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< meta name = "description" content = "动画图解、一键运行的数据结构与算法教程" >
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< title > 9.1 < / title >
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跳转至
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Hello 算法
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9.1 图
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正在初始化搜索引擎
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Hello 算法
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第 0 章 前言
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第 0 章 前言
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0.1 关于本书
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0.2 如何使用本书
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0.3 小结
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第 1 章 初识算法
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第 1 章 初识算法
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1.1 算法无处不在
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1.2 算法是什么
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1.3 小结
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第 2 章 复杂度分析
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第 2 章 复杂度分析
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2.1 算法效率评估
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2.2 迭代与递归
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2.3 时间复杂度
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2.4 空间复杂度
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2.5 小结
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第 3 章 数据结构
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第 3 章 数据结构
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3.1 数据结构分类
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3.2 基本数据类型
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3.3 数字编码 *
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3.4 字符编码 *
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3.5 小结
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第 4 章 数组与链表
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第 4 章 数组与链表
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5.4 小结
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第 6 章 哈希表
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第 6 章 哈希表
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6.1 哈希表
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6.2 哈希冲突
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6.3 哈希算法
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6.4 小结
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第 7 章 树
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第 7 章 树
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7.1 二叉树
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7.3 二叉树数组表示
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7.4 二叉搜索树
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7.5 AVL 树 *
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第 8 章 堆
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8.1 堆
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8.2 建堆操作
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8.3 Top-K 问题
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8.4 小结
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第 9 章 图
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第 9 章 图
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9.1 图
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9.1 图
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9.1.1 图常见类型与术语
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1. 邻接矩阵
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9.2 图基础操作
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9.3 图的遍历
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9.4 小结
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< div class = "md-nav__link md-nav__container" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z" / > < / svg >
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第 10 章 搜索
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第 10 章 搜索
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10.1 二分查找
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10.2 二分查找插入点
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10.3 二分查找边界
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/replace_linear_by_hashing/" class = "md-nav__link" >
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10.4 哈希优化策略
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/searching_algorithm_revisited/" class = "md-nav__link" >
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10.5 重识搜索算法
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10.6 小结
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第 11 章 排序
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第 11 章 排序
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< / label >
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11.1 排序算法
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11.2 选择排序
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11.3 冒泡排序
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11.4 插入排序
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11.5 快速排序
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< li class = "md-nav__item" >
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11.6 归并排序
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< li class = "md-nav__item" >
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11.7 堆排序
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< / span >
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< li class = "md-nav__item" >
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11.8 桶排序
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11.9 计数排序
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11.10 基数排序
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11.11 小结
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第 12 章 分治
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第 12 章 分治
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12.1 分治算法
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12.2 分治搜索策略
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12.3 构建树问题
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12.4 汉诺塔问题
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12.5 小结
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第 13 章 回溯
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第 13 章 回溯
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13.1 回溯算法
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< li class = "md-nav__item" >
< a href = "../../chapter_backtracking/permutations_problem/" class = "md-nav__link" >
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13.2 全排列问题
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14.5 完全背包问题
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14.6 编辑距离问题
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15.1 贪心算法
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1. 邻接矩阵
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< h1 id = "91" > 9.1 图< a class = "headerlink" href = "#91" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > 「图 graph」是一种非线性数据结构， < 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 >
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< div class = "arithmatex" > \[
\begin{aligned}
V & = \{ 1, 2, 3, 4, 5 \} \newline
E & = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline
G & = \{ V, E \} \newline
\end{aligned}
\]< / div >
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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 >
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< 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 >
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< p align = "center" > 图 9-1 链表、树、图之间的关系 < / p >
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< h2 id = "911" > 9.1.1 图常见类型与术语< a class = "headerlink" href = "#911" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 根据边是否具有方向，可分为图 9-2 所示的「无向图 undirected graph」和「有向图 directed graph」。< / p >
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< ul >
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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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< / ul >
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< 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 >
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< p align = "center" > 图 9-2 有向图与无向图 < / p >
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< 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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< ul >
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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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< / ul >
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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" src = "../graph.assets/connected_graph.png" / > < / a > < / p >
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< p align = "center" > 图 9-3 连通图与非连通图 < / p >
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< 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， ， < / p >
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< 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 >
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< p align = "center" > 图 9-4 有权图与无权图 < / p >
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< p > 图数据结构包含以下常用术语。< / p >
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< ul >
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< li > 「邻接 adjacency」： ， < / li >
< li > 「路径 path」： < / li >
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< li > 「度 degree」： ， ， < / li >
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< / ul >
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< h2 id = "912" > 9.1.2 图的表示< a class = "headerlink" href = "#912" title = "Permanent link" > ¶ < / 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 >
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< h3 id = "1" > 1. 邻接矩阵< a class = "headerlink" href = "#1" title = "Permanent link" > ¶ < / a > < / h3 >
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< 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 >
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< 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 >
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< 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 >
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< p align = "center" > 图 9-5 图的邻接矩阵表示 < / p >
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< p > 邻接矩阵具有以下特性。< / p >
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< ul >
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< li > 顶点不能与自身相连，因此邻接矩阵主对角线元素没有意义。< / li >
< li > 对于无向图，两个方向的边等价，此时邻接矩阵关于主对角线对称。< / li >
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< li > 将邻接矩阵的元素从 < span class = "arithmatex" > \(1\)< / span > 和 < span class = "arithmatex" > \(0\)< / span > 替换为权重，则可表示有权图。< / li >
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< / ul >
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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 >
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< h3 id = "2" > 2. 邻接表< a class = "headerlink" href = "#2" title = "Permanent link" > ¶ < / a > < / h3 >
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< 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 >
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< 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 >
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< p align = "center" > 图 9-6 图的邻接表表示 < / p >
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< 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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< p > 观察图 9-6 ， < strong > 邻接表结构与哈希表中的“链式地址”非常相似，因此我们也可以采用类似方法来优化效率< / strong > 。比如当链表较长时，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 > ；还可以把链表转换为哈希表，从而将时间复杂度降低至 < span class = "arithmatex" > \(O(1)\)< / span > 。< / p >
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< h2 id = "913" > 9.1.3 图常见应用< a class = "headerlink" href = "#913" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 如表 9-1 所示，许多现实系统都可以用图来建模，相应的问题也可以约化为图计算问题。< / p >
< p align = "center" > 表 9-1 现实生活中常见的图 < / p >
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< div class = "center-table" >
< table >
< thead >
< tr >
< th > < / th >
< th > 顶点< / th >
< th > 边< / th >
< th > 图计算问题< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > 社交网络< / td >
< td > 用户< / td >
< td > 好友关系< / td >
< td > 潜在好友推荐< / td >
< / tr >
< tr >
< td > 地铁线路< / td >
< td > 站点< / td >
< td > 站点间的连通性< / td >
< td > 最短路线推荐< / td >
< / tr >
< tr >
< td > 太阳系< / td >
< td > 星体< / td >
< td > 星体间的万有引力作用< / td >
< td > 行星轨道计算< / td >
< / tr >
< / tbody >
< / table >
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< a href = "https://twitter.com/krahets" target = "_blank" rel = "noopener" title = "twitter.com" class = "md-social__link" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > <!-- ! Font Awesome Free 6.4.2 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 = "M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z" / > < / svg >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 640 512" > <!-- ! Font Awesome Free 6.4.2 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 = "M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z" / > < / svg >
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< / a >
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< / footer >
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< div class = "md-dialog" data-md-component = "dialog" >
< div class = "md-dialog__inner md-typeset" > < / div >
< / div >
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< script id = "__config" type = "application/json" > { "base" : "../.." , "features" : [ "content.action.edit" , "content.code.annotate" , "content.code.copy" , "content.tabs.link" , "content.tooltips" , "navigation.indexes" , "navigation.instant" , "navigation.top" , "navigation.footer" , "navigation.tracking" , "search.highlight" , "search.share" , "search.suggest" , "toc.follow" ] , "search" : "../../assets/javascripts/workers/search.a264c092.min.js" , "translations" : { "clipboard.copied" : "\u5df2\u590d\u5236" , "clipboard.copy" : "\u590d\u5236" , "search.result.more.one" : "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.more.other" : "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.none" : "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.one" : "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.other" : "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.placeholder" : "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22" , "search.result.term.missing" : "\u7f3a\u5c11" , "select.version" : "\u9009\u62e9\u5f53\u524d\u7248\u672c" } } < / script >
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< script src = "../../assets/javascripts/bundle.726fbb30.min.js" > < / script >
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< script src = "../../javascripts/mathjax.js" > < / script >
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< script src = "https://polyfill.io/v3/polyfill.min.js?features=es6" > < / script >
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< script src = "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" > < / script >
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< script > document $ . subscribe ( ( ) => { const lightbox = GLightbox ( { "touchNavigation" : true , "loop" : false , "zoomable" : true , "draggable" : false , "openEffect" : "zoom" , "closeEffect" : "zoom" , "slideEffect" : "none" } ) ; } ) < / script > < / body >
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< / html >
