xmengnet/hello-algo
1
0
Fork 0
mirror of https://github.com/krahets/hello-algo.git synced 2024-12-26 14:46:28 +08:00
Code Issues Projects Releases Packages Wiki Activity Actions
hello-algo/en/chapter_greedy/max_capacity_problem/index.html
krahets 28380aa3ce deploy
2024-09-28 16:52:45 +08:00

4061 lines
No EOL
122 KiB
HTML
Raw Blame History

<!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_greedy/max_capacity_problem/">
<link rel="prev" href="../fractional_knapsack_problem/">
<link rel="next" href="../max_product_cutting_problem/">
<link rel="icon" href="../../assets/images/favicon.png">
<meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.5">
<title>15.3 Maximum capacity problem - 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%7CRoboto+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto 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>
<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="teal">
<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="#153-maximum-capacity-problem" class="md-skip">
Skip to content
</a>
</div>
<div data-md-component="announce">
<aside class="md-banner">
<div class="md-banner__inner md-grid md-typeset">
<button class="md-banner__button md-icon" aria-label="Don't show this again">
<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>
<div class="banner-svg">
<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 Copyright 2024 Fonticons, Inc.-->
<path
d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
</svg>
<span>Welcome to contribute to Chinese-to-English translation! Please visit <a href="https://github.com/krahets/hello-algo/issues/914">#914</a> for more details.</span>
</div>
</div>
<script>var content,el=document.querySelector("[data-md-component=announce]");el&&(content=el.querySelector(".md-typeset"),__md_hash(content.innerHTML)===__md_get("__announce")&&(el.hidden=!0))</script>
</aside>
</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">
15.3 Maximum capacity problem
</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="teal" aria-label="Dark mode" type="radio" name="__palette" id="__palette_0">
<label class="md-header__button md-icon" title="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="black" data-md-color-accent="teal" aria-label="Light mode" type="radio" name="__palette" id="__palette_1">
<label class="md-header__button md-icon" title="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="/chapter_greedy/max_capacity_problem/" hreflang="zh" class="md-select__link">
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/chapter_greedy/max_capacity_problem/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
<li class="md-select__item">
<a href="/en/chapter_greedy/max_capacity_problem/" 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_hello_algo/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
<span class="md-ellipsis">
Before starting
</span>
</a>
</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>
Before starting
</label>
<ul class="md-nav__list" data-md-scrollfix>
</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_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_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 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 About this 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_3" >
<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. Encounter with algorithms
</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="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter with 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 an algorithm
</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--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_computational_complexity/" 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. Complexity analysis
</span>
</a>
<label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<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>
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/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="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
<span class="md-ellipsis">
2.3 Time complexity
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
<span class="md-ellipsis">
2.4 Space complexity
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
<span class="md-ellipsis">
2.5 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_5" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_data_structure/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data structures
</span>
</a>
<label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
<span class="md-ellipsis">
3.1 Classification of data structures
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">
<span class="md-ellipsis">
3.2 Basic data types
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
<span class="md-ellipsis">
3.3 Number encoding *
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
<span class="md-ellipsis">
3.4 Character encoding *
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_data_structure/summary/" class="md-nav__link">
<span class="md-ellipsis">
3.5 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_6" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and linked list
</span>
</a>
<label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
<span class="md-ellipsis">
4.1 Array
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
<span class="md-ellipsis">
4.2 Linked list
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
<span class="md-ellipsis">
4.3 List
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/ram_and_cache/" class="md-nav__link">
<span class="md-ellipsis">
4.4 Memory and cache *
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
<span class="md-ellipsis">
4.5 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_7" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_stack_and_queue/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and queue
</span>
</a>
<label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
<span class="md-ellipsis">
5.1 Stack
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
<span class="md-ellipsis">
5.2 Queue
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
<span class="md-ellipsis">
5.3 Double-ended queue
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
<span class="md-ellipsis">
5.4 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_8" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_hashing/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash table
</span>
</a>
<label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_8">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
<span class="md-ellipsis">
6.1 Hash table
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
<span class="md-ellipsis">
6.2 Hash collision
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
<span class="md-ellipsis">
6.3 Hash algorithm
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
<span class="md-ellipsis">
6.4 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_9" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_tree/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 7. Tree
</span>
</a>
<label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_9">
<span class="md-nav__icon md-icon"></span>
Chapter 7. Tree
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
<span class="md-ellipsis">
7.1 Binary tree
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
<span class="md-ellipsis">
7.2 Binary tree traversal
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
<span class="md-ellipsis">
7.3 Array Representation of tree
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
<span class="md-ellipsis">
7.4 Binary Search tree
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
<span class="md-ellipsis">
7.5 AVL tree *
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
<span class="md-ellipsis">
7.6 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_10" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_heap/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
<span class="md-ellipsis">
Chapter 8. Heap
</span>
</a>
<label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_10">
<span class="md-nav__icon md-icon"></span>
Chapter 8. Heap
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
<span class="md-ellipsis">
8.1 Heap
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
<span class="md-ellipsis">
8.2 Building a heap
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_heap/top_k/" class="md-nav__link">
<span class="md-ellipsis">
8.3 Top-k problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
<span class="md-ellipsis">
8.4 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_11" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_graph/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
<span class="md-ellipsis">
Chapter 9. Graph
</span>
</a>
<label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_11">
<span class="md-nav__icon md-icon"></span>
Chapter 9. Graph
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
<span class="md-ellipsis">
9.1 Graph
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
<span class="md-ellipsis">
9.2 Basic graph operations
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
<span class="md-ellipsis">
9.3 Graph traversal
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_graph/summary/" class="md-nav__link">
<span class="md-ellipsis">
9.4 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_12" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_searching/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
<span class="md-ellipsis">
Chapter 10. Searching
</span>
</a>
<label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_12">
<span class="md-nav__icon md-icon"></span>
Chapter 10. Searching
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
<span class="md-ellipsis">
10.1 Binary search
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
<span class="md-ellipsis">
10.2 Binary search insertion
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
<span class="md-ellipsis">
10.3 Binary search boundaries
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
<span class="md-ellipsis">
10.4 Hashing optimization strategies
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
<span class="md-ellipsis">
10.5 Search algorithms revisited
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
<span class="md-ellipsis">
10.6 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_13" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_sorting/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
<span class="md-ellipsis">
Chapter 11. Sorting
</span>
</a>
<label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_13">
<span class="md-nav__icon md-icon"></span>
Chapter 11. Sorting
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
<span class="md-ellipsis">
11.1 Sorting algorithms
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.2 Selection sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.3 Bubble sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.4 Insertion sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.5 Quick sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.6 Merge sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.7 Heap sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.8 Bucket sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.9 Counting sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
<span class="md-ellipsis">
11.10 Radix sort
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/summary/" class="md-nav__link">
<span class="md-ellipsis">
11.11 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_14" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
<span class="md-ellipsis">
Chapter 12. Divide and conquer
</span>
</a>
<label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_14">
<span class="md-nav__icon md-icon"></span>
Chapter 12. Divide and conquer
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
<span class="md-ellipsis">
12.1 Divide and conquer algorithms
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
<span class="md-ellipsis">
12.2 Divide and conquer search strategy
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
<span class="md-ellipsis">
12.3 Building binary tree problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
<span class="md-ellipsis">
12.4 Tower of Hanoi Problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">
<span class="md-ellipsis">
12.5 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_15" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_backtracking/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
<span class="md-ellipsis">
Chapter 13. Backtracking
</span>
</a>
<label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_15">
<span class="md-nav__icon md-icon"></span>
Chapter 13. Backtracking
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
<span class="md-ellipsis">
13.1 Backtracking algorithms
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
<span class="md-ellipsis">
13.2 Permutation problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
<span class="md-ellipsis">
13.3 Subset sum problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
<span class="md-ellipsis">
13.4 n queens problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/summary/" class="md-nav__link">
<span class="md-ellipsis">
13.5 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_16" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_dynamic_programming/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M22 15h-2v3c0 1.11-.89 2-2 2h-3v2l-3-3 3-3v2h3v-3h-2l3-3 3 3m0-11v4c0 1.1-.9 2-2 2H10v10c0 1.1-.9 2-2 2H4c-1.1 0-2-.9-2-2V4c0-1.1.9-2 2-2h16c1.1 0 2 .9 2 2M4 8h4V4H4v4m0 2v4h4v-4H4m4 10v-4H4v4h4m6-12V4h-4v4h4m6-4h-4v4h4V4Z"/></svg>
<span class="md-ellipsis">
Chapter 14. Dynamic programming
</span>
</a>
<label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_16">
<span class="md-nav__icon md-icon"></span>
Chapter 14. Dynamic programming
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
<span class="md-ellipsis">
14.1 Introduction to dynamic programming
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
<span class="md-ellipsis">
14.2 Characteristics of DP problems
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">
<span class="md-ellipsis">
14.3 DP problem-solving approach¶
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_dynamic_programming/knapsack_problem/" class="md-nav__link">
<span class="md-ellipsis">
14.4 0-1 Knapsack problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
<span class="md-ellipsis">
14.5 Unbounded knapsack problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
<span class="md-ellipsis">
14.6 Edit distance problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_dynamic_programming/summary/" class="md-nav__link">
<span class="md-ellipsis">
14.7 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_17" 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="M13 3c3.88 0 7 3.14 7 7 0 2.8-1.63 5.19-4 6.31V21H9v-3H8c-1.11 0-2-.89-2-2v-3H4.5c-.42 0-.66-.5-.42-.81L6 9.66A7.003 7.003 0 0 1 13 3m0-2C8.41 1 4.61 4.42 4.06 8.9L2.5 11h-.03l-.02.03c-.55.76-.62 1.76-.19 2.59.36.69 1 1.17 1.74 1.32V16c0 1.85 1.28 3.42 3 3.87V23h11v-5.5c2.5-1.67 4-4.44 4-7.5 0-4.97-4.04-9-9-9m4 7.83c0 1.54-1.36 2.77-3.42 4.64L13 14l-.58-.53C10.36 11.6 9 10.37 9 8.83c0-1.2.96-2.19 2.16-2.2h.04c.69 0 1.35.31 1.8.83.45-.52 1.11-.83 1.8-.83 1.2-.01 2.2.96 2.2 2.16v.04Z"/></svg>
<span class="md-ellipsis">
Chapter 15. Greedy
</span>
</a>
<label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_17">
<span class="md-nav__icon md-icon"></span>
Chapter 15. Greedy
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../greedy_algorithm/" class="md-nav__link">
<span class="md-ellipsis">
15.1 Greedy algorithms
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../fractional_knapsack_problem/" class="md-nav__link">
<span class="md-ellipsis">
15.2 Fractional knapsack problem
</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">
15.3 Maximum capacity problem
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
<span class="md-ellipsis">
15.3 Maximum capacity problem
</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-determination-of-a-greedy-strategy" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Determination of a greedy strategy
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#2-implementation" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Implementation
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#3-proof-of-correctness" class="md-nav__link">
<span class="md-ellipsis">
3. &nbsp; Proof of correctness
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="../max_product_cutting_problem/" class="md-nav__link">
<span class="md-ellipsis">
15.4 Maximum product cutting problem
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
<span class="md-ellipsis">
15.5 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_18" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_appendix/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 18h2v-2h-2v2m1-16A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10A10 10 0 0 0 12 2m0 18c-4.41 0-8-3.59-8-8s3.59-8 8-8 8 3.59 8 8-3.59 8-8 8m0-14a4 4 0 0 0-4 4h2a2 2 0 0 1 2-2 2 2 0 0 1 2 2c0 2-3 1.75-3 5h2c0-2.25 3-2.5 3-5a4 4 0 0 0-4-4Z"/></svg>
<span class="md-ellipsis">
Chapter 16. Appendix
</span>
</a>
<label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_18">
<span class="md-nav__icon md-icon"></span>
Chapter 16. Appendix
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
<span class="md-ellipsis">
16.1 Installation
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
<span class="md-ellipsis">
16.2 Contributing
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/terminology/" class="md-nav__link">
<span class="md-ellipsis">
16.3 Terminology
</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_19" >
<div class="md-nav__link md-nav__container">
<a href="../../chapter_reference/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9 3v15h3V3H9m3 2 4 13 3-1-4-13-3 1M5 5v13h3V5H5M3 19v2h18v-2H3Z"/></svg>
<span class="md-ellipsis">
References
</span>
</a>
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_19">
<span class="md-nav__icon md-icon"></span>
References
</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="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-determination-of-a-greedy-strategy" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Determination of a greedy strategy
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#2-implementation" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Implementation
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#3-proof-of-correctness" class="md-nav__link">
<span class="md-ellipsis">
3. &nbsp; Proof of correctness
</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/en/docs/chapter_greedy/max_capacity_problem.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="153-maximum-capacity-problem">15.3 &nbsp; Maximum capacity problem<a class="headerlink" href="#153-maximum-capacity-problem" title="Permanent link">&para;</a></h1>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>Input an array <span class="arithmatex">\(ht\)</span>, where each element represents the height of a vertical partition. Any two partitions in the array, along with the space between them, can form a container.</p>
<p>The capacity of the container is the product of the height and the width (area), where the height is determined by the shorter partition, and the width is the difference in array indices between the two partitions.</p>
<p>Please select two partitions in the array that maximize the container's capacity and return this maximum capacity. An example is shown in Figure 15-7.</p>
</div>
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example data for the maximum capacity problem" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_example.png" /></a></p>
<p align="center"> Figure 15-7 &nbsp; Example data for the maximum capacity problem </p>
<p>The container is formed by any two partitions, <strong>therefore the state of this problem is represented by the indices of the two partitions, denoted as <span class="arithmatex">\([i, j]\)</span></strong>.</p>
<p>According to the problem statement, the capacity equals the product of height and width, where the height is determined by the shorter partition, and the width is the difference in array indices between the two partitions. The formula for capacity <span class="arithmatex">\(cap[i, j]\)</span> is:</p>
<div class="arithmatex">\[
cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
\]</div>
<p>Assuming the length of the array is <span class="arithmatex">\(n\)</span>, the number of combinations of two partitions (total number of states) is <span class="arithmatex">\(C_n^2 = \frac{n(n - 1)}{2}\)</span>. The most straightforward approach is to <strong>enumerate all possible states</strong>, resulting in a time complexity of <span class="arithmatex">\(O(n^2)\)</span>.</p>
<h3 id="1-determination-of-a-greedy-strategy">1. &nbsp; Determination of a greedy strategy<a class="headerlink" href="#1-determination-of-a-greedy-strategy" title="Permanent link">&para;</a></h3>
<p>There is a more efficient solution to this problem. As shown in Figure 15-8, we select a state <span class="arithmatex">\([i, j]\)</span> where the indices <span class="arithmatex">\(i &lt; j\)</span> and the height <span class="arithmatex">\(ht[i] &lt; ht[j]\)</span>, meaning <span class="arithmatex">\(i\)</span> is the shorter partition, and <span class="arithmatex">\(j\)</span> is the taller one.</p>
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_initial_state.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Initial state" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_initial_state.png" /></a></p>
<p align="center"> Figure 15-8 &nbsp; Initial state </p>
<p>As shown in Figure 15-9, <strong>if we move the taller partition <span class="arithmatex">\(j\)</span> closer to the shorter partition <span class="arithmatex">\(i\)</span>, the capacity will definitely decrease</strong>.</p>
<p>This is because when moving the taller partition <span class="arithmatex">\(j\)</span>, the width <span class="arithmatex">\(j-i\)</span> definitely decreases; and since the height is determined by the shorter partition, the height can only remain the same (if <span class="arithmatex">\(i\)</span> remains the shorter partition) or decrease (if the moved <span class="arithmatex">\(j\)</span> becomes the shorter partition).</p>
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_moving_long_board.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="State after moving the taller partition inward" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_moving_long_board.png" /></a></p>
<p align="center"> Figure 15-9 &nbsp; State after moving the taller partition inward </p>
<p>Conversely, <strong>we can only possibly increase the capacity by moving the shorter partition <span class="arithmatex">\(i\)</span> inward</strong>. Although the width will definitely decrease, <strong>the height may increase</strong> (if the moved shorter partition <span class="arithmatex">\(i\)</span> becomes taller). For example, in Figure 15-10, the area increases after moving the shorter partition.</p>
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_moving_short_board.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="State after moving the shorter partition inward" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_moving_short_board.png" /></a></p>
<p align="center"> Figure 15-10 &nbsp; State after moving the shorter partition inward </p>
<p>This leads us to the greedy strategy for this problem: initialize two pointers at the ends of the container, and in each round, move the pointer corresponding to the shorter partition inward until the two pointers meet.</p>
<p>Figure 15-11 illustrate the execution of the greedy strategy.</p>
<ol>
<li>Initially, the pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> are positioned at the ends of the array.</li>
<li>Calculate the current state's capacity <span class="arithmatex">\(cap[i, j]\)</span> and update the maximum capacity.</li>
<li>Compare the heights of partitions <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span>, and move the shorter partition inward by one step.</li>
<li>Repeat steps <code>2.</code> and <code>3.</code> until <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> meet.</li>
</ol>
<div class="tabbed-set tabbed-alternate" data-tabs="1:9"><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" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="The greedy process for maximum capacity problem" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step1.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="max_capacity_greedy_step2" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step2.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="max_capacity_greedy_step3" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step3.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="max_capacity_greedy_step4" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step4.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="max_capacity_greedy_step5" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step5.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="max_capacity_greedy_step6" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step6.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step7.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="max_capacity_greedy_step7" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step7.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step8.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="max_capacity_greedy_step8" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step8.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_greedy_step9.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="max_capacity_greedy_step9" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step9.png" /></a></p>
</div>
</div>
</div>
<p align="center"> Figure 15-11 &nbsp; The greedy process for maximum capacity problem </p>
<h3 id="2-implementation">2. &nbsp; Implementation<a class="headerlink" href="#2-implementation" title="Permanent link">&para;</a></h3>
<p>The code loops at most <span class="arithmatex">\(n\)</span> times, <strong>thus the time complexity is <span class="arithmatex">\(O(n)\)</span></strong>.</p>
<p>The variables <span class="arithmatex">\(i\)</span>, <span class="arithmatex">\(j\)</span>, and <span class="arithmatex">\(res\)</span> use a constant amount of extra space, <strong>thus the space complexity is <span class="arithmatex">\(O(1)\)</span></strong>.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:14"><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" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><input id="__tabbed_2_14" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label><label for="__tabbed_2_14">Zig</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span> <span class="nf">max_capacity</span><span class="p">(</span><span class="n">ht</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Maximum capacity: Greedy&quot;&quot;&quot;</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="c1"># Initialize i, j, making them split the array at both ends</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="c1"># Initial maximum capacity is 0</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="c1"># Loop for greedy selection until the two boards meet</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="c1"># Update maximum capacity</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="n">cap</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span> <span class="o">*</span> <span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="n">res</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">cap</span><span class="p">)</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="c1"># Move the shorter board inward</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="k">if</span> <span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="k">return</span> <span class="n">res</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Maximum capacity: Greedy */</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">maxCapacity</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">ht</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="c1">// Initialize i, j, making them split the array at both ends</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </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">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ht</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>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="c1">// Initial maximum capacity is 0</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="c1">// Update maximum capacity</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">ht</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="p">(</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>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">);</span>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="c1">// Move the shorter board inward</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">;</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Maximum capacity: Greedy */</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">maxCapacity</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">ht</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w"> </span><span class="c1">// Initialize i, j, making them split the array at both ends</span>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w"> </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">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ht</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>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w"> </span><span class="c1">// Initial maximum capacity is 0</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w"> </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w"> </span><span class="c1">// Update maximum capacity</span>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">ht</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</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>
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">);</span>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w"> </span><span class="c1">// Move the shorter board inward</span>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ht</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">;</span>
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">max_capacity</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">MaxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">maxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">maxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">maxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">maxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">maxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">max_capacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">maxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">maxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">max_capacity</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">max_capacity.zig</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">maxCapacity</span><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<h3 id="3-proof-of-correctness">3. &nbsp; Proof of correctness<a class="headerlink" href="#3-proof-of-correctness" title="Permanent link">&para;</a></h3>
<p>The reason why the greedy method is faster than enumeration is that each round of greedy selection "skips" some states.</p>
<p>For example, under the state <span class="arithmatex">\(cap[i, j]\)</span> where <span class="arithmatex">\(i\)</span> is the shorter partition and <span class="arithmatex">\(j\)</span> is the taller partition, greedily moving the shorter partition <span class="arithmatex">\(i\)</span> inward by one step leads to the "skipped" states shown in Figure 15-12. <strong>This means that these states' capacities cannot be verified later</strong>.</p>
<div class="arithmatex">\[
cap[i, i+1], cap[i, i+2], \dots, cap[i, j-2], cap[i, j-1]
\]</div>
<p><a class="glightbox" href="../max_capacity_problem.assets/max_capacity_skipped_states.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="States skipped by moving the shorter partition" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_skipped_states.png" /></a></p>
<p align="center"> Figure 15-12 &nbsp; States skipped by moving the shorter partition </p>
<p>It is observed that <strong>these skipped states are actually all states where the taller partition <span class="arithmatex">\(j\)</span> is moved inward</strong>. We have already proven that moving the taller partition inward will definitely decrease the capacity. Therefore, the skipped states cannot possibly be the optimal solution, <strong>and skipping them does not lead to missing the optimal solution</strong>.</p>
<p>The analysis shows that the operation of moving the shorter partition is "safe", and the greedy strategy is effective.</p>
<!-- 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="../fractional_knapsack_problem/"
class="md-footer__link md-footer__link--prev"
aria-label="Previous: 15.2 Fractional knapsack problem"
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">
15.2 Fractional knapsack problem
</div>
</div>
</a>
<!-- Link to next page -->
<a
href="../max_product_cutting_problem/"
class="md-footer__link md-footer__link--next"
aria-label="Next: 15.4 Maximum product cutting problem"
rel="next"
>
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
15.4 Maximum product cutting problem
</div>
</div>
<div class="md-footer__button md-icon">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M4 11v2h12l-5.5 5.5 1.42 1.42L19.84 12l-7.92-7.92L10.5 5.5 16 11H4Z"/></svg>
</div>
</a>
</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="light"
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="../fractional_knapsack_problem/" class="md-footer__link md-footer__link--prev" aria-label="Previous: 15.2 Fractional knapsack problem">
<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">
15.2 Fractional knapsack problem
</div>
</div>
</a>
<a href="../max_product_cutting_problem/" class="md-footer__link md-footer__link--next" aria-label="Next: 15.4 Maximum product cutting problem">
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
15.4 Maximum product cutting problem
</div>
</div>
<div class="md-footer__button md-icon">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M4 11v2h12l-5.5 5.5 1.42 1.42L19.84 12l-7.92-7.92L10.5 5.5 16 11H4Z"/></svg>
</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 &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
</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="M389.2 48h70.6L305.6 224.2 487 464H345L233.7 318.6 106.5 464H35.8l164.9-188.5L26.8 48h145.6l100.5 132.9L389.2 48zm-24.8 373.8h39.1L151.1 88h-42l255.3 333.8z"/></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": ["announce.dismiss", "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.b8dbb3d2.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.c18c5fb9.min.js"></script>
<script src="../../javascripts/mathjax.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.2/es5/tex-mml-chtml.min.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>
Reference in a new issue View git blame Copy permalink