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krahets
2026-03-29 05:07:03 +08:00
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chapter_computational_complexity/time_complexity/index.html
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@@ -35,18 +35,23 @@
<link rel="icon" href="../../assets/images/favicon.png">
<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.7.1">
<meta name="generator" content="zensical-0.0.29">
<title>2.3   时间复杂度 - Hello 算法</title>
<title>2.3 时间复杂度 - Hello 算法</title>
<link rel="stylesheet" href="../../assets/stylesheets/main.484c7ddc.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/palette.ab4e12ef.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/modern/main.6f2fbb28.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/modern/palette.dfe2e883.min.css">
@@ -60,29 +65,20 @@
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans+SC:300,300i,400,400i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans+SC:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Noto Sans SC";--md-code-font:"JetBrains Mono"}</style>
<link rel="stylesheet" href="../../stylesheets/extra.css">
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,t)=>(e<<5)-e+t.charCodeAt(0)),0),__md_get=(e,t=localStorage,a=__md_scope)=>JSON.parse(t.getItem(("__palette"===e?"/":a.pathname)+"."+e)),__md_set=(e,t,a=localStorage,_=__md_scope)=>{try{a.setItem(("__palette"===e?"/":_.pathname)+"."+e,JSON.stringify(t))}catch(e){}},document.documentElement.setAttribute("data-platform",navigator.platform)</script>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
.gslide-desc { color: #666; user-select: text; }
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.gscrollbar-fixer { padding-right: 15px; }
.gdesc-inner { font-size: 0.75rem; }
body[data-md-color-scheme="slate"] .gdesc-inner { background: var(--md-default-bg-color);}
body[data-md-color-scheme="slate"] .gslide-title { color: var(--md-default-fg-color);}
body[data-md-color-scheme="slate"] .gslide-desc { color: var(--md-default-fg-color);}</style> <script src="../../assets/javascripts/glightbox.min.js"></script></head>
</head>
@@ -97,7 +93,7 @@
<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
<label class="md-overlay" for="__drawer"></label>
<label class="md-overlay" for="__drawer" aria-label="导航栏"></label>
<div data-md-component="skip">
@@ -139,12 +135,12 @@
<nav class="md-header__inner md-grid" aria-label="页眉">
<a href="../.." title="Hello 算法" class="md-header__button md-logo" aria-label="Hello 算法" data-md-component="logo">
<img src="../../assets/images/logo.svg" alt="logo">
<img src="../../assets/images/logo.svg" alt="Hello 算法">
</a>
<label class="md-header__button md-icon" for="__drawer">
<label class="md-header__button md-icon" for="__drawer" aria-label="导航栏">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-menu" viewBox="0 0 24 24"><path d="M4 5h16M4 12h16M4 19h16"/></svg>
</label>
<div class="md-header__title" data-md-component="header-title">
<div class="md-header__ellipsis">
@@ -169,7 +165,7 @@
<input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal" aria-label="深色模式" type="radio" name="__palette" id="__palette_0">
<input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal" aria-label="深色模式" type="radio" name="__palette" id="__palette_0">
<label class="md-header__button md-icon" title="深色模式" for="__palette_1" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
@@ -179,7 +175,7 @@
<input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal" aria-label="浅色模式" type="radio" name="__palette" id="__palette_1">
<input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal" aria-label="浅色模式" type="radio" name="__palette" id="__palette_1">
<label class="md-header__button md-icon" title="浅色模式" for="__palette_0" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
@@ -197,7 +193,7 @@
<div class="md-select">
<button class="md-header__button md-icon" aria-label="选择当前语言">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12.87 15.07-2.54-2.51.03-.03A17.5 17.5 0 0 0 14.07 6H17V4h-7V2H8v2H1v2h11.17C11.5 7.92 10.44 9.75 9 11.35 8.07 10.32 7.3 9.19 6.69 8h-2c.73 1.63 1.73 3.17 2.98 4.56l-5.09 5.02L4 19l5-5 3.11 3.11zM18.5 10h-2L12 22h2l1.12-3h4.75L21 22h2zm-2.62 7 1.62-4.33L19.12 17z"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-languages" viewBox="0 0 24 24"><path d="m5 8 6 6M4 14l6-6 2-3M2 5h12M7 2h1M22 22l-5-10-5 10M14 18h6"/></svg>
</button>
<div class="md-select__inner">
<ul class="md-select__list">
@@ -240,64 +236,30 @@
<label class="md-header__button md-icon" for="__search">
<label class="md-header__button md-icon" for="__search" aria-label="查找">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-search" viewBox="0 0 24 24"><path d="m21 21-4.34-4.34"/><circle cx="11" cy="11" r="8"/></svg>
</label>
<div class="md-search" data-md-component="search" role="dialog">
<label class="md-search__overlay" for="__search"></label>
<div class="md-search__inner" role="search">
<form class="md-search__form" name="search">
<input type="text" class="md-search__input" name="query" aria-label="搜索" placeholder="搜索" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
<label class="md-search__icon md-icon" for="__search">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11z"/></svg>
</label>
<nav class="md-search__options" aria-label="查找">
<a href="javascript:void(0)" class="md-search__icon md-icon" title="分享" aria-label="分享" data-clipboard data-clipboard-text="" data-md-component="search-share" tabindex="-1">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 16.08c-.76 0-1.44.3-1.96.77L8.91 12.7c.05-.23.09-.46.09-.7s-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91s2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08"/></svg>
</a>
<button type="reset" class="md-search__icon md-icon" title="清空当前内容" aria-label="清空当前内容" tabindex="-1">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12z"/></svg>
</button>
</nav>
<div class="md-search__suggest" data-md-component="search-suggest"></div>
</form>
<div class="md-search__output">
<div class="md-search__scrollwrap" tabindex="0" data-md-scrollfix>
<div class="md-search-result" data-md-component="search-result">
<div class="md-search-result__meta">
正在初始化搜索引擎
</div>
<ol class="md-search-result__list" role="presentation"></ol>
</div>
</div>
</div>
</div>
<div class="md-search" data-md-component="search" role="dialog" aria-label="查找">
<button type="button" class="md-search__button">
查找
</button>
</div>
<div class="md-header__source">
<div class="md-header__source">
<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
<div class="md-source__icon md-icon">
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</div>
<div class="md-source__repository">
krahets/hello-algo
</div>
</a>
</div>
</div>
</nav>
</header>
@@ -325,7 +287,7 @@
<label class="md-nav__title" for="__drawer">
<a href="../.." title="Hello 算法" class="md-nav__button md-logo" aria-label="Hello 算法" data-md-component="logo">
<img src="../../assets/images/logo.svg" alt="logo">
<img src="../../assets/images/logo.svg" alt="Hello 算法">
</a>
Hello 算法
@@ -335,7 +297,7 @@
<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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krahets/hello-algo
@@ -899,7 +861,7 @@
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
<a href="././" class="md-nav__link md-nav__link--active">
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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<nav class="md-nav" aria-label="2.3.3   推算方法">
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<nav class="md-nav" aria-label="2.3.4   常见类型">
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<div class="md-sidebar__inner">
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<nav class="md-nav" aria-label="2.3.3 推算方法">
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</span>
</a>
<nav class="md-nav" aria-label="2.3.4   常见类型">
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>
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</a>
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<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> 增大呈线性增长。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，虽然运行时间很长，但它与输入数据大小 <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>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_simple_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="算法 A、B 和 C 的时间增长趋势" class="animation-figure" src="../time_complexity.assets/time_complexity_simple_example.png" /></a></p>
<p><img alt="算法 A、B 和 C 的时间增长趋势" class="animation-figure" src="../time_complexity.assets/time_complexity_simple_example.png" /></p>
<p align="center"> 图 2-7 &nbsp; 算法 A、B 和 C 的时间增长趋势 </p>
<p>相较于直接统计算法的运行时间，时间复杂度分析有哪些特点呢？</p>
@@ -5373,7 +5337,7 @@ T(n) = 3 + 2n
<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">\(n &gt; n_0\)</span> ，均有 <span class="arithmatex">\(T(n) \leq c \cdot f(n)\)</span> ，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 <span class="arithmatex">\(T(n) = O(f(n))\)</span> 。</p>
</div>
<p>如图 2-8 所示，计算渐近上界就是寻找一个函数 <span class="arithmatex">\(f(n)\)</span> ，使得当 <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，仅相差一个常数系数 <span class="arithmatex">\(c\)</span>。</p>
<p><a class="glightbox" href="../time_complexity.assets/asymptotic_upper_bound.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="函数的渐近上界" class="animation-figure" src="../time_complexity.assets/asymptotic_upper_bound.png" /></a></p>
<p><img alt="函数的渐近上界" class="animation-figure" src="../time_complexity.assets/asymptotic_upper_bound.png" /></p>
<p align="center"> 图 2-8 &nbsp; 函数的渐近上界 </p>
<h2 id="233">2.3.3 &nbsp; 推算方法<a class="headerlink" href="#233" title="Permanent link">&para;</a></h2>
@@ -5659,7 +5623,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
\text{常数阶} &lt; \text{对数阶} &lt; \text{线性阶} &lt; \text{线性对数阶} &lt; \text{平方阶} &lt; \text{指数阶} &lt; \text{阶乘阶}
\end{aligned}
\]</div>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="常见的时间复杂度类型" class="animation-figure" src="../time_complexity.assets/time_complexity_common_types.png" /></a></p>
<p><img alt="常见的时间复杂度类型" class="animation-figure" src="../time_complexity.assets/time_complexity_common_types.png" /></p>
<p align="center"> 图 2-9 &nbsp; 常见的时间复杂度类型 </p>
<h3 id="1-o1">1. &nbsp; 常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-o1" title="Permanent link">&para;</a></h3>
@@ -6324,7 +6288,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20quadratic%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%B9%B3%E6%96%B9%E9%98%B6%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E6%AC%A1%E6%95%B0%E4%B8%8E%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%E6%88%90%E5%B9%B3%E6%96%B9%E5%85%B3%E7%B3%BB%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20quadratic%28n%29%0A%20%20%20%20print%28%22%E5%B9%B3%E6%96%B9%E9%98%B6%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
</details>
<p>图 2-10 对比了常数阶、线性阶和平方阶三种时间复杂度。</p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="常数阶、线性阶和平方阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></a></p>
<p><img alt="常数阶、线性阶和平方阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></p>
<p align="center"> 图 2-10 &nbsp; 常数阶、线性阶和平方阶的时间复杂度 </p>
<p>以冒泡排序为例，外层循环执行 <span class="arithmatex">\(n - 1\)</span> 次，内层循环执行 <span class="arithmatex">\(n-1\)</span>、<span class="arithmatex">\(n-2\)</span>、<span class="arithmatex">\(\dots\)</span>、<span class="arithmatex">\(2\)</span>、<span class="arithmatex">\(1\)</span> 次，平均为 <span class="arithmatex">\(n / 2\)</span> 次，因此时间复杂度为 <span class="arithmatex">\(O((n - 1) n / 2) = O(n^2)\)</span> ：</p>
@@ -6831,7 +6795,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<p><div style="height: 531px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20exponential%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20base%20%3D%201%0A%20%20%20%20%23%20%E7%BB%86%E8%83%9E%E6%AF%8F%E8%BD%AE%E4%B8%80%E5%88%86%E4%B8%BA%E4%BA%8C%EF%BC%8C%E5%BD%A2%E6%88%90%E6%95%B0%E5%88%97%201,%202,%204,%208,%20...,%202%5E%28n-1%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20_%20in%20range%28base%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20%20%20%20%20base%20*%3D%202%0A%20%20%20%20%23%20count%20%3D%201%20%2B%202%20%2B%204%20%2B%208%20%2B%20..%20%2B%202%5E%28n-1%29%20%3D%202%5En%20-%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20exponential%28n%29%0A%20%20%20%20print%28%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20exponential%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20base%20%3D%201%0A%20%20%20%20%23%20%E7%BB%86%E8%83%9E%E6%AF%8F%E8%BD%AE%E4%B8%80%E5%88%86%E4%B8%BA%E4%BA%8C%EF%BC%8C%E5%BD%A2%E6%88%90%E6%95%B0%E5%88%97%201,%202,%204,%208,%20...,%202%5E%28n-1%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20_%20in%20range%28base%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20%20%20%20%20base%20*%3D%202%0A%20%20%20%20%23%20count%20%3D%201%20%2B%202%20%2B%204%20%2B%208%20%2B%20..%20%2B%202%5E%28n-1%29%20%3D%202%5En%20-%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20exponential%28n%29%0A%20%20%20%20print%28%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
</details>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_exponential.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="指数阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_exponential.png" /></a></p>
<p><img alt="指数阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_exponential.png" /></p>
<p align="center"> 图 2-11 &nbsp; 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中。例如在以下代码中，其递归地一分为二，经过 <span class="arithmatex">\(n\)</span> 次分裂后停止：</p>
@@ -7130,7 +7094,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20logarithmic%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20logarithmic%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
</details>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_logarithmic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="对数阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic.png" /></a></p>
<p><img alt="对数阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic.png" /></p>
<p align="center"> 图 2-12 &nbsp; 对数阶的时间复杂度 </p>
<p>与指数阶类似，对数阶也常出现于递归函数中。以下代码形成了一棵高度为 <span class="arithmatex">\(\log_2 n\)</span> 的递归树：</p>
@@ -7445,7 +7409,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20linear_log_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%BA%BF%E6%80%A7%E5%AF%B9%E6%95%B0%E9%98%B6%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%20linear_log_recur%28n%20//%202%29%20%2B%20linear_log_recur%28n%20//%202%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20linear_log_recur%28n%29%0A%20%20%20%20print%28%22%E7%BA%BF%E6%80%A7%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
</details>
<p>图 2-13 展示了线性对数阶的生成方式。二叉树的每一层的操作总数都为 <span class="arithmatex">\(n\)</span> ，树共有 <span class="arithmatex">\(\log_2 n + 1\)</span> 层，因此时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_logarithmic_linear.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="线性对数阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></a></p>
<p><img alt="线性对数阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></p>
<p align="center"> 图 2-13 &nbsp; 线性对数阶的时间复杂度 </p>
<p>主流排序算法的时间复杂度通常为 <span class="arithmatex">\(O(n \log n)\)</span> ，例如快速排序、归并排序、堆排序等。</p>
@@ -7641,7 +7605,7 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
<p><div style="height: 495px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20factorial_recur%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%98%B6%E4%B9%98%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20%E4%BB%8E%201%20%E4%B8%AA%E5%88%86%E8%A3%82%E5%87%BA%20n%20%E4%B8%AA%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20factorial_recur%28n%20-%201%29%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20factorial_recur%28n%29%0A%20%20%20%20print%28%22%E9%98%B6%E4%B9%98%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20factorial_recur%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%98%B6%E4%B9%98%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20%E4%BB%8E%201%20%E4%B8%AA%E5%88%86%E8%A3%82%E5%87%BA%20n%20%E4%B8%AA%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20factorial_recur%28n%20-%201%29%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20factorial_recur%28n%29%0A%20%20%20%20print%28%22%E9%98%B6%E4%B9%98%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
</details>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_factorial.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="阶乘阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_factorial.png" /></a></p>
<p><img alt="阶乘阶的时间复杂度" class="animation-figure" src="../time_complexity.assets/time_complexity_factorial.png" /></p>
<p align="center"> 图 2-14 &nbsp; 阶乘阶的时间复杂度 </p>
<p>请注意，因为当 <span class="arithmatex">\(n \geq 4\)</span> 时恒有 <span class="arithmatex">\(n! &gt; 2^n\)</span> ，所以阶乘阶比指数阶增长得更快，在 <span class="arithmatex">\(n\)</span> 较大时也是不可接受的。</p>
@@ -8044,7 +8008,7 @@ aria-label="页脚"
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