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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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12.1 &nbsp; 分治算法
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12.2 &nbsp; 分治搜索策略
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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15.1 &nbsp; 贪心算法
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15.4 &nbsp; 最大切分乘积问题
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<p>如果将顶点看作节点，将边看作连接各个节点的引用（指针），我们就可以将图看作是一种从链表拓展而来的数据结构。如图 9-1 所示，<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</strong>，从而更为复杂。</p>
<p><img alt="链表、树、图之间的关系" src="../graph.assets/linkedlist_tree_graph.png" /></p>
<p><a class="glightbox" href="../graph.assets/linkedlist_tree_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="链表、树、图之间的关系" src="../graph.assets/linkedlist_tree_graph.png" /></a></p>
<p align="center"> 图 9-1 &nbsp; 链表、树、图之间的关系 </p>
<h2 id="911">9.1.1 &nbsp; 图常见类型与术语<a class="headerlink" href="#911" title="Permanent link">&para;</a></h2>
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<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>
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<p><img alt="有向图与无向图" src="../graph.assets/directed_graph.png" /></p>
<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" 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，可分为图 9-3 所示的「连通图 connected graph」和「非连通图 disconnected graph」。</p>
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<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><img alt="连通图与非连通图" src="../graph.assets/connected_graph.png" /></p>
<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" 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><img alt="有权图与无权图" src="../graph.assets/weighted_graph.png" /></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" 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>
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<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>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><img alt="图的邻接矩阵表示" src="../graph.assets/adjacency_matrix.png" /></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" src="../graph.assets/adjacency_matrix.png" /></a></p>
<p align="center"> 图 9-5 &nbsp; 图的邻接矩阵表示 </p>
<p>邻接矩阵具有以下特性。</p>
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<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> 条链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（即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><img alt="图的邻接表表示" src="../graph.assets/adjacency_list.png" /></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" src="../graph.assets/adjacency_list.png" /></a></p>
<p align="center"> 图 9-6 &nbsp; 图的邻接表表示 </p>
<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>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:5"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">初始化邻接矩阵</label><label for="__tabbed_1_2">添加边</label><label for="__tabbed_1_3">删除边</label><label for="__tabbed_1_4">添加顶点</label><label for="__tabbed_1_5">删除顶点</label></div>
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<p><img alt="邻接矩阵的初始化、增删边、增删顶点" src="../graph_operations.assets/adjacency_matrix_initialization.png" /></p>
<p><a class="glightbox" href="../graph_operations.assets/adjacency_matrix_initialization.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="邻接矩阵的初始化、增删边、增删顶点" src="../graph_operations.assets/adjacency_matrix_initialization.png" /></a></p>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:5"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">初始化邻接表</label><label for="__tabbed_3_2">添加边</label><label for="__tabbed_3_3">删除边</label><label for="__tabbed_3_4">添加顶点</label><label for="__tabbed_3_5">删除顶点</label></div>
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<p>图和树都都需要应用搜索算法来实现遍历操作。图的遍历方式可分为两种：「广度优先遍历 breadth-first traversal」和「深度优先遍历 depth-first traversal」。它们也常被称为「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>
<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，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><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" src="../graph_traversal.assets/graph_bfs.png" /></a></p>
<p align="center"> 图 9-9 &nbsp; 图的广度优先遍历 </p>
<h3 id="1">1. &nbsp; 算法实现<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
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<p><img alt="图的广度优先遍历步骤" src="../graph_traversal.assets/graph_bfs_step1.png" /></p>
<p><a class="glightbox" href="../graph_traversal.assets/graph_bfs_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的广度优先遍历步骤" src="../graph_traversal.assets/graph_bfs_step1.png" /></a></p>
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<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>
<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>。如图 9-11 所示，从左上角顶点出发，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><a class="glightbox" href="../graph_traversal.assets/graph_dfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的深度优先遍历" src="../graph_traversal.assets/graph_dfs.png" /></a></p>
<p align="center"> 图 9-11 &nbsp; 图的深度优先遍历 </p>
<h3 id="1_1">1. &nbsp; 算法实现<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
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</head>
<link href="../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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</span>
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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12.2 &nbsp; 分治搜索策略
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</span>
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12.3 &nbsp; 构建树问题
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</span>
</a>
</li>
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12.4 &nbsp; 汉诺塔问题
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</span>
</a>
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12.5 &nbsp; 小结
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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</span>
</a>
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14.2 &nbsp; DP 问题特性
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</span>
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</li>
@@ -2817,14 +2740,6 @@
14.3 &nbsp; DP 解题思路
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</a>
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14.4 &nbsp; 0-1 背包问题
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14.5 &nbsp; 完全背包问题
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14.6 &nbsp; 编辑距离问题
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14.7 &nbsp; 小结
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第 15 章 &nbsp; 贪心
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15.1 &nbsp; 贪心算法
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15.2 &nbsp; 分数背包问题
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15.3 &nbsp; 最大容量问题
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15.4 &nbsp; 最大切分乘积问题
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15.5 &nbsp; 小结
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<!-- Page content -->
<h1 id="9">第 9 章 &nbsp; 图<a class="headerlink" href="#9" title="Permanent link">&para;</a></h1>
<div class="center-table">
<p><img alt="图" src="../assets/covers/chapter_graph.jpg" width="600" /></p>
<p><a class="glightbox" href="../assets/covers/chapter_graph.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图" src="../assets/covers/chapter_graph.jpg" width="600" /></a></p>
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<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
@@ -3591,10 +3426,15 @@ aria-label="页脚"
<div class="md-copyright">
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Copyright &copy; 2023 Krahets
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@@ -60,7 +60,18 @@
</head>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
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10.2 &nbsp; 二分查找插入点
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</span>
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10.3 &nbsp; 二分查找边界
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<span class="md-status md-status--new" title="最近添加">
</span>
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@@ -2386,14 +2381,6 @@
第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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<span class="md-status md-status--new" title="最近添加">
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12.2 &nbsp; 分治搜索策略
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<span class="md-status md-status--new" title="最近添加">
</span>
</a>
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@@ -2481,14 +2452,6 @@
12.3 &nbsp; 构建树问题
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<span class="md-status md-status--new" title="最近添加">
</span>
</a>
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@@ -2509,14 +2472,6 @@
12.4 &nbsp; 汉诺塔问题
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@@ -2537,14 +2492,6 @@
12.5 &nbsp; 小结
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<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@@ -2776,14 +2723,6 @@
第 14 章 &nbsp; 动态规划
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
@@ -2815,14 +2754,6 @@
14.1 &nbsp; 初探动态规划
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@@ -2843,14 +2774,6 @@
14.2 &nbsp; DP 问题特性
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@@ -2871,14 +2794,6 @@
14.3 &nbsp; DP 解题思路
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
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@@ -2899,14 +2814,6 @@
14.4 &nbsp; 0-1 背包问题
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<span class="md-status md-status--new" title="最近添加">
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14.5 &nbsp; 完全背包问题
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<span class="md-status md-status--new" title="最近添加">
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14.6 &nbsp; 编辑距离问题
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<span class="md-status md-status--new" title="最近添加">
</span>
</a>
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@@ -2983,14 +2874,6 @@
14.7 &nbsp; 小结
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<span class="md-status md-status--new" title="最近添加">
</span>
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@@ -3049,14 +2932,6 @@
第 15 章 &nbsp; 贪心
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<span class="md-status md-status--new" title="最近添加">
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@@ -3088,14 +2963,6 @@
15.1 &nbsp; 贪心算法
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15.2 &nbsp; 分数背包问题
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<span class="md-status md-status--new" title="最近添加">
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</a>
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15.3 &nbsp; 最大容量问题
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<span class="md-status md-status--new" title="最近添加">
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15.4 &nbsp; 最大切分乘积问题
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<span class="md-status md-status--new" title="最近添加">
</span>
</a>
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@@ -3200,14 +3043,6 @@
15.5 &nbsp; 小结
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<span class="md-status md-status--new" title="最近添加">
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@@ -3665,10 +3500,15 @@ aria-label="页脚"
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Copyright &copy; 2023 Krahets
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@@ -3737,5 +3577,5 @@ aria-label="页脚"
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