2994 lines
256 KiB
HTML
2994 lines
256 KiB
HTML
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<!doctype html>
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<meta name="description" content="Your first book to learn Data Structure And Algorithm.">
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<meta name="author" content="Krahets">
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<link rel="canonical" href="https://www.hello-algo.com/chapter_computational_complexity/time_complexity/">
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<title>时间复杂度 - Hello 算法</title>
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Hello 算法
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时间复杂度
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</span>
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</div>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.2.1 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 2022 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
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<div class="md-source__repository">
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krahets/hello-algo
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<label class="md-nav__link" for="__nav_1">
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写在前面
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" aria-label="写在前面" data-md-level="1">
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<label class="md-nav__title" for="__nav_1">
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<span class="md-nav__icon md-icon"></span>
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写在前面
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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关于本书
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_preface/suggestions/" class="md-nav__link">
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如何使用本书
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</a>
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编程环境安装
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_preface/contribution/" class="md-nav__link">
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一起参与创作
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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 class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_2" type="checkbox" id="__nav_2" >
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<label class="md-nav__link" for="__nav_2">
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引言
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" aria-label="引言" data-md-level="1">
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<label class="md-nav__title" for="__nav_2">
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<span class="md-nav__icon md-icon"></span>
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引言
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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_introduction/algorithms_are_everywhere/" class="md-nav__link">
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算法无处不在
|
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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_introduction/what_is_dsa/" class="md-nav__link">
|
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|
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算法是什么
|
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|
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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--active md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_3" type="checkbox" id="__nav_3" checked>
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|
|||
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|
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|
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<label class="md-nav__link" for="__nav_3">
|
|||
|
|
计算复杂度
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
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|
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|
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<nav class="md-nav" aria-label="计算复杂度" data-md-level="1">
|
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<label class="md-nav__title" for="__nav_3">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
计算复杂度
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
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|
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|
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<li class="md-nav__item">
|
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|
|
<a href="../performance_evaluation/" class="md-nav__link">
|
|||
|
|
算法效率评估
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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<li class="md-nav__item md-nav__item--active">
|
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<input class="md-nav__toggle md-toggle" data-md-toggle="toc" type="checkbox" id="__toc">
|
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|
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|
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<label class="md-nav__link md-nav__link--active" for="__toc">
|
|||
|
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时间复杂度
|
|||
|
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<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
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|
|||
|
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<a href="./" class="md-nav__link md-nav__link--active">
|
|||
|
|
时间复杂度
|
|||
|
|
</a>
|
|||
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|
|||
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|
|||
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|
|||
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<nav class="md-nav md-nav--secondary" aria-label="目录">
|
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|
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|
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<label class="md-nav__title" for="__toc">
|
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|
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<span class="md-nav__icon md-icon"></span>
|
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|
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目录
|
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|
|
</label>
|
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|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_2" class="md-nav__link">
|
|||
|
|
统计算法运行时间
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_3" class="md-nav__link">
|
|||
|
|
统计时间增长趋势
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_4" class="md-nav__link">
|
|||
|
|
函数渐近上界
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_5" class="md-nav__link">
|
|||
|
|
推算方法
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="推算方法">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#1" class="md-nav__link">
|
|||
|
|
1. 统计操作数量
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#2" class="md-nav__link">
|
|||
|
|
2. 判断渐近上界
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_6" class="md-nav__link">
|
|||
|
|
常见类型
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="常见类型">
|
|||
|
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<ul class="md-nav__list">
|
|||
|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#o1" class="md-nav__link">
|
|||
|
|
常数阶 \(O(1)\)
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
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|
|||
|
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<li class="md-nav__item">
|
|||
|
|
<a href="#on" class="md-nav__link">
|
|||
|
|
线性阶 \(O(n)\)
|
|||
|
|
</a>
|
|||
|
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|
|||
|
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</li>
|
|||
|
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|
|||
|
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<li class="md-nav__item">
|
|||
|
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<a href="#on2" class="md-nav__link">
|
|||
|
|
平方阶 \(O(n^2)\)
|
|||
|
|
</a>
|
|||
|
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|
|||
|
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</li>
|
|||
|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#o2n" class="md-nav__link">
|
|||
|
|
指数阶 \(O(2^n)\)
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
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|
|||
|
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<li class="md-nav__item">
|
|||
|
|
<a href="#olog-n" class="md-nav__link">
|
|||
|
|
对数阶 \(O(\log n)\)
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
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</li>
|
|||
|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#on-log-n" class="md-nav__link">
|
|||
|
|
线性对数阶 \(O(n \log n)\)
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#on_1" class="md-nav__link">
|
|||
|
|
阶乘阶 \(O(n!)\)
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
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<a href="#_7" class="md-nav__link">
|
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最差、最佳、平均时间复杂度
|
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</a>
|
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|
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空间复杂度
|
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</a>
|
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权衡时间与空间
|
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</a>
|
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|
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|
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|
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<span class="md-nav__icon md-icon"></span>
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<span class="md-nav__icon md-icon"></span>
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数组与链表
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<ul class="md-nav__list" data-md-scrollfix>
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|
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|
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|
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|
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数组（Array）
|
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|
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</a>
|
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|
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|
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|
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|
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
|
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|
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链表（LinkedList）
|
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|
|
</a>
|
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|
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|
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|
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|
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|
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|
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
|
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|
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列表（List）
|
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|
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</a>
|
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|
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|
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|
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|
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
|
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|
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小结
|
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|
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</a>
|
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|
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|
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|
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|
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|
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|
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|
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|
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</nav>
|
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|
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<label class="md-nav__link" for="__nav_6">
|
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|
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栈与队列
|
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|
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" aria-label="栈与队列" data-md-level="1">
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<span class="md-nav__icon md-icon"></span>
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栈与队列
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|
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
|
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|
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栈（Stack）
|
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|
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</a>
|
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|
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
|
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|
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队列（Queue）
|
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|
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</a>
|
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|
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
|
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|
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双向队列（Deque）
|
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|
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</a>
|
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|
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
|
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|
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小结
|
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|
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</a>
|
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|
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|
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<label class="md-nav__link" for="__nav_7">
|
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散列表
|
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" aria-label="散列表" data-md-level="1">
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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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散列表
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</label>
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|
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哈希表（HashMap）
|
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|
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</a>
|
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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哈希冲突处理
|
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|
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</a>
|
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小结
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<label class="md-nav__link" for="__nav_8">
|
|||
|
|
二叉树
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="二叉树" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_8">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
二叉树
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
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|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
|
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|
|||
|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|||
|
|
二叉树（Binary Tree）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
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|
|||
|
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|
|||
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|
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|
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|
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|
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|
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|
|||
|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/binary_tree_types/" class="md-nav__link">
|
|||
|
|
二叉树常见类型
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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|
|||
|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
|||
|
|
二叉搜索树
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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|
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
|||
|
|
AVL 树 *
|
|||
|
|
</a>
|
|||
|
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</li>
|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
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|
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|
|||
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|
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|
|||
|
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|
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|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
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|
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|
|||
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|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_9" type="checkbox" id="__nav_9" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link" for="__nav_9">
|
|||
|
|
查找算法
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="查找算法" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_9">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
查找算法
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
|
|||
|
|
线性查找
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
|||
|
|
二分查找
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
|
|||
|
|
哈希查找
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_10" type="checkbox" id="__nav_10" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link" for="__nav_10">
|
|||
|
|
排序算法
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="排序算法" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_10">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
排序算法
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/intro_to_sort/" class="md-nav__link">
|
|||
|
|
排序简介
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
|||
|
|
冒泡排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
|||
|
|
插入排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
|||
|
|
快速排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
|||
|
|
归并排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_11" type="checkbox" id="__nav_11" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_reference/">参考文献</a>
|
|||
|
|
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="参考文献" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_11">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
参考文献
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
|
|||
|
|
<div class="md-sidebar__scrollwrap">
|
|||
|
|
<div class="md-sidebar__inner">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<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="#_2" class="md-nav__link">
|
|||
|
|
统计算法运行时间
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_3" class="md-nav__link">
|
|||
|
|
统计时间增长趋势
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_4" class="md-nav__link">
|
|||
|
|
函数渐近上界
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_5" class="md-nav__link">
|
|||
|
|
推算方法
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="推算方法">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#1" class="md-nav__link">
|
|||
|
|
1. 统计操作数量
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#2" class="md-nav__link">
|
|||
|
|
2. 判断渐近上界
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_6" class="md-nav__link">
|
|||
|
|
常见类型
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="常见类型">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#o1" class="md-nav__link">
|
|||
|
|
常数阶 \(O(1)\)
|
|||
|
|
</a>
|
|||
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线性阶 \(O(n)\)
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<a href="#on2" class="md-nav__link">
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平方阶 \(O(n^2)\)
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指数阶 \(O(2^n)\)
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<a href="#olog-n" class="md-nav__link">
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对数阶 \(O(\log n)\)
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线性对数阶 \(O(n \log n)\)
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<a href="#on_1" class="md-nav__link">
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阶乘阶 \(O(n!)\)
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</a>
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<a href="#_7" class="md-nav__link">
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最差、最佳、平均时间复杂度
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<a href="https://github.com/krahets/hello-algo/tree/master/docs/chapter_computational_complexity/time_complexity.md" title="编辑此页" class="md-content__button md-icon">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20.71 7.04c.39-.39.39-1.04 0-1.41l-2.34-2.34c-.37-.39-1.02-.39-1.41 0l-1.84 1.83 3.75 3.75M3 17.25V21h3.75L17.81 9.93l-3.75-3.75L3 17.25Z"/></svg>
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<h1 id="_1">时间复杂度<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
|||
|
|
<h2 id="_2">统计算法运行时间<a class="headerlink" href="#_2" title="Permanent link">¶</a></h2>
|
|||
|
|
<p>运行时间能够直观且准确地体现出算法的效率水平。如果我们想要 <strong>准确预估一段代码的运行时间</strong> ，该如何做呢？</p>
|
|||
|
|
<ol>
|
|||
|
|
<li>首先需要 <strong>确定运行平台</strong> ，包括硬件配置、编程语言、系统环境等，这些都会影响到代码的运行效率。</li>
|
|||
|
|
<li>评估 <strong>各种计算操作的所需运行时间</strong> ，例如加法操作 <code>+</code> 需要 1 ns ，乘法操作 <code>*</code> 需要 10 ns ，打印操作需要 5 ns 等。</li>
|
|||
|
|
<li>根据代码 <strong>统计所有计算操作的数量</strong> ，并将所有操作的执行时间求和，即可得到运行时间。</li>
|
|||
|
|
</ol>
|
|||
|
|
<p>例如以下代码，输入数据大小为 <span class="arithmatex">\(n\)</span> ，根据以上方法，可以得到算法运行时间为 <span class="arithmatex">\(6n + 12\)</span> ns 。</p>
|
|||
|
|
<div class="arithmatex">\[
|
|||
|
|
1 + 1 + 10 + (1 + 5) \times n = 6n + 12
|
|||
|
|
\]</div>
|
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|
<div class="tabbed-set tabbed-alternate" data-tabs="1:8"><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" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1">// 在某运行平台下</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 10 ns</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="c1">// 循环 n 次</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 1 ns ，每轮都要执行 i++</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// 5 ns</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="c1">// 在某运行平台下</span>
|
|||
|
|
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
|
|||
|
|
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
|
|||
|
|
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 10 ns</span>
|
|||
|
|
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="c1">// 循环 n 次</span>
|
|||
|
|
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 1 ns ，每轮都要执行 i++</span>
|
|||
|
|
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w"> </span><span class="c1">// 5 ns</span>
|
|||
|
|
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="c1"># 在某运行平台下</span>
|
|||
|
|
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="k">def</span> <span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="n">a</span> <span class="o">=</span> <span class="mi">2</span> <span class="c1"># 1 ns</span>
|
|||
|
|
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span> <span class="c1"># 1 ns</span>
|
|||
|
|
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="mi">2</span> <span class="c1"># 10 ns</span>
|
|||
|
|
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="c1"># 循环 n 次</span>
|
|||
|
|
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span> <span class="c1"># 1 ns</span>
|
|||
|
|
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="c1"># 5 ns</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p>但实际上， <strong>统计算法的运行时间既不合理也不现实。</strong> 首先，我们不希望预估时间和运行平台绑定，毕竟算法需要跑在各式各样的平台之上。其次，我们很难获知每一种操作的运行时间，这为预估过程带来了极大的难度。</p>
|
|||
|
|
<h2 id="_3">统计时间增长趋势<a class="headerlink" href="#_3" title="Permanent link">¶</a></h2>
|
|||
|
|
<p>「时间复杂度分析」采取了不同的做法，其统计的不是算法运行时间，而是 <strong>算法运行时间随着数据量变大时的增长趋势</strong> 。</p>
|
|||
|
|
<p>“时间增长趋势” 这个概念比较抽象，我们借助一个例子来理解。设输入数据大小为 <span class="arithmatex">\(n\)</span> ，给定三个算法 <code>A</code> , <code>B</code> , <code>C</code> 。</p>
|
|||
|
|
<ul>
|
|||
|
|
<li>算法 <code>A</code> 只有 <span class="arithmatex">\(1\)</span> 个打印操作，算法运行时间不随着 <span class="arithmatex">\(n\)</span> 增大而增长。我们称此算法的时间复杂度为「常数阶」。</li>
|
|||
|
|
<li>算法 <code>B</code> 中的打印操作需要循环 <span class="arithmatex">\(n\)</span> 次，算法运行时间随着 <span class="arithmatex">\(n\)</span> 增大成线性增长。此算法的时间复杂度被称为「线性阶」。</li>
|
|||
|
|
<li>算法 <code>C</code> 中的打印操作需要循环 <span class="arithmatex">\(1000000\)</span> 次，但运行时间仍与输入数据大小 <span class="arithmatex">\(n\)</span> 无关。因此 <code>C</code> 的时间复杂度和 <code>A</code> 相同，仍为「常数阶」。</li>
|
|||
|
|
</ul>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="2:8"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label></div>
|
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<div class="tabbed-content">
|
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<div class="tabbed-block">
|
|||
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<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="c1">// 算法 A 时间复杂度：常数阶</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_A</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"></span>
|
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|
|
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="c1">// 算法 B 时间复杂度：线性阶</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="c1">// 算法 C 时间复杂度：常数阶</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">1000000</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="c1">// 算法 A 时间复杂度：常数阶</span>
|
|||
|
|
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_A</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="c1">// 算法 B 时间复杂度：线性阶</span>
|
|||
|
|
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="c1">// 算法 C 时间复杂度：常数阶</span>
|
|||
|
|
<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">1000000</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1"># 算法 A 时间复杂度：常数阶</span>
|
|||
|
|
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="k">def</span> <span class="nf">algorithm_A</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a> <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="c1"># 算法 B 时间复杂度：线性阶</span>
|
|||
|
|
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="k">def</span> <span class="nf">algorithm_B</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a> <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="c1"># 算法 C 时间复杂度：常数阶</span>
|
|||
|
|
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="k">def</span> <span class="nf">algorithm_C</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000000</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a> <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p><img alt="time_complexity_first_example" src="../time_complexity.assets/time_complexity_first_example.png" /></p>
|
|||
|
|
<p align="center"> Fig. 算法 A, B, C 的时间增长趋势 </p>
|
|||
|
|
|
|||
|
|
<p>相比直接统计算法运行时间，时间复杂度分析的做法有什么好处呢？以及有什么不足？</p>
|
|||
|
|
<p><strong>时间复杂度可以有效评估算法效率。</strong> 算法 <code>B</code> 运行时间的增长是线性的，在 <span class="arithmatex">\(n > 1\)</span> 时慢于算法 <code>A</code> ，在 <span class="arithmatex">\(n > 1000000\)</span> 时慢于算法 <code>C</code> 。实质上，只要输入数据大小 <span class="arithmatex">\(n\)</span> 足够大，复杂度为「常数阶」的算法一定优于「线性阶」的算法，这也正是时间增长趋势的含义。</p>
|
|||
|
|
<p><strong>时间复杂度分析将统计「计算操作的运行时间」简化为统计「计算操作的数量」。</strong> 这是因为，无论是运行平台、还是计算操作类型，都与算法运行时间的增长趋势无关。因此，我们可以简单地将所有计算操作的执行时间统一看作是相同的 “单位时间” 。</p>
|
|||
|
|
<p><strong>时间复杂度也存在一定的局限性。</strong> 比如，虽然算法 <code>A</code> 和 <code>C</code> 的时间复杂度相同，但是实际的运行时间有非常大的差别。再比如，虽然算法 <code>B</code> 比 <code>C</code> 的时间复杂度要更高，但在输入数据大小 <span class="arithmatex">\(n\)</span> 比较小时，算法 <code>B</code> 是要明显优于算法 <code>C</code> 的。即使存在这些问题，计算复杂度仍然是评判算法效率的最有效、最常用方法。</p>
|
|||
|
|
<h2 id="_4">函数渐近上界<a class="headerlink" href="#_4" title="Permanent link">¶</a></h2>
|
|||
|
|
<p>设算法「计算操作数量」为 <span class="arithmatex">\(T(n)\)</span> ，其是一个关于输入数据大小 <span class="arithmatex">\(n\)</span> 的函数。例如，以下算法的操作数量为</p>
|
|||
|
|
<div class="arithmatex">\[
|
|||
|
|
T(n) = 3 + 2n
|
|||
|
|
\]</div>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="3:8"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label></div>
|
|||
|
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<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w"> </span><span class="c1">// 循环 n 次</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1（每轮都执行 i ++）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// +1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
|
|||
|
|
<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
|
|||
|
|
<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
|
|||
|
|
<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w"> </span><span class="c1">// 循环 n 次</span>
|
|||
|
|
<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1（每轮都执行 i ++）</span>
|
|||
|
|
<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
|
|||
|
|
<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="k">def</span> <span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a> <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># +1</span>
|
|||
|
|
<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a> <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span> <span class="c1"># +1</span>
|
|||
|
|
<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a> <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="mi">2</span> <span class="c1"># +1</span>
|
|||
|
|
<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a> <span class="c1"># 循环 n 次</span>
|
|||
|
|
<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span> <span class="c1"># +1</span>
|
|||
|
|
<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a> <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="c1"># +1</span>
|
|||
|
|
<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p><span class="arithmatex">\(T(n)\)</span> 是个一次函数，说明时间增长趋势是线性的，因此易得时间复杂度是线性阶。</p>
|
|||
|
|
<p>我们将线性阶的时间复杂度记为 <span class="arithmatex">\(O(n)\)</span> ，这个数学符号被称为「大 <span class="arithmatex">\(O\)</span> 记号 Big-<span class="arithmatex">\(O\)</span> Notation」，代表函数 <span class="arithmatex">\(T(n)\)</span> 的「渐近上界 asymptotic upper bound」。</p>
|
|||
|
|
<p>我们要推算时间复杂度，本质上是在计算「操作数量函数 <span class="arithmatex">\(T(n)\)</span> 」的渐近上界。下面我们先来看看函数渐近上界的数学定义。</p>
|
|||
|
|
<div class="admonition abstract">
|
|||
|
|
<p class="admonition-title">函数渐近上界</p>
|
|||
|
|
<p>若存在正实数 <span class="arithmatex">\(c\)</span> 和实数 <span class="arithmatex">\(n_0\)</span> ，使得对于所有的 <span class="arithmatex">\(n > n_0\)</span> ，均有
|
|||
|
|
$$
|
|||
|
|
T(n) \leq c \cdot f(n)
|
|||
|
|
$$
|
|||
|
|
则可认为 <span class="arithmatex">\(f(n)\)</span> 给出了 <span class="arithmatex">\(T(n)\)</span> 的一个渐近上界，记为
|
|||
|
|
$$
|
|||
|
|
T(n) = O(f(n))
|
|||
|
|
$$</p>
|
|||
|
|
</div>
|
|||
|
|
<p><img alt="asymptotic_upper_bound" src="../time_complexity.assets/asymptotic_upper_bound.png" /></p>
|
|||
|
|
<p align="center"> Fig. 函数的渐近上界 </p>
|
|||
|
|
|
|||
|
|
<p>本质上看，计算渐近上界就是在找一个函数 <span class="arithmatex">\(f(n)\)</span> ，<strong>使得在 <span class="arithmatex">\(n\)</span> 趋向于无穷大时，<span class="arithmatex">\(T(n)\)</span> 和 <span class="arithmatex">\(f(n)\)</span> 处于相同的增长级别（仅相差一个常数项 <span class="arithmatex">\(c\)</span> 的倍数）</strong>。</p>
|
|||
|
|
<div class="admonition tip">
|
|||
|
|
<p class="admonition-title">Tip</p>
|
|||
|
|
<p>渐近上界的数学味儿有点重，如果你感觉没有完全理解，无需担心，因为在实际使用中我们只需要会推算即可，数学意义可以慢慢领悟。</p>
|
|||
|
|
</div>
|
|||
|
|
<h2 id="_5">推算方法<a class="headerlink" href="#_5" title="Permanent link">¶</a></h2>
|
|||
|
|
<p>推算出 <span class="arithmatex">\(f(n)\)</span> 后，我们就得到时间复杂度 <span class="arithmatex">\(O(f(n))\)</span> 。那么，如何来确定渐近上界 <span class="arithmatex">\(f(n)\)</span> 呢？总体分为两步，首先「统计操作数量」，然后「判断渐近上界」。</p>
|
|||
|
|
<h3 id="1">1. 统计操作数量<a class="headerlink" href="#1" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>对着代码，从上到下一行一行地计数即可。然而，<strong>由于上述 <span class="arithmatex">\(c \cdot f(n)\)</span> 中的常数项 <span class="arithmatex">\(c\)</span> 可以取任意大小，因此操作数量 <span class="arithmatex">\(T(n)\)</span> 中的各种系数、常数项都可以被忽略</strong>。根据此原则，可以总结出以下计数偷懒技巧：</p>
|
|||
|
|
<ol>
|
|||
|
|
<li><strong>跳过数量与 <span class="arithmatex">\(n\)</span> 无关的操作。</strong> 因为他们都是 <span class="arithmatex">\(T(n)\)</span> 中的常数项，对时间复杂度不产生影响。</li>
|
|||
|
|
<li><strong>省略所有系数。</strong> 例如，循环 <span class="arithmatex">\(2n\)</span> 次、<span class="arithmatex">\(5n + 1\)</span> 次、……，都可以化简记为 <span class="arithmatex">\(n\)</span> 次，因为 <span class="arithmatex">\(n\)</span> 前面的系数对时间复杂度也不产生影响。</li>
|
|||
|
|
<li><strong>循环嵌套时使用乘法。</strong> 总操作数量等于外层循环和内层循环操作数量之积，每一层循环依然可以分别套用上述 <code>1.</code> 和 <code>2.</code> 技巧。</li>
|
|||
|
|
</ol>
|
|||
|
|
<p>根据以下示例，使用上述技巧前、后的统计结果分别为</p>
|
|||
|
|
<div class="arithmatex">\[
|
|||
|
|
\begin{aligned}
|
|||
|
|
T(n) & = 2n(n + 1) + (5n + 1) + 2 & \text{完整统计 (-.-|||)} \newline
|
|||
|
|
& = 2n^2 + 7n + 3 \newline
|
|||
|
|
T(n) & = n^2 + n & \text{偷懒统计 (o.O)}
|
|||
|
|
\end{aligned}
|
|||
|
|
\]</div>
|
|||
|
|
<p>最终，两者都能推出相同的时间复杂度结果，即 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="4:8"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JavaScript</label><label for="__tabbed_4_6">TypeScript</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0（技巧 1）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0（技巧 1）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w"> </span><span class="c1">// +n（技巧 2）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w"> </span><span class="c1">// +n*n（技巧 3）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0（技巧 1）</span>
|
|||
|
|
<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0（技巧 1）</span>
|
|||
|
|
<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w"> </span><span class="c1">// +n（技巧 2）</span>
|
|||
|
|
<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w"> </span><span class="c1">// +n*n（技巧 3）</span>
|
|||
|
|
<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span> <span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a> <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># +0（技巧 1）</span>
|
|||
|
|
<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a> <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">n</span> <span class="c1"># +0（技巧 1）</span>
|
|||
|
|
<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a> <span class="c1"># +n（技巧 2）</span>
|
|||
|
|
<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a> <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a> <span class="c1"># +n*n（技巧 3）</span>
|
|||
|
|
<a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a> <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="2">2. 判断渐近上界<a class="headerlink" href="#2" title="Permanent link">¶</a></h3>
|
|||
|
|
<p><strong>时间复杂度由多项式 <span class="arithmatex">\(T(n)\)</span> 中最高阶的项来决定</strong>。这是因为在 <span class="arithmatex">\(n\)</span> 趋于无穷大时，最高阶的项将处于主导作用，其它项的影响都可以被忽略。</p>
|
|||
|
|
<p>以下表格给出了一些例子，其中有一些夸张的值，是想要向大家强调 <strong>系数无法撼动阶数</strong> 这一结论。在 <span class="arithmatex">\(n\)</span> 趋于无穷大时，这些常数都是 “浮云” 。</p>
|
|||
|
|
<div class="center-table">
|
|||
|
|
<table>
|
|||
|
|
<thead>
|
|||
|
|
<tr>
|
|||
|
|
<th>操作数量 <span class="arithmatex">\(T(n)\)</span></th>
|
|||
|
|
<th>时间复杂度 <span class="arithmatex">\(O(f(n))\)</span></th>
|
|||
|
|
</tr>
|
|||
|
|
</thead>
|
|||
|
|
<tbody>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(100000\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(O(1)\)</span></td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(3n + 2\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(O(n)\)</span></td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(2n^2 + 3n + 2\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(O(n^2)\)</span></td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(n^3 + 10000n^2\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(O(n^3)\)</span></td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(2^n + 10000n^{10000}\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(O(2^n)\)</span></td>
|
|||
|
|
</tr>
|
|||
|
|
</tbody>
|
|||
|
|
</table>
|
|||
|
|
</div>
|
|||
|
|
<h2 id="_6">常见类型<a class="headerlink" href="#_6" title="Permanent link">¶</a></h2>
|
|||
|
|
<p>设输入数据大小为 <span class="arithmatex">\(n\)</span> ，常见的时间复杂度类型有（从低到高排列）</p>
|
|||
|
|
<div class="arithmatex">\[
|
|||
|
|
\begin{aligned}
|
|||
|
|
O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!) \newline
|
|||
|
|
\text{常数阶} < \text{对数阶} < \text{线性阶} < \text{线性对数阶} < \text{平方阶} < \text{指数阶} < \text{阶乘阶}
|
|||
|
|
\end{aligned}
|
|||
|
|
\]</div>
|
|||
|
|
<p><img alt="time_complexity_common_types" src="../time_complexity.assets/time_complexity_common_types.png" /></p>
|
|||
|
|
<p align="center"> Fig. 时间复杂度的常见类型 </p>
|
|||
|
|
|
|||
|
|
<div class="admonition tip">
|
|||
|
|
<p class="admonition-title">Tip</p>
|
|||
|
|
<p>部分示例代码需要一些前置知识，包括数组、递归算法等。如果遇到看不懂的地方无需担心，可以在学习完后面章节后再来复习，现阶段先聚焦在理解时间复杂度含义和推算方法上。</p>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="o1">常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#o1" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>常数阶的操作数量与输入数据大小 <span class="arithmatex">\(n\)</span> 无关，即不随着 <span class="arithmatex">\(n\)</span> 的变化而变化。</p>
|
|||
|
|
<p>对于以下算法，无论操作数量 <code>size</code> 有多大，只要与数据大小 <span class="arithmatex">\(n\)</span> 无关，时间复杂度就仍为 <span class="arithmatex">\(O(1)\)</span> 。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="5:8"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JavaScript</label><label for="__tabbed_5_6">TypeScript</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* 常数阶 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100000</span><span class="p">;</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
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</div>
|
|||
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<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* 常数阶 */</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100000</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
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<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="sd">""" 常数阶 """</span>
|
|||
|
|
<a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="k">def</span> <span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
|
|||
|
|
<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a> <span class="n">size</span> <span class="o">=</span> <span class="mi">100000</span>
|
|||
|
|
<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a> <span class="k">return</span> <span class="n">count</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="on">线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#on" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>线性阶的操作数量相对输入数据大小成线性级别增长。线性阶常出现于单层循环。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="6:8"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JavaScript</label><label for="__tabbed_6_6">TypeScript</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label></div>
|
|||
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<div class="tabbed-content">
|
|||
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* 线性阶 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* 线性阶 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="sd">""" 线性阶 """</span>
|
|||
|
|
<a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="k">def</span> <span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
|
|||
|
|
<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a> <span class="k">return</span> <span class="n">count</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p>「遍历数组」和「遍历链表」等操作，时间复杂度都为 <span class="arithmatex">\(O(n)\)</span> ，其中 <span class="arithmatex">\(n\)</span> 为数组或链表的长度。</p>
|
|||
|
|
<div class="admonition tip">
|
|||
|
|
<p class="admonition-title">Tip</p>
|
|||
|
|
<p><strong>数据大小 <span class="arithmatex">\(n\)</span> 是根据输入数据的类型来确定的。</strong> 比如，在上述示例中，我们直接将 <span class="arithmatex">\(n\)</span> 看作输入数据大小；以下遍历数组示例中，数据大小 <span class="arithmatex">\(n\)</span> 为数组的长度。</p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="7:8"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JavaScript</label><label for="__tabbed_7_6">TypeScript</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label></div>
|
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<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* 线性阶（遍历数组） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">arrayTraversal</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w"> </span><span class="c1">// 循环次数与数组长度成正比</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* 线性阶（遍历数组） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">arrayTraversal</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>&</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w"> </span><span class="c1">// 循环次数与数组长度成正比</span>
|
|||
|
|
<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="sd">""" 线性阶（遍历数组）"""</span>
|
|||
|
|
<a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="k">def</span> <span class="nf">array_traversal</span><span class="p">(</span><span class="n">nums</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
|
|||
|
|
<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a> <span class="c1"># 循环次数与数组长度成正比</span>
|
|||
|
|
<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a> <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a> <span class="k">return</span> <span class="n">count</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="on2">平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#on2" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>平方阶的操作数量相对输入数据大小成平方级别增长。平方阶常出现于嵌套循环，外层循环和内层循环都为 <span class="arithmatex">\(O(n)\)</span> ，总体为 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
|||
|
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<div class="tabbed-set tabbed-alternate" data-tabs="8:8"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Java</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Python</label><label for="__tabbed_8_4">Go</label><label for="__tabbed_8_5">JavaScript</label><label for="__tabbed_8_6">TypeScript</label><label for="__tabbed_8_7">C</label><label for="__tabbed_8_8">C#</label></div>
|
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<div class="tabbed-content">
|
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<div class="tabbed-block">
|
|||
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<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* 平方阶 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w"> </span><span class="c1">// 循环次数与数组长度成平方关系</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* 平方阶 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w"> </span><span class="c1">// 循环次数与数组长度成平方关系</span>
|
|||
|
|
<a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="sd">""" 平方阶 """</span>
|
|||
|
|
<a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="k">def</span> <span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
|
|||
|
|
<a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a> <span class="c1"># 循环次数与数组长度成平方关系</span>
|
|||
|
|
<a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a> <span class="k">return</span> <span class="n">count</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a>
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</code></pre></div>
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</div>
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</div>
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</div>
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<p><img alt="time_complexity_constant_linear_quadratic" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></p>
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<p align="center"> Fig. 常数阶、线性阶、平方阶的时间复杂度 </p>
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<p>以「冒泡排序」为例，外层循环 <span class="arithmatex">\(n - 1\)</span> 次，内层循环 <span class="arithmatex">\(n-1, n-2, \cdots, 2, 1\)</span> 次，平均为 <span class="arithmatex">\(\frac{n}{2}\)</span> 次，因此时间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
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<div class="arithmatex">\[
|
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O((n - 1) \frac{n}{2}) = O(n^2)
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\]</div>
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<div class="tabbed-set tabbed-alternate" data-tabs="9:8"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Java</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Python</label><label for="__tabbed_9_4">Go</label><label for="__tabbed_9_5">JavaScript</label><label for="__tabbed_9_6">TypeScript</label><label for="__tabbed_9_7">C</label><label for="__tabbed_9_8">C#</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="cm">/* 平方阶（冒泡排序） */</span><span class="w"></span>
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<a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
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<a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计数器</span><span class="w"></span>
|
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<a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w"> </span><span class="c1">// 外循环：待排序元素数量为 n-1, n-2, ..., 1</span><span class="w"></span>
|
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<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
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|
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<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w"> </span><span class="c1">// 内循环：冒泡操作</span><span class="w"></span>
|
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|
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<a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
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<a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="w"> </span><span class="c1">// 交换 nums[j] 与 nums[j + 1]</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-64-10" name="__codelineno-64-10" href="#__codelineno-64-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span><span class="w"></span>
|
|||
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<a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-64-12" name="__codelineno-64-12" href="#__codelineno-64-12"></a><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-64-13" name="__codelineno-64-13" href="#__codelineno-64-13"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// 元素交换包含 3 个单元操作</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-64-14" name="__codelineno-64-14" href="#__codelineno-64-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-64-15" name="__codelineno-64-15" href="#__codelineno-64-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-64-16" name="__codelineno-64-16" href="#__codelineno-64-16"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-64-17" name="__codelineno-64-17" href="#__codelineno-64-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-64-18" name="__codelineno-64-18" href="#__codelineno-64-18"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="cm">/* 平方阶（冒泡排序） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>&</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计数器</span>
|
|||
|
|
<a id="__codelineno-65-4" name="__codelineno-65-4" href="#__codelineno-65-4"></a><span class="w"> </span><span class="c1">// 外循环：待排序元素数量为 n-1, n-2, ..., 1</span>
|
|||
|
|
<a id="__codelineno-65-5" name="__codelineno-65-5" href="#__codelineno-65-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-6" name="__codelineno-65-6" href="#__codelineno-65-6"></a><span class="w"> </span><span class="c1">// 内循环：冒泡操作</span>
|
|||
|
|
<a id="__codelineno-65-7" name="__codelineno-65-7" href="#__codelineno-65-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-8" name="__codelineno-65-8" href="#__codelineno-65-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-9" name="__codelineno-65-9" href="#__codelineno-65-9"></a><span class="w"> </span><span class="c1">// 交换 nums[j] 与 nums[j + 1]</span>
|
|||
|
|
<a id="__codelineno-65-10" name="__codelineno-65-10" href="#__codelineno-65-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-11" name="__codelineno-65-11" href="#__codelineno-65-11"></a><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-12" name="__codelineno-65-12" href="#__codelineno-65-12"></a><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-13" name="__codelineno-65-13" href="#__codelineno-65-13"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// 元素交换包含 3 个单元操作</span>
|
|||
|
|
<a id="__codelineno-65-14" name="__codelineno-65-14" href="#__codelineno-65-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-15" name="__codelineno-65-15" href="#__codelineno-65-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-16" name="__codelineno-65-16" href="#__codelineno-65-16"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-17" name="__codelineno-65-17" href="#__codelineno-65-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-65-18" name="__codelineno-65-18" href="#__codelineno-65-18"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="sd">""" 平方阶（冒泡排序）"""</span>
|
|||
|
|
<a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="k">def</span> <span class="nf">bubble_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># 计数器</span>
|
|||
|
|
<a id="__codelineno-66-4" name="__codelineno-66-4" href="#__codelineno-66-4"></a> <span class="c1"># 外循环：待排序元素数量为 n-1, n-2, ..., 1</span>
|
|||
|
|
<a id="__codelineno-66-5" name="__codelineno-66-5" href="#__codelineno-66-5"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-66-6" name="__codelineno-66-6" href="#__codelineno-66-6"></a> <span class="c1"># 内循环：冒泡操作</span>
|
|||
|
|
<a id="__codelineno-66-7" name="__codelineno-66-7" href="#__codelineno-66-7"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-66-8" name="__codelineno-66-8" href="#__codelineno-66-8"></a> <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">></span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]:</span>
|
|||
|
|
<a id="__codelineno-66-9" name="__codelineno-66-9" href="#__codelineno-66-9"></a> <span class="c1"># 交换 nums[j] 与 nums[j + 1]</span>
|
|||
|
|
<a id="__codelineno-66-10" name="__codelineno-66-10" href="#__codelineno-66-10"></a> <span class="n">tmp</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
|
|||
|
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<a id="__codelineno-66-11" name="__codelineno-66-11" href="#__codelineno-66-11"></a> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
|
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<a id="__codelineno-66-12" name="__codelineno-66-12" href="#__codelineno-66-12"></a> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span>
|
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|
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<a id="__codelineno-66-13" name="__codelineno-66-13" href="#__codelineno-66-13"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">3</span> <span class="c1"># 元素交换包含 3 个单元操作</span>
|
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<a id="__codelineno-66-14" name="__codelineno-66-14" href="#__codelineno-66-14"></a> <span class="k">return</span> <span class="n">count</span>
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<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a>
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<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a>
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</code></pre></div>
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<h3 id="o2n">指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#o2n" title="Permanent link">¶</a></h3>
|
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|
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<div class="admonition note">
|
|||
|
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<p class="admonition-title">Note</p>
|
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<p>生物学科中的 “细胞分裂” 即是指数阶增长：初始状态为 <span class="arithmatex">\(1\)</span> 个细胞，分裂一轮后为 <span class="arithmatex">\(2\)</span> 个，分裂两轮后为 <span class="arithmatex">\(4\)</span> 个，……，分裂 <span class="arithmatex">\(n\)</span> 轮后有 <span class="arithmatex">\(2^n\)</span> 个细胞。</p>
|
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|
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</div>
|
|||
|
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<p>指数阶增长得非常快，在实际应用中一般是不能被接受的。若一个问题使用「暴力枚举」求解的时间复杂度是 <span class="arithmatex">\(O(2^n)\)</span> ，那么一般都需要使用「动态规划」或「贪心算法」等算法来求解。</p>
|
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<div class="tabbed-set tabbed-alternate" data-tabs="10:8"><input checked="checked" id="__tabbed_10_1" name="__tabbed_10" type="radio" /><input id="__tabbed_10_2" name="__tabbed_10" type="radio" /><input id="__tabbed_10_3" name="__tabbed_10" type="radio" /><input id="__tabbed_10_4" name="__tabbed_10" type="radio" /><input id="__tabbed_10_5" name="__tabbed_10" type="radio" /><input id="__tabbed_10_6" name="__tabbed_10" type="radio" /><input id="__tabbed_10_7" name="__tabbed_10" type="radio" /><input id="__tabbed_10_8" name="__tabbed_10" type="radio" /><div class="tabbed-labels"><label for="__tabbed_10_1">Java</label><label for="__tabbed_10_2">C++</label><label for="__tabbed_10_3">Python</label><label for="__tabbed_10_4">Go</label><label for="__tabbed_10_5">JavaScript</label><label for="__tabbed_10_6">TypeScript</label><label for="__tabbed_10_7">C</label><label for="__tabbed_10_8">C#</label></div>
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<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* 指数阶（循环实现） */</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-6" name="__codelineno-72-6" href="#__codelineno-72-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-7" name="__codelineno-72-7" href="#__codelineno-72-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-8" name="__codelineno-72-8" href="#__codelineno-72-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-9" name="__codelineno-72-9" href="#__codelineno-72-9"></a><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-10" name="__codelineno-72-10" href="#__codelineno-72-10"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-11" name="__codelineno-72-11" href="#__codelineno-72-11"></a><span class="w"> </span><span class="c1">// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-12" name="__codelineno-72-12" href="#__codelineno-72-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-72-13" name="__codelineno-72-13" href="#__codelineno-72-13"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* 指数阶（循环实现） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
|||
|
|
<a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-6" name="__codelineno-73-6" href="#__codelineno-73-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-7" name="__codelineno-73-7" href="#__codelineno-73-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-8" name="__codelineno-73-8" href="#__codelineno-73-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-9" name="__codelineno-73-9" href="#__codelineno-73-9"></a><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-10" name="__codelineno-73-10" href="#__codelineno-73-10"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-11" name="__codelineno-73-11" href="#__codelineno-73-11"></a><span class="w"> </span><span class="c1">// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1</span>
|
|||
|
|
<a id="__codelineno-73-12" name="__codelineno-73-12" href="#__codelineno-73-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-73-13" name="__codelineno-73-13" href="#__codelineno-73-13"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="sd">""" 指数阶（循环实现）"""</span>
|
|||
|
|
<a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="k">def</span> <span class="nf">exponential</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a> <span class="n">count</span><span class="p">,</span> <span class="n">base</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a> <span class="c1"># cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
|||
|
|
<a id="__codelineno-74-5" name="__codelineno-74-5" href="#__codelineno-74-5"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-74-6" name="__codelineno-74-6" href="#__codelineno-74-6"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-74-7" name="__codelineno-74-7" href="#__codelineno-74-7"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-74-8" name="__codelineno-74-8" href="#__codelineno-74-8"></a> <span class="n">base</span> <span class="o">*=</span> <span class="mi">2</span>
|
|||
|
|
<a id="__codelineno-74-9" name="__codelineno-74-9" href="#__codelineno-74-9"></a> <span class="c1"># count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1</span>
|
|||
|
|
<a id="__codelineno-74-10" name="__codelineno-74-10" href="#__codelineno-74-10"></a> <span class="k">return</span> <span class="n">count</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p><img alt="time_complexity_exponential" src="../time_complexity.assets/time_complexity_exponential.png" /></p>
|
|||
|
|
<p align="center"> Fig. 指数阶的时间复杂度 </p>
|
|||
|
|
|
|||
|
|
<p>在实际算法中，指数阶常出现于递归函数。例如以下代码，不断地一分为二，分裂 <span class="arithmatex">\(n\)</span> 次后停止。</p>
|
|||
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<div class="tabbed-set tabbed-alternate" data-tabs="11:8"><input checked="checked" id="__tabbed_11_1" name="__tabbed_11" type="radio" /><input id="__tabbed_11_2" name="__tabbed_11" type="radio" /><input id="__tabbed_11_3" name="__tabbed_11" type="radio" /><input id="__tabbed_11_4" name="__tabbed_11" type="radio" /><input id="__tabbed_11_5" name="__tabbed_11" type="radio" /><input id="__tabbed_11_6" name="__tabbed_11" type="radio" /><input id="__tabbed_11_7" name="__tabbed_11" type="radio" /><input id="__tabbed_11_8" name="__tabbed_11" type="radio" /><div class="tabbed-labels"><label for="__tabbed_11_1">Java</label><label for="__tabbed_11_2">C++</label><label for="__tabbed_11_3">Python</label><label for="__tabbed_11_4">Go</label><label for="__tabbed_11_5">JavaScript</label><label for="__tabbed_11_6">TypeScript</label><label for="__tabbed_11_7">C</label><label for="__tabbed_11_8">C#</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* 指数阶（递归实现） */</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">expRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
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<a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">expRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">expRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
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<a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="p">}</span><span class="w"></span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* 指数阶（递归实现） */</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">expRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
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<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
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<a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">expRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">expRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
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|
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<a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="p">}</span><span class="w"></span>
|
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</code></pre></div>
|
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</div>
|
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="sd">""" 指数阶（递归实现）"""</span>
|
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<a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="k">def</span> <span class="nf">exp_recur</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
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|
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<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a> <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">1</span>
|
|||
|
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<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a> <span class="k">return</span> <span class="n">exp_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">exp_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
|
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</code></pre></div>
|
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</div>
|
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a>
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</code></pre></div>
|
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a>
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</code></pre></div>
|
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</div>
|
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a>
|
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</code></pre></div>
|
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a>
|
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|
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</code></pre></div>
|
|||
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</div>
|
|||
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</div>
|
|||
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</div>
|
|||
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<h3 id="olog-n">对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#olog-n" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>对数阶与指数阶正好相反，后者反映 “每轮增加到两倍的情况” ，而前者反映 “每轮缩减到一半的情况” 。对数阶仅次于常数阶，时间增长的很慢，是理想的时间复杂度。</p>
|
|||
|
|
<p>对数阶常出现于「二分查找」和「分治算法」中，体现 “一分为多” 、“化繁为简” 的算法思想。</p>
|
|||
|
|
<p>设输入数据大小为 <span class="arithmatex">\(n\)</span> ，由于每轮缩减到一半，因此循环次数是 <span class="arithmatex">\(\log_2 n\)</span> ，即 <span class="arithmatex">\(2^n\)</span> 的反函数。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="12:8"><input checked="checked" id="__tabbed_12_1" name="__tabbed_12" type="radio" /><input id="__tabbed_12_2" name="__tabbed_12" type="radio" /><input id="__tabbed_12_3" name="__tabbed_12" type="radio" /><input id="__tabbed_12_4" name="__tabbed_12" type="radio" /><input id="__tabbed_12_5" name="__tabbed_12" type="radio" /><input id="__tabbed_12_6" name="__tabbed_12" type="radio" /><input id="__tabbed_12_7" name="__tabbed_12" type="radio" /><input id="__tabbed_12_8" name="__tabbed_12" type="radio" /><div class="tabbed-labels"><label for="__tabbed_12_1">Java</label><label for="__tabbed_12_2">C++</label><label for="__tabbed_12_3">Python</label><label for="__tabbed_12_4">Go</label><label for="__tabbed_12_5">JavaScript</label><label for="__tabbed_12_6">TypeScript</label><label for="__tabbed_12_7">C</label><label for="__tabbed_12_8">C#</label></div>
|
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<div class="tabbed-content">
|
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<div class="tabbed-block">
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|||
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|
<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* 对数阶（循环实现） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-88-2" name="__codelineno-88-2" href="#__codelineno-88-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-88-7" name="__codelineno-88-7" href="#__codelineno-88-7"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-88-8" name="__codelineno-88-8" href="#__codelineno-88-8"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-88-9" name="__codelineno-88-9" href="#__codelineno-88-9"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* 对数阶（循环实现） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-89-2" name="__codelineno-89-2" href="#__codelineno-89-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-89-7" name="__codelineno-89-7" href="#__codelineno-89-7"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-89-8" name="__codelineno-89-8" href="#__codelineno-89-8"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-89-9" name="__codelineno-89-9" href="#__codelineno-89-9"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="sd">""" 对数阶（循环实现）"""</span>
|
|||
|
|
<a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="k">def</span> <span class="nf">logarithmic</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
|
|||
|
|
<a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a> <span class="k">while</span> <span class="n">n</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a> <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">/</span> <span class="mi">2</span>
|
|||
|
|
<a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a> <span class="k">return</span> <span class="n">count</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-91-1" name="__codelineno-91-1" href="#__codelineno-91-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-92-1" name="__codelineno-92-1" href="#__codelineno-92-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-93-1" name="__codelineno-93-1" href="#__codelineno-93-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-94-1" name="__codelineno-94-1" href="#__codelineno-94-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p><img alt="time_complexity_logarithmic" src="../time_complexity.assets/time_complexity_logarithmic.png" /></p>
|
|||
|
|
<p align="center"> Fig. 对数阶的时间复杂度 </p>
|
|||
|
|
|
|||
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<p>与指数阶类似，对数阶也常出现于递归函数。以下代码形成了一个高度为 <span class="arithmatex">\(\log_2 n\)</span> 的递归树。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="13:8"><input checked="checked" id="__tabbed_13_1" name="__tabbed_13" type="radio" /><input id="__tabbed_13_2" name="__tabbed_13" type="radio" /><input id="__tabbed_13_3" name="__tabbed_13" type="radio" /><input id="__tabbed_13_4" name="__tabbed_13" type="radio" /><input id="__tabbed_13_5" name="__tabbed_13" type="radio" /><input id="__tabbed_13_6" name="__tabbed_13" type="radio" /><input id="__tabbed_13_7" name="__tabbed_13" type="radio" /><input id="__tabbed_13_8" name="__tabbed_13" type="radio" /><div class="tabbed-labels"><label for="__tabbed_13_1">Java</label><label for="__tabbed_13_2">C++</label><label for="__tabbed_13_3">Python</label><label for="__tabbed_13_4">Go</label><label for="__tabbed_13_5">JavaScript</label><label for="__tabbed_13_6">TypeScript</label><label for="__tabbed_13_7">C</label><label for="__tabbed_13_8">C#</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* 对数阶（递归实现） */</span><span class="w"></span>
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<a id="__codelineno-96-2" name="__codelineno-96-2" href="#__codelineno-96-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logRecur</span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
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<a id="__codelineno-96-3" name="__codelineno-96-3" href="#__codelineno-96-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
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<a id="__codelineno-96-4" name="__codelineno-96-4" href="#__codelineno-96-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">logRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
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<a id="__codelineno-96-5" name="__codelineno-96-5" href="#__codelineno-96-5"></a><span class="p">}</span><span class="w"></span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-97-1" name="__codelineno-97-1" href="#__codelineno-97-1"></a><span class="cm">/* 对数阶（递归实现） */</span><span class="w"></span>
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<a id="__codelineno-97-2" name="__codelineno-97-2" href="#__codelineno-97-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logRecur</span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
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<a id="__codelineno-97-3" name="__codelineno-97-3" href="#__codelineno-97-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
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<a id="__codelineno-97-4" name="__codelineno-97-4" href="#__codelineno-97-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">logRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
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<a id="__codelineno-97-5" name="__codelineno-97-5" href="#__codelineno-97-5"></a><span class="p">}</span><span class="w"></span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-98-1" name="__codelineno-98-1" href="#__codelineno-98-1"></a><span class="sd">""" 对数阶（递归实现）"""</span>
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<a id="__codelineno-98-2" name="__codelineno-98-2" href="#__codelineno-98-2"></a><span class="k">def</span> <span class="nf">log_recur</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
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<a id="__codelineno-98-3" name="__codelineno-98-3" href="#__codelineno-98-3"></a> <span class="k">if</span> <span class="n">n</span> <span class="o"><=</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">0</span>
|
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<a id="__codelineno-98-4" name="__codelineno-98-4" href="#__codelineno-98-4"></a> <span class="k">return</span> <span class="n">log_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-99-1" name="__codelineno-99-1" href="#__codelineno-99-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-100-1" name="__codelineno-100-1" href="#__codelineno-100-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-101-1" name="__codelineno-101-1" href="#__codelineno-101-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-102-1" name="__codelineno-102-1" href="#__codelineno-102-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a>
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</code></pre></div>
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<h3 id="on-log-n">线性对数阶 <span class="arithmatex">\(O(n \log n)\)</span><a class="headerlink" href="#on-log-n" title="Permanent link">¶</a></h3>
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<p>线性对数阶常出现于嵌套循环中，两层循环的时间复杂度分别为 <span class="arithmatex">\(O(\log n)\)</span> 和 <span class="arithmatex">\(O(n)\)</span> 。</p>
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<p>主流排序算法的时间复杂度都是 <span class="arithmatex">\(O(n \log n )\)</span> ，例如快速排序、归并排序、堆排序等。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="14:8"><input checked="checked" id="__tabbed_14_1" name="__tabbed_14" type="radio" /><input id="__tabbed_14_2" name="__tabbed_14" type="radio" /><input id="__tabbed_14_3" name="__tabbed_14" type="radio" /><input id="__tabbed_14_4" name="__tabbed_14" type="radio" /><input id="__tabbed_14_5" name="__tabbed_14" type="radio" /><input id="__tabbed_14_6" name="__tabbed_14" type="radio" /><input id="__tabbed_14_7" name="__tabbed_14" type="radio" /><input id="__tabbed_14_8" name="__tabbed_14" type="radio" /><div class="tabbed-labels"><label for="__tabbed_14_1">Java</label><label for="__tabbed_14_2">C++</label><label for="__tabbed_14_3">Python</label><label for="__tabbed_14_4">Go</label><label for="__tabbed_14_5">JavaScript</label><label for="__tabbed_14_6">TypeScript</label><label for="__tabbed_14_7">C</label><label for="__tabbed_14_8">C#</label></div>
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<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-104-1" name="__codelineno-104-1" href="#__codelineno-104-1"></a><span class="cm">/* 线性对数阶 */</span><span class="w"></span>
|
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<a id="__codelineno-104-2" name="__codelineno-104-2" href="#__codelineno-104-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLogRecur</span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-104-3" name="__codelineno-104-3" href="#__codelineno-104-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-104-4" name="__codelineno-104-4" href="#__codelineno-104-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span>
|
|||
|
|
<a id="__codelineno-104-5" name="__codelineno-104-5" href="#__codelineno-104-5"></a><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-104-6" name="__codelineno-104-6" href="#__codelineno-104-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-104-7" name="__codelineno-104-7" href="#__codelineno-104-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-104-8" name="__codelineno-104-8" href="#__codelineno-104-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-104-9" name="__codelineno-104-9" href="#__codelineno-104-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-104-10" name="__codelineno-104-10" href="#__codelineno-104-10"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-105-1" name="__codelineno-105-1" href="#__codelineno-105-1"></a><span class="cm">/* 线性对数阶 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-105-2" name="__codelineno-105-2" href="#__codelineno-105-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLogRecur</span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-105-3" name="__codelineno-105-3" href="#__codelineno-105-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-105-4" name="__codelineno-105-4" href="#__codelineno-105-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span>
|
|||
|
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<a id="__codelineno-105-5" name="__codelineno-105-5" href="#__codelineno-105-5"></a><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-105-6" name="__codelineno-105-6" href="#__codelineno-105-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-105-7" name="__codelineno-105-7" href="#__codelineno-105-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-105-8" name="__codelineno-105-8" href="#__codelineno-105-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-105-9" name="__codelineno-105-9" href="#__codelineno-105-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-105-10" name="__codelineno-105-10" href="#__codelineno-105-10"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-106-1" name="__codelineno-106-1" href="#__codelineno-106-1"></a><span class="sd">""" 线性对数阶 """</span>
|
|||
|
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<a id="__codelineno-106-2" name="__codelineno-106-2" href="#__codelineno-106-2"></a><span class="k">def</span> <span class="nf">linear_log_recur</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-106-3" name="__codelineno-106-3" href="#__codelineno-106-3"></a> <span class="k">if</span> <span class="n">n</span> <span class="o"><=</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-106-4" name="__codelineno-106-4" href="#__codelineno-106-4"></a> <span class="n">count</span> <span class="o">=</span> <span class="n">linear_log_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> \
|
|||
|
|
<a id="__codelineno-106-5" name="__codelineno-106-5" href="#__codelineno-106-5"></a> <span class="n">linear_log_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-106-6" name="__codelineno-106-6" href="#__codelineno-106-6"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-106-7" name="__codelineno-106-7" href="#__codelineno-106-7"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-106-8" name="__codelineno-106-8" href="#__codelineno-106-8"></a> <span class="k">return</span> <span class="n">count</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-107-1" name="__codelineno-107-1" href="#__codelineno-107-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
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</div>
|
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<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-108-1" name="__codelineno-108-1" href="#__codelineno-108-1"></a>
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<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-109-1" name="__codelineno-109-1" href="#__codelineno-109-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-110-1" name="__codelineno-110-1" href="#__codelineno-110-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-111-1" name="__codelineno-111-1" href="#__codelineno-111-1"></a>
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</code></pre></div>
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<p><img alt="time_complexity_logarithmic_linear" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></p>
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<p align="center"> Fig. 线性对数阶的时间复杂度 </p>
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<h3 id="on_1">阶乘阶 <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#on_1" title="Permanent link">¶</a></h3>
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<p>阶乘阶对应数学上的「全排列」。即给定 <span class="arithmatex">\(n\)</span> 个互不重复的元素，求其所有可能的排列方案，则方案数量为</p>
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<div class="arithmatex">\[
|
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n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1
|
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\]</div>
|
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<p>阶乘常使用递归实现。例如以下代码，第一层分裂出 <span class="arithmatex">\(n\)</span> 个，第二层分裂出 <span class="arithmatex">\(n - 1\)</span> 个，…… ，直至到第 <span class="arithmatex">\(n\)</span> 层时终止分裂。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="15:8"><input checked="checked" id="__tabbed_15_1" name="__tabbed_15" type="radio" /><input id="__tabbed_15_2" name="__tabbed_15" type="radio" /><input id="__tabbed_15_3" name="__tabbed_15" type="radio" /><input id="__tabbed_15_4" name="__tabbed_15" type="radio" /><input id="__tabbed_15_5" name="__tabbed_15" type="radio" /><input id="__tabbed_15_6" name="__tabbed_15" type="radio" /><input id="__tabbed_15_7" name="__tabbed_15" type="radio" /><input id="__tabbed_15_8" name="__tabbed_15" type="radio" /><div class="tabbed-labels"><label for="__tabbed_15_1">Java</label><label for="__tabbed_15_2">C++</label><label for="__tabbed_15_3">Python</label><label for="__tabbed_15_4">Go</label><label for="__tabbed_15_5">JavaScript</label><label for="__tabbed_15_6">TypeScript</label><label for="__tabbed_15_7">C</label><label for="__tabbed_15_8">C#</label></div>
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<div class="highlight"><span class="filename">time_complexity_types.java</span><pre><span></span><code><a id="__codelineno-112-1" name="__codelineno-112-1" href="#__codelineno-112-1"></a><span class="cm">/* 阶乘阶（递归实现） */</span><span class="w"></span>
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<a id="__codelineno-112-2" name="__codelineno-112-2" href="#__codelineno-112-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">factorialRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
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<a id="__codelineno-112-3" name="__codelineno-112-3" href="#__codelineno-112-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
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|
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<a id="__codelineno-112-4" name="__codelineno-112-4" href="#__codelineno-112-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
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<a id="__codelineno-112-5" name="__codelineno-112-5" href="#__codelineno-112-5"></a><span class="w"> </span><span class="c1">// 从 1 个分裂出 n 个</span><span class="w"></span>
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<a id="__codelineno-112-6" name="__codelineno-112-6" href="#__codelineno-112-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
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<a id="__codelineno-112-7" name="__codelineno-112-7" href="#__codelineno-112-7"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"></span>
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<a id="__codelineno-112-8" name="__codelineno-112-8" href="#__codelineno-112-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
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<a id="__codelineno-112-9" name="__codelineno-112-9" href="#__codelineno-112-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
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<a id="__codelineno-112-10" name="__codelineno-112-10" href="#__codelineno-112-10"></a><span class="p">}</span><span class="w"></span>
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</code></pre></div>
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</div>
|
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity_types.cpp</span><pre><span></span><code><a id="__codelineno-113-1" name="__codelineno-113-1" href="#__codelineno-113-1"></a><span class="cm">/* 阶乘阶（递归实现） */</span><span class="w"></span>
|
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<a id="__codelineno-113-2" name="__codelineno-113-2" href="#__codelineno-113-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">factorialRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-113-3" name="__codelineno-113-3" href="#__codelineno-113-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-113-4" name="__codelineno-113-4" href="#__codelineno-113-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-113-5" name="__codelineno-113-5" href="#__codelineno-113-5"></a><span class="w"> </span><span class="c1">// 从 1 个分裂出 n 个</span>
|
|||
|
|
<a id="__codelineno-113-6" name="__codelineno-113-6" href="#__codelineno-113-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-113-7" name="__codelineno-113-7" href="#__codelineno-113-7"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-113-8" name="__codelineno-113-8" href="#__codelineno-113-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-113-9" name="__codelineno-113-9" href="#__codelineno-113-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-113-10" name="__codelineno-113-10" href="#__codelineno-113-10"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
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</div>
|
|||
|
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<div class="tabbed-block">
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|||
|
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<div class="highlight"><span class="filename">time_complexity_types.py</span><pre><span></span><code><a id="__codelineno-114-1" name="__codelineno-114-1" href="#__codelineno-114-1"></a><span class="sd">""" 阶乘阶（递归实现）"""</span>
|
|||
|
|
<a id="__codelineno-114-2" name="__codelineno-114-2" href="#__codelineno-114-2"></a><span class="k">def</span> <span class="nf">factorial_recur</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-114-3" name="__codelineno-114-3" href="#__codelineno-114-3"></a> <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="k">return</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-114-4" name="__codelineno-114-4" href="#__codelineno-114-4"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
|
|||
|
|
<a id="__codelineno-114-5" name="__codelineno-114-5" href="#__codelineno-114-5"></a> <span class="c1"># 从 1 个分裂出 n 个</span>
|
|||
|
|
<a id="__codelineno-114-6" name="__codelineno-114-6" href="#__codelineno-114-6"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-114-7" name="__codelineno-114-7" href="#__codelineno-114-7"></a> <span class="n">count</span> <span class="o">+=</span> <span class="n">factorial_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-114-8" name="__codelineno-114-8" href="#__codelineno-114-8"></a> <span class="k">return</span> <span class="n">count</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
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|
|||
|
|
<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.go</span><pre><span></span><code><a id="__codelineno-115-1" name="__codelineno-115-1" href="#__codelineno-115-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.js</span><pre><span></span><code><a id="__codelineno-116-1" name="__codelineno-116-1" href="#__codelineno-116-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
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</div>
|
|||
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.ts</span><pre><span></span><code><a id="__codelineno-117-1" name="__codelineno-117-1" href="#__codelineno-117-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
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</div>
|
|||
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.c</span><pre><span></span><code><a id="__codelineno-118-1" name="__codelineno-118-1" href="#__codelineno-118-1"></a>
|
|||
|
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</code></pre></div>
|
|||
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</div>
|
|||
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">time_complexity_types.cs</span><pre><span></span><code><a id="__codelineno-119-1" name="__codelineno-119-1" href="#__codelineno-119-1"></a>
|
|||
|
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</code></pre></div>
|
|||
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</div>
|
|||
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|
</div>
|
|||
|
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</div>
|
|||
|
|
<p><img alt="time_complexity_factorial" src="../time_complexity.assets/time_complexity_factorial.png" /></p>
|
|||
|
|
<p align="center"> Fig. 阶乘阶的时间复杂度 </p>
|
|||
|
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|
|||
|
|
<h2 id="_7">最差、最佳、平均时间复杂度<a class="headerlink" href="#_7" title="Permanent link">¶</a></h2>
|
|||
|
|
<p><strong>某些算法的时间复杂度不是恒定的，而是与输入数据的分布有关。</strong> 举一个例子，输入一个长度为 <span class="arithmatex">\(n\)</span> 数组 <code>nums</code> ，其中 <code>nums</code> 由从 <span class="arithmatex">\(1\)</span> 至 <span class="arithmatex">\(n\)</span> 的数字组成，但元素顺序是随机打乱的；算法的任务是返回元素 <span class="arithmatex">\(1\)</span> 的索引。我们可以得出以下结论：</p>
|
|||
|
|
<ul>
|
|||
|
|
<li>当 <code>nums = [?, ?, ..., 1]</code>，即当末尾元素是 <span class="arithmatex">\(1\)</span> 时，则需完整遍历数组，此时达到 <strong>最差时间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> ；</li>
|
|||
|
|
<li>当 <code>nums = [1, ?, ?, ...]</code> ，即当首个数字为 <span class="arithmatex">\(1\)</span> 时，无论数组多长都不需要继续遍历，此时达到 <strong>最佳时间复杂度 <span class="arithmatex">\(\Omega(1)\)</span></strong> ；</li>
|
|||
|
|
</ul>
|
|||
|
|
<p>「函数渐近上界」使用大 <span class="arithmatex">\(O\)</span> 记号表示，代表「最差时间复杂度」。与之对应，「函数渐近下界」用 <span class="arithmatex">\(\Omega\)</span> 记号（Omega Notation）来表示，代表「最佳时间复杂度」。</p>
|
|||
|
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<div class="tabbed-set tabbed-alternate" data-tabs="16:8"><input checked="checked" id="__tabbed_16_1" name="__tabbed_16" type="radio" /><input id="__tabbed_16_2" name="__tabbed_16" type="radio" /><input id="__tabbed_16_3" name="__tabbed_16" type="radio" /><input id="__tabbed_16_4" name="__tabbed_16" type="radio" /><input id="__tabbed_16_5" name="__tabbed_16" type="radio" /><input id="__tabbed_16_6" name="__tabbed_16" type="radio" /><input id="__tabbed_16_7" name="__tabbed_16" type="radio" /><input id="__tabbed_16_8" name="__tabbed_16" type="radio" /><div class="tabbed-labels"><label for="__tabbed_16_1">Java</label><label for="__tabbed_16_2">C++</label><label for="__tabbed_16_3">Python</label><label for="__tabbed_16_4">Go</label><label for="__tabbed_16_5">JavaScript</label><label for="__tabbed_16_6">TypeScript</label><label for="__tabbed_16_7">C</label><label for="__tabbed_16_8">C#</label></div>
|
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<div class="tabbed-content">
|
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<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">worst_best_time_complexity.java</span><pre><span></span><code><a id="__codelineno-120-1" name="__codelineno-120-1" href="#__codelineno-120-1"></a><span class="kd">public</span><span class="w"> </span><span class="kd">class</span> <span class="nc">worst_best_time_complexity</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-2" name="__codelineno-120-2" href="#__codelineno-120-2"></a><span class="w"> </span><span class="cm">/* 生成一个数组，元素为 { 1, 2, ..., n }，顺序被打乱 */</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-120-3" name="__codelineno-120-3" href="#__codelineno-120-3"></a><span class="w"> </span><span class="kd">static</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">randomNumbers</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-4" name="__codelineno-120-4" href="#__codelineno-120-4"></a><span class="w"> </span><span class="n">Integer</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Integer</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-5" name="__codelineno-120-5" href="#__codelineno-120-5"></a><span class="w"> </span><span class="c1">// 生成数组 nums = { 1, 2, 3, ..., n }</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-6" name="__codelineno-120-6" href="#__codelineno-120-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-7" name="__codelineno-120-7" href="#__codelineno-120-7"></a><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-8" name="__codelineno-120-8" href="#__codelineno-120-8"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-9" name="__codelineno-120-9" href="#__codelineno-120-9"></a><span class="w"> </span><span class="c1">// 随机打乱数组元素</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-10" name="__codelineno-120-10" href="#__codelineno-120-10"></a><span class="w"> </span><span class="n">Collections</span><span class="p">.</span><span class="na">shuffle</span><span class="p">(</span><span class="n">Arrays</span><span class="p">.</span><span class="na">asList</span><span class="p">(</span><span class="n">nums</span><span class="p">));</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-11" name="__codelineno-120-11" href="#__codelineno-120-11"></a><span class="w"> </span><span class="c1">// Integer[] -> int[]</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-12" name="__codelineno-120-12" href="#__codelineno-120-12"></a><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-13" name="__codelineno-120-13" href="#__codelineno-120-13"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-14" name="__codelineno-120-14" href="#__codelineno-120-14"></a><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-15" name="__codelineno-120-15" href="#__codelineno-120-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-16" name="__codelineno-120-16" href="#__codelineno-120-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-17" name="__codelineno-120-17" href="#__codelineno-120-17"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-18" name="__codelineno-120-18" href="#__codelineno-120-18"></a>
|
|||
|
|
<a id="__codelineno-120-19" name="__codelineno-120-19" href="#__codelineno-120-19"></a><span class="w"> </span><span class="cm">/* 查找数组 nums 中数字 1 所在索引 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-20" name="__codelineno-120-20" href="#__codelineno-120-20"></a><span class="w"> </span><span class="kd">static</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">findOne</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-21" name="__codelineno-120-21" href="#__codelineno-120-21"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-22" name="__codelineno-120-22" href="#__codelineno-120-22"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-23" name="__codelineno-120-23" href="#__codelineno-120-23"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-24" name="__codelineno-120-24" href="#__codelineno-120-24"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-25" name="__codelineno-120-25" href="#__codelineno-120-25"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-26" name="__codelineno-120-26" href="#__codelineno-120-26"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-27" name="__codelineno-120-27" href="#__codelineno-120-27"></a>
|
|||
|
|
<a id="__codelineno-120-28" name="__codelineno-120-28" href="#__codelineno-120-28"></a><span class="w"> </span><span class="cm">/* Driver Code */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-29" name="__codelineno-120-29" href="#__codelineno-120-29"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="kd">static</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">main</span><span class="p">(</span><span class="n">String</span><span class="o">[]</span><span class="w"> </span><span class="n">args</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-30" name="__codelineno-120-30" href="#__codelineno-120-30"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-31" name="__codelineno-120-31" href="#__codelineno-120-31"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-32" name="__codelineno-120-32" href="#__codelineno-120-32"></a><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">randomNumbers</span><span class="p">(</span><span class="n">n</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-33" name="__codelineno-120-33" href="#__codelineno-120-33"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findOne</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-34" name="__codelineno-120-34" href="#__codelineno-120-34"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"打乱后的数组为 "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">toString</span><span class="p">(</span><span class="n">nums</span><span class="p">));</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-35" name="__codelineno-120-35" href="#__codelineno-120-35"></a><span class="w"> </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"数字 1 的索引为 "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">index</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-36" name="__codelineno-120-36" href="#__codelineno-120-36"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-37" name="__codelineno-120-37" href="#__codelineno-120-37"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-120-38" name="__codelineno-120-38" href="#__codelineno-120-38"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">worst_best_time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-121-1" name="__codelineno-121-1" href="#__codelineno-121-1"></a><span class="cm">/* 生成一个数组，元素为 { 1, 2, ..., n }，顺序被打乱 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-2" name="__codelineno-121-2" href="#__codelineno-121-2"></a><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">randomNumbers</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-3" name="__codelineno-121-3" href="#__codelineno-121-3"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">(</span><span class="n">n</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-4" name="__codelineno-121-4" href="#__codelineno-121-4"></a><span class="w"> </span><span class="c1">// 生成数组 nums = { 1, 2, 3, ..., n }</span>
|
|||
|
|
<a id="__codelineno-121-5" name="__codelineno-121-5" href="#__codelineno-121-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-6" name="__codelineno-121-6" href="#__codelineno-121-6"></a><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-7" name="__codelineno-121-7" href="#__codelineno-121-7"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-8" name="__codelineno-121-8" href="#__codelineno-121-8"></a><span class="w"> </span><span class="c1">// 使用系统时间生成随机种子</span>
|
|||
|
|
<a id="__codelineno-121-9" name="__codelineno-121-9" href="#__codelineno-121-9"></a><span class="w"> </span><span class="kt">unsigned</span><span class="w"> </span><span class="n">seed</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">chrono</span><span class="o">::</span><span class="n">system_clock</span><span class="o">::</span><span class="n">now</span><span class="p">().</span><span class="n">time_since_epoch</span><span class="p">().</span><span class="n">count</span><span class="p">();</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-10" name="__codelineno-121-10" href="#__codelineno-121-10"></a><span class="w"> </span><span class="c1">// 随机打乱数组元素</span>
|
|||
|
|
<a id="__codelineno-121-11" name="__codelineno-121-11" href="#__codelineno-121-11"></a><span class="w"> </span><span class="n">shuffle</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">(),</span><span class="w"> </span><span class="n">default_random_engine</span><span class="p">(</span><span class="n">seed</span><span class="p">));</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-12" name="__codelineno-121-12" href="#__codelineno-121-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-13" name="__codelineno-121-13" href="#__codelineno-121-13"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-14" name="__codelineno-121-14" href="#__codelineno-121-14"></a>
|
|||
|
|
<a id="__codelineno-121-15" name="__codelineno-121-15" href="#__codelineno-121-15"></a><span class="cm">/* 查找数组 nums 中数字 1 所在索引 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-16" name="__codelineno-121-16" href="#__codelineno-121-16"></a><span class="kt">int</span><span class="w"> </span><span class="n">findOne</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>&</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-17" name="__codelineno-121-17" href="#__codelineno-121-17"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-18" name="__codelineno-121-18" href="#__codelineno-121-18"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-19" name="__codelineno-121-19" href="#__codelineno-121-19"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-20" name="__codelineno-121-20" href="#__codelineno-121-20"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-21" name="__codelineno-121-21" href="#__codelineno-121-21"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-22" name="__codelineno-121-22" href="#__codelineno-121-22"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-23" name="__codelineno-121-23" href="#__codelineno-121-23"></a>
|
|||
|
|
<a id="__codelineno-121-24" name="__codelineno-121-24" href="#__codelineno-121-24"></a>
|
|||
|
|
<a id="__codelineno-121-25" name="__codelineno-121-25" href="#__codelineno-121-25"></a><span class="cm">/* Driver Code */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-26" name="__codelineno-121-26" href="#__codelineno-121-26"></a><span class="kt">int</span><span class="w"> </span><span class="n">main</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-27" name="__codelineno-121-27" href="#__codelineno-121-27"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">1000</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-28" name="__codelineno-121-28" href="#__codelineno-121-28"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-29" name="__codelineno-121-29" href="#__codelineno-121-29"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">randomNumbers</span><span class="p">(</span><span class="n">n</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-30" name="__codelineno-121-30" href="#__codelineno-121-30"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findOne</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-31" name="__codelineno-121-31" href="#__codelineno-121-31"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="s">"</span><span class="se">\n</span><span class="s">数组 [ 1, 2, ..., n ] 被打乱后 = "</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-32" name="__codelineno-121-32" href="#__codelineno-121-32"></a><span class="w"> </span><span class="n">PrintUtil</span><span class="o">::</span><span class="n">printVector</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-33" name="__codelineno-121-33" href="#__codelineno-121-33"></a><span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="s">"数字 1 的索引为 "</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-34" name="__codelineno-121-34" href="#__codelineno-121-34"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-35" name="__codelineno-121-35" href="#__codelineno-121-35"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-121-36" name="__codelineno-121-36" href="#__codelineno-121-36"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">worst_best_time_complexity.py</span><pre><span></span><code><a id="__codelineno-122-1" name="__codelineno-122-1" href="#__codelineno-122-1"></a><span class="sd">""" 生成一个数组，元素为: 1, 2, ..., n ，顺序被打乱 """</span>
|
|||
|
|
<a id="__codelineno-122-2" name="__codelineno-122-2" href="#__codelineno-122-2"></a><span class="k">def</span> <span class="nf">random_numbers</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-122-3" name="__codelineno-122-3" href="#__codelineno-122-3"></a> <span class="c1"># 生成数组 nums =: 1, 2, 3, ..., n </span>
|
|||
|
|
<a id="__codelineno-122-4" name="__codelineno-122-4" href="#__codelineno-122-4"></a> <span class="n">nums</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
|
|||
|
|
<a id="__codelineno-122-5" name="__codelineno-122-5" href="#__codelineno-122-5"></a> <span class="c1"># 随机打乱数组元素</span>
|
|||
|
|
<a id="__codelineno-122-6" name="__codelineno-122-6" href="#__codelineno-122-6"></a> <span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-122-7" name="__codelineno-122-7" href="#__codelineno-122-7"></a> <span class="k">return</span> <span class="n">nums</span>
|
|||
|
|
<a id="__codelineno-122-8" name="__codelineno-122-8" href="#__codelineno-122-8"></a>
|
|||
|
|
<a id="__codelineno-122-9" name="__codelineno-122-9" href="#__codelineno-122-9"></a><span class="sd">""" 查找数组 nums 中数字 1 所在索引 """</span>
|
|||
|
|
<a id="__codelineno-122-10" name="__codelineno-122-10" href="#__codelineno-122-10"></a><span class="k">def</span> <span class="nf">find_one</span><span class="p">(</span><span class="n">nums</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-122-11" name="__codelineno-122-11" href="#__codelineno-122-11"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
|
|||
|
|
<a id="__codelineno-122-12" name="__codelineno-122-12" href="#__codelineno-122-12"></a> <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-122-13" name="__codelineno-122-13" href="#__codelineno-122-13"></a> <span class="k">return</span> <span class="n">i</span>
|
|||
|
|
<a id="__codelineno-122-14" name="__codelineno-122-14" href="#__codelineno-122-14"></a> <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-122-15" name="__codelineno-122-15" href="#__codelineno-122-15"></a>
|
|||
|
|
<a id="__codelineno-122-16" name="__codelineno-122-16" href="#__codelineno-122-16"></a><span class="sd">""" Driver Code """</span>
|
|||
|
|
<a id="__codelineno-122-17" name="__codelineno-122-17" href="#__codelineno-122-17"></a><span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"__main__"</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-122-18" name="__codelineno-122-18" href="#__codelineno-122-18"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-122-19" name="__codelineno-122-19" href="#__codelineno-122-19"></a> <span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
|||
|
|
<a id="__codelineno-122-20" name="__codelineno-122-20" href="#__codelineno-122-20"></a> <span class="n">nums</span> <span class="o">=</span> <span class="n">random_numbers</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-122-21" name="__codelineno-122-21" href="#__codelineno-122-21"></a> <span class="n">index</span> <span class="o">=</span> <span class="n">find_one</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-122-22" name="__codelineno-122-22" href="#__codelineno-122-22"></a> <span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">数组 [ 1, 2, ..., n ] 被打乱后 ="</span><span class="p">,</span> <span class="n">nums</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-122-23" name="__codelineno-122-23" href="#__codelineno-122-23"></a> <span class="nb">print</span><span class="p">(</span><span class="s2">"数字 1 的索引为"</span><span class="p">,</span> <span class="n">index</span><span class="p">)</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">worst_best_time_complexity.go</span><pre><span></span><code><a id="__codelineno-123-1" name="__codelineno-123-1" href="#__codelineno-123-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">worst_best_time_complexity.js</span><pre><span></span><code><a id="__codelineno-124-1" name="__codelineno-124-1" href="#__codelineno-124-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">worst_best_time_complexity.ts</span><pre><span></span><code><a id="__codelineno-125-1" name="__codelineno-125-1" href="#__codelineno-125-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">worst_best_time_complexity.c</span><pre><span></span><code><a id="__codelineno-126-1" name="__codelineno-126-1" href="#__codelineno-126-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">worst_best_time_complexity.cs</span><pre><span></span><code><a id="__codelineno-127-1" name="__codelineno-127-1" href="#__codelineno-127-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<div class="admonition tip">
|
|||
|
|
<p class="admonition-title">Tip</p>
|
|||
|
|
<p>我们在实际应用中很少使用「最佳时间复杂度」，因为往往只有很小概率下才能达到，会带来一定的误导性。反之，「最差时间复杂度」最为实用，因为它给出了一个 “效率安全值” ，让我们可以放心地使用算法。</p>
|
|||
|
|
</div>
|
|||
|
|
<p>从上述示例可以看出，最差或最佳时间复杂度只出现在 “特殊分布的数据” 中，这些情况的出现概率往往很小，因此并不能最真实地反映算法运行效率。<strong>相对地，「平均时间复杂度」可以体现算法在随机输入数据下的运行效率，用 <span class="arithmatex">\(\Theta\)</span> 记号（Theta Notation）来表示</strong>。</p>
|
|||
|
|
<p>对于部分算法，我们可以简单地推算出随机数据分布下的平均情况。比如上述示例，由于输入数组是被打乱的，因此元素 <span class="arithmatex">\(1\)</span> 出现在任意索引的概率都是相等的，那么算法的平均循环次数则是数组长度的一半 <span class="arithmatex">\(\frac{n}{2}\)</span> ，平均时间复杂度为 <span class="arithmatex">\(\Theta(\frac{n}{2}) = \Theta(n)\)</span> 。</p>
|
|||
|
|
<p>但在实际应用中，尤其是较为复杂的算法，计算平均时间复杂度比较困难，因为很难简便地分析出在数据分布下的整体数学期望。这种情况下，我们一般使用最差时间复杂度来作为算法效率的评判标准。</p>
|
|||
|
|
<div class="admonition question">
|
|||
|
|
<p class="admonition-title">为什么很少看到 <span class="arithmatex">\(\Theta\)</span> 符号？</p>
|
|||
|
|
<p>实际中我们经常使用「大 <span class="arithmatex">\(O\)</span> 符号」来表示「平均复杂度」，这样严格意义上来说是不规范的。这可能是因为 <span class="arithmatex">\(O\)</span> 符号实在是太朗朗上口了。</br>如果在本书和其他资料中看到类似 <strong>平均时间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> 的表述，请你直接理解为 <span class="arithmatex">\(\Theta(n)\)</span> 即可。</p>
|
|||
|
|
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|
|||
|
|
|
|||
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