mirror of
https://github.com/mfocko/blog.git
synced 2024-11-22 21:13:47 +01:00
31 lines
32 KiB
HTML
31 lines
32 KiB
HTML
|
<!doctype html>
|
|||
|
|
<html lang="en" dir="ltr" class="docs-wrapper docs-doc-page docs-version-current plugin-docs plugin-id-ib002 docs-doc-id-graphs/bfs-tree">
|
|||
|
|
<head>
|
|||
|
|
<meta charset="UTF-8">
|
|||
|
|
<meta name="generator" content="Docusaurus v2.4.1">
|
|||
|
|
<title data-rh="true">Distance boundaries from BFS tree on undirected graphs | 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/ib002/graphs/bfs-tree"><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-ib002-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-ib002-current"><meta data-rh="true" property="og:title" content="Distance boundaries from BFS tree on undirected graphs | mf"><meta data-rh="true" name="description" content="Short explanation of distance boundaries deduced from a BFS tree.
|
|||
|
|
"><meta data-rh="true" property="og:description" content="Short explanation of distance boundaries deduced from a BFS tree.
|
|||
|
|
"><link data-rh="true" rel="icon" href="/img/favicon.ico"><link data-rh="true" rel="canonical" href="https://blog.mfocko.xyz/ib002/graphs/bfs-tree"><link data-rh="true" rel="alternate" href="https://blog.mfocko.xyz/ib002/graphs/bfs-tree" hreflang="en"><link data-rh="true" rel="alternate" href="https://blog.mfocko.xyz/ib002/graphs/bfs-tree" 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.8480cb83.css">
|
|||
|
|
<link rel="preload" href="/assets/js/runtime~main.be793c46.js" as="script">
|
|||
|
|
<link rel="preload" href="/assets/js/main.9283b308.js" as="script">
|
|||
|
|
</head>
|
|||
|
|
<body class="navigation-with-keyboard">
|
|||
|
|
<script>!function(){function t(t){document.documentElement.setAttribute("data-theme",t)}var e=function(){var t=null;try{t=new URLSearchParams(window.location.search).get("docusaurus-theme")}catch(t){}return t}()||function(){var t=null;try{t=localStorage.getItem("theme")}catch(t){}return t}();t(null!==e?e:"light")}()</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="/ib002/">IB002: Algorithms</a></li><li><a class="dropdown__link" href="/pb071/">PB071: C</a></li><li><a class="dropdown__link" href="/pb161/">PB161: 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="searchBox_ZlJk"><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_
|
|||
|
|
<img loading="lazy" src="/assets/images/bfs_graph_dark-f7a3a78eaf9de049469b4c64e0712867.png#gh-dark-mode-only" width="252" height="539" class="img_ev3q"></p><p>We run BFS from the vertex <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">a</span></span></span></span></span> and obtain the following BFS tree:</p><p><img loading="lazy" src="/assets/images/bfs_tree_light-61d6723c3c587d565b6280b8b8eca211.png#gh-light-mode-only" width="275" height="347" class="img_ev3q">
|
|||
|
|
<img loading="lazy" src="/assets/images/bfs_tree_dark-34f32262c6e4ffc14983b3ebf9a2f5a9.png#gh-dark-mode-only" width="275" height="347" class="img_ev3q"></p><p>Let's consider pair of vertices <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">e</span></span></span></span></span> and <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">h</span></span></span></span></span>. For them we can safely lay, from the BFS tree, following properties:</p><ul><li>lower bound: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span></span></li><li>upper bound: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">4</span></span></span></span></span></li></ul><p>By having a look at the graph we started from, we can see that we have a path ‹<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo separator="true">,</mo><mi>j</mi><mo separator="true">,</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">e, j, h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.05724em">j</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">h</span></span></span></span></span>› that has a length 2. Apart from that we can also notice there is another path from <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">e</span></span></span></span></span> to <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">h</span></span></span></span></span> and that is ‹<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo separator="true">,</mo><mi>a</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi
|
|||
|
|
<img loading="lazy" src="/assets/images/bfs_graph_with_additional_edge_dark-c2da6c052b067785e877b4654a13f328.png#gh-dark-mode-only" width="252" height="539" class="img_ev3q"></p><p>Okay, so we have a graph that breaks the rule we have laid. However, we need to run BFS to obtain the new BFS tree, since we have changed the graph.</p><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><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_S0QG"><p>Do we need to run BFS after <strong>every</strong> change?</p><p>I am leaving that as an exercise ;)</p></div></div><p><img loading="lazy" src="/assets/images/bfs_tree_with_additional_edge_light-93df97e1f9f1883467248532809374d0.png#gh-light-mode-only" width="371" height="347" class="img_ev3q">
|
|||
|
|
<img loading="lazy" src="/assets/images/bfs_tree_with_additional_edge_dark-127aa5b00988d4569669a92f5d841dbf.png#gh-dark-mode-only" width="371" height="347" class="img_ev3q"></p><p>Oops, we have gotten a new BFS tree, that has a height difference of 1.</p><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><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_S0QG"><p>Try to think about a way this can be generalized for shortening of minimal length 3 to minimal length 2 ;)</p></div></div></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="/ib002/tags/graphs">graphs</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/bfs">bfs</a></li></ul></div></div><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://gitlab.com/mfocko/blog/tree/main/ib002/10-graphs/2022-04-30-bfs-tree.md" target="_blank" rel="noreferrer noopener" 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="2022-04-30T00:00:00.000Z">Apr 30, 2022</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="/ib002/graphs/iterative-and-iterators"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Iterative algorithms via iterators</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="#introduction" class="table-of-contents__link toc-highlight">Introduction</a></li><li><a href="#lower-bound" class="table-of-contents__link toc-highlight">Lower bound</a></li><li><a href="#proof-by-contradiction" class="table-of-contents__link toc-highlight">Proof by contradiction</a></li></ul></div></div></div></div></main></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__
|
|||
|
|
<script src="/assets/js/runtime~main.be793c46.js"></script>
|
|||
|
|
<script src="/assets/js/main.9283b308.js"></script>
|
|||
|
|
</body>
|
|||
|
|
</html>
|