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< title > 9.4 < / title >
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跳转至
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Hello 算法
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9.4 小结
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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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3.3 数字编码 *
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第 4 章 数组与链表
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第 5 章 栈与队列
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第 6 章 哈希表
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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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7.6 小结
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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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9.2 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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9.4 小结
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目录
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1. 重点回顾
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" >
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< div class = "md-nav__link md-nav__container" >
< 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 >
< span class = "md-ellipsis" >
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第 10 章 搜索
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< / span >
< / a >
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< label class = "md-nav__title" for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
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第 10 章 搜索
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< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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10.1 二分查找
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search_insertion/" class = "md-nav__link" >
< span class = "md-ellipsis" >
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10.2 二分查找插入点
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< / span >
< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/binary_search_edge/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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10.3 二分查找边界
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< / span >
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< / a >
< / li >
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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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< span class = "md-ellipsis" >
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10.4 哈希优化策略
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< / span >
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< / a >
< / li >
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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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< span class = "md-ellipsis" >
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10.5 重识搜索算法
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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10.6 小结
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< / span >
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< / a >
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< / ul >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_12" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_sorting/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z" / > < / svg >
< span class = "md-ellipsis" >
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第 11 章 排序
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< / span >
< / a >
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_12_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_12" >
< span class = "md-nav__icon md-icon" > < / span >
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第 11 章 排序
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< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
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< span class = "md-ellipsis" >
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11.1 排序算法
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/selection_sort/" class = "md-nav__link" >
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11.2 选择排序
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
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11.3 冒泡排序
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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11.4 插入排序
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< / span >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/quick_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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11.5 快速排序
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/merge_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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11.6 归并排序
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< / span >
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< / a >
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< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/heap_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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11.7 堆排序
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/bucket_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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11.8 桶排序
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/counting_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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11.9 计数排序
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/radix_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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11.10 基数排序
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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11.11 小结
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< / span >
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< / a >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_13" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_divide_and_conquer/" class = "md-nav__link " >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z" / > < / svg >
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第 12 章 分治
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< / span >
< / a >
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< label class = "md-nav__title" for = "__nav_13" >
< span class = "md-nav__icon md-icon" > < / span >
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第 12 章 分治
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< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_divide_and_conquer/divide_and_conquer/" class = "md-nav__link" >
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12.1 分治算法
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_divide_and_conquer/binary_search_recur/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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12.2 分治搜索策略
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_divide_and_conquer/build_binary_tree_problem/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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12.3 构建树问题
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< / span >
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< / a >
< / li >
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第 13 章 回溯
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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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第 16 章 附录
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< h1 id = "94" > 9.4 小结< a class = "headerlink" href = "#94" title = "Permanent link" > ¶ < / a > < / h1 >
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< ul >
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< li > 图由顶点和边组成，可以被表示为一组顶点和一组边构成的集合。< / li >
< li > 相较于线性关系（链表）和分治关系（树），网络关系（图）具有更高的自由度，因而更为复杂。< / li >
< li > 有向图的边具有方向性，连通图中的任意顶点均可达，有权图的每条边都包含权重变量。< / li >
< li > 邻接矩阵利用矩阵来表示图，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 > 表示两个顶点之间有边或无边。邻接矩阵在增删查操作上效率很高，但空间占用较多。< / li >
< li > 邻接表使用多个链表来表示图，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。邻接表相对于邻接矩阵更加节省空间，但由于需要遍历链表来查找边，时间效率较低。< / li >
< li > 当邻接表中的链表过长时，可以将其转换为红黑树或哈希表，从而提升查询效率。< / li >
< li > 从算法思想角度分析，邻接矩阵体现“以空间换时间”，邻接表体现“以时间换空间”。< / li >
< li > 图可用于建模各类现实系统，如社交网络、地铁线路等。< / li >
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< li > 树是图的一种特例，树的遍历也是图的遍历的一种特例。< / li >
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< li > 图的广度优先遍历是一种由近及远、层层扩张的搜索方式，通常借助队列实现。< / li >
< li > 图的深度优先遍历是一种优先走到底、无路可走时再回溯的搜索方式，常基于递归来实现。< / li >
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< p class = "admonition-title" > 路径的定义是顶点序列还是边序列？< / p >
< p > 维基百科上不同语言版本的定义不一致： ， ：
在本文中，路径被认为是一个边序列，而不是一个顶点序列。这是因为两个顶点之间可能存在多条边连接，此时每条边都对应一条路径。< / p >
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< p class = "admonition-title" > 在邻接表中，“与该顶点相连的所有顶点”的顶点顺序是否有要求？< / p >
< p > 可以是任意顺序。但在实际应用中，可能会需要按照指定规则来排序，比如按照顶点添加的次序、或者按照顶点值大小的顺序等等，这样可以有助于快速查找“带有某种极值”的顶点。< / p >
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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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