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@@ -7864,15 +7864,17 @@ $$</p>
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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
<h3 id="Where-can-we-extract-parallelism?">Where can we extract parallelism?<a class="anchor-link" href="#Where-can-we-extract-parallelism?"></a></h3><div class="highlight"><pre><span></span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
<span class="k">for</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">m</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">t</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">t</span><span class="p">]</span><span class="o">/</span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">]</span>
<span class="k">end</span>
<span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
<span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">n</span><span class="w"> </span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">m</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">j</span><span class="p">]</span>
<span class="k">for</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">n</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">m</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">t</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">t</span><span class="p">]</span><span class="o">/</span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">]</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">n</span><span class="w"> </span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">m</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">j</span><span class="p">]</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
<span class="k">end</span>
</pre></div>
</div>
@@ -8056,15 +8058,17 @@ d) only the first inner loop</code></pre>
<h3 id="Data-dependencies">Data dependencies<a class="anchor-link" href="#Data-dependencies"></a></h3><p>Using a cyclic partition, we managed to distribute the work uniformly. But we still need to study the data dependencies in order to implement this algorithm in parallel efficiently.</p>
<p>Look again to the algorithm</p>
<div class="highlight"><pre><span></span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
<span class="k">for</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">m</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">t</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">t</span><span class="p">]</span><span class="o">/</span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">]</span>
<span class="k">end</span>
<span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
<span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">n</span><span class="w"> </span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">m</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">j</span><span class="p">]</span>
<span class="k">for</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">n</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">m</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">t</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">t</span><span class="p">]</span><span class="o">/</span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">]</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">n</span><span class="w"> </span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">m</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">k</span><span class="p">,</span><span class="n">j</span><span class="p">]</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
<span class="k">end</span>
</pre></div>
<p>Note that all updates on the loop over i and j we do the following:</p>