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krahets/hello-algo
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 4H3a2 2 0 0 0-2 2v13a2 2 0 0 0 2 2h18a2 2 0 0 0 2-2V6a2 2 0 0 0-2-2M3 19V6h8v13H3m18 0h-8V6h8v13m-7-9.5h6V11h-6V9.5m0 2.5h6v1.5h-6V12m0 2.5h6V16h-6v-1.5Z"/></svg>
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第 0 章 前言
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<span class="md-ellipsis">
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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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<span class="md-ellipsis">
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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
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第 2 章 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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</a>
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2.3 空间复杂度
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</a>
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2.4 小结
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</a>
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<span class="md-ellipsis">
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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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<li class="md-nav__item">
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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<span class="md-ellipsis">
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3.1 数据结构分类
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</a>
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<a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">
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<span class="md-ellipsis">
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3.2 基本数据类型
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<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.3 数字编码 *
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<span class="md-ellipsis">
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3.4 字符编码 *
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<span class="md-ellipsis">
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3.5 小结
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
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<span class="md-ellipsis">
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第 4 章 数组与链表
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</a>
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<label class="md-nav__link " for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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第 4 章 数组与链表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
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<span class="md-ellipsis">
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4.1 数组
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.2 链表
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.3 列表
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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4.4 小结
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</a>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_stack_and_queue/" class="md-nav__link ">
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<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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<span class="md-ellipsis">
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第 5 章 栈与队列
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</span>
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</a>
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<label class="md-nav__link " for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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第 5 章 栈与队列
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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<span class="md-ellipsis">
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5.1 栈
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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<span class="md-ellipsis">
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5.2 队列
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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<span class="md-ellipsis">
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5.3 双向队列
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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5.4 小结
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</span>
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</a>
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</li>
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</nav>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<a href="../../chapter_hashing/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 6 章 散列表
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</span>
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</a>
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<label class="md-nav__link " for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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第 6 章 散列表
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</label>
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<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">
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6.1 哈希表
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</span>
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</a>
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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">
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6.2 哈希冲突
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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6.3 哈希算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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6.4 小结
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</span>
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</a>
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</li>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
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<span class="md-ellipsis">
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第 7 章 树
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第 7 章 树
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.1 二叉树
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<span class="md-ellipsis">
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7.2 二叉树遍历
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<span class="md-ellipsis">
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7.3 二叉树数组表示
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.4 二叉搜索树
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.5 AVL 树 *
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</a>
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7.6 小结
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</a>
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<a href="../../chapter_heap/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 8 章 堆
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</span>
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</a>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 8 章 堆
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</label>
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<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">
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8.1 堆
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</span>
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</a>
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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">
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8.2 建堆操作
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</span>
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</a>
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<a href="../../chapter_heap/top_k/" class="md-nav__link">
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8.3 Top-K 问题
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8.4 小结
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</span>
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</a>
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</li>
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<a href="../../chapter_graph/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 9 章 图
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</a>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 9 章 图
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<span class="md-ellipsis">
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9.1 图
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</span>
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</a>
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9.2 图基础操作
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</span>
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</a>
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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9.3 图的遍历
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</span>
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</a>
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9.4 小结
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</span>
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</a>
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</li>
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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">
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<a href="../../chapter_searching/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
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<span class="md-ellipsis">
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第 10 章 搜索
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</span>
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</a>
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<label class="md-nav__link " for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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第 10 章 搜索
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</label>
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<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>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
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<span class="md-ellipsis">
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10.2 二分查找插入点
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</span>
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<span class="md-status md-status--new" title="最近添加">
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</span>
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</a>
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</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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<span class="md-status md-status--new" title="最近添加">
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</span>
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</a>
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</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>
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</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>
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</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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</li>
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</ul>
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</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">
|
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|
<a href="../../chapter_sorting/" class="md-nav__link ">
|
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|
|
<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>
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<span class="md-ellipsis">
|
|
|
第 11 章 排序
|
|
|
</span>
|
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|
</a>
|
|
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|
<label class="md-nav__link " for="__nav_12">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
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|
</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>
|
|
|
第 11 章 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>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
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<span class="md-ellipsis">
|
|
|
11.2 选择排序
|
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<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>
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</li>
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<li class="md-nav__item">
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<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>
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</li>
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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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|
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|
|
<span class="md-ellipsis">
|
|
|
11.5 快速排序
|
|
|
</span>
|
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</a>
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</li>
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<li class="md-nav__item">
|
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<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>
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</li>
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<li class="md-nav__item">
|
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<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>
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</li>
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<li class="md-nav__item">
|
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|
<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>
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</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">
|
|
|
11.9 计数排序
|
|
|
</span>
|
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</a>
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</li>
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<li class="md-nav__item">
|
|
|
<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>
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</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>
|
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</li>
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</ul>
|
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</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_13" >
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__container">
|
|
|
<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
|
|
|
|
|
|
|
|
|
<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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|
<span class="md-ellipsis">
|
|
|
第 12 章 分治
|
|
|
</span>
|
|
|
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|
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|
|
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|
|
<span class="md-status md-status--new" title="最近添加">
|
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</span>
|
|
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|
|
|
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|
|
|
</a>
|
|
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|
|
<label class="md-nav__link " for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
第 12 章 分治
|
|
|
</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_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
|
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|
<span class="md-ellipsis">
|
|
|
12.1 分治算法
|
|
|
</span>
|
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<span class="md-status md-status--new" title="最近添加">
|
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</span>
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
|
|
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|
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|
|
<span class="md-ellipsis">
|
|
|
12.2 分治搜索策略
|
|
|
</span>
|
|
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|
<span class="md-status md-status--new" title="最近添加">
|
|
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</span>
|
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12.3 构建树问题
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12.4 汉诺塔问题
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12.5 小结
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第 13 章 回溯
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第 13 章 回溯
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13.1 回溯算法
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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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<span class="md-ellipsis">
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第 14 章 动态规划
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<span class="md-nav__icon md-icon"></span>
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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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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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第 16 章 附录
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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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<h1 id="11">1.1 算法无处不在<a class="headerlink" href="#11" title="Permanent link">¶</a></h1>
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<p>当我们听到“算法”这个词时，很自然地会想到数学。然而实际上，许多算法并不涉及复杂数学，而是更多地依赖于基本逻辑，这些逻辑在我们的日常生活中处处可见。</p>
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<p>在正式探讨算法之前，有一个有趣的事实值得分享：<strong>你已经在不知不觉中学会了许多算法，并习惯将它们应用到日常生活中了</strong>。下面，我将举几个具体例子来证实这一点。</p>
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<p><strong>例一：查阅字典</strong>。在字典里，每个汉字都对应一个拼音，而字典是按照拼音的英文字母顺序排列的。假设我们需要查找一个拼音首字母为 <span class="arithmatex">\(r\)</span> 的字，通常会这样操作：</p>
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<ol>
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<li>翻开字典约一半的页数，查看该页首字母是什么，假设首字母为 <span class="arithmatex">\(m\)</span> 。</li>
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<li>由于在英文字母表中 <span class="arithmatex">\(r\)</span> 位于 <span class="arithmatex">\(m\)</span> 之后，所以排除字典前半部分，查找范围缩小到后半部分。</li>
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<li>不断重复步骤 1-2 ，直至找到拼音首字母为 <span class="arithmatex">\(r\)</span> 的页码为止。</li>
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</ol>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:5"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1"><1></label><label for="__tabbed_1_2"><2></label><label for="__tabbed_1_3"><3></label><label for="__tabbed_1_4"><4></label><label for="__tabbed_1_5"><5></label></div>
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<p><img alt="查字典步骤" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step_1.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="binary_search_dictionary_step_2" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step_2.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="binary_search_dictionary_step_3" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step_3.png" /></p>
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<div class="tabbed-block">
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<p><img alt="binary_search_dictionary_step_4" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step_4.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="binary_search_dictionary_step_5" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step_5.png" /></p>
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</div>
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<p align="center"> 图：查字典步骤 </p>
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<p>查阅字典这个小学生必备技能，实际上就是著名的「二分查找」。从数据结构的角度，我们可以把字典视为一个已排序的「数组」；从算法的角度，我们可以将上述查字典的一系列操作看作是「二分查找」算法。</p>
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<p><strong>例二：整理扑克</strong>。我们在打牌时，每局都需要整理扑克牌，使其从小到大排列，实现流程如下：</p>
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<ol>
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<li>将扑克牌划分为“有序”和“无序”两部分，并假设初始状态下最左 1 张扑克牌已经有序。</li>
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<li>在无序部分抽出一张扑克牌，插入至有序部分的正确位置；完成后最左 2 张扑克已经有序。</li>
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<li>不断循环步骤 <code>2.</code> ，每一轮将一张扑克牌从无序部分插入至有序部分，直至所有扑克牌都有序。</li>
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</ol>
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<p><img alt="扑克排序步骤" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></p>
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<p align="center"> 图：扑克排序步骤 </p>
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<p>上述整理扑克牌的方法本质上是「插入排序」算法，它在处理小型数据集时非常高效。许多编程语言的排序库函数中都存在插入排序的身影。</p>
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<p><strong>例三：货币找零</strong>。假设我们在超市购买了 <span class="arithmatex">\(69\)</span> 元的商品，给收银员付了 <span class="arithmatex">\(100\)</span> 元，则收银员需要给我们找 <span class="arithmatex">\(31\)</span> 元。他会很自然地完成以下思考：</p>
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<ol>
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<li>可选项是比 <span class="arithmatex">\(31\)</span> 元面值更小的货币，包括 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(5\)</span> , <span class="arithmatex">\(10\)</span> , <span class="arithmatex">\(20\)</span> 元。</li>
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<li>从可选项中拿出最大的 <span class="arithmatex">\(20\)</span> 元，剩余 <span class="arithmatex">\(31 - 20 = 11\)</span> 元。</li>
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<li>从剩余可选项中拿出最大的 <span class="arithmatex">\(10\)</span> 元，剩余 <span class="arithmatex">\(11 - 10 = 1\)</span> 元。</li>
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<li>从剩余可选项中拿出最大的 <span class="arithmatex">\(1\)</span> 元，剩余 <span class="arithmatex">\(1 - 1 = 0\)</span> 元。</li>
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<li>完成找零，方案为 <span class="arithmatex">\(20 + 10 + 1 = 31\)</span> 元。</li>
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</ol>
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<p><img alt="货币找零过程" src="../algorithms_are_everywhere.assets/greedy_change.png" /></p>
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<p align="center"> 图：货币找零过程 </p>
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<p>在以上步骤中，我们每一步都采取当前看来最好的选择（尽可能用大面额的货币），最终得到了可行的找零方案。从数据结构与算法的角度看，这种方法本质上是「贪心算法」。</p>
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<p>小到烹饪一道菜，大到星际航行，几乎所有问题的解决都离不开算法。计算机的出现使我们能够通过编程将数据结构存储在内存中，同时编写代码调用 CPU 和 GPU 执行算法。这样一来，我们就能把生活中的问题转移到计算机上，以更高效的方式解决各种复杂问题。</p>
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<div class="admonition tip">
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<p class="admonition-title">Tip</p>
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<p>阅读至此，如果你对数据结构、算法、数组和二分查找等概念仍感到一知半解，那么太好了！因为这正是本书存在的意义。接下来，这本书将引导你一步步深入数据结构与算法的知识殿堂。</p>
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