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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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< meta name = "author" content = "Krahets" >
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< title > 10.4. < / title >
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< style > : root { --md-text-font : "Noto Sans SC" ; --md-code-font : "Fira Code" } < / style >
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< div data-md-component = "skip" >
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跳转至
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Hello 算法
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< div class = "md-header__topic" data-md-component = "header-topic" >
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10.4. 重识搜索算法
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< label class = "md-header__button md-icon" title = "Switch to light mode" for = "__palette_1" hidden >
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< label class = "md-header__button md-icon" for = "__search" >
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< div class = "md-search" data-md-component = "search" role = "dialog" >
< label class = "md-search__overlay" for = "__search" > < / label >
< div class = "md-search__inner" role = "search" >
< form class = "md-search__form" name = "search" >
< input type = "text" class = "md-search__input" name = "query" aria-label = "搜索" placeholder = "搜索" autocapitalize = "off" autocorrect = "off" autocomplete = "off" spellcheck = "false" data-md-component = "search-query" required >
< label class = "md-search__icon md-icon" for = "__search" >
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< / label >
< nav class = "md-search__options" aria-label = "查找" >
< a href = "javascript:void(0)" class = "md-search__icon md-icon" title = "分享" aria-label = "分享" data-clipboard data-clipboard-text = "" data-md-component = "search-share" tabindex = "-1" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M18 16.08c-.76 0-1.44.3-1.96.77L8.91 12.7c.05-.23.09-.46.09-.7 0-.24-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91 1.61 0 2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08Z" / > < / svg >
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< button type = "reset" class = "md-search__icon md-icon" title = "清空当前内容" aria-label = "清空当前内容" tabindex = "-1" >
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< / button >
< / nav >
< div class = "md-search__suggest" data-md-component = "search-suggest" > < / div >
< / form >
< div class = "md-search__output" >
< div class = "md-search__scrollwrap" data-md-scrollfix >
< div class = "md-search-result" data-md-component = "search-result" >
< div class = "md-search-result__meta" >
正在初始化搜索引擎
< / div >
< ol class = "md-search-result__list" role = "presentation" > < / ol >
< / div >
< / div >
< / div >
< / div >
< / div >
< div class = "md-header__source" >
< a href = "https://github.com/krahets/hello-algo" title = "前往仓库" class = "md-source" data-md-component = "source" >
< div class = "md-source__icon md-icon" >
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< / div >
< div class = "md-source__repository" >
krahets/hello-algo
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< / a >
< / div >
< / nav >
< / header >
< div class = "md-container" data-md-component = "container" >
< main class = "md-main" data-md-component = "main" >
< div class = "md-main__inner md-grid" >
< div class = "md-sidebar md-sidebar--primary" data-md-component = "sidebar" data-md-type = "navigation" >
< div class = "md-sidebar__scrollwrap" >
< div class = "md-sidebar__inner" >
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< nav class = "md-nav md-nav--primary" aria-label = "导航栏" data-md-level = "0" >
< label class = "md-nav__title" for = "__drawer" >
< a href = "../.." title = "Hello 算法" class = "md-nav__button md-logo" aria-label = "Hello 算法" data-md-component = "logo" >
< img src = "../../assets/images/logo.png" alt = "logo" >
< / a >
Hello 算法
< / label >
< div class = "md-nav__source" >
< a href = "https://github.com/krahets/hello-algo" title = "前往仓库" class = "md-source" data-md-component = "source" >
< div class = "md-source__icon md-icon" >
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krahets/hello-algo
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< / a >
< / div >
< ul class = "md-nav__list" data-md-scrollfix >
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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_1" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_preface/" class = "md-nav__link " >
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0. 前言
< / span >
< / a >
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< label class = "md-nav__link " for = "__nav_1" >
< span class = "md-nav__icon md-icon" > < / span >
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0. 前言
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0.1. 关于本书
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< a href = "../../chapter_preface/suggestions/" class = "md-nav__link" >
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0.2. 如何使用本书
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0.3. 小结
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< a href = "../../chapter_introduction/" class = "md-nav__link " >
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1. 初识算法
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1. 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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2. 复杂度
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2. 复杂度
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2.1. 算法效率评估
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2.2. 时间复杂度
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2.3. 空间复杂度
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2.4. 小结
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< a href = "../../chapter_data_structure/" class = "md-nav__link " >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M12 3C7.58 3 4 4.79 4 7v10c0 2.21 3.59 4 8 4s8-1.79 8-4V7c0-2.21-3.58-4-8-4m6 14c0 .5-2.13 2-6 2s-6-1.5-6-2v-2.23c1.61.78 3.72 1.23 6 1.23s4.39-.45 6-1.23V17m0-4.55c-1.3.95-3.58 1.55-6 1.55s-4.7-.6-6-1.55V9.64c1.47.83 3.61 1.36 6 1.36s4.53-.53 6-1.36v2.81M12 9C8.13 9 6 7.5 6 7s2.13-2 6-2 6 1.5 6 2-2.13 2-6 2Z" / > < / svg >
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3. 数据结构
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3. 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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4. 数组与链表
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4. 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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< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/list/" class = "md-nav__link" >
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4.3. 列表
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4.4. 小结
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< div class = "md-nav__link md-nav__container" >
< 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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5. 栈与队列
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< / label >
< / div >
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< label class = "md-nav__title" for = "__nav_6" >
< span class = "md-nav__icon md-icon" > < / span >
5. 栈与队列
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/stack/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
5.1. 栈
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/queue/" class = "md-nav__link" >
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5.2. 队列
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/deque/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
5.3. 双向队列
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
5.4. 小结
< / 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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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_hashing/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z" / > < / svg >
< span class = "md-ellipsis" >
6. 散列表
< / span >
< / a >
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< / label >
< / div >
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< label class = "md-nav__title" for = "__nav_7" >
< span class = "md-nav__icon md-icon" > < / span >
6. 散列表
< / 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_hashing/hash_map/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.1. 哈希表
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_hashing/hash_collision/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.2. 哈希冲突
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_algorithm/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.3. 哈希算法
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.4. 小结
< / span >
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< / a >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_tree/" class = "md-nav__link " >
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< span class = "md-ellipsis" >
7. 树
< / span >
< / a >
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< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_8_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_8" >
< span class = "md-nav__icon md-icon" > < / span >
7. 树
< / 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_tree/binary_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.1. 二叉树
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_tree/binary_tree_traversal/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.2. 二叉树遍历
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_tree/array_representation_of_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.3. 二叉树数组表示
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_search_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.4. 二叉搜索树
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/avl_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.5. AVL 树 *
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.6. 小结
< / span >
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< / a >
< / li >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_9" >
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< div class = "md-nav__link md-nav__container" >
< 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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8. 堆
< / span >
< / a >
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< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_9_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_9" >
< span class = "md-nav__icon md-icon" > < / span >
8. 堆
< / 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_heap/heap/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
8.1. 堆
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
8.2. 建堆操作
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_heap/top_k/" class = "md-nav__link" >
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8.3. Top-K 问题
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_heap/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
8.4. 小结
< / span >
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< / a >
< / li >
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< / ul >
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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_10" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_graph/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z" / > < / svg >
< span class = "md-ellipsis" >
9. 图
< / span >
< / a >
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< label class = "md-nav__link " for = "__nav_10" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_10_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_10" >
< span class = "md-nav__icon md-icon" > < / span >
9. 图
< / 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_graph/graph/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.1. 图
< / 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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< span class = "md-ellipsis" >
9.2. 图基础操作
< / 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_traversal/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.3. 图的遍历
< / 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" >
9.4. 小结
< / 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--active md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" checked >
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< div class = "md-nav__link md-nav__container" >
< a href = "../" 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" >
10. 搜索
< / span >
< / a >
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< label class = "md-nav__link " for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_11_label" aria-expanded = "true" >
< label class = "md-nav__title" for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
10. 搜索
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
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< a href = "../binary_search/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.1. 二分查找
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../binary_search_edge/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.2. 二分查找边界
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../replace_linear_by_hashing/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.3. 哈希优化策略
< / span >
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< / a >
< / li >
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< li class = "md-nav__item md-nav__item--active" >
< input class = "md-nav__toggle md-toggle" type = "checkbox" id = "__toc" >
< label class = "md-nav__link md-nav__link--active" for = "__toc" >
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< span class = "md-ellipsis" >
10.4. 重识搜索算法
< / span >
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< span class = "md-nav__icon md-icon" > < / span >
< / label >
< a href = "./" class = "md-nav__link md-nav__link--active" >
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< span class = "md-ellipsis" >
10.4. 重识搜索算法
< / span >
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< / a >
< nav class = "md-nav md-nav--secondary" aria-label = "目录" >
< label class = "md-nav__title" for = "__toc" >
< span class = "md-nav__icon md-icon" > < / span >
目录
< / label >
< ul class = "md-nav__list" data-md-component = "toc" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "#1041" class = "md-nav__link" >
10.4.1. 暴力搜索
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#1042" class = "md-nav__link" >
10.4.2. 自适应搜索
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#1043" class = "md-nav__link" >
10.4.3. 搜索方法选取
< / a >
< / li >
< / ul >
< / nav >
< / li >
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< li class = "md-nav__item" >
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< a href = "../summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.5. 小结
< / 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" >
11. 排序
< / span >
< / a >
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< label class = "md-nav__link " for = "__nav_12" >
< span class = "md-nav__icon md-icon" > < / span >
< / 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 >
11. 排序
< / 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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< span class = "md-ellipsis" >
11.1. 排序算法
< / 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" >
11.2. 选择排序
< / 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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< span class = "md-ellipsis" >
11.3. 冒泡排序
< / 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" >
11.4. 插入排序
< / 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" >
11.5. 快速排序
< / 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" >
11.6. 归并排序
< / 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" >
11.7. 堆排序
< / 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" >
11.8. 桶排序
< / 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" >
11.9. 计数排序
< / 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" >
11.10. 基数排序
< / 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" >
11.11. 小结
< / 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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12.2. 分治搜索策略
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13.3. 子集和问题
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14. 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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15.2. 一起参与创作
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参考文献
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目录
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10.4.1. 暴力搜索
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10.4.2. 自适应搜索
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10.4.3. 搜索方法选取
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< h1 id = "104" > 10.4. 重识搜索算法< a class = "headerlink" href = "#104" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > 「搜索算法 Searching Algorithm」用于在数据结构（ ） < / p >
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< p > 根据实现思路，搜索算法总体可分为两种：< / p >
< ul >
< li > < strong > 通过遍历数据结构来定位目标元素< / strong > ，例如数组、链表、树和图的遍历等。< / li >
< li > < strong > 利用数据组织结构或数据包含的先验信息，实现高效元素查找< / strong > ，例如二分查找、哈希查找和二叉搜索树查找等。< / li >
< / ul >
< p > 不难发现，这些知识点都已在前面的章节中介绍过，因此搜索算法对于我们来说并不陌生。在本节中，我们将从更加系统的视角切入，重新审视搜索算法。< / p >
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< h2 id = "1041" > 10.4.1. 暴力搜索< a class = "headerlink" href = "#1041" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 暴力搜索通过遍历数据结构的每个元素来定位目标元素。< / p >
< ul >
< li > 「线性搜索」适用于数组和链表等线性数据结构。它从数据结构的一端开始，逐个访问元素，直到找到目标元素或到达另一端仍没有找到目标元素为止。< / li >
< li > 「广度优先搜索」和「深度优先搜索」是图和树的两种遍历策略。广度优先搜索从初始节点开始逐层搜索，由近及远地访问各个节点。深度优先搜索是从初始节点开始，沿着一条路径走到头为止，再回溯并尝试其他路径，直到遍历完整个数据结构。< / li >
< / ul >
< p > 暴力搜索的优点是简单且通用性好，< strong > 无需对数据做预处理和借助额外的数据结构< / strong > 。< / p >
< p > 然而，< strong > 此类算法的时间复杂度为 < span class = "arithmatex" > \(O(n)\)< / span > < / strong > ，其中 < span class = "arithmatex" > \(n\)< / span > 为元素数量，因此在数据量较大的情况下性能较差。< / p >
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< h2 id = "1042" > 10.4.2. 自适应搜索< a class = "headerlink" href = "#1042" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 自适应搜索利用数据的特有属性（例如有序性）来优化搜索过程，从而更高效地定位目标元素。< / p >
< ul >
< li > 「二分查找」利用数据的有序性实现高效查找，仅适用于数组。< / li >
< li > 「哈希查找」利用哈希表将搜索数据和目标数据建立为键值对映射，从而实现查询操作。< / li >
< li > 「树查找」在特定的树结构（例如二叉搜索树）中，基于比较节点值来快速排除节点，从而定位目标元素。< / li >
< / ul >
< p > 此类算法的优点是效率高，< strong > 时间复杂度可达到 < span class = "arithmatex" > \(O(\log n)\)< / span > 甚至 < span class = "arithmatex" > \(O(1)\)< / span > < / strong > 。< / p >
< p > 然而，< strong > 使用这些算法往往需要对数据进行预处理< / strong > 。例如，二分查找需要预先对数组进行排序，哈希查找和树查找都需要借助额外的数据结构，维护这些数据结构也需要额外的时间和空间开支。< / p >
< div class = "admonition note" >
< p class = "admonition-title" > Note< / p >
< p > 自适应搜索算法常被称为查找算法，< strong > 主要关注在特定数据结构中快速检索目标元素< / strong > 。< / p >
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< h2 id = "1043" > 10.4.3. 搜索方法选取< a class = "headerlink" href = "#1043" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 给定大小为 < span class = "arithmatex" > \(n\)< / span > 的一组数据，我们可以使用线性搜索、二分查找、树查找、哈希查找等多种方法在该数据中搜索目标元素。各个方法的工作原理如下图所示。< / p >
< p > < img alt = "多种搜索策略" src = "../searching_algorithm_revisited.assets/searching_algorithms.png" / > < / p >
< p align = "center" > Fig. 多种搜索策略 < / p >
< p > 上述几种方法的操作效率与特性如下表所示。< / p >
< div class = "center-table" >
< table >
< thead >
< tr >
< th > < / th >
< th > 线性搜索< / th >
< th > 二分查找< / th >
< th > 树查找< / th >
< th > 哈希查找< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > 查找元素< / td >
< td > < span class = "arithmatex" > \(O(n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(\log n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(\log n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(1)\)< / span > < / td >
< / tr >
< tr >
< td > 插入元素< / td >
< td > < span class = "arithmatex" > \(O(1)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(\log n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(1)\)< / span > < / td >
< / tr >
< tr >
< td > 删除元素< / td >
< td > < span class = "arithmatex" > \(O(n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(\log n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(1)\)< / span > < / td >
< / tr >
< tr >
< td > 额外空间< / td >
< td > < span class = "arithmatex" > \(O(1)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(1)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(n)\)< / span > < / td >
< td > < span class = "arithmatex" > \(O(n)\)< / span > < / td >
< / tr >
< tr >
< td > 数据预处理< / td >
< td > /< / td >
< td > 排序 < span class = "arithmatex" > \(O(n \log n)\)< / span > < / td >
< td > 建树 < span class = "arithmatex" > \(O(n \log n)\)< / span > < / td >
< td > 建哈希表 < span class = "arithmatex" > \(O(n)\)< / span > < / td >
< / tr >
< tr >
< td > 数据是否有序< / td >
< td > 无序< / td >
< td > 有序< / td >
< td > 有序< / td >
< td > 无序< / td >
< / tr >
< / tbody >
< / table >
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< / div >
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< p > 除了以上表格内容，搜索算法的选择还取决于数据体量、搜索性能要求、数据查询与更新频率等。< / p >
< p > < strong > 线性搜索< / strong > < / p >
< ul >
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< li > 通用性较好，无需任何数据预处理操作。假如我们仅需查询一次数据，那么其他三种方法的数据预处理的时间比线性搜索的时间还要更长。< / li >
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< li > 适用于体量较小的数据，此情况下时间复杂度对效率影响较小。< / li >
< li > 适用于数据更新频率较高的场景，因为该方法不需要对数据进行任何额外维护。< / li >
< / ul >
< p > < strong > 二分查找< / strong > < / p >
< ul >
< li > 适用于大数据量的情况，效率表现稳定，最差时间复杂度为 < span class = "arithmatex" > \(O(\log n)\)< / span > 。< / li >
< li > 数据量不能过大，因为存储数组需要连续的内存空间。< / li >
< li > 不适用于高频增删数据的场景，因为维护有序数组的开销较大。< / li >
< / ul >
< p > < strong > 哈希查找< / strong > < / p >
< ul >
< li > 适合对查询性能要求很高的场景，平均时间复杂度为 < span class = "arithmatex" > \(O(1)\)< / span > 。< / li >
< li > 不适合需要有序数据或范围查找的场景，因为哈希表无法维护数据的有序性。< / li >
< li > 对哈希函数和哈希冲突处理策略的依赖性较高，具有较大的性能劣化风险。< / li >
< li > 不适合数据量过大的情况，因为哈希表需要额外空间来最大程度地减少冲突，从而提供良好的查询性能。< / li >
< / ul >
< p > < strong > 树查找< / strong > < / p >
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< ul >
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< li > 适用于海量数据，因为树节点在内存中是离散存储的。< / li >
< li > 适合需要维护有序数据或范围查找的场景。< / li >
< li > 在持续增删节点的过程中，二叉搜索树可能产生倾斜，时间复杂度劣化至 < span class = "arithmatex" > \(O(n)\)< / span > 。< / li >
< li > 若使用 AVL 树或红黑树，则各项操作可在 < span class = "arithmatex" > \(O(\log n)\)< / span > 效率下稳定运行，但维护树平衡的操作会增加额外开销。< / li >
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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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< a href = "https://leetcode.cn/u/jyd/" target = "_blank" rel = "noopener" title = "leetcode.cn" class = "md-social__link" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 640 512" > <!-- ! Font Awesome Free 6.4.0 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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< div class = "md-dialog" data-md-component = "dialog" >
< div class = "md-dialog__inner md-typeset" > < / div >
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< script src = "https://polyfill.io/v3/polyfill.min.js?features=es6" > < / script >
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