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<li>在页面底部填写修改说明，然后点击“Propose file change”按钮。页面跳转后，点击“Create pull request”按钮即可发起拉取请求。</li>
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</ol>
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<p><img alt="页面编辑按键" src="../contribution.assets/edit_markdown.png" /></p>
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<p align="center"> Fig. 页面编辑按键 </p>
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<p align="center"> 图：页面编辑按键 </p>
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<p>图片无法直接修改，需要通过新建 <a href="https://github.com/krahets/hello-algo/issues">Issue</a> 或评论留言来描述问题，我们会尽快重新绘制并替换图片。</p>
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<h2 id="1622">16.2.2. 内容创作<a class="headerlink" href="#1622" title="Permanent link">¶</a></h2>
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</li>
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<li class="md-nav__item">
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<a href="#1619-rust" class="md-nav__link">
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16.1.9. Rust 环境
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<a href="#1619-dart" class="md-nav__link">
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16.1.9. Dart 环境
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#16110-rust" class="md-nav__link">
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16.1.10. Rust 环境
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</a>
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</li>
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</li>
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<li class="md-nav__item">
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<a href="#1619-rust" class="md-nav__link">
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16.1.9. Rust 环境
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<a href="#1619-dart" class="md-nav__link">
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16.1.9. Dart 环境
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#16110-rust" class="md-nav__link">
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16.1.10. Rust 环境
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</a>
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</li>
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<li>下载并安装 <a href="https://www.swift.org/download/">Swift</a>。</li>
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<li>在 VSCode 的插件市场中搜索 <code>swift</code> ，安装 <a href="https://marketplace.visualstudio.com/items?itemName=sswg.swift-lang">Swift for Visual Studio Code</a>。</li>
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</ol>
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<h2 id="1619-rust">16.1.9. Rust 环境<a class="headerlink" href="#1619-rust" title="Permanent link">¶</a></h2>
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<h2 id="1619-dart">16.1.9. Dart 环境<a class="headerlink" href="#1619-dart" title="Permanent link">¶</a></h2>
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<ol>
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<li>下载并安装 <a href="https://dart.dev/get-dart">Dart</a> 。</li>
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<li>在 VSCode 的插件市场中搜索 <code>dart</code> ，安装 <a href="https://marketplace.visualstudio.com/items?itemName=Dart-Code.dart-code">Dart</a> 。</li>
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</ol>
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<h2 id="16110-rust">16.1.10. Rust 环境<a class="headerlink" href="#16110-rust" title="Permanent link">¶</a></h2>
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<ol>
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<li>下载并安装 <a href="https://www.rust-lang.org/tools/install">Rust</a>。</li>
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<li>在 VSCode 的插件市场中搜索 <code>rust</code> ，安装 <a href="https://marketplace.visualstudio.com/items?itemName=rust-lang.rust-analyzer">rust-analyzer</a>。</li>
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Load diff
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<li class="md-nav__item">
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<a href="#421" class="md-nav__link">
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4.2.1. 链表优点
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4.2.1. 链表常用操作
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</a>
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<nav class="md-nav" aria-label="4.2.1. 链表常用操作">
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<ul class="md-nav__list">
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<li class="md-nav__item">
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<a href="#_1" class="md-nav__link">
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初始化链表
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#_2" class="md-nav__link">
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插入节点
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#_3" class="md-nav__link">
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删除节点
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#_4" class="md-nav__link">
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访问节点
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#_5" class="md-nav__link">
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查找节点
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
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<a href="#422" class="md-nav__link">
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4.2.2. 链表缺点
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<a href="#422-vs" class="md-nav__link">
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4.2.2. 数组 VS 链表
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#423" class="md-nav__link">
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4.2.3. 链表常用操作
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4.2.3. 常见链表类型
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#424" class="md-nav__link">
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4.2.4. 常见链表类型
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#425" class="md-nav__link">
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4.2.5. 链表典型应用
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4.2.4. 链表典型应用
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#421" class="md-nav__link">
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4.2.1. 链表优点
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4.2.1. 链表常用操作
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</a>
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<nav class="md-nav" aria-label="4.2.1. 链表常用操作">
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<ul class="md-nav__list">
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<li class="md-nav__item">
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<a href="#_1" class="md-nav__link">
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初始化链表
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#_2" class="md-nav__link">
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插入节点
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#_3" class="md-nav__link">
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删除节点
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#_4" class="md-nav__link">
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访问节点
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#_5" class="md-nav__link">
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查找节点
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
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<a href="#422" class="md-nav__link">
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4.2.2. 链表缺点
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<a href="#422-vs" class="md-nav__link">
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4.2.2. 数组 VS 链表
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#423" class="md-nav__link">
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4.2.3. 链表常用操作
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4.2.3. 常见链表类型
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#424" class="md-nav__link">
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4.2.4. 常见链表类型
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#425" class="md-nav__link">
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4.2.5. 链表典型应用
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4.2.4. 链表典型应用
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</a>
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</li>
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@ -3454,19 +3522,25 @@
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<h1 id="42">4.2. 链表<a class="headerlink" href="#42" title="Permanent link">¶</a></h1>
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<p>内存空间是所有程序的公共资源，排除已被占用的内存空间，空闲内存空间通常散落在内存各处。在上一节中，我们提到存储数组的内存空间必须是连续的，而当需要申请一个非常大的数组时，空闲内存中可能没有这么大的连续空间。与数组相比，链表更具灵活性，它可以被存储在非连续的内存空间中。</p>
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<p>「链表 Linked List」是一种线性数据结构，其每个元素都是一个节点对象，各个节点之间通过指针连接，从当前节点通过指针可以访问到下一个节点。<strong>由于指针记录了下个节点的内存地址，因此无需保证内存地址的连续性</strong>，从而可以将各个节点分散存储在内存各处。</p>
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<p>链表中的「节点 Node」包含两项数据，一是节点「值 Value」，二是指向下一节点的「引用 Reference」，或称「指针 Pointer」。</p>
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<p>内存空间是所有程序的公共资源，在一个复杂的系统运行环境下，空闲的内存空间可能散落在内存各处。我们知道，存储数组的内存空间必须是连续的，而当数组非常大时，内存可能无法提供如此大的连续空间。此时链表的灵活性优势就体现出来了。</p>
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<p>「链表 Linked List」是一种线性数据结构，其中的每个元素都是一个节点对象，各个节点通过“引用”相连接。引用记录了下一个节点的内存地址，我们可以通过它从当前节点访问到下一个节点。这意味着链表的各个节点可以被分散存储在内存各处，它们的内存地址是无需连续的。</p>
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<p><img alt="链表定义与存储方式" src="../linked_list.assets/linkedlist_definition.png" /></p>
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<p align="center"> Fig. 链表定义与存储方式 </p>
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<p align="center"> 图：链表定义与存储方式 </p>
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<p>观察上图，链表中的每个「节点 Node」对象都包含两项数据：节点的“值”、指向下一节点的“引用”。</p>
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<ul>
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<li>链表的首个节点被称为“头节点”，最后一个节点被称为“尾节点”。</li>
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<li>尾节点指向的是“空”，它在 Java, C++, Python 中分别被记为 <span class="arithmatex">\(\text{null}\)</span> , <span class="arithmatex">\(\text{nullptr}\)</span> , <span class="arithmatex">\(\text{None}\)</span> 。</li>
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<li>在 C, C++, Go, Rust 等支持指针的语言中，上述的“引用”应被替换为“指针”。</li>
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</ul>
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<p>如以下代码所示，链表以节点对象 <code>ListNode</code> 为单位，每个节点除了包含值，还需额外保存下一节点的引用（指针）。因此在相同数据量下，<strong>链表通常比数组占用更多的内存空间</strong>。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 链表节点类 */</span>
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<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kd">class</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
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<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向下一节点的指针（引用）</span>
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<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向下一节点的引用</span>
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||||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="c1">// 构造函数</span>
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||||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="p">}</span>
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||||
</code></pre></div>
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@ -3475,7 +3549,7 @@
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<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 链表节点结构体 */</span>
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||||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向下一节点的指针</span>
|
||||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="w"> </span><span class="n">next</span><span class="p">(</span><span class="k">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{}</span><span class="w"> </span><span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="p">};</span>
|
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</code></pre></div>
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@ -3485,14 +3559,14 @@
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<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="w"> </span><span class="sd">"""链表节点类"""</span>
|
||||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
|
||||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span> <span class="c1"># 节点值</span>
|
||||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">ListNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">ListNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向下一节点的引用</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 链表节点结构体 */</span>
|
||||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">ListNode</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w"> </span><span class="nx">Val</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="nx">Next</span><span class="w"> </span><span class="o">*</span><span class="nx">ListNode</span><span class="w"> </span><span class="c1">// 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="nx">Next</span><span class="w"> </span><span class="o">*</span><span class="nx">ListNode</span><span class="w"> </span><span class="c1">// 指向下一节点的指针</span>
|
||||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="p">}</span>
|
||||
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a>
|
||||
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="c1">// NewListNode 构造函数，创建一个新的链表</span>
|
||||
|
|
@ -3532,7 +3606,7 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* 链表节点结构体 */</span>
|
||||
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向下一节点的指针</span>
|
||||
<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="p">};</span>
|
||||
<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a>
|
||||
<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="n">ListNode</span><span class="p">;</span>
|
||||
|
|
@ -3560,7 +3634,7 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 链表节点类 */</span>
|
||||
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">class</span> <span class="nc">ListNode</span> <span class="p">{</span>
|
||||
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a> <span class="kd">var</span> <span class="nv">val</span><span class="p">:</span> <span class="nb">Int</span> <span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="kd">var</span> <span class="nv">next</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="kd">var</span> <span class="nv">next</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 指向下一节点的引用</span>
|
||||
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a>
|
||||
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="kd">init</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">{</span> <span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="n">val</span> <span class="p">=</span> <span class="n">x</span>
|
||||
|
|
@ -3575,7 +3649,7 @@
|
|||
<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w"> </span><span class="kr">const</span><span class="w"> </span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@This</span><span class="p">();</span>
|
||||
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a>
|
||||
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w"> </span><span class="n">val</span><span class="o">:</span><span class="w"> </span><span class="n">T</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w"> </span><span class="n">next</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w"> </span><span class="n">next</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向下一节点的指针</span>
|
||||
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a>
|
||||
<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w"> </span><span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w"> </span><span class="kr">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">init</span><span class="p">(</span><span class="n">self</span><span class="o">:</span><span class="w"> </span><span class="o">*</span><span class="n">Self</span><span class="p">,</span><span class="w"> </span><span class="n">x</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="p">{</span>
|
||||
|
|
@ -3590,7 +3664,7 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 链表节点类 */</span>
|
||||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向下一节点的引用</span>
|
||||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">next</span><span class="p">]);</span><span class="w"> </span><span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -3602,19 +3676,20 @@
|
|||
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="cp">#[derive(Debug)]</span>
|
||||
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="k">struct</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w"> </span><span class="n">val</span>: <span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="n">next</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o">>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="n">next</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o">>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向下一节点的指针</span>
|
||||
<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>我们将链表的首个节点称为「头节点」，最后一个节点称为「尾节点」。尾节点指向的是“空”，在 Java, C++, Python 中分别记为 <span class="arithmatex">\(\text{null}\)</span> , <span class="arithmatex">\(\text{nullptr}\)</span> , <span class="arithmatex">\(\text{None}\)</span> 。在不引起歧义的前提下，本书都使用 <span class="arithmatex">\(\text{None}\)</span> 来表示空。</p>
|
||||
<p><strong>链表初始化方法</strong>。建立链表分为两步，第一步是初始化各个节点对象，第二步是构建引用指向关系。完成后，即可以从链表的头节点（即首个节点）出发，通过指针 <code>next</code> 依次访问所有节点。</p>
|
||||
<h2 id="421">4.2.1. 链表常用操作<a class="headerlink" href="#421" title="Permanent link">¶</a></h2>
|
||||
<h3 id="_1">初始化链表<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<p>建立链表分为两步，第一步是初始化各个节点对象，第二步是构建引用指向关系。初始化完成后，我们就可以从链表的头节点出发，通过引用指向 <code>next</code> 依次访问所有节点。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">linked_list.java</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="cm">/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */</span>
|
||||
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1">// 初始化各个节点 </span>
|
||||
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1">// 初始化各个节点</span>
|
||||
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
|
||||
<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
|
||||
|
|
@ -3629,7 +3704,7 @@
|
|||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">linked_list.cpp</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="cm">/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */</span>
|
||||
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1">// 初始化各个节点 </span>
|
||||
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1">// 初始化各个节点</span>
|
||||
<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
|
||||
<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
|
||||
|
|
@ -3644,7 +3719,7 @@
|
|||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">linked_list.py</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="c1"># 初始化链表 1 -> 3 -> 2 -> 5 -> 4</span>
|
||||
<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1"># 初始化各个节点 </span>
|
||||
<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1"># 初始化各个节点</span>
|
||||
<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="n">n0</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
|
||||
<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="n">n1</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
|
||||
<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="n">n2</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
|
||||
|
|
@ -3704,7 +3779,7 @@
|
|||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">linked_list.c</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */</span>
|
||||
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// 初始化各个节点 </span>
|
||||
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// 初始化各个节点</span>
|
||||
<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
|
||||
<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
|
||||
|
|
@ -3719,7 +3794,7 @@
|
|||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">linked_list.cs</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */</span>
|
||||
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// 初始化各个节点 </span>
|
||||
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// 初始化各个节点</span>
|
||||
<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">3</span><span class="p">);</span>
|
||||
<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">2</span><span class="p">);</span>
|
||||
|
|
@ -3749,7 +3824,7 @@
|
|||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">linked_list.zig</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="c1">// 初始化链表</span>
|
||||
<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// 初始化各个节点 </span>
|
||||
<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// 初始化各个节点</span>
|
||||
<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="kr">var</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inc</span><span class="p">.</span><span class="n">ListNode</span><span class="p">(</span><span class="kt">i32</span><span class="p">){.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">};</span>
|
||||
<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="kr">var</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inc</span><span class="p">.</span><span class="n">ListNode</span><span class="p">(</span><span class="kt">i32</span><span class="p">){.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">};</span>
|
||||
<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="kr">var</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inc</span><span class="p">.</span><span class="n">ListNode</span><span class="p">(</span><span class="kt">i32</span><span class="p">){.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">};</span>
|
||||
|
|
@ -3795,11 +3870,12 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>在编程语言中，数组整体是一个变量，比如数组 <code>nums</code> 包含元素 <code>nums[0]</code> , <code>nums[1]</code> 等。而链表是由多个分散的节点对象组成，<strong>我们通常将头节点当作链表的代称</strong>，比如以上代码中的链表可被记做链表 <code>n0</code> 。</p>
|
||||
<h2 id="421">4.2.1. 链表优点<a class="headerlink" href="#421" title="Permanent link">¶</a></h2>
|
||||
<p><strong>链表中插入与删除节点的操作效率高</strong>。如果我们想在链表中间的两个节点 <code>A</code> , <code>B</code> 之间插入一个新节点 <code>P</code> ，我们只需要改变两个节点指针即可，时间复杂度为 <span class="arithmatex">\(O(1)\)</span> ；相比之下，数组的插入操作效率要低得多。</p>
|
||||
<p>数组整体是一个变量，比如数组 <code>nums</code> 包含元素 <code>nums[0]</code> , <code>nums[1]</code> 等，而链表是由多个独立的节点对象组成的。<strong>我们通常将头节点当作链表的代称</strong>，比如以上代码中的链表可被记做链表 <code>n0</code> 。</p>
|
||||
<h3 id="_2">插入节点<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p><strong>在链表中插入节点非常容易</strong>。假设我们想在相邻的两个节点 <code>n0</code> , <code>n1</code> 之间插入一个新节点 <code>P</code> ，则只需要改变两个节点引用（指针）即可，时间复杂度为 <span class="arithmatex">\(O(1)\)</span> 。</p>
|
||||
<p>相比之下，在数组中插入元素的时间复杂度为 <span class="arithmatex">\(O(n)\)</span> ，在大数据量下的效率较低。</p>
|
||||
<p><img alt="链表插入节点" src="../linked_list.assets/linkedlist_insert_node.png" /></p>
|
||||
<p align="center"> Fig. 链表插入节点 </p>
|
||||
<p align="center"> 图：链表插入节点 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3913,9 +3989,11 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>在链表中删除节点也非常方便，只需改变一个节点的指针即可。如下图所示，尽管在删除操作完成后，节点 <code>P</code> 仍然指向 <code>n1</code> ，但实际上 <code>P</code> 已经不再属于此链表，因为遍历此链表时无法访问到 <code>P</code> 。</p>
|
||||
<h3 id="_3">删除节点<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>在链表中删除节点也非常简便，只需改变一个节点的引用（指针）即可。</p>
|
||||
<p>请注意，尽管在删除操作完成后节点 <code>P</code> 仍然指向 <code>n1</code> ，但实际上遍历此链表已经无法访问到 <code>P</code> ，这意味着 <code>P</code> 已经不再属于该链表了。</p>
|
||||
<p><img alt="链表删除节点" src="../linked_list.assets/linkedlist_remove_node.png" /></p>
|
||||
<p align="center"> Fig. 链表删除节点 </p>
|
||||
<p align="center"> 图：链表删除节点 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="4:12"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JS</label><label for="__tabbed_4_6">TS</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label><label for="__tabbed_4_11">Dart</label><label for="__tabbed_4_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -4072,8 +4150,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<h2 id="422">4.2.2. 链表缺点<a class="headerlink" href="#422" title="Permanent link">¶</a></h2>
|
||||
<p><strong>链表访问节点效率较低</strong>。如上节所述，数组可以在 <span class="arithmatex">\(O(1)\)</span> 时间下访问任意元素。然而链表无法直接访问任意节点，因为程序需要从头节点出发，逐个向后遍历，直至找到目标节点。也就是说，如果想要访问链表中第 <span class="arithmatex">\(i\)</span> 个节点，则需要向后遍历 <span class="arithmatex">\(i - 1\)</span> 轮。</p>
|
||||
<h3 id="_4">访问节点<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p><strong>在链表访问节点的效率较低</strong>。如上节所述，我们可以在 <span class="arithmatex">\(O(1)\)</span> 时间下访问数组中的任意元素。链表则不然，程序需要从头节点出发，逐个向后遍历，直至找到目标节点。也就是说，访问链表的第 <span class="arithmatex">\(i\)</span> 个节点需要循环 <span class="arithmatex">\(i - 1\)</span> 轮，时间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -4223,9 +4301,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>链表的内存占用较大</strong>。链表以节点为单位，每个节点除了包含值，还需额外保存下一节点的引用（指针）。这意味着在相同数据量的情况下，链表比数组需要占用更多的内存空间。</p>
|
||||
<h2 id="423">4.2.3. 链表常用操作<a class="headerlink" href="#423" title="Permanent link">¶</a></h2>
|
||||
<p><strong>遍历链表查找</strong>。遍历链表，查找链表内值为 <code>target</code> 的节点，输出节点在链表中的索引。</p>
|
||||
<h3 id="_5">查找节点<a class="headerlink" href="#_5" title="Permanent link">¶</a></h3>
|
||||
<p>遍历链表，查找链表内值为 <code>target</code> 的节点，输出节点在链表中的索引。此过程也属于「线性查找」。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JS</label><label for="__tabbed_6_6">TS</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -4399,18 +4476,68 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<h2 id="424">4.2.4. 常见链表类型<a class="headerlink" href="#424" title="Permanent link">¶</a></h2>
|
||||
<p><strong>单向链表</strong>。即上述介绍的普通链表。单向链表的节点包含值和指向下一节点的指针（引用）两项数据。我们将首个节点称为头节点，将最后一个节点成为尾节点，尾节点指向空 <span class="arithmatex">\(\text{None}\)</span> 。</p>
|
||||
<h2 id="422-vs">4.2.2. 数组 VS 链表<a class="headerlink" href="#422-vs" title="Permanent link">¶</a></h2>
|
||||
<p>下表总结对比了数组和链表的各项特点与操作效率。由于它们采用两种相反的存储策略，因此各种性质和操作效率也呈现对立的特点。</p>
|
||||
<div class="center-table">
|
||||
<table>
|
||||
<thead>
|
||||
<tr>
|
||||
<th></th>
|
||||
<th>数组</th>
|
||||
<th>链表</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>存储方式</td>
|
||||
<td>连续内存空间</td>
|
||||
<td>离散内存空间</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>缓存局部性</td>
|
||||
<td>友好</td>
|
||||
<td>不友好</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>容量扩展</td>
|
||||
<td>长度不可变</td>
|
||||
<td>可灵活扩展</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>内存效率</td>
|
||||
<td>占用内存少、浪费部分空间</td>
|
||||
<td>占用内存多</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>访问元素</td>
|
||||
<td><span class="arithmatex">\(O(1)\)</span></td>
|
||||
<td><span class="arithmatex">\(O(n)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>添加元素</td>
|
||||
<td><span class="arithmatex">\(O(n)\)</span></td>
|
||||
<td><span class="arithmatex">\(O(1)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>删除元素</td>
|
||||
<td><span class="arithmatex">\(O(n)\)</span></td>
|
||||
<td><span class="arithmatex">\(O(1)\)</span></td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
<h2 id="423">4.2.3. 常见链表类型<a class="headerlink" href="#423" title="Permanent link">¶</a></h2>
|
||||
<p><strong>单向链表</strong>。即上述介绍的普通链表。单向链表的节点包含值和指向下一节点的引用两项数据。我们将首个节点称为头节点，将最后一个节点成为尾节点，尾节点指向空 <span class="arithmatex">\(\text{None}\)</span> 。</p>
|
||||
<p><strong>环形链表</strong>。如果我们令单向链表的尾节点指向头节点（即首尾相接），则得到一个环形链表。在环形链表中，任意节点都可以视作头节点。</p>
|
||||
<p><strong>双向链表</strong>。与单向链表相比，双向链表记录了两个方向的指针（引用）。双向链表的节点定义同时包含指向后继节点（下一节点）和前驱节点（上一节点）的指针。相较于单向链表，双向链表更具灵活性，可以朝两个方向遍历链表，但相应地也需要占用更多的内存空间。</p>
|
||||
<p><strong>双向链表</strong>。与单向链表相比，双向链表记录了两个方向的引用。双向链表的节点定义同时包含指向后继节点（下一个节点）和前驱节点（上一个节点）的引用（指针）。相较于单向链表，双向链表更具灵活性，可以朝两个方向遍历链表，但相应地也需要占用更多的内存空间。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JS</label><label for="__tabbed_7_6">TS</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* 双向链表节点类 */</span>
|
||||
<a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="kd">class</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的引用</span>
|
||||
<a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的引用</span>
|
||||
<a id="__codelineno-72-6" name="__codelineno-72-6" href="#__codelineno-72-6"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-72-7" name="__codelineno-72-7" href="#__codelineno-72-7"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -4419,8 +4546,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* 双向链表节点结构体 */</span>
|
||||
<a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针</span>
|
||||
<a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针</span>
|
||||
<a id="__codelineno-73-6" name="__codelineno-73-6" href="#__codelineno-73-6"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="w"> </span><span class="n">next</span><span class="p">(</span><span class="k">nullptr</span><span class="p">),</span><span class="w"> </span><span class="n">prev</span><span class="p">(</span><span class="k">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{}</span><span class="w"> </span><span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-73-7" name="__codelineno-73-7" href="#__codelineno-73-7"></a><span class="p">};</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -4430,16 +4557,16 @@
|
|||
<a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="w"> </span><span class="sd">"""双向链表节点类"""</span>
|
||||
<a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
|
||||
<a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span> <span class="c1"># 节点值</span>
|
||||
<a id="__codelineno-74-5" name="__codelineno-74-5" href="#__codelineno-74-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">ListNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-74-6" name="__codelineno-74-6" href="#__codelineno-74-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">ListNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-74-5" name="__codelineno-74-5" href="#__codelineno-74-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">ListNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向后继节点的引用</span>
|
||||
<a id="__codelineno-74-6" name="__codelineno-74-6" href="#__codelineno-74-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">ListNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向前驱节点的引用</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* 双向链表节点结构体 */</span>
|
||||
<a id="__codelineno-75-2" name="__codelineno-75-2" href="#__codelineno-75-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w"> </span><span class="nx">Val</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w"> </span><span class="nx">Next</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-75-5" name="__codelineno-75-5" href="#__codelineno-75-5"></a><span class="w"> </span><span class="nx">Prev</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w"> </span><span class="nx">Next</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="c1">// 指向后继节点的指针</span>
|
||||
<a id="__codelineno-75-5" name="__codelineno-75-5" href="#__codelineno-75-5"></a><span class="w"> </span><span class="nx">Prev</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针</span>
|
||||
<a id="__codelineno-75-6" name="__codelineno-75-6" href="#__codelineno-75-6"></a><span class="p">}</span>
|
||||
<a id="__codelineno-75-7" name="__codelineno-75-7" href="#__codelineno-75-7"></a>
|
||||
<a id="__codelineno-75-8" name="__codelineno-75-8" href="#__codelineno-75-8"></a><span class="c1">// NewDoublyListNode 初始化</span>
|
||||
|
|
@ -4460,8 +4587,8 @@
|
|||
<a id="__codelineno-76-5" name="__codelineno-76-5" href="#__codelineno-76-5"></a><span class="w"> </span><span class="nx">prev</span><span class="p">;</span>
|
||||
<a id="__codelineno-76-6" name="__codelineno-76-6" href="#__codelineno-76-6"></a><span class="w"> </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">next</span><span class="p">,</span><span class="w"> </span><span class="nx">prev</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-76-7" name="__codelineno-76-7" href="#__codelineno-76-7"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-76-8" name="__codelineno-76-8" href="#__codelineno-76-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">next</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-76-9" name="__codelineno-76-9" href="#__codelineno-76-9"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">prev</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-76-8" name="__codelineno-76-8" href="#__codelineno-76-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">next</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的引用</span>
|
||||
<a id="__codelineno-76-9" name="__codelineno-76-9" href="#__codelineno-76-9"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">prev</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的引用</span>
|
||||
<a id="__codelineno-76-10" name="__codelineno-76-10" href="#__codelineno-76-10"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-76-11" name="__codelineno-76-11" href="#__codelineno-76-11"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -4474,8 +4601,8 @@
|
|||
<a id="__codelineno-77-5" name="__codelineno-77-5" href="#__codelineno-77-5"></a><span class="w"> </span><span class="nx">prev</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
|
||||
<a id="__codelineno-77-6" name="__codelineno-77-6" href="#__codelineno-77-6"></a><span class="w"> </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">next?</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="nx">prev?</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-77-7" name="__codelineno-77-7" href="#__codelineno-77-7"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-77-8" name="__codelineno-77-8" href="#__codelineno-77-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">next</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-77-9" name="__codelineno-77-9" href="#__codelineno-77-9"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">prev</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-77-8" name="__codelineno-77-8" href="#__codelineno-77-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">next</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的引用</span>
|
||||
<a id="__codelineno-77-9" name="__codelineno-77-9" href="#__codelineno-77-9"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">prev</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的引用</span>
|
||||
<a id="__codelineno-77-10" name="__codelineno-77-10" href="#__codelineno-77-10"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-77-11" name="__codelineno-77-11" href="#__codelineno-77-11"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -4484,8 +4611,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a><span class="cm">/* 双向链表节点结构体 */</span>
|
||||
<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针</span>
|
||||
<a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针</span>
|
||||
<a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a><span class="p">};</span>
|
||||
<a id="__codelineno-78-7" name="__codelineno-78-7" href="#__codelineno-78-7"></a>
|
||||
<a id="__codelineno-78-8" name="__codelineno-78-8" href="#__codelineno-78-8"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="n">ListNode</span><span class="p">;</span>
|
||||
|
|
@ -4505,8 +4632,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* 双向链表节点类 */</span>
|
||||
<a id="__codelineno-79-2" name="__codelineno-79-2" href="#__codelineno-79-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的引用</span>
|
||||
<a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的引用</span>
|
||||
<a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">=></span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-79-7" name="__codelineno-79-7" href="#__codelineno-79-7"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -4515,8 +4642,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* 双向链表节点类 */</span>
|
||||
<a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kd">class</span> <span class="nc">ListNode</span> <span class="p">{</span>
|
||||
<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a> <span class="kd">var</span> <span class="nv">val</span><span class="p">:</span> <span class="nb">Int</span> <span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a> <span class="kd">var</span> <span class="nv">next</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a> <span class="kd">var</span> <span class="nv">prev</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a> <span class="kd">var</span> <span class="nv">next</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 指向后继节点的引用</span>
|
||||
<a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a> <span class="kd">var</span> <span class="nv">prev</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 指向前驱节点的引用</span>
|
||||
<a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a>
|
||||
<a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a> <span class="kd">init</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">{</span> <span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-80-8" name="__codelineno-80-8" href="#__codelineno-80-8"></a> <span class="n">val</span> <span class="p">=</span> <span class="n">x</span>
|
||||
|
|
@ -4531,8 +4658,8 @@
|
|||
<a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w"> </span><span class="kr">const</span><span class="w"> </span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@This</span><span class="p">();</span>
|
||||
<a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a>
|
||||
<a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w"> </span><span class="n">val</span><span class="o">:</span><span class="w"> </span><span class="n">T</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="w"> </span><span class="n">next</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w"> </span><span class="n">prev</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="w"> </span><span class="n">next</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向后继节点的指针</span>
|
||||
<a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w"> </span><span class="n">prev</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针</span>
|
||||
<a id="__codelineno-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a>
|
||||
<a id="__codelineno-81-10" name="__codelineno-81-10" href="#__codelineno-81-10"></a><span class="w"> </span><span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-81-11" name="__codelineno-81-11" href="#__codelineno-81-11"></a><span class="w"> </span><span class="kr">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">init</span><span class="p">(</span><span class="n">self</span><span class="o">:</span><span class="w"> </span><span class="o">*</span><span class="n">Self</span><span class="p">,</span><span class="w"> </span><span class="n">x</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="p">{</span>
|
||||
|
|
@ -4548,8 +4675,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* 双向链表节点类 */</span>
|
||||
<a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向后继节点的引用</span>
|
||||
<a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 指向前驱节点的引用</span>
|
||||
<a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">next</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">prev</span><span class="p">]);</span><span class="w"> </span><span class="c1">// 构造函数</span>
|
||||
<a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -4562,8 +4689,8 @@
|
|||
<a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a><span class="cp">#[derive(Debug)]</span>
|
||||
<a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a><span class="k">struct</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a><span class="w"> </span><span class="n">val</span>: <span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-83-8" name="__codelineno-83-8" href="#__codelineno-83-8"></a><span class="w"> </span><span class="n">next</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o">>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向后继节点的指针（引用）</span>
|
||||
<a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a><span class="w"> </span><span class="n">prev</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o">>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针（引用）</span>
|
||||
<a id="__codelineno-83-8" name="__codelineno-83-8" href="#__codelineno-83-8"></a><span class="w"> </span><span class="n">next</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o">>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向后继节点的指针</span>
|
||||
<a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a><span class="w"> </span><span class="n">prev</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o">>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 指向前驱节点的指针</span>
|
||||
<a id="__codelineno-83-10" name="__codelineno-83-10" href="#__codelineno-83-10"></a><span class="p">}</span>
|
||||
<a id="__codelineno-83-11" name="__codelineno-83-11" href="#__codelineno-83-11"></a>
|
||||
<a id="__codelineno-83-12" name="__codelineno-83-12" href="#__codelineno-83-12"></a><span class="cm">/* 构造函数 */</span>
|
||||
|
|
@ -4581,9 +4708,9 @@
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="常见链表种类" src="../linked_list.assets/linkedlist_common_types.png" /></p>
|
||||
<p align="center"> Fig. 常见链表种类 </p>
|
||||
<p align="center"> 图：常见链表种类 </p>
|
||||
|
||||
<h2 id="425">4.2.5. 链表典型应用<a class="headerlink" href="#425" title="Permanent link">¶</a></h2>
|
||||
<h2 id="424">4.2.4. 链表典型应用<a class="headerlink" href="#424" title="Permanent link">¶</a></h2>
|
||||
<p>单向链表通常用于实现栈、队列、散列表和图等数据结构。</p>
|
||||
<ul>
|
||||
<li><strong>栈与队列</strong>：当插入和删除操作都在链表的一端进行时，它表现出先进后出的的特性，对应栈；当插入操作在链表的一端进行，删除操作在链表的另一端进行，它表现出先进先出的特性，对应队列。</li>
|
||||
|
|
@ -4592,7 +4719,7 @@
|
|||
</ul>
|
||||
<p>双向链表常被用于需要快速查找前一个和下一个元素的场景。</p>
|
||||
<ul>
|
||||
<li><strong>高级数据结构</strong>：比如在红黑树、B 树中，我们需要知道一个节点的父节点，这可以通过在节点中保存一个指向父节点的指针来实现，类似于双向链表。</li>
|
||||
<li><strong>高级数据结构</strong>：比如在红黑树、B 树中，我们需要访问节点的父节点，这可以通过在节点中保存一个指向父节点的引用来实现，类似于双向链表。</li>
|
||||
<li><strong>浏览器历史</strong>：在网页浏览器中，当用户点击前进或后退按钮时，浏览器需要知道用户访问过的前一个和后一个网页。双向链表的特性使得这种操作变得简单。</li>
|
||||
<li><strong>LRU 算法</strong>：在缓存淘汰算法（LRU）中，我们需要快速找到最近最少使用的数据，以及支持快速地添加和删除节点。这时候使用双向链表就非常合适。</li>
|
||||
</ul>
|
||||
|
|
|
|||
|
|
@ -980,11 +980,59 @@
|
|||
4.3.1. 列表常用操作
|
||||
</a>
|
||||
|
||||
<nav class="md-nav" aria-label="4.3.1. 列表常用操作">
|
||||
<ul class="md-nav__list">
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_1" class="md-nav__link">
|
||||
初始化列表
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_2" class="md-nav__link">
|
||||
访问元素
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_3" class="md-nav__link">
|
||||
插入与删除元素
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_4" class="md-nav__link">
|
||||
遍历列表
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_5" class="md-nav__link">
|
||||
拼接列表
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_6" class="md-nav__link">
|
||||
排序列表
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
</ul>
|
||||
</nav>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#432" class="md-nav__link">
|
||||
4.3.2. 列表实现 *
|
||||
4.3.2. 列表实现
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
|
@ -3379,11 +3427,59 @@
|
|||
4.3.1. 列表常用操作
|
||||
</a>
|
||||
|
||||
<nav class="md-nav" aria-label="4.3.1. 列表常用操作">
|
||||
<ul class="md-nav__list">
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_1" class="md-nav__link">
|
||||
初始化列表
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_2" class="md-nav__link">
|
||||
访问元素
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_3" class="md-nav__link">
|
||||
插入与删除元素
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_4" class="md-nav__link">
|
||||
遍历列表
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_5" class="md-nav__link">
|
||||
拼接列表
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_6" class="md-nav__link">
|
||||
排序列表
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
</ul>
|
||||
</nav>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#432" class="md-nav__link">
|
||||
4.3.2. 列表实现 *
|
||||
4.3.2. 列表实现
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
|
@ -3412,10 +3508,11 @@
|
|||
|
||||
|
||||
<h1 id="43">4.3. 列表<a class="headerlink" href="#43" title="Permanent link">¶</a></h1>
|
||||
<p><strong>数组长度不可变导致实用性降低</strong>。在许多情况下，我们事先无法确定需要存储多少数据，这使数组长度的选择变得困难。若长度过小，需要在持续添加数据时频繁扩容数组；若长度过大，则会造成内存空间的浪费。</p>
|
||||
<p>为解决此问题，出现了一种被称为「动态数组 Dynamic Array」的数据结构，即长度可变的数组，也常被称为「列表 List」。列表基于数组实现，继承了数组的优点，并且可以在程序运行过程中动态扩容。在列表中，我们可以自由添加元素，而无需担心超过容量限制。</p>
|
||||
<p><strong>数组长度不可变导致实用性降低</strong>。在实际中，我们可能事先无法确定需要存储多少数据，这使数组长度的选择变得困难。若长度过小，需要在持续添加数据时频繁扩容数组；若长度过大，则会造成内存空间的浪费。</p>
|
||||
<p>为解决此问题，出现了一种被称为「动态数组 Dynamic Array」的数据结构，即长度可变的数组，也常被称为「列表 List」。列表基于数组实现，继承了数组的优点，并且可以在程序运行过程中动态扩容。我们可以在列表中自由地添加元素，而无需担心超过容量限制。</p>
|
||||
<h2 id="431">4.3.1. 列表常用操作<a class="headerlink" href="#431" title="Permanent link">¶</a></h2>
|
||||
<p><strong>初始化列表</strong>。通常我们会使用“无初始值”和“有初始值”的两种初始化方法。</p>
|
||||
<h3 id="_1">初始化列表<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<p>我们通常使用“无初始值”和“有初始值”这两种初始化方法。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -3514,7 +3611,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>访问与更新元素</strong>。由于列表的底层数据结构是数组，因此可以在 <span class="arithmatex">\(O(1)\)</span> 时间内访问和更新元素，效率很高。</p>
|
||||
<h3 id="_2">访问元素<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p>列表本质上是数组，因此可以在 <span class="arithmatex">\(O(1)\)</span> 时间内访问和更新元素，效率很高。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -3610,7 +3708,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>在列表中添加、插入、删除元素</strong>。相较于数组，列表可以自由地添加与删除元素。在列表尾部添加元素的时间复杂度为 <span class="arithmatex">\(O(1)\)</span> ，但插入和删除元素的效率仍与数组相同，时间复杂度为 <span class="arithmatex">\(O(N)\)</span> 。</p>
|
||||
<h3 id="_3">插入与删除元素<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>相较于数组，列表可以自由地添加与删除元素。在列表尾部添加元素的时间复杂度为 <span class="arithmatex">\(O(1)\)</span> ，但插入和删除元素的效率仍与数组相同，时间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -3817,7 +3916,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>遍历列表</strong>。与数组一样，列表可以根据索引遍历，也可以直接遍历各元素。</p>
|
||||
<h3 id="_4">遍历列表<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>与数组一样，列表可以根据索引遍历，也可以直接遍历各元素。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="4:12"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JS</label><label for="__tabbed_4_6">TS</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label><label for="__tabbed_4_11">Dart</label><label for="__tabbed_4_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -3979,7 +4079,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>拼接两个列表</strong>。给定一个新列表 <code>list1</code> ，我们可以将该列表拼接到原列表的尾部。</p>
|
||||
<h3 id="_5">拼接列表<a class="headerlink" href="#_5" title="Permanent link">¶</a></h3>
|
||||
<p>给定一个新列表 <code>list1</code> ，我们可以将该列表拼接到原列表的尾部。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -4057,7 +4158,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>排序列表</strong>。排序也是常用的方法之一。完成列表排序后，我们便可以使用在数组类算法题中经常考察的「二分查找」和「双指针」算法。</p>
|
||||
<h3 id="_6">排序列表<a class="headerlink" href="#_6" title="Permanent link">¶</a></h3>
|
||||
<p>完成列表排序后，我们便可以使用在数组类算法题中经常考察的“二分查找”和“双指针”算法。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JS</label><label for="__tabbed_6_6">TS</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -4121,14 +4223,14 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<h2 id="432">4.3.2. 列表实现 *<a class="headerlink" href="#432" title="Permanent link">¶</a></h2>
|
||||
<p>为了帮助加深对列表的理解，我们在此提供一个简易版列表实现。需要关注三个核心点：</p>
|
||||
<h2 id="432">4.3.2. 列表实现<a class="headerlink" href="#432" title="Permanent link">¶</a></h2>
|
||||
<p>许多编程语言都提供内置的列表，例如 Java, C++, Python 等。它们的实现比较复杂，各个参数的设定也非常有考究，例如初始容量、扩容倍数等。感兴趣的读者可以查阅源码进行学习。</p>
|
||||
<p>为了帮助你理解列表的工作原理，我们在此提供一个简易版列表实现，重点包括：</p>
|
||||
<ul>
|
||||
<li><strong>初始容量</strong>：选取一个合理的数组初始容量。在本示例中，我们选择 10 作为初始容量。</li>
|
||||
<li><strong>数量记录</strong>：声明一个变量 size，用于记录列表当前元素数量，并随着元素插入和删除实时更新。根据此变量，我们可以定位列表尾部，以及判断是否需要扩容。</li>
|
||||
<li><strong>扩容机制</strong>：插入元素时可能超出列表容量，此时需要扩容列表。扩容方法是根据扩容倍数创建一个更大的数组，并将当前数组的所有元素依次移动至新数组。在本示例中，我们规定每次将数组扩容至之前的 2 倍。</li>
|
||||
<li><strong>扩容机制</strong>：若插入元素时列表容量已满，则需要进行扩容。首先根据扩容倍数创建一个更大的数组，再将当前数组的所有元素依次移动至新数组。在本示例中，我们规定每次将数组扩容至之前的 2 倍。</li>
|
||||
</ul>
|
||||
<p>本示例旨在帮助读者直观理解列表的工作机制。实际编程语言中，列表实现更加标准和复杂，各个参数的设定也非常有考究，例如初始容量、扩容倍数等。感兴趣的读者可以查阅源码进行学习。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JS</label><label for="__tabbed_7_6">TS</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
|
|
|
|||
|
|
@ -3399,64 +3399,11 @@
|
|||
|
||||
<h1 id="44">4.4. 小结<a class="headerlink" href="#44" title="Permanent link">¶</a></h1>
|
||||
<ul>
|
||||
<li>数组和链表是两种基本数据结构，分别代表数据在计算机内存中的连续空间存储和离散空间存储方式。两者的优缺点呈现出互补的特性。</li>
|
||||
<li>数组和链表是两种基本的数据结构，分别代表数据在计算机内存中的两种存储方式：连续空间存储和离散空间存储。两者的特点呈现出互补的特性。</li>
|
||||
<li>数组支持随机访问、占用内存较少；但插入和删除元素效率低，且初始化后长度不可变。</li>
|
||||
<li>链表通过更改引用（指针）实现高效的节点插入与删除，且可以灵活调整长度；但节点访问效率低、占用内存较多。常见的链表类型包括单向链表、循环链表、双向链表。</li>
|
||||
<li>动态数组，又称列表，是基于数组实现的一种数据结构。它保留了数组的优势，同时可以灵活调整长度。列表的出现极大地提高了数组的易用性，但可能导致部分内存空间浪费。</li>
|
||||
<li>下表总结并对比了数组与链表的各项特性与操作效率。</li>
|
||||
</ul>
|
||||
<div class="center-table">
|
||||
<table>
|
||||
<thead>
|
||||
<tr>
|
||||
<th></th>
|
||||
<th>数组</th>
|
||||
<th>链表</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>存储方式</td>
|
||||
<td>连续内存空间</td>
|
||||
<td>离散内存空间</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>数据结构长度</td>
|
||||
<td>长度不可变</td>
|
||||
<td>长度可变</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>内存使用率</td>
|
||||
<td>占用内存少、缓存局部性好</td>
|
||||
<td>占用内存多</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>优势操作</td>
|
||||
<td>随机访问</td>
|
||||
<td>插入、删除</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>访问元素</td>
|
||||
<td><span class="arithmatex">\(O(1)\)</span></td>
|
||||
<td><span class="arithmatex">\(O(N)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>添加元素</td>
|
||||
<td><span class="arithmatex">\(O(N)\)</span></td>
|
||||
<td><span class="arithmatex">\(O(1)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>删除元素</td>
|
||||
<td><span class="arithmatex">\(O(N)\)</span></td>
|
||||
<td><span class="arithmatex">\(O(1)\)</span></td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
<div class="admonition note">
|
||||
<p class="admonition-title">缓存局部性</p>
|
||||
<p>在计算机中，数据读写速度排序是“硬盘 < 内存 < CPU 缓存”。当我们访问数组元素时，计算机不仅会加载它，还会缓存其周围的其他数据，从而借助高速缓存来提升后续操作的执行速度。链表则不然，计算机只能挨个地缓存各个节点，这样的多次“搬运”降低了整体效率。</p>
|
||||
</div>
|
||||
<h2 id="441-q-a">4.4.1. Q & A<a class="headerlink" href="#441-q-a" title="Permanent link">¶</a></h2>
|
||||
<div class="admonition question">
|
||||
<p class="admonition-title">数组存储在栈上和存储在堆上，对时间效率和空间效率是否有影响？</p>
|
||||
|
|
@ -3468,7 +3415,7 @@
|
|||
</ol>
|
||||
</div>
|
||||
<div class="admonition question">
|
||||
<p class="admonition-title">为什么数组会强调要求相同类型的元素，而在链表中却没有强调同类型呢？</p>
|
||||
<p class="admonition-title">为什么数组要求相同类型的元素，而在链表中却没有强调同类型呢？</p>
|
||||
<p>链表由结点组成，结点之间通过引用（指针）连接，各个结点可以存储不同类型的数据，例如 int, double, string, object 等。</p>
|
||||
<p>相对地，数组元素则必须是相同类型的，这样才能通过计算偏移量来获取对应元素位置。例如，如果数组同时包含 int 和 long 两种类型，单个元素分别占用 4 bytes 和 8 bytes ，那么此时就不能用以下公式计算偏移量了，因为数组中包含了两种 <code>elementLength</code> 。</p>
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a>// 元素内存地址 = 数组内存地址 + 元素长度 * 元素索引
|
||||
|
|
|
|||
|
|
@ -3637,7 +3637,7 @@
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="在前序遍历中搜索节点" src="../backtracking_algorithm.assets/preorder_find_nodes.png" /></p>
|
||||
<p align="center"> Fig. 在前序遍历中搜索节点 </p>
|
||||
<p align="center"> 图：在前序遍历中搜索节点 </p>
|
||||
|
||||
<h2 id="1311">13.1.1. 尝试与回退<a class="headerlink" href="#1311" title="Permanent link">¶</a></h2>
|
||||
<p><strong>之所以称之为回溯算法，是因为该算法在搜索解空间时会采用“尝试”与“回退”的策略</strong>。当算法在搜索过程中遇到某个状态无法继续前进或无法得到满足条件的解时，它会撤销上一步的选择，退回到之前的状态，并尝试其他可能的选择。</p>
|
||||
|
|
@ -3918,6 +3918,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：尝试与回退 </p>
|
||||
|
||||
<h2 id="1312">13.1.2. 剪枝<a class="headerlink" href="#1312" title="Permanent link">¶</a></h2>
|
||||
<p>复杂的回溯问题通常包含一个或多个约束条件，<strong>约束条件通常可用于“剪枝”</strong>。</p>
|
||||
<div class="admonition question">
|
||||
|
|
@ -4189,7 +4191,7 @@
|
|||
</div>
|
||||
<p>剪枝是一个非常形象的名词。在搜索过程中，<strong>我们“剪掉”了不满足约束条件的搜索分支</strong>，避免许多无意义的尝试，从而实现搜索效率的提高。</p>
|
||||
<p><img alt="根据约束条件剪枝" src="../backtracking_algorithm.assets/preorder_find_constrained_paths.png" /></p>
|
||||
<p align="center"> Fig. 根据约束条件剪枝 </p>
|
||||
<p align="center"> 图：根据约束条件剪枝 </p>
|
||||
|
||||
<h2 id="1313">13.1.3. 框架代码<a class="headerlink" href="#1313" title="Permanent link">¶</a></h2>
|
||||
<p>接下来，我们尝试将回溯的“尝试、回退、剪枝”的主体框架提炼出来，提升代码的通用性。</p>
|
||||
|
|
@ -5003,7 +5005,7 @@
|
|||
</div>
|
||||
<p>根据题意，当找到值为 7 的节点后应该继续搜索，<strong>因此我们需要将记录解之后的 <code>return</code> 语句删除</strong>。下图对比了保留或删除 <code>return</code> 语句的搜索过程。</p>
|
||||
<p><img alt="保留与删除 return 的搜索过程对比" src="../backtracking_algorithm.assets/backtrack_remove_return_or_not.png" /></p>
|
||||
<p align="center"> Fig. 保留与删除 return 的搜索过程对比 </p>
|
||||
<p align="center"> 图：保留与删除 return 的搜索过程对比 </p>
|
||||
|
||||
<p>相比基于前序遍历的代码实现，基于回溯算法框架的代码实现虽然显得啰嗦，但通用性更好。实际上，<strong>许多回溯问题都可以在该框架下解决</strong>。我们只需根据具体问题来定义 <code>state</code> 和 <code>choices</code> ，并实现框架中的各个方法即可。</p>
|
||||
<h2 id="1314">13.1.4. 常用术语<a class="headerlink" href="#1314" title="Permanent link">¶</a></h2>
|
||||
|
|
|
|||
|
|
@ -3432,18 +3432,18 @@
|
|||
</div>
|
||||
<p>如下图所示，当 <span class="arithmatex">\(n = 4\)</span> 时，共可以找到两个解。从回溯算法的角度看，<span class="arithmatex">\(n \times n\)</span> 大小的棋盘共有 <span class="arithmatex">\(n^2\)</span> 个格子，给出了所有的选择 <code>choices</code> 。在逐个放置皇后的过程中，棋盘状态在不断地变化，每个时刻的棋盘就是状态 <code>state</code> 。</p>
|
||||
<p><img alt="4 皇后问题的解" src="../n_queens_problem.assets/solution_4_queens.png" /></p>
|
||||
<p align="center"> Fig. 4 皇后问题的解 </p>
|
||||
<p align="center"> 图：4 皇后问题的解 </p>
|
||||
|
||||
<p>本题共包含三个约束条件：<strong>多个皇后不能在同一行、同一列、同一对角线</strong>。值得注意的是，对角线分为主对角线 <code>\</code> 和次对角线 <code>/</code> 两种。</p>
|
||||
<p><img alt="n 皇后问题的约束条件" src="../n_queens_problem.assets/n_queens_constraints.png" /></p>
|
||||
<p align="center"> Fig. n 皇后问题的约束条件 </p>
|
||||
<p align="center"> 图：n 皇后问题的约束条件 </p>
|
||||
|
||||
<h3 id="_1">逐行放置策略<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<p>皇后的数量和棋盘的行数都为 <span class="arithmatex">\(n\)</span> ，因此我们容易得到一个推论：<strong>棋盘每行都允许且只允许放置一个皇后</strong>。</p>
|
||||
<p>也就是说，我们可以采取逐行放置策略：从第一行开始，在每行放置一个皇后，直至最后一行结束。</p>
|
||||
<p>如下图所示，为 <span class="arithmatex">\(4\)</span> 皇后问题的逐行放置过程。受画幅限制，下图仅展开了第一行的其中一个搜索分支，并且将不满足列约束和对角线约束的方案都进行了剪枝。</p>
|
||||
<p><img alt="逐行放置策略" src="../n_queens_problem.assets/n_queens_placing.png" /></p>
|
||||
<p align="center"> Fig. 逐行放置策略 </p>
|
||||
<p align="center"> 图：逐行放置策略 </p>
|
||||
|
||||
<p>本质上看，<strong>逐行放置策略起到了剪枝的作用</strong>，它避免了同一行出现多个皇后的所有搜索分支。</p>
|
||||
<h3 id="_2">列与对角线剪枝<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
|
|
@ -3452,7 +3452,7 @@
|
|||
<p>也就是说，如果两个格子满足 <span class="arithmatex">\(row_1 - col_1 = row_2 - col_2\)</span> ，则它们一定处在同一条主对角线上。利用该规律，我们可以借助一个数组 <code>diag1</code> 来记录每条主对角线上是否有皇后。</p>
|
||||
<p>同理，<strong>次对角线上的所有格子的 <span class="arithmatex">\(row + col\)</span> 是恒定值</strong>。我们可以使用相同方法，借助数组 <code>diag2</code> 来处理次对角线约束。</p>
|
||||
<p><img alt="处理列约束和对角线约束" src="../n_queens_problem.assets/n_queens_cols_diagonals.png" /></p>
|
||||
<p align="center"> Fig. 处理列约束和对角线约束 </p>
|
||||
<p align="center"> 图：处理列约束和对角线约束 </p>
|
||||
|
||||
<h3 id="_3">代码实现<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>请注意，<span class="arithmatex">\(n\)</span> 维方阵中 <span class="arithmatex">\(row - col\)</span> 的范围是 <span class="arithmatex">\([-n + 1, n - 1]\)</span> ，<span class="arithmatex">\(row + col\)</span> 的范围是 <span class="arithmatex">\([0, 2n - 2]\)</span> ，所以主对角线和次对角线的数量都为 <span class="arithmatex">\(2n - 1\)</span> ，即数组 <code>diag1</code> 和 <code>diag2</code> 的长度都为 <span class="arithmatex">\(2n - 1\)</span> 。</p>
|
||||
|
|
|
|||
|
|
@ -3541,7 +3541,7 @@
|
|||
<p>从回溯代码的角度看，候选集合 <code>choices</code> 是输入数组中的所有元素，状态 <code>state</code> 是直至目前已被选择的元素。请注意，每个元素只允许被选择一次，<strong>因此 <code>state</code> 中的所有元素都应该是唯一的</strong>。</p>
|
||||
<p>如下图所示，我们可以将搜索过程展开成一个递归树，树中的每个节点代表当前状态 <code>state</code> 。从根节点开始，经过三轮选择后到达叶节点，每个叶节点都对应一个排列。</p>
|
||||
<p><img alt="全排列的递归树" src="../permutations_problem.assets/permutations_i.png" /></p>
|
||||
<p align="center"> Fig. 全排列的递归树 </p>
|
||||
<p align="center"> 图：全排列的递归树 </p>
|
||||
|
||||
<h3 id="_1">重复选择剪枝<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<p>为了实现每个元素只被选择一次，我们考虑引入一个布尔型数组 <code>selected</code> ，其中 <code>selected[i]</code> 表示 <code>choices[i]</code> 是否已被选择。剪枝的实现原理为：</p>
|
||||
|
|
@ -3551,7 +3551,7 @@
|
|||
</ul>
|
||||
<p>如下图所示，假设我们第一轮选择 1 ，第二轮选择 3 ，第三轮选择 2 ，则需要在第二轮剪掉元素 1 的分支，在第三轮剪掉元素 1, 3 的分支。</p>
|
||||
<p><img alt="全排列剪枝示例" src="../permutations_problem.assets/permutations_i_pruning.png" /></p>
|
||||
<p align="center"> Fig. 全排列剪枝示例 </p>
|
||||
<p align="center"> 图：全排列剪枝示例 </p>
|
||||
|
||||
<p>观察上图发现，该剪枝操作将搜索空间大小从 <span class="arithmatex">\(O(n^n)\)</span> 降低至 <span class="arithmatex">\(O(n!)\)</span> 。</p>
|
||||
<h3 id="_2">代码实现<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
|
|
@ -3962,7 +3962,7 @@
|
|||
<p>假设输入数组为 <span class="arithmatex">\([1, 1, 2]\)</span> 。为了方便区分两个重复元素 <span class="arithmatex">\(1\)</span> ，我们将第二个 <span class="arithmatex">\(1\)</span> 记为 <span class="arithmatex">\(\hat{1}\)</span> 。</p>
|
||||
<p>如下图所示，上述方法生成的排列有一半都是重复的。</p>
|
||||
<p><img alt="重复排列" src="../permutations_problem.assets/permutations_ii.png" /></p>
|
||||
<p align="center"> Fig. 重复排列 </p>
|
||||
<p align="center"> 图：重复排列 </p>
|
||||
|
||||
<p>那么如何去除重复的排列呢？最直接地，考虑借助一个哈希表，直接对排列结果进行去重。然而这样做不够优雅，<strong>因为生成重复排列的搜索分支是没有必要的，应当被提前识别并剪枝</strong>，这样可以进一步提升算法效率。</p>
|
||||
<h3 id="_3">相等元素剪枝<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
|
|
@ -3970,7 +3970,7 @@
|
|||
<p>同理，在第一轮选择 <span class="arithmatex">\(2\)</span> 后，第二轮选择中的 <span class="arithmatex">\(1\)</span> 和 <span class="arithmatex">\(\hat{1}\)</span> 也会产生重复分支，因此也应将第二轮的 <span class="arithmatex">\(\hat{1}\)</span> 剪枝。</p>
|
||||
<p>本质上看，<strong>我们的目标是在某一轮选择中，保证多个相等的元素仅被选择一次</strong>。</p>
|
||||
<p><img alt="重复排列剪枝" src="../permutations_problem.assets/permutations_ii_pruning.png" /></p>
|
||||
<p align="center"> Fig. 重复排列剪枝 </p>
|
||||
<p align="center"> 图：重复排列剪枝 </p>
|
||||
|
||||
<h3 id="_4">代码实现<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>在上一题的代码的基础上，我们考虑在每一轮选择中开启一个哈希表 <code>duplicated</code> ，用于记录该轮中已经尝试过的元素，并将重复元素剪枝。</p>
|
||||
|
|
@ -4360,7 +4360,7 @@
|
|||
</ul>
|
||||
<p>下图展示了两个剪枝条件的生效范围。注意，树中的每个节点代表一个选择，从根节点到叶节点的路径上的各个节点构成一个排列。</p>
|
||||
<p><img alt="两种剪枝条件的作用范围" src="../permutations_problem.assets/permutations_ii_pruning_summary.png" /></p>
|
||||
<p align="center"> Fig. 两种剪枝条件的作用范围 </p>
|
||||
<p align="center"> 图：两种剪枝条件的作用范围 </p>
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -3913,7 +3913,7 @@
|
|||
<p>向以上代码输入数组 <span class="arithmatex">\([3, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> ，输出结果为 <span class="arithmatex">\([3, 3, 3], [4, 5], [5, 4]\)</span> 。<strong>虽然成功找出了所有和为 <span class="arithmatex">\(9\)</span> 的子集，但其中存在重复的子集 <span class="arithmatex">\([4, 5]\)</span> 和 <span class="arithmatex">\([5, 4]\)</span></strong> 。</p>
|
||||
<p>这是因为搜索过程是区分选择顺序的，然而子集不区分选择顺序。如下图所示，先选 <span class="arithmatex">\(4\)</span> 后选 <span class="arithmatex">\(5\)</span> 与先选 <span class="arithmatex">\(5\)</span> 后选 <span class="arithmatex">\(4\)</span> 是两个不同的分支，但两者对应同一个子集。</p>
|
||||
<p><img alt="子集搜索与越界剪枝" src="../subset_sum_problem.assets/subset_sum_i_naive.png" /></p>
|
||||
<p align="center"> Fig. 子集搜索与越界剪枝 </p>
|
||||
<p align="center"> 图：子集搜索与越界剪枝 </p>
|
||||
|
||||
<p>为了去除重复子集，<strong>一种直接的思路是对结果列表进行去重</strong>。但这个方法效率很低，因为：</p>
|
||||
<ul>
|
||||
|
|
@ -3933,7 +3933,7 @@
|
|||
<li>若第一轮选择 <span class="arithmatex">\(5\)</span> ，<strong>则第二轮应该跳过 <span class="arithmatex">\(3\)</span> 和 <span class="arithmatex">\(4\)</span></strong> ，因为子集 <span class="arithmatex">\([5, 3, \cdots]\)</span> 和子集 <span class="arithmatex">\([5, 4, \cdots]\)</span> 和 <code>1.</code> , <code>2.</code> 中生成的子集完全重复。</li>
|
||||
</ol>
|
||||
<p><img alt="不同选择顺序导致的重复子集" src="../subset_sum_problem.assets/subset_sum_i_pruning.png" /></p>
|
||||
<p align="center"> Fig. 不同选择顺序导致的重复子集 </p>
|
||||
<p align="center"> 图：不同选择顺序导致的重复子集 </p>
|
||||
|
||||
<p>总结来看，给定输入数组 <span class="arithmatex">\([x_1, x_2, \cdots, x_n]\)</span> ，设搜索过程中的选择序列为 <span class="arithmatex">\([x_{i_1}, x_{i_2}, \cdots , x_{i_m}]\)</span> ，则该选择序列需要满足 <span class="arithmatex">\(i_1 \leq i_2 \leq \cdots \leq i_m\)</span> ，<strong>不满足该条件的选择序列都会造成重复，应当剪枝</strong>。</p>
|
||||
<h3 id="_3">代码实现<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
|
|
@ -4364,7 +4364,7 @@
|
|||
</div>
|
||||
<p>如下图所示，为将数组 <span class="arithmatex">\([3, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> 输入到以上代码后的整体回溯过程。</p>
|
||||
<p><img alt="子集和 I 回溯过程" src="../subset_sum_problem.assets/subset_sum_i.png" /></p>
|
||||
<p align="center"> Fig. 子集和 I 回溯过程 </p>
|
||||
<p align="center"> 图：子集和 I 回溯过程 </p>
|
||||
|
||||
<h2 id="1332">13.3.2. 考虑重复元素的情况<a class="headerlink" href="#1332" title="Permanent link">¶</a></h2>
|
||||
<div class="admonition question">
|
||||
|
|
@ -4374,7 +4374,7 @@
|
|||
<p>相比于上题，<strong>本题的输入数组可能包含重复元素</strong>，这引入了新的问题。例如，给定数组 <span class="arithmatex">\([4, \hat{4}, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> ，则现有代码的输出结果为 <span class="arithmatex">\([4, 5], [\hat{4}, 5]\)</span> ，出现了重复子集。</p>
|
||||
<p><strong>造成这种重复的原因是相等元素在某轮中被多次选择</strong>。如下图所示，第一轮共有三个选择，其中两个都为 <span class="arithmatex">\(4\)</span> ，会产生两个重复的搜索分支，从而输出重复子集；同理，第二轮的两个 <span class="arithmatex">\(4\)</span> 也会产生重复子集。</p>
|
||||
<p><img alt="相等元素导致的重复子集" src="../subset_sum_problem.assets/subset_sum_ii_repeat.png" /></p>
|
||||
<p align="center"> Fig. 相等元素导致的重复子集 </p>
|
||||
<p align="center"> 图：相等元素导致的重复子集 </p>
|
||||
|
||||
<h3 id="_4">相等元素剪枝<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>为解决此问题，<strong>我们需要限制相等元素在每一轮中只被选择一次</strong>。实现方式比较巧妙：由于数组是已排序的，因此相等元素都是相邻的。这意味着在某轮选择中，若当前元素与其左边元素相等，则说明它已经被选择过，因此直接跳过当前元素。</p>
|
||||
|
|
@ -4856,7 +4856,7 @@
|
|||
</div>
|
||||
<p>下图展示了数组 <span class="arithmatex">\([4, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> 的回溯过程，共包含四种剪枝操作。请你将图示与代码注释相结合，理解整个搜索过程，以及每种剪枝操作是如何工作的。</p>
|
||||
<p><img alt="子集和 II 回溯过程" src="../subset_sum_problem.assets/subset_sum_ii.png" /></p>
|
||||
<p align="center"> Fig. 子集和 II 回溯过程 </p>
|
||||
<p align="center"> 图：子集和 II 回溯过程 </p>
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -3433,10 +3433,10 @@
|
|||
</ol>
|
||||
<p>因此在能够解决问题的前提下，算法效率成为主要的评价维度，包括：</p>
|
||||
<ul>
|
||||
<li><strong>时间效率</strong>，即算法运行速度的快慢。</li>
|
||||
<li><strong>空间效率</strong>，即算法占用内存空间的大小。</li>
|
||||
<li><strong>时间效率</strong>：算法运行速度的快慢。</li>
|
||||
<li><strong>空间效率</strong>：算法占用内存空间的大小。</li>
|
||||
</ul>
|
||||
<p>简而言之，<strong>我们的目标是设计“既快又省”的数据结构与算法</strong>。而有效地评估算法效率至关重要，因为只有了解评价标准，我们才能对比分析各种算法，从而指导算法设计与优化过程。</p>
|
||||
<p>简而言之，<strong>我们的目标是设计“既快又省”的数据结构与算法</strong>。而有效地评估算法效率至关重要，因为只有这样我们才能将各种算法进行对比，从而指导算法设计与优化过程。</p>
|
||||
<p>效率评估方法主要分为两种：实际测试和理论估算。</p>
|
||||
<h2 id="211">2.1.1. 实际测试<a class="headerlink" href="#211" title="Permanent link">¶</a></h2>
|
||||
<p>假设我们现在有算法 <code>A</code> 和算法 <code>B</code> ，它们都能解决同一问题，现在需要对比这两个算法的效率。最直接的方法是找一台计算机，运行这两个算法，并监控记录它们的运行时间和内存占用情况。这种评估方式能够反映真实情况，但也存在较大局限性。</p>
|
||||
|
|
@ -3445,12 +3445,12 @@
|
|||
<h2 id="212">2.1.2. 理论估算<a class="headerlink" href="#212" title="Permanent link">¶</a></h2>
|
||||
<p>由于实际测试具有较大的局限性，我们可以考虑仅通过一些计算来评估算法的效率。这种估算方法被称为「渐近复杂度分析 Asymptotic Complexity Analysis」，简称为「复杂度分析」。</p>
|
||||
<p><strong>复杂度分析评估的是算法运行效率随着输入数据量增多时的增长趋势</strong>。这个定义有些拗口，我们可以将其分为三个重点来理解：</p>
|
||||
<ul>
|
||||
<ol>
|
||||
<li>“算法运行效率”可分为运行时间和占用空间两部分，与之对应地，复杂度可分为「时间复杂度 Time Complexity」和「空间复杂度 Space Complexity」。</li>
|
||||
<li>“随着输入数据量增多时”表示复杂度与输入数据量有关，反映了算法运行效率与输入数据量之间的关系。</li>
|
||||
<li>“随着输入数据量增多时”意味着复杂度反映了算法运行效率与输入数据量之间的关系。</li>
|
||||
<li>“增长趋势”表示复杂度分析关注的是算法时间与空间的增长趋势，而非具体的运行时间或占用空间。</li>
|
||||
</ul>
|
||||
<p><strong>复杂度分析克服了实际测试方法的弊端</strong>。首先，它独立于测试环境，因此分析结果适用于所有运行平台。其次，它可以体现不同数据量下的算法效率，尤其是在大数据量下的算法性能。</p>
|
||||
</ol>
|
||||
<p><strong>复杂度分析克服了实际测试方法的弊端</strong>。首先，它独立于测试环境，分析结果适用于所有运行平台。其次，它可以体现不同数据量下的算法效率，尤其是在大数据量下的算法性能。</p>
|
||||
<p>如果你对复杂度分析的概念仍感到困惑，无需担心，我们会在后续章节详细介绍。</p>
|
||||
<h2 id="213">2.1.3. 复杂度的重要性<a class="headerlink" href="#213" title="Permanent link">¶</a></h2>
|
||||
<p>复杂度分析为我们提供了一把评估算法效率的“标尺”，帮助我们衡量了执行某个算法所需的时间和空间资源，并使我们能够对比不同算法之间的效率。</p>
|
||||
|
|
|
|||
|
|
@ -3539,7 +3539,7 @@
|
|||
</ul>
|
||||
<p>因此在分析一段程序的空间复杂度时，<strong>我们通常统计暂存数据、输出数据、栈帧空间三部分</strong>。</p>
|
||||
<p><img alt="算法使用的相关空间" src="../space_complexity.assets/space_types.png" /></p>
|
||||
<p align="center"> Fig. 算法使用的相关空间 </p>
|
||||
<p align="center"> 图：算法使用的相关空间 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3594,7 +3594,7 @@
|
|||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="w"> </span><span class="sd">"""类"""</span>
|
||||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
|
||||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">x</span> <span class="c1"># 节点值</span>
|
||||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Node</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向下一节点的指针（引用）</span>
|
||||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Node</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向下一节点的引用</span>
|
||||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
|
||||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="k">def</span> <span class="nf">function</span><span class="p">()</span> <span class="o">-></span> <span class="nb">int</span><span class="p">:</span>
|
||||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w"> </span><span class="sd">"""函数"""</span>
|
||||
|
|
@ -4115,7 +4115,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
|||
\end{aligned}
|
||||
\]</div>
|
||||
<p><img alt="空间复杂度的常见类型" src="../space_complexity.assets/space_complexity_common_types.png" /></p>
|
||||
<p align="center"> Fig. 空间复杂度的常见类型 </p>
|
||||
<p align="center"> 图：空间复杂度的常见类型 </p>
|
||||
|
||||
<div class="admonition tip">
|
||||
<p class="admonition-title">Tip</p>
|
||||
|
|
@ -4390,8 +4390,8 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
|||
<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w"> </span><span class="c1">// 常量、变量、对象占用 O(1) 空间</span>
|
||||
<a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a>
|
||||
<a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">10000</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">10000</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w"> </span><span class="c1">// 循环中的变量占用 O(1) 空间</span>
|
||||
<a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
|
||||
|
|
@ -4796,7 +4796,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="递归函数产生的线性阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></p>
|
||||
<p align="center"> Fig. 递归函数产生的线性阶空间复杂度 </p>
|
||||
<p align="center"> 图：递归函数产生的线性阶空间复杂度 </p>
|
||||
|
||||
<h3 id="on2">平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#on2" title="Permanent link">¶</a></h3>
|
||||
<p>平方阶常见于矩阵和图，元素数量与 <span class="arithmatex">\(n\)</span> 成平方关系。</p>
|
||||
|
|
@ -4962,15 +4962,14 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
|||
<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">numMatrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">_</span><span class="p">)</span><span class="w"> </span><span class="o">=></span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
|
||||
<a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w"> </span><span class="c1">// 二维列表占用 O(n^2) 空间</span>
|
||||
<a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">numList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
|
||||
<a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a>
|
||||
<a id="__codelineno-82-8" name="__codelineno-82-8" href="#__codelineno-82-8"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
|
||||
<a id="__codelineno-82-10" name="__codelineno-82-10" href="#__codelineno-82-10"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-82-12" name="__codelineno-82-12" href="#__codelineno-82-12"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-82-13" name="__codelineno-82-13" href="#__codelineno-82-13"></a><span class="w"> </span><span class="n">numList</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">tmp</span><span class="p">);</span>
|
||||
<a id="__codelineno-82-14" name="__codelineno-82-14" href="#__codelineno-82-14"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-82-15" name="__codelineno-82-15" href="#__codelineno-82-15"></a><span class="p">}</span>
|
||||
<a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-82-8" name="__codelineno-82-8" href="#__codelineno-82-8"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
|
||||
<a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-82-10" name="__codelineno-82-10" href="#__codelineno-82-10"></a><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-82-12" name="__codelineno-82-12" href="#__codelineno-82-12"></a><span class="w"> </span><span class="n">numList</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">tmp</span><span class="p">);</span>
|
||||
<a id="__codelineno-82-13" name="__codelineno-82-13" href="#__codelineno-82-13"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-82-14" name="__codelineno-82-14" href="#__codelineno-82-14"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -5132,7 +5131,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="递归函数产生的平方阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></p>
|
||||
<p align="center"> Fig. 递归函数产生的平方阶空间复杂度 </p>
|
||||
<p align="center"> 图：递归函数产生的平方阶空间复杂度 </p>
|
||||
|
||||
<h3 id="o2n">指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#o2n" title="Permanent link">¶</a></h3>
|
||||
<p>指数阶常见于二叉树。高度为 <span class="arithmatex">\(n\)</span> 的「满二叉树」的节点数量为 <span class="arithmatex">\(2^n - 1\)</span> ，占用 <span class="arithmatex">\(O(2^n)\)</span> 空间。</p>
|
||||
|
|
@ -5281,7 +5280,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="满二叉树产生的指数阶空间复杂度" src="../space_complexity.assets/space_complexity_exponential.png" /></p>
|
||||
<p align="center"> Fig. 满二叉树产生的指数阶空间复杂度 </p>
|
||||
<p align="center"> 图：满二叉树产生的指数阶空间复杂度 </p>
|
||||
|
||||
<h3 id="olog-n">对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#olog-n" title="Permanent link">¶</a></h3>
|
||||
<p>对数阶常见于分治算法和数据类型转换等。</p>
|
||||
|
|
|
|||
|
|
@ -3985,7 +3985,7 @@
|
|||
<p>算法 <code>B</code> 中的打印操作需要循环 <span class="arithmatex">\(n\)</span> 次，算法运行时间随着 <span class="arithmatex">\(n\)</span> 增大呈线性增长。此算法的时间复杂度被称为「线性阶」。</p>
|
||||
<p>算法 <code>C</code> 中的打印操作需要循环 <span class="arithmatex">\(1000000\)</span> 次，虽然运行时间很长，但它与输入数据大小 <span class="arithmatex">\(n\)</span> 无关。因此 <code>C</code> 的时间复杂度和 <code>A</code> 相同，仍为「常数阶」。</p>
|
||||
<p><img alt="算法 A, B, C 的时间增长趋势" src="../time_complexity.assets/time_complexity_simple_example.png" /></p>
|
||||
<p align="center"> Fig. 算法 A, B, C 的时间增长趋势 </p>
|
||||
<p align="center"> 图：算法 A, B, C 的时间增长趋势 </p>
|
||||
|
||||
<p>相较于直接统计算法运行时间，时间复杂度分析有哪些特点呢？</p>
|
||||
<p><strong>时间复杂度能够有效评估算法效率</strong>。例如，算法 <code>B</code> 的运行时间呈线性增长，在 <span class="arithmatex">\(n > 1\)</span> 时比算法 <code>A</code> 更慢，在 <span class="arithmatex">\(n > 1000000\)</span> 时比算法 <code>C</code> 更慢。事实上，只要输入数据大小 <span class="arithmatex">\(n\)</span> 足够大，复杂度为“常数阶”的算法一定优于“线性阶”的算法，这正是时间增长趋势所表达的含义。</p>
|
||||
|
|
@ -4151,7 +4151,7 @@ T(n) = O(f(n))
|
|||
$$</p>
|
||||
</div>
|
||||
<p><img alt="函数的渐近上界" src="../time_complexity.assets/asymptotic_upper_bound.png" /></p>
|
||||
<p align="center"> Fig. 函数的渐近上界 </p>
|
||||
<p align="center"> 图：函数的渐近上界 </p>
|
||||
|
||||
<p>也就是说，计算渐近上界就是寻找一个函数 <span class="arithmatex">\(f(n)\)</span> ，使得当 <span class="arithmatex">\(n\)</span> 趋向于无穷大时，<span class="arithmatex">\(T(n)\)</span> 和 <span class="arithmatex">\(f(n)\)</span> 处于相同的增长级别，仅相差一个常数项 <span class="arithmatex">\(c\)</span> 的倍数。</p>
|
||||
<h2 id="223">2.2.3. 推算方法<a class="headerlink" href="#223" title="Permanent link">¶</a></h2>
|
||||
|
|
@ -4409,7 +4409,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
|||
\end{aligned}
|
||||
\]</div>
|
||||
<p><img alt="时间复杂度的常见类型" src="../time_complexity.assets/time_complexity_common_types.png" /></p>
|
||||
<p align="center"> Fig. 时间复杂度的常见类型 </p>
|
||||
<p align="center"> 图：时间复杂度的常见类型 </p>
|
||||
|
||||
<div class="admonition tip">
|
||||
<p class="admonition-title">Tip</p>
|
||||
|
|
@ -5013,7 +5013,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="常数阶、线性阶、平方阶的时间复杂度" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></p>
|
||||
<p align="center"> Fig. 常数阶、线性阶、平方阶的时间复杂度 </p>
|
||||
<p align="center"> 图：常数阶、线性阶、平方阶的时间复杂度 </p>
|
||||
|
||||
<p>以「冒泡排序」为例，外层循环执行 <span class="arithmatex">\(n - 1\)</span> 次，内层循环执行 <span class="arithmatex">\(n-1, n-2, \cdots, 2, 1\)</span> 次，平均为 <span class="arithmatex">\(\frac{n}{2}\)</span> 次，因此时间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
||||
<div class="arithmatex">\[
|
||||
|
|
@ -5477,7 +5477,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="指数阶的时间复杂度" src="../time_complexity.assets/time_complexity_exponential.png" /></p>
|
||||
<p align="center"> Fig. 指数阶的时间复杂度 </p>
|
||||
<p align="center"> 图：指数阶的时间复杂度 </p>
|
||||
|
||||
<p>在实际算法中，指数阶常出现于递归函数。例如以下代码，其递归地一分为二，经过 <span class="arithmatex">\(n\)</span> 次分裂后停止。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="11:12"><input checked="checked" id="__tabbed_11_1" name="__tabbed_11" type="radio" /><input id="__tabbed_11_2" name="__tabbed_11" type="radio" /><input id="__tabbed_11_3" name="__tabbed_11" type="radio" /><input id="__tabbed_11_4" name="__tabbed_11" type="radio" /><input id="__tabbed_11_5" name="__tabbed_11" type="radio" /><input id="__tabbed_11_6" name="__tabbed_11" type="radio" /><input id="__tabbed_11_7" name="__tabbed_11" type="radio" /><input id="__tabbed_11_8" name="__tabbed_11" type="radio" /><input id="__tabbed_11_9" name="__tabbed_11" type="radio" /><input id="__tabbed_11_10" name="__tabbed_11" type="radio" /><input id="__tabbed_11_11" name="__tabbed_11" type="radio" /><input id="__tabbed_11_12" name="__tabbed_11" type="radio" /><div class="tabbed-labels"><label for="__tabbed_11_1">Java</label><label for="__tabbed_11_2">C++</label><label for="__tabbed_11_3">Python</label><label for="__tabbed_11_4">Go</label><label for="__tabbed_11_5">JS</label><label for="__tabbed_11_6">TS</label><label for="__tabbed_11_7">C</label><label for="__tabbed_11_8">C#</label><label for="__tabbed_11_9">Swift</label><label for="__tabbed_11_10">Zig</label><label for="__tabbed_11_11">Dart</label><label for="__tabbed_11_12">Rust</label></div>
|
||||
|
|
@ -5742,7 +5742,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="对数阶的时间复杂度" src="../time_complexity.assets/time_complexity_logarithmic.png" /></p>
|
||||
<p align="center"> Fig. 对数阶的时间复杂度 </p>
|
||||
<p align="center"> 图：对数阶的时间复杂度 </p>
|
||||
|
||||
<p>与指数阶类似，对数阶也常出现于递归函数。以下代码形成了一个高度为 <span class="arithmatex">\(\log_2 n\)</span> 的递归树。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="13:12"><input checked="checked" id="__tabbed_13_1" name="__tabbed_13" type="radio" /><input id="__tabbed_13_2" name="__tabbed_13" type="radio" /><input id="__tabbed_13_3" name="__tabbed_13" type="radio" /><input id="__tabbed_13_4" name="__tabbed_13" type="radio" /><input id="__tabbed_13_5" name="__tabbed_13" type="radio" /><input id="__tabbed_13_6" name="__tabbed_13" type="radio" /><input id="__tabbed_13_7" name="__tabbed_13" type="radio" /><input id="__tabbed_13_8" name="__tabbed_13" type="radio" /><input id="__tabbed_13_9" name="__tabbed_13" type="radio" /><input id="__tabbed_13_10" name="__tabbed_13" type="radio" /><input id="__tabbed_13_11" name="__tabbed_13" type="radio" /><input id="__tabbed_13_12" name="__tabbed_13" type="radio" /><div class="tabbed-labels"><label for="__tabbed_13_1">Java</label><label for="__tabbed_13_2">C++</label><label for="__tabbed_13_3">Python</label><label for="__tabbed_13_4">Go</label><label for="__tabbed_13_5">JS</label><label for="__tabbed_13_6">TS</label><label for="__tabbed_13_7">C</label><label for="__tabbed_13_8">C#</label><label for="__tabbed_13_9">Swift</label><label for="__tabbed_13_10">Zig</label><label for="__tabbed_13_11">Dart</label><label for="__tabbed_13_12">Rust</label></div>
|
||||
|
|
@ -6021,7 +6021,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="线性对数阶的时间复杂度" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></p>
|
||||
<p align="center"> Fig. 线性对数阶的时间复杂度 </p>
|
||||
<p align="center"> 图：线性对数阶的时间复杂度 </p>
|
||||
|
||||
<h3 id="on_1">阶乘阶 <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#on_1" title="Permanent link">¶</a></h3>
|
||||
<p>阶乘阶对应数学上的“全排列”问题。给定 <span class="arithmatex">\(n\)</span> 个互不重复的元素，求其所有可能的排列方案，方案数量为：</p>
|
||||
|
|
@ -6198,7 +6198,7 @@ n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="阶乘阶的时间复杂度" src="../time_complexity.assets/time_complexity_factorial.png" /></p>
|
||||
<p align="center"> Fig. 阶乘阶的时间复杂度 </p>
|
||||
<p align="center"> 图：阶乘阶的时间复杂度 </p>
|
||||
|
||||
<p>请注意，因为 <span class="arithmatex">\(n! > 2^n\)</span> ，所以阶乘阶比指数阶增长地更快，在 <span class="arithmatex">\(n\)</span> 较大时也是不可接受的。</p>
|
||||
<h2 id="225">2.2.5. 最差、最佳、平均时间复杂度<a class="headerlink" href="#225" title="Permanent link">¶</a></h2>
|
||||
|
|
|
|||
|
|
@ -3458,7 +3458,7 @@
|
|||
<h2 id="341-ascii">3.4.1. ASCII 字符集<a class="headerlink" href="#341-ascii" title="Permanent link">¶</a></h2>
|
||||
<p>「ASCII 码」是最早出现的字符集，全称为“美国标准信息交换代码”。它使用 7 位二进制数（即一个字节的低 7 位）表示一个字符，最多能够表示 128 个不同的字符。这包括英文字母的大小写、数字 0-9 、一些标点符号，以及一些控制字符（如换行符和制表符）。</p>
|
||||
<p><img alt="ASCII 码" src="../character_encoding.assets/ascii_table.png" /></p>
|
||||
<p align="center"> Fig. ASCII 码 </p>
|
||||
<p align="center"> 图：ASCII 码 </p>
|
||||
|
||||
<p>然而，<strong>ASCII 码仅能够表示英文</strong>。随着计算机的全球化，诞生了一种能够表示更多语言的字符集「EASCII」。它在 ASCII 的 7 位基础上扩展到 8 位，能够表示 256 个不同的字符。</p>
|
||||
<p>在世界范围内，陆续出现了一批适用于不同地区的 EASCII 字符集。这些字符集的前 128 个字符统一为 ASCII 码，后 128 个字符定义不同，以适应不同语言的需求。</p>
|
||||
|
|
@ -3473,7 +3473,7 @@
|
|||
<p>Unicode 是一种字符集标准，本质上是给每个字符分配一个编号（称为“码点”），<strong>但它并没有规定在计算机中如何存储这些字符码点</strong>。我们不禁会问：当多种长度的 Unicode 码点同时出现在同一个文本中时，系统如何解析字符？例如给定一个长度为 2 字节的编码，系统如何确认它是一个 2 字节的字符还是两个 1 字节的字符？</p>
|
||||
<p>对于以上问题，<strong>一种直接的解决方案是将所有字符存储为等长的编码</strong>。如下图所示，“Hello”中的每个字符占用 1 字节，“算法”中的每个字符占用 2 字节。我们可以通过高位填 0 ，将“Hello 算法”中的所有字符都编码为 2 字节长度。这样系统就可以每隔 2 字节解析一个字符，恢复出这个短语的内容了。</p>
|
||||
<p><img alt="Unicode 编码示例" src="../character_encoding.assets/unicode_hello_algo.png" /></p>
|
||||
<p align="center"> Fig. Unicode 编码示例 </p>
|
||||
<p align="center"> 图：Unicode 编码示例 </p>
|
||||
|
||||
<p>然而 ASCII 码已经向我们证明，编码英文只需要 1 字节。若采用上述方案，英文文本占用空间的大小将会是 ASCII 编码下大小的两倍，非常浪费内存空间。因此，我们需要一种更加高效的 Unicode 编码方法。</p>
|
||||
<h2 id="344-utf-8">3.4.4. UTF-8 编码<a class="headerlink" href="#344-utf-8" title="Permanent link">¶</a></h2>
|
||||
|
|
@ -3487,7 +3487,7 @@
|
|||
<p>但为什么要将其余所有字节的高 2 位都设置为 <span class="arithmatex">\(10\)</span> 呢？实际上，这个 <span class="arithmatex">\(10\)</span> 能够起到校验符的作用。假设系统从一个错误的字节开始解析文本，字节头部的 <span class="arithmatex">\(10\)</span> 能够帮助系统快速的判断出异常。</p>
|
||||
<p>之所以将 <span class="arithmatex">\(10\)</span> 当作校验符，是因为在 UTF-8 编码规则下，不可能有字符的最高两位是 <span class="arithmatex">\(10\)</span> 。这个结论可以用反证法来证明：假设一个字符的最高两位是 <span class="arithmatex">\(10\)</span> ，说明该字符的长度为 <span class="arithmatex">\(1\)</span> ，对应 ASCII 码。而 ASCII 码的最高位应该是 <span class="arithmatex">\(0\)</span> ，与假设矛盾。</p>
|
||||
<p><img alt="UTF-8 编码示例" src="../character_encoding.assets/utf-8_hello_algo.png" /></p>
|
||||
<p align="center"> Fig. UTF-8 编码示例 </p>
|
||||
<p align="center"> 图：UTF-8 编码示例 </p>
|
||||
|
||||
<p>除了 UTF-8 之外，常见的编码方式还包括：</p>
|
||||
<ul>
|
||||
|
|
|
|||
|
|
@ -3421,7 +3421,7 @@
|
|||
<li><strong>非线性数据结构</strong>：树、堆、图、哈希表。</li>
|
||||
</ul>
|
||||
<p><img alt="线性与非线性数据结构" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></p>
|
||||
<p align="center"> Fig. 线性与非线性数据结构 </p>
|
||||
<p align="center"> 图：线性与非线性数据结构 </p>
|
||||
|
||||
<p>非线性数据结构可以进一步被划分为树形结构和网状结构。</p>
|
||||
<ul>
|
||||
|
|
@ -3434,12 +3434,12 @@
|
|||
<p><strong>在算法运行过程中，相关数据都存储在内存中</strong>。下图展示了一个计算机内存条，其中每个黑色方块都包含一块内存空间。我们可以将内存想象成一个巨大的 Excel 表格，其中每个单元格都可以存储一定大小的数据，在算法运行时，所有数据都被存储在这些单元格中。</p>
|
||||
<p><strong>系统通过内存地址来访问目标位置的数据</strong>。计算机根据特定规则为表格中的每个单元格分配编号，确保每个内存空间都有唯一的内存地址。有了这些地址，程序便可以访问内存中的数据。</p>
|
||||
<p><img alt="内存条、内存空间、内存地址" src="../classification_of_data_structure.assets/computer_memory_location.png" /></p>
|
||||
<p align="center"> Fig. 内存条、内存空间、内存地址 </p>
|
||||
<p align="center"> 图：内存条、内存空间、内存地址 </p>
|
||||
|
||||
<p>内存是所有程序的共享资源，当某块内存被某个程序占用时，则无法被其他程序同时使用了。<strong>因此在数据结构与算法的设计中，内存资源是一个重要的考虑因素</strong>。比如，算法所占用的内存峰值不应超过系统剩余空闲内存；如果缺少连续大块的内存空间，那么所选用的数据结构必须能够存储在离散的内存空间内。</p>
|
||||
<p><strong>「物理结构」反映了数据在计算机内存中的存储方式</strong>，可分为连续空间存储（数组）和离散空间存储（链表）。物理结构从底层决定了数据的访问、更新、增删等操作方法，同时在时间效率和空间效率方面呈现出互补的特点。</p>
|
||||
<p><img alt="连续空间存储与离散空间存储" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></p>
|
||||
<p align="center"> Fig. 连续空间存储与离散空间存储 </p>
|
||||
<p align="center"> 图：连续空间存储与离散空间存储 </p>
|
||||
|
||||
<p>值得说明的是，<strong>所有数据结构都是基于数组、链表或二者的组合实现的</strong>。例如，栈和队列既可以使用数组实现，也可以使用链表实现；而哈希表的实现可能同时包含数组和链表。</p>
|
||||
<ul>
|
||||
|
|
|
|||
|
|
@ -3425,7 +3425,7 @@
|
|||
<li><strong>补码</strong>：正数的补码与其原码相同，负数的补码是在其反码的基础上加 <span class="arithmatex">\(1\)</span> 。</li>
|
||||
</ul>
|
||||
<p><img alt="原码、反码与补码之间的相互转换" src="../number_encoding.assets/1s_2s_complement.png" /></p>
|
||||
<p align="center"> Fig. 原码、反码与补码之间的相互转换 </p>
|
||||
<p align="center"> 图：原码、反码与补码之间的相互转换 </p>
|
||||
|
||||
<p>显然「原码」最为直观。但实际上，<strong>数字是以「补码」的形式存储在计算机中的</strong>。这是因为原码存在一些局限性。</p>
|
||||
<p>一方面，<strong>负数的原码不能直接用于运算</strong>。例如，我们在原码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，得到的结果是 <span class="arithmatex">\(-3\)</span> ，这显然是不对的。</p>
|
||||
|
|
@ -3508,7 +3508,7 @@ b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
|
|||
\end{aligned}
|
||||
\]</div>
|
||||
<p><img alt="IEEE 754 标准下的 float 表示方式" src="../number_encoding.assets/ieee_754_float.png" /></p>
|
||||
<p align="center"> Fig. IEEE 754 标准下的 float 表示方式 </p>
|
||||
<p align="center"> 图：IEEE 754 标准下的 float 表示方式 </p>
|
||||
|
||||
<p>给定一个示例数据 <span class="arithmatex">\(\mathrm{S} = 0\)</span> ， <span class="arithmatex">\(\mathrm{E} = 124\)</span> ，<span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span> ，则有：</p>
|
||||
<div class="arithmatex">\[
|
||||
|
|
|
|||
|
|
@ -3438,7 +3438,7 @@
|
|||
</ol>
|
||||
<p>下图展示了在数组中二分查找元素 <span class="arithmatex">\(6\)</span> 的分治过程。</p>
|
||||
<p><img alt="二分查找的分治过程" src="../binary_search_recur.assets/binary_search_recur.png" /></p>
|
||||
<p align="center"> Fig. 二分查找的分治过程 </p>
|
||||
<p align="center"> 图：二分查找的分治过程 </p>
|
||||
|
||||
<p>在实现代码中，我们声明一个递归函数 <code>dfs()</code> 来求解问题 <span class="arithmatex">\(f(i, j)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
|
|
|
|||
|
|
@ -3453,7 +3453,7 @@
|
|||
<p>给定一个二叉树的前序遍历 <code>preorder</code> 和中序遍历 <code>inorder</code> ，请从中构建二叉树，返回二叉树的根节点。</p>
|
||||
</div>
|
||||
<p><img alt="构建二叉树的示例数据" src="../build_binary_tree_problem.assets/build_tree_example.png" /></p>
|
||||
<p align="center"> Fig. 构建二叉树的示例数据 </p>
|
||||
<p align="center"> 图：构建二叉树的示例数据 </p>
|
||||
|
||||
<h3 id="_1">判断是否为分治问题<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<p>原问题定义为从 <code>preorder</code> 和 <code>inorder</code> 构建二叉树。我们首先从分治的角度分析这道题：</p>
|
||||
|
|
@ -3476,7 +3476,7 @@
|
|||
<li>根据 <code>inorder</code> 划分结果，易得左子树和右子树的节点数量分别为 1 和 3 ，从而可将 <code>preorder</code> 划分为 <code>[ 3 | 9 | 2 1 7 ]</code> 。</li>
|
||||
</ol>
|
||||
<p><img alt="在前序和中序遍历中划分子树" src="../build_binary_tree_problem.assets/build_tree_preorder_inorder_division.png" /></p>
|
||||
<p align="center"> Fig. 在前序和中序遍历中划分子树 </p>
|
||||
<p align="center"> 图：在前序和中序遍历中划分子树 </p>
|
||||
|
||||
<h3 id="_3">基于变量描述子树区间<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>根据以上划分方法，<strong>我们已经得到根节点、左子树、右子树在 <code>preorder</code> 和 <code>inorder</code> 中的索引区间</strong>。而为了描述这些索引区间，我们需要借助几个指针变量：</p>
|
||||
|
|
@ -3516,7 +3516,7 @@
|
|||
</div>
|
||||
<p>请注意，右子树根节点索引中的 <span class="arithmatex">\((m-l)\)</span> 的含义是“左子树的节点数量”，建议配合下图理解。</p>
|
||||
<p><img alt="根节点和左右子树的索引区间表示" src="../build_binary_tree_problem.assets/build_tree_division_pointers.png" /></p>
|
||||
<p align="center"> Fig. 根节点和左右子树的索引区间表示 </p>
|
||||
<p align="center"> 图：根节点和左右子树的索引区间表示 </p>
|
||||
|
||||
<h3 id="_4">代码实现<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>为了提升查询 <span class="arithmatex">\(m\)</span> 的效率，我们借助一个哈希表 <code>hmap</code> 来存储数组 <code>inorder</code> 中元素到索引的映射。</p>
|
||||
|
|
@ -3863,6 +3863,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：构建二叉树的递归过程 </p>
|
||||
|
||||
<p>设树的节点数量为 <span class="arithmatex">\(n\)</span> ，初始化每一个节点（执行一个递归函数 <code>dfs()</code> ）使用 <span class="arithmatex">\(O(1)\)</span> 时间。<strong>因此总体时间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> 。</p>
|
||||
<p>哈希表存储 <code>inorder</code> 元素到索引的映射，空间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。最差情况下，即二叉树退化为链表时，递归深度达到 <span class="arithmatex">\(n\)</span> ，使用 <span class="arithmatex">\(O(n)\)</span> 的栈帧空间。<strong>因此总体空间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> 。</p>
|
||||
|
||||
|
|
|
|||
|
|
@ -3485,7 +3485,7 @@
|
|||
<li><strong>治</strong>：从底至顶地将有序的子数组（子问题的解）进行合并，从而得到有序的原数组（原问题的解）。</li>
|
||||
</ol>
|
||||
<p><img alt="归并排序的分治策略" src="../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" /></p>
|
||||
<p align="center"> Fig. 归并排序的分治策略 </p>
|
||||
<p align="center"> 图：归并排序的分治策略 </p>
|
||||
|
||||
<h2 id="1211">12.1.1. 如何判断分治问题<a class="headerlink" href="#1211" title="Permanent link">¶</a></h2>
|
||||
<p>一个问题是否适合使用分治解决，通常可以参考以下几个判断依据：</p>
|
||||
|
|
@ -3509,7 +3509,7 @@
|
|||
O(n + (\frac{n}{2})^2 \times 2 + n) = O(\frac{n^2}{2} + 2n)
|
||||
\]</div>
|
||||
<p><img alt="划分数组前后的冒泡排序" src="../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" /></p>
|
||||
<p align="center"> Fig. 划分数组前后的冒泡排序 </p>
|
||||
<p align="center"> 图：划分数组前后的冒泡排序 </p>
|
||||
|
||||
<p>接下来，我们计算以下不等式，其左边和右边分别为划分前和划分后的操作总数：</p>
|
||||
<div class="arithmatex">\[
|
||||
|
|
@ -3527,7 +3527,7 @@ n(n - 4) & > 0
|
|||
<p>并行优化在多核或多处理器的环境中尤其有效，因为系统可以同时处理多个子问题，更加充分地利用计算资源，从而显著减少总体的运行时间。</p>
|
||||
<p>比如在桶排序中，我们将海量的数据平均分配到各个桶中，则可所有桶的排序任务分散到各个计算单元，完成后再进行结果合并。</p>
|
||||
<p><img alt="桶排序的并行计算" src="../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" /></p>
|
||||
<p align="center"> Fig. 桶排序的并行计算 </p>
|
||||
<p align="center"> 图：桶排序的并行计算 </p>
|
||||
|
||||
<h2 id="1213">12.1.3. 分治常见应用<a class="headerlink" href="#1213" title="Permanent link">¶</a></h2>
|
||||
<p>一方面，分治可以用来解决许多经典算法问题：</p>
|
||||
|
|
|
|||
|
|
@ -3445,7 +3445,7 @@
|
|||
</ol>
|
||||
</div>
|
||||
<p><img alt="汉诺塔问题示例" src="../hanota_problem.assets/hanota_example.png" /></p>
|
||||
<p align="center"> Fig. 汉诺塔问题示例 </p>
|
||||
<p align="center"> 图：汉诺塔问题示例 </p>
|
||||
|
||||
<p><strong>我们将规模为 <span class="arithmatex">\(i\)</span> 的汉诺塔问题记做 <span class="arithmatex">\(f(i)\)</span></strong> 。例如 <span class="arithmatex">\(f(3)\)</span> 代表将 <span class="arithmatex">\(3\)</span> 个圆盘从 <code>A</code> 移动至 <code>C</code> 的汉诺塔问题。</p>
|
||||
<h3 id="_1">考虑基本情况<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
|
|
@ -3460,6 +3460,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：规模为 1 问题的解 </p>
|
||||
|
||||
<p>对于问题 <span class="arithmatex">\(f(2)\)</span> ，即当有两个圆盘时，<strong>由于要时刻满足小圆盘在大圆盘之上，因此需要借助 <code>B</code> 来完成移动</strong>，包括三步：</p>
|
||||
<ol>
|
||||
<li>先将上面的小圆盘从 <code>A</code> 移至 <code>B</code> 。</li>
|
||||
|
|
@ -3483,6 +3485,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：规模为 2 问题的解 </p>
|
||||
|
||||
<h3 id="_2">子问题分解<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p>对于问题 <span class="arithmatex">\(f(3)\)</span> ，即当有三个圆盘时，情况变得稍微复杂了一些。由于已知 <span class="arithmatex">\(f(1)\)</span> 和 <span class="arithmatex">\(f(2)\)</span> 的解，因此可从分治角度思考，<strong>将 <code>A</code> 顶部的两个圆盘看做一个整体</strong>，执行以下步骤：</p>
|
||||
<ol>
|
||||
|
|
@ -3507,6 +3511,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：规模为 3 问题的解 </p>
|
||||
|
||||
<p>本质上看，<strong>我们将问题 <span class="arithmatex">\(f(3)\)</span> 划分为两个子问题 <span class="arithmatex">\(f(2)\)</span> 和子问题 <span class="arithmatex">\(f(1)\)</span></strong> 。按顺序解决这三个子问题之后，原问题随之得到解决。这说明子问题是独立的，而且解是可以合并的。</p>
|
||||
<p>至此，我们可总结出汉诺塔问题的分治策略：将原问题 <span class="arithmatex">\(f(n)\)</span> 划分为两个子问题 <span class="arithmatex">\(f(n-1)\)</span> 和一个子问题 <span class="arithmatex">\(f(1)\)</span> 。子问题的解决顺序为：</p>
|
||||
<ol>
|
||||
|
|
@ -3516,7 +3522,7 @@
|
|||
</ol>
|
||||
<p>对于这两个子问题 <span class="arithmatex">\(f(n-1)\)</span> ，<strong>可以通过相同的方式进行递归划分</strong>，直至达到最小子问题 <span class="arithmatex">\(f(1)\)</span> 。而 <span class="arithmatex">\(f(1)\)</span> 的解是已知的，只需一次移动操作即可。</p>
|
||||
<p><img alt="汉诺塔问题的分治策略" src="../hanota_problem.assets/hanota_divide_and_conquer.png" /></p>
|
||||
<p align="center"> Fig. 汉诺塔问题的分治策略 </p>
|
||||
<p align="center"> 图：汉诺塔问题的分治策略 </p>
|
||||
|
||||
<h3 id="_3">代码实现<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>在代码中，我们声明一个递归函数 <code>dfs(i, src, buf, tar)</code> ，它的作用是将柱 <code>src</code> 顶部的 <span class="arithmatex">\(i\)</span> 个圆盘借助缓冲柱 <code>buf</code> 移动至目标柱 <code>tar</code> 。</p>
|
||||
|
|
@ -3838,7 +3844,7 @@
|
|||
</div>
|
||||
<p>如下图所示，汉诺塔问题形成一个高度为 <span class="arithmatex">\(n\)</span> 的递归树，每个节点代表一个子问题、对应一个开启的 <code>dfs()</code> 函数，<strong>因此时间复杂度为 <span class="arithmatex">\(O(2^n)\)</span> ，空间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> 。</p>
|
||||
<p><img alt="汉诺塔问题的递归树" src="../hanota_problem.assets/hanota_recursive_tree.png" /></p>
|
||||
<p align="center"> Fig. 汉诺塔问题的递归树 </p>
|
||||
<p align="center"> 图：汉诺塔问题的递归树 </p>
|
||||
|
||||
<div class="admonition quote">
|
||||
<p class="admonition-title">Quote</p>
|
||||
|
|
|
|||
|
|
@ -3435,7 +3435,7 @@
|
|||
</div>
|
||||
<p>如下图所示，若第 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> , <span class="arithmatex">\(3\)</span> 阶的代价分别为 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(10\)</span> , <span class="arithmatex">\(1\)</span> ，则从地面爬到第 <span class="arithmatex">\(3\)</span> 阶的最小代价为 <span class="arithmatex">\(2\)</span> 。</p>
|
||||
<p><img alt="爬到第 3 阶的最小代价" src="../dp_problem_features.assets/min_cost_cs_example.png" /></p>
|
||||
<p align="center"> Fig. 爬到第 3 阶的最小代价 </p>
|
||||
<p align="center"> 图：爬到第 3 阶的最小代价 </p>
|
||||
|
||||
<p>设 <span class="arithmatex">\(dp[i]\)</span> 为爬到第 <span class="arithmatex">\(i\)</span> 阶累计付出的代价，由于第 <span class="arithmatex">\(i\)</span> 阶只可能从 <span class="arithmatex">\(i - 1\)</span> 阶或 <span class="arithmatex">\(i - 2\)</span> 阶走来，因此 <span class="arithmatex">\(dp[i]\)</span> 只可能等于 <span class="arithmatex">\(dp[i - 1] + cost[i]\)</span> 或 <span class="arithmatex">\(dp[i - 2] + cost[i]\)</span> 。为了尽可能减少代价，我们应该选择两者中较小的那一个，即：</p>
|
||||
<div class="arithmatex">\[
|
||||
|
|
@ -3631,7 +3631,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="爬楼梯最小代价的动态规划过程" src="../dp_problem_features.assets/min_cost_cs_dp.png" /></p>
|
||||
<p align="center"> Fig. 爬楼梯最小代价的动态规划过程 </p>
|
||||
<p align="center"> 图：爬楼梯最小代价的动态规划过程 </p>
|
||||
|
||||
<p>本题也可以进行状态压缩，将一维压缩至零维，使得空间复杂度从 <span class="arithmatex">\(O(n)\)</span> 降低至 <span class="arithmatex">\(O(1)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
|
|
@ -3803,7 +3803,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
|
|||
</div>
|
||||
<p>例如，爬上第 <span class="arithmatex">\(3\)</span> 阶仅剩 <span class="arithmatex">\(2\)</span> 种可行方案，其中连续三次跳 <span class="arithmatex">\(1\)</span> 阶的方案不满足约束条件，因此被舍弃。</p>
|
||||
<p><img alt="带约束爬到第 3 阶的方案数量" src="../dp_problem_features.assets/climbing_stairs_constraint_example.png" /></p>
|
||||
<p align="center"> Fig. 带约束爬到第 3 阶的方案数量 </p>
|
||||
<p align="center"> 图：带约束爬到第 3 阶的方案数量 </p>
|
||||
|
||||
<p>在该问题中，如果上一轮是跳 <span class="arithmatex">\(1\)</span> 阶上来的，那么下一轮就必须跳 <span class="arithmatex">\(2\)</span> 阶。这意味着，<strong>下一步选择不能由当前状态（当前楼梯阶数）独立决定，还和前一个状态（上轮楼梯阶数）有关</strong>。</p>
|
||||
<p>不难发现，此问题已不满足无后效性，状态转移方程 <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> 也失效了，因为 <span class="arithmatex">\(dp[i-1]\)</span> 代表本轮跳 <span class="arithmatex">\(1\)</span> 阶，但其中包含了许多“上一轮跳 <span class="arithmatex">\(1\)</span> 阶上来的”方案，而为了满足约束，我们就不能将 <span class="arithmatex">\(dp[i-1]\)</span> 直接计入 <span class="arithmatex">\(dp[i]\)</span> 中。</p>
|
||||
|
|
@ -3820,7 +3820,7 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
|
|||
\end{cases}
|
||||
\]</div>
|
||||
<p><img alt="考虑约束下的递推关系" src="../dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png" /></p>
|
||||
<p align="center"> Fig. 考虑约束下的递推关系 </p>
|
||||
<p align="center"> 图：考虑约束下的递推关系 </p>
|
||||
|
||||
<p>最终，返回 <span class="arithmatex">\(dp[n, 1] + dp[n, 2]\)</span> 即可，两者之和代表爬到第 <span class="arithmatex">\(n\)</span> 阶的方案总数。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
|
|
|
|||
|
|
@ -3517,14 +3517,14 @@
|
|||
</div>
|
||||
<p>例如以下示例数据，给定网格的最小路径和为 <span class="arithmatex">\(13\)</span> 。</p>
|
||||
<p><img alt="最小路径和示例数据" src="../dp_solution_pipeline.assets/min_path_sum_example.png" /></p>
|
||||
<p align="center"> Fig. 最小路径和示例数据 </p>
|
||||
<p align="center"> 图：最小路径和示例数据 </p>
|
||||
|
||||
<p><strong>第一步：思考每轮的决策，定义状态，从而得到 <span class="arithmatex">\(dp\)</span> 表</strong></p>
|
||||
<p>本题的每一轮的决策就是从当前格子向下或向右一步。设当前格子的行列索引为 <span class="arithmatex">\([i, j]\)</span> ，则向下或向右走一步后，索引变为 <span class="arithmatex">\([i+1, j]\)</span> 或 <span class="arithmatex">\([i, j+1]\)</span> 。因此，状态应包含行索引和列索引两个变量，记为 <span class="arithmatex">\([i, j]\)</span> 。</p>
|
||||
<p>状态 <span class="arithmatex">\([i, j]\)</span> 对应的子问题为：从起始点 <span class="arithmatex">\([0, 0]\)</span> 走到 <span class="arithmatex">\([i, j]\)</span> 的最小路径和，解记为 <span class="arithmatex">\(dp[i, j]\)</span> 。</p>
|
||||
<p>至此，我们就得到了一个二维 <span class="arithmatex">\(dp\)</span> 矩阵，其尺寸与输入网格 <span class="arithmatex">\(grid\)</span> 相同。</p>
|
||||
<p><img alt="状态定义与 dp 表" src="../dp_solution_pipeline.assets/min_path_sum_solution_step1.png" /></p>
|
||||
<p align="center"> Fig. 状态定义与 dp 表 </p>
|
||||
<p align="center"> 图：状态定义与 dp 表 </p>
|
||||
|
||||
<div class="admonition note">
|
||||
<p class="admonition-title">Note</p>
|
||||
|
|
@ -3538,7 +3538,7 @@
|
|||
dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
|
||||
\]</div>
|
||||
<p><img alt="最优子结构与状态转移方程" src="../dp_solution_pipeline.assets/min_path_sum_solution_step2.png" /></p>
|
||||
<p align="center"> Fig. 最优子结构与状态转移方程 </p>
|
||||
<p align="center"> 图：最优子结构与状态转移方程 </p>
|
||||
|
||||
<div class="admonition note">
|
||||
<p class="admonition-title">Note</p>
|
||||
|
|
@ -3549,7 +3549,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
|
|||
<p>在本题中，处在首行的状态只能向右转移，首列状态只能向下转移，因此首行 <span class="arithmatex">\(i = 0\)</span> 和首列 <span class="arithmatex">\(j = 0\)</span> 是边界条件。</p>
|
||||
<p>每个格子是由其左方格子和上方格子转移而来，因此我们使用采用循环来遍历矩阵，外循环遍历各行、内循环遍历各列。</p>
|
||||
<p><img alt="边界条件与状态转移顺序" src="../dp_solution_pipeline.assets/min_path_sum_solution_step3.png" /></p>
|
||||
<p align="center"> Fig. 边界条件与状态转移顺序 </p>
|
||||
<p align="center"> 图：边界条件与状态转移顺序 </p>
|
||||
|
||||
<div class="admonition note">
|
||||
<p class="admonition-title">Note</p>
|
||||
|
|
@ -3753,7 +3753,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
|
|||
<p>下图给出了以 <span class="arithmatex">\(dp[2, 1]\)</span> 为根节点的递归树，其中包含一些重叠子问题，其数量会随着网格 <code>grid</code> 的尺寸变大而急剧增多。</p>
|
||||
<p>本质上看，造成重叠子问题的原因为：<strong>存在多条路径可以从左上角到达某一单元格</strong>。</p>
|
||||
<p><img alt="暴力搜索递归树" src="../dp_solution_pipeline.assets/min_path_sum_dfs.png" /></p>
|
||||
<p align="center"> Fig. 暴力搜索递归树 </p>
|
||||
<p align="center"> 图：暴力搜索递归树 </p>
|
||||
|
||||
<p>每个状态都有向下和向右两种选择，从左上角走到右下角总共需要 <span class="arithmatex">\(m + n - 2\)</span> 步，所以最差时间复杂度为 <span class="arithmatex">\(O(2^{m + n})\)</span> 。请注意，这种计算方式未考虑临近网格边界的情况，当到达网络边界时只剩下一种选择。因此实际的路径数量会少一些。</p>
|
||||
<h3 id="_2">方法二：记忆化搜索<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
|
|
@ -3992,7 +3992,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
|
|||
</div>
|
||||
<p>引入记忆化后，所有子问题的解只需计算一次，因此时间复杂度取决于状态总数，即网格尺寸 <span class="arithmatex">\(O(nm)\)</span> 。</p>
|
||||
<p><img alt="记忆化搜索递归树" src="../dp_solution_pipeline.assets/min_path_sum_dfs_mem.png" /></p>
|
||||
<p align="center"> Fig. 记忆化搜索递归树 </p>
|
||||
<p align="center"> 图：记忆化搜索递归树 </p>
|
||||
|
||||
<h3 id="_3">方法三：动态规划<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>基于迭代实现动态规划解法。</p>
|
||||
|
|
@ -4279,6 +4279,8 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：最小路径和的动态规划过程 </p>
|
||||
|
||||
<h3 id="_4">状态压缩<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>由于每个格子只与其左边和上边的格子有关，因此我们可以只用一个单行数组来实现 <span class="arithmatex">\(dp\)</span> 表。</p>
|
||||
<p>请注意，因为数组 <code>dp</code> 只能表示一行的状态，所以我们无法提前初始化首列状态，而是在遍历每行中更新它。</p>
|
||||
|
|
|
|||
|
|
@ -3428,13 +3428,13 @@
|
|||
</div>
|
||||
<p>如下图所示，将 <code>kitten</code> 转换为 <code>sitting</code> 需要编辑 3 步，包括 2 次替换操作与 1 次添加操作；将 <code>hello</code> 转换为 <code>algo</code> 需要 3 步，包括 2 次替换操作和 1 次删除操作。</p>
|
||||
<p><img alt="编辑距离的示例数据" src="../edit_distance_problem.assets/edit_distance_example.png" /></p>
|
||||
<p align="center"> Fig. 编辑距离的示例数据 </p>
|
||||
<p align="center"> 图：编辑距离的示例数据 </p>
|
||||
|
||||
<p><strong>编辑距离问题可以很自然地用决策树模型来解释</strong>。字符串对应树节点，一轮决策（一次编辑操作）对应树的一条边。</p>
|
||||
<p>如下图所示，在不限制操作的情况下，每个节点都可以派生出许多条边，每条边对应一种操作，这意味着从 <code>hello</code> 转换到 <code>algo</code> 有许多种可能的路径。</p>
|
||||
<p>从决策树的角度看，本题的目标是求解节点 <code>hello</code> 和节点 <code>algo</code> 之间的最短路径。</p>
|
||||
<p><img alt="基于决策树模型表示编辑距离问题" src="../edit_distance_problem.assets/edit_distance_decision_tree.png" /></p>
|
||||
<p align="center"> Fig. 基于决策树模型表示编辑距离问题 </p>
|
||||
<p align="center"> 图：基于决策树模型表示编辑距离问题 </p>
|
||||
|
||||
<p><strong>第一步：思考每轮的决策，定义状态，从而得到 <span class="arithmatex">\(dp\)</span> 表</strong></p>
|
||||
<p>每一轮的决策是对字符串 <span class="arithmatex">\(s\)</span> 进行一次编辑操作。</p>
|
||||
|
|
@ -3454,7 +3454,7 @@
|
|||
<li>将 <span class="arithmatex">\(s[i-1]\)</span> 替换为 <span class="arithmatex">\(t[j-1]\)</span> ，则剩余子问题 <span class="arithmatex">\(dp[i-1, j-1]\)</span> 。</li>
|
||||
</ol>
|
||||
<p><img alt="编辑距离的状态转移" src="../edit_distance_problem.assets/edit_distance_state_transfer.png" /></p>
|
||||
<p align="center"> Fig. 编辑距离的状态转移 </p>
|
||||
<p align="center"> 图：编辑距离的状态转移 </p>
|
||||
|
||||
<p>根据以上分析，可得最优子结构：<span class="arithmatex">\(dp[i, j]\)</span> 的最少编辑步数等于 <span class="arithmatex">\(dp[i, j-1]\)</span> , <span class="arithmatex">\(dp[i-1, j]\)</span> , <span class="arithmatex">\(dp[i-1, j-1]\)</span> 三者中的最少编辑步数，再加上本次的编辑步数 <span class="arithmatex">\(1\)</span> 。对应的状态转移方程为：</p>
|
||||
<div class="arithmatex">\[
|
||||
|
|
@ -3786,6 +3786,8 @@ dp[i, j] = dp[i-1, j-1]
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：编辑距离的动态规划过程 </p>
|
||||
|
||||
<h3 id="_2">状态压缩<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p>由于 <span class="arithmatex">\(dp[i,j]\)</span> 是由上方 <span class="arithmatex">\(dp[i-1, j]\)</span> 、左方 <span class="arithmatex">\(dp[i, j-1]\)</span> 、左上方状态 <span class="arithmatex">\(dp[i-1, j-1]\)</span> 转移而来，而正序遍历会丢失左上方 <span class="arithmatex">\(dp[i-1, j-1]\)</span> ，倒序遍历无法提前构建 <span class="arithmatex">\(dp[i, j-1]\)</span> ，因此两种遍历顺序都不可取。</p>
|
||||
<p>为此，我们可以使用一个变量 <code>leftup</code> 来暂存左上方的解 <span class="arithmatex">\(dp[i-1, j-1]\)</span> ，从而只需考虑左方和上方的解。此时的情况与完全背包问题相同，可使用正序遍历。</p>
|
||||
|
|
|
|||
|
|
@ -3456,7 +3456,7 @@
|
|||
</div>
|
||||
<p>如下图所示，对于一个 <span class="arithmatex">\(3\)</span> 阶楼梯，共有 <span class="arithmatex">\(3\)</span> 种方案可以爬到楼顶。</p>
|
||||
<p><img alt="爬到第 3 阶的方案数量" src="../intro_to_dynamic_programming.assets/climbing_stairs_example.png" /></p>
|
||||
<p align="center"> Fig. 爬到第 3 阶的方案数量 </p>
|
||||
<p align="center"> 图：爬到第 3 阶的方案数量 </p>
|
||||
|
||||
<p>本题的目标是求解方案数量，<strong>我们可以考虑通过回溯来穷举所有可能性</strong>。具体来说，将爬楼梯想象为一个多轮选择的过程：从地面出发，每轮选择上 <span class="arithmatex">\(1\)</span> 阶或 <span class="arithmatex">\(2\)</span> 阶，每当到达楼梯顶部时就将方案数量加 <span class="arithmatex">\(1\)</span> ，当越过楼梯顶部时就将其剪枝。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
|
|
@ -3579,7 +3579,7 @@
|
|||
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w"> </span><span class="c1">// 当爬到第 n 阶时，方案数量加 1</span>
|
||||
<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w"> </span><span class="c1">// 遍历所有选择</span>
|
||||
<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w"> </span><span class="c1">// 剪枝：不允许越过第 n 阶</span>
|
||||
<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新状态</span>
|
||||
|
|
@ -3610,7 +3610,7 @@
|
|||
<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w"> </span><span class="c1">// 当爬到第 n 阶时，方案数量加 1</span>
|
||||
<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w"> </span><span class="c1">// 遍历所有选择</span>
|
||||
<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w"> </span><span class="c1">// 剪枝：不允许越过第 n 阶</span>
|
||||
<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新状态</span>
|
||||
|
|
@ -3791,7 +3791,7 @@ dp[i] = dp[i-1] + dp[i-2]
|
|||
\]</div>
|
||||
<p>这意味着在爬楼梯问题中，各个子问题之间存在递推关系，<strong>原问题的解可以由子问题的解构建得来</strong>。</p>
|
||||
<p><img alt="方案数量递推关系" src="../intro_to_dynamic_programming.assets/climbing_stairs_state_transfer.png" /></p>
|
||||
<p align="center"> Fig. 方案数量递推关系 </p>
|
||||
<p align="center"> 图：方案数量递推关系 </p>
|
||||
|
||||
<p>我们可以根据递推公式得到暴力搜索解法：</p>
|
||||
<ul>
|
||||
|
|
@ -3995,7 +3995,7 @@ dp[i] = dp[i-1] + dp[i-2]
|
|||
</div>
|
||||
<p>下图展示了暴力搜索形成的递归树。对于问题 <span class="arithmatex">\(dp[n]\)</span> ，其递归树的深度为 <span class="arithmatex">\(n\)</span> ，时间复杂度为 <span class="arithmatex">\(O(2^n)\)</span> 。指数阶属于爆炸式增长，如果我们输入一个比较大的 <span class="arithmatex">\(n\)</span> ，则会陷入漫长的等待之中。</p>
|
||||
<p><img alt="爬楼梯对应递归树" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_tree.png" /></p>
|
||||
<p align="center"> Fig. 爬楼梯对应递归树 </p>
|
||||
<p align="center"> 图：爬楼梯对应递归树 </p>
|
||||
|
||||
<p>观察上图发现，<strong>指数阶的时间复杂度是由于「重叠子问题」导致的</strong>。例如：<span class="arithmatex">\(dp[9]\)</span> 被分解为 <span class="arithmatex">\(dp[8]\)</span> 和 <span class="arithmatex">\(dp[7]\)</span> ，<span class="arithmatex">\(dp[8]\)</span> 被分解为 <span class="arithmatex">\(dp[7]\)</span> 和 <span class="arithmatex">\(dp[6]\)</span> ，两者都包含子问题 <span class="arithmatex">\(dp[7]\)</span> 。</p>
|
||||
<p>以此类推，子问题中包含更小的重叠子问题，子子孙孙无穷尽也。绝大部分计算资源都浪费在这些重叠的问题上。</p>
|
||||
|
|
@ -4282,7 +4282,7 @@ dp[i] = dp[i-1] + dp[i-2]
|
|||
</div>
|
||||
<p>观察下图，<strong>经过记忆化处理后，所有重叠子问题都只需被计算一次，时间复杂度被优化至 <span class="arithmatex">\(O(n)\)</span></strong> ，这是一个巨大的飞跃。</p>
|
||||
<p><img alt="记忆化搜索对应递归树" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png" /></p>
|
||||
<p align="center"> Fig. 记忆化搜索对应递归树 </p>
|
||||
<p align="center"> 图：记忆化搜索对应递归树 </p>
|
||||
|
||||
<h2 id="1413">14.1.3. 方法三：动态规划<a class="headerlink" href="#1413" title="Permanent link">¶</a></h2>
|
||||
<p><strong>记忆化搜索是一种“从顶至底”的方法</strong>：我们从原问题（根节点）开始，递归地将较大子问题分解为较小子问题，直至解已知的最小子问题（叶节点）。之后，通过回溯将子问题的解逐层收集，构建出原问题的解。</p>
|
||||
|
|
@ -4500,7 +4500,7 @@ dp[i] = dp[i-1] + dp[i-2]
|
|||
<li>将递推公式 <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> 称为「状态转移方程」。</li>
|
||||
</ul>
|
||||
<p><img alt="爬楼梯的动态规划过程" src="../intro_to_dynamic_programming.assets/climbing_stairs_dp.png" /></p>
|
||||
<p align="center"> Fig. 爬楼梯的动态规划过程 </p>
|
||||
<p align="center"> 图：爬楼梯的动态规划过程 </p>
|
||||
|
||||
<h2 id="1414">14.1.4. 状态压缩<a class="headerlink" href="#1414" title="Permanent link">¶</a></h2>
|
||||
<p>细心的你可能发现，<strong>由于 <span class="arithmatex">\(dp[i]\)</span> 只与 <span class="arithmatex">\(dp[i-1]\)</span> 和 <span class="arithmatex">\(dp[i-2]\)</span> 有关，因此我们无需使用一个数组 <code>dp</code> 来存储所有子问题的解</strong>，而只需两个变量滚动前进即可。</p>
|
||||
|
|
|
|||
|
|
@ -3456,7 +3456,7 @@
|
|||
</div>
|
||||
<p>请注意，物品编号 <span class="arithmatex">\(i\)</span> 从 <span class="arithmatex">\(1\)</span> 开始计数，数组索引从 <span class="arithmatex">\(0\)</span> 开始计数，因此物品 <span class="arithmatex">\(i\)</span> 对应重量 <span class="arithmatex">\(wgt[i-1]\)</span> 和价值 <span class="arithmatex">\(val[i-1]\)</span> 。</p>
|
||||
<p><img alt="0-1 背包的示例数据" src="../knapsack_problem.assets/knapsack_example.png" /></p>
|
||||
<p align="center"> Fig. 0-1 背包的示例数据 </p>
|
||||
<p align="center"> 图：0-1 背包的示例数据 </p>
|
||||
|
||||
<p>我们可以将 0-1 背包问题看作是一个由 <span class="arithmatex">\(n\)</span> 轮决策组成的过程，每个物体都有不放入和放入两种决策，因此该问题是满足决策树模型的。</p>
|
||||
<p>该问题的目标是求解“在限定背包容量下的最大价值”，因此较大概率是个动态规划问题。</p>
|
||||
|
|
@ -3674,7 +3674,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
|
|||
<p>如下图所示，由于每个物品都会产生不选和选两条搜索分支，因此时间复杂度为 <span class="arithmatex">\(O(2^n)\)</span> 。</p>
|
||||
<p>观察递归树，容易发现其中存在重叠子问题，例如 <span class="arithmatex">\(dp[1, 10]\)</span> 等。而当物品较多、背包容量较大，尤其是相同重量的物品较多时，重叠子问题的数量将会大幅增多。</p>
|
||||
<p><img alt="0-1 背包的暴力搜索递归树" src="../knapsack_problem.assets/knapsack_dfs.png" /></p>
|
||||
<p align="center"> Fig. 0-1 背包的暴力搜索递归树 </p>
|
||||
<p align="center"> 图：0-1 背包的暴力搜索递归树 </p>
|
||||
|
||||
<h3 id="_2">方法二：记忆化搜索<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p>为了保证重叠子问题只被计算一次，我们借助记忆列表 <code>mem</code> 来记录子问题的解，其中 <code>mem[i][c]</code> 对应 <span class="arithmatex">\(dp[i, c]\)</span> 。</p>
|
||||
|
|
@ -3916,7 +3916,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="0-1 背包的记忆化搜索递归树" src="../knapsack_problem.assets/knapsack_dfs_mem.png" /></p>
|
||||
<p align="center"> Fig. 0-1 背包的记忆化搜索递归树 </p>
|
||||
<p align="center"> 图：0-1 背包的记忆化搜索递归树 </p>
|
||||
|
||||
<h3 id="_3">方法三：动态规划<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>动态规划实质上就是在状态转移中填充 <span class="arithmatex">\(dp\)</span> 表的过程，代码如下所示。</p>
|
||||
|
|
@ -4180,6 +4180,8 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：0-1 背包的动态规划过程 </p>
|
||||
|
||||
<h3 id="_4">状态压缩<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>由于每个状态都只与其上一行的状态有关，因此我们可以使用两个数组滚动前进，将空间复杂度从 <span class="arithmatex">\(O(n^2)\)</span> 将低至 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
||||
<p>进一步思考，我们是否可以仅用一个数组实现状态压缩呢？观察可知，每个状态都是由正上方或左上方的格子转移过来的。假设只有一个数组，当开始遍历第 <span class="arithmatex">\(i\)</span> 行时，该数组存储的仍然是第 <span class="arithmatex">\(i-1\)</span> 行的状态。</p>
|
||||
|
|
@ -4210,6 +4212,8 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：0-1 背包的状态压缩后的动态规划过程 </p>
|
||||
|
||||
<p>在代码实现中，我们仅需将数组 <code>dp</code> 的第一维 <span class="arithmatex">\(i\)</span> 直接删除，并且把内循环更改为倒序遍历即可。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JS</label><label for="__tabbed_6_6">TS</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
|
|||
|
|
@ -3561,7 +3561,7 @@
|
|||
<p>给定 <span class="arithmatex">\(n\)</span> 个物品，第 <span class="arithmatex">\(i\)</span> 个物品的重量为 <span class="arithmatex">\(wgt[i-1]\)</span> 、价值为 <span class="arithmatex">\(val[i-1]\)</span> ，和一个容量为 <span class="arithmatex">\(cap\)</span> 的背包。<strong>每个物品可以重复选取</strong>，问在不超过背包容量下能放入物品的最大价值。</p>
|
||||
</div>
|
||||
<p><img alt="完全背包问题的示例数据" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_example.png" /></p>
|
||||
<p align="center"> Fig. 完全背包问题的示例数据 </p>
|
||||
<p align="center"> 图：完全背包问题的示例数据 </p>
|
||||
|
||||
<p>完全背包和 0-1 背包问题非常相似，<strong>区别仅在于不限制物品的选择次数</strong>。</p>
|
||||
<ul>
|
||||
|
|
@ -3817,6 +3817,8 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：完全背包的状态压缩后的动态规划过程 </p>
|
||||
|
||||
<p>代码实现比较简单，仅需将数组 <code>dp</code> 的第一维删除。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -4036,7 +4038,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
|
|||
<p>给定 <span class="arithmatex">\(n\)</span> 种硬币，第 <span class="arithmatex">\(i\)</span> 种硬币的面值为 <span class="arithmatex">\(coins[i - 1]\)</span> ，目标金额为 <span class="arithmatex">\(amt\)</span> ，<strong>每种硬币可以重复选取</strong>，问能够凑出目标金额的最少硬币个数。如果无法凑出目标金额则返回 <span class="arithmatex">\(-1\)</span> 。</p>
|
||||
</div>
|
||||
<p><img alt="零钱兑换问题的示例数据" src="../unbounded_knapsack_problem.assets/coin_change_example.png" /></p>
|
||||
<p align="center"> Fig. 零钱兑换问题的示例数据 </p>
|
||||
<p align="center"> 图：零钱兑换问题的示例数据 </p>
|
||||
|
||||
<p><strong>零钱兑换可以看作是完全背包的一种特殊情况</strong>，两者具有以下联系与不同点：</p>
|
||||
<ul>
|
||||
|
|
@ -4377,6 +4379,8 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：零钱兑换问题的动态规划过程 </p>
|
||||
|
||||
<h3 id="_4">状态压缩<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>零钱兑换的状态压缩的处理方式和完全背包一致。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JS</label><label for="__tabbed_6_6">TS</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
|
|
@ -4628,7 +4632,7 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
|
|||
<p>给定 <span class="arithmatex">\(n\)</span> 种硬币，第 <span class="arithmatex">\(i\)</span> 种硬币的面值为 <span class="arithmatex">\(coins[i - 1]\)</span> ，目标金额为 <span class="arithmatex">\(amt\)</span> ，每种硬币可以重复选取，<strong>问在凑出目标金额的硬币组合数量</strong>。</p>
|
||||
</div>
|
||||
<p><img alt="零钱兑换问题 II 的示例数据" src="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" /></p>
|
||||
<p align="center"> Fig. 零钱兑换问题 II 的示例数据 </p>
|
||||
<p align="center"> 图：零钱兑换问题 II 的示例数据 </p>
|
||||
|
||||
<p>相比于上一题，本题目标是组合数量，因此子问题变为：<strong>前 <span class="arithmatex">\(i\)</span> 种硬币能够凑出金额 <span class="arithmatex">\(a\)</span> 的组合数量</strong>。而 <span class="arithmatex">\(dp\)</span> 表仍然是尺寸为 <span class="arithmatex">\((n+1) \times (amt + 1)\)</span> 的二维矩阵。</p>
|
||||
<p>当前状态的组合数量等于不选当前硬币与选当前硬币这两种决策的组合数量之和。状态转移方程为：</p>
|
||||
|
|
|
|||
|
|
@ -3489,7 +3489,7 @@ G & = \{ V, E \} \newline
|
|||
\end{aligned}
|
||||
\]</div>
|
||||
<p><img alt="链表、树、图之间的关系" src="../graph.assets/linkedlist_tree_graph.png" /></p>
|
||||
<p align="center"> Fig. 链表、树、图之间的关系 </p>
|
||||
<p align="center"> 图：链表、树、图之间的关系 </p>
|
||||
|
||||
<p>那么，图与其他数据结构的关系是什么？如果我们把「顶点」看作节点，把「边」看作连接各个节点的指针，则可将「图」看作是一种从「链表」拓展而来的数据结构。<strong>相较于线性关系（链表）和分治关系（树），网络关系（图）的自由度更高，从而更为复杂</strong>。</p>
|
||||
<h2 id="911">9.1.1. 图常见类型<a class="headerlink" href="#911" title="Permanent link">¶</a></h2>
|
||||
|
|
@ -3499,7 +3499,7 @@ G & = \{ V, E \} \newline
|
|||
<li>在有向图中，边具有方向性，即 <span class="arithmatex">\(A \rightarrow B\)</span> 和 <span class="arithmatex">\(A \leftarrow B\)</span> 两个方向的边是相互独立的，例如微博或抖音上的“关注”与“被关注”关系。</li>
|
||||
</ul>
|
||||
<p><img alt="有向图与无向图" src="../graph.assets/directed_graph.png" /></p>
|
||||
<p align="center"> Fig. 有向图与无向图 </p>
|
||||
<p align="center"> 图：有向图与无向图 </p>
|
||||
|
||||
<p>根据所有顶点是否连通，可分为「连通图 Connected Graph」和「非连通图 Disconnected Graph」。</p>
|
||||
<ul>
|
||||
|
|
@ -3507,11 +3507,11 @@ G & = \{ V, E \} \newline
|
|||
<li>对于非连通图，从某个顶点出发，至少有一个顶点无法到达。</li>
|
||||
</ul>
|
||||
<p><img alt="连通图与非连通图" src="../graph.assets/connected_graph.png" /></p>
|
||||
<p align="center"> Fig. 连通图与非连通图 </p>
|
||||
<p align="center"> 图：连通图与非连通图 </p>
|
||||
|
||||
<p>我们还可以为边添加“权重”变量，从而得到「有权图 Weighted Graph」。例如，在王者荣耀等手游中，系统会根据共同游戏时间来计算玩家之间的“亲密度”，这种亲密度网络就可以用有权图来表示。</p>
|
||||
<p><img alt="有权图与无权图" src="../graph.assets/weighted_graph.png" /></p>
|
||||
<p align="center"> Fig. 有权图与无权图 </p>
|
||||
<p align="center"> 图：有权图与无权图 </p>
|
||||
|
||||
<h2 id="912">9.1.2. 图常用术语<a class="headerlink" href="#912" title="Permanent link">¶</a></h2>
|
||||
<ul>
|
||||
|
|
@ -3525,7 +3525,7 @@ G & = \{ V, E \} \newline
|
|||
<p>设图的顶点数量为 <span class="arithmatex">\(n\)</span> ，「邻接矩阵 Adjacency Matrix」使用一个 <span class="arithmatex">\(n \times n\)</span> 大小的矩阵来表示图，每一行（列）代表一个顶点，矩阵元素代表边，用 <span class="arithmatex">\(1\)</span> 或 <span class="arithmatex">\(0\)</span> 表示两个顶点之间是否存在边。</p>
|
||||
<p>如下图所示，设邻接矩阵为 <span class="arithmatex">\(M\)</span> 、顶点列表为 <span class="arithmatex">\(V\)</span> ，那么矩阵元素 <span class="arithmatex">\(M[i][j] = 1\)</span> 表示顶点 <span class="arithmatex">\(V[i]\)</span> 到顶点 <span class="arithmatex">\(V[j]\)</span> 之间存在边，反之 <span class="arithmatex">\(M[i][j] = 0\)</span> 表示两顶点之间无边。</p>
|
||||
<p><img alt="图的邻接矩阵表示" src="../graph.assets/adjacency_matrix.png" /></p>
|
||||
<p align="center"> Fig. 图的邻接矩阵表示 </p>
|
||||
<p align="center"> 图：图的邻接矩阵表示 </p>
|
||||
|
||||
<p>邻接矩阵具有以下特性：</p>
|
||||
<ul>
|
||||
|
|
@ -3537,7 +3537,7 @@ G & = \{ V, E \} \newline
|
|||
<h3 id="_2">邻接表<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p>「邻接表 Adjacency List」使用 <span class="arithmatex">\(n\)</span> 个链表来表示图，链表节点表示顶点。第 <span class="arithmatex">\(i\)</span> 条链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（即与该顶点相连的顶点）。</p>
|
||||
<p><img alt="图的邻接表表示" src="../graph.assets/adjacency_list.png" /></p>
|
||||
<p align="center"> Fig. 图的邻接表表示 </p>
|
||||
<p align="center"> 图：图的邻接表表示 </p>
|
||||
|
||||
<p>邻接表仅存储实际存在的边，而边的总数通常远小于 <span class="arithmatex">\(n^2\)</span> ，因此它更加节省空间。然而，在邻接表中需要通过遍历链表来查找边，因此其时间效率不如邻接矩阵。</p>
|
||||
<p>观察上图可发现，<strong>邻接表结构与哈希表中的「链地址法」非常相似，因此我们也可以采用类似方法来优化效率</strong>。例如，当链表较长时，可以将链表转化为 AVL 树或红黑树，从而将时间效率从 <span class="arithmatex">\(O(n)\)</span> 优化至 <span class="arithmatex">\(O(\log n)\)</span> ，还可以通过中序遍历获取有序序列；此外，还可以将链表转换为哈希表，将时间复杂度降低至 <span class="arithmatex">\(O(1)\)</span> 。</p>
|
||||
|
|
|
|||
|
|
@ -3454,6 +3454,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：邻接矩阵的初始化、增删边、增删顶点 </p>
|
||||
|
||||
<p>以下是基于邻接矩阵表示图的实现代码。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -4553,6 +4555,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：邻接表的初始化、增删边、增删顶点 </p>
|
||||
|
||||
<p>以下是基于邻接表实现图的代码示例。细心的同学可能注意到，<strong>我们在邻接表中使用 <code>Vertex</code> 节点类来表示顶点</strong>，这样做的原因有：</p>
|
||||
<ul>
|
||||
<li>如果我们选择通过顶点值来区分不同顶点，那么值重复的顶点将无法被区分。</li>
|
||||
|
|
@ -4931,7 +4935,7 @@
|
|||
<a id="__codelineno-16-62" name="__codelineno-16-62" href="#__codelineno-16-62"></a><span class="w"> </span><span class="c1">// 在邻接表中删除顶点 vet 对应的链表</span>
|
||||
<a id="__codelineno-16-63" name="__codelineno-16-63" href="#__codelineno-16-63"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="ow">delete</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span>
|
||||
<a id="__codelineno-16-64" name="__codelineno-16-64" href="#__codelineno-16-64"></a><span class="w"> </span><span class="c1">// 遍历其他顶点的链表，删除所有包含 vet 的边</span>
|
||||
<a id="__codelineno-16-65" name="__codelineno-16-65" href="#__codelineno-16-65"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">set</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-16-65" name="__codelineno-16-65" href="#__codelineno-16-65"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">set</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-16-66" name="__codelineno-16-66" href="#__codelineno-16-66"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">set</span><span class="p">.</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span>
|
||||
<a id="__codelineno-16-67" name="__codelineno-16-67" href="#__codelineno-16-67"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-16-68" name="__codelineno-16-68" href="#__codelineno-16-68"></a><span class="w"> </span><span class="nx">set</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
|
||||
|
|
@ -5018,7 +5022,7 @@
|
|||
<a id="__codelineno-17-62" name="__codelineno-17-62" href="#__codelineno-17-62"></a><span class="w"> </span><span class="c1">// 在邻接表中删除顶点 vet 对应的链表</span>
|
||||
<a id="__codelineno-17-63" name="__codelineno-17-63" href="#__codelineno-17-63"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="ow">delete</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span>
|
||||
<a id="__codelineno-17-64" name="__codelineno-17-64" href="#__codelineno-17-64"></a><span class="w"> </span><span class="c1">// 遍历其他顶点的链表，删除所有包含 vet 的边</span>
|
||||
<a id="__codelineno-17-65" name="__codelineno-17-65" href="#__codelineno-17-65"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">set</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-17-65" name="__codelineno-17-65" href="#__codelineno-17-65"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">set</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-17-66" name="__codelineno-17-66" href="#__codelineno-17-66"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">set</span><span class="p">.</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span>
|
||||
<a id="__codelineno-17-67" name="__codelineno-17-67" href="#__codelineno-17-67"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-17-68" name="__codelineno-17-68" href="#__codelineno-17-68"></a><span class="w"> </span><span class="nx">set</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
|
||||
|
|
|
|||
|
|
@ -3501,7 +3501,7 @@
|
|||
<h2 id="931">9.3.1. 广度优先遍历<a class="headerlink" href="#931" title="Permanent link">¶</a></h2>
|
||||
<p><strong>广度优先遍历是一种由近及远的遍历方式，从距离最近的顶点开始访问，并一层层向外扩张</strong>。具体来说，从某个顶点出发，先遍历该顶点的所有邻接顶点，然后遍历下一个顶点的所有邻接顶点，以此类推，直至所有顶点访问完毕。</p>
|
||||
<p><img alt="图的广度优先遍历" src="../graph_traversal.assets/graph_bfs.png" /></p>
|
||||
<p align="center"> Fig. 图的广度优先遍历 </p>
|
||||
<p align="center"> 图：图的广度优先遍历 </p>
|
||||
|
||||
<h3 id="_1">算法实现<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<p>BFS 通常借助「队列」来实现。队列具有“先入先出”的性质，这与 BFS 的“由近及远”的思想异曲同工。</p>
|
||||
|
|
@ -3891,6 +3891,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：图的广度优先遍历步骤 </p>
|
||||
|
||||
<div class="admonition question">
|
||||
<p class="admonition-title">广度优先遍历的序列是否唯一？</p>
|
||||
<p>不唯一。广度优先遍历只要求按“由近及远”的顺序遍历，<strong>而多个相同距离的顶点的遍历顺序是允许被任意打乱的</strong>。以上图为例，顶点 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(3\)</span> 的访问顺序可以交换、顶点 <span class="arithmatex">\(2\)</span> , <span class="arithmatex">\(4\)</span> , <span class="arithmatex">\(6\)</span> 的访问顺序也可以任意交换。</p>
|
||||
|
|
@ -3901,7 +3903,7 @@
|
|||
<h2 id="932">9.3.2. 深度优先遍历<a class="headerlink" href="#932" title="Permanent link">¶</a></h2>
|
||||
<p><strong>深度优先遍历是一种优先走到底、无路可走再回头的遍历方式</strong>。具体地，从某个顶点出发，访问当前顶点的某个邻接顶点，直到走到尽头时返回，再继续走到尽头并返回，以此类推，直至所有顶点遍历完成。</p>
|
||||
<p><img alt="图的深度优先遍历" src="../graph_traversal.assets/graph_dfs.png" /></p>
|
||||
<p align="center"> Fig. 图的深度优先遍历 </p>
|
||||
<p align="center"> 图：图的深度优先遍历 </p>
|
||||
|
||||
<h3 id="_3">算法实现<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>这种“走到尽头 + 回溯”的算法形式通常基于递归来实现。与 BFS 类似，在 DFS 中我们也需要借助一个哈希表 <code>visited</code> 来记录已被访问的顶点，以避免重复访问顶点。</p>
|
||||
|
|
@ -4274,6 +4276,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：图的深度优先遍历步骤 </p>
|
||||
|
||||
<div class="admonition question">
|
||||
<p class="admonition-title">深度优先遍历的序列是否唯一？</p>
|
||||
<p>与广度优先遍历类似，深度优先遍历序列的顺序也不是唯一的。给定某顶点，先往哪个方向探索都可以，即邻接顶点的顺序可以任意打乱，都是深度优先遍历。</p>
|
||||
|
|
|
|||
|
|
@ -3440,7 +3440,7 @@
|
|||
<p>给定 <span class="arithmatex">\(n\)</span> 个物品，第 <span class="arithmatex">\(i\)</span> 个物品的重量为 <span class="arithmatex">\(wgt[i-1]\)</span> 、价值为 <span class="arithmatex">\(val[i-1]\)</span> ，和一个容量为 <span class="arithmatex">\(cap\)</span> 的背包。每个物品只能选择一次，<strong>但可以选择物品的一部分，价值根据选择的重量比例计算</strong>，问在不超过背包容量下背包中物品的最大价值。</p>
|
||||
</div>
|
||||
<p><img alt="分数背包问题的示例数据" src="../fractional_knapsack_problem.assets/fractional_knapsack_example.png" /></p>
|
||||
<p align="center"> Fig. 分数背包问题的示例数据 </p>
|
||||
<p align="center"> 图：分数背包问题的示例数据 </p>
|
||||
|
||||
<p>本题和 0-1 背包整体上非常相似，状态包含当前物品 <span class="arithmatex">\(i\)</span> 和容量 <span class="arithmatex">\(c\)</span> ，目标是求不超过背包容量下的最大价值。</p>
|
||||
<p>不同点在于，本题允许只选择物品的一部分，<strong>这意味着可以对物品任意地进行切分，并按照重量比例来计算物品价值</strong>，因此有：</p>
|
||||
|
|
@ -3449,7 +3449,7 @@
|
|||
<li>假设放入一部分物品 <span class="arithmatex">\(i\)</span> ，重量为 <span class="arithmatex">\(w\)</span> ，则背包增加的价值为 <span class="arithmatex">\(w \times val[i-1] / wgt[i-1]\)</span> 。</li>
|
||||
</ol>
|
||||
<p><img alt="物品在单位重量下的价值" src="../fractional_knapsack_problem.assets/fractional_knapsack_unit_value.png" /></p>
|
||||
<p align="center"> Fig. 物品在单位重量下的价值 </p>
|
||||
<p align="center"> 图：物品在单位重量下的价值 </p>
|
||||
|
||||
<h3 id="_1">贪心策略确定<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<p>最大化背包内物品总价值，<strong>本质上是要最大化单位重量下的物品价值</strong>。由此便可推出本题的贪心策略：</p>
|
||||
|
|
@ -3459,7 +3459,7 @@
|
|||
<li>若剩余背包容量不足，则使用当前物品的一部分填满背包即可。</li>
|
||||
</ol>
|
||||
<p><img alt="分数背包的贪心策略" src="../fractional_knapsack_problem.assets/fractional_knapsack_greedy_strategy.png" /></p>
|
||||
<p align="center"> Fig. 分数背包的贪心策略 </p>
|
||||
<p align="center"> 图：分数背包的贪心策略 </p>
|
||||
|
||||
<h3 id="_2">代码实现<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p>我们建立了一个物品类 <code>Item</code> ，以便将物品按照单位价值进行排序。循环进行贪心选择，当背包已满时跳出并返回解。</p>
|
||||
|
|
@ -3765,7 +3765,7 @@
|
|||
<p>如下图所示，如果将物品重量和物品单位价值分别看作一个 2D 图表的横轴和纵轴，则分数背包问题可被转化为“求在有限横轴区间下的最大围成面积”。</p>
|
||||
<p>通过这个类比，我们可以从几何角度理解贪心策略的有效性。</p>
|
||||
<p><img alt="分数背包问题的几何表示" src="../fractional_knapsack_problem.assets/fractional_knapsack_area_chart.png" /></p>
|
||||
<p align="center"> Fig. 分数背包问题的几何表示 </p>
|
||||
<p align="center"> 图：分数背包问题的几何表示 </p>
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -3461,7 +3461,7 @@
|
|||
</div>
|
||||
<p>这道题的贪心策略在生活中很常见：给定目标金额，<strong>我们贪心地选择不大于且最接近它的硬币</strong>，不断循环该步骤，直至凑出目标金额为止。</p>
|
||||
<p><img alt="零钱兑换的贪心策略" src="../greedy_algorithm.assets/coin_change_greedy_strategy.png" /></p>
|
||||
<p align="center"> Fig. 零钱兑换的贪心策略 </p>
|
||||
<p align="center"> 图：零钱兑换的贪心策略 </p>
|
||||
|
||||
<p>实现代码如下所示。你可能会不由地发出感叹：So Clean ！贪心算法仅用十行代码就解决了零钱兑换问题。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
|
|
@ -3648,7 +3648,7 @@
|
|||
<li><strong>反例 <span class="arithmatex">\(coins = [1, 49, 50]\)</span></strong>：假设 <span class="arithmatex">\(amt = 98\)</span> ，贪心算法只能找到 <span class="arithmatex">\(50 + 1 \times 48\)</span> 的兑换组合，共计 <span class="arithmatex">\(49\)</span> 枚硬币，但动态规划可以找到最优解 <span class="arithmatex">\(49 + 49\)</span> ，仅需 <span class="arithmatex">\(2\)</span> 枚硬币。</li>
|
||||
</ul>
|
||||
<p><img alt="贪心无法找出最优解的示例" src="../greedy_algorithm.assets/coin_change_greedy_vs_dp.png" /></p>
|
||||
<p align="center"> Fig. 贪心无法找出最优解的示例 </p>
|
||||
<p align="center"> 图：贪心无法找出最优解的示例 </p>
|
||||
|
||||
<p>也就是说，对于零钱兑换问题，贪心算法无法保证找到全局最优解，并且有可能找到非常差的解。它更适合用动态规划解决。</p>
|
||||
<p>一般情况下，贪心算法适用于以下两类问题：</p>
|
||||
|
|
|
|||
|
|
@ -3441,7 +3441,7 @@
|
|||
<p>请在数组中选择两个隔板，使得组成的容器的容量最大，返回最大容量。</p>
|
||||
</div>
|
||||
<p><img alt="最大容量问题的示例数据" src="../max_capacity_problem.assets/max_capacity_example.png" /></p>
|
||||
<p align="center"> Fig. 最大容量问题的示例数据 </p>
|
||||
<p align="center"> 图：最大容量问题的示例数据 </p>
|
||||
|
||||
<p>容器由任意两个隔板围成，<strong>因此本题的状态为两个隔板的索引，记为 <span class="arithmatex">\([i, j]\)</span></strong> 。</p>
|
||||
<p>根据题意，容量等于高度乘以宽度，其中高度由短板决定，宽度是两隔板的索引之差。设容量为 <span class="arithmatex">\(cap[i, j]\)</span> ，则可得计算公式：</p>
|
||||
|
|
@ -3452,7 +3452,7 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
|
|||
<h3 id="_1">贪心策略确定<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<p>这道题还有更高效率的解法。如下图所示，现选取一个状态 <span class="arithmatex">\([i, j]\)</span> ，其满足索引 <span class="arithmatex">\(i < j\)</span> 且高度 <span class="arithmatex">\(ht[i] < ht[j]\)</span> ，即 <span class="arithmatex">\(i\)</span> 为短板、 <span class="arithmatex">\(j\)</span> 为长板。</p>
|
||||
<p><img alt="初始状态" src="../max_capacity_problem.assets/max_capacity_initial_state.png" /></p>
|
||||
<p align="center"> Fig. 初始状态 </p>
|
||||
<p align="center"> 图：初始状态 </p>
|
||||
|
||||
<p>我们发现，<strong>如果此时将长板 <span class="arithmatex">\(j\)</span> 向短板 <span class="arithmatex">\(i\)</span> 靠近，则容量一定变小</strong>。这是因为在移动长板 <span class="arithmatex">\(j\)</span> 后：</p>
|
||||
<ul>
|
||||
|
|
@ -3460,11 +3460,11 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
|
|||
<li>高度由短板决定，因此高度只可能不变（ <span class="arithmatex">\(i\)</span> 仍为短板）或变小（移动后的 <span class="arithmatex">\(j\)</span> 成为短板）。</li>
|
||||
</ul>
|
||||
<p><img alt="向内移动长板后的状态" src="../max_capacity_problem.assets/max_capacity_moving_long_board.png" /></p>
|
||||
<p align="center"> Fig. 向内移动长板后的状态 </p>
|
||||
<p align="center"> 图：向内移动长板后的状态 </p>
|
||||
|
||||
<p>反向思考，<strong>我们只有向内收缩短板 <span class="arithmatex">\(i\)</span> ，才有可能使容量变大</strong>。因为虽然宽度一定变小，<strong>但高度可能会变大</strong>（移动后的短板 <span class="arithmatex">\(i\)</span> 可能会变长）。</p>
|
||||
<p><img alt="向内移动长板后的状态" src="../max_capacity_problem.assets/max_capacity_moving_short_board.png" /></p>
|
||||
<p align="center"> Fig. 向内移动长板后的状态 </p>
|
||||
<p align="center"> 图：向内移动长板后的状态 </p>
|
||||
|
||||
<p>由此便可推出本题的贪心策略：</p>
|
||||
<ol>
|
||||
|
|
@ -3504,6 +3504,8 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：最大容量问题的贪心过程 </p>
|
||||
|
||||
<h3 id="_2">代码实现<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p>代码循环最多 <span class="arithmatex">\(n\)</span> 轮，<strong>因此时间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> 。</p>
|
||||
<p>变量 <span class="arithmatex">\(i\)</span> , <span class="arithmatex">\(j\)</span> , <span class="arithmatex">\(res\)</span> 使用常数大小额外空间，<strong>因此空间复杂度为 <span class="arithmatex">\(O(1)\)</span></strong> 。</p>
|
||||
|
|
@ -3697,7 +3699,7 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
|
|||
cap[i, i+1], cap[i, i+2], \cdots, cap[i, j-2], cap[i, j-1]
|
||||
\]</div>
|
||||
<p><img alt="移动短板导致被跳过的状态" src="../max_capacity_problem.assets/max_capacity_skipped_states.png" /></p>
|
||||
<p align="center"> Fig. 移动短板导致被跳过的状态 </p>
|
||||
<p align="center"> 图：移动短板导致被跳过的状态 </p>
|
||||
|
||||
<p>观察发现，<strong>这些被跳过的状态实际上就是将长板 <span class="arithmatex">\(j\)</span> 向内移动的所有状态</strong>。而在第二步中，我们已经证明内移长板一定会导致容量变小。也就是说，被跳过的状态都不可能是最优解，<strong>跳过它们不会导致错过最优解</strong>。</p>
|
||||
<p>以上的分析说明，<strong>移动短板的操作是“安全”的</strong>，贪心策略是有效的。</p>
|
||||
|
|
|
|||
|
|
@ -3439,7 +3439,7 @@
|
|||
<p>给定一个正整数 <span class="arithmatex">\(n\)</span> ，将其切分为至少两个正整数的和，求切分后所有整数的乘积最大是多少。</p>
|
||||
</div>
|
||||
<p><img alt="最大切分乘积的问题定义" src="../max_product_cutting_problem.assets/max_product_cutting_definition.png" /></p>
|
||||
<p align="center"> Fig. 最大切分乘积的问题定义 </p>
|
||||
<p align="center"> 图：最大切分乘积的问题定义 </p>
|
||||
|
||||
<p>假设我们将 <span class="arithmatex">\(n\)</span> 切分为 <span class="arithmatex">\(m\)</span> 个整数因子，其中第 <span class="arithmatex">\(i\)</span> 个因子记为 <span class="arithmatex">\(n_i\)</span> ，即</p>
|
||||
<div class="arithmatex">\[
|
||||
|
|
@ -3462,13 +3462,13 @@ n & \geq 4
|
|||
<p>我们发现当 <span class="arithmatex">\(n \geq 4\)</span> 时，切分出一个 <span class="arithmatex">\(2\)</span> 后乘积会变大，<strong>这说明大于等于 <span class="arithmatex">\(4\)</span> 的整数都应该被切分</strong>。</p>
|
||||
<p><strong>贪心策略一</strong>：如果切分方案中包含 <span class="arithmatex">\(\geq 4\)</span> 的因子，那么它就应该被继续切分。最终的切分方案只应出现 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> , <span class="arithmatex">\(3\)</span> 这三种因子。</p>
|
||||
<p><img alt="切分导致乘积变大" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_infer1.png" /></p>
|
||||
<p align="center"> Fig. 切分导致乘积变大 </p>
|
||||
<p align="center"> 图：切分导致乘积变大 </p>
|
||||
|
||||
<p>接下来思考哪个因子是最优的。在 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> , <span class="arithmatex">\(3\)</span> 这三个因子中，显然 <span class="arithmatex">\(1\)</span> 是最差的，因为 <span class="arithmatex">\(1 \times (n-1) < n\)</span> 恒成立，即切分出 <span class="arithmatex">\(1\)</span> 反而会导致乘积减小。</p>
|
||||
<p>我们发现，当 <span class="arithmatex">\(n = 6\)</span> 时，有 <span class="arithmatex">\(3 \times 3 > 2 \times 2 \times 2\)</span> 。<strong>这意味着切分出 <span class="arithmatex">\(3\)</span> 比切分出 <span class="arithmatex">\(2\)</span> 更优</strong>。</p>
|
||||
<p><strong>贪心策略二</strong>：在切分方案中，最多只应存在两个 <span class="arithmatex">\(2\)</span> 。因为三个 <span class="arithmatex">\(2\)</span> 总是可以被替换为两个 <span class="arithmatex">\(3\)</span> ，从而获得更大乘积。</p>
|
||||
<p><img alt="最优切分因子" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_infer3.png" /></p>
|
||||
<p align="center"> Fig. 最优切分因子 </p>
|
||||
<p align="center"> 图：最优切分因子 </p>
|
||||
|
||||
<p>总结以上，可推出贪心策略：</p>
|
||||
<ol>
|
||||
|
|
@ -3664,7 +3664,7 @@ n = 3 a + b
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="最大切分乘积的计算方法" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_calculation.png" /></p>
|
||||
<p align="center"> Fig. 最大切分乘积的计算方法 </p>
|
||||
<p align="center"> 图：最大切分乘积的计算方法 </p>
|
||||
|
||||
<p><strong>时间复杂度取决于编程语言的幂运算的实现方法</strong>。以 Python 为例，常用的幂计算函数有三种：</p>
|
||||
<ul>
|
||||
|
|
|
|||
|
|
@ -3443,7 +3443,7 @@
|
|||
<p>在上两节中，我们了解了哈希表的工作原理和哈希冲突的处理方法。然而无论是开放寻址还是链地址法，<strong>它们只能保证哈希表可以在发生冲突时正常工作，但无法减少哈希冲突的发生</strong>。</p>
|
||||
<p>如果哈希冲突过于频繁，哈希表的性能则会急剧劣化。对于链地址哈希表，理想情况下键值对平均分布在各个桶中，达到最佳查询效率；最差情况下所有键值对都被存储到同一个桶中，时间复杂度退化至 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
||||
<p><img alt="哈希冲突的最佳与最差情况" src="../hash_algorithm.assets/hash_collision_best_worst_condition.png" /></p>
|
||||
<p align="center"> Fig. 哈希冲突的最佳与最差情况 </p>
|
||||
<p align="center"> 图：哈希冲突的最佳与最差情况 </p>
|
||||
|
||||
<p><strong>键值对的分布情况由哈希函数决定</strong>。回忆哈希函数的计算步骤，先计算哈希值，再对数组长度取模：</p>
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span>hash<span class="o">(</span>key<span class="o">)</span><span class="w"> </span>%<span class="w"> </span>capacity
|
||||
|
|
@ -3844,13 +3844,45 @@
|
|||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">simple_hash.dart</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">add_hash</span><span class="p">}</span>
|
||||
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a>
|
||||
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">mul_hash</span><span class="p">}</span>
|
||||
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a>
|
||||
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">xor_hash</span><span class="p">}</span>
|
||||
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a>
|
||||
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">rot_hash</span><span class="p">}</span>
|
||||
<div class="highlight"><span class="filename">simple_hash.dart</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 加法哈希 */</span>
|
||||
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">addHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">hash</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">codeUnitAt</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="p">}</span>
|
||||
<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a>
|
||||
<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="cm">/* 乘法哈希 */</span>
|
||||
<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">mulHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="m">31</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">codeUnitAt</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="p">}</span>
|
||||
<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a>
|
||||
<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="cm">/* 异或哈希 */</span>
|
||||
<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="kt">int</span><span class="w"> </span><span class="n">xorHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">^=</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">codeUnitAt</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
|
||||
<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">&</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="p">}</span>
|
||||
<a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a>
|
||||
<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="cm">/* 旋转哈希 */</span>
|
||||
<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="kt">int</span><span class="w"> </span><span class="n">rotHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">((</span><span class="n">hash</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="m">4</span><span class="p">)</span><span class="w"> </span><span class="o">^</span><span class="w"> </span><span class="p">(</span><span class="n">hash</span><span class="w"> </span><span class="o">>></span><span class="w"> </span><span class="m">28</span><span class="p">)</span><span class="w"> </span><span class="o">^</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">codeUnitAt</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="p">;</span>
|
||||
<a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -4013,7 +4045,7 @@
|
|||
<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="c1"># 布尔量 True 的哈希值为 1</span>
|
||||
<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a>
|
||||
<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="n">dec</span> <span class="o">=</span> <span class="mf">3.14159</span>
|
||||
<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="n">hash_dec</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">dec</span><span class="p">)</span>
|
||||
<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="n">hash_dec</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">dec</span><span class="p">)</span>
|
||||
<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="c1"># 小数 3.14159 的哈希值为 326484311674566659</span>
|
||||
<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a>
|
||||
<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="nb">str</span> <span class="o">=</span> <span class="s2">"Hello 算法"</span>
|
||||
|
|
|
|||
|
|
@ -3476,7 +3476,7 @@
|
|||
<h2 id="621">6.2.1. 链式地址<a class="headerlink" href="#621" title="Permanent link">¶</a></h2>
|
||||
<p>在原始哈希表中，每个桶仅能存储一个键值对。「链式地址 Separate Chaining」将单个元素转换为链表，将键值对作为链表节点，将所有发生冲突的键值对都存储在同一链表中。</p>
|
||||
<p><img alt="链式地址哈希表" src="../hash_collision.assets/hash_table_chaining.png" /></p>
|
||||
<p align="center"> Fig. 链式地址哈希表 </p>
|
||||
<p align="center"> 图：链式地址哈希表 </p>
|
||||
|
||||
<p>链式地址下，哈希表的操作方法包括：</p>
|
||||
<ul>
|
||||
|
|
@ -3991,7 +3991,7 @@
|
|||
<a id="__codelineno-4-68" name="__codelineno-4-68" href="#__codelineno-4-68"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-4-69" name="__codelineno-4-69" href="#__codelineno-4-69"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">bucket</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-4-70" name="__codelineno-4-70" href="#__codelineno-4-70"></a><span class="w"> </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-4-71" name="__codelineno-4-71" href="#__codelineno-4-71"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="o">--</span><span class="p">;</span>
|
||||
<a id="__codelineno-4-71" name="__codelineno-4-71" href="#__codelineno-4-71"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">size</span><span class="o">--</span><span class="p">;</span>
|
||||
<a id="__codelineno-4-72" name="__codelineno-4-72" href="#__codelineno-4-72"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-4-73" name="__codelineno-4-73" href="#__codelineno-4-73"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-4-74" name="__codelineno-4-74" href="#__codelineno-4-74"></a><span class="w"> </span><span class="p">}</span>
|
||||
|
|
@ -4598,7 +4598,7 @@
|
|||
<li><strong>查找元素</strong>：若发现哈希冲突，则使用相同步长向后线性遍历，直到找到对应元素，返回 <code>value</code> 即可；如果遇到空位，说明目标键值对不在哈希表中，返回 <span class="arithmatex">\(\text{None}\)</span> 。</li>
|
||||
</ul>
|
||||
<p><img alt="线性探测" src="../hash_collision.assets/hash_table_linear_probing.png" /></p>
|
||||
<p align="center"> Fig. 线性探测 </p>
|
||||
<p align="center"> 图：线性探测 </p>
|
||||
|
||||
<p>然而，线性探测存在以下缺陷：</p>
|
||||
<ul>
|
||||
|
|
|
|||
|
|
@ -3429,7 +3429,7 @@
|
|||
<p>散列表，又称「哈希表 Hash Table」，其通过建立键 <code>key</code> 与值 <code>value</code> 之间的映射，实现高效的元素查询。具体而言，我们向哈希表输入一个键 <code>key</code> ，则可以在 <span class="arithmatex">\(O(1)\)</span> 时间内获取对应的值 <code>value</code> 。</p>
|
||||
<p>以一个包含 <span class="arithmatex">\(n\)</span> 个学生的数据库为例，每个学生都有“姓名”和“学号”两项数据。假如我们希望实现“输入一个学号，返回对应的姓名”的查询功能，则可以采用哈希表来实现。</p>
|
||||
<p><img alt="哈希表的抽象表示" src="../hash_map.assets/hash_table_lookup.png" /></p>
|
||||
<p align="center"> Fig. 哈希表的抽象表示 </p>
|
||||
<p align="center"> 图：哈希表的抽象表示 </p>
|
||||
|
||||
<p>除哈希表外，我们还可以使用数组或链表实现查询功能。若将学生数据看作数组（链表）元素，则有：</p>
|
||||
<ul>
|
||||
|
|
@ -3853,7 +3853,7 @@
|
|||
<p>随后，我们就可以利用 <code>index</code> 在哈希表中访问对应的桶，从而获取 <code>value</code> 。</p>
|
||||
<p>设数组长度 <code>capacity = 100</code> 、哈希算法 <code>hash(key) = key</code> ，易得哈希函数为 <code>key % 100</code> 。下图以 <code>key</code> 学号和 <code>value</code> 姓名为例，展示了哈希函数的工作原理。</p>
|
||||
<p><img alt="哈希函数工作原理" src="../hash_map.assets/hash_function.png" /></p>
|
||||
<p align="center"> Fig. 哈希函数工作原理 </p>
|
||||
<p align="center"> 图：哈希函数工作原理 </p>
|
||||
|
||||
<p>以下代码实现了一个简单哈希表。其中，我们将 <code>key</code> 和 <code>value</code> 封装成一个类 <code>Pair</code> ，以表示键值对。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
|
|
@ -4862,11 +4862,11 @@
|
|||
</code></pre></div>
|
||||
<p>如下图所示，两个学号指向了同一个姓名，这显然是不对的。我们将这种多个输入对应同一输出的情况称为「哈希冲突 Hash Collision」。</p>
|
||||
<p><img alt="哈希冲突示例" src="../hash_map.assets/hash_collision.png" /></p>
|
||||
<p align="center"> Fig. 哈希冲突示例 </p>
|
||||
<p align="center"> 图：哈希冲突示例 </p>
|
||||
|
||||
<p>容易想到，哈希表容量 <span class="arithmatex">\(n\)</span> 越大，多个 <code>key</code> 被分配到同一个桶中的概率就越低，冲突就越少。因此，<strong>我们可以通过扩容哈希表来减少哈希冲突</strong>。如下图所示，扩容前键值对 <code>(136, A)</code> 和 <code>(236, D)</code> 发生冲突，扩容后冲突消失。</p>
|
||||
<p><img alt="哈希表扩容" src="../hash_map.assets/hash_table_reshash.png" /></p>
|
||||
<p align="center"> Fig. 哈希表扩容 </p>
|
||||
<p align="center"> 图：哈希表扩容 </p>
|
||||
|
||||
<p>类似于数组扩容，哈希表扩容需将所有键值对从原哈希表迁移至新哈希表，非常耗时。并且由于哈希表容量 <code>capacity</code> 改变，我们需要通过哈希函数来重新计算所有键值对的存储位置，这进一步提高了扩容过程的计算开销。为此，编程语言通常会预留足够大的哈希表容量，防止频繁扩容。</p>
|
||||
<p>「负载因子 Load Factor」是哈希表的一个重要概念，其定义为哈希表的元素数量除以桶数量，用于衡量哈希冲突的严重程度，<strong>也常被作为哈希表扩容的触发条件</strong>。例如在 Java 中，当负载因子超过 <span class="arithmatex">\(0.75\)</span> 时，系统会将哈希表容量扩展为原先的 <span class="arithmatex">\(2\)</span> 倍。</p>
|
||||
|
|
|
|||
|
|
@ -3567,7 +3567,7 @@
|
|||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="n">_maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
|
||||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="c1">// 堆化除叶节点以外的其他所有节点</span>
|
||||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_parent</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w"> </span><span class="n">_siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
|
||||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w"> </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
|
||||
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -3596,7 +3596,7 @@
|
|||
<p>将上述两者相乘，可得到建堆过程的时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。<strong>然而，这个估算结果并不准确，因为我们没有考虑到二叉树底层节点数量远多于顶层节点的特性</strong>。</p>
|
||||
<p>接下来我们来进行更为详细的计算。为了减小计算难度，我们假设树是一个“完美二叉树”，该假设不会影响计算结果的正确性。设二叉树（即堆）节点数量为 <span class="arithmatex">\(n\)</span> ，树高度为 <span class="arithmatex">\(h\)</span> 。上文提到，<strong>节点堆化最大迭代次数等于该节点到叶节点的距离，而该距离正是“节点高度”</strong>。</p>
|
||||
<p><img alt="完美二叉树的各层节点数量" src="../build_heap.assets/heapify_operations_count.png" /></p>
|
||||
<p align="center"> Fig. 完美二叉树的各层节点数量 </p>
|
||||
<p align="center"> 图：完美二叉树的各层节点数量 </p>
|
||||
|
||||
<p>因此，我们可以将各层的“节点数量 <span class="arithmatex">\(\times\)</span> 节点高度”求和，<strong>从而得到所有节点的堆化迭代次数的总和</strong>。</p>
|
||||
<div class="arithmatex">\[
|
||||
|
|
|
|||
|
|
@ -3500,7 +3500,7 @@
|
|||
<li>「小顶堆 Min Heap」，任意节点的值 <span class="arithmatex">\(\leq\)</span> 其子节点的值。</li>
|
||||
</ul>
|
||||
<p><img alt="小顶堆与大顶堆" src="../heap.assets/min_heap_and_max_heap.png" /></p>
|
||||
<p align="center"> Fig. 小顶堆与大顶堆 </p>
|
||||
<p align="center"> 图：小顶堆与大顶堆 </p>
|
||||
|
||||
<p>堆作为完全二叉树的一个特例，具有以下特性：</p>
|
||||
<ul>
|
||||
|
|
@ -3816,7 +3816,7 @@
|
|||
<p>当使用数组表示二叉树时，元素代表节点值，索引代表节点在二叉树中的位置。<strong>节点指针通过索引映射公式来实现</strong>。</p>
|
||||
<p>具体而言，给定索引 <span class="arithmatex">\(i\)</span> ，其左子节点索引为 <span class="arithmatex">\(2i + 1\)</span> ，右子节点索引为 <span class="arithmatex">\(2i + 2\)</span> ，父节点索引为 <span class="arithmatex">\((i - 1) / 2\)</span>（向下取整）。当索引越界时，表示空节点或节点不存在。</p>
|
||||
<p><img alt="堆的表示与存储" src="../heap.assets/representation_of_heap.png" /></p>
|
||||
<p align="center"> Fig. 堆的表示与存储 </p>
|
||||
<p align="center"> 图：堆的表示与存储 </p>
|
||||
|
||||
<p>我们可以将索引映射公式封装成函数，方便后续使用。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
|
|
@ -4149,6 +4149,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：元素入堆步骤 </p>
|
||||
|
||||
<p>设节点总数为 <span class="arithmatex">\(n\)</span> ，则树的高度为 <span class="arithmatex">\(O(\log n)\)</span> 。由此可知，堆化操作的循环轮数最多为 <span class="arithmatex">\(O(\log n)\)</span> ，<strong>元素入堆操作的时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span></strong> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -4414,10 +4416,24 @@
|
|||
<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w"> </span><span class="c1">// 添加节点</span>
|
||||
<a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w"> </span><span class="n">_maxHeap</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
|
||||
<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w"> </span><span class="c1">// 从底至顶堆化</span>
|
||||
<a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w"> </span><span class="n">_siftUp</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w"> </span><span class="n">siftUp</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
|
||||
<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="p">}</span>
|
||||
<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a>
|
||||
<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">MaxHeap</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">siftUp</span><span class="p">}</span>
|
||||
<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="cm">/* 从节点 i 开始，从底至顶堆化 */</span>
|
||||
<a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="kt">void</span><span class="w"> </span><span class="n">siftUp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w"> </span><span class="c1">// 获取节点 i 的父节点</span>
|
||||
<a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_parent</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
|
||||
<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w"> </span><span class="c1">// 当“越过根节点”或“节点无需修复”时，结束堆化</span>
|
||||
<a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">p</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w"> </span><span class="c1">// 交换两节点</span>
|
||||
<a id="__codelineno-46-19" name="__codelineno-46-19" href="#__codelineno-46-19"></a><span class="w"> </span><span class="n">_swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">);</span>
|
||||
<a id="__codelineno-46-20" name="__codelineno-46-20" href="#__codelineno-46-20"></a><span class="w"> </span><span class="c1">// 循环向上堆化</span>
|
||||
<a id="__codelineno-46-21" name="__codelineno-46-21" href="#__codelineno-46-21"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span><span class="p">;</span>
|
||||
<a id="__codelineno-46-22" name="__codelineno-46-22" href="#__codelineno-46-22"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-46-23" name="__codelineno-46-23" href="#__codelineno-46-23"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -4494,6 +4510,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：堆顶元素出堆步骤 </p>
|
||||
|
||||
<p>与元素入堆操作相似，堆顶元素出堆操作的时间复杂度也为 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JS</label><label for="__tabbed_7_6">TS</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -4881,12 +4899,28 @@
|
|||
<a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w"> </span><span class="c1">// 删除节点</span>
|
||||
<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
|
||||
<a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w"> </span><span class="c1">// 从顶至底堆化</span>
|
||||
<a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w"> </span><span class="n">_siftDown</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w"> </span><span class="n">siftDown</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w"> </span><span class="c1">// 返回堆顶元素</span>
|
||||
<a id="__codelineno-58-12" name="__codelineno-58-12" href="#__codelineno-58-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
|
||||
<a id="__codelineno-58-13" name="__codelineno-58-13" href="#__codelineno-58-13"></a><span class="p">}</span>
|
||||
<a id="__codelineno-58-14" name="__codelineno-58-14" href="#__codelineno-58-14"></a>
|
||||
<a id="__codelineno-58-15" name="__codelineno-58-15" href="#__codelineno-58-15"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">MaxHeap</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">siftDown</span><span class="p">}</span>
|
||||
<a id="__codelineno-58-15" name="__codelineno-58-15" href="#__codelineno-58-15"></a><span class="cm">/* 从节点 i 开始，从顶至底堆化 */</span>
|
||||
<a id="__codelineno-58-16" name="__codelineno-58-16" href="#__codelineno-58-16"></a><span class="kt">void</span><span class="w"> </span><span class="n">siftDown</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-58-17" name="__codelineno-58-17" href="#__codelineno-58-17"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-58-18" name="__codelineno-58-18" href="#__codelineno-58-18"></a><span class="w"> </span><span class="c1">// 判断节点 i, l, r 中值最大的节点，记为 ma</span>
|
||||
<a id="__codelineno-58-19" name="__codelineno-58-19" href="#__codelineno-58-19"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_left</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
|
||||
<a id="__codelineno-58-20" name="__codelineno-58-20" href="#__codelineno-58-20"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_right</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
|
||||
<a id="__codelineno-58-21" name="__codelineno-58-21" href="#__codelineno-58-21"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
|
||||
<a id="__codelineno-58-22" name="__codelineno-58-22" href="#__codelineno-58-22"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
|
||||
<a id="__codelineno-58-23" name="__codelineno-58-23" href="#__codelineno-58-23"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
|
||||
<a id="__codelineno-58-24" name="__codelineno-58-24" href="#__codelineno-58-24"></a><span class="w"> </span><span class="c1">// 若节点 i 最大或索引 l, r 越界，则无需继续堆化，跳出</span>
|
||||
<a id="__codelineno-58-25" name="__codelineno-58-25" href="#__codelineno-58-25"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-58-26" name="__codelineno-58-26" href="#__codelineno-58-26"></a><span class="w"> </span><span class="c1">// 交换两节点</span>
|
||||
<a id="__codelineno-58-27" name="__codelineno-58-27" href="#__codelineno-58-27"></a><span class="w"> </span><span class="n">_swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">);</span>
|
||||
<a id="__codelineno-58-28" name="__codelineno-58-28" href="#__codelineno-58-28"></a><span class="w"> </span><span class="c1">// 循环向下堆化</span>
|
||||
<a id="__codelineno-58-29" name="__codelineno-58-29" href="#__codelineno-58-29"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
|
||||
<a id="__codelineno-58-30" name="__codelineno-58-30" href="#__codelineno-58-30"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-58-31" name="__codelineno-58-31" href="#__codelineno-58-31"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
|
|||
|
|
@ -3435,17 +3435,17 @@
|
|||
<p>我们可以进行 <span class="arithmatex">\(k\)</span> 轮遍历，分别在每轮中提取第 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> , <span class="arithmatex">\(\cdots\)</span> , <span class="arithmatex">\(k\)</span> 大的元素，时间复杂度为 <span class="arithmatex">\(O(nk)\)</span> 。</p>
|
||||
<p>该方法只适用于 <span class="arithmatex">\(k \ll n\)</span> 的情况，因为当 <span class="arithmatex">\(k\)</span> 与 <span class="arithmatex">\(n\)</span> 比较接近时，其时间复杂度趋向于 <span class="arithmatex">\(O(n^2)\)</span> ，非常耗时。</p>
|
||||
<p><img alt="遍历寻找最大的 k 个元素" src="../top_k.assets/top_k_traversal.png" /></p>
|
||||
<p align="center"> Fig. 遍历寻找最大的 k 个元素 </p>
|
||||
<p align="center"> 图：遍历寻找最大的 k 个元素 </p>
|
||||
|
||||
<div class="admonition tip">
|
||||
<p class="admonition-title">Tip</p>
|
||||
<p>当 <span class="arithmatex">\(k = n\)</span> 时，我们可以得到从大到小的序列，等价于「选择排序」算法。 </p>
|
||||
<p>当 <span class="arithmatex">\(k = n\)</span> 时，我们可以得到从大到小的序列，等价于「选择排序」算法。</p>
|
||||
</div>
|
||||
<h2 id="832">8.3.2. 方法二：排序<a class="headerlink" href="#832" title="Permanent link">¶</a></h2>
|
||||
<p>我们可以对数组 <code>nums</code> 进行排序，并返回最右边的 <span class="arithmatex">\(k\)</span> 个元素，时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
|
||||
<p>显然，该方法“超额”完成任务了，因为我们只需要找出最大的 <span class="arithmatex">\(k\)</span> 个元素即可，而不需要排序其他元素。</p>
|
||||
<p><img alt="排序寻找最大的 k 个元素" src="../top_k.assets/top_k_sorting.png" /></p>
|
||||
<p align="center"> Fig. 排序寻找最大的 k 个元素 </p>
|
||||
<p align="center"> 图：排序寻找最大的 k 个元素 </p>
|
||||
|
||||
<h2 id="833">8.3.3. 方法三：堆<a class="headerlink" href="#833" title="Permanent link">¶</a></h2>
|
||||
<p>我们可以基于堆更加高效地解决 Top-K 问题，流程如下：</p>
|
||||
|
|
@ -3486,6 +3486,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：基于堆寻找最大的 k 个元素 </p>
|
||||
|
||||
<p>总共执行了 <span class="arithmatex">\(n\)</span> 轮入堆和出堆，堆的最大长度为 <span class="arithmatex">\(k\)</span> ，因此时间复杂度为 <span class="arithmatex">\(O(n \log k)\)</span> 。该方法的效率很高，当 <span class="arithmatex">\(k\)</span> 较小时，时间复杂度趋向 <span class="arithmatex">\(O(n)\)</span> ；当 <span class="arithmatex">\(k\)</span> 较大时，时间复杂度不会超过 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
|
||||
<p>另外，该方法适用于动态数据流的使用场景。在不断加入数据时，我们可以持续维护堆内的元素，从而实现最大 <span class="arithmatex">\(k\)</span> 个元素的动态更新。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
|
|
@ -3621,7 +3623,20 @@
|
|||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">top_k.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">top_k_heap</span><span class="p">}</span>
|
||||
<div class="highlight"><span class="filename">top_k.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 基于堆查找数组中最大的 k 个元素 */</span>
|
||||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">MinHeap</span><span class="w"> </span><span class="n">topKHeap</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="c1">// 将数组的前 k 个元素入堆</span>
|
||||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="n">MinHeap</span><span class="w"> </span><span class="n">heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MinHeap</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">sublist</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">));</span>
|
||||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="c1">// 从第 k+1 个元素开始，保持堆的长度为 k</span>
|
||||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w"> </span><span class="c1">// 若当前元素大于堆顶元素，则将堆顶元素出堆、当前元素入堆</span>
|
||||
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">peek</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
|
||||
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
|
||||
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">heap</span><span class="p">;</span>
|
||||
<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
|
|||
|
|
@ -3373,6 +3373,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：查字典步骤 </p>
|
||||
|
||||
<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」。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」算法。</p>
|
||||
<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，实现流程如下：</p>
|
||||
<ol>
|
||||
|
|
@ -3381,7 +3383,7 @@
|
|||
<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>
|
||||
</ol>
|
||||
<p><img alt="扑克排序步骤" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></p>
|
||||
<p align="center"> Fig. 扑克排序步骤 </p>
|
||||
<p align="center"> 图：扑克排序步骤 </p>
|
||||
|
||||
<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>
|
||||
<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，给收银员付了 <span class="arithmatex">\(100\)</span> 元，则收银员需要给我们找 <span class="arithmatex">\(31\)</span> 元。他会很自然地完成以下思考：</p>
|
||||
|
|
@ -3393,7 +3395,7 @@
|
|||
<li>完成找零，方案为 <span class="arithmatex">\(20 + 10 + 1 = 31\)</span> 元。</li>
|
||||
</ol>
|
||||
<p><img alt="货币找零过程" src="../algorithms_are_everywhere.assets/greedy_change.png" /></p>
|
||||
<p align="center"> Fig. 货币找零过程 </p>
|
||||
<p align="center"> 图：货币找零过程 </p>
|
||||
|
||||
<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>
|
||||
<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>
|
||||
|
|
|
|||
|
|
@ -3453,11 +3453,11 @@
|
|||
<li>特定算法通常有对应最优的数据结构。算法通常可以基于不同的数据结构进行实现，但最终执行效率可能相差很大。</li>
|
||||
</ul>
|
||||
<p><img alt="数据结构与算法的关系" src="../what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png" /></p>
|
||||
<p align="center"> Fig. 数据结构与算法的关系 </p>
|
||||
<p align="center"> 图：数据结构与算法的关系 </p>
|
||||
|
||||
<p>数据结构与算法犹如拼装积木。一套积木，除了包含许多零件之外，还附有详细的组装说明书。我们按照说明书一步步操作，就能组装出精美的积木模型。</p>
|
||||
<p><img alt="拼装积木" src="../what_is_dsa.assets/assembling_blocks.jpg" /></p>
|
||||
<p align="center"> Fig. 拼装积木 </p>
|
||||
<p align="center"> 图：拼装积木 </p>
|
||||
|
||||
<p>两者的详细对应关系如下表所示。</p>
|
||||
<div class="center-table">
|
||||
|
|
|
|||
|
|
@ -3448,7 +3448,7 @@
|
|||
<li><strong>算法</strong>：搜索、排序、分治、回溯、动态规划、贪心等算法的定义、优缺点、效率、应用场景、解题步骤、示例题目等。</li>
|
||||
</ul>
|
||||
<p><img alt="Hello 算法内容结构" src="../about_the_book.assets/hello_algo_mindmap.png" /></p>
|
||||
<p align="center"> Fig. Hello 算法内容结构 </p>
|
||||
<p align="center"> 图：Hello 算法内容结构 </p>
|
||||
|
||||
<h2 id="013">0.1.3. 致谢<a class="headerlink" href="#013" title="Permanent link">¶</a></h2>
|
||||
<p>在本书的创作过程中，我得到了许多人的帮助，包括但不限于：</p>
|
||||
|
|
|
|||
|
|
@ -3598,14 +3598,14 @@
|
|||
<p>相较于文字，视频和图片具有更高的信息密度和结构化程度，更易于理解。在本书中，<strong>重点和难点知识将主要通过动画和图解形式展示</strong>，而文字则作为动画和图片的解释与补充。</p>
|
||||
<p>在阅读本书时，如果发现某段内容提供了动画或图解，<strong>建议以图为主线</strong>，以文字（通常位于图像上方）为辅，综合两者来理解内容。</p>
|
||||
<p><img alt="动画图解示例" src="../../index.assets/animation.gif" /></p>
|
||||
<p align="center"> Fig. 动画图解示例 </p>
|
||||
<p align="center"> 图：动画图解示例 </p>
|
||||
|
||||
<h2 id="023">0.2.3. 在代码实践中加深理解<a class="headerlink" href="#023" title="Permanent link">¶</a></h2>
|
||||
<p>本书的配套代码被托管在 <a href="https://github.com/krahets/hello-algo">GitHub 仓库</a>。<strong>源代码附有测试样例，可一键运行</strong>。</p>
|
||||
<p>如果时间允许，<strong>建议你参照代码自行敲一遍</strong>。如果学习时间有限，请至少通读并运行所有代码。</p>
|
||||
<p>与阅读代码相比，编写代码的过程往往能带来更多收获。<strong>动手学，才是真的学</strong>。</p>
|
||||
<p><img alt="运行代码示例" src="../../index.assets/running_code.gif" /></p>
|
||||
<p align="center"> Fig. 运行代码示例 </p>
|
||||
<p align="center"> 图：运行代码示例 </p>
|
||||
|
||||
<p><strong>第一步：安装本地编程环境</strong>。请参照<a href="https://www.hello-algo.com/chapter_appendix/installation/">附录教程</a>进行安装，如果已安装则可跳过此步骤。</p>
|
||||
<p><strong>第二步：下载代码仓</strong>。如果已经安装 <a href="https://git-scm.com/downloads">Git</a> ，可以通过以下命令克隆本仓库。</p>
|
||||
|
|
@ -3613,17 +3613,17 @@
|
|||
</code></pre></div>
|
||||
<p>当然，你也可以点击“Download ZIP”直接下载代码压缩包，然后在本地解压即可。</p>
|
||||
<p><img alt="克隆仓库与下载代码" src="../suggestions.assets/download_code.png" /></p>
|
||||
<p align="center"> Fig. 克隆仓库与下载代码 </p>
|
||||
<p align="center"> 图：克隆仓库与下载代码 </p>
|
||||
|
||||
<p><strong>第三步：运行源代码</strong>。如果代码块顶部标有文件名称，则可以在仓库的 <code>codes</code> 文件夹中找到相应的源代码文件。源代码文件将帮助你节省不必要的调试时间，让你能够专注于学习内容。</p>
|
||||
<p><img alt="代码块与对应的源代码文件" src="../suggestions.assets/code_md_to_repo.png" /></p>
|
||||
<p align="center"> Fig. 代码块与对应的源代码文件 </p>
|
||||
<p align="center"> 图：代码块与对应的源代码文件 </p>
|
||||
|
||||
<h2 id="024">0.2.4. 在提问讨论中共同成长<a class="headerlink" href="#024" title="Permanent link">¶</a></h2>
|
||||
<p>阅读本书时，请不要“惯着”那些没学明白的知识点。<strong>欢迎在评论区提出你的问题</strong>，我和其他小伙伴们将竭诚为你解答，一般情况下可在两天内得到回复。</p>
|
||||
<p>同时，也希望您能在评论区多花些时间。一方面，您可以了解大家遇到的问题，从而查漏补缺，这将有助于激发更深入的思考。另一方面，希望您能慷慨地回答其他小伙伴的问题、分享您的见解，让大家共同学习和进步。</p>
|
||||
<p><img alt="评论区示例" src="../../index.assets/comment.gif" /></p>
|
||||
<p align="center"> Fig. 评论区示例 </p>
|
||||
<p align="center"> 图：评论区示例 </p>
|
||||
|
||||
<h2 id="025">0.2.5. 算法学习路线<a class="headerlink" href="#025" title="Permanent link">¶</a></h2>
|
||||
<p>从总体上看，我们可以将学习数据结构与算法的过程划分为三个阶段：</p>
|
||||
|
|
@ -3634,7 +3634,7 @@
|
|||
</ol>
|
||||
<p>作为一本入门教程，本书内容主要涵盖“第一阶段”，旨在帮助你更高效地展开第二和第三阶段的学习。</p>
|
||||
<p><img alt="算法学习路线" src="../suggestions.assets/learning_route.png" /></p>
|
||||
<p align="center"> Fig. 算法学习路线 </p>
|
||||
<p align="center"> 图：算法学习路线 </p>
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -3418,7 +3418,7 @@
|
|||
<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的数组 <code>nums</code> ，元素按从小到大的顺序排列，数组不包含重复元素。请查找并返回元素 <code>target</code> 在该数组中的索引。若数组不包含该元素，则返回 <span class="arithmatex">\(-1\)</span> 。</p>
|
||||
</div>
|
||||
<p><img alt="二分查找示例数据" src="../binary_search.assets/binary_search_example.png" /></p>
|
||||
<p align="center"> Fig. 二分查找示例数据 </p>
|
||||
<p align="center"> 图：二分查找示例数据 </p>
|
||||
|
||||
<p>对于上述问题，我们先初始化指针 <span class="arithmatex">\(i = 0\)</span> 和 <span class="arithmatex">\(j = n - 1\)</span> ，分别指向数组首元素和尾元素，代表搜索区间 <span class="arithmatex">\([0, n - 1]\)</span> 。请注意，中括号表示闭区间，其包含边界值本身。</p>
|
||||
<p>接下来，循环执行以下两个步骤：</p>
|
||||
|
|
@ -3457,6 +3457,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：binary_search_step1 </p>
|
||||
|
||||
<p>值得注意的是，由于 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 都是 <code>int</code> 类型，<strong>因此 <span class="arithmatex">\(i + j\)</span> 可能会超出 <code>int</code> 类型的取值范围</strong>。为了避免大数越界，我们通常采用公式 <span class="arithmatex">\(m = \lfloor {i + (j - i) / 2} \rfloor\)</span> 来计算中点。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3988,7 +3990,7 @@
|
|||
<p>如下图所示，在两种区间表示下，二分查找算法的初始化、循环条件和缩小区间操作皆有所不同。</p>
|
||||
<p>在“双闭区间”表示法中，由于左右边界都被定义为闭区间，因此指针 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 缩小区间操作也是对称的。这样更不容易出错。因此，<strong>我们通常采用“双闭区间”的写法</strong>。</p>
|
||||
<p><img alt="两种区间定义" src="../binary_search.assets/binary_search_ranges.png" /></p>
|
||||
<p align="center"> Fig. 两种区间定义 </p>
|
||||
<p align="center"> 图：两种区间定义 </p>
|
||||
|
||||
<h2 id="1012">10.1.2. 优点与局限性<a class="headerlink" href="#1012" title="Permanent link">¶</a></h2>
|
||||
<p>二分查找在时间和空间方面都有较好的性能：</p>
|
||||
|
|
|
|||
|
|
@ -3589,7 +3589,7 @@
|
|||
<p>实际上，我们可以利用查找最左元素的函数来查找最右元素，具体方法为：<strong>将查找最右一个 <code>target</code> 转化为查找最左一个 <code>target + 1</code></strong>。</p>
|
||||
<p>查找完成后，指针 <span class="arithmatex">\(i\)</span> 指向最左一个 <code>target + 1</code>（如果存在），而 <span class="arithmatex">\(j\)</span> 指向最右一个 <code>target</code> ，<strong>因此返回 <span class="arithmatex">\(j\)</span> 即可</strong>。</p>
|
||||
<p><img alt="将查找右边界转化为查找左边界" src="../binary_search_edge.assets/binary_search_right_edge_by_left_edge.png" /></p>
|
||||
<p align="center"> Fig. 将查找右边界转化为查找左边界 </p>
|
||||
<p align="center"> 图：将查找右边界转化为查找左边界 </p>
|
||||
|
||||
<p>请注意，返回的插入点是 <span class="arithmatex">\(i\)</span> ，因此需要将其减 <span class="arithmatex">\(1\)</span> ，从而获得 <span class="arithmatex">\(j\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
|
|
@ -3722,7 +3722,7 @@
|
|||
<li>查找最右一个 <code>target</code> ：可以转化为查找 <code>target + 0.5</code> ，并返回指针 <span class="arithmatex">\(j\)</span> 。</li>
|
||||
</ul>
|
||||
<p><img alt="将查找边界转化为查找元素" src="../binary_search_edge.assets/binary_search_edge_by_element.png" /></p>
|
||||
<p align="center"> Fig. 将查找边界转化为查找元素 </p>
|
||||
<p align="center"> 图：将查找边界转化为查找元素 </p>
|
||||
|
||||
<p>代码在此省略，值得注意的有：</p>
|
||||
<ul>
|
||||
|
|
|
|||
|
|
@ -3427,7 +3427,7 @@
|
|||
<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的有序数组 <code>nums</code> 和一个元素 <code>target</code> ，数组不存在重复元素。现将 <code>target</code> 插入到数组 <code>nums</code> 中，并保持其有序性。若数组中已存在元素 <code>target</code> ，则插入到其左方。请返回插入后 <code>target</code> 在数组中的索引。</p>
|
||||
</div>
|
||||
<p><img alt="二分查找插入点示例数据" src="../binary_search_insertion.assets/binary_search_insertion_example.png" /></p>
|
||||
<p align="center"> Fig. 二分查找插入点示例数据 </p>
|
||||
<p align="center"> 图：二分查找插入点示例数据 </p>
|
||||
|
||||
<p>如果想要复用上节的二分查找代码，则需要回答以下两个问题。</p>
|
||||
<p><strong>问题一</strong>：当数组中包含 <code>target</code> 时，插入点的索引是否是该元素的索引？</p>
|
||||
|
|
@ -3586,7 +3586,7 @@
|
|||
<li>从索引 <span class="arithmatex">\(k\)</span> 开始，向左进行线性遍历，当找到最左边的 <code>target</code> 时返回。</li>
|
||||
</ol>
|
||||
<p><img alt="线性查找重复元素的插入点" src="../binary_search_insertion.assets/binary_search_insertion_naive.png" /></p>
|
||||
<p align="center"> Fig. 线性查找重复元素的插入点 </p>
|
||||
<p align="center"> 图：线性查找重复元素的插入点 </p>
|
||||
|
||||
<p>此方法虽然可用，但其包含线性查找，因此时间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。当数组中存在很多重复的 <code>target</code> 时，该方法效率很低。</p>
|
||||
<p>现考虑修改二分查找代码。整体流程不变，每轮先计算中点索引 <span class="arithmatex">\(m\)</span> ，再判断 <code>target</code> 和 <code>nums[m]</code> 大小关系：</p>
|
||||
|
|
@ -3623,6 +3623,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：二分查找重复元素的插入点的步骤 </p>
|
||||
|
||||
<p>观察以下代码，判断分支 <code>nums[m] > target</code> 和 <code>nums[m] == target</code> 的操作相同，因此两者可以合并。</p>
|
||||
<p>即便如此，我们仍然可以将判断条件保持展开，因为其逻辑更加清晰、可读性更好。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
|
|
|
|||
|
|
@ -3420,7 +3420,7 @@
|
|||
<h2 id="1041">10.4.1. 线性查找：以时间换空间<a class="headerlink" href="#1041" title="Permanent link">¶</a></h2>
|
||||
<p>考虑直接遍历所有可能的组合。开启一个两层循环，在每轮中判断两个整数的和是否为 <code>target</code> ，若是，则返回它们的索引。</p>
|
||||
<p><img alt="线性查找求解两数之和" src="../replace_linear_by_hashing.assets/two_sum_brute_force.png" /></p>
|
||||
<p align="center"> Fig. 线性查找求解两数之和 </p>
|
||||
<p align="center"> 图：线性查找求解两数之和 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3583,13 +3583,14 @@
|
|||
<div class="highlight"><span class="filename">two_sum.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 方法一： 暴力枚举 */</span>
|
||||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">twoSumBruteForce</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
|
||||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">];</span>
|
||||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="m">0</span><span class="p">];</span>
|
||||
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="p">}</span>
|
||||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
|
||||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">];</span>
|
||||
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="m">0</span><span class="p">];</span>
|
||||
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -3630,6 +3631,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：辅助哈希表求解两数之和 </p>
|
||||
|
||||
<p>实现代码如下所示，仅需单层循环即可。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3834,15 +3837,17 @@
|
|||
<div class="highlight"><span class="filename">two_sum.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 方法二： 辅助哈希表 */</span>
|
||||
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">twoSumHashTable</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="n">Map</span><span class="o"><</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">();</span>
|
||||
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">dic</span><span class="p">[</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">!</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">];</span>
|
||||
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="n">putIfAbsent</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="o">=></span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="m">0</span><span class="p">];</span>
|
||||
<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="p">}</span>
|
||||
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="c1">// 辅助哈希表，空间复杂度 O(n)</span>
|
||||
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="n">Map</span><span class="o"><</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">();</span>
|
||||
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w"> </span><span class="c1">// 单层循环，时间复杂度 O(n)</span>
|
||||
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">dic</span><span class="p">[</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">!</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">];</span>
|
||||
<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="n">putIfAbsent</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="o">=></span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="m">0</span><span class="p">];</span>
|
||||
<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
|
|||
|
|
@ -3457,7 +3457,7 @@
|
|||
<h2 id="1053">10.5.3. 搜索方法选取<a class="headerlink" href="#1053" title="Permanent link">¶</a></h2>
|
||||
<p>给定大小为 <span class="arithmatex">\(n\)</span> 的一组数据，我们可以使用线性搜索、二分查找、树查找、哈希查找等多种方法在该数据中搜索目标元素。各个方法的工作原理如下图所示。</p>
|
||||
<p><img alt="多种搜索策略" src="../searching_algorithm_revisited.assets/searching_algorithms.png" /></p>
|
||||
<p align="center"> Fig. 多种搜索策略 </p>
|
||||
<p align="center"> 图：多种搜索策略 </p>
|
||||
|
||||
<p>上述几种方法的操作效率与特性如下表所示。</p>
|
||||
<div class="center-table">
|
||||
|
|
|
|||
|
|
@ -3453,6 +3453,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：利用元素交换操作模拟冒泡 </p>
|
||||
|
||||
<h2 id="1131">11.3.1. 算法流程<a class="headerlink" href="#1131" title="Permanent link">¶</a></h2>
|
||||
<p>设数组的长度为 <span class="arithmatex">\(n\)</span> ，冒泡排序的步骤为：</p>
|
||||
<ol>
|
||||
|
|
@ -3462,7 +3464,7 @@
|
|||
<li>仅剩的一个元素必定是最小元素，无需排序，因此数组排序完成。</li>
|
||||
</ol>
|
||||
<p><img alt="冒泡排序流程" src="../bubble_sort.assets/bubble_sort_overview.png" /></p>
|
||||
<p align="center"> Fig. 冒泡排序流程 </p>
|
||||
<p align="center"> 图：冒泡排序流程 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
|
|||
|
|
@ -3436,7 +3436,7 @@
|
|||
<li>按照桶的从小到大的顺序，合并结果。</li>
|
||||
</ol>
|
||||
<p><img alt="桶排序算法流程" src="../bucket_sort.assets/bucket_sort_overview.png" /></p>
|
||||
<p align="center"> Fig. 桶排序算法流程 </p>
|
||||
<p align="center"> 图：桶排序算法流程 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3805,11 +3805,11 @@
|
|||
<p>桶排序的时间复杂度理论上可以达到 <span class="arithmatex">\(O(n)\)</span> ，<strong>关键在于将元素均匀分配到各个桶中</strong>，因为实际数据往往不是均匀分布的。例如，我们想要将淘宝上的所有商品按价格范围平均分配到 10 个桶中，但商品价格分布不均，低于 100 元的非常多，高于 1000 元的非常少。若将价格区间平均划分为 10 份，各个桶中的商品数量差距会非常大。</p>
|
||||
<p>为实现平均分配，我们可以先设定一个大致的分界线，将数据粗略地分到 3 个桶中。<strong>分配完毕后，再将商品较多的桶继续划分为 3 个桶，直至所有桶中的元素数量大致相等</strong>。这种方法本质上是创建一个递归树，使叶节点的值尽可能平均。当然，不一定要每轮将数据划分为 3 个桶，具体划分方式可根据数据特点灵活选择。</p>
|
||||
<p><img alt="递归划分桶" src="../bucket_sort.assets/scatter_in_buckets_recursively.png" /></p>
|
||||
<p align="center"> Fig. 递归划分桶 </p>
|
||||
<p align="center"> 图：递归划分桶 </p>
|
||||
|
||||
<p>如果我们提前知道商品价格的概率分布，<strong>则可以根据数据概率分布设置每个桶的价格分界线</strong>。值得注意的是，数据分布并不一定需要特意统计，也可以根据数据特点采用某种概率模型进行近似。如下图所示，我们假设商品价格服从正态分布，这样就可以合理地设定价格区间，从而将商品平均分配到各个桶中。</p>
|
||||
<p><img alt="根据概率分布划分桶" src="../bucket_sort.assets/scatter_in_buckets_distribution.png" /></p>
|
||||
<p align="center"> Fig. 根据概率分布划分桶 </p>
|
||||
<p align="center"> 图：根据概率分布划分桶 </p>
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -3449,7 +3449,7 @@
|
|||
<li><strong>由于 <code>counter</code> 的各个索引天然有序，因此相当于所有数字已经被排序好了</strong>。接下来，我们遍历 <code>counter</code> ，根据各数字的出现次数，将它们按从小到大的顺序填入 <code>nums</code> 即可。</li>
|
||||
</ol>
|
||||
<p><img alt="计数排序流程" src="../counting_sort.assets/counting_sort_overview.png" /></p>
|
||||
<p align="center"> Fig. 计数排序流程 </p>
|
||||
<p align="center"> 图：计数排序流程 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3774,6 +3774,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：计数排序步骤 </p>
|
||||
|
||||
<p>计数排序的实现代码如下所示。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
|
|||
|
|
@ -3471,6 +3471,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：堆排序步骤 </p>
|
||||
|
||||
<p>在代码实现中，我们使用了与堆章节相同的从顶至底堆化（Sift Down）的函数。值得注意的是，由于堆的长度会随着提取最大元素而减小，因此我们需要给 Sift Down 函数添加一个长度参数 <span class="arithmatex">\(n\)</span> ，用于指定堆的当前有效长度。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
|
|||
|
|
@ -3430,7 +3430,7 @@
|
|||
<p>具体来说，我们在未排序区间选择一个基准元素，将该元素与其左侧已排序区间的元素逐一比较大小，并将该元素插入到正确的位置。</p>
|
||||
<p>回忆数组的元素插入操作，设基准元素为 <code>base</code> ，我们需要将从目标索引到 <code>base</code> 之间的所有元素向右移动一位，然后再将 <code>base</code> 赋值给目标索引。</p>
|
||||
<p><img alt="单次插入操作" src="../insertion_sort.assets/insertion_operation.png" /></p>
|
||||
<p align="center"> Fig. 单次插入操作 </p>
|
||||
<p align="center"> 图：单次插入操作 </p>
|
||||
|
||||
<h2 id="1141">11.4.1. 算法流程<a class="headerlink" href="#1141" title="Permanent link">¶</a></h2>
|
||||
<p>插入排序的整体流程如下：</p>
|
||||
|
|
@ -3441,7 +3441,7 @@
|
|||
<li>以此类推，在最后一轮中，选取最后一个元素作为 <code>base</code> ，将其插入到正确位置后，<strong>所有元素均已排序</strong>。</li>
|
||||
</ol>
|
||||
<p><img alt="插入排序流程" src="../insertion_sort.assets/insertion_sort_overview.png" /></p>
|
||||
<p align="center"> Fig. 插入排序流程 </p>
|
||||
<p align="center"> 图：插入排序流程 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
|
|||
|
|
@ -3432,7 +3432,7 @@
|
|||
<li><strong>合并阶段</strong>：当子数组长度为 1 时终止划分，开始合并，持续地将左右两个较短的有序数组合并为一个较长的有序数组，直至结束。</li>
|
||||
</ol>
|
||||
<p><img alt="归并排序的划分与合并阶段" src="../merge_sort.assets/merge_sort_overview.png" /></p>
|
||||
<p align="center"> Fig. 归并排序的划分与合并阶段 </p>
|
||||
<p align="center"> 图：归并排序的划分与合并阶段 </p>
|
||||
|
||||
<h2 id="1161">11.6.1. 算法流程<a class="headerlink" href="#1161" title="Permanent link">¶</a></h2>
|
||||
<p>“划分阶段”从顶至底递归地将数组从中点切为两个子数组：</p>
|
||||
|
|
@ -3475,6 +3475,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：归并排序步骤 </p>
|
||||
|
||||
<p>观察发现，归并排序的递归顺序与二叉树的后序遍历相同，具体来看：</p>
|
||||
<ul>
|
||||
<li><strong>后序遍历</strong>：先递归左子树，再递归右子树，最后处理根节点。</li>
|
||||
|
|
|
|||
|
|
@ -3493,6 +3493,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：哨兵划分步骤 </p>
|
||||
|
||||
<div class="admonition note">
|
||||
<p class="admonition-title">快速排序的分治思想</p>
|
||||
<p>哨兵划分的实质是将一个较长数组的排序问题简化为两个较短数组的排序问题。</p>
|
||||
|
|
@ -3797,7 +3799,7 @@
|
|||
<li>持续递归，直至子数组长度为 1 时终止，从而完成整个数组的排序。</li>
|
||||
</ol>
|
||||
<p><img alt="快速排序流程" src="../quick_sort.assets/quick_sort_overview.png" /></p>
|
||||
<p align="center"> Fig. 快速排序流程 </p>
|
||||
<p align="center"> 图：快速排序流程 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
|
|||
|
|
@ -3422,7 +3422,7 @@
|
|||
<li>将 <span class="arithmatex">\(k\)</span> 增加 <span class="arithmatex">\(1\)</span> ，然后返回步骤 <code>2.</code> 继续迭代，直到所有位都排序完成后结束。</li>
|
||||
</ol>
|
||||
<p><img alt="基数排序算法流程" src="../radix_sort.assets/radix_sort_overview.png" /></p>
|
||||
<p align="center"> Fig. 基数排序算法流程 </p>
|
||||
<p align="center"> 图：基数排序算法流程 </p>
|
||||
|
||||
<p>下面来剖析代码实现。对于一个 <span class="arithmatex">\(d\)</span> 进制的数字 <span class="arithmatex">\(x\)</span> ，要获取其第 <span class="arithmatex">\(k\)</span> 位 <span class="arithmatex">\(x_k\)</span> ，可以使用以下计算公式：</p>
|
||||
<div class="arithmatex">\[
|
||||
|
|
|
|||
|
|
@ -3444,6 +3444,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：选择排序步骤 </p>
|
||||
|
||||
<p>在代码中，我们用 <span class="arithmatex">\(k\)</span> 来记录未排序区间内的最小元素。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3665,7 +3667,7 @@
|
|||
<li><strong>非稳定排序</strong>：在交换元素时，有可能将 <code>nums[i]</code> 交换至其相等元素的右边，导致两者的相对顺序发生改变。</li>
|
||||
</ul>
|
||||
<p><img alt="选择排序非稳定示例" src="../selection_sort.assets/selection_sort_instability.png" /></p>
|
||||
<p align="center"> Fig. 选择排序非稳定示例 </p>
|
||||
<p align="center"> 图：选择排序非稳定示例 </p>
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -3415,7 +3415,7 @@
|
|||
<p>「排序算法 Sorting Algorithm」用于对一组数据按照特定顺序进行排列。排序算法有着广泛的应用，因为有序数据通常能够被更有效地查找、分析和处理。</p>
|
||||
<p>在排序算法中，数据类型可以是整数、浮点数、字符或字符串等；顺序的判断规则可根据需求设定，如数字大小、字符 ASCII 码顺序或自定义规则。</p>
|
||||
<p><img alt="数据类型和判断规则示例" src="../sorting_algorithm.assets/sorting_examples.png" /></p>
|
||||
<p align="center"> Fig. 数据类型和判断规则示例 </p>
|
||||
<p align="center"> 图：数据类型和判断规则示例 </p>
|
||||
|
||||
<h2 id="1111">11.1.1. 评价维度<a class="headerlink" href="#1111" title="Permanent link">¶</a></h2>
|
||||
<p><strong>运行效率</strong>：我们期望排序算法的时间复杂度尽量低，且总体操作数量较少（即时间复杂度中的常数项降低）。对于大数据量情况，运行效率显得尤为重要。</p>
|
||||
|
|
|
|||
|
|
@ -3409,7 +3409,7 @@
|
|||
<li>总的来说，我们希望找到一种排序算法，具有高效率、稳定、原地以及正向自适应性等优点。然而，正如其他数据结构和算法一样，没有一种排序算法能够同时满足所有这些条件。在实际应用中，我们需要根据数据的特性来选择合适的排序算法。</li>
|
||||
</ul>
|
||||
<p><img alt="排序算法对比" src="../summary.assets/sorting_algorithms_comparison.png" /></p>
|
||||
<p align="center"> Fig. 排序算法对比 </p>
|
||||
<p align="center"> 图：排序算法对比 </p>
|
||||
|
||||
<h2 id="11111-q-a">11.11.1. Q & A<a class="headerlink" href="#11111-q-a" title="Permanent link">¶</a></h2>
|
||||
<div class="admonition question">
|
||||
|
|
|
|||
|
|
@ -3468,7 +3468,7 @@
|
|||
<h1 id="53">5.3. 双向队列<a class="headerlink" href="#53" title="Permanent link">¶</a></h1>
|
||||
<p>对于队列，我们仅能在头部删除或在尾部添加元素。然而，「双向队列 Deque」提供了更高的灵活性，允许在头部和尾部执行元素的添加或删除操作。</p>
|
||||
<p><img alt="双向队列的操作" src="../deque.assets/deque_operations.png" /></p>
|
||||
<p align="center"> Fig. 双向队列的操作 </p>
|
||||
<p align="center"> 图：双向队列的操作 </p>
|
||||
|
||||
<h2 id="531">5.3.1. 双向队列常用操作<a class="headerlink" href="#531" title="Permanent link">¶</a></h2>
|
||||
<p>双向队列的常用操作如下表所示，具体的方法名称需要根据所使用的编程语言来确定。</p>
|
||||
|
|
@ -3816,6 +3816,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：基于链表实现双向队列的入队出队操作 </p>
|
||||
|
||||
<p>以下是具体实现代码。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3823,8 +3825,8 @@
|
|||
<div class="highlight"><span class="filename">linkedlist_deque.java</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="cm">/* 双向链表节点 */</span>
|
||||
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="kd">class</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 后继节点引用（指针）</span>
|
||||
<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 前驱节点引用（指针）</span>
|
||||
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 后继节点引用</span>
|
||||
<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 前驱节点引用</span>
|
||||
<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>
|
||||
<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
|
||||
|
|
@ -4094,8 +4096,8 @@
|
|||
<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
|
||||
<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w"> </span><span class="sd">"""构造方法"""</span>
|
||||
<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span>
|
||||
<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 后继节点引用（指针）</span>
|
||||
<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a> <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 前驱节点引用（指针）</span>
|
||||
<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 后继节点引用</span>
|
||||
<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a> <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 前驱节点引用</span>
|
||||
<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a>
|
||||
<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="k">class</span> <span class="nc">LinkedListDeque</span><span class="p">:</span>
|
||||
<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w"> </span><span class="sd">"""基于双向链表实现的双向队列"""</span>
|
||||
|
|
@ -4690,8 +4692,8 @@
|
|||
<div class="highlight"><span class="filename">linkedlist_deque.cs</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 双向链表节点 */</span>
|
||||
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 后继节点引用（指针）</span>
|
||||
<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 前驱节点引用（指针）</span>
|
||||
<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 后继节点引用</span>
|
||||
<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 前驱节点引用</span>
|
||||
<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a>
|
||||
<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="nf">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
|
||||
|
|
@ -4831,8 +4833,8 @@
|
|||
<div class="highlight"><span class="filename">linkedlist_deque.swift</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* 双向链表节点 */</span>
|
||||
<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">class</span> <span class="nc">ListNode</span> <span class="p">{</span>
|
||||
<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a> <span class="kd">var</span> <span class="nv">val</span><span class="p">:</span> <span class="nb">Int</span> <span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a> <span class="kd">var</span> <span class="nv">next</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 后继节点引用（指针）</span>
|
||||
<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a> <span class="kr">weak</span> <span class="kd">var</span> <span class="nv">prev</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 前驱节点引用（指针）</span>
|
||||
<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a> <span class="kd">var</span> <span class="nv">next</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 后继节点引用</span>
|
||||
<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a> <span class="kr">weak</span> <span class="kd">var</span> <span class="nv">prev</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// 前驱节点引用</span>
|
||||
<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a>
|
||||
<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a> <span class="kd">init</span><span class="p">(</span><span class="n">val</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">{</span>
|
||||
<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a> <span class="kc">self</span><span class="p">.</span><span class="n">val</span> <span class="p">=</span> <span class="n">val</span>
|
||||
|
|
@ -4966,8 +4968,8 @@
|
|||
<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w"> </span><span class="kr">const</span><span class="w"> </span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@This</span><span class="p">();</span>
|
||||
<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a>
|
||||
<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w"> </span><span class="n">val</span><span class="o">:</span><span class="w"> </span><span class="n">T</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">undefined</span><span class="p">,</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w"> </span><span class="n">next</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 后继节点引用（指针）</span>
|
||||
<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w"> </span><span class="n">prev</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 前驱节点引用（指针）</span>
|
||||
<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w"> </span><span class="n">next</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 后继节点指针</span>
|
||||
<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w"> </span><span class="n">prev</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// 前驱节点指针</span>
|
||||
<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a>
|
||||
<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w"> </span><span class="c1">// Initialize a list node with specific value</span>
|
||||
<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w"> </span><span class="kr">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">init</span><span class="p">(</span><span class="n">self</span><span class="o">:</span><span class="w"> </span><span class="o">*</span><span class="n">Self</span><span class="p">,</span><span class="w"> </span><span class="n">x</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="p">{</span>
|
||||
|
|
@ -5121,8 +5123,8 @@
|
|||
<div class="highlight"><span class="filename">linkedlist_deque.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 双向链表节点 */</span>
|
||||
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 后继节点引用（指针）</span>
|
||||
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 前驱节点引用（指针）</span>
|
||||
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// 后继节点引用</span>
|
||||
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// 前驱节点引用</span>
|
||||
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a>
|
||||
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="p">{</span><span class="k">this</span><span class="p">.</span><span class="n">next</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">prev</span><span class="p">});</span>
|
||||
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="p">}</span>
|
||||
|
|
@ -5249,8 +5251,8 @@
|
|||
<div class="highlight"><span class="filename">linkedlist_deque.rs</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* 双向链表节点 */</span>
|
||||
<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">struct</span> <span class="nc">ListNode</span><span class="o"><</span><span class="n">T</span><span class="o">></span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w"> </span><span class="k">pub</span><span class="w"> </span><span class="n">val</span>: <span class="nc">T</span><span class="p">,</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w"> </span><span class="k">pub</span><span class="w"> </span><span class="n">next</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o"><</span><span class="n">T</span><span class="o">>>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 后继节点引用（指针）</span>
|
||||
<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w"> </span><span class="k">pub</span><span class="w"> </span><span class="n">prev</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o"><</span><span class="n">T</span><span class="o">>>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 前驱节点引用（指针）</span>
|
||||
<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w"> </span><span class="k">pub</span><span class="w"> </span><span class="n">next</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o"><</span><span class="n">T</span><span class="o">>>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 后继节点指针</span>
|
||||
<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w"> </span><span class="k">pub</span><span class="w"> </span><span class="n">prev</span>: <span class="nb">Option</span><span class="o"><</span><span class="n">Rc</span><span class="o"><</span><span class="n">RefCell</span><span class="o"><</span><span class="n">ListNode</span><span class="o"><</span><span class="n">T</span><span class="o">>>>></span><span class="p">,</span><span class="w"> </span><span class="c1">// 前驱节点指针</span>
|
||||
<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="p">}</span>
|
||||
<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a>
|
||||
<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="k">impl</span><span class="o"><</span><span class="n">T</span><span class="o">></span><span class="w"> </span><span class="n">ListNode</span><span class="o"><</span><span class="n">T</span><span class="o">></span><span class="w"> </span><span class="p">{</span>
|
||||
|
|
@ -5434,6 +5436,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：基于数组实现双向队列的入队出队操作 </p>
|
||||
|
||||
<p>以下是具体实现代码。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
|
|||
|
|
@ -3469,7 +3469,7 @@
|
|||
<p>「队列 Queue」是一种遵循先入先出（First In, First Out）规则的线性数据结构。顾名思义，队列模拟了排队现象，即新来的人不断加入队列的尾部，而位于队列头部的人逐个离开。</p>
|
||||
<p>我们把队列的头部称为「队首」，尾部称为「队尾」，把将元素加入队尾的操作称为「入队」，删除队首元素的操作称为「出队」。</p>
|
||||
<p><img alt="队列的先入先出规则" src="../queue.assets/queue_operations.png" /></p>
|
||||
<p align="center"> Fig. 队列的先入先出规则 </p>
|
||||
<p align="center"> 图：队列的先入先出规则 </p>
|
||||
|
||||
<h2 id="521">5.2.1. 队列常用操作<a class="headerlink" href="#521" title="Permanent link">¶</a></h2>
|
||||
<p>队列的常见操作如下表所示。需要注意的是，不同编程语言的方法名称可能会有所不同。我们在此采用与栈相同的方法命名。</p>
|
||||
|
|
@ -3762,6 +3762,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：基于链表实现队列的入队出队操作 </p>
|
||||
|
||||
<p>以下是用链表实现队列的示例代码。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -4633,6 +4635,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：基于数组实现队列的入队出队操作 </p>
|
||||
|
||||
<p>你可能会发现一个问题：在不断进行入队和出队的过程中，<code>front</code> 和 <code>rear</code> 都在向右移动，<strong>当它们到达数组尾部时就无法继续移动了</strong>。为解决此问题，我们可以将数组视为首尾相接的「环形数组」。</p>
|
||||
<p>对于环形数组，我们需要让 <code>front</code> 或 <code>rear</code> 在越过数组尾部时，直接回到数组头部继续遍历。这种周期性规律可以通过“取余操作”来实现，代码如下所示。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
|
|
|
|||
|
|
@ -3538,7 +3538,7 @@
|
|||
<p>我们可以将栈类比为桌面上的一摞盘子，如果需要拿出底部的盘子，则需要先将上面的盘子依次取出。我们将盘子替换为各种类型的元素（如整数、字符、对象等），就得到了栈数据结构。</p>
|
||||
<p>在栈中，我们把堆叠元素的顶部称为「栈顶」，底部称为「栈底」。将把元素添加到栈顶的操作叫做「入栈」，而删除栈顶元素的操作叫做「出栈」。</p>
|
||||
<p><img alt="栈的先入后出规则" src="../stack.assets/stack_operations.png" /></p>
|
||||
<p align="center"> Fig. 栈的先入后出规则 </p>
|
||||
<p align="center"> 图：栈的先入后出规则 </p>
|
||||
|
||||
<h2 id="511">5.1.1. 栈常用操作<a class="headerlink" href="#511" title="Permanent link">¶</a></h2>
|
||||
<p>栈的常用操作如下表所示，具体的方法名需要根据所使用的编程语言来确定。在此，我们以常见的 <code>push()</code> , <code>pop()</code> , <code>peek()</code> 命名为例。</p>
|
||||
|
|
@ -3829,6 +3829,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：基于链表实现栈的入栈出栈操作 </p>
|
||||
|
||||
<p>以下是基于链表实现栈的示例代码。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -4577,6 +4579,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：基于数组实现栈的入栈出栈操作 </p>
|
||||
|
||||
<p>由于入栈的元素可能会源源不断地增加，因此我们可以使用动态数组，这样就无需自行处理数组扩容问题。以下为示例代码。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
|
|||
|
|
@ -3432,13 +3432,13 @@
|
|||
<p>先分析一个简单案例。给定一个完美二叉树，我们将所有节点按照层序遍历的顺序存储在一个数组中，则每个节点都对应唯一的数组索引。</p>
|
||||
<p>根据层序遍历的特性，我们可以推导出父节点索引与子节点索引之间的“映射公式”：<strong>若节点的索引为 <span class="arithmatex">\(i\)</span> ，则该节点的左子节点索引为 <span class="arithmatex">\(2i + 1\)</span> ，右子节点索引为 <span class="arithmatex">\(2i + 2\)</span></strong> 。</p>
|
||||
<p><img alt="完美二叉树的数组表示" src="../array_representation_of_tree.assets/array_representation_binary_tree.png" /></p>
|
||||
<p align="center"> Fig. 完美二叉树的数组表示 </p>
|
||||
<p align="center"> 图：完美二叉树的数组表示 </p>
|
||||
|
||||
<p><strong>映射公式的角色相当于链表中的指针</strong>。给定数组中的任意一个节点，我们都可以通过映射公式来访问它的左（右）子节点。</p>
|
||||
<h2 id="732">7.3.2. 表示任意二叉树<a class="headerlink" href="#732" title="Permanent link">¶</a></h2>
|
||||
<p>然而完美二叉树是一个特例，在二叉树的中间层，通常存在许多 <span class="arithmatex">\(\text{None}\)</span> 。由于层序遍历序列并不包含这些 <span class="arithmatex">\(\text{None}\)</span> ，因此我们无法仅凭该序列来推测 <span class="arithmatex">\(\text{None}\)</span> 的数量和分布位置。<strong>这意味着存在多种二叉树结构都符合该层序遍历序列</strong>。显然在这种情况下，上述的数组表示方法已经失效。</p>
|
||||
<p><img alt="层序遍历序列对应多种二叉树可能性" src="../array_representation_of_tree.assets/array_representation_without_empty.png" /></p>
|
||||
<p align="center"> Fig. 层序遍历序列对应多种二叉树可能性 </p>
|
||||
<p align="center"> 图：层序遍历序列对应多种二叉树可能性 </p>
|
||||
|
||||
<p>为了解决此问题，<strong>我们可以考虑在层序遍历序列中显式地写出所有 <span class="arithmatex">\(\text{None}\)</span></strong> 。如下图所示，这样处理后，层序遍历序列就可以唯一表示二叉树了。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
|
|
@ -3514,11 +3514,11 @@
|
|||
</div>
|
||||
</div>
|
||||
<p><img alt="任意类型二叉树的数组表示" src="../array_representation_of_tree.assets/array_representation_with_empty.png" /></p>
|
||||
<p align="center"> Fig. 任意类型二叉树的数组表示 </p>
|
||||
<p align="center"> 图：任意类型二叉树的数组表示 </p>
|
||||
|
||||
<p>值得说明的是，<strong>完全二叉树非常适合使用数组来表示</strong>。回顾完全二叉树的定义，<span class="arithmatex">\(\text{None}\)</span> 只出现在最底层且靠右的位置，<strong>因此所有 <span class="arithmatex">\(\text{None}\)</span> 一定出现在层序遍历序列的末尾</strong>。这意味着使用数组表示完全二叉树时，可以省略存储所有 <span class="arithmatex">\(\text{None}\)</span> ，非常方便。</p>
|
||||
<p><img alt="完全二叉树的数组表示" src="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" /></p>
|
||||
<p align="center"> Fig. 完全二叉树的数组表示 </p>
|
||||
<p align="center"> 图：完全二叉树的数组表示 </p>
|
||||
|
||||
<p>如下代码给出了数组表示下的二叉树的简单实现，包括以下操作：</p>
|
||||
<ul>
|
||||
|
|
@ -4328,7 +4328,96 @@
|
|||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">array_binary_tree.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{}</span>
|
||||
<div class="highlight"><span class="filename">array_binary_tree.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 数组表示下的二叉树类 */</span>
|
||||
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">_tree</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a>
|
||||
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="cm">/* 构造方法 */</span>
|
||||
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w"> </span><span class="n">ArrayBinaryTree</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">_tree</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a>
|
||||
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w"> </span><span class="cm">/* 节点数量 */</span>
|
||||
<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">_tree</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a>
|
||||
<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w"> </span><span class="cm">/* 获取索引为 i 节点的值 */</span>
|
||||
<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w"> </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w"> </span><span class="c1">// 若索引越界，则返回 null ，代表空位</span>
|
||||
<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">_tree</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
|
||||
<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a>
|
||||
<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w"> </span><span class="cm">/* 获取索引为 i 节点的左子节点的索引 */</span>
|
||||
<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w"> </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a>
|
||||
<a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w"> </span><span class="cm">/* 获取索引为 i 节点的右子节点的索引 */</span>
|
||||
<a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a><span class="w"> </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-29" name="__codelineno-22-29" href="#__codelineno-22-29"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-30" name="__codelineno-22-30" href="#__codelineno-22-30"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-31" name="__codelineno-22-31" href="#__codelineno-22-31"></a>
|
||||
<a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="w"> </span><span class="cm">/* 获取索引为 i 节点的父节点的索引 */</span>
|
||||
<a id="__codelineno-22-33" name="__codelineno-22-33" href="#__codelineno-22-33"></a><span class="w"> </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-34" name="__codelineno-22-34" href="#__codelineno-22-34"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></a>
|
||||
<a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w"> </span><span class="cm">/* 层序遍历 */</span>
|
||||
<a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">levelOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-39" name="__codelineno-22-39" href="#__codelineno-22-39"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
|
||||
<a id="__codelineno-22-40" name="__codelineno-22-40" href="#__codelineno-22-40"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-41" name="__codelineno-22-41" href="#__codelineno-22-41"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-42" name="__codelineno-22-42" href="#__codelineno-22-42"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">!</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-44" name="__codelineno-22-44" href="#__codelineno-22-44"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-45" name="__codelineno-22-45" href="#__codelineno-22-45"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-46" name="__codelineno-22-46" href="#__codelineno-22-46"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-47" name="__codelineno-22-47" href="#__codelineno-22-47"></a>
|
||||
<a id="__codelineno-22-48" name="__codelineno-22-48" href="#__codelineno-22-48"></a><span class="w"> </span><span class="cm">/* 深度优先遍历 */</span>
|
||||
<a id="__codelineno-22-49" name="__codelineno-22-49" href="#__codelineno-22-49"></a><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-50" name="__codelineno-22-50" href="#__codelineno-22-50"></a><span class="w"> </span><span class="c1">// 若为空位，则返回</span>
|
||||
<a id="__codelineno-22-51" name="__codelineno-22-51" href="#__codelineno-22-51"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-52" name="__codelineno-22-52" href="#__codelineno-22-52"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-53" name="__codelineno-22-53" href="#__codelineno-22-53"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-54" name="__codelineno-22-54" href="#__codelineno-22-54"></a><span class="w"> </span><span class="c1">// 前序遍历</span>
|
||||
<a id="__codelineno-22-55" name="__codelineno-22-55" href="#__codelineno-22-55"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">order</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="s1">'pre'</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-56" name="__codelineno-22-56" href="#__codelineno-22-56"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
|
||||
<a id="__codelineno-22-57" name="__codelineno-22-57" href="#__codelineno-22-57"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-58" name="__codelineno-22-58" href="#__codelineno-22-58"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">!</span><span class="p">,</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-59" name="__codelineno-22-59" href="#__codelineno-22-59"></a><span class="w"> </span><span class="c1">// 中序遍历</span>
|
||||
<a id="__codelineno-22-60" name="__codelineno-22-60" href="#__codelineno-22-60"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">order</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="s1">'in'</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-61" name="__codelineno-22-61" href="#__codelineno-22-61"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
|
||||
<a id="__codelineno-22-62" name="__codelineno-22-62" href="#__codelineno-22-62"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-63" name="__codelineno-22-63" href="#__codelineno-22-63"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">!</span><span class="p">,</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-64" name="__codelineno-22-64" href="#__codelineno-22-64"></a><span class="w"> </span><span class="c1">// 后序遍历</span>
|
||||
<a id="__codelineno-22-65" name="__codelineno-22-65" href="#__codelineno-22-65"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">order</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="s1">'post'</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-66" name="__codelineno-22-66" href="#__codelineno-22-66"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
|
||||
<a id="__codelineno-22-67" name="__codelineno-22-67" href="#__codelineno-22-67"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-68" name="__codelineno-22-68" href="#__codelineno-22-68"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-69" name="__codelineno-22-69" href="#__codelineno-22-69"></a>
|
||||
<a id="__codelineno-22-70" name="__codelineno-22-70" href="#__codelineno-22-70"></a><span class="w"> </span><span class="cm">/* 前序遍历 */</span>
|
||||
<a id="__codelineno-22-71" name="__codelineno-22-71" href="#__codelineno-22-71"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">preOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-72" name="__codelineno-22-72" href="#__codelineno-22-72"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
|
||||
<a id="__codelineno-22-73" name="__codelineno-22-73" href="#__codelineno-22-73"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="s1">'pre'</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-74" name="__codelineno-22-74" href="#__codelineno-22-74"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-75" name="__codelineno-22-75" href="#__codelineno-22-75"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-76" name="__codelineno-22-76" href="#__codelineno-22-76"></a>
|
||||
<a id="__codelineno-22-77" name="__codelineno-22-77" href="#__codelineno-22-77"></a><span class="w"> </span><span class="cm">/* 中序遍历 */</span>
|
||||
<a id="__codelineno-22-78" name="__codelineno-22-78" href="#__codelineno-22-78"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">inOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-79" name="__codelineno-22-79" href="#__codelineno-22-79"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
|
||||
<a id="__codelineno-22-80" name="__codelineno-22-80" href="#__codelineno-22-80"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="s1">'in'</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-81" name="__codelineno-22-81" href="#__codelineno-22-81"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-82" name="__codelineno-22-82" href="#__codelineno-22-82"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-83" name="__codelineno-22-83" href="#__codelineno-22-83"></a>
|
||||
<a id="__codelineno-22-84" name="__codelineno-22-84" href="#__codelineno-22-84"></a><span class="w"> </span><span class="cm">/* 后序遍历 */</span>
|
||||
<a id="__codelineno-22-85" name="__codelineno-22-85" href="#__codelineno-22-85"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">postOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-86" name="__codelineno-22-86" href="#__codelineno-22-86"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
|
||||
<a id="__codelineno-22-87" name="__codelineno-22-87" href="#__codelineno-22-87"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="s1">'post'</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-88" name="__codelineno-22-88" href="#__codelineno-22-88"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-89" name="__codelineno-22-89" href="#__codelineno-22-89"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-90" name="__codelineno-22-90" href="#__codelineno-22-90"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
|
|||
|
|
@ -3619,11 +3619,11 @@
|
|||
<p>在二叉搜索树章节中，我们提到了在多次插入和删除操作后，二叉搜索树可能退化为链表。这种情况下，所有操作的时间复杂度将从 <span class="arithmatex">\(O(\log n)\)</span> 恶化为 <span class="arithmatex">\(O(n)\)</span>。</p>
|
||||
<p>如下图所示，经过两次删除节点操作，这个二叉搜索树便会退化为链表。</p>
|
||||
<p><img alt="AVL 树在删除节点后发生退化" src="../avl_tree.assets/avltree_degradation_from_removing_node.png" /></p>
|
||||
<p align="center"> Fig. AVL 树在删除节点后发生退化 </p>
|
||||
<p align="center"> 图：AVL 树在删除节点后发生退化 </p>
|
||||
|
||||
<p>再例如，在以下完美二叉树中插入两个节点后，树将严重向左倾斜，查找操作的时间复杂度也随之恶化。</p>
|
||||
<p><img alt="AVL 树在插入节点后发生退化" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></p>
|
||||
<p align="center"> Fig. AVL 树在插入节点后发生退化 </p>
|
||||
<p align="center"> 图：AVL 树在插入节点后发生退化 </p>
|
||||
|
||||
<p>G. M. Adelson-Velsky 和 E. M. Landis 在其 1962 年发表的论文 "An algorithm for the organization of information" 中提出了「AVL 树」。论文中详细描述了一系列操作，确保在持续添加和删除节点后，AVL 树不会退化，从而使得各种操作的时间复杂度保持在 <span class="arithmatex">\(O(\log n)\)</span> 级别。换句话说，在需要频繁进行增删查改操作的场景中，AVL 树能始终保持高效的数据操作性能，具有很好的应用价值。</p>
|
||||
<h2 id="751-avl">7.5.1. AVL 树常见术语<a class="headerlink" href="#751-avl" title="Permanent link">¶</a></h2>
|
||||
|
|
@ -3945,14 +3945,15 @@
|
|||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 获取节点高度 */</span>
|
||||
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">height</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="p">}</span>
|
||||
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a>
|
||||
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="cm">/* 更新节点高度 */</span>
|
||||
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="kt">void</span><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w"> </span><span class="c1">// 节点高度等于最高子树高度 + 1</span>
|
||||
<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w"> </span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="p">}</span>
|
||||
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="c1">// 空节点高度为 -1 ，叶节点高度为 0</span>
|
||||
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">height</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="p">}</span>
|
||||
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a>
|
||||
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="cm">/* 更新节点高度 */</span>
|
||||
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="kt">void</span><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w"> </span><span class="c1">// 节点高度等于最高子树高度 + 1</span>
|
||||
<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w"> </span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -4139,9 +4140,11 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：右旋操作步骤 </p>
|
||||
|
||||
<p>此外，如果节点 <code>child</code> 本身有右子节点（记为 <code>grandChild</code> ），则需要在「右旋」中添加一步：将 <code>grandChild</code> 作为 <code>node</code> 的左子节点。</p>
|
||||
<p><img alt="有 grandChild 的右旋操作" src="../avl_tree.assets/avltree_right_rotate_with_grandchild.png" /></p>
|
||||
<p align="center"> Fig. 有 grandChild 的右旋操作 </p>
|
||||
<p align="center"> 图：有 grandChild 的右旋操作 </p>
|
||||
|
||||
<p>“向右旋转”是一种形象化的说法，实际上需要通过修改节点指针来实现，代码如下所示。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
|
|
@ -4348,11 +4351,11 @@
|
|||
<h3 id="_4">左旋<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>相应的，如果考虑上述失衡二叉树的“镜像”，则需要执行「左旋」操作。</p>
|
||||
<p><img alt="左旋操作" src="../avl_tree.assets/avltree_left_rotate.png" /></p>
|
||||
<p align="center"> Fig. 左旋操作 </p>
|
||||
<p align="center"> 图：左旋操作 </p>
|
||||
|
||||
<p>同理，若节点 <code>child</code> 本身有左子节点（记为 <code>grandChild</code> ），则需要在「左旋」中添加一步：将 <code>grandChild</code> 作为 <code>node</code> 的右子节点。</p>
|
||||
<p><img alt="有 grandChild 的左旋操作" src="../avl_tree.assets/avltree_left_rotate_with_grandchild.png" /></p>
|
||||
<p align="center"> Fig. 有 grandChild 的左旋操作 </p>
|
||||
<p align="center"> 图：有 grandChild 的左旋操作 </p>
|
||||
|
||||
<p>可以观察到，<strong>右旋和左旋操作在逻辑上是镜像对称的，它们分别解决的两种失衡情况也是对称的</strong>。基于对称性，我们可以轻松地从右旋的代码推导出左旋的代码。具体地，只需将「右旋」代码中的把所有的 <code>left</code> 替换为 <code>right</code> ，将所有的 <code>right</code> 替换为 <code>left</code> ，即可得到「左旋」代码。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JS</label><label for="__tabbed_6_6">TS</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
|
|
@ -4559,17 +4562,17 @@
|
|||
<h3 id="_5">先左旋后右旋<a class="headerlink" href="#_5" title="Permanent link">¶</a></h3>
|
||||
<p>对于下图中的失衡节点 3，仅使用左旋或右旋都无法使子树恢复平衡。此时需要先左旋后右旋，即先对 <code>child</code> 执行「左旋」，再对 <code>node</code> 执行「右旋」。</p>
|
||||
<p><img alt="先左旋后右旋" src="../avl_tree.assets/avltree_left_right_rotate.png" /></p>
|
||||
<p align="center"> Fig. 先左旋后右旋 </p>
|
||||
<p align="center"> 图：先左旋后右旋 </p>
|
||||
|
||||
<h3 id="_6">先右旋后左旋<a class="headerlink" href="#_6" title="Permanent link">¶</a></h3>
|
||||
<p>同理，对于上述失衡二叉树的镜像情况，需要先右旋后左旋，即先对 <code>child</code> 执行「右旋」，然后对 <code>node</code> 执行「左旋」。</p>
|
||||
<p><img alt="先右旋后左旋" src="../avl_tree.assets/avltree_right_left_rotate.png" /></p>
|
||||
<p align="center"> Fig. 先右旋后左旋 </p>
|
||||
<p align="center"> 图：先右旋后左旋 </p>
|
||||
|
||||
<h3 id="_7">旋转的选择<a class="headerlink" href="#_7" title="Permanent link">¶</a></h3>
|
||||
<p>下图展示的四种失衡情况与上述案例逐个对应，分别需要采用右旋、左旋、先右后左、先左后右的旋转操作。</p>
|
||||
<p><img alt="AVL 树的四种旋转情况" src="../avl_tree.assets/avltree_rotation_cases.png" /></p>
|
||||
<p align="center"> Fig. AVL 树的四种旋转情况 </p>
|
||||
<p align="center"> 图：AVL 树的四种旋转情况 </p>
|
||||
|
||||
<p>在代码中，我们通过判断失衡节点的平衡因子以及较高一侧子节点的平衡因子的正负号，来确定失衡节点属于上图中的哪种情况。</p>
|
||||
<div class="center-table">
|
||||
|
|
|
|||
|
|
@ -3500,7 +3500,7 @@
|
|||
<li>任意节点的左、右子树也是二叉搜索树，即同样满足条件 <code>1.</code> 。</li>
|
||||
</ol>
|
||||
<p><img alt="二叉搜索树" src="../binary_search_tree.assets/binary_search_tree.png" /></p>
|
||||
<p align="center"> Fig. 二叉搜索树 </p>
|
||||
<p align="center"> 图：二叉搜索树 </p>
|
||||
|
||||
<h2 id="741">7.4.1. 二叉搜索树的操作<a class="headerlink" href="#741" title="Permanent link">¶</a></h2>
|
||||
<p>我们将二叉搜索树封装为一个类 <code>ArrayBinaryTree</code> ，并声明一个成员变量 <code>root</code> ，指向树的根节点。</p>
|
||||
|
|
@ -3527,6 +3527,8 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：二叉搜索树查找节点示例 </p>
|
||||
|
||||
<p>二叉搜索树的查找操作与二分查找算法的工作原理一致，都是每轮排除一半情况。循环次数最多为二叉树的高度，当二叉树平衡时，使用 <span class="arithmatex">\(O(\log n)\)</span> 时间。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3739,7 +3741,24 @@
|
|||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">search</span><span class="p">}</span>
|
||||
<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 查找节点 */</span>
|
||||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">search</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="kt">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_root</span><span class="p">;</span>
|
||||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="c1">// 循环查找，越过叶节点后跳出</span>
|
||||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="c1">// 目标节点在 cur 的右子树中</span>
|
||||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="kt">num</span><span class="p">)</span>
|
||||
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
|
||||
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w"> </span><span class="c1">// 目标节点在 cur 的左子树中</span>
|
||||
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="kt">num</span><span class="p">)</span>
|
||||
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
|
||||
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w"> </span><span class="c1">// 找到目标节点，跳出循环</span>
|
||||
<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w"> </span><span class="k">else</span>
|
||||
<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w"> </span><span class="c1">// 返回目标节点</span>
|
||||
<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
|
||||
<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -3777,7 +3796,7 @@
|
|||
</ol>
|
||||
<p>二叉搜索树不允许存在重复节点，否则将违反其定义。因此，若待插入节点在树中已存在，则不执行插入，直接返回。</p>
|
||||
<p><img alt="在二叉搜索树中插入节点" src="../binary_search_tree.assets/bst_insert.png" /></p>
|
||||
<p align="center"> Fig. 在二叉搜索树中插入节点 </p>
|
||||
<p align="center"> 图：在二叉搜索树中插入节点 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -4086,7 +4105,31 @@
|
|||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
|
||||
<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 插入节点 */</span>
|
||||
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="kt">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="c1">// 若树为空，直接提前返回</span>
|
||||
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_root</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w"> </span><span class="c1">// 循环查找，越过叶节点后跳出</span>
|
||||
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w"> </span><span class="c1">// 找到重复节点，直接返回</span>
|
||||
<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kt">num</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w"> </span><span class="c1">// 插入位置在 cur 的右子树中</span>
|
||||
<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="kt">num</span><span class="p">)</span>
|
||||
<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w"> </span><span class="c1">// 插入位置在 cur 的左子树中</span>
|
||||
<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w"> </span><span class="k">else</span>
|
||||
<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="c1">// 插入节点</span>
|
||||
<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="kt">num</span><span class="p">);</span>
|
||||
<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="kt">num</span><span class="p">)</span>
|
||||
<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w"> </span><span class="k">else</span>
|
||||
<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
|
||||
<a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -4133,11 +4176,11 @@
|
|||
<p>与插入节点类似，我们需要在删除操作后维持二叉搜索树的“左子树 < 根节点 < 右子树”的性质。首先，我们需要在二叉树中执行查找操作，获取待删除节点。接下来，根据待删除节点的子节点数量，删除操作需分为三种情况：</p>
|
||||
<p>当待删除节点的度为 <span class="arithmatex">\(0\)</span> 时，表示待删除节点是叶节点，可以直接删除。</p>
|
||||
<p><img alt="在二叉搜索树中删除节点（度为 0）" src="../binary_search_tree.assets/bst_remove_case1.png" /></p>
|
||||
<p align="center"> Fig. 在二叉搜索树中删除节点（度为 0） </p>
|
||||
<p align="center"> 图：在二叉搜索树中删除节点（度为 0） </p>
|
||||
|
||||
<p>当待删除节点的度为 <span class="arithmatex">\(1\)</span> 时，将待删除节点替换为其子节点即可。</p>
|
||||
<p><img alt="在二叉搜索树中删除节点（度为 1）" src="../binary_search_tree.assets/bst_remove_case2.png" /></p>
|
||||
<p align="center"> Fig. 在二叉搜索树中删除节点（度为 1） </p>
|
||||
<p align="center"> 图：在二叉搜索树中删除节点（度为 1） </p>
|
||||
|
||||
<p>当待删除节点的度为 <span class="arithmatex">\(2\)</span> 时，我们无法直接删除它，而需要使用一个节点替换该节点。由于要保持二叉搜索树“左 <span class="arithmatex">\(<\)</span> 根 <span class="arithmatex">\(<\)</span> 右”的性质，因此这个节点可以是右子树的最小节点或左子树的最大节点。</p>
|
||||
<p>假设我们选择右子树的最小节点（即中序遍历的下一个节点），则删除操作为：</p>
|
||||
|
|
@ -4161,8 +4204,10 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：二叉搜索树删除节点示例 </p>
|
||||
|
||||
<p>删除节点操作同样使用 <span class="arithmatex">\(O(\log n)\)</span> 时间，其中查找待删除节点需要 <span class="arithmatex">\(O(\log n)\)</span> 时间，获取中序遍历后继节点需要 <span class="arithmatex">\(O(\log n)\)</span> 时间。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:11"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* 删除节点 */</span>
|
||||
|
|
@ -4703,75 +4748,152 @@
|
|||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
<p>```dart title="binary_search_tree.dart"
|
||||
/* 插入节点 */
|
||||
void insert(int num) {
|
||||
// 若树为空，直接提前返回
|
||||
if (_root == null) return;
|
||||
TreeNode? cur = _root;
|
||||
TreeNode? pre = null;
|
||||
// 循环查找，越过叶节点后跳出
|
||||
while (cur != null) {
|
||||
// 找到重复节点，直接返回
|
||||
if (cur.val == num) return;
|
||||
pre = cur;
|
||||
// 插入位置在 cur 的右子树中
|
||||
if (cur.val < num)
|
||||
cur = cur.right;
|
||||
// 插入位置在 cur 的左子树中
|
||||
else
|
||||
cur = cur.left;
|
||||
}
|
||||
// 插入节点
|
||||
TreeNode? node = TreeNode(num);
|
||||
if (pre!.val < num)
|
||||
pre.right = node;
|
||||
else
|
||||
pre.left = node;
|
||||
}</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>/* 删除节点 */
|
||||
void remove(int num) {
|
||||
// 若树为空，直接提前返回
|
||||
if (_root == null) return;</p>
|
||||
<div class="highlight"><pre><span></span><code> TreeNode? cur = _root;
|
||||
TreeNode? pre = null;
|
||||
// 循环查找，越过叶节点后跳出
|
||||
while (cur != null) {
|
||||
// 找到待删除节点，跳出循环
|
||||
if (cur.val == num) break;
|
||||
pre = cur;
|
||||
// 待删除节点在 cur 的右子树中
|
||||
if (cur.val < num)
|
||||
cur = cur.right;
|
||||
// 待删除节点在 cur 的左子树中
|
||||
else
|
||||
cur = cur.left;
|
||||
}
|
||||
// 若无待删除节点，直接返回
|
||||
if (cur == null) return;
|
||||
// 子节点数量 = 0 or 1
|
||||
if (cur.left == null || cur.right == null) {
|
||||
// 当子节点数量 = 0 / 1 时， child = null / 该子节点
|
||||
TreeNode? child = cur.left ?? cur.right;
|
||||
// 删除节点 cur
|
||||
if (cur != _root) {
|
||||
if (pre!.left == cur)
|
||||
pre.left = child;
|
||||
else
|
||||
pre.right = child;
|
||||
} else {
|
||||
// 若删除节点为根节点，则重新指定根节点
|
||||
_root = child;
|
||||
}
|
||||
} else {
|
||||
// 子节点数量 = 2
|
||||
// 获取中序遍历中 cur 的下一个节点
|
||||
TreeNode? tmp = cur.right;
|
||||
while (tmp!.left != null) {
|
||||
tmp = tmp.left;
|
||||
}
|
||||
// 递归删除节点 tmp
|
||||
remove(tmp.val);
|
||||
// 用 tmp 覆盖 cur
|
||||
cur.val = tmp.val;
|
||||
}
|
||||
}
|
||||
```
|
||||
</code></pre></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:1"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* 删除节点 */</span>
|
||||
<a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span> <span class="nf">remove</span><span class="p">(</span><span class="o">&</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w"> </span><span class="c1">// 若树为空，直接提前返回</span>
|
||||
<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"> </span>
|
||||
<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span>
|
||||
<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w"> </span><span class="c1">// 循环查找，越过叶节点后跳出</span>
|
||||
<a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w"> </span><span class="c1">// 找到待删除节点，跳出循环</span>
|
||||
<a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w"> </span><span class="c1">// 待删除节点在 cur 的右子树中</span>
|
||||
<a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w"> </span><span class="c1">// 待删除节点在 cur 的左子树中</span>
|
||||
<a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-24" name="__codelineno-35-24" href="#__codelineno-35-24"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-25" name="__codelineno-35-25" href="#__codelineno-35-25"></a><span class="w"> </span><span class="c1">// 若无待删除节点，则直接返回</span>
|
||||
<a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-27" name="__codelineno-35-27" href="#__codelineno-35-27"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-28" name="__codelineno-35-28" href="#__codelineno-35-28"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-29" name="__codelineno-35-29" href="#__codelineno-35-29"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w"> </span><span class="c1">// 子节点数量 = 0 or 1</span>
|
||||
<a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-32" name="__codelineno-35-32" href="#__codelineno-35-32"></a><span class="w"> </span><span class="c1">// 当子节点数量 = 0 / 1 时， child = nullptr / 该子节点</span>
|
||||
<a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">().</span><span class="n">or_else</span><span class="p">(</span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
|
||||
<a id="__codelineno-35-34" name="__codelineno-35-34" href="#__codelineno-35-34"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-35" name="__codelineno-35-35" href="#__codelineno-35-35"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-36" name="__codelineno-35-36" href="#__codelineno-35-36"></a><span class="w"> </span><span class="c1">// 删除节点 cur</span>
|
||||
<a id="__codelineno-35-37" name="__codelineno-35-37" href="#__codelineno-35-37"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">Rc</span>::<span class="n">ptr_eq</span><span class="p">(</span><span class="o">&</span><span class="n">cur</span><span class="p">,</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">as_ref</span><span class="p">().</span><span class="n">unwrap</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-38" name="__codelineno-35-38" href="#__codelineno-35-38"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">Rc</span>::<span class="n">ptr_eq</span><span class="p">(</span><span class="o">&</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-39" name="__codelineno-35-39" href="#__codelineno-35-39"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-40" name="__codelineno-35-40" href="#__codelineno-35-40"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-42" name="__codelineno-35-42" href="#__codelineno-35-42"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-43" name="__codelineno-35-43" href="#__codelineno-35-43"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-44" name="__codelineno-35-44" href="#__codelineno-35-44"></a><span class="w"> </span><span class="c1">// 若删除节点为根节点，则重新指定根节点</span>
|
||||
<a id="__codelineno-35-45" name="__codelineno-35-45" href="#__codelineno-35-45"></a><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-46" name="__codelineno-35-46" href="#__codelineno-35-46"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-47" name="__codelineno-35-47" href="#__codelineno-35-47"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-48" name="__codelineno-35-48" href="#__codelineno-35-48"></a><span class="w"> </span><span class="c1">// 子节点数量 = 2</span>
|
||||
<a id="__codelineno-35-49" name="__codelineno-35-49" href="#__codelineno-35-49"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-50" name="__codelineno-35-50" href="#__codelineno-35-50"></a><span class="w"> </span><span class="c1">// 获取中序遍历中 cur 的下一个节点</span>
|
||||
<a id="__codelineno-35-51" name="__codelineno-35-51" href="#__codelineno-35-51"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-52" name="__codelineno-35-52" href="#__codelineno-35-52"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-53" name="__codelineno-35-53" href="#__codelineno-35-53"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-54" name="__codelineno-35-54" href="#__codelineno-35-54"></a><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-35-55" name="__codelineno-35-55" href="#__codelineno-35-55"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-56" name="__codelineno-35-56" href="#__codelineno-35-56"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-57" name="__codelineno-35-57" href="#__codelineno-35-57"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-58" name="__codelineno-35-58" href="#__codelineno-35-58"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-59" name="__codelineno-35-59" href="#__codelineno-35-59"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">tmpval</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">unwrap</span><span class="p">().</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-60" name="__codelineno-35-60" href="#__codelineno-35-60"></a><span class="w"> </span><span class="c1">// 递归删除节点 tmp</span>
|
||||
<a id="__codelineno-35-61" name="__codelineno-35-61" href="#__codelineno-35-61"></a><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmpval</span><span class="p">);</span>
|
||||
<a id="__codelineno-35-62" name="__codelineno-35-62" href="#__codelineno-35-62"></a><span class="w"> </span><span class="c1">// 用 tmp 覆盖 cur</span>
|
||||
<a id="__codelineno-35-63" name="__codelineno-35-63" href="#__codelineno-35-63"></a><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmpval</span><span class="p">;</span>
|
||||
<a id="__codelineno-35-64" name="__codelineno-35-64" href="#__codelineno-35-64"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-65" name="__codelineno-35-65" href="#__codelineno-35-65"></a><span class="p">}</span>
|
||||
<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* 删除节点 */</span>
|
||||
<a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span> <span class="nf">remove</span><span class="p">(</span><span class="o">&</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w"> </span><span class="c1">// 若树为空，直接提前返回</span>
|
||||
<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"> </span>
|
||||
<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span>
|
||||
<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w"> </span><span class="c1">// 循环查找，越过叶节点后跳出</span>
|
||||
<a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w"> </span><span class="c1">// 找到待删除节点，跳出循环</span>
|
||||
<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w"> </span><span class="c1">// 待删除节点在 cur 的右子树中</span>
|
||||
<a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w"> </span><span class="c1">// 待删除节点在 cur 的左子树中</span>
|
||||
<a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-22" name="__codelineno-34-22" href="#__codelineno-34-22"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><span class="w"> </span><span class="c1">// 若无待删除节点，则直接返回</span>
|
||||
<a id="__codelineno-34-26" name="__codelineno-34-26" href="#__codelineno-34-26"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-27" name="__codelineno-34-27" href="#__codelineno-34-27"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-28" name="__codelineno-34-28" href="#__codelineno-34-28"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-30" name="__codelineno-34-30" href="#__codelineno-34-30"></a><span class="w"> </span><span class="c1">// 子节点数量 = 0 or 1</span>
|
||||
<a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w"> </span><span class="c1">// 当子节点数量 = 0 / 1 时， child = nullptr / 该子节点</span>
|
||||
<a id="__codelineno-34-33" name="__codelineno-34-33" href="#__codelineno-34-33"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">().</span><span class="n">or_else</span><span class="p">(</span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
|
||||
<a id="__codelineno-34-34" name="__codelineno-34-34" href="#__codelineno-34-34"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-35" name="__codelineno-34-35" href="#__codelineno-34-35"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-36" name="__codelineno-34-36" href="#__codelineno-34-36"></a><span class="w"> </span><span class="c1">// 删除节点 cur</span>
|
||||
<a id="__codelineno-34-37" name="__codelineno-34-37" href="#__codelineno-34-37"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">Rc</span>::<span class="n">ptr_eq</span><span class="p">(</span><span class="o">&</span><span class="n">cur</span><span class="p">,</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">as_ref</span><span class="p">().</span><span class="n">unwrap</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-38" name="__codelineno-34-38" href="#__codelineno-34-38"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">Rc</span>::<span class="n">ptr_eq</span><span class="p">(</span><span class="o">&</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-39" name="__codelineno-34-39" href="#__codelineno-34-39"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-40" name="__codelineno-34-40" href="#__codelineno-34-40"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-42" name="__codelineno-34-42" href="#__codelineno-34-42"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-43" name="__codelineno-34-43" href="#__codelineno-34-43"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-44" name="__codelineno-34-44" href="#__codelineno-34-44"></a><span class="w"> </span><span class="c1">// 若删除节点为根节点，则重新指定根节点</span>
|
||||
<a id="__codelineno-34-45" name="__codelineno-34-45" href="#__codelineno-34-45"></a><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-46" name="__codelineno-34-46" href="#__codelineno-34-46"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-47" name="__codelineno-34-47" href="#__codelineno-34-47"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-48" name="__codelineno-34-48" href="#__codelineno-34-48"></a><span class="w"> </span><span class="c1">// 子节点数量 = 2</span>
|
||||
<a id="__codelineno-34-49" name="__codelineno-34-49" href="#__codelineno-34-49"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-50" name="__codelineno-34-50" href="#__codelineno-34-50"></a><span class="w"> </span><span class="c1">// 获取中序遍历中 cur 的下一个节点</span>
|
||||
<a id="__codelineno-34-51" name="__codelineno-34-51" href="#__codelineno-34-51"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-52" name="__codelineno-34-52" href="#__codelineno-34-52"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-53" name="__codelineno-34-53" href="#__codelineno-34-53"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-54" name="__codelineno-34-54" href="#__codelineno-34-54"></a><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
|
||||
<a id="__codelineno-34-55" name="__codelineno-34-55" href="#__codelineno-34-55"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-34-56" name="__codelineno-34-56" href="#__codelineno-34-56"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-57" name="__codelineno-34-57" href="#__codelineno-34-57"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-58" name="__codelineno-34-58" href="#__codelineno-34-58"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-59" name="__codelineno-34-59" href="#__codelineno-34-59"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">tmpval</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">unwrap</span><span class="p">().</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-60" name="__codelineno-34-60" href="#__codelineno-34-60"></a><span class="w"> </span><span class="c1">// 递归删除节点 tmp</span>
|
||||
<a id="__codelineno-34-61" name="__codelineno-34-61" href="#__codelineno-34-61"></a><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmpval</span><span class="p">);</span>
|
||||
<a id="__codelineno-34-62" name="__codelineno-34-62" href="#__codelineno-34-62"></a><span class="w"> </span><span class="c1">// 用 tmp 覆盖 cur</span>
|
||||
<a id="__codelineno-34-63" name="__codelineno-34-63" href="#__codelineno-34-63"></a><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmpval</span><span class="p">;</span>
|
||||
<a id="__codelineno-34-64" name="__codelineno-34-64" href="#__codelineno-34-64"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-34-65" name="__codelineno-34-65" href="#__codelineno-34-65"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -4780,7 +4902,7 @@
|
|||
<p>我们知道，二叉树的中序遍历遵循“左 <span class="arithmatex">\(\rightarrow\)</span> 根 <span class="arithmatex">\(\rightarrow\)</span> 右”的遍历顺序，而二叉搜索树满足“左子节点 <span class="arithmatex">\(<\)</span> 根节点 <span class="arithmatex">\(<\)</span> 右子节点”的大小关系。因此，在二叉搜索树中进行中序遍历时，总是会优先遍历下一个最小节点，从而得出一个重要性质：<strong>二叉搜索树的中序遍历序列是升序的</strong>。</p>
|
||||
<p>利用中序遍历升序的性质，我们在二叉搜索树中获取有序数据仅需 <span class="arithmatex">\(O(n)\)</span> 时间，无需额外排序，非常高效。</p>
|
||||
<p><img alt="二叉搜索树的中序遍历序列" src="../binary_search_tree.assets/bst_inorder_traversal.png" /></p>
|
||||
<p align="center"> Fig. 二叉搜索树的中序遍历序列 </p>
|
||||
<p align="center"> 图：二叉搜索树的中序遍历序列 </p>
|
||||
|
||||
<h2 id="742">7.4.2. 二叉搜索树的效率<a class="headerlink" href="#742" title="Permanent link">¶</a></h2>
|
||||
<p>给定一组数据，我们考虑使用数组或二叉搜索树存储。</p>
|
||||
|
|
@ -4816,7 +4938,7 @@
|
|||
<p>在理想情况下，二叉搜索树是“平衡”的，这样就可以在 <span class="arithmatex">\(\log n\)</span> 轮循环内查找任意节点。</p>
|
||||
<p>然而，如果我们在二叉搜索树中不断地插入和删除节点，可能导致二叉树退化为链表，这时各种操作的时间复杂度也会退化为 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
||||
<p><img alt="二叉搜索树的平衡与退化" src="../binary_search_tree.assets/bst_degradation.png" /></p>
|
||||
<p align="center"> Fig. 二叉搜索树的平衡与退化 </p>
|
||||
<p align="center"> 图：二叉搜索树的平衡与退化 </p>
|
||||
|
||||
<h2 id="743">7.4.3. 二叉搜索树常见应用<a class="headerlink" href="#743" title="Permanent link">¶</a></h2>
|
||||
<ul>
|
||||
|
|
|
|||
|
|
@ -3508,15 +3508,15 @@
|
|||
|
||||
|
||||
<h1 id="71">7.1. 二叉树<a class="headerlink" href="#71" title="Permanent link">¶</a></h1>
|
||||
<p>「二叉树 Binary Tree」是一种非线性数据结构，代表着祖先与后代之间的派生关系，体现着“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含一个「值」和两个「指针」。</p>
|
||||
<p>「二叉树 Binary Tree」是一种非线性数据结构，代表着祖先与后代之间的派生关系，体现着“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含：值、左子节点引用、右子节点引用。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 二叉树节点类 */</span>
|
||||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kd">class</span> <span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
|
||||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
|
||||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点引用</span>
|
||||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点引用</span>
|
||||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -3536,8 +3536,8 @@
|
|||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="w"> </span><span class="sd">"""二叉树节点类"""</span>
|
||||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
|
||||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span> <span class="c1"># 节点值</span>
|
||||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 左子节点指针</span>
|
||||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 右子节点指针</span>
|
||||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 左子节点引用</span>
|
||||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 右子节点引用</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
|
|
@ -3550,9 +3550,9 @@
|
|||
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* 节点初始化方法 */</span>
|
||||
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="nx">v</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">&</span><span class="nx">TreeNode</span><span class="p">{</span>
|
||||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w"> </span><span class="nx">Left</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span>
|
||||
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w"> </span><span class="nx">Right</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span>
|
||||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w"> </span><span class="nx">Val</span><span class="p">:</span><span class="w"> </span><span class="nx">v</span><span class="p">,</span>
|
||||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w"> </span><span class="nx">Left</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
|
||||
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w"> </span><span class="nx">Right</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
|
||||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w"> </span><span class="nx">Val</span><span class="p">:</span><span class="w"> </span><span class="nx">v</span><span class="p">,</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -3561,8 +3561,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 二叉树节点类 */</span>
|
||||
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
|
||||
<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
|
||||
<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// 左子节点引用</span>
|
||||
<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 右子节点引用</span>
|
||||
<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
|
|
@ -3575,8 +3575,8 @@
|
|||
<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a>
|
||||
<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w"> </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">left?</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="nx">right?</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
|
||||
<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
|
||||
<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点引用</span>
|
||||
<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点引用</span>
|
||||
<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -3609,8 +3609,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 二叉树节点类 */</span>
|
||||
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
|
||||
<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
|
||||
<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点引用</span>
|
||||
<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点引用</span>
|
||||
<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -3619,8 +3619,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 二叉树节点类 */</span>
|
||||
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">class</span> <span class="nc">TreeNode</span> <span class="p">{</span>
|
||||
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a> <span class="kd">var</span> <span class="nv">val</span><span class="p">:</span> <span class="nb">Int</span> <span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="kd">var</span> <span class="nv">left</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?</span> <span class="c1">// 左子节点指针</span>
|
||||
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="kd">var</span> <span class="nv">right</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?</span> <span class="c1">// 右子节点指针</span>
|
||||
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="kd">var</span> <span class="nv">left</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?</span> <span class="c1">// 左子节点引用</span>
|
||||
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="kd">var</span> <span class="nv">right</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?</span> <span class="c1">// 右子节点引用</span>
|
||||
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a>
|
||||
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="kd">init</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">{</span>
|
||||
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <span class="n">val</span> <span class="p">=</span> <span class="n">x</span>
|
||||
|
|
@ -3636,8 +3636,8 @@
|
|||
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 二叉树节点类 */</span>
|
||||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
||||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
|
||||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
|
||||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点引用</span>
|
||||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点引用</span>
|
||||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">right</span><span class="p">]);</span>
|
||||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
|
|
@ -3651,7 +3651,7 @@
|
|||
<p>节点的两个指针分别指向「左子节点」和「右子节点」，同时该节点被称为这两个子节点的「父节点」。当给定一个二叉树的节点时，我们将该节点的左子节点及其以下节点形成的树称为该节点的「左子树」，同理可得「右子树」。</p>
|
||||
<p><strong>在二叉树中，除叶节点外，其他所有节点都包含子节点和非空子树</strong>。例如，在以下示例中，若将“节点 2”视为父节点，则其左子节点和右子节点分别是“节点 4”和“节点 5”，左子树是“节点 4 及其以下节点形成的树”，右子树是“节点 5 及其以下节点形成的树”。</p>
|
||||
<p><img alt="父节点、子节点、子树" src="../binary_tree.assets/binary_tree_definition.png" /></p>
|
||||
<p align="center"> Fig. 父节点、子节点、子树 </p>
|
||||
<p align="center"> 图：父节点、子节点、子树 </p>
|
||||
|
||||
<h2 id="711">7.1.1. 二叉树常见术语<a class="headerlink" href="#711" title="Permanent link">¶</a></h2>
|
||||
<p>二叉树涉及的术语较多，建议尽量理解并记住。</p>
|
||||
|
|
@ -3666,7 +3666,7 @@
|
|||
<li>节点的「高度 Height」：从最远叶节点到该节点所经过的边的数量。</li>
|
||||
</ul>
|
||||
<p><img alt="二叉树的常用术语" src="../binary_tree.assets/binary_tree_terminology.png" /></p>
|
||||
<p align="center"> Fig. 二叉树的常用术语 </p>
|
||||
<p align="center"> 图：二叉树的常用术语 </p>
|
||||
|
||||
<div class="admonition tip">
|
||||
<p class="admonition-title">高度与深度的定义</p>
|
||||
|
|
@ -3836,7 +3836,7 @@
|
|||
</div>
|
||||
<p><strong>插入与删除节点</strong>。与链表类似，通过修改指针来实现插入与删除节点。</p>
|
||||
<p><img alt="在二叉树中插入与删除节点" src="../binary_tree.assets/binary_tree_add_remove.png" /></p>
|
||||
<p align="center"> Fig. 在二叉树中插入与删除节点 </p>
|
||||
<p align="center"> 图：在二叉树中插入与删除节点 </p>
|
||||
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
|
|
@ -3960,22 +3960,22 @@
|
|||
<p>在中文社区中，完美二叉树常被称为「满二叉树」，请注意区分。</p>
|
||||
</div>
|
||||
<p><img alt="完美二叉树" src="../binary_tree.assets/perfect_binary_tree.png" /></p>
|
||||
<p align="center"> Fig. 完美二叉树 </p>
|
||||
<p align="center"> 图：完美二叉树 </p>
|
||||
|
||||
<h3 id="_2">完全二叉树<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<p>「完全二叉树 Complete Binary Tree」只有最底层的节点未被填满，且最底层节点尽量靠左填充。</p>
|
||||
<p><img alt="完全二叉树" src="../binary_tree.assets/complete_binary_tree.png" /></p>
|
||||
<p align="center"> Fig. 完全二叉树 </p>
|
||||
<p align="center"> 图：完全二叉树 </p>
|
||||
|
||||
<h3 id="_3">完满二叉树<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||||
<p>「完满二叉树 Full Binary Tree」除了叶节点之外，其余所有节点都有两个子节点。</p>
|
||||
<p><img alt="完满二叉树" src="../binary_tree.assets/full_binary_tree.png" /></p>
|
||||
<p align="center"> Fig. 完满二叉树 </p>
|
||||
<p align="center"> 图：完满二叉树 </p>
|
||||
|
||||
<h3 id="_4">平衡二叉树<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||||
<p>「平衡二叉树 Balanced Binary Tree」中任意节点的左子树和右子树的高度之差的绝对值不超过 1 。</p>
|
||||
<p><img alt="平衡二叉树" src="../binary_tree.assets/balanced_binary_tree.png" /></p>
|
||||
<p align="center"> Fig. 平衡二叉树 </p>
|
||||
<p align="center"> 图：平衡二叉树 </p>
|
||||
|
||||
<h2 id="714">7.1.4. 二叉树的退化<a class="headerlink" href="#714" title="Permanent link">¶</a></h2>
|
||||
<p>当二叉树的每层节点都被填满时，达到「完美二叉树」；而当所有节点都偏向一侧时，二叉树退化为「链表」。</p>
|
||||
|
|
@ -3984,7 +3984,7 @@
|
|||
<li>链表则是另一个极端，各项操作都变为线性操作，时间复杂度退化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
|
||||
</ul>
|
||||
<p><img alt="二叉树的最佳与最差结构" src="../binary_tree.assets/binary_tree_best_worst_cases.png" /></p>
|
||||
<p align="center"> Fig. 二叉树的最佳与最差结构 </p>
|
||||
<p align="center"> 图：二叉树的最佳与最差结构 </p>
|
||||
|
||||
<p>如下表所示，在最佳和最差结构下，二叉树的叶节点数量、节点总数、高度等达到极大或极小值。</p>
|
||||
<div class="center-table">
|
||||
|
|
|
|||
|
|
@ -3418,7 +3418,7 @@
|
|||
<p>「层序遍历 Level-Order Traversal」从顶部到底部逐层遍历二叉树，并在每一层按照从左到右的顺序访问节点。</p>
|
||||
<p>层序遍历本质上属于「广度优先搜索 Breadth-First Traversal」，它体现了一种“一圈一圈向外扩展”的逐层搜索方式。</p>
|
||||
<p><img alt="二叉树的层序遍历" src="../binary_tree_traversal.assets/binary_tree_bfs.png" /></p>
|
||||
<p align="center"> Fig. 二叉树的层序遍历 </p>
|
||||
<p align="center"> 图：二叉树的层序遍历 </p>
|
||||
|
||||
<p>广度优先遍历通常借助「队列」来实现。队列遵循“先进先出”的规则，而广度优先遍历则遵循“逐层推进”的规则，两者背后的思想是一致的。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
|
|
@ -3709,7 +3709,7 @@
|
|||
<p>相应地，前序、中序和后序遍历都属于「深度优先遍历 Depth-First Traversal」，它体现了一种“先走到尽头，再回溯继续”的遍历方式。</p>
|
||||
<p>如下图所示，左侧是深度优先遍历的示意图，右上方是对应的递归代码。深度优先遍历就像是绕着整个二叉树的外围“走”一圈，在这个过程中，在每个节点都会遇到三个位置，分别对应前序遍历、中序遍历和后序遍历。</p>
|
||||
<p><img alt="二叉搜索树的前、中、后序遍历" src="../binary_tree_traversal.assets/binary_tree_dfs.png" /></p>
|
||||
<p align="center"> Fig. 二叉搜索树的前、中、后序遍历 </p>
|
||||
<p align="center"> 图：二叉搜索树的前、中、后序遍历 </p>
|
||||
|
||||
<p>以下给出了实现代码，请配合上图理解深度优先遍历的递归过程。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
|
|
@ -4151,6 +4151,7 @@
|
|||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p align="center"> 图：前序遍历的递归过程 </p>
|
||||
|
||||
|
||||
|
||||
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<lastmod>2023-08-16</lastmod>
|
||||
<changefreq>daily</changefreq>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://www.hello-algo.com/chapter_tree/binary_tree_traversal/</loc>
|
||||
<lastmod>2023-08-14</lastmod>
|
||||
<lastmod>2023-08-16</lastmod>
|
||||
<changefreq>daily</changefreq>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://www.hello-algo.com/chapter_tree/summary/</loc>
|
||||
<lastmod>2023-08-14</lastmod>
|
||||
<lastmod>2023-08-16</lastmod>
|
||||
<changefreq>daily</changefreq>
|
||||
</url>
|
||||
</urlset>
|
||||
BIN
sitemap.xml.gz
BIN
sitemap.xml.gz
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Reference in a new issue