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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/paths-in-graphs/"><span itemprop="name">Paths in Graphs</span></a><meta itemprop="position" content="1"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link" itemprop="name">From BF to A*</span><meta itemprop="position" content="2"></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"><header><h1>From BF to A*</h1></header><h2 class="anchor anchorWithStickyNavbar_LWe7" id="intro">Intro<a href="#intro" class="hash-link" aria-label="Direct link to Intro" title="Direct link to Intro"></a></h2>
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<p>We will delve into the details and ideas of the most common path-finding
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algorithms. For the purpose of demonstrating some “features” of the improved
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algorithms, we will use a 2D map with some rules that will allow us to show cons
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and pros of the shown algorithms.</p>
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<p>Let's have a look at the example map:</p>
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<div class="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-text 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 plain">#############</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#..#..*.*.**#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">##***.....**#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#..########.#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#...###...#.#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#..#...##.#.#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#..#.*.#..#.#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#....#....#.#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">########*.*.#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#...........#</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#############</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>
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<p>We can see three different kinds of cells:</p>
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<ol>
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<li><code>#</code> which represent walls, that cannot be entered at all</li>
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<li><code>*</code> which represent vortices that can be entered at the cost of 5 coins</li>
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<li><code>.</code> which represent normal cells that can be entered for 1 coin (which is the
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base price of moving around the map)</li>
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</ol>
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<p>Let's dissect a specific position on the map to get a better grasp of the rules:</p>
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<div class="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-text 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 plain"> .</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain">#S*</span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> .</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>
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<p>We are standing in the cell marked with <code>S</code> and we have the following options</p>
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<ul>
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<li>move to the north (<code>.</code>) with the cost of 1 coin,</li>
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<li>move to the west (<code>#</code>) <strong>is not</strong> allowed because of the wall,</li>
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<li>move to the east (<code>*</code>) is allowed with the cost of 5 coins, and finally</li>
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<li>move to the south (<code>.</code>) with the cost of 1 coin.</li>
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</ul>
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<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>info</div><div class="admonitionContent_BuS1"><p>Further on I will follow the same scheme for marking cells with an addition of
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<code>D</code> to denote the <em>destination</em> to which we will be finding the shortest path.</p></div></div>
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<h2 class="anchor anchorWithStickyNavbar_LWe7" id="boilerplate">Boilerplate<a href="#boilerplate" class="hash-link" aria-label="Direct link to Boilerplate" title="Direct link to Boilerplate"></a></h2>
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<p>For working with this map I have prepared a basic structure for the graph in C++
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that will abstract some of the internal workings of our map, namely:</p>
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<ul>
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<li>remembers the costs of moving around</li>
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<li>provides a simple function that returns price for moving <strong>directly</strong> between
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two positions on the map</li>
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<li>allows us to print the map out, just in case we'd need some adjustments to be
|
||
made</li>
|
||
</ul>
|
||
<p>We can see the <code>graph</code> header here:</p>
|
||
<div class="language-cpp 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-cpp 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 macro property directive-hash" style="color:hsl(5, 74%, 59%)">#</span><span class="token macro property directive keyword" style="color:hsl(301, 63%, 40%)">ifndef</span><span class="token macro property" style="color:hsl(5, 74%, 59%)"> </span><span class="token macro property expression" style="color:hsl(5, 74%, 59%)">_GRAPH_HPP</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 macro property directive-hash" style="color:hsl(5, 74%, 59%)">#</span><span class="token macro property directive keyword" style="color:hsl(301, 63%, 40%)">define</span><span class="token macro property" style="color:hsl(5, 74%, 59%)"> </span><span class="token macro property macro-name" style="color:hsl(5, 74%, 59%)">_GRAPH_HPP</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 macro property directive-hash" style="color:hsl(5, 74%, 59%)">#</span><span class="token macro property directive keyword" style="color:hsl(301, 63%, 40%)">include</span><span class="token macro property" style="color:hsl(5, 74%, 59%)"> </span><span class="token macro property string" style="color:hsl(119, 34%, 47%)"><cmath></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 macro property directive-hash" style="color:hsl(5, 74%, 59%)">#</span><span class="token macro property directive keyword" style="color:hsl(301, 63%, 40%)">include</span><span class="token macro property" style="color:hsl(5, 74%, 59%)"> </span><span class="token macro property string" style="color:hsl(119, 34%, 47%)"><limits></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 macro property directive-hash" style="color:hsl(5, 74%, 59%)">#</span><span class="token macro property directive keyword" style="color:hsl(301, 63%, 40%)">include</span><span class="token macro property" style="color:hsl(5, 74%, 59%)"> </span><span class="token macro property string" style="color:hsl(119, 34%, 47%)"><ostream></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 macro property directive-hash" style="color:hsl(5, 74%, 59%)">#</span><span class="token macro property directive keyword" style="color:hsl(301, 63%, 40%)">include</span><span class="token macro property" style="color:hsl(5, 74%, 59%)"> </span><span class="token macro property string" style="color:hsl(119, 34%, 47%)"><utility></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 macro property directive-hash" style="color:hsl(5, 74%, 59%)">#</span><span class="token macro property directive keyword" style="color:hsl(301, 63%, 40%)">include</span><span class="token macro property" style="color:hsl(5, 74%, 59%)"> </span><span class="token macro property string" style="color:hsl(119, 34%, 47%)"><vector></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 keyword" style="color:hsl(301, 63%, 40%)">using</span><span class="token plain"> vertex_t </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">pair</span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</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"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token operator" style="color:hsl(221, 87%, 60%)">></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 keyword" style="color:hsl(301, 63%, 40%)">struct</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">graph</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 function" style="color:hsl(221, 87%, 60%)">graph</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">vector</span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">vector</span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">char</span><span class="token operator" style="color:hsl(221, 87%, 60%)">>></span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> map</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 function" style="color:hsl(221, 87%, 60%)">map</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">map</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 function" style="color:hsl(221, 87%, 60%)">_height</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token generic-function function" style="color:hsl(221, 87%, 60%)">static_cast</span><span class="token generic-function generic class-name operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token generic-function generic class-name keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token generic-function generic class-name operator" style="color:hsl(221, 87%, 60%)">></span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">map</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token function" style="color:hsl(221, 87%, 60%)">size</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 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 function" style="color:hsl(221, 87%, 60%)">_width</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">map</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token function" style="color:hsl(221, 87%, 60%)">empty</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><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 plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</span><span class="token plain"> </span><span class="token generic-function function" style="color:hsl(221, 87%, 60%)">static_cast</span><span class="token generic-function generic class-name operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token generic-function generic class-name keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token generic-function generic class-name operator" style="color:hsl(221, 87%, 60%)">></span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">map</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 function" style="color:hsl(221, 87%, 60%)">size</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><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 keyword" style="color:hsl(301, 63%, 40%)">static</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">unreachable</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><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%)">int</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> UNREACHABLE</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 keyword" style="color:hsl(301, 63%, 40%)">static</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">normal_cost</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><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%)">int</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> NORMAL_COST</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 keyword" style="color:hsl(301, 63%, 40%)">static</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">vortex_cost</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><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%)">int</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> VORTEX_COST</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" 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 keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">cost</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</span><span class="token plain"> vertex_t</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> u</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</span><span class="token plain"> vertex_t</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> v</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</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 keyword" style="color:hsl(301, 63%, 40%)">int</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 keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">ux</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> uy</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"> u</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 keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">vx</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> vy</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"> v</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 keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> md </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token function" style="color:hsl(221, 87%, 60%)">abs</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">ux </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> vx</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"> std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token function" style="color:hsl(221, 87%, 60%)">abs</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">uy </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> vy</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 keyword" style="color:hsl(301, 63%, 40%)">switch</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">md</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%)">// ‹u = v›; staying on the same 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%)">case</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</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"> </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><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%)">// ‹u› and ‹v› are 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%)">case</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</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%)">break</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%)">// ‹u› and ‹v› are not neighbouring cells</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%)">default</span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</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"> UNREACHABLE</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%)">// boundary check</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%)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">vy </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 plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"> vy </span><span class="token operator" style="color:hsl(221, 87%, 60%)">>=</span><span class="token plain"> _height </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"> vx </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 plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"> vx </span><span class="token operator" style="color:hsl(221, 87%, 60%)">>=</span><span class="token plain"> _width</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 keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> UNREACHABLE</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 keyword" style="color:hsl(301, 63%, 40%)">switch</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">map</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">vy</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">vx</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><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 keyword" style="color:hsl(301, 63%, 40%)">case</span><span class="token plain"> </span><span class="token char" style="color:hsl(119, 34%, 47%)">'#'</span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</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"> UNREACHABLE</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 keyword" style="color:hsl(301, 63%, 40%)">case</span><span class="token plain"> </span><span class="token char" style="color:hsl(119, 34%, 47%)">'*'</span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</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"> VORTEX_COST</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 keyword" style="color:hsl(301, 63%, 40%)">default</span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</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"> NORMAL_COST</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"> </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 keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">width</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><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</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 keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> _width</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 keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">height</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><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</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 keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> _height</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 keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">has</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</span><span class="token plain"> vertex_t</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> v</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</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 keyword" style="color:hsl(301, 63%, 40%)">bool</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 keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token plain"> </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"> 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"> v</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 keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> </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 plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)"><=</span><span class="token plain"> y </span><span class="token operator" style="color:hsl(221, 87%, 60%)">&&</span><span class="token plain"> y </span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token plain"> _height</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 punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token number" style="color:hsl(35, 99%, 36%)">0</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 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"> _width</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 keyword" style="color:hsl(301, 63%, 40%)">friend</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">ostream</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%)">operator</span><span class="token operator" style="color:hsl(221, 87%, 60%)"><<</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">ostream</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> os</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</span><span class="token plain"> graph</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> g</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 keyword" style="color:hsl(301, 63%, 40%)">private</span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">vector</span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">vector</span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">char</span><span class="token operator" style="color:hsl(221, 87%, 60%)">>></span><span class="token plain"> map</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 keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> _height</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> _width</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 keyword" style="color:hsl(301, 63%, 40%)">const</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"> UNREACHABLE </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token class-name" style="color:hsl(35, 99%, 36%)">numeric_limits</span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token operator" style="color:hsl(221, 87%, 60%)">></span><span class="token double-colon 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 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 comment" style="color:hsl(230, 4%, 64%)">// XXX: modify here to change the price of entering the vortex</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%)">const</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"> VORTEX_COST </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%)">5</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 keyword" style="color:hsl(301, 63%, 40%)">const</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"> NORMAL_COST </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 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">std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">ostream</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%)">operator</span><span class="token operator" style="color:hsl(221, 87%, 60%)"><<</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:hsl(119, 34%, 47%)">::</span><span class="token plain">ostream</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> os</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">const</span><span class="token plain"> graph</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> g</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 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%)">const</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">auto</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&</span><span class="token plain"> row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">:</span><span class="token plain"> g</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">map</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 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%)">const</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">char</span><span class="token plain"> cell </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><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"> os </span><span class="token operator" style="color:hsl(221, 87%, 60%)"><<</span><span class="token plain"> cell</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"> os </span><span class="token operator" style="color:hsl(221, 87%, 60%)"><<</span><span class="token plain"> </span><span class="token string" style="color:hsl(119, 34%, 47%)">"\n"</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 keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> os</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 macro property directive-hash" style="color:hsl(5, 74%, 59%)">#</span><span class="token macro property directive keyword" style="color:hsl(301, 63%, 40%)">endif</span><span class="token macro property" style="color:hsl(5, 74%, 59%)"> </span><span class="token macro property comment" style="color:hsl(230, 4%, 64%)">/* _GRAPH_HPP */</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" 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<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 code</div><div class="admonitionContent_BuS1"><p>You can find all the source code referenced in this series
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<a href="/files/algorithms/paths/bf-to-astar.tar.gz" target="_blank" rel="noopener noreferrer">here</a>.</p></div></div>
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<p>Let's finally start with some 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/brute-force/">brute force</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/bellman-ford/">bellman ford</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/dijkstra/">dijkstra</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/a-star/">a star</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/index.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="2024-01-01T00:00:00.000Z">Jan 1, 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/category/paths-in-graphs/"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Paths in Graphs</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/algorithms/paths/bf-to-astar/bf/"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">BF</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="#boilerplate" class="table-of-contents__link toc-highlight">Boilerplate</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" 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