mirror of
https://github.com/krahets/hello-algo.git
synced 2024-12-27 16:06:29 +08:00
1204 lines
No EOL
44 KiB
HTML
1204 lines
No EOL
44 KiB
HTML
|
|
<!doctype html>
|
|
<html lang="en" class="no-js">
|
|
<head>
|
|
|
|
<meta charset="utf-8">
|
|
<meta name="viewport" content="width=device-width,initial-scale=1">
|
|
|
|
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
|
|
|
|
|
|
<meta name="author" content="Krahets">
|
|
|
|
|
|
<link rel="canonical" href="https://www.hello-algo.com/en/chapter_computational_complexity/summary/">
|
|
|
|
|
|
<link rel="prev" href="../space_complexity/">
|
|
|
|
|
|
|
|
<link rel="icon" href="../../assets/images/favicon.png">
|
|
<meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.2">
|
|
|
|
|
|
|
|
<title>2.5 Summary - Hello Algo</title>
|
|
|
|
|
|
|
|
<link rel="stylesheet" href="../../assets/stylesheets/main.50c56a3b.min.css">
|
|
|
|
|
|
<link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
|
|
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CFira+Code:400,400i,700,700i&display=fallback">
|
|
<style>:root{--md-text-font:"Roboto";--md-code-font:"Fira Code"}</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>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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; }
|
|
.gslide-image img { background: white; }
|
|
|
|
.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>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<body dir="ltr" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo">
|
|
|
|
|
|
<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>
|
|
<div data-md-component="skip">
|
|
|
|
|
|
<a href="#25-summary" class="md-skip">
|
|
Skip to content
|
|
</a>
|
|
|
|
</div>
|
|
<div data-md-component="announce">
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<header class="md-header md-header--shadow" data-md-component="header">
|
|
<nav class="md-header__inner md-grid" aria-label="Header">
|
|
<a href="../.." title="Hello Algo" class="md-header__button md-logo" aria-label="Hello Algo" data-md-component="logo">
|
|
|
|
<img src="../../assets/images/logo.svg" alt="logo">
|
|
|
|
</a>
|
|
<label class="md-header__button md-icon" for="__drawer">
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3V6m0 5h18v2H3v-2m0 5h18v2H3v-2Z"/></svg>
|
|
</label>
|
|
<div class="md-header__title" data-md-component="header-title">
|
|
<div class="md-header__ellipsis">
|
|
<div class="md-header__topic">
|
|
<span class="md-ellipsis">
|
|
Hello Algo
|
|
</span>
|
|
</div>
|
|
<div class="md-header__topic" data-md-component="header-topic">
|
|
<span class="md-ellipsis">
|
|
|
|
2.5 Summary
|
|
|
|
</span>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<form class="md-header__option" data-md-component="palette">
|
|
|
|
|
|
|
|
|
|
<input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo" aria-label="Switch to dark mode" type="radio" name="__palette" id="__palette_0">
|
|
|
|
<label class="md-header__button md-icon" title="Switch to dark mode" 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 19.07 19.07 3.5m-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 1.13.51 1.67m-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 1.36.03-1.05.87.4 1.31M19 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 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo" aria-label="Switch to light mode" type="radio" name="__palette" id="__palette_1">
|
|
|
|
<label class="md-header__button md-icon" title="Switch to light mode" 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 19.07 19.07 3.5m-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 1.13.51 1.67m-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 1.36.03-1.05.87.4 1.31M19 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 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
|
|
</label>
|
|
|
|
|
|
</form>
|
|
|
|
|
|
|
|
<script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
|
|
|
|
|
|
<div class="md-header__option">
|
|
<div class="md-select">
|
|
|
|
<button class="md-header__button md-icon" aria-label="Select language">
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12.87 15.07-2.54-2.51.03-.03A17.52 17.52 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 3.11.76-2.04M18.5 10h-2L12 22h2l1.12-3h4.75L21 22h2l-4.5-12m-2.62 7 1.62-4.33L19.12 17h-3.24Z"/></svg>
|
|
</button>
|
|
<div class="md-select__inner">
|
|
<ul class="md-select__list">
|
|
|
|
<li class="md-select__item">
|
|
<a href="/" hreflang="zh" class="md-select__link">
|
|
中文
|
|
</a>
|
|
</li>
|
|
|
|
<li class="md-select__item">
|
|
<a href="/en/" hreflang="en" class="md-select__link">
|
|
English
|
|
</a>
|
|
</li>
|
|
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<label class="md-header__button 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-.27A6.516 6.516 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 5Z"/></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="Search" placeholder="Search" 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-.27A6.516 6.516 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 5Z"/></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 11h12Z"/></svg>
|
|
</label>
|
|
<nav class="md-search__options" aria-label="Search">
|
|
|
|
<a href="javascript:void(0)" class="md-search__icon md-icon" title="Share" aria-label="Share" 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-.7 0-.24-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91 1.61 0 2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08Z"/></svg>
|
|
</a>
|
|
|
|
<button type="reset" class="md-search__icon md-icon" title="Clear" aria-label="Clear" 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 12 19 6.41Z"/></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" data-md-scrollfix>
|
|
<div class="md-search-result" data-md-component="search-result">
|
|
<div class="md-search-result__meta">
|
|
Initializing search
|
|
</div>
|
|
<ol class="md-search-result__list" role="presentation"></ol>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div class="md-header__source">
|
|
<a href="https://github.com/krahets/hello-algo" title="Go to repository" class="md-source" data-md-component="source">
|
|
<div class="md-source__icon md-icon">
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.5.1 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 2023 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>
|
|
</div>
|
|
<div class="md-source__repository">
|
|
krahets/hello-algo
|
|
</div>
|
|
</a>
|
|
</div>
|
|
|
|
</nav>
|
|
|
|
</header>
|
|
|
|
<div class="md-container" data-md-component="container">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<main class="md-main" data-md-component="main">
|
|
<div class="md-main__inner md-grid">
|
|
|
|
|
|
|
|
<div class="md-sidebar md-sidebar--primary" data-md-component="sidebar" data-md-type="navigation" >
|
|
<div class="md-sidebar__scrollwrap">
|
|
<div class="md-sidebar__inner">
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav md-nav--primary" aria-label="Navigation" data-md-level="0">
|
|
<label class="md-nav__title" for="__drawer">
|
|
<a href="../.." title="Hello Algo" class="md-nav__button md-logo" aria-label="Hello Algo" data-md-component="logo">
|
|
|
|
<img src="../../assets/images/logo.svg" alt="logo">
|
|
|
|
</a>
|
|
Hello Algo
|
|
</label>
|
|
|
|
<div class="md-nav__source">
|
|
<a href="https://github.com/krahets/hello-algo" title="Go to repository" class="md-source" data-md-component="source">
|
|
<div class="md-source__icon md-icon">
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.5.1 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 2023 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>
|
|
</div>
|
|
<div class="md-source__repository">
|
|
krahets/hello-algo
|
|
</div>
|
|
</a>
|
|
</div>
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__container">
|
|
<a href="../../chapter_preface/" class="md-nav__link ">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 4H3a2 2 0 0 0-2 2v13a2 2 0 0 0 2 2h18a2 2 0 0 0 2-2V6a2 2 0 0 0-2-2M3 19V6h8v13H3m18 0h-8V6h8v13m-7-9.5h6V11h-6V9.5m0 2.5h6v1.5h-6V12m0 2.5h6V16h-6v-1.5Z"/></svg>
|
|
|
|
<span class="md-ellipsis">
|
|
Chapter 0. Preface
|
|
</span>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
<label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
</label>
|
|
|
|
</div>
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
|
|
<label class="md-nav__title" for="__nav_1">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
Chapter 0. Preface
|
|
</label>
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
0.1 The Book
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_preface/suggestions/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
0.2 How to Read
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_preface/summary/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
0.3 Summary
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
</nav>
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__container">
|
|
<a href="../../chapter_introduction/" class="md-nav__link ">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
|
|
|
|
<span class="md-ellipsis">
|
|
Chapter 1. Introduction to Algorithms
|
|
</span>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
<label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
</label>
|
|
|
|
</div>
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
|
|
<label class="md-nav__title" for="__nav_2">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
Chapter 1. Introduction to Algorithms
|
|
</label>
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
1.1 Algorithms Are Everywhere
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
1.2 What is Algorithms
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_introduction/summary/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
1.3 Summary
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
</nav>
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--active md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" checked>
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__container">
|
|
<a href="../" class="md-nav__link ">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
|
|
|
|
<span class="md-ellipsis">
|
|
Chapter 2. Computational Complexity
|
|
</span>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
<label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
</label>
|
|
|
|
</div>
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
|
|
<label class="md-nav__title" for="__nav_3">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
Chapter 2. Computational Complexity
|
|
</label>
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../performance_evaluation/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
2.1 Evaluating Algorithm Efficiency
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../iteration_and_recursion/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
2.2 Iteration and Recursion
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../time_complexity/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
2.3 Time Complexity
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../space_complexity/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
2.4 Space Complexity
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--active">
|
|
|
|
<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-nav__link md-nav__link--active" for="__toc">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
2.5 Summary
|
|
</span>
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
</label>
|
|
|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
2.5 Summary
|
|
</span>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-nav__title" for="__toc">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
Table of contents
|
|
</label>
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
|
|
|
<li class="md-nav__item">
|
|
<a href="#1-highlights" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
1. Highlights
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
<li class="md-nav__item">
|
|
<a href="#2-q-a" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
2. Q & A
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
|
|
</nav>
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
</nav>
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
|
|
|
|
|
|
<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
References
|
|
</span>
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
</label>
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
|
|
<label class="md-nav__title" for="__nav_4">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
References
|
|
</label>
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_reference/index.md" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
None
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
</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="Table of contents">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-nav__title" for="__toc">
|
|
<span class="md-nav__icon md-icon"></span>
|
|
Table of contents
|
|
</label>
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
|
|
|
<li class="md-nav__item">
|
|
<a href="#1-highlights" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
1. Highlights
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
<li class="md-nav__item">
|
|
<a href="#2-q-a" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
2. Q & A
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
|
|
</nav>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
|
|
<div class="md-content" data-md-component="content">
|
|
<article class="md-content__inner md-typeset">
|
|
|
|
<!-- Tags -->
|
|
|
|
|
|
<!-- Actions -->
|
|
<!-- Actions -->
|
|
|
|
|
|
<!-- Edit button -->
|
|
|
|
<a
|
|
href="https://github.com/krahets/hello-algo/tree/main/docs-en/chapter_computational_complexity/summary.md"
|
|
title="Edit this page"
|
|
class="md-content__button md-icon"
|
|
>
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.5.1 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 2023 Fonticons, Inc.--><path d="M441 58.9 453.1 71c9.4 9.4 9.4 24.6 0 33.9L424 134.1 377.9 88 407 58.9c9.4-9.4 24.6-9.4 33.9 0zM209.8 256.2 344 121.9l46.1 46.1-134.3 134.2c-2.9 2.9-6.5 5-10.4 6.1L186.9 325l16.7-58.5c1.1-3.9 3.2-7.5 6.1-10.4zM373.1 25 175.8 222.2c-8.7 8.7-15 19.4-18.3 31.1l-28.6 100c-2.4 8.4-.1 17.4 6.1 23.6s15.2 8.5 23.6 6.1l100-28.6c11.8-3.4 22.5-9.7 31.1-18.3L487 138.9c28.1-28.1 28.1-73.7 0-101.8L474.9 25c-28.1-28.1-73.7-28.1-101.8 0zM88 64c-48.6 0-88 39.4-88 88v272c0 48.6 39.4 88 88 88h272c48.6 0 88-39.4 88-88V312c0-13.3-10.7-24-24-24s-24 10.7-24 24v112c0 22.1-17.9 40-40 40H88c-22.1 0-40-17.9-40-40V152c0-22.1 17.9-40 40-40h112c13.3 0 24-10.7 24-24s-10.7-24-24-24H88z"/></svg>
|
|
</a>
|
|
|
|
|
|
<!-- View button -->
|
|
|
|
|
|
|
|
<!-- Page content -->
|
|
<h1 id="25-summary">2.5 Summary<a class="headerlink" href="#25-summary" title="Permanent link">¶</a></h1>
|
|
<h3 id="1-highlights">1. Highlights<a class="headerlink" href="#1-highlights" title="Permanent link">¶</a></h3>
|
|
<p><strong>Evaluation of Algorithm Efficiency</strong></p>
|
|
<ul>
|
|
<li>Time and space efficiency are the two leading evaluation indicators to measure an algorithm.</li>
|
|
<li>We can evaluate the efficiency of an algorithm through real-world testing. Still, it isn't easy to eliminate the side effects from the testing environment, and it consumes a lot of computational resources.</li>
|
|
<li>Complexity analysis overcomes the drawbacks of real-world testing. The analysis results can apply to all operating platforms and reveal the algorithm's efficiency under variant data scales.</li>
|
|
</ul>
|
|
<p><strong>Time Complexity</strong></p>
|
|
<ul>
|
|
<li>Time complexity is used to measure the trend of algorithm running time as the data size grows., which can effectively evaluate the algorithm's efficiency. However, it may fail in some cases, such as when the input volume is small or the time complexities are similar, making it difficult to precisely compare the efficiency of algorithms.</li>
|
|
<li>The worst time complexity is denoted by big <span class="arithmatex">\(O\)</span> notation, which corresponds to the asymptotic upper bound of the function, reflecting the growth rate in the number of operations <span class="arithmatex">\(T(n)\)</span> as <span class="arithmatex">\(n\)</span> tends to positive infinity.</li>
|
|
<li>The estimation of time complexity involves two steps: first, counting the number of operations, and then determining the asymptotic upper bound.</li>
|
|
<li>Common time complexities, from lowest to highest, are <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n \log n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, <span class="arithmatex">\(O(2^n)\)</span>, and <span class="arithmatex">\(O(n!)\)</span>.</li>
|
|
<li>The time complexity of certain algorithms is not fixed and depends on the distribution of the input data. The time complexity can be categorized into worst-case, best-case, and average. The best-case time complexity is rarely used because the input data must meet strict conditions to achieve the best-case.</li>
|
|
<li>The average time complexity reflects the efficiency of an algorithm with random data inputs, which is closest to the performance of algorithms in real-world scenarios. Calculating the average time complexity requires statistical analysis of input data and a synthesized mathematical expectation.</li>
|
|
</ul>
|
|
<p><strong>Space Complexity</strong></p>
|
|
<ul>
|
|
<li>Space complexity serves a similar purpose to time complexity and is used to measure the trend of space occupied by an algorithm as the data volume increases.</li>
|
|
<li>The memory space associated with the operation of an algorithm can be categorized into input space, temporary space, and output space. Normally, the input space is not considered when determining space complexity. The temporary space can be classified into instruction space, data space, and stack frame space, and the stack frame space usually only affects the space complexity for recursion functions.</li>
|
|
<li>We mainly focus on the worst-case space complexity, which refers to the measurement of an algorithm's space usage when given the worst-case input data and during the worst-case execution scenario.</li>
|
|
<li>Common space complexities are <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span> and <span class="arithmatex">\(O(2^n)\)</span> from lowest to highest.</li>
|
|
</ul>
|
|
<h3 id="2-q-a">2. Q & A<a class="headerlink" href="#2-q-a" title="Permanent link">¶</a></h3>
|
|
<div class="admonition question">
|
|
<p class="admonition-title">Is the space complexity of tail recursion <span class="arithmatex">\(O(1)\)</span>?</p>
|
|
<p>Theoretically, the space complexity of a tail recursion function can be optimized to <span class="arithmatex">\(O(1)\)</span>. However, most programming languages (e.g., Java, Python, C++, Go, C#, etc.) do not support auto-optimization for tail recursion, so the space complexity is usually considered as <span class="arithmatex">\(O(n)\)</span>.</p>
|
|
</div>
|
|
<div class="admonition question">
|
|
<p class="admonition-title">What is the difference between the terms function and method?</p>
|
|
<p>A <em>function</em> can be executed independently, and all arguments are passed explicitly. A <em>method</em> is associated with an object and is implicitly passed to the object that calls it, allowing it to operate on the data contained within an instance of a class.</p>
|
|
<p>Let's illustrate with a few common programming languages.</p>
|
|
<ul>
|
|
<li>C is a procedural programming language without object-oriented concepts, so it has only functions. However, we can simulate object-oriented programming by creating structures (struct), and the functions associated with structures are equivalent to methods in other languages.</li>
|
|
<li>Java and C# are object-oriented programming languages, and blocks of code (methods) are typically part of a class. Static methods behave like a function because it is bound to the class and cannot access specific instance variables.</li>
|
|
<li>Both C++ and Python support both procedural programming (functions) and object-oriented programming (methods).</li>
|
|
</ul>
|
|
</div>
|
|
<div class="admonition question">
|
|
<p class="admonition-title">Does the figure "Common Types of Space Complexity" reflect the absolute size of the occupied space?</p>
|
|
<p>No, that figure shows the space complexity, which reflects the growth trend, not the absolute size of the space occupied.</p>
|
|
<p>For example, if you take <span class="arithmatex">\(n = 8\)</span> , the values of each curve do not align with the function because each curve contains a constant term used to compress the range of values to a visually comfortable range.</p>
|
|
<p>In practice, since we usually don't know each method's "constant term" complexity, it is generally impossible to choose the optimal solution for <span class="arithmatex">\(n = 8\)</span> based on complexity alone. But it's easier to choose for <span class="arithmatex">\(n = 8^5\)</span> as the growth trend is already dominant.</p>
|
|
</div>
|
|
|
|
<!-- Source file information -->
|
|
|
|
|
|
<!-- Was this page helpful? -->
|
|
|
|
|
|
|
|
|
|
<!-- Previous and next pages link -->
|
|
<nav
|
|
class="md-footer__inner md-grid"
|
|
aria-label="Footer"
|
|
|
|
>
|
|
|
|
<!-- Link to previous page -->
|
|
|
|
|
|
<a
|
|
href="../space_complexity/"
|
|
class="md-footer__link md-footer__link--prev"
|
|
aria-label="Previous: 2.4 Space Complexity"
|
|
rel="prev"
|
|
>
|
|
<div class="md-footer__button md-icon">
|
|
|
|
<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 11h12Z"/></svg>
|
|
</div>
|
|
<div class="md-footer__title">
|
|
<span class="md-footer__direction">
|
|
Previous
|
|
</span>
|
|
<div class="md-ellipsis">
|
|
2.4 Space Complexity
|
|
</div>
|
|
</div>
|
|
</a>
|
|
|
|
|
|
<!-- Link to next page -->
|
|
|
|
</nav>
|
|
|
|
<!-- Comment system -->
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
|
|
|
|
<!-- Insert generated snippet here -->
|
|
<script
|
|
src="https://giscus.app/client.js"
|
|
data-repo="krahets/hello-algo"
|
|
data-repo-id="R_kgDOIXtSqw"
|
|
data-category="Announcements"
|
|
data-category-id="DIC_kwDOIXtSq84CSZk_"
|
|
data-mapping="pathname"
|
|
data-strict="1"
|
|
data-reactions-enabled="1"
|
|
data-emit-metadata="0"
|
|
data-input-position="top"
|
|
data-theme="preferred_color_scheme"
|
|
data-lang="en"
|
|
crossorigin="anonymous"
|
|
async
|
|
>
|
|
</script>
|
|
<!-- Synchronize Giscus theme with palette -->
|
|
<script>
|
|
var giscus = document.querySelector("script[src*=giscus]")
|
|
|
|
/* Set palette on initial load */
|
|
var palette = __md_get("__palette")
|
|
if (palette && typeof palette.color === "object") {
|
|
var theme = palette.color.scheme === "slate" ? "dark_dimmed" : "light"
|
|
giscus.setAttribute("data-theme", theme)
|
|
}
|
|
|
|
/* Register event handlers after documented loaded */
|
|
document.addEventListener("DOMContentLoaded", function() {
|
|
var ref = document.querySelector("[data-md-component=palette]")
|
|
ref.addEventListener("change", function() {
|
|
var palette = __md_get("__palette")
|
|
if (palette && typeof palette.color === "object") {
|
|
var theme = palette.color.scheme === "slate" ? "dark_dimmed" : "light"
|
|
|
|
/* Instruct Giscus to change theme */
|
|
var frame = document.querySelector(".giscus-frame")
|
|
frame.contentWindow.postMessage(
|
|
{ giscus: { setConfig: { theme } } },
|
|
"https://giscus.app"
|
|
)
|
|
}
|
|
})
|
|
})
|
|
</script>
|
|
|
|
|
|
</article>
|
|
</div>
|
|
|
|
|
|
<script>var tabs=__md_get("__tabs");if(Array.isArray(tabs))e:for(var set of document.querySelectorAll(".tabbed-set")){var tab,labels=set.querySelector(".tabbed-labels");for(tab of tabs)for(var label of labels.getElementsByTagName("label"))if(label.innerText.trim()===tab){var input=document.getElementById(label.htmlFor);input.checked=!0;continue e}}</script>
|
|
|
|
<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
|
|
</div>
|
|
|
|
<button type="button" class="md-top md-icon" data-md-component="top" hidden>
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
|
|
Back to top
|
|
</button>
|
|
|
|
</main>
|
|
|
|
<footer class="md-footer">
|
|
|
|
|
|
|
|
<nav class="md-footer__inner md-grid" aria-label="Footer" >
|
|
|
|
|
|
<a href="../space_complexity/" class="md-footer__link md-footer__link--prev" aria-label="Previous: 2.4 Space Complexity">
|
|
<div class="md-footer__button md-icon">
|
|
|
|
<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 11h12Z"/></svg>
|
|
</div>
|
|
<div class="md-footer__title">
|
|
<span class="md-footer__direction">
|
|
Previous
|
|
</span>
|
|
<div class="md-ellipsis">
|
|
2.4 Space Complexity
|
|
</div>
|
|
</div>
|
|
</a>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
<div class="md-footer-meta md-typeset">
|
|
<div class="md-footer-meta__inner md-grid">
|
|
<div class="md-copyright">
|
|
|
|
<div class="md-copyright__highlight">
|
|
Copyright © 2022 - 2023 Krahets
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
<div class="md-social">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="https://github.com/krahets" target="_blank" rel="noopener" title="github.com" class="md-social__link">
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.5.1 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 2023 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>
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="https://twitter.com/krahets" target="_blank" rel="noopener" title="twitter.com" class="md-social__link">
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.5.1 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 2023 Fonticons, Inc.--><path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"/></svg>
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="https://leetcode.cn/u/jyd/" target="_blank" rel="noopener" title="leetcode.cn" class="md-social__link">
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><!--! Font Awesome Free 6.5.1 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 2023 Fonticons, Inc.--><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg>
|
|
</a>
|
|
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
</footer>
|
|
|
|
</div>
|
|
<div class="md-dialog" data-md-component="dialog">
|
|
<div class="md-dialog__inner md-typeset"></div>
|
|
</div>
|
|
|
|
|
|
<script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "Copied to clipboard", "clipboard.copy": "Copy to clipboard", "search.result.more.one": "1 more on this page", "search.result.more.other": "# more on this page", "search.result.none": "No matching documents", "search.result.one": "1 matching document", "search.result.other": "# matching documents", "search.result.placeholder": "Type to start searching", "search.result.term.missing": "Missing", "select.version": "Select version"}}</script>
|
|
|
|
|
|
<script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
|
|
|
|
<script src="../../javascripts/mathjax.js"></script>
|
|
|
|
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
|
|
|
|
<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
|
|
|
|
|
<script>document$.subscribe(() => {const lightbox = GLightbox({"touchNavigation": true, "loop": false, "zoomable": true, "draggable": false, "openEffect": "zoom", "closeEffect": "zoom", "slideEffect": "none"});})</script></body>
|
|
</html> |