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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>
<ul class="md-nav__list" data-md-scrollfix>
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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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>
</nav>
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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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<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>
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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>
</label>
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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>
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@@ -2035,7 +2058,7 @@ G &amp; = \{ V, E \} \newline
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