<!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_introduction/algorithms_are_everywhere/" >
< link rel = "prev" href = "../" >
< link rel = "next" href = "../what_is_dsa/" >
< link rel = "icon" href = "../../assets/images/favicon.png" >
< meta name = "generator" content = "mkdocs-1.5.3, mkdocs-material-9.5.2" >
< title > 1.1 Algorithms are Everywhere - Hello Algo< / title >
< link rel = "stylesheet" href = "../../assets/stylesheets/main.50c56a3b.min.css" >
< link rel = "stylesheet" href = "../../assets/stylesheets/palette.06af60db.min.css" >
< link rel = "preconnect" href = "https://fonts.gstatic.com" crossorigin >
< link rel = "stylesheet" href = "https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CFira+Code:400,400i,700,700i&display=fallback" >
< style > : root { --md-text-font : "Roboto" ; --md-code-font : "Fira Code" } < / style >
< link rel = "stylesheet" href = "../../stylesheets/extra.css" >
< script > _ _md _scope = new URL ( "../.." , location ) , _ _md _hash = e => [ ... e ] . reduce ( ( e , _ ) => ( e << 5 ) - e + _ . charCodeAt ( 0 ) , 0 ) , _ _md _get = ( e , _ = localStorage , t = _ _md _scope ) => JSON . parse ( _ . getItem ( t . pathname + "." + e ) ) , _ _md _set = ( e , _ , t = localStorage , a = _ _md _scope ) => { try { t . setItem ( a . pathname + "." + e , JSON . stringify ( _ ) ) } catch ( e ) { } } < / script >
< link href = "../../assets/stylesheets/glightbox.min.css" rel = "stylesheet" / > < style >
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
.gslide-desc { color: #666; user-select: text; }
.gslide-image img { background: white; }
.gscrollbar-fixer { padding-right: 15px; }
.gdesc-inner { font-size: 0.75rem; }
body[data-md-color-scheme="slate"] .gdesc-inner { background: var(--md-default-bg-color);}
body[data-md-color-scheme="slate"] .gslide-title { color: var(--md-default-fg-color);}
body[data-md-color-scheme="slate"] .gslide-desc { color: var(--md-default-fg-color);}
< / style > < script src = "../../assets/javascripts/glightbox.min.js" > < / script > < / head >
< body dir = "ltr" data-md-color-scheme = "default" data-md-color-primary = "white" data-md-color-accent = "indigo" >
< input class = "md-toggle" data-md-toggle = "drawer" type = "checkbox" id = "__drawer" autocomplete = "off" >
< input class = "md-toggle" data-md-toggle = "search" type = "checkbox" id = "__search" autocomplete = "off" >
< label class = "md-overlay" for = "__drawer" > < / label >
< div data-md-component = "skip" >
< a href = "#11-algorithms-are-everywhere" class = "md-skip" >
Skip to content
< / a >
< / div >
< div data-md-component = "announce" >
< / div >
< header class = "md-header md-header--shadow" data-md-component = "header" >
< nav class = "md-header__inner md-grid" aria-label = "Header" >
< a href = "../.." title = "Hello Algo" class = "md-header__button md-logo" aria-label = "Hello Algo" data-md-component = "logo" >
< img src = "../../assets/images/logo.svg" alt = "logo" >
< / a >
< label class = "md-header__button md-icon" for = "__drawer" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M3 6h18v2H3V6m0 5h18v2H3v-2m0 5h18v2H3v-2Z" / > < / svg >
< / label >
< div class = "md-header__title" data-md-component = "header-title" >
< div class = "md-header__ellipsis" >
< div class = "md-header__topic" >
< span class = "md-ellipsis" >
Hello Algo
< / span >
< / div >
< div class = "md-header__topic" data-md-component = "header-topic" >
< span class = "md-ellipsis" >
1.1 Algorithms are Everywhere
< / span >
< / div >
< / div >
< / div >
< form class = "md-header__option" data-md-component = "palette" >
< input class = "md-option" data-md-color-media = "" data-md-color-scheme = "default" data-md-color-primary = "white" data-md-color-accent = "indigo" aria-label = "Switch to dark mode" type = "radio" name = "__palette" id = "__palette_0" >
< label class = "md-header__button md-icon" title = "Switch to dark mode" for = "__palette_1" hidden >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z" / > < / svg >
< / label >
< input class = "md-option" data-md-color-media = "" data-md-color-scheme = "slate" data-md-color-primary = "grey" data-md-color-accent = "indigo" aria-label = "Switch to light mode" type = "radio" name = "__palette" id = "__palette_1" >
< label class = "md-header__button md-icon" title = "Switch to light mode" for = "__palette_0" hidden >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z" / > < / svg >
< / label >
< / form >
< script > var media , input , key , value , palette = _ _md _get ( "__palette" ) ; if ( palette && palette . color ) { "(prefers-color-scheme)" === palette . color . media && ( media = matchMedia ( "(prefers-color-scheme: light)" ) , input = document . querySelector ( media . matches ? "[data-md-color-media='(prefers-color-scheme: light)']" : "[data-md-color-media='(prefers-color-scheme: dark)']" ) , palette . color . media = input . getAttribute ( "data-md-color-media" ) , palette . color . scheme = input . getAttribute ( "data-md-color-scheme" ) , palette . color . primary = input . getAttribute ( "data-md-color-primary" ) , palette . color . accent = input . getAttribute ( "data-md-color-accent" ) ) ; for ( [ key , value ] of Object . entries ( palette . color ) ) document . body . setAttribute ( "data-md-color-" + key , value ) } < / script >
< div class = "md-header__option" >
< div class = "md-select" >
< button class = "md-header__button md-icon" aria-label = "Select language" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m12.87 15.07-2.54-2.51.03-.03A17.52 17.52 0 0 0 14.07 6H17V4h-7V2H8v2H1v2h11.17C11.5 7.92 10.44 9.75 9 11.35 8.07 10.32 7.3 9.19 6.69 8h-2c.73 1.63 1.73 3.17 2.98 4.56l-5.09 5.02L4 19l5-5 3.11 3.11.76-2.04M18.5 10h-2L12 22h2l1.12-3h4.75L21 22h2l-4.5-12m-2.62 7 1.62-4.33L19.12 17h-3.24Z" / > < / svg >
< / button >
< div class = "md-select__inner" >
< ul class = "md-select__list" >
< li class = "md-select__item" >
< a href = "/" hreflang = "zh" class = "md-select__link" >
中文
< / a >
< / li >
< li class = "md-select__item" >
< a href = "/en/" hreflang = "en" class = "md-select__link" >
English
< / a >
< / li >
< / ul >
< / div >
< / div >
< / div >
< label class = "md-header__button md-icon" for = "__search" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5Z" / > < / svg >
< / label >
< div class = "md-search" data-md-component = "search" role = "dialog" >
< label class = "md-search__overlay" for = "__search" > < / label >
< div class = "md-search__inner" role = "search" >
< form class = "md-search__form" name = "search" >
< input type = "text" class = "md-search__input" name = "query" aria-label = "Search" placeholder = "Search" autocapitalize = "off" autocorrect = "off" autocomplete = "off" spellcheck = "false" data-md-component = "search-query" required >
< label class = "md-search__icon md-icon" for = "__search" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5Z" / > < / svg >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z" / > < / svg >
< / label >
< nav class = "md-search__options" aria-label = "Search" >
< a href = "javascript:void(0)" class = "md-search__icon md-icon" title = "Share" aria-label = "Share" data-clipboard data-clipboard-text = "" data-md-component = "search-share" tabindex = "-1" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M18 16.08c-.76 0-1.44.3-1.96.77L8.91 12.7c.05-.23.09-.46.09-.7 0-.24-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91 1.61 0 2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08Z" / > < / svg >
< / a >
< button type = "reset" class = "md-search__icon md-icon" title = "Clear" aria-label = "Clear" tabindex = "-1" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12 19 6.41Z" / > < / svg >
< / button >
< / nav >
< div class = "md-search__suggest" data-md-component = "search-suggest" > < / div >
< / form >
< div class = "md-search__output" >
< div class = "md-search__scrollwrap" data-md-scrollfix >
< div class = "md-search-result" data-md-component = "search-result" >
< div class = "md-search-result__meta" >
Initializing search
< / div >
< ol class = "md-search-result__list" role = "presentation" > < / ol >
< / div >
< / div >
< / div >
< / div >
< / div >
< div class = "md-header__source" >
< a href = "https://github.com/krahets/hello-algo" title = "Go to repository" class = "md-source" data-md-component = "source" >
< div class = "md-source__icon md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 496 512" > <!-- ! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" / > < / svg >
< / div >
< div class = "md-source__repository" >
krahets/hello-algo
< / div >
< / a >
< / div >
< / nav >
< / header >
< div class = "md-container" data-md-component = "container" >
< main class = "md-main" data-md-component = "main" >
< div class = "md-main__inner md-grid" >
< div class = "md-sidebar md-sidebar--primary" data-md-component = "sidebar" data-md-type = "navigation" >
< div class = "md-sidebar__scrollwrap" >
< div class = "md-sidebar__inner" >
< nav class = "md-nav md-nav--primary" aria-label = "Navigation" data-md-level = "0" >
< label class = "md-nav__title" for = "__drawer" >
< a href = "../.." title = "Hello Algo" class = "md-nav__button md-logo" aria-label = "Hello Algo" data-md-component = "logo" >
< img src = "../../assets/images/logo.svg" alt = "logo" >
< / a >
Hello Algo
< / label >
< div class = "md-nav__source" >
< a href = "https://github.com/krahets/hello-algo" title = "Go to repository" class = "md-source" data-md-component = "source" >
< div class = "md-source__icon md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 496 512" > <!-- ! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" / > < / svg >
< / div >
< div class = "md-source__repository" >
krahets/hello-algo
< / div >
< / a >
< / div >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_1" >
< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_preface/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M21 4H3a2 2 0 0 0-2 2v13a2 2 0 0 0 2 2h18a2 2 0 0 0 2-2V6a2 2 0 0 0-2-2M3 19V6h8v13H3m18 0h-8V6h8v13m-7-9.5h6V11h-6V9.5m0 2.5h6v1.5h-6V12m0 2.5h6V16h-6v-1.5Z" / > < / svg >
< span class = "md-ellipsis" >
Chapter 0. Preface
< / span >
< / a >
< label class = "md-nav__link " for = "__nav_1" id = "__nav_1_label" tabindex = "0" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_1_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_1" >
< span class = "md-nav__icon md-icon" > < / span >
Chapter 0. Preface
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_preface/about_the_book/" class = "md-nav__link" >
< span class = "md-ellipsis" >
0.1 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--active md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_2" 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 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z" / > < / svg >
< span class = "md-ellipsis" >
Chapter 1. Introduction to Algorithms
< / span >
< / a >
< label class = "md-nav__link " for = "__nav_2" id = "__nav_2_label" tabindex = "0" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_2_label" aria-expanded = "true" >
< label class = "md-nav__title" for = "__nav_2" >
< span class = "md-nav__icon md-icon" > < / span >
Chapter 1. Introduction to Algorithms
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item md-nav__item--active" >
< input class = "md-nav__toggle md-toggle" type = "checkbox" id = "__toc" >
< a href = "./" class = "md-nav__link md-nav__link--active" >
< span class = "md-ellipsis" >
1.1 Algorithms are Everywhere
< / span >
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../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 = "../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_3" >
< 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_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 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_4" >
< 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_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 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 Fundamental 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_5" >
< 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_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 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 >
< / 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" >
< / nav >
< / div >
< / div >
< / div >
< div class = "md-content" data-md-component = "content" >
< article class = "md-content__inner md-typeset" >
<!-- Tags -->
<!-- Actions -->
<!-- Actions -->
<!-- Edit button -->
< a
href="https://github.com/krahets/hello-algo/tree/main/docs-en/chapter_introduction/algorithms_are_everywhere.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 = "11-algorithms-are-everywhere" > 1.1 Algorithms are Everywhere< a class = "headerlink" href = "#11-algorithms-are-everywhere" title = "Permanent link" > ¶ < / a > < / h1 >
< p > When we hear the word "algorithm," we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.< / p >
< p > Before formally discussing algorithms, there's an interesting fact worth sharing: < strong > you have already unconsciously learned many algorithms and have become accustomed to applying them in your daily life< / strong > . Here, I will give a few specific examples to prove this point.< / p >
< p > < strong > Example 1: Looking Up a Dictionary< / strong > . In an English dictionary, words are listed alphabetically. Suppose we're searching for a word that starts with the letter < span class = "arithmatex" > \(r\)< / span > . This is typically done in the following way:< / p >
< ol >
< li > Open the dictionary to about halfway and check the first letter on the page, let's say the letter is < span class = "arithmatex" > \(m\)< / span > .< / li >
< li > Since < span class = "arithmatex" > \(r\)< / span > comes after < span class = "arithmatex" > \(m\)< / span > in the alphabet, we can ignore the first half of the dictionary and focus on the latter half.< / li >
< li > Repeat steps < code > 1.< / code > and < code > 2.< / code > until you find the page where the word starts with < span class = "arithmatex" > \(r\)< / span > .< / li >
< / ol >
< div class = "tabbed-set tabbed-alternate" data-tabs = "1:5" > < 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" / > < div class = "tabbed-labels" > < label for = "__tabbed_1_1" > < 1> < / label > < label for = "__tabbed_1_2" > < 2> < / label > < label for = "__tabbed_1_3" > < 3> < / label > < label for = "__tabbed_1_4" > < 4> < / label > < label for = "__tabbed_1_5" > < 5> < / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Process of Looking Up a Dictionary" class = "animation-figure" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" / > < / a > < / p >
< / div >
< div class = "tabbed-block" >
< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Binary Search in Dictionary Step 2" class = "animation-figure" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" / > < / a > < / p >
< / div >
< div class = "tabbed-block" >
< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Binary Search in Dictionary Step 3" class = "animation-figure" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" / > < / a > < / p >
< / div >
< div class = "tabbed-block" >
< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Binary Search in Dictionary Step 4" class = "animation-figure" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" / > < / a > < / p >
< / div >
< div class = "tabbed-block" >
< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Binary Search in Dictionary Step 5" class = "animation-figure" src = "../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" / > < / a > < / p >
< / div >
< / div >
< / div >
< p align = "center" > Figure 1-1 Process of Looking Up a Dictionary < / p >
< p > This essential skill for elementary students, looking up a dictionary, is actually the famous "Binary Search" algorithm. From a data structure perspective, we can consider the dictionary as a sorted "array"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as "Binary Search."< / p >
< p > < strong > Example 2: Organizing Playing Cards< / strong > . When playing cards, we need to arrange the cards in our hand in ascending order, as shown in the following process.< / p >
< ol >
< li > Divide the playing cards into "ordered" and "unordered" sections, assuming initially the leftmost card is already in order.< / li >
< li > Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.< / li >
< li > Continue to repeat step < code > 2.< / code > until all cards are in order.< / li >
< / ol >
< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/playing_cards_sorting.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Playing Cards Sorting Process" class = "animation-figure" src = "../algorithms_are_everywhere.assets/playing_cards_sorting.png" / > < / a > < / p >
< p align = "center" > Figure 1-2 Playing Cards Sorting Process < / p >
< p > The above method of organizing playing cards is essentially the "Insertion Sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.< / p >
< p > < strong > Example 3: Making Change< / strong > . Suppose we buy goods worth < span class = "arithmatex" > \(69\)< / span > yuan at a supermarket and give the cashier < span class = "arithmatex" > \(100\)< / span > yuan, then the cashier needs to give us < span class = "arithmatex" > \(31\)< / span > yuan in change. They would naturally complete the thought process as shown below.< / p >
< ol >
< li > The options are currencies smaller than < span class = "arithmatex" > \(31\)< / span > , including < span class = "arithmatex" > \(1\)< / span > , < span class = "arithmatex" > \(5\)< / span > , < span class = "arithmatex" > \(10\)< / span > , and < span class = "arithmatex" > \(20\)< / span > .< / li >
< li > Take out the largest < span class = "arithmatex" > \(20\)< / span > from the options, leaving < span class = "arithmatex" > \(31 - 20 = 11\)< / span > .< / li >
< li > Take out the largest < span class = "arithmatex" > \(10\)< / span > from the remaining options, leaving < span class = "arithmatex" > \(11 - 10 = 1\)< / span > .< / li >
< li > Take out the largest < span class = "arithmatex" > \(1\)< / span > from the remaining options, leaving < span class = "arithmatex" > \(1 - 1 = 0\)< / span > .< / li >
< li > Complete the change-making, with the solution being < span class = "arithmatex" > \(20 + 10 + 1 = 31\)< / span > .< / li >
< / ol >
< p > < a class = "glightbox" href = "../algorithms_are_everywhere.assets/greedy_change.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Change making process" class = "animation-figure" src = "../algorithms_are_everywhere.assets/greedy_change.png" / > < / a > < / p >
< p align = "center" > Figure 1-3 Change making process < / p >
< p > In the above steps, we make the best choice at each step (using the largest denomination possible), ultimately resulting in a feasible change-making plan. From the perspective of data structures and algorithms, this method is essentially a "Greedy" algorithm.< / p >
< p > From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers, solving various complex issues more efficiently.< / p >
< div class = "admonition tip" >
< p class = "admonition-title" > Tip< / p >
< p > If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obsecure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms.< / p >
< / div >
<!-- Source file information -->
<!-- Was this page helpful? -->
<!-- Previous and next pages link -->
< nav
class="md-footer__inner md-grid"
aria-label="Footer"
>
<!-- Link to previous page -->
< a
href="../"
class="md-footer__link md-footer__link--prev"
aria-label="Previous: Chapter 1. & nbsp; Introduction to Algorithms"
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" >
Chapter 1. Introduction to Algorithms
< / div >
< / div >
< / a >
<!-- Link to next page -->
< a
href="../what_is_dsa/"
class="md-footer__link md-footer__link--next"
aria-label="Next: 1.2 What is an Algorithm"
rel="next"
>
< div class = "md-footer__title" >
< span class = "md-footer__direction" >
Next
< / span >
< div class = "md-ellipsis" >
1.2 What is an Algorithm
< / 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="preferred_color_scheme"
data-lang="en"
crossorigin="anonymous"
async
>
< / script >
<!-- Synchronize Giscus theme with palette -->
< script >
var giscus = document.querySelector("script[src*=giscus]")
/* Set palette on initial load */
var palette = __md_get("__palette")
if (palette & & typeof palette.color === "object") {
var theme = palette.color.scheme === "slate" ? "dark_dimmed" : "light"
giscus.setAttribute("data-theme", theme)
}
/* Register event handlers after documented loaded */
document.addEventListener("DOMContentLoaded", function() {
var ref = document.querySelector("[data-md-component=palette]")
ref.addEventListener("change", function() {
var palette = __md_get("__palette")
if (palette & & typeof palette.color === "object") {
var theme = palette.color.scheme === "slate" ? "dark_dimmed" : "light"
/* Instruct Giscus to change theme */
var frame = document.querySelector(".giscus-frame")
frame.contentWindow.postMessage(
{ giscus: { setConfig: { theme } } },
"https://giscus.app"
)
}
})
})
< / script >
< / article >
< / div >
< script > var tabs = _ _md _get ( "__tabs" ) ; if ( Array . isArray ( tabs ) ) e : for ( var set of document . querySelectorAll ( ".tabbed-set" ) ) { var tab , labels = set . querySelector ( ".tabbed-labels" ) ; for ( tab of tabs ) for ( var label of labels . getElementsByTagName ( "label" ) ) if ( label . innerText . trim ( ) === tab ) { var input = document . getElementById ( label . htmlFor ) ; input . checked = ! 0 ; continue e } } < / script >
< script > var target = document . getElementById ( location . hash . slice ( 1 ) ) ; target && target . name && ( target . checked = target . name . startsWith ( "__tabbed_" ) ) < / script >
< / div >
< button type = "button" class = "md-top md-icon" data-md-component = "top" hidden >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z" / > < / svg >
Back to top
< / button >
< / main >
< footer class = "md-footer" >
< nav class = "md-footer__inner md-grid" aria-label = "Footer" >
< a href = "../" class = "md-footer__link md-footer__link--prev" aria-label = "Previous: Chapter 1. &nbsp; Introduction to Algorithms" >
< 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" >
Chapter 1. Introduction to Algorithms
< / div >
< / div >
< / a >
< a href = "../what_is_dsa/" class = "md-footer__link md-footer__link--next" aria-label = "Next: 1.2 What is an Algorithm" >
< div class = "md-footer__title" >
< span class = "md-footer__direction" >
Next
< / span >
< div class = "md-ellipsis" >
1.2 What is an Algorithm
< / 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 © 2022 - 2024 Krahets
< / div >
< / div >
< div class = "md-social" >
< a href = "https://github.com/krahets" target = "_blank" rel = "noopener" title = "github.com" class = "md-social__link" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 496 512" > <!-- ! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" / > < / svg >
< / a >
< a href = "https://twitter.com/krahets" target = "_blank" rel = "noopener" title = "twitter.com" class = "md-social__link" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > <!-- ! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z" / > < / svg >
< / a >
< a href = "https://leetcode.cn/u/jyd/" target = "_blank" rel = "noopener" title = "leetcode.cn" class = "md-social__link" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 640 512" > <!-- ! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z" / > < / svg >
< / a >
< / div >
< / div >
< / div >
< / footer >
< / div >
< div class = "md-dialog" data-md-component = "dialog" >
< div class = "md-dialog__inner md-typeset" > < / div >
< / div >
< script id = "__config" type = "application/json" > { "base" : "../.." , "features" : [ "content.action.edit" , "content.code.annotate" , "content.code.copy" , "content.tabs.link" , "content.tooltips" , "navigation.indexes" , "navigation.top" , "navigation.footer" , "navigation.tracking" , "search.highlight" , "search.share" , "search.suggest" , "toc.follow" ] , "search" : "../../assets/javascripts/workers/search.f886a092.min.js" , "translations" : { "clipboard.copied" : "Copied to clipboard" , "clipboard.copy" : "Copy to clipboard" , "search.result.more.one" : "1 more on this page" , "search.result.more.other" : "# more on this page" , "search.result.none" : "No matching documents" , "search.result.one" : "1 matching document" , "search.result.other" : "# matching documents" , "search.result.placeholder" : "Type to start searching" , "search.result.term.missing" : "Missing" , "select.version" : "Select version" } } < / script >
< script src = "../../assets/javascripts/bundle.d7c377c4.min.js" > < / script >
< script src = "../../javascripts/mathjax.js" > < / script >
< script src = "https://polyfill.io/v3/polyfill.min.js?features=es6" > < / script >
< script src = "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" > < / script >
< script > document $ . subscribe ( ( ) => { const lightbox = GLightbox ( { "touchNavigation" : true , "loop" : false , "zoomable" : true , "draggable" : false , "openEffect" : "zoom" , "closeEffect" : "zoom" , "slideEffect" : "none" } ) ; } ) < / script > < / body >
< / html >
