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Hello 算法
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9.1 图
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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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< a href = "../../chapter_computational_complexity/" class = "md-nav__link " >
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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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< a href = "../../chapter_data_structure/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z" / > < / svg >
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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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< a href = "../../chapter_array_and_linkedlist/" class = "md-nav__link " >
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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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4.1 数组
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4.2 链表
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4.3 列表
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4.4 小结
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< a href = "../../chapter_stack_and_queue/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z" / > < / svg >
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第 5 章 栈与队列
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第 5 章 栈与队列
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5.1 栈
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5.2 队列
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5.3 双向队列
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5.4 小结
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< a href = "../../chapter_hashing/" class = "md-nav__link " >
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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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< a href = "../../chapter_tree/" class = "md-nav__link " >
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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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7.2 二叉树遍历
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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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7.6 小结
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< a href = "../../chapter_heap/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z" / > < / svg >
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第 8 章 堆
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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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目录
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9.1.1 图常见类型与术语
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9.1.2 图的表示
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1. 邻接矩阵
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2. 邻接表
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9.1.3 图常见应用
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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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< a href = "../../chapter_searching/" class = "md-nav__link " >
< 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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10.4 哈希优化策略
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10.5 重识搜索算法
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10.6 小结
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< a href = "../../chapter_sorting/" class = "md-nav__link " >
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第 11 章 排序
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第 11 章 排序
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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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11.6 归并排序
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11.7 堆排序
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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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< a href = "../../chapter_backtracking/" class = "md-nav__link " >
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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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13.2 全排列问题
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13.3 子集和问题
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13.4 N 皇后问题
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13.5 小结
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< a href = "../../chapter_dynamic_programming/" class = "md-nav__link " >
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第 14 章 动态规划
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第 14 章 动态规划
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14.1 初探动态规划
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14.2 DP 问题特性
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14.3 DP 解题思路
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14.4 0-1 背包问题
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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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14.7 小结
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< a href = "../../chapter_greedy/" class = "md-nav__link " >
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第 15 章 贪心
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第 15 章 贪心
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9.1.1 图常见类型与术语
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9.1.2 图的表示
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1. 邻接矩阵
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2. 邻接表
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9.1.3 图常见应用
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< h1 id = "91" > 9.1 图< a class = "headerlink" href = "#91" title = "Permanent link" > ¶ < / a > < / h1 >
< 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 >
< 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 >
< 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 > < 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 链表、树、图之间的关系 < / p >
< h2 id = "911" > 9.1.1 图常见类型与术语< a class = "headerlink" href = "#911" title = "Permanent link" > ¶ < / a > < / h2 >
< 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 >
< 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 >
< / ul >
< 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 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 >
< 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 >
< / ul >
< 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 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， ， < / 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 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」： ， < / li >
< li > 「路径 path」： < / li >
< li > 「度 degree」： ， ， < / li >
< / ul >
< 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 >
< h3 id = "1" > 1. 邻接矩阵< a class = "headerlink" href = "#1" title = "Permanent link" > ¶ < / 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 > < 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 图的邻接矩阵表示 < / p >
< p > 邻接矩阵具有以下特性。< / p >
< ul >
< 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 >
< 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 >
< / 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. 邻接表< a class = "headerlink" href = "#2" title = "Permanent link" > ¶ < / 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 > < 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 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 >
< 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 >
< h2 id = "913" > 9.1.3 图常见应用< a class = "headerlink" href = "#913" title = "Permanent link" > ¶ < / a > < / h2 >
< 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 >
< div class = "center-table" >
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< tr >
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< th > 顶点< / th >
< th > 边< / th >
< th > 图计算问题< / th >
< / tr >
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< 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 >
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