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序
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 10 章 &nbsp; 搜索
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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<a href="../../chapter_paperbook/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M96 0C43 0 0 43 0 96v320c0 53 43 96 96 96h320c17.7 0 32-14.3 32-32s-14.3-32-32-32v-64c17.7 0 32-14.3 32-32V32c0-17.7-14.3-32-32-32H96zm0 384h256v64H96c-17.7 0-32-14.3-32-32s14.3-32 32-32zm32-240c0-8.8 7.2-16 16-16h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16zm16 48h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16s7.2-16 16-16z"/></svg>
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纸质书
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纸质书
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<!-- Page content -->
<h1 id="91">9.1 &nbsp; 图<a class="headerlink" href="#91" 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>
<p><u>图（graph）</u>是一种非线性数据结构，由<u>顶点（vertex）</u>和<u>边（edge）</u>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}
V &amp; = \{ 1, 2, 3, 4, 5 \} \newline
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<p align="center"> 图 9-1 &nbsp; 链表、树、图之间的关系 </p>
<h2 id="911">9.1.1 &nbsp; 图的常见类型与术语<a class="headerlink" href="#911" 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」，如图 9-2 所示。</p>
<p>根据边是否具有方向，可分为<u>无向图（undirected graph）</u>和<u>有向图（directed graph）</u>，如图 9-2 所示。</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>
<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>
@@ -3688,7 +3745,7 @@ G &amp; = \{ V, E \} \newline
<p><a class="glightbox" href="../graph.assets/directed_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有向图与无向图" class="animation-figure" src="../graph.assets/directed_graph.png" /></a></p>
<p align="center"> 图 9-2 &nbsp; 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」，如图 9-3 所示。</p>
<p>根据所有顶点是否连通，可分为<u>连通图（connected graph）</u>和<u>非连通图（disconnected graph）</u>，如图 9-3 所示。</p>
<ul>
<li>对于连通图，从某个顶点出发，可以到达其余任意顶点。</li>
<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>
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<p><a class="glightbox" href="../graph.assets/connected_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="连通图与非连通图" class="animation-figure" src="../graph.assets/connected_graph.png" /></a></p>
<p align="center"> 图 9-3 &nbsp; 连通图与非连通图 </p>
<p>我们还可以为边添加“权重”变量，从而得到如图 9-4 所示的「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>我们还可以为边添加“权重”变量，从而得到如图 9-4 所示的<u>有权图（weighted graph）</u>。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><a class="glightbox" href="../graph.assets/weighted_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有权图与无权图" class="animation-figure" src="../graph.assets/weighted_graph.png" /></a></p>
<p align="center"> 图 9-4 &nbsp; 有权图与无权图 </p>
<p>图数据结构包含以下常用术语。</p>
<ul>
<li>「邻接 adjacency」：当两顶点之间存在边相连时，称这两顶点“邻接”。在图 9-4 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”。在图 9-4 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」：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>
<li><u>邻接（adjacency）</u>：当两顶点之间存在边相连时，称这两顶点“邻接”。在图 9-4 中，顶点 1 的邻接顶点为顶点 2、3、5。</li>
<li><u>路径（path）</u>：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”。在图 9-4 中，边序列 1-5-2-4 是顶点 1 到顶点 4 的一条路径。</li>
<li><u>度（degree）</u>：一个顶点拥有的边数。对于有向图，<u>入度（in-degree）</u>表示有多少条边指向该顶点，<u>出度（out-degree）</u>表示有多少条边从该顶点指出。</li>
</ul>
<h2 id="912">9.1.2 &nbsp; 图的表示<a class="headerlink" href="#912" title="Permanent link">&para;</a></h2>
<p>图的常用表示方式包括“邻接矩阵”和“邻接表”。以下使用无向图进行举例。</p>
<h3 id="1">1. &nbsp; 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>
<p>设图的顶点数量为 <span class="arithmatex">\(n\)</span> ，<u>邻接矩阵（adjacency matrix）</u>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>如图 9-5 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><a class="glightbox" href="../graph.assets/adjacency_matrix.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的邻接矩阵表示" class="animation-figure" src="../graph.assets/adjacency_matrix.png" /></a></p>
<p align="center"> 图 9-5 &nbsp; 图的邻接矩阵表示 </p>
@@ -3722,7 +3779,7 @@ G &amp; = \{ V, E \} \newline
</ul>
<p>使用邻接矩阵表示图时，我们可以直接访问矩阵元素以获取边，因此增删查改操作的效率很高，时间复杂度均为 <span class="arithmatex">\(O(1)\)</span> 。然而，矩阵的空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> ，内存占用较多。</p>
<h3 id="2">2. &nbsp; 邻接表<a class="headerlink" href="#2" title="Permanent link">&para;</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> 个链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（与该顶点相连的顶点）。图 9-6 展示了一个使用邻接表存储的图的示例。</p>
<p><u>邻接表（adjacency list）</u>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> 个链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（与该顶点相连的顶点）。图 9-6 展示了一个使用邻接表存储的图的示例。</p>
<p><a class="glightbox" href="../graph.assets/adjacency_list.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的邻接表表示" class="animation-figure" src="../graph.assets/adjacency_list.png" /></a></p>
<p align="center"> 图 9-6 &nbsp; 图的邻接表表示 </p>
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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纸质书
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Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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序
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 10 章 &nbsp; 搜索
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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纸质书
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<!-- Page content -->
<h1 id="93">9.3 &nbsp; 图的遍历<a class="headerlink" href="#93" title="Permanent link">&para;</a></h1>
<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>。</p>
<p>图和树都需要应用搜索算法来实现遍历操作。图的遍历方式也可分为两种：「广度优先遍历」和「深度优先遍历」。</p>
<p>图和树都需要应用搜索算法来实现遍历操作。图的遍历方式也可分为两种：<u>广度优先遍历</u>和<u>深度优先遍历</u>。</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>
<p><strong>广度优先遍历是一种由近及远的遍历方式，从某个节点出发，始终优先访问距离最近的顶点，并一层层向外扩张</strong>。如图 9-9 所示，从左上角顶点出发，首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><a class="glightbox" href="../graph_traversal.assets/graph_bfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的广度优先遍历" class="animation-figure" src="../graph_traversal.assets/graph_bfs.png" /></a></p>
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Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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序
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 10 章 &nbsp; 搜索
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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纸质书
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Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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序
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 10 章 &nbsp; 搜索
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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纸质书
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