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<html lang="en" dir="ltr" class="docs-wrapper plugin-docs plugin-id-algorithms docs-version-current docs-doc-page docs-doc-id-paths/2024-01-01-bf-to-astar/index" data-has-hydrated="false">
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<title data-rh="true">From BF to A* | mf</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://blog.mfocko.xyz/algorithms/paths/bf-to-astar/"><meta data-rh="true" property="og:locale" content="en"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-algorithms-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-algorithms-current"><meta data-rh="true" property="og:title" content="From BF to A* | mf"><meta data-rh="true" name="description" content="Figuring out shortest-path problem from the BF to the A* algorithm.
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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-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#..*.*.**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">##***.....**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..########.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#...###...#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#...##.#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#.*.#..#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#....#....#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">########*.*.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#...........#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span></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-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain"> .</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#S*</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> .</span><br></span></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
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made</li>
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</ul>
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<p>We can see the <code>graph</code> header here:</p>
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<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">ifndef</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression" style="color:#36acaa">_GRAPH_HPP</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">_GRAPH_HPP</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property string" style="color:#e3116c"><cmath></span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property string" style="color:#e3116c"><limits></span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property string" style="color:#e3116c"><ostream></span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property string" style="color:#e3116c"><utility></span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property string" style="color:#e3116c"><vector></span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">using</span><span class="token plain"> vertex_t </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">pair</span><span class="token operator" style="color:#393A34"><</span><span
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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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