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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"><a class="breadcrumbs__link" itemprop="item" href="/algorithms/paths/bf-to-astar/"><span itemprop="name">From BF to A*</span></a><meta itemprop="position" content="2"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link" itemprop="name">A* algorithm</span><meta itemprop="position" content="3"></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>A* algorithm</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>Let's start by the recap of what we've achieved so far:</p>
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<ol>
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<li>We have implemented a naïve brute-force algorithm that tries to relax paths
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as long as there are any paths to be relaxed.</li>
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<li>Then we have fixed an issue caused by negative loops that can result in
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a non-terminating run of our brute-force method. At this moment we have made
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some small arguments why are bounding is enough and doesn't prevent any
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shortest path to <em>not be</em> discovered.</li>
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<li>Finally we have converted our bounded brute-force algorithm into the
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Bellman-Ford algorithm.</li>
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<li>We have mentioned the worst-case time complexity of our bounded naïve
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approach and also the Bellman-Ford algorithm. Our worst-case depended on the
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fact that we assumed the worst possible ordering of the relaxations. However
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we could also try to relax in the most ideal ordering which could result in a
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faster algorithm and that's how we got to the Dijkstra's algorithm.</li>
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</ol>
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<p>Now the question is, could we improve the Dijkstra's algorithm to get even
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better results? And the answer is <em>maybe</em>!</p>
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<p>Dijkstra's algorithm chooses the next cheapest vertex for relaxing. This is good
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as long as there is no additional information. However, imagine a roadmap of
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some country. If you're in the middle of the map and you want to go south, it
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doesn't make much sense for you to go to the north (in the opposite direction),
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but a little bit might make sense, so that you can switch to highway and go much
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faster.</p>
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<p>The important question here is how to <em>influence</em> the algorithm, so that it does
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choose the path that <em>makes more sense</em> rather than the one that costs the
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least.</p>
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<h2 class="anchor anchorWithStickyNavbar_LWe7" id="a-description">A* description<a href="#a-description" class="hash-link" aria-label="Direct link to A* description" title="Direct link to A* description"></a></h2>
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<p>The <em>A* algorithm</em> can be considered a modification of Dijkstra's algorithm. The
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cost is still the same, we cannot change it, right? However when we pick the
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vertices from the heap, we can influence the order by some <em>heuristic</em>. In this
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case, we introduce a function that can suggest how feasible the vertex is.</p>
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<h2 class="anchor anchorWithStickyNavbar_LWe7" id="roadmap-heuristic">Roadmap heuristic<a href="#roadmap-heuristic" class="hash-link" aria-label="Direct link to Roadmap heuristic" title="Direct link to Roadmap heuristic"></a></h2>
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<p>Let's have a look at the heuristic we could use for the roadmap example. There
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are roads (the edges) and towns (the vertices). Cost could be an average time to
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travel the road. What heuristic could we use to influence our algorithm to
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choose a better ordering of the vertices when relaxing?</p>
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<p>In the former example we've said that it doesn't make much sense to go in the
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opposite direction than our goal is… We could choose the distance from our goal
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as the heuristic, e.g. right now we're 100 km away from our goal, using this
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road makes us 50 km away and using the other road we will be 200 km away.</p>
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<h2 class="anchor anchorWithStickyNavbar_LWe7" id="heuristic-for-our-map">Heuristic for our map<a href="#heuristic-for-our-map" class="hash-link" aria-label="Direct link to Heuristic for our map" title="Direct link to Heuristic for our map"></a></h2>
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<p>Our map is a bit simpler, but we can use a very similar principle. We will use
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the <em>Manhattan distance</em>, which is defined in a following way:</p>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>a</mi></msub><mo>−</mo><msub><mi>x</mi><mi>b</mi></msub><mi mathvariant="normal">∣</mi><mo>+</mo><mi mathvariant="normal">∣</mi><msub><mi>y</mi><mi>a</mi></msub><mo>−</mo><msub><mi>y</mi><mi>b</mi></msub><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\vert x_a - x_b \vert + \vert y_a - y_b \vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span></span>
|
||
<p>Since we cannot move in diagonals, it makes sense to maintain the distance in
|
||
the actual steps from the goal.</p>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="passing-the-heuristic">Passing the heuristic<a href="#passing-the-heuristic" class="hash-link" aria-label="Direct link to Passing the heuristic" title="Direct link to Passing the heuristic"></a></h2>
|
||
<p>In our case, when we're using C++, we can just template the function that will
|
||
calculate the shortest path and pass the heuristic as a parameter.</p>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="implementation">Implementation<a href="#implementation" class="hash-link" aria-label="Direct link to Implementation" title="Direct link to Implementation"></a></h2>
|
||
<p>Actual implementation is very easy once we have the Dijkstra's algorithm:</p>
|
||
<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">astar</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> graph</span><span class="token operator" style="color:#393A34">&</span><span class="token plain"> g</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> vertex_t</span><span class="token operator" style="color:#393A34">&</span><span class="token plain"> source</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token operator" style="color:#393A34">&</span><span class="token plain"> h</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token operator" style="color:#393A34">-></span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token keyword" style="color:#00009f">int</span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// make sure that ‹source› exists</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token function" style="color:#d73a49">assert</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">g</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">has</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">source</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// initialize the distances</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token keyword" style="color:#00009f">int</span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token function" style="color:#d73a49">distances</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> g</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">height</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">vector</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">g</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">width</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> graph</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">unreachable</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// initialize the visited</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">vector</span><span class="token operator" style="color:#393A34"><</span><span class="token keyword" style="color:#00009f">bool</span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token function" style="color:#d73a49">visited</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">g</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">height</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">vector</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">g</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">width</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token boolean" style="color:#36acaa">false</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// ‹source› destination denotes the beginning where the cost is 0</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">sx</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> sy</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> source</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> distances</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">sy</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">sx</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> pqueue_t priority_queue</span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain">std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">make_pair</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">0</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">h</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">source</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> source</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">}</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">optional</span><span class="token operator" style="color:#393A34"><</span><span class="token plain">pqueue_item_t</span><span class="token operator" style="color:#393A34">></span><span class="token plain"> item</span><span class="token punctuation" style="color:#393A34">{</span><span class="token punctuation" style="color:#393A34">}</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">while</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">item </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">popq</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">priority_queue</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">cost</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> u</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">*</span><span class="token plain">item</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> y</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> u</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// we have already found the shortest path</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">if</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">visited</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">y</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">continue</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> visited</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">y</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token boolean" style="color:#36acaa">true</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">for</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token operator" style="color:#393A34">&</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">dx</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> dy</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">:</span><span class="token plain"> DIRECTIONS</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> v </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 function" style="color:#d73a49">make_pair</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> dx</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> y </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> dy</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> cost </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> g</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">cost</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">u</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> v</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// if we can move to the cell and it's better, relax¹ it and update queue</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">if</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">cost </span><span class="token operator" style="color:#393A34">!=</span><span class="token plain"> graph</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">unreachable</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">&&</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> distances</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">y</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> cost </span><span class="token operator" style="color:#393A34"><</span><span class="token plain"> distances</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">y </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> dy</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> dx</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> distances</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">y </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> dy</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> dx</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> distances</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">y</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> cost</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token function" style="color:#d73a49">pushq</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">priority_queue</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">make_pair</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">distances</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">y </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> dy</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> dx</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">h</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">v</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> v</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> distances</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</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>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="running-on-our-map">Running on our map<a href="#running-on-our-map" class="hash-link" aria-label="Direct link to Running on our map" title="Direct link to Running on our map"></a></h2>
|
||
<p>For this algorithm I will also show the example of a call:</p>
|
||
<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">distances </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">astar</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">g</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">make_pair</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">9</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">[</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token operator" style="color:#393A34">&</span><span class="token plain"> u</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> y</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> u</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">abs</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">1</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">-</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">abs</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">7</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">-</span><span class="token plain"> y</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">std</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">cout </span><span class="token operator" style="color:#393A34"><<</span><span class="token plain"> </span><span class="token string" style="color:#e3116c">"[A*] Cost: "</span><span class="token plain"> </span><span class="token operator" style="color:#393A34"><<</span><span class="token plain"> distances</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">7</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34"><<</span><span class="token plain"> </span><span class="token string" style="color:#e3116c">"\n"</span><span class="token punctuation" style="color:#393A34">;</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>First argument to the function is the graph itself. Second argument is the
|
||
source vertex where we start. And finally the lambda returns
|
||
<em>Manhattan distance</em> to the goal.</p>
|
||
<p>And we get the following result:</p>
|
||
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">Normal cost: 1</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Vortex cost: 5</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Graph:</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#..*.*.**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">##***.....**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..########.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#...###...#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#...##.#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#.*.#..#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#D...#....#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">########*.*.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#S..........#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Finite BF] Cost: 22</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Bellman-Ford] Cost: 22</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Dijkstra] Cost: 22</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[A*] Cost: 22</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="comparison">Comparison<a href="#comparison" class="hash-link" aria-label="Direct link to Comparison" title="Direct link to Comparison"></a></h2>
|
||
<p>Now you may wonder how does it compare to the previous algorithms. Supposedly it
|
||
should be faster. Let's add counters and debugging output when we update
|
||
distance to our goal. And now if we run our code, we get the following output:</p>
|
||
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">Normal cost: 1</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Vortex cost: 5</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Graph:</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#..*.*.**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">##***.....**#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..########.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#...###...#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#...##.#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#..#.*.#..#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#D...#....#.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">########*.*.#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#S..........#</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">#############</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Relaxing path to goal in 40. relaxation</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Relaxing path to goal in 68. relaxation</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Relaxing path to goal in 89. relaxation</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Finite BF] Cost: 22</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Relaxing path to goal in 40. relaxation</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Relaxing path to goal in 68. relaxation</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Relaxing path to goal in 89. relaxation</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Bellman-Ford] Cost: 22</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Relaxing path to goal in 41. iteration</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[Dijkstra] Cost: 22</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">Relaxing path to goal in 31. iteration</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">[A*] Cost: 22</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p>From the output we can easily deduce that for both brute-force and Bellman-Ford,
|
||
which are in our case identical, we actually relax three times and for the last
|
||
time in the 89th iteration.</p>
|
||
<p>Dijkstra's algorithm manages to find the shortest path to our goal already in
|
||
the 41st iteration.</p>
|
||
<p>And finally after introducing some heuristic, we could find the shortest path
|
||
in the 31st iteration.</p>
|
||
<div class="theme-admonition theme-admonition-danger admonition_xJq3 alert alert--danger"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>danger</div><div class="admonitionContent_BuS1"><p>Please keep in mind that choosing bad heuristic can actually lead to worse
|
||
results than using no heuristic at all.</p></div></div>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="summary">Summary<a href="#summary" class="hash-link" aria-label="Direct link to Summary" title="Direct link to Summary"></a></h2>
|
||
<p>And there we have it. We have made our way from the brute-force algorithm all
|
||
the way to more optimal ones. Hopefully we could notice how the small
|
||
improvements of the already existing algorithms made them much better.</p></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-tags-row row margin-bottom--sm"><div class="col"><b>Tags:</b><ul class="tags_jXut padding--none margin-left--sm"><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/cpp/">cpp</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/dynamic-programming/">dynamic programming</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/astar/">astar</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/03-astar.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 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