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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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2.5 小结
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目录
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1. 重点回顾
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2. Q & A
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第 3 章 数据结构
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4.1 数组
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5.3 双向队列
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5.4 小结
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第 6 章 哈希表
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6.3 哈希算法
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6.4 小结
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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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9.1 图
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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_graph/graph_operations/" class = "md-nav__link" >
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9.2 图基础操作
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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9.3 图的遍历
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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9.4 小结
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< / span >
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< / a >
< / li >
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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_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 >
< / div >
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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" >
< 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 >
< / li >
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< / ul >
< / nav >
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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 class = "md-nav__link " for = "__nav_12" id = "__nav_12_label" tabindex = "0" >
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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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< a href = "../../chapter_sorting/sorting_algorithm/" class = "md-nav__link" >
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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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< span class = "md-ellipsis" >
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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" >
< 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" >
< a href = "../../chapter_sorting/insertion_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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11.4 插入排序
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< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/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 >
< / 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" >
< 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" >
< 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 >
< / li >
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< / ul >
< / nav >
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< / li >
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12.2 分治搜索策略
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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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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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14.2 DP 问题特性
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14.6 编辑距离问题
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14.7 小结
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第 15 章 贪心
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15.1 贪心算法
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15.2 分数背包问题
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15.3 最大容量问题
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15.4 最大切分乘积问题
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15.5 小结
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16.1 编程环境安装
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16.2 一起参与创作
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16.3 术语表
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1. 重点回顾
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2. Q & A
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< h1 id = "25" > 2.5 小结< a class = "headerlink" href = "#25" title = "Permanent link" > ¶ < / a > < / h1 >
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< h3 id = "1" > 1. 重点回顾< a class = "headerlink" href = "#1" title = "Permanent link" > ¶ < / a > < / h3 >
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< p > < strong > 算法效率评估< / strong > < / p >
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< ul >
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< li > 时间效率和空间效率是衡量算法优劣的两个主要评价指标。< / li >
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< li > 我们可以通过实际测试来评估算法效率，但难以消除测试环境的影响，且会耗费大量计算资源。< / li >
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< li > 复杂度分析可以消除实际测试的弊端，分析结果适用于所有运行平台，并且能够揭示算法在不同数据规模下的效率。< / li >
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< / ul >
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< p > < strong > 时间复杂度< / strong > < / p >
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< ul >
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< li > 时间复杂度用于衡量算法运行时间随数据量增长的趋势，可以有效评估算法效率，但在某些情况下可能失效，如在输入的数据量较小或时间复杂度相同时，无法精确对比算法效率的优劣。< / li >
< li > 最差时间复杂度使用大 < span class = "arithmatex" > \(O\)< / span > 符号表示，对应函数渐近上界，反映当 < span class = "arithmatex" > \(n\)< / span > 趋向正无穷时，操作数量 < span class = "arithmatex" > \(T(n)\)< / span > 的增长级别。< / li >
< li > 推算时间复杂度分为两步，首先统计操作数量，然后判断渐近上界。< / li >
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< li > 常见时间复杂度从低到高排列有 < span class = "arithmatex" > \(O(1)\)< / span > 、< span class = "arithmatex" > \(O(\log n)\)< / span > 、< span class = "arithmatex" > \(O(n)\)< / span > 、< span class = "arithmatex" > \(O(n \log n)\)< / span > 、< span class = "arithmatex" > \(O(n^2)\)< / span > 、< span class = "arithmatex" > \(O(2^n)\)< / span > 和 < span class = "arithmatex" > \(O(n!)\)< / span > 等。< / li >
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< li > 某些算法的时间复杂度非固定，而是与输入数据的分布有关。时间复杂度分为最差、最佳、平均时间复杂度，最佳时间复杂度几乎不用，因为输入数据一般需要满足严格条件才能达到最佳情况。< / li >
< li > 平均时间复杂度反映算法在随机数据输入下的运行效率，最接近实际应用中的算法性能。计算平均时间复杂度需要统计输入数据分布以及综合后的数学期望。< / li >
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< / ul >
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< p > < strong > 空间复杂度< / strong > < / p >
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< ul >
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< li > 空间复杂度的作用类似于时间复杂度，用于衡量算法占用内存空间随数据量增长的趋势。< / li >
< li > 算法运行过程中的相关内存空间可分为输入空间、暂存空间、输出空间。通常情况下，输入空间不纳入空间复杂度计算。暂存空间可分为暂存数据、栈帧空间和指令空间，其中栈帧空间通常仅在递归函数中影响空间复杂度。< / li >
< li > 我们通常只关注最差空间复杂度，即统计算法在最差输入数据和最差运行时刻下的空间复杂度。< / li >
< li > 常见空间复杂度从低到高排列有 < span class = "arithmatex" > \(O(1)\)< / span > 、< span class = "arithmatex" > \(O(\log n)\)< / span > 、< span class = "arithmatex" > \(O(n)\)< / span > 、< span class = "arithmatex" > \(O(n^2)\)< / span > 和 < span class = "arithmatex" > \(O(2^n)\)< / span > 等。< / li >
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< h3 id = "2-q-a" > 2. Q & A< a class = "headerlink" href = "#2-q-a" title = "Permanent link" > ¶ < / a > < / h3 >
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< div class = "admonition question" >
< p class = "admonition-title" > 尾递归的空间复杂度是 < span class = "arithmatex" > \(O(1)\)< / span > 吗？< / p >
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< p > 理论上，尾递归函数的空间复杂度可以优化至 < span class = "arithmatex" > \(O(1)\)< / span > 。不过绝大多数编程语言（例如 Java、Python、C++、Go、C# 等）不支持自动优化尾递归，因此通常认为空间复杂度是 < span class = "arithmatex" > \(O(n)\)< / span > 。< / p >
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< / div >
< div class = "admonition question" >
< p class = "admonition-title" > 函数和方法这两个术语的区别是什么？< / p >
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< p > 「函数 function」可以被独立执行， ， ， < / p >
< p > 下面以几种常见的编程语言为例来说明。< / p >
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< ul >
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< li > C 语言是过程式编程语言， ， （ ） ， < / li >
< li > Java 和 C# 是面向对象的编程语言，代码块（方法）通常作为某个类的一部分。静态方法的行为类似于函数，因为它被绑定在类上，不能访问特定的实例变量。< / li >
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< li > C++ 和 Python 既支持过程式编程（函数），也支持面向对象编程（方法）。< / li >
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< / ul >
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< div class = "admonition question" >
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< p class = "admonition-title" > 图解“常见的空间复杂度类型”反映的是否是占用空间的绝对大小？< / p >
< p > 不是，该图展示的是空间复杂度，其反映的是增长趋势，而不是占用空间的绝对大小。< / p >
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< p > 假设取 < span class = "arithmatex" > \(n = 8\)< / span > ，你可能会发现每条曲线的值与函数对应不上。这是因为每条曲线都包含一个常数项，用于将取值范围压缩到一个视觉舒适的范围内。< / p >
< p > 在实际中，因为我们通常不知道每个方法的“常数项”复杂度是多少，所以一般无法仅凭复杂度来选择 < span class = "arithmatex" > \(n = 8\)< / span > 之下的最优解法。但对于 < span class = "arithmatex" > \(n = 8^5\)< / span > 就很好选了，这时增长趋势已经占主导了。< / p >
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