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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown"> </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
<h2 id="Contents">Contents<a class="anchor-link" href="#Contents"></a></h2><p>In this notebook, we will learn</p> <h2 id="Contents">Contents<a class="anchor-link" href="#Contents"></a></h2><p>In this notebook, we will learn</p>
<ul> <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> <li>How communication can affect the correctness of a distributed algorithm</li>
</ul> </ul>
</div> </div>
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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> <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>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> <li>Compute the length of the shortest path between any two nodes in $G$</li>
</ul> </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> <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> <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> <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
100000 9 8 10 100000 0 4 100000 4
5 10 9 100000 6 100000 0 5 100000
3 10 7 10 3 4 100000 0 100000
10 100000 9 4 6 3 100000 7 0</pre>
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<div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">MPIClusterManagers</span> <div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">MPIClusterManagers</span>
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<pre>MPI.ThreadLevel(2)</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">floyd_dist!</span><span class="p">(</span><span class="n">C</span><span class="p">)</span> <div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">floyd_dist!</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
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<pre>floyd_dist! (generic function with 1 method)</pre>
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<div class="highlight hl-julia"><pre><span></span><span class="nd">@everywhere</span><span class="w"> </span><span class="k">function</span><span class="w"> </span><span class="n">floyd_worker!</span><span class="p">(</span><span class="n">Cw</span><span class="p">,</span><span class="n">rows_w</span><span class="p">)</span> <div class="highlight hl-julia"><pre><span></span><span class="nd">@everywhere</span><span class="w"> </span><span class="k">function</span><span class="w"> </span><span class="n">floyd_worker!</span><span class="p">(</span><span class="n">Cw</span><span class="p">,</span><span class="n">rows_w</span><span class="p">)</span>
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<div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Test</span> <div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Test</span>
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<pre><span class="ansi-green-intense-fg ansi-bold">Test Passed</span></pre>
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"\n", "\n",
"In this notebook, we will learn\n", "In this notebook, we will learn\n",
"\n", "\n",
"- How to parallelize the Floyd's algorithm\n", "- How to parallelize Floyd's algorithm\n",
"- How communication can affect the correctness of a distributed algorithm\n" "- How communication can affect the correctness of a distributed algorithm\n"
] ]
}, },
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"source": [ "source": [
"## The all pairs of shortest paths (ASP) problem\n", "## The all pairs of shortest paths (ASP) problem\n",
"\n", "\n",
"Let us start by presenting the all pairs of shortest path (ASP) problem and its solution with the [Floyd–Warshall algorithm](https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm).\n", "Let us start by presenting the all pairs of shortest paths (ASP) problem and its solution with the [Floyd–Warshall algorithm](https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm).\n",
"\n", "\n",
"### Problem statement\n", "### Problem statement\n",
"\n", "\n",
"- Given a graph $G$ with a distance table $C$\n", "- Given a graph $G$ with a distance table $C$\n",
"- Compute the length of the shortest path between any two nodes in $G$\n", "- Compute the length of the shortest path between any two nodes in $G$\n",
"\n", "\n",
"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.\n" "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.\n"
] ]
}, },
{ {
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"function floyd!(C)\n", "function floyd!(C)\n",
" n = size(C,1)\n", " n = size(C,1)\n",
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}, },
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"function rand_distance_table(n)\n", "function rand_distance_table(n)\n",
" threshold = 0.4\n", " threshold = 0.4\n",
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"10×10 Matrix{Int64}:\n",
" 0 3 100000 100000 100000 … 3 100000 7 100000\n",
" 8 0 3 10 5 9 9 10 10\n",
" 100000 7 0 100000 8 100000 100000 7 7\n",
" 3 100000 6 0 7 9 5 10 10\n",
" 10 5 100000 7 0 100000 100000 8 100000\n",
" 5 10 100000 3 9 … 4 10 7 100000\n",
" 100000 9 8 10 100000 0 4 100000 4\n",
" 5 10 9 100000 6 100000 0 5 100000\n",
" 3 10 7 10 3 4 100000 0 100000\n",
" 10 100000 9 4 6 3 100000 7 0"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"rand_distance_table(10)" "rand_distance_table(10)"
] ]
@ -500,21 +457,10 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 2, "execution_count": null,
"id": "6bc0122b", "id": "6bc0122b",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [],
{
"data": {
"text/plain": [
"MPI.ThreadLevel(2)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"using MPIClusterManagers\n", "using MPIClusterManagers\n",
"using Distributed\n", "using Distributed\n",
@ -536,21 +482,10 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 3, "execution_count": null,
"id": "22234b2f", "id": "22234b2f",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [],
{
"data": {
"text/plain": [
"floyd_dist! (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"function floyd_dist!(C)\n", "function floyd_dist!(C)\n",
" m = size(C,1)\n", " m = size(C,1)\n",
@ -567,7 +502,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 23, "execution_count": null,
"id": "d5a8d3a8", "id": "d5a8d3a8",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
@ -603,21 +538,10 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 24, "execution_count": null,
"id": "dd77ee3d", "id": "dd77ee3d",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [],
{
"data": {
"text/plain": [
"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"using Test\n", "using Test\n",
"load = 10\n", "load = 10\n",

4
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@ -1,5 +1,5 @@
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@ -18,4 +18,4 @@
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iFrameResize({log:true}, myIframe); iFrameResize({log:true}, myIframe);
}); });
</script></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 17 August 2023 13:09">Thursday 17 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> </script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi</a><a class="docs-footer-nextpage" href="../solutions/">Solutions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Friday 18 August 2023 08:10">Friday 18 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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