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<title>10.3.   图的遍历 - Hello 算法</title>
<title>9.3.   图的遍历 - Hello 算法</title>
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<div data-md-component="skip">
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
10.3. &nbsp; 图的遍历
9.3. &nbsp; 图的遍历
</span>
</div>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 二分查找
6. &nbsp; &nbsp; 散列表
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6. &nbsp; &nbsp; 二分查找
6. &nbsp; &nbsp; 散列表
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<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. &nbsp; 哈希表
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search_edge/" class="md-nav__link">
6.2. &nbsp; 二分查找边界
<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>
</li>
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 散列表
7. &nbsp; &nbsp; 树
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7. &nbsp; &nbsp; 树
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
7.1. &nbsp; 哈希表
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7.1. &nbsp; 二叉树
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
7.2. &nbsp; 哈希冲突处理
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
7.3. &nbsp; 小结
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
7.3. &nbsp; 二叉树数组表示
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.4. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.5. &nbsp; AVL 树 *
</a>
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<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.6. &nbsp; 小结
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 树
8. &nbsp; &nbsp; 堆
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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/array_representation_of_tree/" class="md-nav__link">
8.3. &nbsp; 二叉树数组表示
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
8.4. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.5. &nbsp; AVL 树 *
</a>
</li>
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8.6. &nbsp; 小结
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<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
9.1. &nbsp; 堆
8.1. &nbsp; 堆
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
9.2. &nbsp; 建堆操作 *
8.2. &nbsp; 建堆操作 *
</a>
</li>
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<li class="md-nav__item">
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9.3. &nbsp; 小结
8.3. &nbsp; 小结
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. &nbsp; &nbsp; 图
<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 图
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10. &nbsp; &nbsp; 图
9. &nbsp; &nbsp; 图
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<li class="md-nav__item">
<a href="../graph/" class="md-nav__link">
10.1. &nbsp; 图
9.1. &nbsp; 图
</a>
</li>
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<li class="md-nav__item">
<a href="../graph_operations/" class="md-nav__link">
10.2. &nbsp; 图基础操作
9.2. &nbsp; 图基础操作
</a>
</li>
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<label class="md-nav__link md-nav__link--active" for="__toc">
10.3. &nbsp; 图的遍历
9.3. &nbsp; 图的遍历
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
10.3. &nbsp; 图的遍历
9.3. &nbsp; 图的遍历
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1031" class="md-nav__link">
10.3.1. &nbsp; 广度优先遍历
<a href="#931" class="md-nav__link">
9.3.1. &nbsp; 广度优先遍历
</a>
<nav class="md-nav" aria-label="10.3.1. &nbsp; 广度优先遍历">
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</li>
<li class="md-nav__item">
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10.3.2. &nbsp; 深度优先遍历
<a href="#932" class="md-nav__link">
9.3.2. &nbsp; 深度优先遍历
</a>
<nav class="md-nav" aria-label="10.3.2. &nbsp; 深度优先遍历">
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<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
10.4. &nbsp; 小结
9.4. &nbsp; 小结
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</ul>
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</li>
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10. &nbsp; &nbsp; 搜索算法
<span class="md-nav__icon md-icon"></span>
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10. &nbsp; &nbsp; 搜索算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.1. &nbsp; 二分查找（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
10.2. &nbsp; 二分查找边界（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
10.3. &nbsp; 哈希优化策略
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
10.4. &nbsp; 重识搜索算法
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.5. &nbsp; 小结
</a>
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 搜索算法
12. &nbsp; &nbsp; 回溯算法
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<li class="md-nav__item">
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
12.1. &nbsp; 搜索算法（New）
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12.1. &nbsp; 回溯算法（New）
</a>
</li>
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<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 href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
12.2. &nbsp; 全排列问题（New）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
12.3. &nbsp; 小结
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.3. &nbsp; N 皇后问题（New）
</a>
</li>
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<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
13. &nbsp; &nbsp; 回溯算法
13. &nbsp; &nbsp; 附录
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13. &nbsp; &nbsp; 附录
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<li class="md-nav__item">
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
13.1. &nbsp; 回溯算法（New）
<a href="../../chapter_appendix/installation/" class="md-nav__link">
13.1. &nbsp; 编程环境安装
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
13.2. &nbsp; 全排列问题（New）
</a>
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<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
13.3. &nbsp; N 皇后问题（New）
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
13.2. &nbsp; 一起参与创作
</a>
</li>
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<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
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<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
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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="#1031" class="md-nav__link">
10.3.1. &nbsp; 广度优先遍历
<a href="#931" class="md-nav__link">
9.3.1. &nbsp; 广度优先遍历
</a>
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<h1 id="103">10.3. &nbsp; 图的遍历<a class="headerlink" href="#103" title="Permanent link">&para;</a></h1>
<h1 id="93">9.3. &nbsp; 图的遍历<a class="headerlink" href="#93" 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="1031">10.3.1. &nbsp; 广度优先遍历<a class="headerlink" href="#1031" title="Permanent link">&para;</a></h2>
<h2 id="931">9.3.1. &nbsp; 广度优先遍历<a class="headerlink" href="#931" 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>
@@ -2299,7 +2260,7 @@
<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="1032">10.3.2. &nbsp; 深度优先遍历<a class="headerlink" href="#1032" title="Permanent link">&para;</a></h2>
<h2 id="932">9.3.2. &nbsp; 深度优先遍历<a class="headerlink" href="#932" 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>
@@ -2667,7 +2628,7 @@
<nav class="md-footer__inner md-grid" aria-label="页脚" >
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@@ -2676,20 +2637,20 @@
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