mirror of
https://github.com/mfocko/blog.git
synced 2024-12-23 21:01:29 +01:00
82 lines
No EOL
53 KiB
HTML
82 lines
No EOL
53 KiB
HTML
<!doctype html>
|
||
<html lang="en" dir="ltr" class="docs-wrapper plugin-docs plugin-id-algorithms docs-version-current docs-doc-page docs-doc-id-recursion/2023-08-17-pyramid-slide-down/bottom-up-dp" data-has-hydrated="false">
|
||
<head>
|
||
<meta charset="UTF-8">
|
||
<meta name="generator" content="Docusaurus v3.1.1">
|
||
<title data-rh="true">Bottom-up DP solution | mf</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://blog.mfocko.xyz/algorithms/recursion/pyramid-slide-down/bottom-up-dp/"><meta data-rh="true" property="og:locale" content="en"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-algorithms-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-algorithms-current"><meta data-rh="true" property="og:title" content="Bottom-up DP solution | mf"><meta data-rh="true" name="description" content="Bottom-up DP solution of the Pyramid Slide Down.
|
||
"><meta data-rh="true" property="og:description" content="Bottom-up DP solution of the Pyramid Slide Down.
|
||
"><link data-rh="true" rel="icon" href="/img/favicon.ico"><link data-rh="true" rel="canonical" href="https://blog.mfocko.xyz/algorithms/recursion/pyramid-slide-down/bottom-up-dp/"><link data-rh="true" rel="alternate" href="https://blog.mfocko.xyz/algorithms/recursion/pyramid-slide-down/bottom-up-dp/" hreflang="en"><link data-rh="true" rel="alternate" href="https://blog.mfocko.xyz/algorithms/recursion/pyramid-slide-down/bottom-up-dp/" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://0VXRFPR4QF-dsn.algolia.net" crossorigin="anonymous"><link rel="search" type="application/opensearchdescription+xml" title="mf" href="/opensearch.xml">
|
||
|
||
|
||
|
||
<link rel="alternate" type="application/rss+xml" href="/blog/rss.xml" title="mf RSS Feed">
|
||
<link rel="alternate" type="application/atom+xml" href="/blog/atom.xml" title="mf Atom Feed">
|
||
<link rel="alternate" type="application/json" href="/blog/feed.json" title="mf JSON Feed">
|
||
|
||
|
||
|
||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous"><link rel="stylesheet" href="/assets/css/styles.cc45d56f.css">
|
||
<script src="/assets/js/runtime~main.8ecd1e7a.js" defer="defer"></script>
|
||
<script src="/assets/js/main.ca43560f.js" defer="defer"></script>
|
||
</head>
|
||
<body class="navigation-with-keyboard">
|
||
<script>!function(){function t(t){document.documentElement.setAttribute("data-theme",t)}var e=function(){try{return new URLSearchParams(window.location.search).get("docusaurus-theme")}catch(t){}}()||function(){try{return localStorage.getItem("theme")}catch(t){}}();t(null!==e?e:"light")}(),function(){try{const c=new URLSearchParams(window.location.search).entries();for(var[t,e]of c)if(t.startsWith("docusaurus-data-")){var a=t.replace("docusaurus-data-","data-");document.documentElement.setAttribute(a,e)}}catch(t){}}()</script><div id="__docusaurus"><div role="region" aria-label="Skip to main content"><a class="skipToContent_fXgn" href="#__docusaurus_skipToContent_fallback">Skip to main content</a></div><nav aria-label="Main" class="navbar navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><button aria-label="Toggle navigation bar" aria-expanded="false" class="navbar__toggle clean-btn" type="button"><svg width="30" height="30" viewBox="0 0 30 30" aria-hidden="true"><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></button><a class="navbar__brand" href="/"><b class="navbar__title text--truncate">mf</b></a><div class="navbar__item dropdown dropdown--hoverable"><a href="#" aria-haspopup="true" aria-expanded="false" role="button" class="navbar__link">Additional FI MU materials</a><ul class="dropdown__menu"><li><a aria-current="page" class="dropdown__link dropdown__link--active" href="/algorithms/">Algorithms</a></li><li><a class="dropdown__link" href="/c/">C</a></li><li><a class="dropdown__link" href="/cpp/">C++</a></li></ul></div><a class="navbar__item navbar__link" href="/contributions/">Contributions</a><a class="navbar__item navbar__link" href="/talks/">Talks</a></div><div class="navbar__items navbar__items--right"><a class="navbar__item navbar__link" href="/blog/">Blog</a><div class="toggle_vylO colorModeToggle_DEke"><button class="clean-btn toggleButton_gllP toggleButtonDisabled_aARS" type="button" disabled="" title="Switch between dark and light mode (currently light mode)" aria-label="Switch between dark and light mode (currently light mode)" aria-live="polite"><svg viewBox="0 0 24 24" width="24" height="24" class="lightToggleIcon_pyhR"><path fill="currentColor" d="M12,9c1.65,0,3,1.35,3,3s-1.35,3-3,3s-3-1.35-3-3S10.35,9,12,9 M12,7c-2.76,0-5,2.24-5,5s2.24,5,5,5s5-2.24,5-5 S14.76,7,12,7L12,7z M2,13l2,0c0.55,0,1-0.45,1-1s-0.45-1-1-1l-2,0c-0.55,0-1,0.45-1,1S1.45,13,2,13z M20,13l2,0c0.55,0,1-0.45,1-1 s-0.45-1-1-1l-2,0c-0.55,0-1,0.45-1,1S19.45,13,20,13z M11,2v2c0,0.55,0.45,1,1,1s1-0.45,1-1V2c0-0.55-0.45-1-1-1S11,1.45,11,2z M11,20v2c0,0.55,0.45,1,1,1s1-0.45,1-1v-2c0-0.55-0.45-1-1-1C11.45,19,11,19.45,11,20z M5.99,4.58c-0.39-0.39-1.03-0.39-1.41,0 c-0.39,0.39-0.39,1.03,0,1.41l1.06,1.06c0.39,0.39,1.03,0.39,1.41,0s0.39-1.03,0-1.41L5.99,4.58z M18.36,16.95 c-0.39-0.39-1.03-0.39-1.41,0c-0.39,0.39-0.39,1.03,0,1.41l1.06,1.06c0.39,0.39,1.03,0.39,1.41,0c0.39-0.39,0.39-1.03,0-1.41 L18.36,16.95z M19.42,5.99c0.39-0.39,0.39-1.03,0-1.41c-0.39-0.39-1.03-0.39-1.41,0l-1.06,1.06c-0.39,0.39-0.39,1.03,0,1.41 s1.03,0.39,1.41,0L19.42,5.99z M7.05,18.36c0.39-0.39,0.39-1.03,0-1.41c-0.39-0.39-1.03-0.39-1.41,0l-1.06,1.06 c-0.39,0.39-0.39,1.03,0,1.41s1.03,0.39,1.41,0L7.05,18.36z"></path></svg><svg viewBox="0 0 24 24" width="24" height="24" class="darkToggleIcon_wfgR"><path fill="currentColor" d="M9.37,5.51C9.19,6.15,9.1,6.82,9.1,7.5c0,4.08,3.32,7.4,7.4,7.4c0.68,0,1.35-0.09,1.99-0.27C17.45,17.19,14.93,19,12,19 c-3.86,0-7-3.14-7-7C5,9.07,6.81,6.55,9.37,5.51z M12,3c-4.97,0-9,4.03-9,9s4.03,9,9,9s9-4.03,9-9c0-0.46-0.04-0.92-0.1-1.36 c-0.98,1.37-2.58,2.26-4.4,2.26c-2.98,0-5.4-2.42-5.4-5.4c0-1.81,0.89-3.42,2.26-4.4C12.92,3.04,12.46,3,12,3L12,3z"></path></svg></button></div><div class="navbarSearchContainer_Bca1"><button type="button" class="DocSearch DocSearch-Button" aria-label="Search"><span class="DocSearch-Button-Container"><svg width="20" height="20" class="DocSearch-Search-Icon" viewBox="0 0 20 20"><path d="M14.386 14.386l4.0877 4.0877-4.0877-4.0877c-2.9418 2.9419-7.7115 2.9419-10.6533 0-2.9419-2.9418-2.9419-7.7115 0-10.6533 2.9418-2.9419 7.7115-2.9419 10.6533 0 2.9419 2.9418 2.9419 7.7115 0 10.6533z" stroke="currentColor" fill="none" fill-rule="evenodd" stroke-linecap="round" stroke-linejoin="round"></path></svg><span class="DocSearch-Button-Placeholder">Search</span></span><span class="DocSearch-Button-Keys"></span></button></div></div></div><div role="presentation" class="navbar-sidebar__backdrop"></div></nav><div id="__docusaurus_skipToContent_fallback" class="main-wrapper mainWrapper_z2l0"><div class="docsWrapper_hBAB"><button aria-label="Scroll back to top" class="clean-btn theme-back-to-top-button backToTopButton_sjWU" type="button"></button><div class="docRoot_UBD9"><aside class="theme-doc-sidebar-container docSidebarContainer_YfHR"><div class="sidebarViewport_aRkj"><div class="sidebar_njMd"><nav aria-label="Docs sidebar" class="menu thin-scrollbar menu_SIkG"><ul class="theme-doc-sidebar-menu menu__list"><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-1 menu__list-item"><a class="menu__link" href="/algorithms/">Introduction</a></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist" aria-expanded="false" href="/algorithms/category/algorithms-and-correctness/">Algorithms and Correctness</a><button aria-label="Expand sidebar category 'Algorithms and Correctness'" type="button" class="clean-btn menu__caret"></button></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist" aria-expanded="false" href="/algorithms/category/asymptotic-notation-and-time-complexity/">Asymptotic Notation and Time Complexity</a><button aria-label="Expand sidebar category 'Asymptotic Notation and Time Complexity'" type="button" class="clean-btn menu__caret"></button></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--active" aria-expanded="true" href="/algorithms/category/recursion/">Recursion</a><button aria-label="Collapse sidebar category 'Recursion'" type="button" class="clean-btn menu__caret"></button></div><ul style="display:block;overflow:visible;height:auto" class="menu__list"><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-2 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist" aria-expanded="false" tabindex="0" href="/algorithms/recursion/karel/">Recursion and backtracking with Robot Karel</a><button aria-label="Expand sidebar category 'Recursion and backtracking with Robot Karel'" type="button" class="clean-btn menu__caret"></button></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-2 menu__list-item"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--active" aria-expanded="true" tabindex="0" href="/algorithms/recursion/pyramid-slide-down/">Introduction to dynamic programming</a><button aria-label="Collapse sidebar category 'Introduction to dynamic programming'" type="button" class="clean-btn menu__caret"></button></div><ul style="display:block;overflow:visible;height:auto" class="menu__list"><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="/algorithms/recursion/pyramid-slide-down/naive/">Naïve solution</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="/algorithms/recursion/pyramid-slide-down/greedy/">Greedy solution</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="/algorithms/recursion/pyramid-slide-down/top-down-dp/">Top-down DP solution</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link menu__link--active" aria-current="page" tabindex="0" href="/algorithms/recursion/pyramid-slide-down/bottom-up-dp/">Bottom-up DP solution</a></li></ul></li></ul></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist" aria-expanded="false" href="/algorithms/category/red-black-trees/">Red-Black Trees</a><button aria-label="Expand sidebar category 'Red-Black Trees'" type="button" class="clean-btn menu__caret"></button></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist" aria-expanded="false" href="/algorithms/category/graphs/">Graphs</a><button aria-label="Expand sidebar category 'Graphs'" type="button" class="clean-btn menu__caret"></button></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist" aria-expanded="false" href="/algorithms/category/paths-in-graphs/">Paths in Graphs</a><button aria-label="Expand sidebar category 'Paths in Graphs'" type="button" class="clean-btn menu__caret"></button></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist" aria-expanded="false" href="/algorithms/category/hash-tables/">Hash Tables</a><button aria-label="Expand sidebar category 'Hash Tables'" type="button" class="clean-btn menu__caret"></button></div></li></ul></nav><button type="button" title="Collapse sidebar" aria-label="Collapse sidebar" class="button button--secondary button--outline collapseSidebarButton_PEFL"><svg width="20" height="20" aria-hidden="true" class="collapseSidebarButtonIcon_kv0_"><g fill="#7a7a7a"><path d="M9.992 10.023c0 .2-.062.399-.172.547l-4.996 7.492a.982.982 0 01-.828.454H1c-.55 0-1-.453-1-1 0-.2.059-.403.168-.551l4.629-6.942L.168 3.078A.939.939 0 010 2.528c0-.548.45-.997 1-.997h2.996c.352 0 .649.18.828.45L9.82 9.472c.11.148.172.347.172.55zm0 0"></path><path d="M19.98 10.023c0 .2-.058.399-.168.547l-4.996 7.492a.987.987 0 01-.828.454h-3c-.547 0-.996-.453-.996-1 0-.2.059-.403.168-.551l4.625-6.942-4.625-6.945a.939.939 0 01-.168-.55 1 1 0 01.996-.997h3c.348 0 .649.18.828.45l4.996 7.492c.11.148.168.347.168.55zm0 0"></path></g></svg></button></div></div></aside><main class="docMainContainer_TBSr"><div class="container padding-top--md padding-bottom--lg"><div class="row"><div class="col docItemCol_VOVn"><div class="docItemContainer_Djhp"><article><nav class="theme-doc-breadcrumbs breadcrumbsContainer_Z_bl" aria-label="Breadcrumbs"><ul class="breadcrumbs" itemscope="" itemtype="https://schema.org/BreadcrumbList"><li class="breadcrumbs__item"><a aria-label="Home page" class="breadcrumbs__link" href="/"><svg viewBox="0 0 24 24" class="breadcrumbHomeIcon_YNFT"><path d="M10 19v-5h4v5c0 .55.45 1 1 1h3c.55 0 1-.45 1-1v-7h1.7c.46 0 .68-.57.33-.87L12.67 3.6c-.38-.34-.96-.34-1.34 0l-8.36 7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 1-1z" fill="currentColor"></path></svg></a></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="/algorithms/category/recursion/"><span itemprop="name">Recursion</span></a><meta itemprop="position" content="1"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="/algorithms/recursion/pyramid-slide-down/"><span itemprop="name">Introduction to dynamic programming</span></a><meta itemprop="position" content="2"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link" itemprop="name">Bottom-up DP solution</span><meta itemprop="position" content="3"></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><h1>Bottom-up dynamic programming</h1>
|
||
<p>If you try to think in depth about the top-down DP solution, you might notice
|
||
that the <em>core</em> of it stands on caching the calculations that have been already
|
||
done on the lower “levels” of the pyramid. Our bottom-up implementation will be
|
||
using this fact!</p>
|
||
<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_BuS1"><p>As I have said in the <em>top-down DP</em> section, it is the easiest way to implement
|
||
DP (unless the cached function has complicated parameters, in that case it might
|
||
get messy).</p><p>Bottom-up dynamic programming can be more effective, but may be more complicated
|
||
to implement right from the beginning.</p></div></div>
|
||
<p>Let's see how we can implement it:</p>
|
||
<div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token keyword" style="color:hsl(301, 63%, 40%)">public</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">static</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// In the beginning we declare new array. At this point it is easier to just</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// work with the one dimension, i.e. just allocating the space for the rows.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> slideDowns </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">new</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// Bottom row gets just copied, there's nothing else to do… It's the base</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// case.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> slideDowns</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">Arrays</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token function" style="color:hsl(221, 87%, 60%)">copyOf</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// Then we need to propagate the found slide downs for each of the levels</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// above.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">for</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> y </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">2</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"> y </span><span class="token operator" style="color:hsl(221, 87%, 60%)">>=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">--</span><span class="token plain">y</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// We start by copying the values lying in the row we're processing.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// They get included in the final sum and we need to allocate the space</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// for the precalculated slide downs anyways.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">Arrays</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token function" style="color:hsl(221, 87%, 60%)">copyOf</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">y</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">y</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// At this we just need to “fetch” the partial results from “neighbours”</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">for</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> x </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"> x </span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token plain"> row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">++</span><span class="token plain">x</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// We look under our position, since we expect the rows to get</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// shorter, we can safely assume such position exists.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> under </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> slideDowns</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">y </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">x</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// Then we have a look to the right, such position doesn't have to</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// exist, e.g. on the right edge, so we validate the index, and if</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// it doesn't exist, we just assign minimum of the ‹int› which makes</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// sure that it doesn't get picked in the ‹Math.max()› call.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> toRight </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> x </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token plain"> slideDowns</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">y </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">?</span><span class="token plain"> slideDowns</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">y </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">x </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">Integer</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token constant" style="color:hsl(35, 99%, 36%)">MIN_VALUE</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// Finally we add the best choice at this point.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">x</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+=</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">Math</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token function" style="color:hsl(221, 87%, 60%)">max</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">under</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> toRight</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// And save the row we've just calculated partial results for to the</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// “table”.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> slideDowns</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">y</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// At the end we can find our seeked slide down at the top cell.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> slideDowns</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p>I've tried to explain the code as much as possible within the comments, since it
|
||
might be more beneficial to see right next to the “offending” lines.</p>
|
||
<p>As you can see, in this approach we go from the other side<sup><a href="#user-content-fn-1" id="user-content-fnref-1" data-footnote-ref="true" aria-describedby="footnote-label">1</a></sup>, the bottom of
|
||
the pyramid and propagate the partial results up.</p>
|
||
<div class="theme-admonition theme-admonition-info admonition_xJq3 alert alert--info"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>How is this different from the <em>greedy</em> solution???</div><div class="admonitionContent_BuS1"><p>If you try to compare them, you might find a very noticable difference. The
|
||
greedy approach is going from the top to the bottom without <strong>any</strong> knowledge of
|
||
what's going on below. On the other hand, bottom-up DP is going from the bottom
|
||
(<em>DUH…</em>) and <strong>propagates</strong> the partial results to the top. The propagation is
|
||
what makes sure that at the top I don't choose the best <strong>local</strong> choice, but
|
||
the best <strong>overall</strong> result I can achieve.</p></div></div>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="time-complexity">Time complexity<a href="#time-complexity" class="hash-link" aria-label="Direct link to Time complexity" title="Direct link to Time complexity"></a></h2>
|
||
<p>Time complexity of this solution is rather simple. We allocate an array for the
|
||
rows and then for each row, we copy it and adjust the partial results. Doing
|
||
this we get:</p>
|
||
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>r</mi><mi>o</mi><mi>w</mi><mi>s</mi><mo>+</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(rows + 2n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">ro</span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span>
|
||
<p>Of course, this is an upper bound, since we iterate through the bottom row only
|
||
once.</p>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="memory-complexity">Memory complexity<a href="#memory-complexity" class="hash-link" aria-label="Direct link to Memory complexity" title="Direct link to Memory complexity"></a></h2>
|
||
<p>We're allocating an array for the pyramid <strong>again</strong> for our partial results, so
|
||
we get:</p>
|
||
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span>
|
||
<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_BuS1"><p>If we were writing this in C++ or Rust, we could've avoided that, but not
|
||
really.</p><p>C++ would allow us to <strong>copy</strong> the pyramid rightaway into the parameter, so we
|
||
would be able to directly change it. However it's still a copy, even though we
|
||
don't need to allocate anything ourselves. It's just implicitly done for us.</p><p>Rust is more funny in this case. If the pyramids weren't used after the call of
|
||
<code>longest_slide_down</code>, it would simply <strong>move</strong> them into the functions. If they
|
||
were used afterwards, the compiler would force you to either borrow it, or
|
||
<em>clone-and-move</em> for the function.</p><hr><p>Since we're doing it in Java, we get a reference to the <em>original</em> array and we
|
||
can't do whatever we want with it.</p></div></div>
|
||
<h1>Summary</h1>
|
||
<p>And we've finally reached the end. We have seen 4 different “solutions”<sup><a href="#user-content-fn-2" id="user-content-fnref-2" data-footnote-ref="true" aria-describedby="footnote-label">2</a></sup> of
|
||
the same problem using different approaches. Different approaches follow the
|
||
order in which you might come up with them, each approach influences its
|
||
successor and represents the way we can enhance the existing implementation.</p>
|
||
<hr>
|
||
<div class="theme-admonition theme-admonition-info admonition_xJq3 alert alert--info"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>source</div><div class="admonitionContent_BuS1"><p>You can find source code referenced in the text
|
||
<a href="/files/algorithms/recursion/pyramid-slide-down.tar.gz" target="_blank" rel="noopener noreferrer">here</a>.</p></div></div>
|
||
<section data-footnotes="true" class="footnotes"><h2 class="anchor anchorWithStickyNavbar_LWe7 sr-only" id="footnote-label">Footnotes<a href="#footnote-label" class="hash-link" aria-label="Direct link to Footnotes" title="Direct link to Footnotes"></a></h2>
|
||
<ol>
|
||
<li id="user-content-fn-1">
|
||
<p>definitely not an RHCP reference <!-- -->😉 <a href="#user-content-fnref-1" data-footnote-backref="" aria-label="Back to reference 1" class="data-footnote-backref">↩</a></p>
|
||
</li>
|
||
<li id="user-content-fn-2">
|
||
<p>one was not correct, thus the quotes <a href="#user-content-fnref-2" data-footnote-backref="" aria-label="Back to reference 2" class="data-footnote-backref">↩</a></p>
|
||
</li>
|
||
</ol>
|
||
</section></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-tags-row row margin-bottom--sm"><div class="col"><b>Tags:</b><ul class="tags_jXut padding--none margin-left--sm"><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/java/">java</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/dynamic-programming/">dynamic-programming</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/bottom-up-dp/">bottom-up-dp</a></li></ul></div></div><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://github.com/mfocko/blog/tree/main/algorithms/04-recursion/2023-08-17-pyramid-slide-down/04-bottom-up-dp.md" target="_blank" rel="noopener noreferrer" class="theme-edit-this-page"><svg fill="currentColor" height="20" width="20" viewBox="0 0 40 40" class="iconEdit_Z9Sw" aria-hidden="true"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="col lastUpdated_vwxv"><span class="theme-last-updated">Last updated<!-- --> on <b><time datetime="2023-08-17T00:00:00.000Z">Aug 17, 2023</time></b></span></div></div></footer></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/algorithms/recursion/pyramid-slide-down/top-down-dp/"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Top-down DP solution</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/algorithms/category/red-black-trees/"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Red-Black Trees</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#time-complexity" class="table-of-contents__link toc-highlight">Time complexity</a></li><li><a href="#memory-complexity" class="table-of-contents__link toc-highlight">Memory complexity</a></li></ul></div></div></div></div></main></div></div></div><footer class="footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="col footer__col"><div class="footer__title">Git</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitHub<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://gitlab.com/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitLab<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://git.mfocko.xyz/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">Gitea (self-hosted)<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li></ul></div><div class="col footer__col"><div class="footer__title">Social #1</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://www.linkedin.com/in/mfocko/" target="_blank" rel="noopener noreferrer" class="footer__link-item">LinkedIn<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://fosstodon.org/@m4tt_314" target="_blank" rel="noopener noreferrer" class="footer__link-item">Fosstodon<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://hachyderm.io/@m4tt_314" target="_blank" rel="noopener noreferrer" class="footer__link-item">Hachyderm.io<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li></ul></div><div class="col footer__col"><div class="footer__title">Social #2</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://twitter.com/m4tt_314" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://twitch.tv/m4tt_314" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitch<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://ko-fi.com/m4tt_314" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ko-fi<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2024 Matej Focko.</div></div></div></footer></div>
|
||
</body>
|
||
</html> |