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<h2 id="Contents">Contents<a class="anchor-link" href="#Contents"></a></h2><p>In this notebook, we will learn</p>
<ul>
<li>How to parallelize the Floyd's algorithm</li>
<li>How to parallelize Floyd's algorithm</li>
<li>How communication can affect the correctness of a distributed algorithm</li>
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<h2 id="The-all-pairs-of-shortest-paths-(ASP)-problem">The all pairs of shortest paths (ASP) problem<a class="anchor-link" href="#The-all-pairs-of-shortest-paths-(ASP)-problem"></a></h2><p>Let us start by presenting the all pairs of shortest path (ASP) problem and its solution with the <a href="https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm">Floyd–Warshall algorithm</a>.</p>
<h2 id="The-all-pairs-of-shortest-paths-(ASP)-problem">The all pairs of shortest paths (ASP) problem<a class="anchor-link" href="#The-all-pairs-of-shortest-paths-(ASP)-problem"></a></h2><p>Let us start by presenting the all pairs of shortest paths (ASP) problem and its solution with the <a href="https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm">Floyd–Warshall algorithm</a>.</p>
<h3 id="Problem-statement">Problem statement<a class="anchor-link" href="#Problem-statement"></a></h3><ul>
<li>Given a graph $G$ with a distance table $C$</li>
<li>Compute the length of the shortest path between any two nodes in $G$</li>
</ul>
<p>We represent the distance table as a matrix, where $C_{ij}$ is the distance from node $i$ to node $j$. Next figure shows the input and solution (output) of the ASP problem for a simple 4-node directed graph. For example, the solution of the problem contains the minimum distance from node 2 to node 3, which is $C_{23}=8$ as highlighted in the figure.</p>
<p>We represent the distance table as a matrix, where $C_{ij}$ is the distance from node $i$ to node $j$. Next figure shows the input and solution (output) of the ASP problem for a simple 4-node directed graph. Note that the minimum distance from node 2 to node 3, which is $C_{23}=8$ as highlighted in the figure.</p>
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<div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">floyd!</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
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<pre>floyd! (generic function with 1 method)</pre>
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<div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">rand_distance_table</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
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<pre>rand_distance_table (generic function with 1 method)</pre>
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<div class="highlight hl-julia"><pre><span></span><span class="n">rand_distance_table</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
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<pre>10×10 Matrix{Int64}:
0 3 100000 100000 100000 … 3 100000 7 100000
8 0 3 10 5 9 9 10 10
100000 7 0 100000 8 100000 100000 7 7
3 100000 6 0 7 9 5 10 10
10 5 100000 7 0 100000 100000 8 100000
5 10 100000 3 9 … 4 10 7 100000
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10 100000 9 4 6 3 100000 7 0</pre>
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<pre>MPI.ThreadLevel(2)</pre>
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<pre>floyd_dist! (generic function with 1 method)</pre>
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<pre><span class="ansi-green-intense-fg ansi-bold">Test Passed</span></pre>
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