mirror of
https://github.com/mfocko/blog.git
synced 2024-11-23 05:23:47 +01:00
145 lines
85 KiB
HTML
145 lines
85 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-paths/2024-01-01-bf-to-astar/dijkstra" data-has-hydrated="false">
|
|||
|
|
<head>
|
|||
|
|
<meta charset="UTF-8">
|
|||
|
|
<meta name="generator" content="Docusaurus v3.0.1">
|
|||
|
|
<title data-rh="true">Dijkstra's algorithm | 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/paths/bf-to-astar/dijkstra/"><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="Dijkstra's algorithm | mf"><meta data-rh="true" name="description" content="Moving from Bellman-Ford into the Dijsktra's algorithm.
|
|||
|
|
"><meta data-rh="true" property="og:description" content="Moving from Bellman-Ford into the Dijsktra's algorithm.
|
|||
|
|
"><link data-rh="true" rel="icon" href="/img/favicon.ico"><link data-rh="true" rel="canonical" href="https://blog.mfocko.xyz/algorithms/paths/bf-to-astar/dijkstra/"><link data-rh="true" rel="alternate" href="https://blog.mfocko.xyz/algorithms/paths/bf-to-astar/dijkstra/" hreflang="en"><link data-rh="true" rel="alternate" href="https://blog.mfocko.xyz/algorithms/paths/bf-to-astar/dijkstra/" 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.e1ac7597.css">
|
|||
|
|
<script src="/assets/js/runtime~main.65095f6f.js" defer="defer"></script>
|
|||
|
|
<script src="/assets/js/main.2a166133.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.
|
|||
|
|
<p>Let's rewind back to the small argument in the previous post about the fact that
|
|||
|
|
we can safely bound the amount of iterations with relaxations being done.</p>
|
|||
|
|
<p>We have said that assuming the worst-case scenario (bad order of relaxations) we
|
|||
|
|
<strong>need</strong> at most <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>V</mi><mi mathvariant="normal">∣</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\vert V \vert - 1</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">∣</span><span class="mord mathnormal" style="margin-right:0.22222em">V</span><span class="mord">∣</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:0.6444em"></span><span class="mord">1</span></span></span></span> iterations over all edges. We've used that
|
|||
|
|
to our advantage to <em>bound</em> the iterations instead of the <code>do-while</code> loop that
|
|||
|
|
was a risk given the possibility of the infinite loop (when negative loops are
|
|||
|
|
present in the graph).</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>We could've possibly used both <em>boolean flag</em> to denote that some relaxation has
|
|||
|
|
happened and the upper bound of iterations, for some graphs that would result in
|
|||
|
|
faster termination.</p><p>Using only the upper bound we try to relax edges even though we can't.</p></div></div>
|
|||
|
|
<p>Now the question arises, could we leverage this somehow in a different way? What
|
|||
|
|
if we used it to improve the algorithm instead of just bounding the iterations?
|
|||
|
|
Would that be even possible?</p>
|
|||
|
|
<p><strong>Yes, it would!</strong> And that's when <em>Dijkstra's algorithm</em> comes in.</p>
|
|||
|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="dijkstras-algorithm">Dijkstra's algorithm<a href="#dijkstras-algorithm" class="hash-link" aria-label="Direct link to Dijkstra's algorithm" title="Direct link to Dijkstra's algorithm"></a></h2>
|
|||
|
|
<p>I'll start with a well-known meme about Dijkstra's algorithm:
|
|||
|
|
<img loading="lazy" alt="Dijkstra&#39;s algorithm meme" src="/assets/images/dijkstra-meme-405d6b8dcc7aec5846fef402abfa8317.jpg" width="960" height="724" class="img_ev3q"></p>
|
|||
|
|
<p>And then follow up on that with the actual backstory from Dijkstra himself:</p>
|
|||
|
|
<blockquote>
|
|||
|
|
<p>What is the shortest way to travel from Rotterdam to Groningen, in general:
|
|||
|
|
from given city to given city. It is the algorithm for the shortest path,
|
|||
|
|
which I designed in about twenty minutes. One morning I was shopping in
|
|||
|
|
Amsterdam with my young fiancée, and tired, we sat down on the café terrace to
|
|||
|
|
drink a cup of coffee and I was just thinking about whether I could do this,
|
|||
|
|
and I then designed the algorithm for the shortest path. As I said, it was
|
|||
|
|
a twenty-minute invention. In fact, it was published in '59, three years
|
|||
|
|
later. The publication is still readable, it is, in fact, quite nice. One of
|
|||
|
|
the reasons that it is so nice was that I designed it without pencil and
|
|||
|
|
paper. I learned later that one of the advantages of designing without pencil
|
|||
|
|
and paper is that you are almost forced to avoid all avoidable complexities.
|
|||
|
|
Eventually, that algorithm became to my great amazement, one of the
|
|||
|
|
cornerstones of my fame.</p>
|
|||
|
|
<p>— Edsger Dijkstra, in an interview with Philip L. Frana, Communications of the
|
|||
|
|
ACM, 2001</p>
|
|||
|
|
</blockquote>
|
|||
|
|
<div class="theme-admonition theme-admonition-caution admonition_xJq3 alert alert--warning"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Precondition</div><div class="admonitionContent_BuS1"><p>As our own naïve algorithm, Dijkstra's algorithm has a precondition that places
|
|||
|
|
a requirement of <em>no edges with negative weights</em> in the graph. This
|
|||
|
|
precondition is required because of the nature of the algorithm that requires
|
|||
|
|
monotonically non-decreasing changes in the costs of shortest paths.</p></div></div>
|
|||
|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="short-description">Short description<a href="#short-description" class="hash-link" aria-label="Direct link to Short description" title="Direct link to Short description"></a></h2>
|
|||
|
|
<p>Let's have a brief look at the pseudocode taken from the Wikipedia:</p>
|
|||
|
|
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">function Dijkstra(Graph, source):</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> for each vertex v in Graph.Vertices:</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> dist[v] ← INFINITY</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> prev[v] ← UNDEFINED</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> add v to Q</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> dist[source] ← 0</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> while Q is not empty:</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> u ← vertex in Q with min dist[u]</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> remove u from Q</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> for each neighbor v of u still in Q:</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> alt ← dist[u] + Graph.Edges(u, v)</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> if alt < dist[v]:</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> dist[v] ← alt</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> prev[v] ← u</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> return dist[], prev[]</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>Dijkstra's algorithm works in such way that it always tries to find the shortest
|
|||
|
|
paths from a vertex to which it already has a shortest path. This may result in
|
|||
|
|
finding the shortest path to another vertex, or at least some path, till further
|
|||
|
|
relaxation.</p>
|
|||
|
|
<p>Given that we need to <strong>always</strong> choose the <em>cheapest</em> vertex, we can use a min
|
|||
|
|
heap to our advantage which can further improve the time complexity of the
|
|||
|
|
algorithm.</p>
|
|||
|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="used-techniques">Used techniques<a href="#used-techniques" class="hash-link" aria-label="Direct link to Used techniques" title="Direct link to Used techniques"></a></h2>
|
|||
|
|
<p>This algorithm leverages the <em>dynamic programming</em> technique that has already
|
|||
|
|
been mentioned with regards to the <em>Bellman-Ford</em> algorithm and also <em>greedy</em>
|
|||
|
|
technique. Let's talk about them both!</p>
|
|||
|
|
<p><em>Dynamic programming</em> technique comes from the fact that we are continuously
|
|||
|
|
building on top of the shortest paths that we have found so far. We slowly build
|
|||
|
|
the shortest paths from the given source vertex to all other vertices that are
|
|||
|
|
reachable.</p>
|
|||
|
|
<p><em>Greedy</em> technique is utilized in such way that Dijkstra's algorithm always
|
|||
|
|
improves the shortest paths from the vertex that is the closest, i.e. it tries
|
|||
|
|
extending the shortest path to some vertex by appending an edge, such extended
|
|||
|
|
path may (or may not) be the shortest path to another vertex.</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>Greedy algorithms</div><div class="admonitionContent_BuS1"><p><em>Greedy algorithms</em> are algorithms that choose the most optimal action
|
|||
|
|
<strong>at the moment</strong>.</p></div></div>
|
|||
|
|
<p>The reason why the algorithm requires no edges with negative weights comes from
|
|||
|
|
the fact that it's greedy. By laying the requirement of non-negative weights in
|
|||
|
|
the graph we are guaranteed that at any given moment of processing outgoing
|
|||
|
|
edges from a vertex, we already have a shortest path to the given vertex. This
|
|||
|
|
means that either this is the shortest path, or there is some other vertex that
|
|||
|
|
may have a higher cost, but the outgoing edge compensates for it.</p>
|
|||
|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="implementation">Implementation<a href="#implementation" class="hash-link" aria-label="Direct link to Implementation" title="Direct link to Implementation"></a></h2>
|
|||
|
|
<p>Firstly we need to have some priority queue wrappers. C++ itself offers
|
|||
|
|
functions that can be used for maintaining max heaps. They also have generalized
|
|||
|
|
version with any ordering, in our case we need reverse ordering, because we need
|
|||
|
|
the min heap.</p>
|
|||
|
|
<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token keyword" style="color:#00009f">using</span><span class="token plain"> pqueue_item_t </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">pair</span><span class="token operator" style="color:#393A34"><</span><span class="token keyword" style="color:#00009f">int</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> vertex_t</span><span class="token operator" style="color:#393A34">></span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">using</span><span class="token plain"> pqueue_t </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token plain">pqueue_item_t</span><span class="token operator" style="color:#393A34">></span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">pushq</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">pqueue_t</span><span class="token operator" style="color:#393A34">&</span><span class="token plain"> q</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> pqueue_item_t v</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">-></span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">void</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> q</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">push_back</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">v</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">push_heap</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">q</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">begin</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> q</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">end</span><spa
|
|||
|
|
<p>And now we can finally move to the actual implementation of the Dijkstra's
|
|||
|
|
algorithm:</p>
|
|||
|
|
<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">dijkstra</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> graph</span><span class="token operator" style="color:#393A34">&</span><span class="token plain"> g</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> vertex_t</span><span class="token operator" style="color:#393A34">&</span><span class="token plain"> source</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token operator" style="color:#393A34">-></span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token keyword" style="color:#00009f">int</span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// make sure that ‹source› exists</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token function" style="color:#d73a49">assert</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">g</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">has</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">source</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// initialize the distances</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token keyword" style="color:#00009f">int</span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token function" style="color:#d73a49">distances</span><span class="token punctuation" style="color:#393A34">(</span><span
|
|||
|
|
<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>The time complexity of Dijkstra's algorithm differs based on the backing data
|
|||
|
|
structure.</p>
|
|||
|
|
<p>The original implementation doesn't leverage the heap which results in
|
|||
|
|
repetitive <em>look up</em> of the “closest” vertex, hence we get the following
|
|||
|
|
worst-case time complexity in the <em>Bachmann-Landau</em> notation:</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="normal">Θ</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>V</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(\vert V \vert^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.22222em">V</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>
|
|||
|
|
<p>If we turn our attention to the backing data structure, we always want the
|
|||
|
|
“cheapest” vertex, that's why we can use the min heap, given that we use
|
|||
|
|
Fibonacci heap we can achieve the following amortized time complexity:</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 mathvariant="normal">∣</mi><mi>E</mi><mi mathvariant="normal">∣</mi><mo>+</mo><mi mathvariant="normal">∣</mi><mi>V</mi><mi mathvariant="normal">∣</mi><mo>⋅</mo><mi>log</mi><mo></mo><mrow><mi mathvariant="normal">∣</mi><mi>V</mi><mi mathvariant="normal">∣</mi></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(\vert E \vert + \vert V \vert \cdot \log{\vert V \vert})</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">∣</span><span class="mord mathnormal" style="margin-right:0.05764em">E</span><span class="mord">∣</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">∣</span><span class="mord mathnormal" style="margin-right:0.22222em">V</span><span class="mord">∣</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="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.22222em">V</span><span class="mord">∣</span></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>Fibonacci heap</div><div class="admonitionContent_BuS1"><p>Fibonacci heap is known as the heap that provides <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(1)</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">Θ</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> <strong>amortized</strong>
|
|||
|
|
insertion and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(\log{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="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span><span class="mclose">)</span></span></span></span> <strong>amortized</strong> removal of the top (either
|
|||
|
|
min or max).</p></div></div>
|
|||
|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="running-the-dijkstra">Running the Dijkstra<a href="#running-the-dijkstra" class="hash-link" aria-label="Direct link to Running the Dijkstra" title="Direct link to Running the Dijkstra"></a></h2>
|
|||
|
|
<p>Let's run our code:</p>
|
|||
|
|
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">Normal cost: 1</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Vortex cost: 5</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Graph:</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#..*.*.**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">##***.....**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..########.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#...###...#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#...##.#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#.*.#..#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#D...#....#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">########*.*.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#S..........#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Finite BF] Cost: 22</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Bellman-Ford] Cost: 22</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Dijkstra] Cost: 22</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>OK, so it seems to be working just fine. Now the question arises:</p>
|
|||
|
|
<blockquote>
|
|||
|
|
<p>What happens when we have negative weights in our graph?</p>
|
|||
|
|
</blockquote>
|
|||
|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="busting-the-myth-about-looping-dijkstra">Busting the myth about looping Dijkstra<a href="#busting-the-myth-about-looping-dijkstra" class="hash-link" aria-label="Direct link to Busting the myth about looping Dijkstra" title="Direct link to Busting the myth about looping Dijkstra"></a></h2>
|
|||
|
|
<p>One of the very common misconception about Dijkstra's algorithm is that it loops
|
|||
|
|
infinitely when you have negative weights or loops in the graph. Well, if we use
|
|||
|
|
our <em>propelling vortices</em>, not only we have the negative weights, but also the
|
|||
|
|
negative loops. Let's run our code! Our first naïve approach was actually
|
|||
|
|
looping:</p>
|
|||
|
|
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">Normal cost: 1</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Vortex cost: -1</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Graph:</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#..*.*.**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">##***.....**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..########.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#...###...#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#...##.#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#.*.#..#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#D...#....#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">########*.*.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#S..........#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Finite BF] Cost: -240</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Bellman-Ford] Found a negative loop</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Bellman-Ford] Cost: -240</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Dijkstra] Cost: 14</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>Well, it definitely doesn't loop. How much does <code>14</code> make sense is a different
|
|||
|
|
matter.</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>Variations</div><div class="admonitionContent_BuS1"><p>There are multiple variations of the Dijkstra's algorithm. You <strong>can</strong> implement
|
|||
|
|
it in such way that with negative weights or loops it loops infinitely, but it
|
|||
|
|
can be countered. In our case we keep the track of the vertices that already got
|
|||
|
|
a shortest path established via the <code>visited</code>, that's how even multiple entries
|
|||
|
|
for one vertex in the heap are not an issue.</p></div></div>
|
|||
|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="summary">Summary<a href="#summary" class="hash-link" aria-label="Direct link to Summary" title="Direct link to Summary"></a></h2>
|
|||
|
|
<p>Now we have an algorithm for finding the shortest path that is faster than our
|
|||
|
|
original naïve brute-force or Bellman-Ford. However we need to keep in mind its
|
|||
|
|
requirement of no negative weights for correct functioning.</p>
|
|||
|
|
<p>You can also see how we used our thought process of figuring out the worst-case
|
|||
|
|
time complexity for the naïve or Bellman-Ford algorithm to improve the original
|
|||
|
|
path-finding algorithms.</p></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/cpp/">cpp</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/greedy/">greedy</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/dijkstra/">dijkstra</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/11-paths/2024-01-01-bf-to-astar/02-dijkstra.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="2024-01-03T00:00:00.000Z">Jan 3, 2024</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/paths/bf-to-astar/bf/"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">BF</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/algorithms/paths/bf-to-astar/astar/"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">A* algorithm</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="#intro" class="table-of-contents__link toc-highlight">Intro</a></li><li><a href="#dijkstras-algorithm" class="table-of-contents__link toc-highlight">Dijkstra's algorithm</a></li><li><a href="#short-description" class="table-of-contents__link toc-highlight">Short description</a></li><li><a href="#used-techniques" class="table-of-contents__link toc-highlight">Used techniques</a></li><li><a href="#implementation" class="table-of-contents__link toc-highlight">Implementation</a></li><li><a href="#time-complexity" class="table-of-contents__link toc-highlight">Time complexity</a></li><li><a href="#running-the-dijkstra" class="table-of-contents__link toc-highlight">Running the Dijkstra</a></li><li><a href="#busting-the-myth-about-looping-dijkstra" class="table-of-contents__link toc-highlight">Busting the myth about looping Dijkstra</a></li><li><a href="#summary" class="table-of-contents__link toc-highlight">Summary</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="
|
|||
|
|
</body>
|
|||
|
|
</html>
|