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krahets
2026-03-29 05:07:03 +08:00
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ru/chapter_dynamic_programming/intro_to_dynamic_programming/index.html
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@@ -35,18 +35,23 @@
<link rel="icon" href="../../assets/images/favicon.png">
<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.7.1">
<meta name="generator" content="zensical-0.0.29">
<title>14.1 Введение в динамическое программирование - Hello Algo</title>
<title>14.1 Первое знакомство с динамическим программированием - Hello Algo</title>
<link rel="stylesheet" href="../../assets/stylesheets/main.484c7ddc.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/palette.ab4e12ef.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/modern/main.6f2fbb28.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/modern/palette.dfe2e883.min.css">
@@ -60,29 +65,20 @@
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
<link rel="stylesheet" href="../../stylesheets/extra.css">
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,t)=>(e<<5)-e+t.charCodeAt(0)),0),__md_get=(e,t=localStorage,a=__md_scope)=>JSON.parse(t.getItem(("__palette"===e?"/":a.pathname)+"."+e)),__md_set=(e,t,a=localStorage,_=__md_scope)=>{try{a.setItem(("__palette"===e?"/":_.pathname)+"."+e,JSON.stringify(t))}catch(e){}},document.documentElement.setAttribute("data-platform",navigator.platform)</script>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
.gslide-desc { color: #666; user-select: text; }
.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>
</head>
@@ -97,7 +93,7 @@
<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
<label class="md-overlay" for="__drawer"></label>
<label class="md-overlay" for="__drawer" aria-label="Навигация"></label>
<div data-md-component="skip">
@@ -137,12 +133,12 @@
<nav class="md-header__inner md-grid" aria-label="Верхний колонтитул">
<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">
<img src="../../assets/images/logo.svg" alt="Hello Algo">
</a>
<label class="md-header__button md-icon" for="__drawer">
<label class="md-header__button md-icon" for="__drawer" aria-label="Навигация">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-menu" viewBox="0 0 24 24"><path d="M4 5h16M4 12h16M4 19h16"/></svg>
</label>
<div class="md-header__title" data-md-component="header-title">
<div class="md-header__ellipsis">
@@ -154,7 +150,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
14.1 Введение в динамическое программирование
14.1 &nbsp; Первое знакомство с динамическим программированием
</span>
</div>
@@ -167,7 +163,7 @@
<input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal" aria-label="Темная тема" type="radio" name="__palette" id="__palette_0">
<input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal" aria-label="Темная тема" type="radio" name="__palette" id="__palette_0">
<label class="md-header__button md-icon" title="Темная тема" for="__palette_1" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
@@ -177,7 +173,7 @@
<input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal" aria-label="Светлая тема" type="radio" name="__palette" id="__palette_1">
<input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal" aria-label="Светлая тема" type="radio" name="__palette" id="__palette_1">
<label class="md-header__button md-icon" title="Светлая тема" for="__palette_0" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
@@ -195,7 +191,7 @@
<div class="md-select">
<button class="md-header__button md-icon" aria-label="Выберите язык">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12.87 15.07-2.54-2.51.03-.03A17.5 17.5 0 0 0 14.07 6H17V4h-7V2H8v2H1v2h11.17C11.5 7.92 10.44 9.75 9 11.35 8.07 10.32 7.3 9.19 6.69 8h-2c.73 1.63 1.73 3.17 2.98 4.56l-5.09 5.02L4 19l5-5 3.11 3.11zM18.5 10h-2L12 22h2l1.12-3h4.75L21 22h2zm-2.62 7 1.62-4.33L19.12 17z"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-languages" viewBox="0 0 24 24"><path d="m5 8 6 6M4 14l6-6 2-3M2 5h12M7 2h1M22 22l-5-10-5 10M14 18h6"/></svg>
</button>
<div class="md-select__inner">
<ul class="md-select__list">
@@ -238,64 +234,30 @@
<label class="md-header__button md-icon" for="__search">
<label class="md-header__button md-icon" for="__search" aria-label="Поиск">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-search" viewBox="0 0 24 24"><path d="m21 21-4.34-4.34"/><circle cx="11" cy="11" r="8"/></svg>
</label>
<div class="md-search" data-md-component="search" role="dialog">
<label class="md-search__overlay" for="__search"></label>
<div class="md-search__inner" role="search">
<form class="md-search__form" name="search">
<input type="text" class="md-search__input" name="query" aria-label="Поиск" placeholder="Поиск" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
<label class="md-search__icon md-icon" for="__search">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11z"/></svg>
</label>
<nav class="md-search__options" aria-label="Поиск">
<a href="javascript:void(0)" class="md-search__icon md-icon" title="Поделиться" aria-label="Поделиться" data-clipboard data-clipboard-text="" data-md-component="search-share" tabindex="-1">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 16.08c-.76 0-1.44.3-1.96.77L8.91 12.7c.05-.23.09-.46.09-.7s-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91s2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08"/></svg>
</a>
<button type="reset" class="md-search__icon md-icon" title="Очистить" aria-label="Очистить" tabindex="-1">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12z"/></svg>
</button>
</nav>
<div class="md-search__suggest" data-md-component="search-suggest"></div>
</form>
<div class="md-search__output">
<div class="md-search__scrollwrap" tabindex="0" data-md-scrollfix>
<div class="md-search-result" data-md-component="search-result">
<div class="md-search-result__meta">
Инициализация поиска
</div>
<ol class="md-search-result__list" role="presentation"></ol>
</div>
</div>
</div>
</div>
<div class="md-search" data-md-component="search" role="dialog" aria-label="Поиск">
<button type="button" class="md-search__button">
Поиск
</button>
</div>
<div class="md-header__source">
<div class="md-header__source">
<a href="https://github.com/krahets/hello-algo" title="Перейти к репозиторию" class="md-source" data-md-component="source">
<div class="md-source__icon md-icon">
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</div>
<div class="md-source__repository">
krahets/hello-algo
</div>
</a>
</div>
</div>
</nav>
</header>
@@ -323,7 +285,7 @@
<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">
<img src="../../assets/images/logo.svg" alt="Hello Algo">
</a>
Hello Algo
@@ -333,7 +295,7 @@
<a href="https://github.com/krahets/hello-algo" title="Перейти к репозиторию" class="md-source" data-md-component="source">
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krahets/hello-algo
@@ -3617,7 +3579,7 @@
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<a href="./" class="md-nav__link md-nav__link--active">
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@@ -3636,6 +3598,7 @@
<nav class="md-nav md-nav--secondary" aria-label="Содержание">
@@ -4338,6 +4301,7 @@
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@@ -4423,7 +4387,7 @@
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@@ -4440,7 +4404,7 @@
<p>Дана лестница из <span class="arithmatex">\(n\)</span> ступеней. За один шаг можно подняться на <span class="arithmatex">\(1\)</span> или на <span class="arithmatex">\(2\)</span> ступени. Сколькими способами можно добраться до вершины?</p>
</div>
<p>Как показано на рисунке 14-1, для лестницы из <span class="arithmatex">\(3\)</span> ступеней существует <span class="arithmatex">\(3\)</span> способа добраться до вершины.</p>
<p><a class="glightbox" href="../intro_to_dynamic_programming.assets/climbing_stairs_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Число способов подняться на 3-ю ступень" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_example.png" /></a></p>
<p><img alt="Число способов подняться на 3-ю ступень" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_example.png" /></p>
<p align="center"> Рисунок 14-1 &nbsp; Число способов подняться на 3-ю ступень </p>
<p>Цель этой задачи - вычислить количество способов. <strong>Поэтому можно попробовать грубо перебрать все варианты с помощью backtracking</strong>. Если представить подъем по лестнице как последовательность решений, то мы начинаем от земли и на каждом раунде выбираем прыжок на <span class="arithmatex">\(1\)</span> или на <span class="arithmatex">\(2\)</span> ступени; всякий раз, когда достигаем вершины, увеличиваем число способов на <span class="arithmatex">\(1\)</span> , а если перескакиваем вершину, обрезаем эту ветвь. Код выглядит так:</p>
@@ -4795,56 +4759,29 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">climbing_stairs_backtrack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="cm">=begin</span>
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="cm">File: climbing_stairs_backtrack.rb</span>
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="cm">Created Time: 2024-05-29</span>
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="cm">Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)</span>
<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="cm">=end</span>
<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>
<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="c1"># ## Бэктрекинг ###</span>
<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w"> </span><span class="c1"># Когда подъем достигает n-й ступени, число вариантов увеличивается на 1</span>
<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span>
<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w"> </span><span class="c1"># Перебор всех вариантов выбора</span>
<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span>
<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w"> </span><span class="c1"># Отсечение: нельзя выходить за n-ю ступень</span>
<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w"> </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span>
<div class="highlight"><span class="filename">climbing_stairs_backtrack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Бэктрекинг ###</span>
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w"> </span><span class="c1"># Когда подъем достигает n-й ступени, число вариантов увеличивается на 1</span>
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span>
<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w"> </span><span class="c1"># Перебор всех вариантов выбора</span>
<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span>
<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w"> </span><span class="c1"># Отсечение: нельзя выходить за n-ю ступень</span>
<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w"> </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span>
<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a>
<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w"> </span><span class="c1"># Попытка: сделать выбор и обновить состояние</span>
<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w"> </span><span class="k">end</span>
<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w"> </span><span class="c1"># Откат</span>
<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="k">end</span>
<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a>
<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w"> </span><span class="c1"># Попытка: сделать выбор и обновить состояние</span>
<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w"> </span><span class="k">end</span>
<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w"> </span><span class="c1"># Откат</span>
<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="k">end</span>
<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a>
<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="cm">=begin</span>
<a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="cm">File: climbing_stairs_backtrack.rb</span>
<a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="cm">Created Time: 2024-05-29</span>
<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="cm">Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)</span>
<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="cm">=end</span>
<a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a>
<a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="c1"># ## Бэктрекинг ###</span>
<a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w"> </span><span class="c1"># Когда подъем достигает n-й ступени, число вариантов увеличивается на 1</span>
<a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span>
<a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="w"> </span><span class="c1"># Перебор всех вариантов выбора</span>
<a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span>
<a id="__codelineno-12-34" name="__codelineno-12-34" href="#__codelineno-12-34"></a><span class="w"> </span><span class="c1"># Отсечение: нельзя выходить за n-ю ступень</span>
<a id="__codelineno-12-35" name="__codelineno-12-35" href="#__codelineno-12-35"></a><span class="w"> </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span>
<a id="__codelineno-12-36" name="__codelineno-12-36" href="#__codelineno-12-36"></a>
<a id="__codelineno-12-37" name="__codelineno-12-37" href="#__codelineno-12-37"></a><span class="w"> </span><span class="c1"># Попытка: сделать выбор и обновить состояние</span>
<a id="__codelineno-12-38" name="__codelineno-12-38" href="#__codelineno-12-38"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-12-39" name="__codelineno-12-39" href="#__codelineno-12-39"></a><span class="w"> </span><span class="k">end</span>
<a id="__codelineno-12-40" name="__codelineno-12-40" href="#__codelineno-12-40"></a><span class="w"> </span><span class="c1"># Откат</span>
<a id="__codelineno-12-41" name="__codelineno-12-41" href="#__codelineno-12-41"></a><span class="k">end</span>
<a id="__codelineno-12-42" name="__codelineno-12-42" href="#__codelineno-12-42"></a>
<a id="__codelineno-12-43" name="__codelineno-12-43" href="#__codelineno-12-43"></a><span class="c1"># ## Подъем по лестнице: бэктрекинг ###</span>
<a id="__codelineno-12-44" name="__codelineno-12-44" href="#__codelineno-12-44"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_backtrack</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-12-45" name="__codelineno-12-45" href="#__codelineno-12-45"></a><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="c1"># Можно подняться на 1 или 2 ступени</span>
<a id="__codelineno-12-46" name="__codelineno-12-46" href="#__codelineno-12-46"></a><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Начать подъем с 0-й ступени</span>
<a id="__codelineno-12-47" name="__codelineno-12-47" href="#__codelineno-12-47"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="c1"># Использовать res[0] для хранения числа решений</span>
<a id="__codelineno-12-48" name="__codelineno-12-48" href="#__codelineno-12-48"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-12-49" name="__codelineno-12-49" href="#__codelineno-12-49"></a><span class="w"> </span><span class="n">res</span><span class="o">.</span><span class="n">first</span>
<a id="__codelineno-12-50" name="__codelineno-12-50" href="#__codelineno-12-50"></a><span class="k">end</span>
<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="c1">### Подъем по лестнице: бэктрекинг ###</span>
<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_backtrack</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="c1"># Можно подняться на 1 или 2 ступени</span>
<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Начать подъем с 0-й ступени</span>
<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="c1"># Использовать res[0] для хранения числа решений</span>
<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w"> </span><span class="n">res</span><span class="o">.</span><span class="n">first</span>
<a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="k">end</span>
</code></pre></div>
</div>
</div>
@@ -4866,7 +4803,7 @@ dp[i-1], dp[i-2], \dots, dp[2], dp[1]
dp[i] = dp[i-1] + dp[i-2]
\]</div>
<p>Это означает, что в задаче о подъеме по лестнице между подзадачами существует рекуррентная зависимость, и <strong>решение исходной задачи может быть построено на основе решений подзадач</strong>. Эта связь показана на рисунке 14-2.</p>
<p><a class="glightbox" href="../intro_to_dynamic_programming.assets/climbing_stairs_state_transfer.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Рекуррентная связь числа способов" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_state_transfer.png" /></a></p>
<p><img alt="Рекуррентная связь числа способов" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_state_transfer.png" /></p>
<p align="center"> Рисунок 14-2 &nbsp; Рекуррентная связь числа способов </p>
<p>По рекуррентной формуле можно получить решение полного перебора. Начиная с <span class="arithmatex">\(dp[n]\)</span> , <strong>мы рекурсивно разлагаем большую задачу в сумму двух меньших задач</strong> , пока не дойдем до наименьших подзадач <span class="arithmatex">\(dp[1]\)</span> и <span class="arithmatex">\(dp[2]\)</span> . Их решения уже известны: <span class="arithmatex">\(dp[1] = 1\)</span> и <span class="arithmatex">\(dp[2] = 2\)</span> , что означает <span class="arithmatex">\(1\)</span> и <span class="arithmatex">\(2\)</span> способа подняться соответственно на <span class="arithmatex">\(1\)</span>-ю и <span class="arithmatex">\(2\)</span>-ю ступени.</p>
@@ -5075,38 +5012,18 @@ dp[i] = dp[i-1] + dp[i-2]
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">climbing_stairs_dfs.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="cm">=begin</span>
<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="cm">File: climbing_stairs_dfs.rb</span>
<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="cm">Created Time: 2024-05-29</span>
<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="cm">Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)</span>
<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="cm">=end</span>
<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a>
<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="c1"># ## Поиск ###</span>
<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w"> </span><span class="c1"># dp[1] и dp[2] уже известны, вернуть их</span>
<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</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="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>
<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w"> </span><span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w"> </span><span class="n">dfs</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="n">dfs</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">2</span><span class="p">)</span>
<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="k">end</span>
<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a>
<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="cm">=begin</span>
<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="cm">File: climbing_stairs_dfs.rb</span>
<a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="cm">Created Time: 2024-05-29</span>
<a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="cm">Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)</span>
<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="cm">=end</span>
<a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a>
<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="c1"># ## Поиск ###</span>
<a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w"> </span><span class="c1"># dp[1] и dp[2] уже известны, вернуть их</span>
<a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</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="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>
<a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w"> </span><span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
<a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w"> </span><span class="n">dfs</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="n">dfs</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">2</span><span class="p">)</span>
<a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="k">end</span>
<a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a>
<a id="__codelineno-25-29" name="__codelineno-25-29" href="#__codelineno-25-29"></a><span class="c1"># ## Подъем по лестнице: поиск ###</span>
<a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dfs</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-25-32" name="__codelineno-25-32" href="#__codelineno-25-32"></a><span class="k">end</span>
<div class="highlight"><span class="filename">climbing_stairs_dfs.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Поиск ###</span>
<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w"> </span><span class="c1"># dp[1] и dp[2] уже известны, вернуть их</span>
<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</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="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>
<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w"> </span><span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w"> </span><span class="n">dfs</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="n">dfs</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">2</span><span class="p">)</span>
<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="k">end</span>
<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a>
<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="c1">### Подъем по лестнице: поиск ###</span>
<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dfs</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="k">end</span>
</code></pre></div>
</div>
</div>
@@ -5117,7 +5034,7 @@ dp[i] = dp[i-1] + dp[i-2]
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%D0%9F%D0%BE%D0%B8%D1%81%D0%BA%22%22%22%0A%20%20%20%20%23%20dp%5B1%5D%20%D0%B8%20dp%5B2%5D%20%D1%83%D0%B6%D0%B5%20%D0%B8%D0%B7%D0%B2%D0%B5%D1%81%D1%82%D0%BD%D1%8B%2C%20%D0%B2%D0%B5%D1%80%D0%BD%D1%83%D1%82%D1%8C%20%D0%B8%D1%85%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201%29%20%2B%20dfs%28i%20-%202%29%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%D0%9F%D0%BE%D0%B4%D1%8A%D0%B5%D0%BC%20%D0%BF%D0%BE%20%D0%BB%D0%B5%D1%81%D1%82%D0%BD%D0%B8%D1%86%D0%B5%3A%20%D0%BF%D0%BE%D0%B8%D1%81%D0%BA%22%22%22%0A%20%20%20%20return%20dfs%28n%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs%28n%29%0A%20%20%20%20print%28f%22%D0%9A%D0%BE%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D1%82%D0%B2%D0%BE%20%D1%81%D0%BF%D0%BE%D1%81%D0%BE%D0%B1%D0%BE%D0%B2%20%D0%BF%D0%BE%D0%B4%D0%BD%D1%8F%D1%82%D1%8C%D1%81%D1%8F%20%D0%BF%D0%BE%20%D0%BB%D0%B5%D1%81%D1%82%D0%BD%D0%B8%D1%86%D0%B5%20%D0%B8%D0%B7%20%7Bn%7D%20%D1%81%D1%82%D1%83%D0%BF%D0%B5%D0%BD%D0%B5%D0%B9%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Во весь экран &gt;</a></div></p>
</details>
<p>На рисунке 14-3 показано дерево рекурсии, возникающее при полном переборе. Для задачи <span class="arithmatex">\(dp[n]\)</span> глубина дерева рекурсии равна <span class="arithmatex">\(n\)</span> , а временная сложность равна <span class="arithmatex">\(O(2^n)\)</span> . Экспоненциальный рост взрывообразен: если подать на вход достаточно большое значение <span class="arithmatex">\(n\)</span> , ожидание станет очень долгим.</p>
<p><a class="glightbox" href="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Дерево рекурсии для подъема по лестнице" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_tree.png" /></a></p>
<p><img alt="Дерево рекурсии для подъема по лестнице" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_tree.png" /></p>
<p align="center"> Рисунок 14-3 &nbsp; Дерево рекурсии для подъема по лестнице </p>
<p>Если посмотреть на рисунок 14-3, то видно, что <strong>экспоненциальная временная сложность порождается "перекрывающимися подзадачами"</strong>. Например, <span class="arithmatex">\(dp[9]\)</span> раскладывается в <span class="arithmatex">\(dp[8]\)</span> и <span class="arithmatex">\(dp[7]\)</span> , а <span class="arithmatex">\(dp[8]\)</span> - в <span class="arithmatex">\(dp[7]\)</span> и <span class="arithmatex">\(dp[6]\)</span> ; обе ветви содержат подзадачу <span class="arithmatex">\(dp[7]\)</span> .</p>
@@ -5427,50 +5344,25 @@ dp[i] = dp[i-1] + dp[i-2]
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="cm">=begin</span>
<a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="cm">File: climbing_stairs_dfs_mem.rb</span>
<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="cm">Created Time: 2024-05-29</span>
<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="cm">Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)</span>
<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="cm">=end</span>
<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a>
<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="c1"># ## Поиск с мемоизацией ###</span>
<a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w"> </span><span class="c1"># dp[1] и dp[2] уже известны, вернуть их</span>
<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</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="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>
<a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w"> </span><span class="c1"># Если запись dp[i] существует, сразу вернуть ее</span>
<a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Поиск с мемоизацией ###</span>
<a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w"> </span><span class="c1"># dp[1] и dp[2] уже известны, вернуть их</span>
<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</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="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>
<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w"> </span><span class="c1"># Если запись dp[i] существует, сразу вернуть ее</span>
<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a>
<a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w"> </span><span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</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="n">mem</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dfs</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">2</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w"> </span><span class="c1"># Сохранить dp[i]</span>
<a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">count</span>
<a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="k">end</span>
<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a>
<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w"> </span><span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
<a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</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="n">mem</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dfs</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">2</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w"> </span><span class="c1"># Сохранить dp[i]</span>
<a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">count</span>
<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="k">end</span>
<a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a>
<a id="__codelineno-38-20" name="__codelineno-38-20" href="#__codelineno-38-20"></a><span class="cm">=begin</span>
<a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="cm">File: climbing_stairs_dfs_mem.rb</span>
<a id="__codelineno-38-22" name="__codelineno-38-22" href="#__codelineno-38-22"></a><span class="cm">Created Time: 2024-05-29</span>
<a id="__codelineno-38-23" name="__codelineno-38-23" href="#__codelineno-38-23"></a><span class="cm">Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)</span>
<a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a><span class="cm">=end</span>
<a id="__codelineno-38-25" name="__codelineno-38-25" href="#__codelineno-38-25"></a>
<a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a><span class="c1"># ## Поиск с мемоизацией ###</span>
<a id="__codelineno-38-27" name="__codelineno-38-27" href="#__codelineno-38-27"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
<a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w"> </span><span class="c1"># dp[1] и dp[2] уже известны, вернуть их</span>
<a id="__codelineno-38-29" name="__codelineno-38-29" href="#__codelineno-38-29"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</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="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>
<a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w"> </span><span class="c1"># Если запись dp[i] существует, сразу вернуть ее</span>
<a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
<a id="__codelineno-38-32" name="__codelineno-38-32" href="#__codelineno-38-32"></a>
<a id="__codelineno-38-33" name="__codelineno-38-33" href="#__codelineno-38-33"></a><span class="w"> </span><span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
<a id="__codelineno-38-34" name="__codelineno-38-34" href="#__codelineno-38-34"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</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="n">mem</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dfs</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">2</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
<a id="__codelineno-38-35" name="__codelineno-38-35" href="#__codelineno-38-35"></a><span class="w"> </span><span class="c1"># Сохранить dp[i]</span>
<a id="__codelineno-38-36" name="__codelineno-38-36" href="#__codelineno-38-36"></a><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">count</span>
<a id="__codelineno-38-37" name="__codelineno-38-37" href="#__codelineno-38-37"></a><span class="k">end</span>
<a id="__codelineno-38-38" name="__codelineno-38-38" href="#__codelineno-38-38"></a>
<a id="__codelineno-38-39" name="__codelineno-38-39" href="#__codelineno-38-39"></a><span class="c1"># ## Подъем по лестнице: поиск с мемоизацией ###</span>
<a id="__codelineno-38-40" name="__codelineno-38-40" href="#__codelineno-38-40"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dfs_mem</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-38-41" name="__codelineno-38-41" href="#__codelineno-38-41"></a><span class="w"> </span><span class="c1"># mem[i] хранит число способов подняться на i-ю ступень, -1 означает отсутствие записи</span>
<a id="__codelineno-38-42" name="__codelineno-38-42" href="#__codelineno-38-42"></a><span class="w"> </span><span class="n">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</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="mi">1</span><span class="p">)</span>
<a id="__codelineno-38-43" name="__codelineno-38-43" href="#__codelineno-38-43"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
<a id="__codelineno-38-44" name="__codelineno-38-44" href="#__codelineno-38-44"></a><span class="k">end</span>
<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="c1">### Подъем по лестнице: поиск с мемоизацией ###</span>
<a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dfs_mem</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w"> </span><span class="c1"># mem[i] хранит число способов подняться на i-ю ступень, -1 означает отсутствие записи</span>
<a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a><span class="w"> </span><span class="n">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</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="mi">1</span><span class="p">)</span>
<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
<a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="k">end</span>
</code></pre></div>
</div>
</div>
@@ -5481,7 +5373,7 @@ dp[i] = dp[i-1] + dp[i-2]
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28i%3A%20int%2C%20mem%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%D0%9F%D0%BE%D0%B8%D1%81%D0%BA%20%D1%81%20%D0%BC%D0%B5%D0%BC%D0%BE%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D0%B5%D0%B9%22%22%22%0A%20%20%20%20%23%20dp%5B1%5D%20%D0%B8%20dp%5B2%5D%20%D1%83%D0%B6%D0%B5%20%D0%B8%D0%B7%D0%B2%D0%B5%D1%81%D1%82%D0%BD%D1%8B%2C%20%D0%B2%D0%B5%D1%80%D0%BD%D1%83%D1%82%D1%8C%20%D0%B8%D1%85%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20%D0%95%D1%81%D0%BB%D0%B8%20%D0%B7%D0%B0%D0%BF%D0%B8%D1%81%D1%8C%20dp%5Bi%5D%20%D1%81%D1%83%D1%89%D0%B5%D1%81%D1%82%D0%B2%D1%83%D0%B5%D1%82%2C%20%D1%81%D1%80%D0%B0%D0%B7%D1%83%20%D0%B2%D0%B5%D1%80%D0%BD%D1%83%D1%82%D1%8C%20%D0%B5%D0%B5%0A%20%20%20%20if%20mem%5Bi%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201%2C%20mem%29%20%2B%20dfs%28i%20-%202%2C%20mem%29%0A%20%20%20%20%23%20%D0%A1%D0%BE%D1%85%D1%80%D0%B0%D0%BD%D0%B8%D1%82%D1%8C%20dp%5Bi%5D%0A%20%20%20%20mem%5Bi%5D%20%3D%20count%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs_mem%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%D0%9F%D0%BE%D0%B4%D1%8A%D0%B5%D0%BC%20%D0%BF%D0%BE%20%D0%BB%D0%B5%D1%81%D1%82%D0%BD%D0%B8%D1%86%D0%B5%3A%20%D0%BF%D0%BE%D0%B8%D1%81%D0%BA%20%D1%81%20%D0%BC%D0%B5%D0%BC%D0%BE%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D0%B5%D0%B9%22%22%22%0A%20%20%20%20%23%20mem%5Bi%5D%20%D1%85%D1%80%D0%B0%D0%BD%D0%B8%D1%82%20%D1%87%D0%B8%D1%81%D0%BB%D0%BE%20%D1%81%D0%BF%D0%BE%D1%81%D0%BE%D0%B1%D0%BE%D0%B2%20%D0%BF%D0%BE%D0%B4%D0%BD%D1%8F%D1%82%D1%8C%D1%81%D1%8F%20%D0%BD%D0%B0%20i-%D1%8E%20%D1%81%D1%82%D1%83%D0%BF%D0%B5%D0%BD%D1%8C%2C%20-1%20%D0%BE%D0%B7%D0%BD%D0%B0%D1%87%D0%B0%D0%B5%D1%82%20%D0%BE%D1%82%D1%81%D1%83%D1%82%D1%81%D1%82%D0%B2%D0%B8%D0%B5%20%D0%B7%D0%B0%D0%BF%D0%B8%D1%81%D0%B8%0A%20%20%20%20mem%20%3D%20%5B-1%5D%20%2A%20%28n%20%2B%201%29%0A%20%20%20%20return%20dfs%28n%2C%20mem%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs_mem%28n%29%0A%20%20%20%20print%28f%22%D0%9A%D0%BE%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D1%82%D0%B2%D0%BE%20%D1%81%D0%BF%D0%BE%D1%81%D0%BE%D0%B1%D0%BE%D0%B2%20%D0%BF%D0%BE%D0%B4%D0%BD%D1%8F%D1%82%D1%8C%D1%81%D1%8F%20%D0%BF%D0%BE%20%D0%BB%D0%B5%D1%81%D1%82%D0%BD%D0%B8%D1%86%D0%B5%20%D0%B8%D0%B7%20%7Bn%7D%20%D1%81%D1%82%D1%83%D0%BF%D0%B5%D0%BD%D0%B5%D0%B9%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Во весь экран &gt;</a></div></p>
</details>
<p>Как показано на рисунке 14-4, <strong>после введения мемоизации каждая перекрывающаяся подзадача вычисляется только один раз, и временная сложность оптимизируется до <span class="arithmatex">\(O(n)\)</span></strong> . Это огромный скачок в эффективности.</p>
<p><a class="glightbox" href="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Дерево рекурсии для поиска с мемоизацией" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png" /></a></p>
<p><img alt="Дерево рекурсии для поиска с мемоизацией" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png" /></p>
<p align="center"> Рисунок 14-4 &nbsp; Дерево рекурсии для поиска с мемоизацией </p>
<h2 id="1413-3">14.1.3 &nbsp; Метод 3: динамическое программирование<a class="headerlink" href="#1413-3" title="Permanent link">&para;</a></h2>
@@ -5706,25 +5598,19 @@ dp[i] = dp[i-1] + dp[i-2]
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="cm">=begin</span>
<a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="cm">File: climbing_stairs_dp.rb</span>
<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="cm">Created Time: 2024-05-29</span>
<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="cm">Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)</span>
<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="cm">=end</span>
<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a>
<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="c1"># ## Подъем по лестнице: динамическое программирование ###</span>
<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a>
<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w"> </span><span class="c1"># Инициализация таблицы dp для хранения решений подзадач</span>
<a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</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="mi">0</span><span class="p">)</span>
<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w"> </span><span class="c1"># Начальное состояние: заранее задать решения наименьших подзадач</span>
<a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">]</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="mi">2</span>
<a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w"> </span><span class="c1"># Переход состояний: постепенное решение больших подзадач через меньшие</span>
<a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</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="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</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="o">]</span><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-51-17" name="__codelineno-51-17" href="#__codelineno-51-17"></a>
<a id="__codelineno-51-18" name="__codelineno-51-18" href="#__codelineno-51-18"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">n</span><span class="o">]</span>
<a id="__codelineno-51-19" name="__codelineno-51-19" href="#__codelineno-51-19"></a><span class="k">end</span>
<div class="highlight"><span class="filename">climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Подъем по лестнице: динамическое программирование ###</span>
<a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a>
<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w"> </span><span class="c1"># Инициализация таблицы dp для хранения решений подзадач</span>
<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</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="mi">0</span><span class="p">)</span>
<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w"> </span><span class="c1"># Начальное состояние: заранее задать решения наименьших подзадач</span>
<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">]</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="mi">2</span>
<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w"> </span><span class="c1"># Переход состояний: постепенное решение больших подзадач через меньшие</span>
<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</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="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</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="o">]</span><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a>
<a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">n</span><span class="o">]</span>
<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="k">end</span>
</code></pre></div>
</div>
</div>
@@ -5735,7 +5621,7 @@ dp[i] = dp[i-1] + dp[i-2]
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20climbing_stairs_dp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%D0%9F%D0%BE%D0%B4%D1%8A%D0%B5%D0%BC%20%D0%BF%D0%BE%20%D0%BB%D0%B5%D1%81%D1%82%D0%BD%D0%B8%D1%86%D0%B5%3A%20%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5%20%D0%BF%D1%80%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%0A%20%20%20%20%23%20%D0%98%D0%BD%D0%B8%D1%86%D0%B8%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F%20%D1%82%D0%B0%D0%B1%D0%BB%D0%B8%D1%86%D1%8B%20dp%20%D0%B4%D0%BB%D1%8F%20%D1%85%D1%80%D0%B0%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F%20%D1%80%D0%B5%D1%88%D0%B5%D0%BD%D0%B8%D0%B9%20%D0%BF%D0%BE%D0%B4%D0%B7%D0%B0%D0%B4%D0%B0%D1%87%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28n%20%2B%201%29%0A%20%20%20%20%23%20%D0%9D%D0%B0%D1%87%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5%20%D1%81%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5%3A%20%D0%B7%D0%B0%D1%80%D0%B0%D0%BD%D0%B5%D0%B5%20%D0%B7%D0%B0%D0%B4%D0%B0%D1%82%D1%8C%20%D1%80%D0%B5%D1%88%D0%B5%D0%BD%D0%B8%D1%8F%20%D0%BD%D0%B0%D0%B8%D0%BC%D0%B5%D0%BD%D1%8C%D1%88%D0%B8%D1%85%20%D0%BF%D0%BE%D0%B4%D0%B7%D0%B0%D0%B4%D0%B0%D1%87%0A%20%20%20%20dp%5B1%5D%2C%20dp%5B2%5D%20%3D%201%2C%202%0A%20%20%20%20%23%20%D0%9F%D0%B5%D1%80%D0%B5%D1%85%D0%BE%D0%B4%20%D1%81%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B9%3A%20%D0%BF%D0%BE%D1%81%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5%20%D1%80%D0%B5%D1%88%D0%B5%D0%BD%D0%B8%D0%B5%20%D0%B1%D0%BE%D0%BB%D1%8C%D1%88%D0%B8%D1%85%20%D0%BF%D0%BE%D0%B4%D0%B7%D0%B0%D0%B4%D0%B0%D1%87%20%D1%87%D0%B5%D1%80%D0%B5%D0%B7%20%D0%BC%D0%B5%D0%BD%D1%8C%D1%88%D0%B8%D0%B5%0A%20%20%20%20for%20i%20in%20range%283%2C%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%20%3D%20dp%5Bi%20-%201%5D%20%2B%20dp%5Bi%20-%202%5D%0A%20%20%20%20return%20dp%5Bn%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dp%28n%29%0A%20%20%20%20print%28f%22%D0%9A%D0%BE%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D1%82%D0%B2%D0%BE%20%D1%81%D0%BF%D0%BE%D1%81%D0%BE%D0%B1%D0%BE%D0%B2%20%D0%BF%D0%BE%D0%B4%D0%BD%D1%8F%D1%82%D1%8C%D1%81%D1%8F%20%D0%BF%D0%BE%20%D0%BB%D0%B5%D1%81%D1%82%D0%BD%D0%B8%D1%86%D0%B5%20%D0%B8%D0%B7%20%7Bn%7D%20%D1%81%D1%82%D1%83%D0%BF%D0%B5%D0%BD%D0%B5%D0%B9%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Во весь экран &gt;</a></div></p>
</details>
<p>На рисунке 14-5 смоделирован процесс выполнения этого кода.</p>
<p><a class="glightbox" href="../intro_to_dynamic_programming.assets/climbing_stairs_dp.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Процесс динамического программирования для подъема по лестнице" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dp.png" /></a></p>
<p><img alt="Процесс динамического программирования для подъема по лестнице" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dp.png" /></p>
<p align="center"> Рисунок 14-5 &nbsp; Процесс динамического программирования для подъема по лестнице </p>
<p>Как и в backtracking, в динамическом программировании используется понятие "состояние" для обозначения некоторого этапа решения задачи; каждое состояние соответствует одной подзадаче и ее локально оптимальному решению. Например, в задаче о лестнице состояние определяется текущим номером ступени <span class="arithmatex">\(i\)</span> .</p>
@@ -5926,35 +5812,15 @@ dp[i] = dp[i-1] + dp[i-2]
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="cm">=begin</span>
<a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="cm">File: climbing_stairs_dp.rb</span>
<a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="cm">Created Time: 2024-05-29</span>
<a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="cm">Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)</span>
<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="cm">=end</span>
<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a>
<a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="c1"># ## Подъем по лестнице: динамическое программирование ###</span>
<a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
<a id="__codelineno-64-10" name="__codelineno-64-10" href="#__codelineno-64-10"></a>
<a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="w"> </span><span class="c1"># Инициализация таблицы dp для хранения решений подзадач</span>
<a id="__codelineno-64-12" name="__codelineno-64-12" href="#__codelineno-64-12"></a><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</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="mi">0</span><span class="p">)</span>
<a id="__codelineno-64-13" name="__codelineno-64-13" href="#__codelineno-64-13"></a><span class="w"> </span><span class="c1"># Начальное состояние: заранее задать решения наименьших подзадач</span>
<a id="__codelineno-64-14" name="__codelineno-64-14" href="#__codelineno-64-14"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">]</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="mi">2</span>
<a id="__codelineno-64-15" name="__codelineno-64-15" href="#__codelineno-64-15"></a><span class="w"> </span><span class="c1"># Переход состояний: постепенное решение больших подзадач через меньшие</span>
<a id="__codelineno-64-16" name="__codelineno-64-16" href="#__codelineno-64-16"></a><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</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="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</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="o">]</span><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-64-17" name="__codelineno-64-17" href="#__codelineno-64-17"></a>
<a id="__codelineno-64-18" name="__codelineno-64-18" href="#__codelineno-64-18"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">n</span><span class="o">]</span>
<a id="__codelineno-64-19" name="__codelineno-64-19" href="#__codelineno-64-19"></a><span class="k">end</span>
<a id="__codelineno-64-20" name="__codelineno-64-20" href="#__codelineno-64-20"></a>
<a id="__codelineno-64-21" name="__codelineno-64-21" href="#__codelineno-64-21"></a><span class="c1"># ## Подъем по лестнице: динамическое программирование с оптимизацией памяти ###</span>
<a id="__codelineno-64-22" name="__codelineno-64-22" href="#__codelineno-64-22"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp_comp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-64-23" name="__codelineno-64-23" href="#__codelineno-64-23"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
<a id="__codelineno-64-24" name="__codelineno-64-24" href="#__codelineno-64-24"></a>
<a id="__codelineno-64-25" name="__codelineno-64-25" href="#__codelineno-64-25"></a><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</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="mi">2</span>
<a id="__codelineno-64-26" name="__codelineno-64-26" href="#__codelineno-64-26"></a><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-64-27" name="__codelineno-64-27" href="#__codelineno-64-27"></a>
<a id="__codelineno-64-28" name="__codelineno-64-28" href="#__codelineno-64-28"></a><span class="w"> </span><span class="n">b</span>
<a id="__codelineno-64-29" name="__codelineno-64-29" href="#__codelineno-64-29"></a><span class="k">end</span>
<div class="highlight"><span class="filename">climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Подъем по лестнице: динамическое программирование с оптимизацией памяти ###</span>
<a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp_comp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
<a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a>
<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</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="mi">2</span>
<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a>
<a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w"> </span><span class="n">b</span>
<a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="k">end</span>
</code></pre></div>
</div>
</div>
@@ -5993,7 +5859,7 @@ aria-label="Нижний колонтитул"
>
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@@ -6103,7 +5969,7 @@ aria-label="Нижний колонтитул"
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К началу
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@@ -6111,46 +5977,6 @@ aria-label="Нижний колонтитул"
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