\u500b\u4f8b\u5b50\u3002</p> <p></p> <p> \u5716 7-15 \u00a0 \u5b8c\u5168\u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a </p> <p>\u4ee5\u4e0b\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e86\u4e00\u68f5\u57fa\u65bc\u9663\u5217\u8868\u793a\u7684\u4e8c\u5143\u6a39\uff0c\u5305\u62ec\u4ee5\u4e0b\u5e7e\u7a2e\u64cd\u4f5c\u3002</p> <ul> <li>\u7d66\u5b9a\u67d0\u7bc0\u9ede\uff0c\u7372\u53d6\u5b83\u7684\u503c\u3001\u5de6\uff08\u53f3\uff09\u5b50\u7bc0\u9ede\u3001\u7236\u7bc0\u9ede\u3002</li> <li>\u7372\u53d6\u524d\u5e8f\u8d70\u8a2a\u3001\u4e2d\u5e8f\u8d70\u8a2a\u3001\u5f8c\u5e8f\u8d70\u8a2a\u3001\u5c64\u5e8f\u8d70\u8a2a\u5e8f\u5217\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"\u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"\u5efa\u69cb\u5b50\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"\u4e32\u5217\u5bb9\u91cf\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int | None:\n        \"\"\"\u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c\"\"\"\n        # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"\u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"\u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"\u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"\u5c64\u5e8f\u8d70\u8a2a\"\"\"\n        self.res = []\n        # \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\"\"\"\n        if self.val(i) is None:\n            return\n        # \u524d\u5e8f\u8d70\u8a2a\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # \u4e2d\u5e8f\u8d70\u8a2a\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # \u5f8c\u5e8f\u8d70\u8a2a\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"\u524d\u5e8f\u8d70\u8a2a\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"\u4e2d\u5e8f\u8d70\u8a2a\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"\u5f8c\u5e8f\u8d70\u8a2a\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n  public:\n    /* \u5efa\u69cb\u5b50 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    int size() {\n        return tree.size();\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    int val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (val(i) == INT_MAX)\n            return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* \u5efa\u69cb\u5b50 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    public int size() {\n        return tree.size();\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    public Integer val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (val(i) == null)\n            return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    public int? Val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (!Val(i).HasValue)\n            return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* \u5efa\u69cb\u5b50 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* \u4e32\u5217\u5bb9\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* \u5c64\u5e8f\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n    if abt.val(i) == nil {\n        return\n    }\n    // \u524d\u5e8f\u8d70\u8a2a\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // \u4e2d\u5e8f\u8d70\u8a2a\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // \u5f8c\u5e8f\u8d70\u8a2a\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* \u524d\u5e8f\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* \u5efa\u69cb\u5b50 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    func val(i: Int) -&gt; Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        guard let val = val(i: i) else {\n            return\n        }\n        // \u524d\u5e8f\u8d70\u8a2a\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    val(i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    levelOrder() {\n        let res = [];\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    #dfs(i, order, res) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* \u5efa\u69cb\u5b50 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    val(i: number): number | null {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    levelOrder(): number[] {\n        let res = [];\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* \u5efa\u69cb\u5b50 */\n  ArrayBinaryTree(this._tree);\n\n  /* \u4e32\u5217\u5bb9\u91cf */\n  int size() {\n    return _tree.length;\n  }\n\n  /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n  int? val(int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* \u5c64\u5e8f\u8d70\u8a2a */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n    if (val(i) == null) {\n      return;\n    }\n    // \u524d\u5e8f\u8d70\u8a2a\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // \u4e2d\u5e8f\u8d70\u8a2a\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // \u5f8c\u5e8f\u8d70\u8a2a\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* \u524d\u5e8f\u8d70\u8a2a */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* \u4e2d\u5e8f\u8d70\u8a2a */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* \u5f8c\u5e8f\u8d70\u8a2a */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* \u5efa\u69cb\u5b50 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* \u4e32\u5217\u5bb9\u91cf */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        self.tree.iter().filter_map(|&amp;x| x).collect()\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    fn dfs(&amp;self, i: i32, order: &amp;'static str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // \u524d\u5e8f\u8d70\u8a2a\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u7d50\u69cb\u9ad4 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* \u5efa\u69cb\u5b50 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* \u6790\u69cb\u51fd\u5f0f */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* \u4e32\u5217\u5bb9\u91cf */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* \u5c64\u5e8f\u8d70\u8a2a */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n    if (val(abt, i) == INT_MAX)\n        return;\n    // \u524d\u5e8f\u8d70\u8a2a\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // \u4e2d\u5e8f\u8d70\u8a2a\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // \u5f8c\u5e8f\u8d70\u8a2a\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u8d70\u8a2a */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 */\nclass ArrayBinaryTree(val tree: MutableList&lt;Int?&gt;) {\n    /* \u4e32\u5217\u5bb9\u91cf */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c */\n    fun _val(i: Int): Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* \u5c64\u5e8f\u8d70\u8a2a */\n    fun levelOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        // \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n        for (i in 0..&lt;size()) {\n            if (_val(i) != null)\n                res.add(_val(i))\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // \u82e5\u70ba\u7a7a\u4f4d\uff0c\u5247\u8fd4\u56de\n        if (_val(i) == null)\n            return\n        // \u524d\u5e8f\u8d70\u8a2a\n        if (\"pre\" == order)\n            res.add(_val(i))\n        dfs(left(i), order, res)\n        // \u4e2d\u5e8f\u8d70\u8a2a\n        if (\"in\" == order)\n            res.add(_val(i))\n        dfs(right(i), order, res)\n        // \u5f8c\u5e8f\u8d70\u8a2a\n        if (\"post\" == order)\n            res.add(_val(i))\n    }\n\n    /* \u524d\u5e8f\u8d70\u8a2a */\n    fun preOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u8d70\u8a2a */\n    fun inOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* \u5f8c\u5e8f\u8d70\u8a2a */\n    fun postOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>### \u9663\u5217\u8868\u793a\u4e0b\u7684\u4e8c\u5143\u6a39\u985e\u5225 ###\nclass ArrayBinaryTree\n  ### \u5efa\u69cb\u5b50 ###\n  def initialize(arr)\n    @tree = arr.to_a\n  end\n\n  ### \u4e32\u5217\u5bb9\u91cf ###\n  def size\n    @tree.length\n  end\n\n  ### \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u503c ###\n  def val(i)\n    # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5247\u8fd4\u56de nil \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    return if i &lt; 0 || i &gt;= size\n\n    @tree[i]\n  end\n\n  ### \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 ###\n  def left(i)\n    2 * i + 1\n  end\n\n  ### \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u53f3\u5b50\u7bc0\u9ede\u7684\u7d22\u5f15 ###\n  def right(i)\n    2 * i + 2\n  end\n\n  ### \u7372\u53d6\u7d22\u5f15\u70ba i \u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\u7684\u7d22\u5f15 ###\n  def parent(i)\n    (i - 1) / 2\n  end\n\n  ### \u5c64\u5e8f\u8d70\u8a2a ###\n  def level_order\n    @res = []\n\n    # \u76f4\u63a5\u8d70\u8a2a\u9663\u5217\n    for i in 0...size\n      @res &lt;&lt; val(i) unless val(i).nil?\n    end\n\n    @res\n  end\n\n  ### \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a ###\n  def dfs(i, order)\n    return if val(i).nil?\n    # \u524d\u5e8f\u8d70\u8a2a\n    @res &lt;&lt; val(i) if order == :pre\n    dfs(left(i), order)\n    # \u4e2d\u5e8f\u8d70\u8a2a\n    @res &lt;&lt; val(i) if order == :in\n    dfs(right(i), order)\n    # \u5f8c\u5e8f\u8d70\u8a2a\n    @res &lt;&lt; val(i) if order == :post\n  end\n\n  ### \u524d\u5e8f\u8d70\u8a2a ###\n  def pre_order\n    @res = []\n    dfs(0, :pre)\n    @res\n  end\n\n  ### \u4e2d\u5e8f\u8d70\u8a2a ###\n  def in_order\n    @res = []\n    dfs(0, :in)\n    @res\n  end\n\n  ### \u5f8c\u5e8f\u8d70\u8a2a ###\n  def post_order\n    @res = []\n    dfs(0, :post)\n    @res\n  end\nend\n</code></pre> array_binary_tree.zig<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3 \u00a0 \u512a\u9ede\u8207\u4fb7\u9650\u6027","text":"<p>\u4e8c\u5143\u6a39\u7684\u9663\u5217\u8868\u793a\u4e3b\u8981\u6709\u4ee5\u4e0b\u512a\u9ede\u3002</p> <ul> <li>\u9663\u5217\u5132\u5b58\u5728\u9023\u7e8c\u7684\u8a18\u61b6\u9ad4\u7a7a\u9593\u4e2d\uff0c\u5c0d\u5feb\u53d6\u53cb\u597d\uff0c\u8a2a\u554f\u8207\u8d70\u8a2a\u901f\u5ea6\u8f03\u5feb\u3002</li> <li>\u4e0d\u9700\u8981\u5132\u5b58\u6307\u6a19\uff0c\u6bd4\u8f03\u7bc0\u7701\u7a7a\u9593\u3002</li> <li>\u5141\u8a31\u96a8\u6a5f\u8a2a\u554f\u7bc0\u9ede\u3002</li> </ul> <p>\u7136\u800c\uff0c\u9663\u5217\u8868\u793a\u4e5f\u5b58\u5728\u4e00\u4e9b\u4fb7\u9650\u6027\u3002</p> <ul> <li>\u9663\u5217\u5132\u5b58\u9700\u8981\u9023\u7e8c\u8a18\u61b6\u9ad4\u7a7a\u9593\uff0c\u56e0\u6b64\u4e0d\u9069\u5408\u5132\u5b58\u8cc7\u6599\u91cf\u904e\u5927\u7684\u6a39\u3002</li> <li>\u589e\u522a\u7bc0\u9ede\u9700\u8981\u900f\u904e\u9663\u5217\u63d2\u5165\u8207\u522a\u9664\u64cd\u4f5c\u5be6\u73fe\uff0c\u6548\u7387\u8f03\u4f4e\u3002</li> <li>\u7576\u4e8c\u5143\u6a39\u4e2d\u5b58\u5728\u5927\u91cf <code>None</code> \u6642\uff0c\u9663\u5217\u4e2d\u5305\u542b\u7684\u7bc0\u9ede\u8cc7\u6599\u6bd4\u91cd\u8f03\u4f4e\uff0c\u7a7a\u9593\u5229\u7528\u7387\u8f03\u4f4e\u3002</li> </ul>"},{"location":"chapter_tree/avl_tree/","title":"7.5 \u00a0 AVL \u6a39 *","text":"<p>\u5728\u201c\u4e8c\u5143\u641c\u5c0b\u6a39\u201d\u7ae0\u7bc0\u4e2d\u6211\u5011\u63d0\u5230\uff0c\u5728\u591a\u6b21\u63d2\u5165\u548c\u522a\u9664\u64cd\u4f5c\u5f8c\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u53ef\u80fd\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5c07\u5f9e \\(O(\\log n)\\) \u52a3\u5316\u70ba \\(O(n)\\) \u3002</p> <p>\u5982\u5716 7-24 \u6240\u793a\uff0c\u7d93\u904e\u5169\u6b21\u522a\u9664\u7bc0\u9ede\u64cd\u4f5c\uff0c\u9019\u68f5\u4e8c\u5143\u641c\u5c0b\u6a39\u4fbf\u6703\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u3002</p> <p></p> <p> \u5716 7-24 \u00a0 AVL \u6a39\u5728\u522a\u9664\u7bc0\u9ede\u5f8c\u767c\u751f\u9000\u5316 </p> <p>\u518d\u4f8b\u5982\uff0c\u5728\u5716 7-25 \u6240\u793a\u7684\u5b8c\u7f8e\u4e8c\u5143\u6a39\u4e2d\u63d2\u5165\u5169\u500b\u7bc0\u9ede\u5f8c\uff0c\u6a39\u5c07\u56b4\u91cd\u5411\u5de6\u50be\u659c\uff0c\u67e5\u8a62\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e5f\u96a8\u4e4b\u52a3\u5316\u3002</p> <p></p> <p> \u5716 7-25 \u00a0 AVL \u6a39\u5728\u63d2\u5165\u7bc0\u9ede\u5f8c\u767c\u751f\u9000\u5316 </p> <p>1962 \u5e74 G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u8ad6\u6587\u201cAn algorithm for the organization of information\u201d\u4e2d\u63d0\u51fa\u4e86 AVL \u6a39\u3002\u8ad6\u6587\u4e2d\u8a73\u7d30\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u78ba\u4fdd\u5728\u6301\u7e8c\u65b0\u589e\u548c\u522a\u9664\u7bc0\u9ede\u5f8c\uff0cAVL \u6a39\u4e0d\u6703\u9000\u5316\uff0c\u5f9e\u800c\u4f7f\u5f97\u5404\u7a2e\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7d1a\u5225\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u5728\u9700\u8981\u983b\u7e41\u9032\u884c\u589e\u522a\u67e5\u6539\u64cd\u4f5c\u7684\u5834\u666f\u4e2d\uff0cAVL \u6a39\u80fd\u59cb\u7d42\u4fdd\u6301\u9ad8\u6548\u7684\u8cc7\u6599\u64cd\u4f5c\u6548\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u61c9\u7528\u50f9\u503c\u3002</p>"},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1 \u00a0 AVL \u6a39\u5e38\u898b\u8853\u8a9e","text":"<p>AVL \u6a39\u65e2\u662f\u4e8c\u5143\u641c\u5c0b\u6a39\uff0c\u4e5f\u662f\u5e73\u8861\u4e8c\u5143\u6a39\uff0c\u540c\u6642\u6eff\u8db3\u9019\u5169\u985e\u4e8c\u5143\u6a39\u7684\u6240\u6709\u6027\u8cea\uff0c\u56e0\u6b64\u662f\u4e00\u7a2e\u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39\uff08balanced binary search tree\uff09\u3002</p>"},{"location":"chapter_tree/avl_tree/#1","title":"1. \u00a0 \u7bc0\u9ede\u9ad8\u5ea6","text":"<p>\u7531\u65bc AVL \u6a39\u7684\u76f8\u95dc\u64cd\u4f5c\u9700\u8981\u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6\uff0c\u56e0\u6b64\u6211\u5011\u9700\u8981\u70ba\u7bc0\u9ede\u985e\u5225\u65b0\u589e <code>height</code> \u8b8a\u6578\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"AVL \u6a39\u7bc0\u9ede\u985e\u5225\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # \u7bc0\u9ede\u503c\n        self.height: int = 0                # \u7bc0\u9ede\u9ad8\u5ea6\n        self.left: TreeNode | None = None   # \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n        self.right: TreeNode | None = None  # \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nstruct TreeNode {\n    int val{};          // \u7bc0\u9ede\u503c\n    int height = 0;     // \u7bc0\u9ede\u9ad8\u5ea6\n    TreeNode *left{};   // \u5de6\u5b50\u7bc0\u9ede\n    TreeNode *right{};  // \u53f3\u5b50\u7bc0\u9ede\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    public int val;        // \u7bc0\u9ede\u503c\n    public int height;     // \u7bc0\u9ede\u9ad8\u5ea6\n    public TreeNode left;  // \u5de6\u5b50\u7bc0\u9ede\n    public TreeNode right; // \u53f3\u5b50\u7bc0\u9ede\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u7bc0\u9ede\u503c\n    public int height;      // \u7bc0\u9ede\u9ad8\u5ea6\n    public TreeNode? left;  // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntype TreeNode struct {\n    Val    int       // \u7bc0\u9ede\u503c\n    Height int       // \u7bc0\u9ede\u9ad8\u5ea6\n    Left   *TreeNode // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    Right  *TreeNode // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var height: Int // \u7bc0\u9ede\u9ad8\u5ea6\n    var left: TreeNode? // \u5de6\u5b50\u7bc0\u9ede\n    var right: TreeNode? // \u53f3\u5b50\u7bc0\u9ede\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    val; // \u7bc0\u9ede\u503c\n    height; //\u7bc0\u9ede\u9ad8\u5ea6\n    left; // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    right; // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    val: number;            // \u7bc0\u9ede\u503c\n    height: number;         // \u7bc0\u9ede\u9ad8\u5ea6\n    left: TreeNode | null;  // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    right: TreeNode | null; // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n  int val;         // \u7bc0\u9ede\u503c\n  int height;      // \u7bc0\u9ede\u9ad8\u5ea6\n  TreeNode? left;  // \u5de6\u5b50\u7bc0\u9ede\n  TreeNode? right; // \u53f3\u5b50\u7bc0\u9ede\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL \u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct TreeNode {\n    val: i32,                               // \u7bc0\u9ede\u503c\n    height: i32,                            // \u7bc0\u9ede\u9ad8\u5ea6\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u7bc0\u9ede\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u7bc0\u9ede\n}\n\nimpl TreeNode {\n    /* \u5efa\u69cb\u5b50 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* \u5efa\u69cb\u5b50 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL \u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode(val _val: Int) {  // \u7bc0\u9ede\u503c\n    val height: Int = 0          // \u7bc0\u9ede\u9ad8\u5ea6\n    val left: TreeNode? = null   // \u5de6\u5b50\u7bc0\u9ede\n    val right: TreeNode? = null  // \u53f3\u5b50\u7bc0\u9ede\n}\n</code></pre> <pre><code>### AVL \u6a39\u7bc0\u9ede\u985e\u5225 ###\nclass TreeNode\n  attr_accessor :val    # \u7bc0\u9ede\u503c\n  attr_accessor :height # \u7bc0\u9ede\u9ad8\u5ea6\n  attr_accessor :left   # \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n  attr_accessor :right  # \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n\n  def initialize(val)\n    @val = val\n    @height = 0\n  end\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u201c\u7bc0\u9ede\u9ad8\u5ea6\u201d\u662f\u6307\u5f9e\u8a72\u7bc0\u9ede\u5230\u5b83\u7684\u6700\u9060\u8449\u7bc0\u9ede\u7684\u8ddd\u96e2\uff0c\u5373\u6240\u7d93\u904e\u7684\u201c\u908a\u201d\u7684\u6578\u91cf\u3002\u9700\u8981\u7279\u5225\u6ce8\u610f\u7684\u662f\uff0c\u8449\u7bc0\u9ede\u7684\u9ad8\u5ea6\u70ba \\(0\\) \uff0c\u800c\u7a7a\u7bc0\u9ede\u7684\u9ad8\u5ea6\u70ba \\(-1\\) \u3002\u6211\u5011\u5c07\u5efa\u7acb\u5169\u500b\u5de5\u5177\u51fd\u5f0f\uff0c\u5206\u5225\u7528\u65bc\u7372\u53d6\u548c\u66f4\u65b0\u7bc0\u9ede\u7684\u9ad8\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6\"\"\"\n    # \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"\u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\"\"\"\n    # \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint height(TreeNode node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    node?.height ?? -1\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nheight(node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\n#updateHeight(node) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n  // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n  return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n  // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 */\nfun height(node: TreeNode?): Int {\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return node?.height ?: -1\n}\n\n/* \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 */\nfun updateHeight(node: TreeNode?) {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node?.height = max(height(node?.left), height(node?.right)) + 1\n}\n</code></pre> avl_tree.rb<pre><code>### \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6 ###\ndef height(node)\n  # \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n  return node.height unless node.nil?\n\n  -1\nend\n\n### \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6 ###\ndef update_height(node)\n  # \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n  node.height = [height(node.left), height(node.right)].max + 1\nend\n</code></pre> avl_tree.zig<pre><code>// \u7372\u53d6\u7bc0\u9ede\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    _ = self;\n    // \u7a7a\u7bc0\u9ede\u9ad8\u5ea6\u70ba -1 \uff0c\u8449\u7bc0\u9ede\u9ad8\u5ea6\u70ba 0\n    return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n    // \u7bc0\u9ede\u9ad8\u5ea6\u7b49\u65bc\u6700\u9ad8\u5b50\u6a39\u9ad8\u5ea6 + 1\n    node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2","title":"2. \u00a0 \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50","text":"<p>\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\uff08balance factor\uff09\u5b9a\u7fa9\u70ba\u7bc0\u9ede\u5de6\u5b50\u6a39\u7684\u9ad8\u5ea6\u6e1b\u53bb\u53f3\u5b50\u6a39\u7684\u9ad8\u5ea6\uff0c\u540c\u6642\u898f\u5b9a\u7a7a\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\u70ba \\(0\\) \u3002\u6211\u5011\u540c\u6a23\u5c07\u7372\u53d6\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88dd\u6210\u51fd\u5f0f\uff0c\u65b9\u4fbf\u5f8c\u7e8c\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u7372\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n    # \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if node is None:\n        return 0\n    # \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == nullptr)\n        return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == null)\n        return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == null) return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if node == nil {\n        return 0\n    }\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    guard let node = node else { return 0 }\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node === null) return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node === null) return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n  // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n  if (node == null) return 0;\n  // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n        None =&gt; 0,\n        // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == NULL) {\n        return 0;\n    }\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* \u7372\u53d6\u5e73\u8861\u56e0\u5b50 */\nfun balanceFactor(node: TreeNode?): Int {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == null) return 0\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>### \u7372\u53d6\u5e73\u8861\u56e0\u5b50 ###\ndef balance_factor(node)\n  # \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n  return 0 if node.nil?\n\n  # \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n  height(node.left) - height(node.right)\nend\n</code></pre> avl_tree.zig<pre><code>// \u7372\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    // \u7a7a\u7bc0\u9ede\u5e73\u8861\u56e0\u5b50\u70ba 0\n    if (node == null) return 0;\n    // \u7bc0\u9ede\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6a39\u9ad8\u5ea6 - \u53f3\u5b50\u6a39\u9ad8\u5ea6\n    return self.height(node.?.left) - self.height(node.?.right);\n}\n</code></pre> <p>Tip</p> <p>\u8a2d\u5e73\u8861\u56e0\u5b50\u70ba \\(f\\) \uff0c\u5247\u4e00\u68f5 AVL \u6a39\u7684\u4efb\u610f\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6eff\u8db3 \\(-1 \\le f \\le 1\\) \u3002</p>"},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2 \u00a0 AVL \u6a39\u65cb\u8f49","text":"<p>AVL \u6a39\u7684\u7279\u9ede\u5728\u65bc\u201c\u65cb\u8f49\u201d\u64cd\u4f5c\uff0c\u5b83\u80fd\u5920\u5728\u4e0d\u5f71\u97ff\u4e8c\u5143\u6a39\u7684\u4e2d\u5e8f\u8d70\u8a2a\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u7bc0\u9ede\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u65cb\u8f49\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u201c\u4e8c\u5143\u641c\u5c0b\u6a39\u201d\u7684\u6027\u8cea\uff0c\u4e5f\u80fd\u4f7f\u6a39\u91cd\u65b0\u8b8a\u70ba\u201c\u5e73\u8861\u4e8c\u5143\u6a39\u201d\u3002</p> <p>\u6211\u5011\u5c07\u5e73\u8861\u56e0\u5b50\u7d55\u5c0d\u503c \\(&gt; 1\\) \u7684\u7bc0\u9ede\u7a31\u70ba\u201c\u5931\u8861\u7bc0\u9ede\u201d\u3002\u6839\u64da\u7bc0\u9ede\u5931\u8861\u60c5\u6cc1\u7684\u4e0d\u540c\uff0c\u65cb\u8f49\u64cd\u4f5c\u5206\u70ba\u56db\u7a2e\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\u3002\u4e0b\u9762\u8a73\u7d30\u4ecb\u7d39\u9019\u4e9b\u65cb\u8f49\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/avl_tree/#1_1","title":"1. \u00a0 \u53f3\u65cb","text":"<p>\u5982\u5716 7-26 \u6240\u793a\uff0c\u7bc0\u9ede\u4e0b\u65b9\u70ba\u5e73\u8861\u56e0\u5b50\u3002\u5f9e\u5e95\u81f3\u9802\u770b\uff0c\u4e8c\u5143\u6a39\u4e2d\u9996\u500b\u5931\u8861\u7bc0\u9ede\u662f\u201c\u7bc0\u9ede 3\u201d\u3002\u6211\u5011\u95dc\u6ce8\u4ee5\u8a72\u5931\u8861\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\u7684\u5b50\u6a39\uff0c\u5c07\u8a72\u7bc0\u9ede\u8a18\u70ba <code>node</code> \uff0c\u5176\u5de6\u5b50\u7bc0\u9ede\u8a18\u70ba <code>child</code> \uff0c\u57f7\u884c\u201c\u53f3\u65cb\u201d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u5f8c\uff0c\u5b50\u6a39\u6062\u5fa9\u5e73\u8861\uff0c\u4e26\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u6027\u8cea\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 7-26 \u00a0 \u53f3\u65cb\u64cd\u4f5c\u6b65\u9a5f </p> <p>\u5982\u5716 7-27 \u6240\u793a\uff0c\u7576\u7bc0\u9ede <code>child</code> \u6709\u53f3\u5b50\u7bc0\u9ede\uff08\u8a18\u70ba <code>grand_child</code> \uff09\u6642\uff0c\u9700\u8981\u5728\u53f3\u65cb\u4e2d\u65b0\u589e\u4e00\u6b65\uff1a\u5c07 <code>grand_child</code> \u4f5c\u70ba <code>node</code> \u7684\u5de6\u5b50\u7bc0\u9ede\u3002</p> <p></p> <p> \u5716 7-27 \u00a0 \u6709 grand_child \u7684\u53f3\u65cb\u64cd\u4f5c </p> <p>\u201c\u5411\u53f3\u65cb\u8f49\u201d\u662f\u4e00\u7a2e\u5f62\u8c61\u5316\u7684\u8aaa\u6cd5\uff0c\u5be6\u969b\u4e0a\u9700\u8981\u900f\u904e\u4fee\u6539\u7bc0\u9ede\u6307\u6a19\u4f86\u5be6\u73fe\uff0c\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n    child = node.left\n    grand_child = child.right\n    # \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node\n    node.left = grand_child\n    # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.Right = node\n    node.Left = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child?.right = node\n    node?.left = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n  child.right = node;\n  node.left = grandChild;\n  // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.right = node\n    node.left = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### \u53f3\u65cb\u64cd\u4f5c ###\ndef right_rotate(node)\n  child = node.left\n  grand_child = child.right\n  # \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n  child.right = node\n  node.left = grand_child\n  # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  update_height(node)\n  update_height(child)\n  # \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  child\nend\n</code></pre> avl_tree.zig<pre><code>// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.left;\n    var grandChild = child.?.right;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u53f3\u65cb\u8f49\n    child.?.right = node;\n    node.?.left = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_1","title":"2. \u00a0 \u5de6\u65cb","text":"<p>\u76f8\u61c9\u5730\uff0c\u5982\u679c\u8003\u616e\u4e0a\u8ff0\u5931\u8861\u4e8c\u5143\u6a39\u7684\u201c\u6620\u8c61\u201d\uff0c\u5247\u9700\u8981\u57f7\u884c\u5716 7-28 \u6240\u793a\u7684\u201c\u5de6\u65cb\u201d\u64cd\u4f5c\u3002</p> <p></p> <p> \u5716 7-28 \u00a0 \u5de6\u65cb\u64cd\u4f5c </p> <p>\u540c\u7406\uff0c\u5982\u5716 7-29 \u6240\u793a\uff0c\u7576\u7bc0\u9ede <code>child</code> \u6709\u5de6\u5b50\u7bc0\u9ede\uff08\u8a18\u70ba <code>grand_child</code> \uff09\u6642\uff0c\u9700\u8981\u5728\u5de6\u65cb\u4e2d\u65b0\u589e\u4e00\u6b65\uff1a\u5c07 <code>grand_child</code> \u4f5c\u70ba <code>node</code> \u7684\u53f3\u5b50\u7bc0\u9ede\u3002</p> <p></p> <p> \u5716 7-29 \u00a0 \u6709 grand_child \u7684\u5de6\u65cb\u64cd\u4f5c </p> <p>\u53ef\u4ee5\u89c0\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u908f\u8f2f\u4e0a\u662f\u6620\u8c61\u5c0d\u7a31\u7684\uff0c\u5b83\u5011\u5206\u5225\u89e3\u6c7a\u7684\u5169\u7a2e\u5931\u8861\u60c5\u6cc1\u4e5f\u662f\u5c0d\u7a31\u7684\u3002\u57fa\u65bc\u5c0d\u7a31\u6027\uff0c\u6211\u5011\u53ea\u9700\u5c07\u53f3\u65cb\u7684\u5be6\u73fe\u7a0b\u5f0f\u78bc\u4e2d\u7684\u6240\u6709\u7684 <code>left</code> \u66ff\u63db\u70ba <code>right</code> \uff0c\u5c07\u6240\u6709\u7684 <code>right</code> \u66ff\u63db\u70ba <code>left</code> \uff0c\u5373\u53ef\u5f97\u5230\u5de6\u65cb\u7684\u5be6\u73fe\u7a0b\u5f0f\u78bc\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n    child = node.right\n    grand_child = child.left\n    # \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node\n    node.right = grand_child\n    # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.Left = node\n    node.Right = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child?.left = node\n    node?.right = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n  child.left = node;\n  node.right = grandChild;\n  // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.left = node\n    node.right = grandChild\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### \u5de6\u65cb\u64cd\u4f5c ###\ndef left_rotate(node)\n  child = node.right\n  grand_child = child.left\n  # \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n  child.left = node\n  node.right = grand_child\n  # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  update_height(node)\n  update_height(child)\n  # \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  child\nend\n</code></pre> avl_tree.zig<pre><code>// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.right;\n    var grandChild = child.?.left;\n    // \u4ee5 child \u70ba\u539f\u9ede\uff0c\u5c07 node \u5411\u5de6\u65cb\u8f49\n    child.?.left = node;\n    node.?.right = grandChild;\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f49\u5f8c\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3","title":"3. \u00a0 \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb","text":"<p>\u5c0d\u65bc\u5716 7-30 \u4e2d\u7684\u5931\u8861\u7bc0\u9ede 3 \uff0c\u50c5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u7121\u6cd5\u4f7f\u5b50\u6a39\u6062\u5fa9\u5e73\u8861\u3002\u6b64\u6642\u9700\u8981\u5148\u5c0d <code>child</code> \u57f7\u884c\u201c\u5de6\u65cb\u201d\uff0c\u518d\u5c0d <code>node</code> \u57f7\u884c\u201c\u53f3\u65cb\u201d\u3002</p> <p></p> <p> \u5716 7-30 \u00a0 \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb </p>"},{"location":"chapter_tree/avl_tree/#4","title":"4. \u00a0 \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb","text":"<p>\u5982\u5716 7-31 \u6240\u793a\uff0c\u5c0d\u65bc\u4e0a\u8ff0\u5931\u8861\u4e8c\u5143\u6a39\u7684\u6620\u8c61\u60c5\u6cc1\uff0c\u9700\u8981\u5148\u5c0d <code>child</code> \u57f7\u884c\u201c\u53f3\u65cb\u201d\uff0c\u518d\u5c0d <code>node</code> \u57f7\u884c\u201c\u5de6\u65cb\u201d\u3002</p> <p></p> <p> \u5716 7-31 \u00a0 \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb </p>"},{"location":"chapter_tree/avl_tree/#5","title":"5. \u00a0 \u65cb\u8f49\u7684\u9078\u64c7","text":"<p>\u5716 7-32 \u5c55\u793a\u7684\u56db\u7a2e\u5931\u8861\u60c5\u6cc1\u8207\u4e0a\u8ff0\u6848\u4f8b\u9010\u500b\u5c0d\u61c9\uff0c\u5206\u5225\u9700\u8981\u63a1\u7528\u53f3\u65cb\u3001\u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\u3001\u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\u3001\u5de6\u65cb\u7684\u64cd\u4f5c\u3002</p> <p></p> <p> \u5716 7-32 \u00a0 AVL \u6a39\u7684\u56db\u7a2e\u65cb\u8f49\u60c5\u6cc1 </p> <p>\u5982\u4e0b\u8868\u6240\u793a\uff0c\u6211\u5011\u900f\u904e\u5224\u65b7\u5931\u8861\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f03\u9ad8\u4e00\u5074\u5b50\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8ca0\u865f\uff0c\u4f86\u78ba\u5b9a\u5931\u8861\u7bc0\u9ede\u5c6c\u65bc\u5716 7-32 \u4e2d\u7684\u54ea\u7a2e\u60c5\u6cc1\u3002</p> <p> \u8868 7-3 \u00a0 \u56db\u7a2e\u65cb\u8f49\u60c5\u6cc1\u7684\u9078\u64c7\u689d\u4ef6 </p> \u5931\u8861\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u7bc0\u9ede\u7684\u5e73\u8861\u56e0\u5b50 \u61c9\u63a1\u7528\u7684\u65cb\u8f49\u65b9\u6cd5 \\(&gt; 1\\) \uff08\u5de6\u504f\u6a39\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(&gt; 1\\) \uff08\u5de6\u504f\u6a39\uff09 \\(&lt;0\\) \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6a39\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6a39\uff09 \\(&gt;0\\) \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb <p>\u70ba\u4e86\u4fbf\u65bc\u4f7f\u7528\uff0c\u6211\u5011\u5c07\u65cb\u8f49\u64cd\u4f5c\u5c01\u88dd\u6210\u4e00\u500b\u51fd\u5f0f\u3002\u6709\u4e86\u9019\u500b\u51fd\u5f0f\uff0c\u6211\u5011\u5c31\u80fd\u5c0d\u5404\u7a2e\u5931\u8861\u60c5\u6cc1\u9032\u884c\u65cb\u8f49\uff0c\u4f7f\u5931\u8861\u7bc0\u9ede\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\"\"\"\n    # \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    balance_factor = self.balance_factor(node)\n    # \u5de6\u504f\u6a39\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # \u53f3\u65cb\n            return self.right_rotate(node)\n        else:\n            # \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # \u53f3\u504f\u6a39\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # \u5de6\u65cb\n            return self.left_rotate(node)\n        else:\n            # \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    int _balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactorInt = BalanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return RightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return LeftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    // Go \u63a8\u85a6\u77ed\u8b8a\u6578\uff0c\u9019\u88e1 bf \u6307\u4ee3 t.balanceFactor\n    bf := t.balanceFactor(node)\n    // \u5de6\u504f\u6a39\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // \u53f3\u65cb\n            return t.rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // \u5de6\u65cb\n            return t.leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    let balanceFactor = balanceFactor(node: node)\n    // \u5de6\u504f\u6a39\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // \u53f3\u65cb\n            return rightRotate(node: node)\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // \u5de6\u65cb\n            return leftRotate(node: node)\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n#rotate(node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.#rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.#leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n  // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n  int factor = balanceFactor(node);\n  // \u5de6\u504f\u6a39\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // \u53f3\u65cb\n      return rightRotate(node);\n    } else {\n      // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // \u53f3\u504f\u6a39\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // \u5de6\u65cb\n      return leftRotate(node);\n    } else {\n      // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    let balance_factor = Self::balance_factor(node.clone());\n    // \u5de6\u504f\u6a39\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // \u53f3\u65cb\n            Self::right_rotate(Some(node))\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // \u53f3\u504f\u6a39\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // \u5de6\u65cb\n            Self::left_rotate(Some(node))\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    int bf = balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\nfun rotate(node: TreeNode): TreeNode {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    val balanceFactor = balanceFactor(node)\n    // \u5de6\u504f\u6a39\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 ###\ndef rotate(node)\n  # \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n  balance_factor = balance_factor(node)\n  # \u5de6\u904d\u6a39\n  if balance_factor &gt; 1\n    if balance_factor(node.left) &gt;= 0\n      # \u53f3\u65cb\n      return right_rotate(node)\n    else\n      # \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n      node.left = left_rotate(node.left)\n      return right_rotate(node)\n    end\n  # \u53f3\u904d\u6a39\n  elsif balance_factor &lt; -1\n    if balance_factor(node.right) &lt;= 0\n      # \u5de6\u65cb\n      return left_rotate(node)\n    else\n      # \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n      node.right = right_rotate(node.right)\n      return left_rotate(node)\n    end\n  end\n  # \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n  node\nend\n</code></pre> avl_tree.zig<pre><code>// \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    // \u7372\u53d6\u7bc0\u9ede node \u7684\u5e73\u8861\u56e0\u5b50\n    var balance_factor = self.balanceFactor(node);\n    // \u5de6\u504f\u6a39\n    if (balance_factor &gt; 1) {\n        if (self.balanceFactor(node.?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return self.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u5f8c\u53f3\u65cb\n            node.?.left = self.leftRotate(node.?.left);\n            return self.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6a39\n    if (balance_factor &lt; -1) {\n        if (self.balanceFactor(node.?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return self.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u5f8c\u5de6\u65cb\n            node.?.right = self.rightRotate(node.?.right);\n            return self.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6a39\uff0c\u7121\u9808\u65cb\u8f49\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3 \u00a0 AVL \u6a39\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#1_2","title":"1. \u00a0 \u63d2\u5165\u7bc0\u9ede","text":"<p>AVL \u6a39\u7684\u7bc0\u9ede\u63d2\u5165\u64cd\u4f5c\u8207\u4e8c\u5143\u641c\u5c0b\u6a39\u5728\u4e3b\u9ad4\u4e0a\u985e\u4f3c\u3002\u552f\u4e00\u7684\u5340\u5225\u5728\u65bc\uff0c\u5728 AVL \u6a39\u4e2d\u63d2\u5165\u7bc0\u9ede\u5f8c\uff0c\u5f9e\u8a72\u7bc0\u9ede\u5230\u6839\u7bc0\u9ede\u7684\u8def\u5f91\u4e0a\u53ef\u80fd\u6703\u51fa\u73fe\u4e00\u7cfb\u5217\u5931\u8861\u7bc0\u9ede\u3002\u56e0\u6b64\uff0c\u6211\u5011\u9700\u8981\u5f9e\u9019\u500b\u7bc0\u9ede\u958b\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u7bc0\u9ede\u6062\u5fa9\u5e73\u8861\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"\u63d2\u5165\u7bc0\u9ede\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"\u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    UpdateHeight(node);  // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u51fd\u5f0f\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    }\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    this.#updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n  updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n                }\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n\n            /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u51fd\u5f0f\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node;\n    }\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfun insert(_val: Int) {\n    root = insertHelper(root, _val)\n}\n\n/* \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfun insertHelper(n: TreeNode?, _val: Int): TreeNode {\n    if (n == null)\n        return TreeNode(_val)\n    var node = n\n    /* 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede */\n    if (_val &lt; node._val)\n        node.left = insertHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = insertHelper(node.right, _val)\n    else\n        return node // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node) // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### \u63d2\u5165\u7bc0\u9ede ###\ndef insert(val)\n  @root = insert_helper(@root, val)\nend\n\n### \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09###\ndef insert_helper(node, val)\n  return TreeNode.new(val) if node.nil?\n  # 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede\n  if val &lt; node.val\n    node.left = insert_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = insert_helper(node.right, val)\n  else\n    # \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n  end\n  # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  update_height(node)\n  # 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n  rotate(node)\nend\n</code></pre> avl_tree.zig<pre><code>// \u63d2\u5165\u7bc0\u9ede\nfn insert(self: *Self, val: T) !void {\n    self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u905e\u8ff4\u63d2\u5165\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) {\n        var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n        tmp_node.init(val);\n        return tmp_node;\n    }\n    // 1. \u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\u4e26\u63d2\u5165\u7bc0\u9ede\n    if (val &lt; node.?.val) {\n        node.?.left = try self.insertHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = try self.insertHelper(node.?.right, val);\n    } else {\n        return node;            // \u91cd\u8907\u7bc0\u9ede\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    self.updateHeight(node);    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    // 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_2","title":"2. \u00a0 \u522a\u9664\u7bc0\u9ede","text":"<p>\u985e\u4f3c\u5730\uff0c\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u522a\u9664\u7bc0\u9ede\u65b9\u6cd5\u7684\u57fa\u790e\u4e0a\uff0c\u9700\u8981\u5f9e\u5e95\u81f3\u9802\u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u7bc0\u9ede\u6062\u5fa9\u5e73\u8861\u3002\u7a0b\u5f0f\u78bc\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"\u522a\u9664\u7bc0\u9ede\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"\u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return None\n    # 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if child is None:\n                return None\n            # \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else:\n                node = child\n        else:\n            # \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u51fd\u5f0f\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n                return nil\n            } else {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n                node = child\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if child == nil {\n                return nil\n            }\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else {\n                node = child\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child === null) return null;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else node = child;\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child === null) {\n                return null;\n            } else {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n      if (child == null)\n        return null;\n      // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n      else\n        node = child;\n    } else {\n      // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n                    None =&gt; {\n                        return None;\n                    }\n                    // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n\n            /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u522a\u9664\u7bc0\u9ede */\n// \u7531\u65bc\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u8655\u7121\u6cd5\u4f7f\u7528 remove \u95dc\u9375\u8a5e\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u51fd\u5f0f\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfun remove(_val: Int) {\n    root = removeHelper(root, _val)\n}\n\n/* \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09 */\nfun removeHelper(n: TreeNode?, _val: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664 */\n    if (_val &lt; node._val)\n        node.left = removeHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = removeHelper(node.right, _val)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null)\n                node.left\n            else\n                node.right\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == null)\n                return null\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            else\n                node = child\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp._val)\n            node._val = temp._val\n        }\n    }\n    updateHeight(node) // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    /* 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### \u522a\u9664\u7bc0\u9ede ###\ndef remove(val)\n  @root = remove_helper(@root, val)\nend\n\n### \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09###\ndef remove_helper(node, val)\n  return if node.nil?\n  # 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664\n  if val &lt; node.val\n    node.left = remove_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = remove_helper(node.right, val)\n  else\n    if node.left.nil? || node.right.nil?\n      child = node.left || node.right\n      # \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n      return if child.nil?\n      # \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n      node = child\n    else\n      # \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n      temp = node.right\n      while !temp.left.nil?\n        temp = temp.left\n      end\n      node.right = remove_helper(node.right, temp.val)\n      node.val = temp.val\n    end\n  end\n  # \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n  update_height(node)\n  # 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n  rotate(node)\nend\n</code></pre> avl_tree.zig<pre><code>// \u522a\u9664\u7bc0\u9ede\nfn remove(self: *Self, val: T) void {\n   self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u905e\u8ff4\u522a\u9664\u7bc0\u9ede\uff08\u8f14\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) return null;\n    // 1. \u67e5\u8a62\u7bc0\u9ede\u4e26\u522a\u9664\n    if (val &lt; node.?.val) {\n        node.?.left = self.removeHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = self.removeHelper(node.?.right, val);\n    } else {\n        if (node.?.left == null or node.?.right == null) {\n            var child = if (node.?.left != null) node.?.left else node.?.right;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 \uff0c\u76f4\u63a5\u522a\u9664 node \u4e26\u8fd4\u56de\n            if (child == null) {\n                return null;\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 1 \uff0c\u76f4\u63a5\u522a\u9664 node\n            } else {\n                node = child;\n            }\n        } else {\n            // \u5b50\u7bc0\u9ede\u6578\u91cf = 2 \uff0c\u5247\u5c07\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u500b\u7bc0\u9ede\u522a\u9664\uff0c\u4e26\u7528\u8a72\u7bc0\u9ede\u66ff\u63db\u7576\u524d\u7bc0\u9ede\n            var temp = node.?.right;\n            while (temp.?.left != null) {\n                temp = temp.?.left;\n            }\n            node.?.right = self.removeHelper(node.?.right, temp.?.val);\n            node.?.val = temp.?.val;\n        }\n    }\n    self.updateHeight(node); // \u66f4\u65b0\u7bc0\u9ede\u9ad8\u5ea6\n    // 2. \u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u8a72\u5b50\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3_1","title":"3. \u00a0 \u67e5\u8a62\u7bc0\u9ede","text":"<p>AVL \u6a39\u7684\u7bc0\u9ede\u67e5\u8a62\u64cd\u4f5c\u8207\u4e8c\u5143\u641c\u5c0b\u6a39\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d05\u8ff0\u3002</p>"},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4 \u00a0 AVL \u6a39\u5178\u578b\u61c9\u7528","text":"<ul> <li>\u7d44\u7e54\u548c\u5132\u5b58\u5927\u578b\u8cc7\u6599\uff0c\u9069\u7528\u65bc\u9ad8\u983b\u67e5\u8a62\u3001\u4f4e\u983b\u589e\u522a\u7684\u5834\u666f\u3002</li> <li>\u7528\u65bc\u69cb\u5efa\u8cc7\u6599\u5eab\u4e2d\u7684\u7d22\u5f15\u7cfb\u7d71\u3002</li> <li>\u7d05\u9ed1\u6a39\u4e5f\u662f\u4e00\u7a2e\u5e38\u898b\u7684\u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39\u3002\u76f8\u8f03\u65bc AVL \u6a39\uff0c\u7d05\u9ed1\u6a39\u7684\u5e73\u8861\u689d\u4ef6\u66f4\u5bec\u9b06\uff0c\u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede\u6240\u9700\u7684\u65cb\u8f49\u64cd\u4f5c\u66f4\u5c11\uff0c\u7bc0\u9ede\u589e\u522a\u64cd\u4f5c\u7684\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> </ul>"},{"location":"chapter_tree/binary_search_tree/","title":"7.4 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39","text":"<p>\u5982\u5716 7-16 \u6240\u793a\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\uff08binary search tree\uff09\u6eff\u8db3\u4ee5\u4e0b\u689d\u4ef6\u3002</p> <ol> <li>\u5c0d\u65bc\u6839\u7bc0\u9ede\uff0c\u5de6\u5b50\u6a39\u4e2d\u6240\u6709\u7bc0\u9ede\u7684\u503c \\(&lt;\\) \u6839\u7bc0\u9ede\u7684\u503c \\(&lt;\\) \u53f3\u5b50\u6a39\u4e2d\u6240\u6709\u7bc0\u9ede\u7684\u503c\u3002</li> <li>\u4efb\u610f\u7bc0\u9ede\u7684\u5de6\u3001\u53f3\u5b50\u6a39\u4e5f\u662f\u4e8c\u5143\u641c\u5c0b\u6a39\uff0c\u5373\u540c\u6a23\u6eff\u8db3\u689d\u4ef6 <code>1.</code> \u3002</li> </ol> <p></p> <p> \u5716 7-16 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39 </p>"},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u64cd\u4f5c","text":"<p>\u6211\u5011\u5c07\u4e8c\u5143\u641c\u5c0b\u6a39\u5c01\u88dd\u70ba\u4e00\u500b\u985e\u5225 <code>BinarySearchTree</code> \uff0c\u4e26\u5ba3\u544a\u4e00\u500b\u6210\u54e1\u8b8a\u6578 <code>root</code> \uff0c\u6307\u5411\u6a39\u7684\u6839\u7bc0\u9ede\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#1","title":"1. \u00a0 \u67e5\u8a62\u7bc0\u9ede","text":"<p>\u7d66\u5b9a\u76ee\u6a19\u7bc0\u9ede\u503c <code>num</code> \uff0c\u53ef\u4ee5\u6839\u64da\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u6027\u8cea\u4f86\u67e5\u8a62\u3002\u5982\u5716 7-17 \u6240\u793a\uff0c\u6211\u5011\u5ba3\u544a\u4e00\u500b\u7bc0\u9ede <code>cur</code> \uff0c\u5f9e\u4e8c\u5143\u6a39\u7684\u6839\u7bc0\u9ede <code>root</code> \u51fa\u767c\uff0c\u8ff4\u5708\u6bd4\u8f03\u7bc0\u9ede\u503c <code>cur.val</code> \u548c <code>num</code> \u4e4b\u9593\u7684\u5927\u5c0f\u95dc\u4fc2\u3002</p> <ul> <li>\u82e5 <code>cur.val &lt; num</code> \uff0c\u8aaa\u660e\u76ee\u6a19\u7bc0\u9ede\u5728 <code>cur</code> \u7684\u53f3\u5b50\u6a39\u4e2d\uff0c\u56e0\u6b64\u57f7\u884c <code>cur = cur.right</code> \u3002</li> <li>\u82e5 <code>cur.val &gt; num</code> \uff0c\u8aaa\u660e\u76ee\u6a19\u7bc0\u9ede\u5728 <code>cur</code> \u7684\u5de6\u5b50\u6a39\u4e2d\uff0c\u56e0\u6b64\u57f7\u884c <code>cur = cur.left</code> \u3002</li> <li>\u82e5 <code>cur.val = num</code> \uff0c\u8aaa\u660e\u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\u4e26\u8fd4\u56de\u8a72\u7bc0\u9ede\u3002</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 7-17 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u67e5\u8a62\u7bc0\u9ede\u793a\u4f8b </p> <p>\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u67e5\u8a62\u64cd\u4f5c\u8207\u4e8c\u5206\u641c\u5c0b\u6f14\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f2a\u6392\u9664\u4e00\u534a\u60c5\u6cc1\u3002\u8ff4\u5708\u6b21\u6578\u6700\u591a\u70ba\u4e8c\u5143\u6a39\u7684\u9ad8\u5ea6\uff0c\u7576\u4e8c\u5143\u6a39\u5e73\u8861\u6642\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6642\u9593\u3002\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"\u67e5\u8a62\u7bc0\u9ede\"\"\"\n    cur = self._root\n    # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while cur is not None:\n        # \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        elif cur.val &gt; num:\n            cur = cur.left\n        # \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            node = node.Left\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while cur != nil {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else {\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nsearch(num) {\n    let cur = this.root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  while (cur != null) {\n    // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n    else\n      break;\n  }\n  // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            cur = cur-&gt;left;\n        } else {\n            // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u67e5\u8a62\u7bc0\u9ede */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        cur = if (cur._val &lt; num)\n            cur.right\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else if (cur._val &gt; num)\n            cur.left\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        else\n            break\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>### \u67e5\u8a62\u7bc0\u9ede ###\ndef search(num)\n  cur = @root\n\n  # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  while !cur.nil?\n    # \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if cur.val &lt; num\n      cur = cur.right\n    # \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    elsif cur.val &gt; num\n      cur = cur.left\n    # \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n    else\n      break\n    end\n  end\n\n  cur\nend\n</code></pre> binary_search_tree.zig<pre><code>// \u67e5\u8a62\u7bc0\u9ede\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n    var cur = self.root;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u76ee\u6a19\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        } else if (cur.?.val &gt; num) {\n            cur = cur.?.left;\n        // \u627e\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        } else {\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6a19\u7bc0\u9ede\n    return cur;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#2","title":"2. \u00a0 \u63d2\u5165\u7bc0\u9ede","text":"<p>\u7d66\u5b9a\u4e00\u500b\u5f85\u63d2\u5165\u5143\u7d20 <code>num</code> \uff0c\u70ba\u4e86\u4fdd\u6301\u4e8c\u5143\u641c\u5c0b\u6a39\u201c\u5de6\u5b50\u6a39 &lt; \u6839\u7bc0\u9ede &lt; \u53f3\u5b50\u6a39\u201d\u7684\u6027\u8cea\uff0c\u63d2\u5165\u64cd\u4f5c\u6d41\u7a0b\u5982\u5716 7-18 \u6240\u793a\u3002</p> <ol> <li>\u67e5\u8a62\u63d2\u5165\u4f4d\u7f6e\uff1a\u8207\u67e5\u8a62\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5f9e\u6839\u7bc0\u9ede\u51fa\u767c\uff0c\u6839\u64da\u7576\u524d\u7bc0\u9ede\u503c\u548c <code>num</code> \u7684\u5927\u5c0f\u95dc\u4fc2\u8ff4\u5708\u5411\u4e0b\u641c\u5c0b\uff0c\u76f4\u5230\u8d8a\u904e\u8449\u7bc0\u9ede\uff08\u8d70\u8a2a\u81f3 <code>None</code> \uff09\u6642\u8df3\u51fa\u8ff4\u5708\u3002</li> <li>\u5728\u8a72\u4f4d\u7f6e\u63d2\u5165\u7bc0\u9ede\uff1a\u521d\u59cb\u5316\u7bc0\u9ede <code>num</code> \uff0c\u5c07\u8a72\u7bc0\u9ede\u7f6e\u65bc <code>None</code> \u7684\u4f4d\u7f6e\u3002</li> </ol> <p></p> <p> \u5716 7-18 \u00a0 \u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u63d2\u5165\u7bc0\u9ede </p> <p>\u5728\u7a0b\u5f0f\u78bc\u5be6\u73fe\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u5169\u9ede\u3002</p> <ul> <li>\u4e8c\u5143\u641c\u5c0b\u6a39\u4e0d\u5141\u8a31\u5b58\u5728\u91cd\u8907\u7bc0\u9ede\uff0c\u5426\u5247\u5c07\u9055\u53cd\u5176\u5b9a\u7fa9\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u7bc0\u9ede\u5728\u6a39\u4e2d\u5df2\u5b58\u5728\uff0c\u5247\u4e0d\u57f7\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002</li> <li>\u70ba\u4e86\u5be6\u73fe\u63d2\u5165\u7bc0\u9ede\uff0c\u6211\u5011\u9700\u8981\u85c9\u52a9\u7bc0\u9ede <code>pre</code> \u5132\u5b58\u4e0a\u4e00\u8f2a\u8ff4\u5708\u7684\u7bc0\u9ede\u3002\u9019\u6a23\u5728\u8d70\u8a2a\u81f3 <code>None</code> \u6642\uff0c\u6211\u5011\u53ef\u4ee5\u7372\u53d6\u5230\u5176\u7236\u7bc0\u9ede\uff0c\u5f9e\u800c\u5b8c\u6210\u7bc0\u9ede\u63d2\u5165\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"\u63d2\u5165\u7bc0\u9ede\"\"\"\n    # \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur.val == num:\n            return\n        pre = cur\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else:\n            cur = cur.left\n    # \u63d2\u5165\u7bc0\u9ede\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid Insert(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur.left;\n    }\n\n    // \u63d2\u5165\u7bc0\u9ede\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // \u5f85\u63d2\u5165\u7bc0\u9ede\u4e4b\u524d\u7684\u7bc0\u9ede\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfunc insert(num: Int) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\ninsert(num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\ninsert(num: number): void {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(int _num) {\n  // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n    if (cur.val == _num) return;\n    pre = cur;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u63d2\u5165\u7bc0\u9ede\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\npub fn insert(&amp;mut self, num: i32) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n            Ordering::Equal =&gt; return,\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nvoid insert(BinarySearchTree *bst, int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u63d2\u5165\u7bc0\u9ede */\nfun insert(num: Int) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur._val == num)\n            return\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        cur = if (cur._val &lt; num)\n            cur.right\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur.left\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    val node = TreeNode(num)\n    if (pre?._val!! &lt; num)\n        pre.right = node\n    else\n        pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>### \u63d2\u5165\u7bc0\u9ede ###\ndef insert(num)\n  # \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n  if @root.nil?\n    @root = TreeNode.new(num)\n    return\n  end\n\n  # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  cur, pre = @root, nil\n  while !cur.nil?\n    # \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n    return if cur.val == num\n\n    pre = cur\n    # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if cur.val &lt; num\n      cur = cur.right\n    # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else\n      cur = cur.left\n    end\n  end\n\n  # \u63d2\u5165\u7bc0\u9ede\n  node = TreeNode.new(num)\n  if pre.val &lt; num\n    pre.right = node\n  else\n    pre.left = node\n  end\nend\n</code></pre> binary_search_tree.zig<pre><code>// \u63d2\u5165\u7bc0\u9ede\nfn insert(self: *Self, num: T) !void {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u5247\u521d\u59cb\u5316\u6839\u7bc0\u9ede\n    if (self.root == null) {\n        self.root = try self.mem_allocator.create(inc.TreeNode(T));\n        return;\n    }\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u8907\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.?.val == num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u63d2\u5165\u7bc0\u9ede\n    var node = try self.mem_allocator.create(inc.TreeNode(T));\n    node.init(num);\n    if (pre.?.val &lt; num) {\n        pre.?.right = node;\n    } else {\n        pre.?.left = node;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>\u8207\u67e5\u8a62\u7bc0\u9ede\u76f8\u540c\uff0c\u63d2\u5165\u7bc0\u9ede\u4f7f\u7528 \\(O(\\log n)\\) \u6642\u9593\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#3","title":"3. \u00a0 \u522a\u9664\u7bc0\u9ede","text":"<p>\u5148\u5728\u4e8c\u5143\u6a39\u4e2d\u67e5\u8a62\u5230\u76ee\u6a19\u7bc0\u9ede\uff0c\u518d\u5c07\u5176\u522a\u9664\u3002\u8207\u63d2\u5165\u7bc0\u9ede\u985e\u4f3c\uff0c\u6211\u5011\u9700\u8981\u4fdd\u8b49\u5728\u522a\u9664\u64cd\u4f5c\u5b8c\u6210\u5f8c\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u201c\u5de6\u5b50\u6a39 &lt; \u6839\u7bc0\u9ede &lt; \u53f3\u5b50\u6a39\u201d\u7684\u6027\u8cea\u4ecd\u7136\u6eff\u8db3\u3002\u56e0\u6b64\uff0c\u6211\u5011\u6839\u64da\u76ee\u6a19\u7bc0\u9ede\u7684\u5b50\u7bc0\u9ede\u6578\u91cf\uff0c\u5206 0\u30011 \u548c 2 \u4e09\u7a2e\u60c5\u6cc1\uff0c\u57f7\u884c\u5c0d\u61c9\u7684\u522a\u9664\u7bc0\u9ede\u64cd\u4f5c\u3002</p> <p>\u5982\u5716 7-19 \u6240\u793a\uff0c\u7576\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u5ea6\u70ba \\(0\\) \u6642\uff0c\u8868\u793a\u8a72\u7bc0\u9ede\u662f\u8449\u7bc0\u9ede\uff0c\u53ef\u4ee5\u76f4\u63a5\u522a\u9664\u3002</p> <p></p> <p> \u5716 7-19 \u00a0 \u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u522a\u9664\u7bc0\u9ede\uff08\u5ea6\u70ba 0 \uff09 </p> <p>\u5982\u5716 7-20 \u6240\u793a\uff0c\u7576\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u5ea6\u70ba \\(1\\) \u6642\uff0c\u5c07\u5f85\u522a\u9664\u7bc0\u9ede\u66ff\u63db\u70ba\u5176\u5b50\u7bc0\u9ede\u5373\u53ef\u3002</p> <p></p> <p> \u5716 7-20 \u00a0 \u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u522a\u9664\u7bc0\u9ede\uff08\u5ea6\u70ba 1 \uff09 </p> <p>\u7576\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u5ea6\u70ba \\(2\\) \u6642\uff0c\u6211\u5011\u7121\u6cd5\u76f4\u63a5\u522a\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u500b\u7bc0\u9ede\u66ff\u63db\u8a72\u7bc0\u9ede\u3002\u7531\u65bc\u8981\u4fdd\u6301\u4e8c\u5143\u641c\u5c0b\u6a39\u201c\u5de6\u5b50\u6a39 \\(&lt;\\) \u6839\u7bc0\u9ede \\(&lt;\\) \u53f3\u5b50\u6a39\u201d\u7684\u6027\u8cea\uff0c\u56e0\u6b64\u9019\u500b\u7bc0\u9ede\u53ef\u4ee5\u662f\u53f3\u5b50\u6a39\u7684\u6700\u5c0f\u7bc0\u9ede\u6216\u5de6\u5b50\u6a39\u7684\u6700\u5927\u7bc0\u9ede\u3002</p> <p>\u5047\u8a2d\u6211\u5011\u9078\u64c7\u53f3\u5b50\u6a39\u7684\u6700\u5c0f\u7bc0\u9ede\uff08\u4e2d\u5e8f\u8d70\u8a2a\u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\uff09\uff0c\u5247\u522a\u9664\u64cd\u4f5c\u6d41\u7a0b\u5982\u5716 7-21 \u6240\u793a\u3002</p> <ol> <li>\u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\u5728\u201c\u4e2d\u5e8f\u8d70\u8a2a\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\uff0c\u8a18\u70ba <code>tmp</code> \u3002</li> <li>\u7528 <code>tmp</code> \u7684\u503c\u8986\u84cb\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u503c\uff0c\u4e26\u5728\u6a39\u4e2d\u905e\u8ff4\u522a\u9664\u7bc0\u9ede <code>tmp</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u5716 7-21 \u00a0 \u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u522a\u9664\u7bc0\u9ede\uff08\u5ea6\u70ba 2 \uff09 </p> <p>\u522a\u9664\u7bc0\u9ede\u64cd\u4f5c\u540c\u6a23\u4f7f\u7528 \\(O(\\log n)\\) \u6642\u9593\uff0c\u5176\u4e2d\u67e5\u8a62\u5f85\u522a\u9664\u7bc0\u9ede\u9700\u8981 \\(O(\\log n)\\) \u6642\u9593\uff0c\u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u5f8c\u7e7c\u7bc0\u9ede\u9700\u8981 \\(O(\\log n)\\) \u6642\u9593\u3002\u793a\u4f8b\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"\u522a\u9664\u7bc0\u9ede\"\"\"\n    # \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self._root is None:\n        return\n    # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if cur.val == num:\n            break\n        pre = cur\n        # \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else:\n            cur = cur.left\n    # \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if cur is None:\n        return\n\n    # \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if cur.left is None or cur.right is None:\n        # \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        child = cur.left or cur.right\n        # \u522a\u9664\u7bc0\u9ede cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            self._root = child\n    # \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else:\n        # \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        self.remove(tmp.val)\n        # \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == nullptr)\n        return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = nullptr / \u8a72\u5b50\u7bc0\u9ede\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child;\n        }\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        delete cur;\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        remove(tmp-&gt;val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child;\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        remove(tmp.val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid Remove(int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        TreeNode? child = cur.left ?? cur.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child;\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        Remove(tmp.val!.Value);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5f85\u522a\u9664\u7bc0\u9ede\u4e4b\u524d\u7684\u7bc0\u9ede\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728\u53f3\u5b50\u6a39\u4e2d\n            cur = cur.Right\n        } else {\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728\u5de6\u5b50\u6a39\u4e2d\n            cur = cur.Left\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u70ba 0 \u6216 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // \u53d6\u51fa\u5f85\u522a\u9664\u7bc0\u9ede\u7684\u5b50\u7bc0\u9ede\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // \u522a\u9664\u7bc0\u9ede cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            bst.root = child\n        }\n        // \u5b50\u7bc0\u9ede\u6578\u70ba 2\n    } else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d\u5f85\u522a\u9664\u7bc0\u9ede cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        bst.remove(tmp.Val.(int))\n        // \u7528 tmp \u8986\u84cb cur\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfunc remove(num: Int) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        let child = cur?.left ?? cur?.right\n        // \u522a\u9664\u7bc0\u9ede cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        remove(num: tmp!.val)\n        // \u7528 tmp \u8986\u84cb cur\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nremove(num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        const child = cur.left !== null ? cur.left : cur.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            this.root = child;\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        this.remove(tmp.val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nremove(num: number): void {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            this.root = child;\n        }\n    }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        this.remove(tmp!.val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nvoid remove(int _num) {\n  // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n    if (cur.val == _num) break;\n    pre = cur;\n    // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u76f4\u63a5\u8fd4\u56de\n  if (cur == null) return;\n  // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n    TreeNode? child = cur.left ?? cur.right;\n    // \u522a\u9664\u7bc0\u9ede cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n      _root = child;\n    }\n  } else {\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n    remove(tmp.val);\n    // \u7528 tmp \u8986\u84cb cur\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u522a\u9664\u7bc0\u9ede */\npub fn remove(&amp;mut self, num: i32) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n            Ordering::Equal =&gt; break,\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = nullptr / \u8a72\u5b50\u7bc0\u9ede\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // \u522a\u9664\u7bc0\u9ede cur\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n                self.root = child;\n            }\n        }\n        // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n        (Some(_), Some(_)) =&gt; {\n            // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmp_val = tmp.unwrap().borrow().val;\n            // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n            self.remove(tmp_val);\n            // \u7528 tmp \u8986\u84cb cur\n            cur.borrow_mut().val = tmp_val;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u522a\u9664\u7bc0\u9ede */\n// \u7531\u65bc\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u8655\u7121\u6cd5\u4f7f\u7528 remove \u95dc\u9375\u8a5e\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 root \u7684\u53f3\u5b50\u6a39\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 root \u7684\u5de6\u5b50\u6a39\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == NULL)\n        return;\n    // \u5224\u65b7\u5f85\u522a\u9664\u7bc0\u9ede\u662f\u5426\u5b58\u5728\u5b50\u7bc0\u9ede\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1 */\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = nullptr / \u8a72\u5b50\u7bc0\u9ede\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // \u91cb\u653e\u8a18\u61b6\u9ad4\n        free(cur);\n    } else {\n        /* \u5b50\u7bc0\u9ede\u6578\u91cf = 2 */\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        removeItem(bst, tmp-&gt;val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u522a\u9664\u7bc0\u9ede */\nfun remove(num: Int) {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return\n    var cur = root\n    var pre: TreeNode? = null\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur._val == num)\n            break\n        pre = cur\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        cur = if (cur._val &lt; num)\n            cur.right\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        else\n            cur.left\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        val child = if (cur.left != null)\n            cur.left\n        else\n            cur.right\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (cur != root) {\n            if (pre!!.left == cur)\n                pre.left = child\n            else\n                pre.right = child\n        } else {\n            // \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n            root = child\n        }\n        // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    } else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        remove(tmp._val)\n        // \u7528 tmp \u8986\u84cb cur\n        cur._val = tmp._val\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>### \u522a\u9664\u7bc0\u9ede ###\ndef remove(num)\n  # \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  return if @root.nil?\n\n  # \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n  cur, pre = @root, nil\n  while !cur.nil?\n    # \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n    break if cur.val == num\n\n    pre = cur\n    # \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n    if cur.val &lt; num\n      cur = cur.right\n    # \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n    else\n      cur = cur.left\n    end\n  end\n  # \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n  return if cur.nil?\n\n  # \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n  if cur.left.nil? || cur.right.nil?\n    # \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n    child = cur.left || cur.right\n    # \u522a\u9664\u7bc0\u9ede cur\n    if cur != @root\n      if pre.left == cur\n        pre.left = child\n      else\n        pre.right = child\n      end\n    else\n      # \u82e5\u522a\u9664\u7bc0\u9ede\u70ba\u6839\u7bc0\u9ede\uff0c\u5247\u91cd\u65b0\u6307\u5b9a\u6839\u7bc0\u9ede\n      @root = child\n    end\n  # \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n  else\n    # \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n    tmp = cur.right\n    while !tmp.left.nil?\n      tmp = tmp.left\n    end\n    # \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n    remove(tmp.val)\n    # \u7528 tmp \u8986\u84cb cur\n    cur.val = tmp.val\n  end\nend\n</code></pre> binary_search_tree.zig<pre><code>// \u522a\u9664\u7bc0\u9ede\nfn remove(self: *Self, num: T) void {\n    // \u82e5\u6a39\u70ba\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (self.root == null) return;\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u8ff4\u5708\u67e5\u8a62\uff0c\u8d8a\u904e\u8449\u7bc0\u9ede\u5f8c\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u8df3\u51fa\u8ff4\u5708\n        if (cur.?.val == num) break;\n        pre = cur;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u53f3\u5b50\u6a39\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u5f85\u522a\u9664\u7bc0\u9ede\u5728 cur \u7684\u5de6\u5b50\u6a39\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u82e5\u7121\u5f85\u522a\u9664\u7bc0\u9ede\uff0c\u5247\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return;\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 0 or 1\n    if (cur.?.left == null or cur.?.right == null) {\n        // \u7576\u5b50\u7bc0\u9ede\u6578\u91cf = 0 / 1 \u6642\uff0c child = null / \u8a72\u5b50\u7bc0\u9ede\n        var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n        // \u522a\u9664\u7bc0\u9ede cur\n        if (pre.?.left == cur) {\n            pre.?.left = child;\n        } else {\n            pre.?.right = child;\n        }\n    // \u5b50\u7bc0\u9ede\u6578\u91cf = 2\n    } else {\n        // \u7372\u53d6\u4e2d\u5e8f\u8d70\u8a2a\u4e2d cur \u7684\u4e0b\u4e00\u500b\u7bc0\u9ede\n        var tmp = cur.?.right;\n        while (tmp.?.left != null) {\n            tmp = tmp.?.left;\n        }\n        var tmp_val = tmp.?.val;\n        // \u905e\u8ff4\u522a\u9664\u7bc0\u9ede tmp\n        self.remove(tmp.?.val);\n        // \u7528 tmp \u8986\u84cb cur\n        cur.?.val = tmp_val;\n    }\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#4","title":"4. \u00a0 \u4e2d\u5e8f\u8d70\u8a2a\u6709\u5e8f","text":"<p>\u5982\u5716 7-22 \u6240\u793a\uff0c\u4e8c\u5143\u6a39\u7684\u4e2d\u5e8f\u8d70\u8a2a\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u8d70\u8a2a\u9806\u5e8f\uff0c\u800c\u4e8c\u5143\u641c\u5c0b\u6a39\u6eff\u8db3\u201c\u5de6\u5b50\u7bc0\u9ede \\(&lt;\\) \u6839\u7bc0\u9ede \\(&lt;\\) \u53f3\u5b50\u7bc0\u9ede\u201d\u7684\u5927\u5c0f\u95dc\u4fc2\u3002</p> <p>\u9019\u610f\u5473\u8457\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u9032\u884c\u4e2d\u5e8f\u8d70\u8a2a\u6642\uff0c\u7e3d\u662f\u6703\u512a\u5148\u8d70\u8a2a\u4e0b\u4e00\u500b\u6700\u5c0f\u7bc0\u9ede\uff0c\u5f9e\u800c\u5f97\u51fa\u4e00\u500b\u91cd\u8981\u6027\u8cea\uff1a\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u4e2d\u5e8f\u8d70\u8a2a\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002</p> <p>\u5229\u7528\u4e2d\u5e8f\u8d70\u8a2a\u5347\u5e8f\u7684\u6027\u8cea\uff0c\u6211\u5011\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u7372\u53d6\u6709\u5e8f\u8cc7\u6599\u50c5\u9700 \\(O(n)\\) \u6642\u9593\uff0c\u7121\u9808\u9032\u884c\u984d\u5916\u7684\u6392\u5e8f\u64cd\u4f5c\uff0c\u975e\u5e38\u9ad8\u6548\u3002</p> <p></p> <p> \u5716 7-22 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u4e2d\u5e8f\u8d70\u8a2a\u5e8f\u5217 </p>"},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u6548\u7387","text":"<p>\u7d66\u5b9a\u4e00\u7d44\u8cc7\u6599\uff0c\u6211\u5011\u8003\u616e\u4f7f\u7528\u9663\u5217\u6216\u4e8c\u5143\u641c\u5c0b\u6a39\u5132\u5b58\u3002\u89c0\u5bdf\u8868 7-2 \uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u5404\u9805\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u90fd\u662f\u5c0d\u6578\u968e\uff0c\u5177\u6709\u7a69\u5b9a\u4e14\u9ad8\u6548\u7684\u6548\u80fd\u3002\u53ea\u6709\u5728\u9ad8\u983b\u65b0\u589e\u3001\u4f4e\u983b\u67e5\u8a62\u522a\u9664\u8cc7\u6599\u7684\u5834\u666f\u4e0b\uff0c\u9663\u5217\u6bd4\u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u6548\u7387\u66f4\u9ad8\u3002</p> <p> \u8868 7-2 \u00a0 \u9663\u5217\u8207\u641c\u5c0b\u6a39\u7684\u6548\u7387\u5c0d\u6bd4 </p> \u7121\u5e8f\u9663\u5217 \u4e8c\u5143\u641c\u5c0b\u6a39 \u67e5\u8a62\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u522a\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) <p>\u5728\u7406\u60f3\u60c5\u6cc1\u4e0b\uff0c\u4e8c\u5143\u641c\u5c0b\u6a39\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u9019\u6a23\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f2a\u8ff4\u5708\u5167\u67e5\u8a62\u4efb\u610f\u7bc0\u9ede\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u5011\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\u4e0d\u65b7\u5730\u63d2\u5165\u548c\u522a\u9664\u7bc0\u9ede\uff0c\u53ef\u80fd\u5c0e\u81f4\u4e8c\u5143\u6a39\u9000\u5316\u70ba\u5716 7-23 \u6240\u793a\u7684\u93c8\u7d50\u4e32\u5217\uff0c\u9019\u6642\u5404\u7a2e\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u4e5f\u6703\u9000\u5316\u70ba \\(O(n)\\) \u3002</p> <p></p> <p> \u5716 7-23 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u9000\u5316 </p>"},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u5e38\u898b\u61c9\u7528","text":"<ul> <li>\u7528\u4f5c\u7cfb\u7d71\u4e2d\u7684\u591a\u7d1a\u7d22\u5f15\uff0c\u5be6\u73fe\u9ad8\u6548\u7684\u67e5\u8a62\u3001\u63d2\u5165\u3001\u522a\u9664\u64cd\u4f5c\u3002</li> <li>\u4f5c\u70ba\u67d0\u4e9b\u641c\u5c0b\u6f14\u7b97\u6cd5\u7684\u5e95\u5c64\u8cc7\u6599\u7d50\u69cb\u3002</li> <li>\u7528\u65bc\u5132\u5b58\u8cc7\u6599\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72c0\u614b\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree/","title":"7.1 \u00a0 \u4e8c\u5143\u6a39","text":"<p>\u4e8c\u5143\u6a39\uff08binary tree\uff09\u662f\u4e00\u7a2e\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u4ee3\u8868\u201c\u7956\u5148\u201d\u8207\u201c\u5f8c\u4ee3\u201d\u4e4b\u9593\u7684\u6d3e\u751f\u95dc\u4fc2\uff0c\u9ad4\u73fe\u4e86\u201c\u4e00\u5206\u70ba\u4e8c\u201d\u7684\u5206\u6cbb\u908f\u8f2f\u3002\u8207\u93c8\u7d50\u4e32\u5217\u985e\u4f3c\uff0c\u4e8c\u5143\u6a39\u7684\u57fa\u672c\u55ae\u5143\u662f\u7bc0\u9ede\uff0c\u6bcf\u500b\u7bc0\u9ede\u5305\u542b\u503c\u3001\u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\u548c\u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"\u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u7bc0\u9ede\u503c\n        self.left: TreeNode | None = None  # \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n        self.right: TreeNode | None = None # \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct TreeNode {\n    int val;          // \u7bc0\u9ede\u503c\n    TreeNode *left;   // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    TreeNode *right;  // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    int val;         // \u7bc0\u9ede\u503c\n    TreeNode left;   // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    TreeNode right;  // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u7bc0\u9ede\u503c\n    public TreeNode? left;  // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* \u5efa\u69cb\u5b50 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n        Right: nil, // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n        Val:   v,   // \u7bc0\u9ede\u503c\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    var val: Int // \u7bc0\u9ede\u503c\n    var left: TreeNode? // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    var right: TreeNode? // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    val; // \u7bc0\u9ede\u503c\n    left; // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    right; // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // \u7bc0\u9ede\u503c\n        this.left = left === undefined ? null : left; // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n        this.right = right === undefined ? null : right; // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode {\n  int val;         // \u7bc0\u9ede\u503c\n  TreeNode? left;  // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n  TreeNode? right; // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\nstruct TreeNode {\n    val: i32,                               // \u7bc0\u9ede\u503c\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n\nimpl TreeNode {\n    /* \u5efa\u69cb\u5b50 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u7d50\u69cb\u9ad4 */\ntypedef struct TreeNode {\n    int val;                // \u7bc0\u9ede\u503c\n    int height;             // \u7bc0\u9ede\u9ad8\u5ea6\n    struct TreeNode *left;  // \u5de6\u5b50\u7bc0\u9ede\u6307\u6a19\n    struct TreeNode *right; // \u53f3\u5b50\u7bc0\u9ede\u6307\u6a19\n} TreeNode;\n\n/* \u5efa\u69cb\u5b50 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 */\nclass TreeNode(val _val: Int) {  // \u7bc0\u9ede\u503c\n    val left: TreeNode? = null   // \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n    val right: TreeNode? = null  // \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n}\n</code></pre> <pre><code>### \u4e8c\u5143\u6a39\u7bc0\u9ede\u985e\u5225 ###\nclass TreeNode\n  attr_accessor :val    # \u7bc0\u9ede\u503c\n  attr_accessor :left   # \u5de6\u5b50\u7bc0\u9ede\u5f15\u7528\n  attr_accessor :right  # \u53f3\u5b50\u7bc0\u9ede\u5f15\u7528\n\n  def initialize(val)\n    @val = val\n  end\nend\n</code></pre> <pre><code>\n</code></pre> <p>\u6bcf\u500b\u7bc0\u9ede\u90fd\u6709\u5169\u500b\u5f15\u7528\uff08\u6307\u6a19\uff09\uff0c\u5206\u5225\u6307\u5411\u5de6\u5b50\u7bc0\u9ede\uff08left-child node\uff09\u548c\u53f3\u5b50\u7bc0\u9ede\uff08right-child node\uff09\uff0c\u8a72\u7bc0\u9ede\u88ab\u7a31\u70ba\u9019\u5169\u500b\u5b50\u7bc0\u9ede\u7684\u7236\u7bc0\u9ede\uff08parent node\uff09\u3002\u7576\u7d66\u5b9a\u4e00\u500b\u4e8c\u5143\u6a39\u7684\u7bc0\u9ede\u6642\uff0c\u6211\u5011\u5c07\u8a72\u7bc0\u9ede\u7684\u5de6\u5b50\u7bc0\u9ede\u53ca\u5176\u4ee5\u4e0b\u7bc0\u9ede\u5f62\u6210\u7684\u6a39\u7a31\u70ba\u8a72\u7bc0\u9ede\u7684\u5de6\u5b50\u6a39\uff08left subtree\uff09\uff0c\u540c\u7406\u53ef\u5f97\u53f3\u5b50\u6a39\uff08right subtree\uff09\u3002</p> <p>\u5728\u4e8c\u5143\u6a39\u4e2d\uff0c\u9664\u8449\u7bc0\u9ede\u5916\uff0c\u5176\u4ed6\u6240\u6709\u7bc0\u9ede\u90fd\u5305\u542b\u5b50\u7bc0\u9ede\u548c\u975e\u7a7a\u5b50\u6a39\u3002\u5982\u5716 7-1 \u6240\u793a\uff0c\u5982\u679c\u5c07\u201c\u7bc0\u9ede 2\u201d\u8996\u70ba\u7236\u7bc0\u9ede\uff0c\u5247\u5176\u5de6\u5b50\u7bc0\u9ede\u548c\u53f3\u5b50\u7bc0\u9ede\u5206\u5225\u662f\u201c\u7bc0\u9ede 4\u201d\u548c\u201c\u7bc0\u9ede 5\u201d\uff0c\u5de6\u5b50\u6a39\u662f\u201c\u7bc0\u9ede 4 \u53ca\u5176\u4ee5\u4e0b\u7bc0\u9ede\u5f62\u6210\u7684\u6a39\u201d\uff0c\u53f3\u5b50\u6a39\u662f\u201c\u7bc0\u9ede 5 \u53ca\u5176\u4ee5\u4e0b\u7bc0\u9ede\u5f62\u6210\u7684\u6a39\u201d\u3002</p> <p></p> <p> \u5716 7-1 \u00a0 \u7236\u7bc0\u9ede\u3001\u5b50\u7bc0\u9ede\u3001\u5b50\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1 \u00a0 \u4e8c\u5143\u6a39\u5e38\u898b\u8853\u8a9e","text":"<p>\u4e8c\u5143\u6a39\u7684\u5e38\u7528\u8853\u8a9e\u5982\u5716 7-2 \u6240\u793a\u3002</p> <ul> <li>\u6839\u7bc0\u9ede\uff08root node\uff09\uff1a\u4f4d\u65bc\u4e8c\u5143\u6a39\u9802\u5c64\u7684\u7bc0\u9ede\uff0c\u6c92\u6709\u7236\u7bc0\u9ede\u3002</li> <li>\u8449\u7bc0\u9ede\uff08leaf node\uff09\uff1a\u6c92\u6709\u5b50\u7bc0\u9ede\u7684\u7bc0\u9ede\uff0c\u5176\u5169\u500b\u6307\u6a19\u5747\u6307\u5411 <code>None</code> \u3002</li> <li>\u908a\uff08edge\uff09\uff1a\u9023\u7dda\u5169\u500b\u7bc0\u9ede\u7684\u7dda\u6bb5\uff0c\u5373\u7bc0\u9ede\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002</li> <li>\u7bc0\u9ede\u6240\u5728\u7684\u5c64\uff08level\uff09\uff1a\u5f9e\u9802\u81f3\u5e95\u905e\u589e\uff0c\u6839\u7bc0\u9ede\u6240\u5728\u5c64\u70ba 1 \u3002</li> <li>\u7bc0\u9ede\u7684\u5ea6\uff08degree\uff09\uff1a\u7bc0\u9ede\u7684\u5b50\u7bc0\u9ede\u7684\u6578\u91cf\u3002\u5728\u4e8c\u5143\u6a39\u4e2d\uff0c\u5ea6\u7684\u53d6\u503c\u7bc4\u570d\u662f 0\u30011\u30012 \u3002</li> <li>\u4e8c\u5143\u6a39\u7684\u9ad8\u5ea6\uff08height\uff09\uff1a\u5f9e\u6839\u7bc0\u9ede\u5230\u6700\u9060\u8449\u7bc0\u9ede\u6240\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u3002</li> <li>\u7bc0\u9ede\u7684\u6df1\u5ea6\uff08depth\uff09\uff1a\u5f9e\u6839\u7bc0\u9ede\u5230\u8a72\u7bc0\u9ede\u6240\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u3002</li> <li>\u7bc0\u9ede\u7684\u9ad8\u5ea6\uff08height\uff09\uff1a\u5f9e\u8ddd\u96e2\u8a72\u7bc0\u9ede\u6700\u9060\u7684\u8449\u7bc0\u9ede\u5230\u8a72\u7bc0\u9ede\u6240\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u3002</li> </ul> <p></p> <p> \u5716 7-2 \u00a0 \u4e8c\u5143\u6a39\u7684\u5e38\u7528\u8853\u8a9e </p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u6211\u5011\u901a\u5e38\u5c07\u201c\u9ad8\u5ea6\u201d\u548c\u201c\u6df1\u5ea6\u201d\u5b9a\u7fa9\u70ba\u201c\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u984c\u76ee\u6216\u6559\u6750\u53ef\u80fd\u6703\u5c07\u5176\u5b9a\u7fa9\u70ba\u201c\u7d93\u904e\u7684\u7bc0\u9ede\u7684\u6578\u91cf\u201d\u3002\u5728\u9019\u7a2e\u60c5\u6cc1\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002</p>"},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2 \u00a0 \u4e8c\u5143\u6a39\u57fa\u672c\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_tree/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e8c\u5143\u6a39","text":"<p>\u8207\u93c8\u7d50\u4e32\u5217\u985e\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u7bc0\u9ede\uff0c\u7136\u5f8c\u69cb\u5efa\u5f15\u7528\uff08\u6307\u6a19\uff09\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># \u521d\u59cb\u5316\u4e8c\u5143\u6a39\n# \u521d\u59cb\u5316\u7bc0\u9ede\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// \u521d\u59cb\u5316\u7bc0\u9ede\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// \u521d\u59cb\u5316\u7bc0\u9ede\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* \u521d\u59cb\u5316\u4e8c\u5143\u6a39 */\n// \u521d\u59cb\u5316\u7bc0\u9ede\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// \u521d\u59cb\u5316\u7bc0\u9ede\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code># \u521d\u59cb\u5316\u4e8c\u5143\u6a39\n# \u521d\u59cb\u5316\u7bc0\u9ede\nn1 = TreeNode.new(1)\nn2 = TreeNode.new(2)\nn3 = TreeNode.new(3)\nn4 = TreeNode.new(4)\nn5 = TreeNode.new(5)\n# \u69cb\u5efa\u7bc0\u9ede\u4e4b\u9593\u7684\u5f15\u7528\uff08\u6307\u6a19\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/binary_tree/#2","title":"2. \u00a0 \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede","text":"<p>\u8207\u93c8\u7d50\u4e32\u5217\u985e\u4f3c\uff0c\u5728\u4e8c\u5143\u6a39\u4e2d\u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede\u53ef\u4ee5\u900f\u904e\u4fee\u6539\u6307\u6a19\u4f86\u5be6\u73fe\u3002\u5716 7-3 \u7d66\u51fa\u4e86\u4e00\u500b\u793a\u4f8b\u3002</p> <p></p> <p> \u5716 7-3 \u00a0 \u5728\u4e8c\u5143\u6a39\u4e2d\u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede\np = TreeNode(0)\n# \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = p\np.left = n2\n# \u522a\u9664\u7bc0\u9ede P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nTreeNode P = new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u522a\u9664\u7bc0\u9ede P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P\nP.left = n2\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nlet P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nconst P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P;\nP.left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// \u522a\u9664\u7bc0\u9ede p\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u522a\u9664\u7bc0\u9ede P\nn1-&gt;left = n2;\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede P\nn1.left = P\nP.left = n2\n// \u522a\u9664\u7bc0\u9ede P\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code># \u63d2\u5165\u8207\u522a\u9664\u7bc0\u9ede\n_p = TreeNode.new(0)\n# \u5728 n1 -&gt; n2 \u4e2d\u9593\u63d2\u5165\u7bc0\u9ede _p\nn1.left = _p\n_p.left = n2\n# \u522a\u9664\u7bc0\u9ede\nn1.left = n2\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>Tip</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u7bc0\u9ede\u53ef\u80fd\u6703\u6539\u8b8a\u4e8c\u5143\u6a39\u7684\u539f\u6709\u908f\u8f2f\u7d50\u69cb\uff0c\u800c\u522a\u9664\u7bc0\u9ede\u901a\u5e38\u610f\u5473\u8457\u522a\u9664\u8a72\u7bc0\u9ede\u53ca\u5176\u6240\u6709\u5b50\u6a39\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u5143\u6a39\u4e2d\uff0c\u63d2\u5165\u8207\u522a\u9664\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5be6\u73fe\u6709\u5be6\u969b\u610f\u7fa9\u7684\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3 \u00a0 \u5e38\u898b\u4e8c\u5143\u6a39\u578b\u5225","text":""},{"location":"chapter_tree/binary_tree/#1_1","title":"1. \u00a0 \u5b8c\u7f8e\u4e8c\u5143\u6a39","text":"<p>\u5982\u5716 7-4 \u6240\u793a\uff0c\u5b8c\u7f8e\u4e8c\u5143\u6a39\uff08perfect binary tree\uff09\u6240\u6709\u5c64\u7684\u7bc0\u9ede\u90fd\u88ab\u5b8c\u5168\u586b\u6eff\u3002\u5728\u5b8c\u7f8e\u4e8c\u5143\u6a39\u4e2d\uff0c\u8449\u7bc0\u9ede\u7684\u5ea6\u70ba \\(0\\) \uff0c\u5176\u9918\u6240\u6709\u7bc0\u9ede\u7684\u5ea6\u90fd\u70ba \\(2\\) \uff1b\u82e5\u6a39\u7684\u9ad8\u5ea6\u70ba \\(h\\) \uff0c\u5247\u7bc0\u9ede\u7e3d\u6578\u70ba \\(2^{h+1} - 1\\) \uff0c\u5448\u73fe\u6a19\u6e96\u7684\u6307\u6578\u7d1a\u95dc\u4fc2\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u898b\u7684\u7d30\u80de\u5206\u88c2\u73fe\u8c61\u3002</p> <p>Tip</p> <p>\u8acb\u6ce8\u610f\uff0c\u5728\u4e2d\u6587\u793e\u7fa4\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u5143\u6a39\u5e38\u88ab\u7a31\u70ba\u6eff\u4e8c\u5143\u6a39\u3002</p> <p></p> <p> \u5716 7-4 \u00a0 \u5b8c\u7f8e\u4e8c\u5143\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#2_1","title":"2. \u00a0 \u5b8c\u5168\u4e8c\u5143\u6a39","text":"<p>\u5982\u5716 7-5 \u6240\u793a\uff0c\u5b8c\u5168\u4e8c\u5143\u6a39\uff08complete binary tree\uff09\u53ea\u6709\u6700\u5e95\u5c64\u7684\u7bc0\u9ede\u672a\u88ab\u586b\u6eff\uff0c\u4e14\u6700\u5e95\u5c64\u7bc0\u9ede\u5118\u91cf\u9760\u5de6\u586b\u5145\u3002\u8acb\u6ce8\u610f\uff0c\u5b8c\u7f8e\u4e8c\u5143\u6a39\u4e5f\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u5143\u6a39\u3002</p> <p></p> <p> \u5716 7-5 \u00a0 \u5b8c\u5168\u4e8c\u5143\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#3","title":"3. \u00a0 \u5b8c\u6eff\u4e8c\u5143\u6a39","text":"<p>\u5982\u5716 7-6 \u6240\u793a\uff0c\u5b8c\u6eff\u4e8c\u5143\u6a39\uff08full binary tree\uff09\u9664\u4e86\u8449\u7bc0\u9ede\u4e4b\u5916\uff0c\u5176\u9918\u6240\u6709\u7bc0\u9ede\u90fd\u6709\u5169\u500b\u5b50\u7bc0\u9ede\u3002</p> <p></p> <p> \u5716 7-6 \u00a0 \u5b8c\u6eff\u4e8c\u5143\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#4","title":"4. \u00a0 \u5e73\u8861\u4e8c\u5143\u6a39","text":"<p>\u5982\u5716 7-7 \u6240\u793a\uff0c\u5e73\u8861\u4e8c\u5143\u6a39\uff08balanced binary tree\uff09\u4e2d\u4efb\u610f\u7bc0\u9ede\u7684\u5de6\u5b50\u6a39\u548c\u53f3\u5b50\u6a39\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7d55\u5c0d\u503c\u4e0d\u8d85\u904e 1 \u3002</p> <p></p> <p> \u5716 7-7 \u00a0 \u5e73\u8861\u4e8c\u5143\u6a39 </p>"},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4 \u00a0 \u4e8c\u5143\u6a39\u7684\u9000\u5316","text":"<p>\u5716 7-8 \u5c55\u793a\u4e86\u4e8c\u5143\u6a39\u7684\u7406\u60f3\u7d50\u69cb\u8207\u9000\u5316\u7d50\u69cb\u3002\u7576\u4e8c\u5143\u6a39\u7684\u6bcf\u5c64\u7bc0\u9ede\u90fd\u88ab\u586b\u6eff\u6642\uff0c\u9054\u5230\u201c\u5b8c\u7f8e\u4e8c\u5143\u6a39\u201d\uff1b\u800c\u7576\u6240\u6709\u7bc0\u9ede\u90fd\u504f\u5411\u4e00\u5074\u6642\uff0c\u4e8c\u5143\u6a39\u9000\u5316\u70ba\u201c\u93c8\u7d50\u4e32\u5217\u201d\u3002</p> <ul> <li>\u5b8c\u7f8e\u4e8c\u5143\u6a39\u662f\u7406\u60f3\u60c5\u6cc1\uff0c\u53ef\u4ee5\u5145\u5206\u767c\u63ee\u4e8c\u5143\u6a39\u201c\u5206\u6cbb\u201d\u7684\u512a\u52e2\u3002</li> <li>\u93c8\u7d50\u4e32\u5217\u5247\u662f\u53e6\u4e00\u500b\u6975\u7aef\uff0c\u5404\u9805\u64cd\u4f5c\u90fd\u8b8a\u70ba\u7dda\u6027\u64cd\u4f5c\uff0c\u6642\u9593\u8907\u96dc\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</li> </ul> <p></p> <p> \u5716 7-8 \u00a0 \u4e8c\u5143\u6a39\u7684\u6700\u4f73\u7d50\u69cb\u8207\u6700\u5dee\u7d50\u69cb </p> <p>\u5982\u8868 7-1 \u6240\u793a\uff0c\u5728\u6700\u4f73\u7d50\u69cb\u548c\u6700\u5dee\u7d50\u69cb\u4e0b\uff0c\u4e8c\u5143\u6a39\u7684\u8449\u7bc0\u9ede\u6578\u91cf\u3001\u7bc0\u9ede\u7e3d\u6578\u3001\u9ad8\u5ea6\u7b49\u9054\u5230\u6975\u5927\u503c\u6216\u6975\u5c0f\u503c\u3002</p> <p> \u8868 7-1 \u00a0 \u4e8c\u5143\u6a39\u7684\u6700\u4f73\u7d50\u69cb\u8207\u6700\u5dee\u7d50\u69cb </p> \u5b8c\u7f8e\u4e8c\u5143\u6a39 \u93c8\u7d50\u4e32\u5217 \u7b2c \\(i\\) \u5c64\u7684\u7bc0\u9ede\u6578\u91cf \\(2^{i-1}\\) \\(1\\) \u9ad8\u5ea6\u70ba \\(h\\) \u7684\u6a39\u7684\u8449\u7bc0\u9ede\u6578\u91cf \\(2^h\\) \\(1\\) \u9ad8\u5ea6\u70ba \\(h\\) \u7684\u6a39\u7684\u7bc0\u9ede\u7e3d\u6578 \\(2^{h+1} - 1\\) \\(h + 1\\) \u7bc0\u9ede\u7e3d\u6578\u70ba \\(n\\) \u7684\u6a39\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 \u4e8c\u5143\u6a39\u8d70\u8a2a","text":"<p>\u5f9e\u7269\u7406\u7d50\u69cb\u7684\u89d2\u5ea6\u4f86\u770b\uff0c\u6a39\u662f\u4e00\u7a2e\u57fa\u65bc\u93c8\u7d50\u4e32\u5217\u7684\u8cc7\u6599\u7d50\u69cb\uff0c\u56e0\u6b64\u5176\u8d70\u8a2a\u65b9\u5f0f\u662f\u900f\u904e\u6307\u6a19\u9010\u500b\u8a2a\u554f\u7bc0\u9ede\u3002\u7136\u800c\uff0c\u6a39\u662f\u4e00\u7a2e\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u9019\u4f7f\u5f97\u8d70\u8a2a\u6a39\u6bd4\u8d70\u8a2a\u93c8\u7d50\u4e32\u5217\u66f4\u52a0\u8907\u96dc\uff0c\u9700\u8981\u85c9\u52a9\u641c\u5c0b\u6f14\u7b97\u6cd5\u4f86\u5be6\u73fe\u3002</p> <p>\u4e8c\u5143\u6a39\u5e38\u898b\u7684\u8d70\u8a2a\u65b9\u5f0f\u5305\u62ec\u5c64\u5e8f\u8d70\u8a2a\u3001\u524d\u5e8f\u8d70\u8a2a\u3001\u4e2d\u5e8f\u8d70\u8a2a\u548c\u5f8c\u5e8f\u8d70\u8a2a\u7b49\u3002</p>"},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1 \u00a0 \u5c64\u5e8f\u8d70\u8a2a","text":"<p>\u5982\u5716 7-9 \u6240\u793a\uff0c\u5c64\u5e8f\u8d70\u8a2a\uff08level-order traversal\uff09\u5f9e\u9802\u90e8\u5230\u5e95\u90e8\u9010\u5c64\u8d70\u8a2a\u4e8c\u5143\u6a39\uff0c\u4e26\u5728\u6bcf\u4e00\u5c64\u6309\u7167\u5f9e\u5de6\u5230\u53f3\u7684\u9806\u5e8f\u8a2a\u554f\u7bc0\u9ede\u3002</p> <p>\u5c64\u5e8f\u8d70\u8a2a\u672c\u8cea\u4e0a\u5c6c\u65bc\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\uff08breadth-first traversal\uff09\uff0c\u4e5f\u7a31\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\uff08breadth-first search, BFS\uff09\uff0c\u5b83\u9ad4\u73fe\u4e86\u4e00\u7a2e\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u64f4\u5c55\u201d\u7684\u9010\u5c64\u8d70\u8a2a\u65b9\u5f0f\u3002</p> <p></p> <p> \u5716 7-9 \u00a0 \u4e8c\u5143\u6a39\u7684\u5c64\u5e8f\u8d70\u8a2a </p>"},{"location":"chapter_tree/binary_tree_traversal/#1","title":"1. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u901a\u5e38\u85c9\u52a9\u201c\u4f47\u5217\u201d\u4f86\u5be6\u73fe\u3002\u4f47\u5217\u9075\u5faa\u201c\u5148\u9032\u5148\u51fa\u201d\u7684\u898f\u5247\uff0c\u800c\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u5247\u9075\u5faa\u201c\u9010\u5c64\u63a8\u9032\u201d\u7684\u898f\u5247\uff0c\u5169\u8005\u80cc\u5f8c\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002\u5be6\u73fe\u7a0b\u5f0f\u78bc\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"\u5c64\u5e8f\u8d70\u8a2a\"\"\"\n    # \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # \u968a\u5217\u51fa\u968a\n        res.append(node.val)  # \u5132\u5b58\u7bc0\u9ede\u503c\n        if node.left is not None:\n            queue.append(node.left)  # \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if node.right is not None:\n            queue.append(node.right)  # \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // \u968a\u5217\u51fa\u968a\n        vec.push_back(node-&gt;val); // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // \u968a\u5217\u51fa\u968a\n        list.add(node.val);           // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left != null)\n            queue.offer(node.left);   // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node.right != null)\n            queue.offer(node.right);  // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // \u968a\u5217\u51fa\u968a\n        list.Add(node.val!.Value);       // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left != null)\n            queue.Enqueue(node.left);    // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node.right != null)\n            queue.Enqueue(node.right);   // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfunc levelOrder(root *TreeNode) []any {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    queue := list.New()\n    queue.PushBack(root)\n    // \u521d\u59cb\u5316\u4e00\u500b\u5207\u7247\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // \u968a\u5217\u51fa\u968a\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // \u5132\u5b58\u7bc0\u9ede\u503c\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    var queue: [TreeNode] = [root]\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // \u968a\u5217\u51fa\u968a\n        list.append(node.val) // \u5132\u5b58\u7bc0\u9ede\u503c\n        if let left = node.left {\n            queue.append(left) // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n        if let right = node.right {\n            queue.append(right) // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfunction levelOrder(root) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // \u968a\u5217\u51fa\u968a\n        list.push(node.val); // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left) queue.push(node.left); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node.right) queue.push(node.right); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // \u968a\u5217\u51fa\u968a\n        list.push(node.val); // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left) {\n            queue.push(node.left); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n        if (node.right) {\n            queue.push(node.right); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // \u968a\u5217\u51fa\u968a\n    res.add(node!.val); // \u5132\u5b58\u7bc0\u9ede\u503c\n    if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n    if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    let mut que = VecDeque::new();\n    que.push_back(root.clone());\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // \u968a\u5217\u51fa\u968a\n        vec.push(node.borrow().val); // \u5132\u5b58\u7bc0\u9ede\u503c\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(left.clone()); // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(right.clone()); // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nint *levelOrder(TreeNode *root, int *size) {\n    /* \u8f14\u52a9\u4f47\u5217 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* \u8f14\u52a9\u4f47\u5217 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // \u4f47\u5217\u6307\u6a19\n    front = 0, rear = 0;\n    // \u52a0\u5165\u6839\u7bc0\u9ede\n    queue[rear++] = root;\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    /* \u8f14\u52a9\u9663\u5217 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // \u9663\u5217\u6307\u6a19\n    index = 0;\n    while (front &lt; rear) {\n        // \u968a\u5217\u51fa\u968a\n        node = queue[front++];\n        // \u5132\u5b58\u7bc0\u9ede\u503c\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // \u66f4\u65b0\u9663\u5217\u9577\u5ea6\u7684\u503c\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // \u91cb\u653e\u8f14\u52a9\u9663\u5217\u7a7a\u9593\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* \u5c64\u5e8f\u8d70\u8a2a */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    val list = mutableListOf&lt;Int&gt;()\n    while (queue.isNotEmpty()) {\n        val node = queue.poll()      // \u968a\u5217\u51fa\u968a\n        list.add(node?._val!!)       // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left != null)\n            queue.offer(node.left)   // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        if (node.right != null)\n            queue.offer(node.right)  // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>### \u5c64\u5e8f\u8d70\u8a2a ###\ndef level_order(root)\n  # \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n  queue = [root]\n  # \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n  res = []\n  while !queue.empty?\n    node = queue.shift # \u968a\u5217\u51fa\u968a\n    res &lt;&lt; node.val # \u5132\u5b58\u7bc0\u9ede\u503c\n    queue &lt;&lt; node.left unless node.left.nil? # \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n    queue &lt;&lt; node.right unless node.right.nil? # \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n  end\n  res\nend\n</code></pre> binary_tree_bfs.zig<pre><code>// \u5c64\u5e8f\u8d70\u8a2a\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n    // \u521d\u59cb\u5316\u4f47\u5217\uff0c\u52a0\u5165\u6839\u7bc0\u9ede\n    const L = std.TailQueue(*inc.TreeNode(T));\n    var queue = L{};\n    var root_node = try mem_allocator.create(L.Node);\n    root_node.data = root;\n    queue.append(root_node); \n    // \u521d\u59cb\u5316\u4e00\u500b\u4e32\u5217\uff0c\u7528\u65bc\u5132\u5b58\u8d70\u8a2a\u5e8f\u5217\n    var list = std.ArrayList(T).init(std.heap.page_allocator);\n    while (queue.len &gt; 0) {\n        var queue_node = queue.popFirst().?;    // \u968a\u5217\u51fa\u968a\n        var node = queue_node.data;\n        try list.append(node.val);              // \u5132\u5b58\u7bc0\u9ede\u503c\n        if (node.left != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.left.?;\n            queue.append(tmp_node);             // \u5de6\u5b50\u7bc0\u9ede\u5165\u5217\n        }\n        if (node.right != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.right.?;\n            queue.append(tmp_node);             // \u53f3\u5b50\u7bc0\u9ede\u5165\u5217\n        }        \n    }\n    return list;\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p>"},{"location":"chapter_tree/binary_tree_traversal/#2","title":"2. \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1a\u6240\u6709\u7bc0\u9ede\u88ab\u8a2a\u554f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\uff0c\u5176\u4e2d \\(n\\) \u70ba\u7bc0\u9ede\u6578\u91cf\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u5373\u6eff\u4e8c\u5143\u6a39\u6642\uff0c\u8d70\u8a2a\u5230\u6700\u5e95\u5c64\u4e4b\u524d\uff0c\u4f47\u5217\u4e2d\u6700\u591a\u540c\u6642\u5b58\u5728 \\((n + 1) / 2\\) \u500b\u7bc0\u9ede\uff0c\u4f54\u7528 \\(O(n)\\) \u7a7a\u9593\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2 \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a","text":"<p>\u76f8\u61c9\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u5f8c\u5e8f\u8d70\u8a2a\u90fd\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\uff08depth-first traversal\uff09\uff0c\u4e5f\u7a31\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\uff08depth-first search, DFS\uff09\uff0c\u5b83\u9ad4\u73fe\u4e86\u4e00\u7a2e\u201c\u5148\u8d70\u5230\u76e1\u982d\uff0c\u518d\u56de\u6eaf\u7e7c\u7e8c\u201d\u7684\u8d70\u8a2a\u65b9\u5f0f\u3002</p> <p>\u5716 7-10 \u5c55\u793a\u4e86\u5c0d\u4e8c\u5143\u6a39\u9032\u884c\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u7684\u5de5\u4f5c\u539f\u7406\u3002\u6df1\u5ea6\u512a\u5148\u8d70\u8a2a\u5c31\u50cf\u662f\u7e5e\u8457\u6574\u68f5\u4e8c\u5143\u6a39\u7684\u5916\u570d\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u6bcf\u500b\u7bc0\u9ede\u90fd\u6703\u9047\u5230\u4e09\u500b\u4f4d\u7f6e\uff0c\u5206\u5225\u5c0d\u61c9\u524d\u5e8f\u8d70\u8a2a\u3001\u4e2d\u5e8f\u8d70\u8a2a\u548c\u5f8c\u5e8f\u8d70\u8a2a\u3002</p> <p></p> <p> \u5716 7-10 \u00a0 \u4e8c\u5143\u641c\u5c0b\u6a39\u7684\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a </p>"},{"location":"chapter_tree/binary_tree_traversal/#1_1","title":"1. \u00a0 \u7a0b\u5f0f\u78bc\u5be6\u73fe","text":"<p>\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u901a\u5e38\u57fa\u65bc\u905e\u8ff4\u5be6\u73fe\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"\u524d\u5e8f\u8d70\u8a2a\"\"\"\n    if root is None:\n        return\n    # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"\u4e2d\u5e8f\u8d70\u8a2a\"\"\"\n    if root is None:\n        return\n    # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"\u5f8c\u5e8f\u8d70\u8a2a\"\"\"\n    if root is None:\n        return\n    # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfunction preOrder(root) {\n    if (root === null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunction inOrder(root) {\n    if (root === null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunction postOrder(root) {\n    if (root === null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n            let node = node.borrow();\n            res.push(node.val);\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            res.push(node.val);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n            res.push(node.val);\n        }\n    }\n\n    dfs(root, &amp;mut result);\n\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* \u524d\u5e8f\u8d70\u8a2a */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    list.add(root._val)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* \u4e2d\u5e8f\u8d70\u8a2a */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    inOrder(root.left)\n    list.add(root._val)\n    inOrder(root.right)\n}\n\n/* \u5f8c\u5e8f\u8d70\u8a2a */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root._val)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>### \u524d\u5e8f\u8d70\u8a2a ###\ndef pre_order(root)\n  return if root.nil?\n\n  # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n  $res &lt;&lt; root.val\n  pre_order(root.left)\n  pre_order(root.right)\nend\n\n### \u4e2d\u5e8f\u8d70\u8a2a ###\ndef in_order(root)\n  return if root.nil?\n\n  # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n  in_order(root.left)\n  $res &lt;&lt; root.val\n  in_order(root.right)\nend\n\n### \u5f8c\u5e8f\u8d70\u8a2a ###\ndef post_order(root)\n  return if root.nil?\n\n  # \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n  post_order(root.left)\n  post_order(root.right)\n  $res &lt;&lt; root.val\nend\n</code></pre> binary_tree_dfs.zig<pre><code>// \u524d\u5e8f\u8d70\u8a2a\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u6839\u7bc0\u9ede -&gt; \u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39\n    try list.append(root.?.val);\n    try preOrder(T, root.?.left);\n    try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u8d70\u8a2a\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede -&gt; \u53f3\u5b50\u6a39\n    try inOrder(T, root.?.left);\n    try list.append(root.?.val);\n    try inOrder(T, root.?.right);\n}\n\n// \u5f8c\u5e8f\u8d70\u8a2a\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8a2a\u554f\u512a\u5148\u9806\u5e8f\uff1a\u5de6\u5b50\u6a39 -&gt; \u53f3\u5b50\u6a39 -&gt; \u6839\u7bc0\u9ede\n    try postOrder(T, root.?.left);\n    try postOrder(T, root.?.right);\n    try list.append(root.?.val);\n}\n</code></pre> \u8996\u89ba\u5316\u57f7\u884c <p> \u5168\u87a2\u5e55\u89c0\u770b &gt;</p> <p>Tip</p> <p>\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\u4e5f\u53ef\u4ee5\u57fa\u65bc\u8fed\u4ee3\u5be6\u73fe\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u4ee5\u81ea\u884c\u7814\u7a76\u3002</p> <p>\u5716 7-11 \u5c55\u793a\u4e86\u524d\u5e8f\u8d70\u8a2a\u4e8c\u5143\u6a39\u7684\u905e\u8ff4\u904e\u7a0b\uff0c\u5176\u53ef\u5206\u70ba\u201c\u905e\u201d\u548c\u201c\u8ff4\u201d\u5169\u500b\u9006\u5411\u7684\u90e8\u5206\u3002</p> <ol> <li>\u201c\u905e\u201d\u8868\u793a\u958b\u555f\u65b0\u65b9\u6cd5\uff0c\u7a0b\u5f0f\u5728\u6b64\u904e\u7a0b\u4e2d\u8a2a\u554f\u4e0b\u4e00\u500b\u7bc0\u9ede\u3002</li> <li>\u201c\u8ff4\u201d\u8868\u793a\u51fd\u5f0f\u8fd4\u56de\uff0c\u4ee3\u8868\u7576\u524d\u7bc0\u9ede\u5df2\u7d93\u8a2a\u554f\u5b8c\u7562\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u5716 7-11 \u00a0 \u524d\u5e8f\u8d70\u8a2a\u7684\u905e\u8ff4\u904e\u7a0b </p>"},{"location":"chapter_tree/binary_tree_traversal/#2_1","title":"2. \u00a0 \u8907\u96dc\u5ea6\u5206\u6790","text":"<ul> <li>\u6642\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1a\u6240\u6709\u7bc0\u9ede\u88ab\u8a2a\u554f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u6642\u9593\u3002</li> <li>\u7a7a\u9593\u8907\u96dc\u5ea6\u70ba \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u6cc1\u4e0b\uff0c\u5373\u6a39\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u6642\uff0c\u905e\u8ff4\u6df1\u5ea6\u9054\u5230 \\(n\\) \uff0c\u7cfb\u7d71\u4f54\u7528 \\(O(n)\\) \u5806\u758a\u5e40\u7a7a\u9593\u3002</li> </ul>"},{"location":"chapter_tree/summary/","title":"7.6 \u00a0 \u5c0f\u7d50","text":""},{"location":"chapter_tree/summary/#1","title":"1. \u00a0 \u91cd\u9ede\u56de\u9867","text":"<ul> <li>\u4e8c\u5143\u6a39\u662f\u4e00\u7a2e\u975e\u7dda\u6027\u8cc7\u6599\u7d50\u69cb\uff0c\u9ad4\u73fe\u201c\u4e00\u5206\u70ba\u4e8c\u201d\u7684\u5206\u6cbb\u908f\u8f2f\u3002\u6bcf\u500b\u4e8c\u5143\u6a39\u7bc0\u9ede\u5305\u542b\u4e00\u500b\u503c\u4ee5\u53ca\u5169\u500b\u6307\u6a19\uff0c\u5206\u5225\u6307\u5411\u5176\u5de6\u5b50\u7bc0\u9ede\u548c\u53f3\u5b50\u7bc0\u9ede\u3002</li> <li>\u5c0d\u65bc\u4e8c\u5143\u6a39\u4e2d\u7684\u67d0\u500b\u7bc0\u9ede\uff0c\u5176\u5de6\uff08\u53f3\uff09\u5b50\u7bc0\u9ede\u53ca\u5176\u4ee5\u4e0b\u5f62\u6210\u7684\u6a39\u88ab\u7a31\u70ba\u8a72\u7bc0\u9ede\u7684\u5de6\uff08\u53f3\uff09\u5b50\u6a39\u3002</li> <li>\u4e8c\u5143\u6a39\u7684\u76f8\u95dc\u8853\u8a9e\u5305\u62ec\u6839\u7bc0\u9ede\u3001\u8449\u7bc0\u9ede\u3001\u5c64\u3001\u5ea6\u3001\u908a\u3001\u9ad8\u5ea6\u548c\u6df1\u5ea6\u7b49\u3002</li> <li>\u4e8c\u5143\u6a39\u7684\u521d\u59cb\u5316\u3001\u7bc0\u9ede\u63d2\u5165\u548c\u7bc0\u9ede\u522a\u9664\u64cd\u4f5c\u8207\u93c8\u7d50\u4e32\u5217\u64cd\u4f5c\u65b9\u6cd5\u985e\u4f3c\u3002</li> <li>\u5e38\u898b\u7684\u4e8c\u5143\u6a39\u578b\u5225\u6709\u5b8c\u7f8e\u4e8c\u5143\u6a39\u3001\u5b8c\u5168\u4e8c\u5143\u6a39\u3001\u5b8c\u6eff\u4e8c\u5143\u6a39\u548c\u5e73\u8861\u4e8c\u5143\u6a39\u3002\u5b8c\u7f8e\u4e8c\u5143\u6a39\u662f\u6700\u7406\u60f3\u7684\u72c0\u614b\uff0c\u800c\u93c8\u7d50\u4e32\u5217\u662f\u9000\u5316\u5f8c\u7684\u6700\u5dee\u72c0\u614b\u3002</li> <li>\u4e8c\u5143\u6a39\u53ef\u4ee5\u7528\u9663\u5217\u8868\u793a\uff0c\u65b9\u6cd5\u662f\u5c07\u7bc0\u9ede\u503c\u548c\u7a7a\u4f4d\u6309\u5c64\u5e8f\u8d70\u8a2a\u9806\u5e8f\u6392\u5217\uff0c\u4e26\u6839\u64da\u7236\u7bc0\u9ede\u8207\u5b50\u7bc0\u9ede\u4e4b\u9593\u7684\u7d22\u5f15\u5c0d\u6620\u95dc\u4fc2\u4f86\u5be6\u73fe\u6307\u6a19\u3002</li> <li>\u4e8c\u5143\u6a39\u7684\u5c64\u5e8f\u8d70\u8a2a\u662f\u4e00\u7a2e\u5ee3\u5ea6\u512a\u5148\u641c\u5c0b\u65b9\u6cd5\uff0c\u5b83\u9ad4\u73fe\u4e86\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u64f4\u5c55\u201d\u7684\u9010\u5c64\u8d70\u8a2a\u65b9\u5f0f\uff0c\u901a\u5e38\u900f\u904e\u4f47\u5217\u4f86\u5be6\u73fe\u3002</li> <li>\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a\u7686\u5c6c\u65bc\u6df1\u5ea6\u512a\u5148\u641c\u5c0b\uff0c\u5b83\u5011\u9ad4\u73fe\u4e86\u201c\u5148\u8d70\u5230\u76e1\u982d\uff0c\u518d\u56de\u6eaf\u7e7c\u7e8c\u201d\u7684\u8d70\u8a2a\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u905e\u8ff4\u4f86\u5be6\u73fe\u3002</li> <li>\u4e8c\u5143\u641c\u5c0b\u6a39\u662f\u4e00\u7a2e\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8a62\u8cc7\u6599\u7d50\u69cb\uff0c\u5176\u67e5\u8a62\u3001\u63d2\u5165\u548c\u522a\u9664\u64cd\u4f5c\u7684\u6642\u9593\u8907\u96dc\u5ea6\u5747\u70ba \\(O(\\log n)\\) \u3002\u7576\u4e8c\u5143\u641c\u5c0b\u6a39\u9000\u5316\u70ba\u93c8\u7d50\u4e32\u5217\u6642\uff0c\u5404\u9805\u6642\u9593\u8907\u96dc\u5ea6\u6703\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>AVL \u6a39\uff0c\u4e5f\u7a31\u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39\uff0c\u5b83\u900f\u904e\u65cb\u8f49\u64cd\u4f5c\u78ba\u4fdd\u5728\u4e0d\u65b7\u63d2\u5165\u548c\u522a\u9664\u7bc0\u9ede\u5f8c\u6a39\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002</li> <li>AVL \u6a39\u7684\u65cb\u8f49\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u522a\u9664\u7bc0\u9ede\u5f8c\uff0cAVL \u6a39\u6703\u5f9e\u5e95\u5411\u9802\u57f7\u884c\u65cb\u8f49\u64cd\u4f5c\uff0c\u4f7f\u6a39\u91cd\u65b0\u6062\u5fa9\u5e73\u8861\u3002</li> </ul>"},{"location":"chapter_tree/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u5c0d\u65bc\u53ea\u6709\u4e00\u500b\u7bc0\u9ede\u7684\u4e8c\u5143\u6a39\uff0c\u6a39\u7684\u9ad8\u5ea6\u548c\u6839\u7bc0\u9ede\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u55ce\uff1f</p> <p>\u662f\u7684\uff0c\u56e0\u70ba\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u7fa9\u70ba\u201c\u7d93\u904e\u7684\u908a\u7684\u6578\u91cf\u201d\u3002</p> <p>Q\uff1a\u4e8c\u5143\u6a39\u4e2d\u7684\u63d2\u5165\u8207\u522a\u9664\u4e00\u822c\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\uff0c\u9019\u88e1\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u9ebc\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u70ba\u8cc7\u6e90\u7684\u5b50\u7bc0\u9ede\u7684\u8cc7\u6e90\u91cb\u653e\u55ce\uff1f</p> <p>\u62ff\u4e8c\u5143\u641c\u5c0b\u6a39\u4f86\u8209\u4f8b\uff0c\u522a\u9664\u7bc0\u9ede\u64cd\u4f5c\u8981\u5206\u4e09\u7a2e\u60c5\u6cc1\u8655\u7406\uff0c\u5176\u4e2d\u6bcf\u7a2e\u60c5\u6cc1\u90fd\u9700\u8981\u9032\u884c\u591a\u500b\u6b65\u9a5f\u7684\u7bc0\u9ede\u64cd\u4f5c\u3002</p> <p>Q\uff1a\u70ba\u4ec0\u9ebc DFS \u8d70\u8a2a\u4e8c\u5143\u6a39\u6709\u524d\u3001\u4e2d\u3001\u5f8c\u4e09\u7a2e\u9806\u5e8f\uff0c\u5206\u5225\u6709\u4ec0\u9ebc\u7528\u5462\uff1f</p> <p>\u8207\u9806\u5e8f\u548c\u9006\u5e8f\u8d70\u8a2a\u9663\u5217\u985e\u4f3c\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u5f8c\u5e8f\u8d70\u8a2a\u662f\u4e09\u7a2e\u4e8c\u5143\u6a39\u8d70\u8a2a\u65b9\u6cd5\uff0c\u6211\u5011\u53ef\u4ee5\u4f7f\u7528\u5b83\u5011\u5f97\u5230\u4e00\u500b\u7279\u5b9a\u9806\u5e8f\u7684\u8d70\u8a2a\u7d50\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u4e2d\uff0c\u7531\u65bc\u7bc0\u9ede\u5927\u5c0f\u6eff\u8db3 <code>\u5de6\u5b50\u7bc0\u9ede\u503c &lt; \u6839\u7bc0\u9ede\u503c &lt; \u53f3\u5b50\u7bc0\u9ede\u503c</code> \uff0c\u56e0\u6b64\u6211\u5011\u53ea\u8981\u6309\u7167\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u512a\u5148\u9806\u5e8f\u8d70\u8a2a\u6a39\uff0c\u5c31\u53ef\u4ee5\u7372\u5f97\u6709\u5e8f\u7684\u7bc0\u9ede\u5e8f\u5217\u3002</p> <p>Q\uff1a\u53f3\u65cb\u64cd\u4f5c\u662f\u8655\u7406\u5931\u8861\u7bc0\u9ede <code>node</code>\u3001<code>child</code>\u3001<code>grand_child</code> \u4e4b\u9593\u7684\u95dc\u4fc2\uff0c\u90a3 <code>node</code> \u7684\u7236\u7bc0\u9ede\u548c <code>node</code> \u539f\u4f86\u7684\u9023\u7dda\u4e0d\u9700\u8981\u7dad\u8b77\u55ce\uff1f\u53f3\u65cb\u64cd\u4f5c\u5f8c\u8c48\u4e0d\u662f\u65b7\u6389\u4e86\uff1f</p> <p>\u6211\u5011\u9700\u8981\u5f9e\u905e\u8ff4\u7684\u8996\u89d2\u4f86\u770b\u9019\u500b\u554f\u984c\u3002\u53f3\u65cb\u64cd\u4f5c <code>right_rotate(root)</code> \u50b3\u5165\u7684\u662f\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\uff0c\u6700\u7d42 <code>return child</code> \u8fd4\u56de\u65cb\u8f49\u4e4b\u5f8c\u7684\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\u3002\u5b50\u6a39\u7684\u6839\u7bc0\u9ede\u548c\u5176\u7236\u7bc0\u9ede\u7684\u9023\u7dda\u662f\u5728\u8a72\u51fd\u5f0f\u8fd4\u56de\u5f8c\u5b8c\u6210\u7684\uff0c\u4e0d\u5c6c\u65bc\u53f3\u65cb\u64cd\u4f5c\u7684\u7dad\u8b77\u7bc4\u570d\u3002</p> <p>Q\uff1a\u5728 C++ \u4e2d\uff0c\u51fd\u5f0f\u88ab\u5283\u5206\u5230 <code>private</code> \u548c <code>public</code> \u4e2d\uff0c\u9019\u65b9\u9762\u6709\u4ec0\u9ebc\u8003\u91cf\u55ce\uff1f\u70ba\u4ec0\u9ebc\u8981\u5c07 <code>height()</code> \u51fd\u5f0f\u548c <code>updateHeight()</code> \u51fd\u5f0f\u5206\u5225\u653e\u5728 <code>public</code> \u548c <code>private</code> \u4e2d\u5462\uff1f</p> <p>\u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u7bc4\u570d\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u985e\u5225\u5167\u90e8\u4f7f\u7528\uff0c\u90a3\u9ebc\u5c31\u8a2d\u8a08\u70ba <code>private</code> \u3002\u4f8b\u5982\uff0c\u4f7f\u7528\u8005\u55ae\u7368\u547c\u53eb <code>updateHeight()</code> \u662f\u6c92\u6709\u610f\u7fa9\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u522a\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800c <code>height()</code> \u662f\u8a2a\u554f\u7bc0\u9ede\u9ad8\u5ea6\uff0c\u985e\u4f3c\u65bc <code>vector.size()</code> \uff0c\u56e0\u6b64\u8a2d\u5b9a\u6210 <code>public</code> \u4ee5\u4fbf\u4f7f\u7528\u3002</p> <p>Q\uff1a\u5982\u4f55\u5f9e\u4e00\u7d44\u8f38\u5165\u8cc7\u6599\u69cb\u5efa\u4e00\u68f5\u4e8c\u5143\u641c\u5c0b\u6a39\uff1f\u6839\u7bc0\u9ede\u7684\u9078\u64c7\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f</p> <p>\u662f\u7684\uff0c\u69cb\u5efa\u6a39\u7684\u65b9\u6cd5\u5df2\u5728\u4e8c\u5143\u641c\u5c0b\u6a39\u7a0b\u5f0f\u78bc\u4e2d\u7684 <code>build_tree()</code> \u65b9\u6cd5\u4e2d\u7d66\u51fa\u3002\u81f3\u65bc\u6839\u7bc0\u9ede\u7684\u9078\u64c7\uff0c\u6211\u5011\u901a\u5e38\u6703\u5c07\u8f38\u5165\u8cc7\u6599\u6392\u5e8f\uff0c\u7136\u5f8c\u5c07\u4e2d\u9ede\u5143\u7d20\u4f5c\u70ba\u6839\u7bc0\u9ede\uff0c\u518d\u905e\u8ff4\u5730\u69cb\u5efa\u5de6\u53f3\u5b50\u6a39\u3002\u9019\u6a23\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8b49\u6a39\u7684\u5e73\u8861\u6027\u3002</p> <p>Q\uff1a\u5728 Java \u4e2d\uff0c\u5b57\u4e32\u5c0d\u6bd4\u662f\u5426\u4e00\u5b9a\u8981\u7528 <code>equals()</code> \u65b9\u6cd5\uff1f</p> <p>\u5728 Java \u4e2d\uff0c\u5c0d\u65bc\u57fa\u672c\u8cc7\u6599\u578b\u5225\uff0c<code>==</code> \u7528\u65bc\u5c0d\u6bd4\u5169\u500b\u8b8a\u6578\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002\u5c0d\u65bc\u5f15\u7528\u578b\u5225\uff0c\u5169\u7a2e\u7b26\u865f\u7684\u5de5\u4f5c\u539f\u7406\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li><code>==</code> \uff1a\u7528\u4f86\u6bd4\u8f03\u5169\u500b\u8b8a\u6578\u662f\u5426\u6307\u5411\u540c\u4e00\u500b\u7269\u4ef6\uff0c\u5373\u5b83\u5011\u5728\u8a18\u61b6\u9ad4\u4e2d\u7684\u4f4d\u7f6e\u662f\u5426\u76f8\u540c\u3002</li> <li><code>equals()</code>\uff1a\u7528\u4f86\u5c0d\u6bd4\u5169\u500b\u7269\u4ef6\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002</li> </ul> <p>\u56e0\u6b64\uff0c\u5982\u679c\u8981\u5c0d\u6bd4\u503c\uff0c\u6211\u5011\u61c9\u8a72\u4f7f\u7528 <code>equals()</code> \u3002\u7136\u800c\uff0c\u900f\u904e <code>String a = \"hi\"; String b = \"hi\";</code> \u521d\u59cb\u5316\u7684\u5b57\u4e32\u90fd\u5132\u5b58\u5728\u5b57\u4e32\u5e38\u6578\u6c60\u4e2d\uff0c\u5b83\u5011\u6307\u5411\u540c\u4e00\u500b\u7269\u4ef6\uff0c\u56e0\u6b64\u4e5f\u53ef\u4ee5\u7528 <code>a == b</code> \u4f86\u6bd4\u8f03\u5169\u500b\u5b57\u4e32\u7684\u5167\u5bb9\u3002</p> <p>Q\uff1a\u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a\u5230\u6700\u5e95\u5c64\u4e4b\u524d\uff0c\u4f47\u5217\u4e2d\u7684\u7bc0\u9ede\u6578\u91cf\u662f \\(2^h\\) \u55ce\uff1f</p> <p>\u662f\u7684\uff0c\u4f8b\u5982\u9ad8\u5ea6 \\(h = 2\\) \u7684\u6eff\u4e8c\u5143\u6a39\uff0c\u5176\u7bc0\u9ede\u7e3d\u6578 \\(n = 7\\) \uff0c\u5247\u5e95\u5c64\u7bc0\u9ede\u6578\u91cf \\(4 = 2^h = (n + 1) / 2\\) \u3002</p>"}]}
\ No newline at end of file
diff --git a/zh-hant/sitemap.xml.gz b/zh-hant/sitemap.xml.gz
index c404b49b7c49c1985ecbcafda48e5c46e6bee141..3550fadd161b1200fbb3c74708d643cc10a52fac 100644
GIT binary patch
delta 15
Wcmeyw{)wGUzMF$XPH7|C8)g6~9|V~I

delta 15
Wcmeyw{)wGUzMF$%J=;dMH_QMk`2`LD