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<title>10.2.   图基础操作 - Hello 算法</title>
<title>9.2.   图基础操作 - Hello 算法</title>
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<div data-md-component="skip">
<a href="#102" class="md-skip">
<a href="#92" class="md-skip">
跳转至
</a>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
10.2. &nbsp; 图基础操作
9.2. &nbsp; 图基础操作
</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
<span class="md-nav__icon md-icon"></span>
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<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
6. &nbsp; &nbsp; 二分查找
6. &nbsp; &nbsp; 散列表
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -885,8 +887,8 @@
<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>
@@ -899,8 +901,22 @@
<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
<span class="md-nav__icon md-icon"></span>
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7. &nbsp; &nbsp; 散列表
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
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
</a>
</li>
@@ -972,8 +994,8 @@
<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>
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@@ -986,8 +1008,50 @@
<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>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.6. &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; 树
8. &nbsp; &nbsp; 堆
<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">
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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>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
8.6. &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
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9. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 堆
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<ul class="md-nav__list" data-md-scrollfix>
@@ -1181,7 +1110,7 @@
<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
9.1. &nbsp; 堆
8.1. &nbsp; 堆
</a>
</li>
@@ -1195,7 +1124,7 @@
<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>
@@ -1209,7 +1138,7 @@
<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &nbsp; 小结
8.3. &nbsp; 小结
</a>
</li>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
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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
<span class="md-nav__icon md-icon"></span>
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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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<label class="md-nav__link md-nav__link--active" for="__toc">
10.2. &nbsp; 图基础操作
9.2. &nbsp; 图基础操作
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
10.2. &nbsp; 图基础操作
9.2. &nbsp; 图基础操作
</a>
@@ -1319,22 +1248,22 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1021" class="md-nav__link">
10.2.1. &nbsp; 基于邻接矩阵的实现
<a href="#921" class="md-nav__link">
9.2.1. &nbsp; 基于邻接矩阵的实现
</a>
</li>
<li class="md-nav__item">
<a href="#1022" class="md-nav__link">
10.2.2. &nbsp; 基于邻接表的实现
<a href="#922" class="md-nav__link">
9.2.2. &nbsp; 基于邻接表的实现
</a>
</li>
<li class="md-nav__item">
<a href="#1023" class="md-nav__link">
10.2.3. &nbsp; 效率对比
<a href="#923" class="md-nav__link">
9.2.3. &nbsp; 效率对比
</a>
</li>
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<li class="md-nav__item">
<a href="../graph_traversal/" class="md-nav__link">
10.3. &nbsp; 图的遍历
9.3. &nbsp; 图的遍历
</a>
</li>
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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
</a>
</li>
</ul>
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<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; 搜索算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_11">
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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>
</li>
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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
<span class="md-nav__icon md-icon"></span>
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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）
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
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>
@@ -1675,19 +1723,17 @@
<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
<span class="md-nav__icon md-icon"></span>
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13. &nbsp; &nbsp; 回溯算法
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>
@@ -1711,22 +1757,8 @@
<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>
</li>
<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">
14. &nbsp; &nbsp; 附录
<span class="md-nav__icon md-icon"></span>
</label>
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<label class="md-nav__title" for="__nav_15">
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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
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_16">
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<span class="md-nav__icon md-icon"></span>
参考文献
</label>
@@ -1884,22 +1845,22 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1021" class="md-nav__link">
10.2.1. &nbsp; 基于邻接矩阵的实现
<a href="#921" class="md-nav__link">
9.2.1. &nbsp; 基于邻接矩阵的实现
</a>
</li>
<li class="md-nav__item">
<a href="#1022" class="md-nav__link">
10.2.2. &nbsp; 基于邻接表的实现
<a href="#922" class="md-nav__link">
9.2.2. &nbsp; 基于邻接表的实现
</a>
</li>
<li class="md-nav__item">
<a href="#1023" class="md-nav__link">
10.2.3. &nbsp; 效率对比
<a href="#923" class="md-nav__link">
9.2.3. &nbsp; 效率对比
</a>
</li>
@@ -1927,9 +1888,9 @@
<h1 id="102">10.2. &nbsp; 图基础操作<a class="headerlink" href="#102" title="Permanent link">&para;</a></h1>
<h1 id="92">9.2. &nbsp; 图基础操作<a class="headerlink" href="#92" 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="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>
<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>
<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>
@@ -2671,7 +2632,7 @@
</div>
</div>
</div>
<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>
<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>
<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>
@@ -3347,7 +3308,7 @@
</div>
</div>
</div>
<h2 id="1023">10.2.3. &nbsp; 效率对比<a class="headerlink" href="#1023" title="Permanent link">&para;</a></h2>
<h2 id="923">9.2.3. &nbsp; 效率对比<a class="headerlink" href="#923" 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">
<table>
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<nav class="md-footer__inner md-grid" aria-label="页脚" >
<a href="../graph/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 10.1. &amp;nbsp; 图" rel="prev">
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</div>
@@ -3486,20 +3447,20 @@
上一页
</span>
<div class="md-ellipsis">
10.1. &nbsp; 图
9.1. &nbsp; 图
</div>
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