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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 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>
<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>
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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="n">inf</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1000</span>
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<pre>4×4 Matrix{Int64}:
0 9 6 1
2 0 8 3
5 3 0 6
10 8 5 0</pre>
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<p>TODO explain this part</p>
<p>At each iteration, the processor needs the input values <code>C[i,j]</code>, <code>C[i,k]</code> and <code>C[k,j]</code>. Since we split the data row-wise, the process already has values <code>C[i,j]</code> and <code>C[i,k]</code>. However, <code>C[k,j]</code> may be stored in a different process.</p>
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<p>As we iterate over columns $j$, the process needs input from all values of row $k$. Therefore, at the start of iteration $k$, the whole row $k$ needs to be communicated.</p>
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