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generic-shortest-path-superior.html
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<h1 id="article-title">Generic Shortest Path Superior</h1>
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<h1 id="problem">Problem</h1>
<p>Let <span class="math inline">D=(V,A)</span> be a directed graph with weight <span class="math inline">w:A\to S</span>, where <span class="math inline">(S,\oplus,\otimes,\bar{0},\bar{1})</span> is a semiring with superiority. <span class="math inline">s</span> be a given vertex. <span class="math inline">P(v)</span> be the set of all one way paths from <span class="math inline">s</span> to <span class="math inline">v</span>.</p>
<p>Find <span class="math display">
\delta(v) = \bigoplus_{\pi \in P(v)} \bigotimes_{a \in \pi} w(a)
</span> for each <span class="math inline">v</span>.</p>
<h1 id="algorithms">Algorithms</h1>
<p>Parameters: <span class="math inline">n=|V|</span>, <span class="math inline">m=|A|</span>, <span class="math inline">M</span> is the maximum time required to preform semiring operations.</p>
<h2 id="dijkstra-algorithm">Dijkstra Algorithm</h2>
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<li><span class="math inline">O(M(m+n\log n))</span> time.</li>
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