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第 0 章 前言
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第 0 章 前言
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0.1 关于本书
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0.2 如何使用本书
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0.3 小结
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第 1 章 初识算法
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第 1 章 初识算法
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1.1 算法无处不在
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1.2 算法是什么
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1.3 小结
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第 2 章 复杂度分析
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第 2 章 复杂度分析
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2.1 算法效率评估
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2.2 迭代与递归
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2.3 时间复杂度
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2.4 空间复杂度
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2.5 小结
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第 3 章 数据结构
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第 3 章 数据结构
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3.1 数据结构分类
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3.2 基本数据类型
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3.3 数字编码 *
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3.4 字符编码 *
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3.5 小结
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第 4 章 数组与链表
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第 4 章 数组与链表
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第 5 章 栈与队列
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5.3 双向队列
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5.4 小结
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第 6 章 哈希表
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第 6 章 哈希表
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6.1 哈希表
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6.2 哈希冲突
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6.3 哈希算法
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6.4 小结
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第 7 章 树
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第 7 章 树
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7.1 二叉树
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7.3 二叉树数组表示
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7.4 二叉搜索树
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7.5 AVL 树 *
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7.6 小结
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第 8 章 堆
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第 8 章 堆
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< / label >
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8.1 堆
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8.2 建堆操作
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8.3 Top-K 问题
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8.4 小结
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第 9 章 图
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第 9 章 图
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9.1 图
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9.2 图基础操作
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9.3 图的遍历
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9.4 小结
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z" / > < / svg >
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第 10 章 搜索
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第 10 章 搜索
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< / label >
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10.1 二分查找
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< li class = "md-nav__item" >
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10.2 二分查找插入点
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10.3 二分查找边界
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/replace_linear_by_hashing/" class = "md-nav__link" >
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10.4 哈希优化策略
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< / span >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/searching_algorithm_revisited/" class = "md-nav__link" >
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10.5 重识搜索算法
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< / span >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
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10.6 小结
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第 11 章 排序
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第 11 章 排序
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11.1 排序算法
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11.2 选择排序
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11.3 冒泡排序
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11.4 插入排序
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11.5 快速排序
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11.6 归并排序
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11.7 堆排序
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11.8 桶排序
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11.9 计数排序
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11.10 基数排序
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11.11 小结
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第 12 章 分治
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第 12 章 分治
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12.1 分治算法
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12.2 分治搜索策略
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12.3 构建树问题
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12.4 汉诺塔问题
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12.5 小结
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第 13 章 回溯
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第 13 章 回溯
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13.1 回溯算法
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13.2 全排列问题
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13.3 子集和问题
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13.4 N 皇后问题
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13.5 小结
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第 14 章 动态规划
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14.1 初探动态规划
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14.2 DP 问题特性
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14.3 DP 解题思路
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14.4 0-1 背包问题
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14.5 完全背包问题
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14.6 编辑距离问题
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14.7 小结
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第 15 章 贪心
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15.1 贪心算法
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15.2 分数背包问题
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15.3 最大容量问题
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15.4 最大切分乘积问题
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15.5 小结
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第 16 章 附录
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16.1 编程环境安装
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16.2 一起参与创作
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< h1 id = "11" > 1.1 算法无处不在< a class = "headerlink" href = "#11" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > 当我们听到“算法”这个词时，很自然地会想到数学。然而实际上，许多算法并不涉及复杂数学，而是更多地依赖于基本逻辑，这些逻辑在我们的日常生活中处处可见。< / p >
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< p > 在正式探讨算法之前，有一个有趣的事实值得分享：< strong > 你已经在不知不觉中学会了许多算法，并习惯将它们应用到日常生活中了< / strong > 。下面，我将举几个具体例子来证实这一点。< / p >
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< p > < strong > 例一：查阅字典< / strong > 。在字典里，每个汉字都对应一个拼音，而字典是按照拼音字母顺序排列的。假设我们需要查找一个拼音首字母为 < span class = "arithmatex" > \(r\)< / span > 的字，通常会按照图 1-1 所示的方式实现。< / p >
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< ol >
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< li > 翻开字典约一半的页数，查看该页的首字母是什么，假设首字母为 < span class = "arithmatex" > \(m\)< / span > 。< / li >
< li > 由于在拼音字母表中 < span class = "arithmatex" > \(r\)< / span > 位于 < span class = "arithmatex" > \(m\)< / span > 之后，所以排除字典前半部分，查找范围缩小到后半部分。< / li >
< li > 不断重复步骤 < code > 1.< / code > 和 步骤 < code > 2.< / code > ，直至找到拼音首字母为 < span class = "arithmatex" > \(r\)< / span > 的页码为止。< / li >
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< / ol >
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< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "查字典步骤" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" / > < / a > < / p >
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< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "binary_search_dictionary_step3" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" / > < / a > < / p >
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< div class = "tabbed-block" >
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< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "binary_search_dictionary_step4" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" / > < / a > < / p >
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< / div >
< div class = "tabbed-block" >
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< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "binary_search_dictionary_step5" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" / > < / a > < / p >
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< / div >
< / div >
< / div >
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< p align = "center" > 图 1-1 查字典步骤 < / p >
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< p > 查阅字典这个小学生必备技能，实际上就是著名的二分查找算法。从数据结构的角度，我们可以把字典视为一个已排序的“数组”；从算法的角度，我们可以将上述查字典的一系列操作看作是“二分查找”。< / p >
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< p > < strong > 例二：整理扑克< / strong > 。我们在打牌时，每局都需要整理扑克牌，使其从小到大排列，实现流程如图 1-2 所示。< / p >
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< ol >
< li > 将扑克牌划分为“有序”和“无序”两部分，并假设初始状态下最左 1 张扑克牌已经有序。< / li >
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< li > 在无序部分抽出一张扑克牌，插入至有序部分的正确位置；完成后最左 2 张扑克已经有序。< / li >
< li > 不断循环步骤 < code > 2.< / code > ，每一轮将一张扑克牌从无序部分插入至有序部分，直至所有扑克牌都有序。< / li >
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< / ol >
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< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/playing_cards_sorting.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "扑克排序步骤" src = "../algorithms_are_everywhere.assets/playing_cards_sorting.png" / > < / a > < / p >
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< p align = "center" > 图 1-2 扑克排序步骤 < / p >
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< p > 上述整理扑克牌的方法本质上是“插入排序”算法，它在处理小型数据集时非常高效。许多编程语言的排序库函数中都存在插入排序的身影。< / p >
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< p > < strong > 例三：货币找零< / strong > 。假设我们在超市购买了 < span class = "arithmatex" > \(69\)< / span > 元的商品，给了收银员 < span class = "arithmatex" > \(100\)< / span > 元，则收银员需要找我们 < span class = "arithmatex" > \(31\)< / span > 元。他会很自然地完成如图 1-3 所示的思考。< / p >
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< ol >
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< li > 可选项是比 < span class = "arithmatex" > \(31\)< / span > 元面值更小的货币，包括 < span class = "arithmatex" > \(1\)< / span > 元、< span class = "arithmatex" > \(5\)< / span > 元、< span class = "arithmatex" > \(10\)< / span > 元、< span class = "arithmatex" > \(20\)< / span > 元。< / li >
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< li > 从可选项中拿出最大的 < span class = "arithmatex" > \(20\)< / span > 元，剩余 < span class = "arithmatex" > \(31 - 20 = 11\)< / span > 元。< / li >
< li > 从剩余可选项中拿出最大的 < span class = "arithmatex" > \(10\)< / span > 元，剩余 < span class = "arithmatex" > \(11 - 10 = 1\)< / span > 元。< / li >
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< li > 从剩余可选项中拿出最大的 < span class = "arithmatex" > \(1\)< / span > 元，剩余 < span class = "arithmatex" > \(1 - 1 = 0\)< / span > 元。< / li >
< li > 完成找零，方案为 < span class = "arithmatex" > \(20 + 10 + 1 = 31\)< / span > 元。< / li >
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< / ol >
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< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/greedy_change.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "货币找零过程" src = "../algorithms_are_everywhere.assets/greedy_change.png" / > < / a > < / p >
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< p align = "center" > 图 1-3 货币找零过程 < / p >
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< p > 在以上步骤中，我们每一步都采取当前看来最好的选择（尽可能用大面额的货币），最终得到了可行的找零方案。从数据结构与算法的角度看，这种方法本质上是“贪心”算法。< / p >
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< p > 小到烹饪一道菜，大到星际航行，几乎所有问题的解决都离不开算法。计算机的出现使我们能够通过编程将数据结构存储在内存中，同时编写代码调用 CPU 和 GPU 执行算法。这样一来，我们就能把生活中的问题转移到计算机上，以更高效的方式解决各种复杂问题。< / p >
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< div class = "admonition tip" >
< p class = "admonition-title" > Tip< / p >
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< p > 如果你对数据结构、算法、数组和二分查找等概念仍感到一知半解，请继续往下阅读，这本书将引导你迈入数据结构与算法的知识殿堂。< / p >
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< script src = "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" > < / script >
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< script > document $ . subscribe ( ( ) => { const lightbox = GLightbox ( { "touchNavigation" : true , "loop" : false , "zoomable" : true , "draggable" : false , "openEffect" : "zoom" , "closeEffect" : "zoom" , "slideEffect" : "none" } ) ; } ) < / script > < / body >
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< / html >
