mirror of
https://github.com/krahets/hello-algo.git
synced 2024-12-26 15:36:29 +08:00
4208 lines
No EOL
164 KiB
HTML
4208 lines
No EOL
164 KiB
HTML
|
|
<!doctype html>
|
|
<html lang="en" class="no-js">
|
|
<head>
|
|
|
|
<meta charset="utf-8">
|
|
<meta name="viewport" content="width=device-width,initial-scale=1">
|
|
|
|
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
|
|
|
|
|
|
<meta name="author" content="krahets">
|
|
|
|
|
|
<link rel="canonical" href="https://www.hello-algo.com/en/chapter_tree/array_representation_of_tree/">
|
|
|
|
|
|
<link rel="prev" href="../binary_tree_traversal/">
|
|
|
|
|
|
<link rel="next" href="../binary_search_tree/">
|
|
|
|
|
|
<link rel="icon" href="../../assets/images/favicon.png">
|
|
<meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.5">
|
|
|
|
|
|
|
|
<title>7.3 Array Representation of tree - 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="#73-array-representation-of-binary-trees" 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">
|
|
|
|
7.3 Array Representation of tree
|
|
|
|
</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_tree/array_representation_of_tree/" hreflang="zh" class="md-select__link">
|
|
简体中文
|
|
</a>
|
|
</li>
|
|
|
|
<li class="md-select__item">
|
|
<a href="/zh-hant/chapter_tree/array_representation_of_tree/" hreflang="zh-Hant" class="md-select__link">
|
|
繁體中文
|
|
</a>
|
|
</li>
|
|
|
|
<li class="md-select__item">
|
|
<a href="/en/chapter_tree/array_representation_of_tree/" 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--active md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" 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="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="true">
|
|
<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="../binary_tree/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
7.1 Binary tree
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../binary_tree_traversal/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
7.2 Binary tree traversal
|
|
</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">
|
|
7.3 Array Representation of tree
|
|
</span>
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
</label>
|
|
|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
7.3 Array Representation of tree
|
|
</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="#731-representing-perfect-binary-trees" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
7.3.1 Representing perfect binary trees
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
<li class="md-nav__item">
|
|
<a href="#732-representing-any-binary-tree" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
7.3.2 Representing any binary tree
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
<li class="md-nav__item">
|
|
<a href="#733-advantages-and-limitations" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
7.3.3 Advantages and limitations
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
|
|
</nav>
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../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="../avl_tree/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
7.5 AVL tree *
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../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--nested">
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__container">
|
|
<a href="../../chapter_greedy/" 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="false">
|
|
<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="../../chapter_greedy/greedy_algorithm/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
15.1 Greedy algorithms
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_greedy/fractional_knapsack_problem/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
15.2 Fractional knapsack problem
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_greedy/max_capacity_problem/" class="md-nav__link">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
15.3 Maximum capacity problem
|
|
</span>
|
|
|
|
|
|
</a>
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
<a href="../../chapter_greedy/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="../../chapter_greedy/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="#731-representing-perfect-binary-trees" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
7.3.1 Representing perfect binary trees
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
<li class="md-nav__item">
|
|
<a href="#732-representing-any-binary-tree" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
7.3.2 Representing any binary tree
|
|
</span>
|
|
</a>
|
|
|
|
</li>
|
|
|
|
<li class="md-nav__item">
|
|
<a href="#733-advantages-and-limitations" class="md-nav__link">
|
|
<span class="md-ellipsis">
|
|
7.3.3 Advantages and limitations
|
|
</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_tree/array_representation_of_tree.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="73-array-representation-of-binary-trees">7.3 Array representation of binary trees<a class="headerlink" href="#73-array-representation-of-binary-trees" title="Permanent link">¶</a></h1>
|
|
<p>Under the linked list representation, the storage unit of a binary tree is a node <code>TreeNode</code>, with nodes connected by pointers. The basic operations of binary trees under the linked list representation were introduced in the previous section.</p>
|
|
<p>So, can we use an array to represent a binary tree? The answer is yes.</p>
|
|
<h2 id="731-representing-perfect-binary-trees">7.3.1 Representing perfect binary trees<a class="headerlink" href="#731-representing-perfect-binary-trees" title="Permanent link">¶</a></h2>
|
|
<p>Let's analyze a simple case first. Given a perfect binary tree, we store all nodes in an array according to the order of level-order traversal, where each node corresponds to a unique array index.</p>
|
|
<p>Based on the characteristics of level-order traversal, we can deduce a "mapping formula" between the index of a parent node and its children: <strong>If a node's index is <span class="arithmatex">\(i\)</span>, then the index of its left child is <span class="arithmatex">\(2i + 1\)</span> and the right child is <span class="arithmatex">\(2i + 2\)</span></strong>. Figure 7-12 shows the mapping relationship between the indices of various nodes.</p>
|
|
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array representation of a perfect binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_binary_tree.png" /></a></p>
|
|
<p align="center"> Figure 7-12 Array representation of a perfect binary tree </p>
|
|
|
|
<p><strong>The mapping formula plays a role similar to the node references (pointers) in linked lists</strong>. Given any node in the array, we can access its left (right) child node using the mapping formula.</p>
|
|
<h2 id="732-representing-any-binary-tree">7.3.2 Representing any binary tree<a class="headerlink" href="#732-representing-any-binary-tree" title="Permanent link">¶</a></h2>
|
|
<p>Perfect binary trees are a special case; there are often many <code>None</code> values in the middle levels of a binary tree. Since the sequence of level-order traversal does not include these <code>None</code> values, we cannot solely rely on this sequence to deduce the number and distribution of <code>None</code> values. <strong>This means that multiple binary tree structures can match the same level-order traversal sequence</strong>.</p>
|
|
<p>As shown in Figure 7-13, given a non-perfect binary tree, the above method of array representation fails.</p>
|
|
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_without_empty.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Level-order traversal sequence corresponds to multiple binary tree possibilities" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_without_empty.png" /></a></p>
|
|
<p align="center"> Figure 7-13 Level-order traversal sequence corresponds to multiple binary tree possibilities </p>
|
|
|
|
<p>To solve this problem, <strong>we can consider explicitly writing out all <code>None</code> values in the level-order traversal sequence</strong>. As shown in Figure 7-14, after this treatment, the level-order traversal sequence can uniquely represent a binary tree. Example code is as follows:</p>
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="1:14"><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" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><input id="__tabbed_1_14" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label><label for="__tabbed_1_14">Zig</label></div>
|
|
<div class="tabbed-content">
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Array representation of a binary tree</span>
|
|
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># Using None to represent empty slots</span>
|
|
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">tree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">15</span><span class="p">]</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Using the maximum integer value INT_MAX to mark empty slots</span>
|
|
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="p">};</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Using the Integer wrapper class allows for using null to mark empty slots</span>
|
|
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="n">Integer</span><span class="o">[]</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="w"> </span><span class="p">};</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Using nullable int (int?) allows for using null to mark empty slots</span>
|
|
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kt">int?</span><span class="p">[]</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">,</span><span class="w"> </span><span class="m">7</span><span class="p">,</span><span class="w"> </span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="m">9</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="m">12</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="m">15</span><span class="p">];</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Using an any type slice, allowing for nil to mark empty slots</span>
|
|
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="nx">tree</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="p">}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Using optional Int (Int?) allows for using nil to mark empty slots</span>
|
|
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">let</span> <span class="nv">tree</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">?]</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="mi">15</span><span class="p">]</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// Using null to represent empty slots</span>
|
|
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">,</span><span class="w"> </span><span class="mf">7</span><span class="p">,</span><span class="w"> </span><span class="mf">8</span><span class="p">,</span><span class="w"> </span><span class="mf">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">15</span><span class="p">];</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// Using null to represent empty slots</span>
|
|
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">tree</span><span class="o">:</span><span class="w"> </span><span class="p">(</span><span class="kt">number</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">,</span><span class="w"> </span><span class="mf">7</span><span class="p">,</span><span class="w"> </span><span class="mf">8</span><span class="p">,</span><span class="w"> </span><span class="mf">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">15</span><span class="p">];</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// Using nullable int (int?) allows for using null to mark empty slots</span>
|
|
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">?></span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">,</span><span class="w"> </span><span class="m">7</span><span class="p">,</span><span class="w"> </span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="m">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">15</span><span class="p">];</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Using None to mark empty slots</span>
|
|
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="kd">let</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nb">Some</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">6</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">7</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">8</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">9</span><span class="p">),</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">12</span><span class="p">),</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">15</span><span class="p">)];</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="c1">// Using the maximum int value to mark empty slots, therefore, node values must not be INT_MAX</span>
|
|
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="kt">int</span><span class="w"> </span><span class="n">tree</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="p">};</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Array representation of a binary tree */</span>
|
|
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Using null to represent empty slots</span>
|
|
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="kd">val</span><span class="w"> </span><span class="nv">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="p">(</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">,</span><span class="w"> </span><span class="m">7</span><span class="p">,</span><span class="w"> </span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="m">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">15</span><span class="w"> </span><span class="p">)</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>
|
|
</code></pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_with_empty.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array representation of any type of binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_with_empty.png" /></a></p>
|
|
<p align="center"> Figure 7-14 Array representation of any type of binary tree </p>
|
|
|
|
<p>It's worth noting that <strong>complete binary trees are very suitable for array representation</strong>. Recalling the definition of a complete binary tree, <code>None</code> appears only at the bottom level and towards the right, <strong>meaning all <code>None</code> values definitely appear at the end of the level-order traversal sequence</strong>.</p>
|
|
<p>This means that when using an array to represent a complete binary tree, it's possible to omit storing all <code>None</code> values, which is very convenient. Figure 7-15 gives an example.</p>
|
|
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array representation of a complete binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" /></a></p>
|
|
<p align="center"> Figure 7-15 Array representation of a complete binary tree </p>
|
|
|
|
<p>The following code implements a binary tree based on array representation, including the following operations:</p>
|
|
<ul>
|
|
<li>Given a node, obtain its value, left (right) child node, and parent node.</li>
|
|
<li>Obtain the preorder, inorder, postorder, and level-order traversal sequences.</li>
|
|
</ul>
|
|
<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">array_binary_tree.py</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="k">class</span> <span class="nc">ArrayBinaryTree</span><span class="p">:</span>
|
|
<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="w"> </span><span class="sd">"""Array-based binary tree class"""</span>
|
|
<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a>
|
|
<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arr</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">]):</span>
|
|
<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w"> </span><span class="sd">"""Constructor"""</span>
|
|
<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">_tree</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">arr</span><span class="p">)</span>
|
|
<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a>
|
|
<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a> <span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
|
<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w"> </span><span class="sd">"""List capacity"""</span>
|
|
<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a> <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_tree</span><span class="p">)</span>
|
|
<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a>
|
|
<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a> <span class="k">def</span> <span class="nf">val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
|
|
<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w"> </span><span class="sd">"""Get the value of the node at index i"""</span>
|
|
<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a> <span class="c1"># If the index is out of bounds, return None, representing a vacancy</span>
|
|
<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a> <span class="k">if</span> <span class="n">i</span> <span class="o"><</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">>=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">():</span>
|
|
<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a> <span class="k">return</span> <span class="kc">None</span>
|
|
<a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_tree</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
|
<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a>
|
|
<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a> <span class="k">def</span> <span class="nf">left</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
|
|
<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w"> </span><span class="sd">"""Get the index of the left child of the node at index i"""</span>
|
|
<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a> <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
|
|
<a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a>
|
|
<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a> <span class="k">def</span> <span class="nf">right</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
|
|
<a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w"> </span><span class="sd">"""Get the index of the right child of the node at index i"""</span>
|
|
<a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a> <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">2</span>
|
|
<a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a>
|
|
<a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a> <span class="k">def</span> <span class="nf">parent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
|
|
<a id="__codelineno-14-28" name="__codelineno-14-28" href="#__codelineno-14-28"></a><span class="w"> </span><span class="sd">"""Get the index of the parent of the node at index i"""</span>
|
|
<a id="__codelineno-14-29" name="__codelineno-14-29" href="#__codelineno-14-29"></a> <span class="k">return</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
|
|
<a id="__codelineno-14-30" name="__codelineno-14-30" href="#__codelineno-14-30"></a>
|
|
<a id="__codelineno-14-31" name="__codelineno-14-31" href="#__codelineno-14-31"></a> <span class="k">def</span> <span class="nf">level_order</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
|
|
<a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w"> </span><span class="sd">"""Level-order traversal"""</span>
|
|
<a id="__codelineno-14-33" name="__codelineno-14-33" href="#__codelineno-14-33"></a> <span class="bp">self</span><span class="o">.</span><span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
|
|
<a id="__codelineno-14-34" name="__codelineno-14-34" href="#__codelineno-14-34"></a> <span class="c1"># Traverse array</span>
|
|
<a id="__codelineno-14-35" name="__codelineno-14-35" href="#__codelineno-14-35"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()):</span>
|
|
<a id="__codelineno-14-36" name="__codelineno-14-36" href="#__codelineno-14-36"></a> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
|
<a id="__codelineno-14-37" name="__codelineno-14-37" href="#__codelineno-14-37"></a> <span class="bp">self</span><span class="o">.</span><span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
|
|
<a id="__codelineno-14-38" name="__codelineno-14-38" href="#__codelineno-14-38"></a> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">res</span>
|
|
<a id="__codelineno-14-39" name="__codelineno-14-39" href="#__codelineno-14-39"></a>
|
|
<a id="__codelineno-14-40" name="__codelineno-14-40" href="#__codelineno-14-40"></a> <span class="k">def</span> <span class="nf">dfs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">order</span><span class="p">:</span> <span class="nb">str</span><span class="p">):</span>
|
|
<a id="__codelineno-14-41" name="__codelineno-14-41" href="#__codelineno-14-41"></a><span class="w"> </span><span class="sd">"""Depth-first traversal"""</span>
|
|
<a id="__codelineno-14-42" name="__codelineno-14-42" href="#__codelineno-14-42"></a> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
|
<a id="__codelineno-14-43" name="__codelineno-14-43" href="#__codelineno-14-43"></a> <span class="k">return</span>
|
|
<a id="__codelineno-14-44" name="__codelineno-14-44" href="#__codelineno-14-44"></a> <span class="c1"># Pre-order traversal</span>
|
|
<a id="__codelineno-14-45" name="__codelineno-14-45" href="#__codelineno-14-45"></a> <span class="k">if</span> <span class="n">order</span> <span class="o">==</span> <span class="s2">"pre"</span><span class="p">:</span>
|
|
<a id="__codelineno-14-46" name="__codelineno-14-46" href="#__codelineno-14-46"></a> <span class="bp">self</span><span class="o">.</span><span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
|
|
<a id="__codelineno-14-47" name="__codelineno-14-47" href="#__codelineno-14-47"></a> <span class="bp">self</span><span class="o">.</span><span class="n">dfs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">order</span><span class="p">)</span>
|
|
<a id="__codelineno-14-48" name="__codelineno-14-48" href="#__codelineno-14-48"></a> <span class="c1"># In-order traversal</span>
|
|
<a id="__codelineno-14-49" name="__codelineno-14-49" href="#__codelineno-14-49"></a> <span class="k">if</span> <span class="n">order</span> <span class="o">==</span> <span class="s2">"in"</span><span class="p">:</span>
|
|
<a id="__codelineno-14-50" name="__codelineno-14-50" href="#__codelineno-14-50"></a> <span class="bp">self</span><span class="o">.</span><span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
|
|
<a id="__codelineno-14-51" name="__codelineno-14-51" href="#__codelineno-14-51"></a> <span class="bp">self</span><span class="o">.</span><span class="n">dfs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">order</span><span class="p">)</span>
|
|
<a id="__codelineno-14-52" name="__codelineno-14-52" href="#__codelineno-14-52"></a> <span class="c1"># Post-order traversal</span>
|
|
<a id="__codelineno-14-53" name="__codelineno-14-53" href="#__codelineno-14-53"></a> <span class="k">if</span> <span class="n">order</span> <span class="o">==</span> <span class="s2">"post"</span><span class="p">:</span>
|
|
<a id="__codelineno-14-54" name="__codelineno-14-54" href="#__codelineno-14-54"></a> <span class="bp">self</span><span class="o">.</span><span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
|
|
<a id="__codelineno-14-55" name="__codelineno-14-55" href="#__codelineno-14-55"></a>
|
|
<a id="__codelineno-14-56" name="__codelineno-14-56" href="#__codelineno-14-56"></a> <span class="k">def</span> <span class="nf">pre_order</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
|
|
<a id="__codelineno-14-57" name="__codelineno-14-57" href="#__codelineno-14-57"></a><span class="w"> </span><span class="sd">"""Pre-order traversal"""</span>
|
|
<a id="__codelineno-14-58" name="__codelineno-14-58" href="#__codelineno-14-58"></a> <span class="bp">self</span><span class="o">.</span><span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
|
|
<a id="__codelineno-14-59" name="__codelineno-14-59" href="#__codelineno-14-59"></a> <span class="bp">self</span><span class="o">.</span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">"pre"</span><span class="p">)</span>
|
|
<a id="__codelineno-14-60" name="__codelineno-14-60" href="#__codelineno-14-60"></a> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">res</span>
|
|
<a id="__codelineno-14-61" name="__codelineno-14-61" href="#__codelineno-14-61"></a>
|
|
<a id="__codelineno-14-62" name="__codelineno-14-62" href="#__codelineno-14-62"></a> <span class="k">def</span> <span class="nf">in_order</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
|
|
<a id="__codelineno-14-63" name="__codelineno-14-63" href="#__codelineno-14-63"></a><span class="w"> </span><span class="sd">"""In-order traversal"""</span>
|
|
<a id="__codelineno-14-64" name="__codelineno-14-64" href="#__codelineno-14-64"></a> <span class="bp">self</span><span class="o">.</span><span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
|
|
<a id="__codelineno-14-65" name="__codelineno-14-65" href="#__codelineno-14-65"></a> <span class="bp">self</span><span class="o">.</span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">"in"</span><span class="p">)</span>
|
|
<a id="__codelineno-14-66" name="__codelineno-14-66" href="#__codelineno-14-66"></a> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">res</span>
|
|
<a id="__codelineno-14-67" name="__codelineno-14-67" href="#__codelineno-14-67"></a>
|
|
<a id="__codelineno-14-68" name="__codelineno-14-68" href="#__codelineno-14-68"></a> <span class="k">def</span> <span class="nf">post_order</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
|
|
<a id="__codelineno-14-69" name="__codelineno-14-69" href="#__codelineno-14-69"></a><span class="w"> </span><span class="sd">"""Post-order traversal"""</span>
|
|
<a id="__codelineno-14-70" name="__codelineno-14-70" href="#__codelineno-14-70"></a> <span class="bp">self</span><span class="o">.</span><span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
|
|
<a id="__codelineno-14-71" name="__codelineno-14-71" href="#__codelineno-14-71"></a> <span class="bp">self</span><span class="o">.</span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">"post"</span><span class="p">)</span>
|
|
<a id="__codelineno-14-72" name="__codelineno-14-72" href="#__codelineno-14-72"></a> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">res</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.cpp</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.java</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Array-based binary tree class */</span>
|
|
<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kd">class</span> <span class="nc">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w"> </span><span class="kd">private</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">tree</span><span class="p">;</span>
|
|
<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a>
|
|
<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w"> </span><span class="cm">/* Constructor */</span>
|
|
<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="nf">ArrayBinaryTree</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">arr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">(</span><span class="n">arr</span><span class="p">);</span>
|
|
<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a>
|
|
<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w"> </span><span class="cm">/* List capacity */</span>
|
|
<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
|
|
<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a>
|
|
<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w"> </span><span class="cm">/* Get the value of the node at index i */</span>
|
|
<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">Integer</span><span class="w"> </span><span class="nf">val</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w"> </span><span class="c1">// If the index is out of bounds, return null, representing an empty spot</span>
|
|
<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
|
|
<a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
|
|
<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
|
|
<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a>
|
|
<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w"> </span><span class="cm">/* Get the index of the left child of the node at index i */</span>
|
|
<a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">Integer</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</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-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a>
|
|
<a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w"> </span><span class="cm">/* Get the index of the right child of the node at index i */</span>
|
|
<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">Integer</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
|
|
<a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-32" name="__codelineno-16-32" href="#__codelineno-16-32"></a>
|
|
<a id="__codelineno-16-33" name="__codelineno-16-33" href="#__codelineno-16-33"></a><span class="w"> </span><span class="cm">/* Get the index of the parent of the node at index i */</span>
|
|
<a id="__codelineno-16-34" name="__codelineno-16-34" href="#__codelineno-16-34"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">Integer</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-35" name="__codelineno-16-35" href="#__codelineno-16-35"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
|
|
<a id="__codelineno-16-36" name="__codelineno-16-36" href="#__codelineno-16-36"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-37" name="__codelineno-16-37" href="#__codelineno-16-37"></a>
|
|
<a id="__codelineno-16-38" name="__codelineno-16-38" href="#__codelineno-16-38"></a><span class="w"> </span><span class="cm">/* Level-order traversal */</span>
|
|
<a id="__codelineno-16-39" name="__codelineno-16-39" href="#__codelineno-16-39"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="nf">levelOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
|
|
<a id="__codelineno-16-41" name="__codelineno-16-41" href="#__codelineno-16-41"></a><span class="w"> </span><span class="c1">// Traverse array</span>
|
|
<a id="__codelineno-16-42" name="__codelineno-16-42" href="#__codelineno-16-42"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-43" name="__codelineno-16-43" href="#__codelineno-16-43"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
|
|
<a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
|
|
<a id="__codelineno-16-45" name="__codelineno-16-45" href="#__codelineno-16-45"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></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-16-47" name="__codelineno-16-47" href="#__codelineno-16-47"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-48" name="__codelineno-16-48" href="#__codelineno-16-48"></a>
|
|
<a id="__codelineno-16-49" name="__codelineno-16-49" href="#__codelineno-16-49"></a><span class="w"> </span><span class="cm">/* Depth-first traversal */</span>
|
|
<a id="__codelineno-16-50" name="__codelineno-16-50" href="#__codelineno-16-50"></a><span class="w"> </span><span class="kd">private</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">Integer</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-51" name="__codelineno-16-51" href="#__codelineno-16-51"></a><span class="w"> </span><span class="c1">// If it is an empty spot, return</span>
|
|
<a id="__codelineno-16-52" name="__codelineno-16-52" href="#__codelineno-16-52"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
|
|
<a id="__codelineno-16-53" name="__codelineno-16-53" href="#__codelineno-16-53"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
|
<a id="__codelineno-16-54" name="__codelineno-16-54" href="#__codelineno-16-54"></a><span class="w"> </span><span class="c1">// Pre-order traversal</span>
|
|
<a id="__codelineno-16-55" name="__codelineno-16-55" href="#__codelineno-16-55"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="s">"pre"</span><span class="p">.</span><span class="na">equals</span><span class="p">(</span><span class="n">order</span><span class="p">))</span>
|
|
<a id="__codelineno-16-56" name="__codelineno-16-56" href="#__codelineno-16-56"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
|
|
<a id="__codelineno-16-57" name="__codelineno-16-57" href="#__codelineno-16-57"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|
<a id="__codelineno-16-58" name="__codelineno-16-58" href="#__codelineno-16-58"></a><span class="w"> </span><span class="c1">// In-order traversal</span>
|
|
<a id="__codelineno-16-59" name="__codelineno-16-59" href="#__codelineno-16-59"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="s">"in"</span><span class="p">.</span><span class="na">equals</span><span class="p">(</span><span class="n">order</span><span class="p">))</span>
|
|
<a id="__codelineno-16-60" name="__codelineno-16-60" href="#__codelineno-16-60"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
|
|
<a id="__codelineno-16-61" name="__codelineno-16-61" href="#__codelineno-16-61"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|
<a id="__codelineno-16-62" name="__codelineno-16-62" href="#__codelineno-16-62"></a><span class="w"> </span><span class="c1">// Post-order traversal</span>
|
|
<a id="__codelineno-16-63" name="__codelineno-16-63" href="#__codelineno-16-63"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="s">"post"</span><span class="p">.</span><span class="na">equals</span><span class="p">(</span><span class="n">order</span><span class="p">))</span>
|
|
<a id="__codelineno-16-64" name="__codelineno-16-64" href="#__codelineno-16-64"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
|
|
<a id="__codelineno-16-65" name="__codelineno-16-65" href="#__codelineno-16-65"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-66" name="__codelineno-16-66" href="#__codelineno-16-66"></a>
|
|
<a id="__codelineno-16-67" name="__codelineno-16-67" href="#__codelineno-16-67"></a><span class="w"> </span><span class="cm">/* Pre-order traversal */</span>
|
|
<a id="__codelineno-16-68" name="__codelineno-16-68" href="#__codelineno-16-68"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="nf">preOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-69" name="__codelineno-16-69" href="#__codelineno-16-69"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
|
|
<a id="__codelineno-16-70" name="__codelineno-16-70" href="#__codelineno-16-70"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="s">"pre"</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|
<a id="__codelineno-16-71" name="__codelineno-16-71" href="#__codelineno-16-71"></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-16-72" name="__codelineno-16-72" href="#__codelineno-16-72"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-73" name="__codelineno-16-73" href="#__codelineno-16-73"></a>
|
|
<a id="__codelineno-16-74" name="__codelineno-16-74" href="#__codelineno-16-74"></a><span class="w"> </span><span class="cm">/* In-order traversal */</span>
|
|
<a id="__codelineno-16-75" name="__codelineno-16-75" href="#__codelineno-16-75"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="nf">inOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-76" name="__codelineno-16-76" href="#__codelineno-16-76"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
|
|
<a id="__codelineno-16-77" name="__codelineno-16-77" href="#__codelineno-16-77"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="s">"in"</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|
<a id="__codelineno-16-78" name="__codelineno-16-78" href="#__codelineno-16-78"></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-16-79" name="__codelineno-16-79" href="#__codelineno-16-79"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-80" name="__codelineno-16-80" href="#__codelineno-16-80"></a>
|
|
<a id="__codelineno-16-81" name="__codelineno-16-81" href="#__codelineno-16-81"></a><span class="w"> </span><span class="cm">/* Post-order traversal */</span>
|
|
<a id="__codelineno-16-82" name="__codelineno-16-82" href="#__codelineno-16-82"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="nf">postOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-16-83" name="__codelineno-16-83" href="#__codelineno-16-83"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
|
|
<a id="__codelineno-16-84" name="__codelineno-16-84" href="#__codelineno-16-84"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="s">"post"</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|
<a id="__codelineno-16-85" name="__codelineno-16-85" href="#__codelineno-16-85"></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-16-86" name="__codelineno-16-86" href="#__codelineno-16-86"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-16-87" name="__codelineno-16-87" href="#__codelineno-16-87"></a><span class="p">}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.go</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{</span><span class="nx">arrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.swift</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.js</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="nx">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.ts</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="nx">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.rs</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.c</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.kt</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.rb</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{</span><span class="no">ArrayBinaryTree</span><span class="p">}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">array_binary_tree.zig</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">ArrayBinaryTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{}</span>
|
|
</code></pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<h2 id="733-advantages-and-limitations">7.3.3 Advantages and limitations<a class="headerlink" href="#733-advantages-and-limitations" title="Permanent link">¶</a></h2>
|
|
<p>The array representation of binary trees has the following advantages:</p>
|
|
<ul>
|
|
<li>Arrays are stored in contiguous memory spaces, which is cache-friendly and allows for faster access and traversal.</li>
|
|
<li>It does not require storing pointers, which saves space.</li>
|
|
<li>It allows random access to nodes.</li>
|
|
</ul>
|
|
<p>However, the array representation also has some limitations:</p>
|
|
<ul>
|
|
<li>Array storage requires contiguous memory space, so it is not suitable for storing trees with a large amount of data.</li>
|
|
<li>Adding or deleting nodes requires array insertion and deletion operations, which are less efficient.</li>
|
|
<li>When there are many <code>None</code> values in the binary tree, the proportion of node data contained in the array is low, leading to lower space utilization.</li>
|
|
</ul>
|
|
|
|
<!-- 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="../binary_tree_traversal/"
|
|
class="md-footer__link md-footer__link--prev"
|
|
aria-label="Previous: 7.2 Binary tree traversal"
|
|
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">
|
|
7.2 Binary tree traversal
|
|
</div>
|
|
</div>
|
|
</a>
|
|
|
|
|
|
<!-- Link to next page -->
|
|
|
|
|
|
<a
|
|
href="../binary_search_tree/"
|
|
class="md-footer__link md-footer__link--next"
|
|
aria-label="Next: 7.4 Binary Search tree"
|
|
rel="next"
|
|
>
|
|
<div class="md-footer__title">
|
|
<span class="md-footer__direction">
|
|
Next
|
|
</span>
|
|
<div class="md-ellipsis">
|
|
7.4 Binary Search tree
|
|
</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="../binary_tree_traversal/" class="md-footer__link md-footer__link--prev" aria-label="Previous: 7.2 Binary tree traversal">
|
|
<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">
|
|
7.2 Binary tree traversal
|
|
</div>
|
|
</div>
|
|
</a>
|
|
|
|
|
|
|
|
<a href="../binary_search_tree/" class="md-footer__link md-footer__link--next" aria-label="Next: 7.4 Binary Search tree">
|
|
<div class="md-footer__title">
|
|
<span class="md-footer__direction">
|
|
Next
|
|
</span>
|
|
<div class="md-ellipsis">
|
|
7.4 Binary Search tree
|
|
</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 © 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://polyfill.io/v3/polyfill.min.js?features=es6"></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> |