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<title>9.1.   图 - Hello 算法</title>
<title>10.1.   图 - Hello 算法</title>
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<div data-md-component="skip">
<a href="#91" class="md-skip">
<a href="#101" class="md-skip">
跳转至
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
9.1. &nbsp; 图
10.1. &nbsp; 图
</span>
</div>
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
2.4. &nbsp; 权衡时间与空间
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
2.5. &nbsp; 小结
2.4. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 二分查找
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</label>
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6. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 二分查找
</label>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. &nbsp; 哈希表
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. &nbsp; 哈希冲突处理
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. &nbsp; 小结
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
</a>
</li>
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 树
7. &nbsp; &nbsp; 散列表
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7. &nbsp; &nbsp; 树
7. &nbsp; &nbsp; 散列表
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -962,8 +910,8 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
7.1. &nbsp; 哈希表
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
7.2. &nbsp; 哈希冲突处理
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.3. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. &nbsp; AVL 树 *
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.5. &nbsp; 小结
<a href="../../chapter_hashing/summary/" class="md-nav__link">
7.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_9">
<span class="md-nav__icon md-icon"></span>
8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
8.1. &nbsp; 二叉树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
8.2. &nbsp; 二叉树遍历
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
8.3. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.4. &nbsp; AVL 树 *
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
8.5. &nbsp; 小结
</a>
</li>
</ul>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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9. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
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9. &nbsp; &nbsp; 堆
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
8.1. &nbsp; 堆
9.1. &nbsp; 堆
</a>
</li>
@@ -1092,7 +1131,7 @@
<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
8.2. &nbsp; 建堆操作 *
9.2. &nbsp; 建堆操作 *
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
8.3. &nbsp; 小结
9.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 图
<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. &nbsp; &nbsp; 图
<span class="md-nav__icon md-icon"></span>
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9. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 图
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<label class="md-nav__link md-nav__link--active" for="__toc">
9.1. &nbsp; 图
10.1. &nbsp; 图
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
9.1. &nbsp; 图
10.1. &nbsp; 图
</a>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#911" class="md-nav__link">
9.1.1. &nbsp; 图常见类型
<a href="#1011" class="md-nav__link">
10.1.1. &nbsp; 图常见类型
</a>
</li>
<li class="md-nav__item">
<a href="#912" class="md-nav__link">
9.1.2. &nbsp; 图常用术语
<a href="#1012" class="md-nav__link">
10.1.2. &nbsp; 图常用术语
</a>
</li>
<li class="md-nav__item">
<a href="#913" class="md-nav__link">
9.1.3. &nbsp; 图的表示
<a href="#1013" class="md-nav__link">
10.1.3. &nbsp; 图的表示
</a>
<nav class="md-nav" aria-label="9.1.3. &nbsp; 图的表示">
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</li>
<li class="md-nav__item">
<a href="#914" class="md-nav__link">
9.1.4. &nbsp; 图常见应用
<a href="#1014" class="md-nav__link">
10.1.4. &nbsp; 图常见应用
</a>
</li>
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<li class="md-nav__item">
<a href="../graph_operations/" class="md-nav__link">
9.2. &nbsp; 图基础操作
10.2. &nbsp; 图基础操作
</a>
</li>
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<li class="md-nav__item">
<a href="../graph_traversal/" class="md-nav__link">
9.3. &nbsp; 图的遍历
10.3. &nbsp; 图的遍历
</a>
</li>
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<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
9.4. &nbsp; 小结
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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10. &nbsp; &nbsp; 查找算法
<span class="md-nav__icon md-icon"></span>
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10. &nbsp; &nbsp; 查找算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. &nbsp; 线性查找
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.2. &nbsp; 二分查找
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. &nbsp; 哈希查找
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.4. &nbsp; 小结
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.6. &nbsp; 桶排序（New）
11.6. &nbsp; 桶排序
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
11.7. &nbsp; 计数排序（New）
11.7. &nbsp; 计数排序
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
11.8. &nbsp; 基数排序（New）
11.8. &nbsp; 基数排序
</a>
</li>
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
<span class="md-nav__icon md-icon"></span>
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12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
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<li class="md-nav__item">
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
12.1. &nbsp; 回溯算法（New）
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
12.1. &nbsp; 搜索算法（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
12.2. &nbsp; 哈希优化策略
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
12.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
13. &nbsp; &nbsp; 附录
13. &nbsp; &nbsp; 回溯算法
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13. &nbsp; &nbsp; 附录
13. &nbsp; &nbsp; 回溯算法
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<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
13.1. &nbsp; 编程环境安装
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
13.2. &nbsp; 一起参与创作
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
13.1. &nbsp; 回溯算法（New）
</a>
</li>
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<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
14. &nbsp; &nbsp; 附录
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14. &nbsp; &nbsp; 附录
</label>
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<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
14.1. &nbsp; 编程环境安装
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
</a>
</li>
</ul>
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参考文献
</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#911" class="md-nav__link">
9.1.1. &nbsp; 图常见类型
<a href="#1011" class="md-nav__link">
10.1.1. &nbsp; 图常见类型
</a>
</li>
<li class="md-nav__item">
<a href="#912" class="md-nav__link">
9.1.2. &nbsp; 图常用术语
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10.1.2. &nbsp; 图常用术语
</a>
</li>
<li class="md-nav__item">
<a href="#913" class="md-nav__link">
9.1.3. &nbsp; 图的表示
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10.1.3. &nbsp; 图的表示
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9.1.4. &nbsp; 图常见应用
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10.1.4. &nbsp; 图常见应用
</a>
</li>
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<h1 id="91">9.1. &nbsp; 图<a class="headerlink" href="#91" title="Permanent link">&para;</a></h1>
<h1 id="101">10.1. &nbsp; 图<a class="headerlink" href="#101" title="Permanent link">&para;</a></h1>
<p>「图 Graph」是一种非线性数据结构，由「顶点 Vertex」和「边 Edge」组成。我们可以将图 <span class="arithmatex">\(G\)</span> 抽象地表示为一组顶点 <span class="arithmatex">\(V\)</span> 和一组边 <span class="arithmatex">\(E\)</span> 的集合。以下示例展示了一个包含 5 个顶点和 7 条边的图。</p>
<div class="arithmatex">\[
\begin{aligned}
@@ -1875,7 +1898,7 @@ G &amp; = \{ V, E \} \newline
<p align="center"> Fig. 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. &nbsp; 图常见类型<a class="headerlink" href="#911" title="Permanent link">&para;</a></h2>
<h2 id="1011">10.1.1. &nbsp; 图常见类型<a class="headerlink" href="#1011" title="Permanent link">&para;</a></h2>
<p>根据边是否具有方向，可分为「无向图 Undirected Graph」和「有向图 Directed Graph」。</p>
<ul>
<li>在无向图中，边表示两顶点之间的“双向”连接关系，例如微信或 QQ 中的“好友关系”；</li>
@@ -1896,13 +1919,13 @@ G &amp; = \{ V, E \} \newline
<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>
<h2 id="912">9.1.2. &nbsp; 图常用术语<a class="headerlink" href="#912" title="Permanent link">&para;</a></h2>
<h2 id="1012">10.1.2. &nbsp; 图常用术语<a class="headerlink" href="#1012" title="Permanent link">&para;</a></h2>
<ul>
<li>「邻接 Adjacency」：当两顶点之间存在边相连时，称这两顶点“邻接”。在上图中，顶点 1 的邻接顶点为顶点 2、3、5。</li>
<li>「路径 Path」：从顶点 A 到顶点 B 经过的边构成的序列被称为从 A 到 B 的“路径”。在上图中，边序列 1-5-2-4 是顶点 1 到顶点 4 的一条路径。</li>
<li>「度 Degree」表示一个顶点拥有的边数。对于有向图，「入度 In-Degree」表示有多少条边指向该顶点，「出度 Out-Degree」表示有多少条边从该顶点指出。</li>
</ul>
<h2 id="913">9.1.3. &nbsp; 图的表示<a class="headerlink" href="#913" title="Permanent link">&para;</a></h2>
<h2 id="1013">10.1.3. &nbsp; 图的表示<a class="headerlink" href="#1013" title="Permanent link">&para;</a></h2>
<p>图的常用表示方法包括「邻接矩阵」和「邻接表」。以下使用无向图进行举例。</p>
<h3 id="_1">邻接矩阵<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<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>
@@ -1924,7 +1947,7 @@ G &amp; = \{ V, E \} \newline
<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>
<h2 id="914">9.1.4. &nbsp; 图常见应用<a class="headerlink" href="#914" title="Permanent link">&para;</a></h2>
<h2 id="1014">10.1.4. &nbsp; 图常见应用<a class="headerlink" href="#1014" title="Permanent link">&para;</a></h2>
<p>实际应用中，许多系统都可以用图来建模，相应的待求解问题也可以约化为图计算问题。</p>
<div class="center-table">
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@@ -2035,7 +2058,7 @@ G &amp; = \{ V, E \} \newline
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2.4. &nbsp; 权衡时间与空间
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6.3. &nbsp; 小结
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6.1. &nbsp; 二分查找
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7. &nbsp; &nbsp; 散列表
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7.2. &nbsp; 二叉树遍历
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7.2. &nbsp; 哈希冲突处理
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7.3. &nbsp; 二叉搜索树
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. &nbsp; AVL 树 *
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7.5. &nbsp; 小结
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8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
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8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
8.1. &nbsp; 二叉树
</a>
</li>
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
8.2. &nbsp; 二叉树遍历
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
8.3. &nbsp; 二叉搜索树
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8.4. &nbsp; AVL 树 *
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8.5. &nbsp; 小结
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8.1. &nbsp; 堆
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</a>
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
8.2. &nbsp; 建堆操作 *
9.2. &nbsp; 建堆操作 *
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10. &nbsp; &nbsp; 图
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<a href="../graph/" class="md-nav__link">
9.1. &nbsp; 图
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</a>
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<label class="md-nav__link md-nav__link--active" for="__toc">
9.2. &nbsp; 图基础操作
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<a href="./" class="md-nav__link md-nav__link--active">
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10.2. &nbsp; 图基础操作
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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10.2.1. &nbsp; 基于邻接矩阵的实现
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9.2.2. &nbsp; 基于邻接表的实现
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10.2.2. &nbsp; 基于邻接表的实现
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9.2.3. &nbsp; 效率对比
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10.2.3. &nbsp; 效率对比
</a>
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@@ -1252,7 +1291,7 @@
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9.3. &nbsp; 图的遍历
10.3. &nbsp; 图的遍历
</a>
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10.1. &nbsp; 线性查找
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10.2. &nbsp; 二分查找
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10.3. &nbsp; 哈希查找
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11.6. &nbsp; 桶排序（New）
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11.7. &nbsp; 计数排序（New）
11.7. &nbsp; 计数排序
</a>
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<li class="md-nav__item">
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
11.8. &nbsp; 基数排序（New）
11.8. &nbsp; 基数排序
</a>
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12.1. &nbsp; 回溯算法（New）
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12.1. &nbsp; 搜索算法（New）
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13.2. &nbsp; 一起参与创作
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13.1. &nbsp; 回溯算法（New）
</a>
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14.1. &nbsp; 编程环境安装
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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9.2.1. &nbsp; 基于邻接矩阵的实现
<a href="#1021" class="md-nav__link">
10.2.1. &nbsp; 基于邻接矩阵的实现
</a>
</li>
<li class="md-nav__item">
<a href="#922" class="md-nav__link">
9.2.2. &nbsp; 基于邻接表的实现
<a href="#1022" class="md-nav__link">
10.2.2. &nbsp; 基于邻接表的实现
</a>
</li>
<li class="md-nav__item">
<a href="#923" class="md-nav__link">
9.2.3. &nbsp; 效率对比
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10.2.3. &nbsp; 效率对比
</a>
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@@ -1808,9 +1831,9 @@
<h1 id="92">9.2. &nbsp; 图基础操作<a class="headerlink" href="#92" title="Permanent link">&para;</a></h1>
<h1 id="102">10.2. &nbsp; 图基础操作<a class="headerlink" href="#102" title="Permanent link">&para;</a></h1>
<p>图的基础操作可分为对「边」的操作和对「顶点」的操作。在「邻接矩阵」和「邻接表」两种表示方法下，实现方式有所不同。</p>
<h2 id="921">9.2.1. &nbsp; 基于邻接矩阵的实现<a class="headerlink" href="#921" title="Permanent link">&para;</a></h2>
<h2 id="1021">10.2.1. &nbsp; 基于邻接矩阵的实现<a class="headerlink" href="#1021" title="Permanent link">&para;</a></h2>
<p>给定一个顶点数量为 <span class="arithmatex">\(n\)</span> 的无向图，则有：</p>
<ul>
<li><strong>添加或删除边</strong>：直接在邻接矩阵中修改指定的边即可，使用 <span class="arithmatex">\(O(1)\)</span> 时间。而由于是无向图，因此需要同时更新两个方向的边。</li>
@@ -2565,7 +2588,7 @@
</div>
</div>
</div>
<h2 id="922">9.2.2. &nbsp; 基于邻接表的实现<a class="headerlink" href="#922" title="Permanent link">&para;</a></h2>
<h2 id="1022">10.2.2. &nbsp; 基于邻接表的实现<a class="headerlink" href="#1022" title="Permanent link">&para;</a></h2>
<p>设无向图的顶点总数为 <span class="arithmatex">\(n\)</span> 、边总数为 <span class="arithmatex">\(m\)</span> ，则有：</p>
<ul>
<li><strong>添加边</strong>：在顶点对应链表的末尾添加边即可，使用 <span class="arithmatex">\(O(1)\)</span> 时间。因为是无向图，所以需要同时添加两个方向的边。</li>
@@ -3236,7 +3259,7 @@
</div>
</div>
</div>
<h2 id="923">9.2.3. &nbsp; 效率对比<a class="headerlink" href="#923" title="Permanent link">&para;</a></h2>
<h2 id="1023">10.2.3. &nbsp; 效率对比<a class="headerlink" href="#1023" title="Permanent link">&para;</a></h2>
<p>设图中共有 <span class="arithmatex">\(n\)</span> 个顶点和 <span class="arithmatex">\(m\)</span> 条边，下表为邻接矩阵和邻接表的时间和空间效率对比。</p>
<div class="center-table">
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<nav class="md-footer__inner md-grid" aria-label="页脚" >
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</span>
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9.1. &nbsp; 图
10.1. &nbsp; 图
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</a>
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<title>9.3.   图的遍历 - Hello 算法</title>
<title>10.3.   图的遍历 - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
9.3. &nbsp; 图的遍历
10.3. &nbsp; 图的遍历
</span>
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
2.4. &nbsp; 权衡时间与空间
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
2.5. &nbsp; 小结
2.4. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 二分查找
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6. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 二分查找
</label>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. &nbsp; 哈希表
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. &nbsp; 哈希冲突处理
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. &nbsp; 小结
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
</a>
</li>
@@ -938,21 +890,17 @@
<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 树
7. &nbsp; &nbsp; 散列表
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7. &nbsp; &nbsp; 树
7. &nbsp; &nbsp; 散列表
</label>
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@@ -962,8 +910,8 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
7.1. &nbsp; 哈希表
</a>
</li>
@@ -976,8 +924,8 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
7.2. &nbsp; 哈希冲突处理
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.3. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. &nbsp; AVL 树 *
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.5. &nbsp; 小结
<a href="../../chapter_hashing/summary/" class="md-nav__link">
7.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
8.1. &nbsp; 二叉树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
8.2. &nbsp; 二叉树遍历
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
8.3. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.4. &nbsp; AVL 树 *
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
8.5. &nbsp; 小结
</a>
</li>
</ul>
</nav>
</li>
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9. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
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9. &nbsp; &nbsp; 堆
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1078,7 +1117,7 @@
<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
8.1. &nbsp; 堆
9.1. &nbsp; 堆
</a>
</li>
@@ -1092,7 +1131,7 @@
<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
8.2. &nbsp; 建堆操作 *
9.2. &nbsp; 建堆操作 *
</a>
</li>
@@ -1106,7 +1145,7 @@
<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
8.3. &nbsp; 小结
9.3. &nbsp; 小结
</a>
</li>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" checked>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 图
<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. &nbsp; &nbsp; 图
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
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<label class="md-nav__title" for="__nav_11">
<span class="md-nav__icon md-icon"></span>
9. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 图
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1169,7 +1208,7 @@
<li class="md-nav__item">
<a href="../graph/" class="md-nav__link">
9.1. &nbsp; 图
10.1. &nbsp; 图
</a>
</li>
@@ -1183,7 +1222,7 @@
<li class="md-nav__item">
<a href="../graph_operations/" class="md-nav__link">
9.2. &nbsp; 图基础操作
10.2. &nbsp; 图基础操作
</a>
</li>
@@ -1206,12 +1245,12 @@
<label class="md-nav__link md-nav__link--active" for="__toc">
9.3. &nbsp; 图的遍历
10.3. &nbsp; 图的遍历
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
9.3. &nbsp; 图的遍历
10.3. &nbsp; 图的遍历
</a>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#931" class="md-nav__link">
9.3.1. &nbsp; 广度优先遍历
<a href="#1031" class="md-nav__link">
10.3.1. &nbsp; 广度优先遍历
</a>
<nav class="md-nav" aria-label="9.3.1. &nbsp; 广度优先遍历">
<nav class="md-nav" aria-label="10.3.1. &nbsp; 广度优先遍历">
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@@ -1257,11 +1296,11 @@
</li>
<li class="md-nav__item">
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9.3.2. &nbsp; 深度优先遍历
<a href="#1032" class="md-nav__link">
10.3.2. &nbsp; 深度优先遍历
</a>
<nav class="md-nav" aria-label="9.3.2. &nbsp; 深度优先遍历">
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<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
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</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. &nbsp; &nbsp; 查找算法
<span class="md-nav__icon md-icon"></span>
</label>
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<span class="md-nav__icon md-icon"></span>
10. &nbsp; &nbsp; 查找算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. &nbsp; 线性查找
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.2. &nbsp; 二分查找
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. &nbsp; 哈希查找
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.4. &nbsp; 小结
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.6. &nbsp; 桶排序（New）
11.6. &nbsp; 桶排序
</a>
</li>
@@ -1557,7 +1493,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
11.7. &nbsp; 计数排序（New）
11.7. &nbsp; 计数排序
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
11.8. &nbsp; 基数排序（New）
11.8. &nbsp; 基数排序
</a>
</li>
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
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12.1. &nbsp; 回溯算法（New）
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
12.1. &nbsp; 搜索算法（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
12.2. &nbsp; 哈希优化策略
</a>
</li>
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12.3. &nbsp; 小结
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13. &nbsp; &nbsp; 回溯算法
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<li class="md-nav__item">
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13.1. &nbsp; 编程环境安装
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
13.2. &nbsp; 一起参与创作
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
13.1. &nbsp; 回溯算法（New）
</a>
</li>
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14. &nbsp; &nbsp; 附录
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
14.1. &nbsp; 编程环境安装
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
</a>
</li>
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参考文献
</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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9.3.1. &nbsp; 广度优先遍历
<a href="#1031" class="md-nav__link">
10.3.1. &nbsp; 广度优先遍历
</a>
<nav class="md-nav" aria-label="9.3.1. &nbsp; 广度优先遍历">
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@@ -1825,11 +1848,11 @@
</li>
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10.3.2. &nbsp; 深度优先遍历
</a>
<nav class="md-nav" aria-label="9.3.2. &nbsp; 深度优先遍历">
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@@ -1874,14 +1897,14 @@
<h1 id="93">9.3. &nbsp; 图的遍历<a class="headerlink" href="#93" title="Permanent link">&para;</a></h1>
<h1 id="103">10.3. &nbsp; 图的遍历<a class="headerlink" href="#103" title="Permanent link">&para;</a></h1>
<div class="admonition note">
<p class="admonition-title">图与树的关系</p>
<p>树代表的是“一对多”的关系，而图则具有更高的自由度，可以表示任意的“多对多”关系。因此，我们可以把树看作是图的一种特例。显然，<strong>树的遍历操作也是图的遍历操作的一种特例</strong>，建议你在学习本章节时融会贯通两者的概念与实现方法。</p>
</div>
<p>「图」和「树」都是非线性数据结构，都需要使用「搜索算法」来实现遍历操作。</p>
<p>与树类似，图的遍历方式也可分为两种，即「广度优先遍历 Breadth-First Traversal」和「深度优先遍历 Depth-First Traversal」，也称为「广度优先搜索 Breadth-First Search」和「深度优先搜索 Depth-First Search」，简称 BFS 和 DFS。</p>
<h2 id="931">9.3.1. &nbsp; 广度优先遍历<a class="headerlink" href="#931" title="Permanent link">&para;</a></h2>
<h2 id="1031">10.3.1. &nbsp; 广度优先遍历<a class="headerlink" href="#1031" title="Permanent link">&para;</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>
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<h3 id="_2">复杂度分析<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
<p><strong>时间复杂度：</strong> 所有顶点都会入队并出队一次，使用 <span class="arithmatex">\(O(|V|)\)</span> 时间；在遍历邻接顶点的过程中，由于是无向图，因此所有边都会被访问 <span class="arithmatex">\(2\)</span> 次，使用 <span class="arithmatex">\(O(2|E|)\)</span> 时间；总体使用 <span class="arithmatex">\(O(|V| + |E|)\)</span> 时间。</p>
<p><strong>空间复杂度：</strong> 列表 <code>res</code> ，哈希表 <code>visited</code> ，队列 <code>que</code> 中的顶点数量最多为 <span class="arithmatex">\(|V|\)</span> ，使用 <span class="arithmatex">\(O(|V|)\)</span> 空间。</p>
<h2 id="932">9.3.2. &nbsp; 深度优先遍历<a class="headerlink" href="#932" title="Permanent link">&para;</a></h2>
<h2 id="1032">10.3.2. &nbsp; 深度优先遍历<a class="headerlink" href="#1032" title="Permanent link">&para;</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>
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<li class="md-nav__item">
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<h1 id="104">10.4. &nbsp; 小结<a class="headerlink" href="#104" title="Permanent link">&para;</a></h1>
<ul>
<li>图由顶点和边组成，可以被表示为一组顶点和一组边构成的集合。</li>
<li>相较于线性关系（链表）和分治关系（树），网络关系（图）具有更高的自由度，因而更为复杂。</li>
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