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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>
<p>We will delve into the details and ideas of the most common path-finding
algorithms. For the purpose of demonstrating some “features” of the improved
algorithms, we will use a 2D map with some rules that will allow us to show cons
and pros of the shown algorithms.</p>
<p>Let&#x27;s have a look at the example map:</p>
<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>
<p>We can see three different kinds of cells:</p>
<ol>
<li><code>#</code> which represent walls, that cannot be entered at all</li>
<li><code>*</code> which represent vortices that can be entered at the cost of 5 coins</li>
<li><code>.</code> which represent normal cells that can be entered for 1 coin (which is the
base price of moving around the map)</li>
</ol>
<p>Let&#x27;s dissect a specific position on the map to get a better grasp of the rules:</p>
<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>
<p>We are standing in the cell marked with <code>S</code> and we have the following options</p>
<ul>
<li>move to the north (<code>.</code>) with the cost of 1 coin,</li>
<li>move to the west (<code>#</code>) <strong>is not</strong> allowed because of the wall,</li>
<li>move to the east (<code>*</code>) is allowed with the cost of 5 coins, and finally</li>
<li>move to the south (<code>.</code>) with the cost of 1 coin.</li>
</ul>
<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
<code>D</code> to denote the <em>destination</em> to which we will be finding the shortest path.</p></div></div>
<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>
<p>For working with this map I have prepared a basic structure for the graph in C++
that will abstract some of the internal workings of our map, namely:</p>
<ul>
<li>remembers the costs of moving around</li>
<li>provides a simple function that returns price for moving <strong>directly</strong> between
two positions on the map</li>
<li>allows us to print the map out, just in case we&#x27;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%)">&lt;cmath&gt;</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%)">&lt;limits&gt;</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%)">&lt;ostream&gt;</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%)">&lt;utility&gt;</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%)">&lt;vector&gt;</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%)">&lt;</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%)">&gt;</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%)">&lt;</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%)">&lt;</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">char</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&gt;&gt;</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&amp;</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%)">&lt;</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%)">&gt;</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%)">&lt;</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%)">&gt;</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%)">-&gt;</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%)">-&gt;</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%)">-&gt;</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%)">&amp;</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%)">&amp;</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%)">-&gt;</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%)">&lt;</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%)">&gt;=</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%)">&lt;</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%)">&gt;=</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%)">&#x27;#&#x27;</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%)">&#x27;*&#x27;</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%)">-&gt;</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%)">-&gt;</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%)">&amp;</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%)">-&gt;</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%)">&lt;=</span><span class="token plain"> y </span><span class="token operator" style="color:hsl(221, 87%, 60%)">&amp;&amp;</span><span class="token plain"> y </span><span class="token operator" style="color:hsl(221, 87%, 60%)">&lt;</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%)">&amp;&amp;</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%)">&lt;=</span><span class="token plain"> x </span><span class="token operator" style="color:hsl(221, 87%, 60%)">&amp;&amp;</span><span class="token plain"> x </span><span class="token operator" style="color:hsl(221, 87%, 60%)">&lt;</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%)">&amp;</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%)">&lt;&lt;</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%)">&amp;</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%)">&amp;</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%)">&lt;</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%)">&lt;</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">char</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&gt;&gt;</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%)">&lt;</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token operator" style="color:hsl(221, 87%, 60%)">&gt;</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%)">&amp;</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%)">&lt;&lt;</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%)">&amp;</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%)">&amp;</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%)">&amp;</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%)">&lt;&lt;</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%)">&lt;&lt;</span><span class="token plain"> </span><span class="token string" style="color:hsl(119, 34%, 47%)">&quot;\n&quot;</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 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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
<a href="/files/algorithms/paths/bf-to-astar.tar.gz" target="_blank" rel="noopener noreferrer">here</a>.</p></div></div>
<p>Let&#x27;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" 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