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<head><title>allPairsShortestPath -- computes lengths of shortest paths between all pairs in a directed graph</title>
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<div><h1>allPairsShortestPath -- computes lengths of shortest paths between all pairs in a directed graph</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>D = allPairsShortestPath(G)</tt><br/><tt>D = allPairsShortestPath(A)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>G</tt>, <span>an object of class <a href="___Directed__Graph.html" title="a class for directed graphs">DirectedGraph</a></span></span></li>
<li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, adjacency matrix of a directed graph, whose (i,j) entry is the length of the directed edge from i to j and is infinity (1/0.) if there is no such directed edge.</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>D</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose (i,j) entry is the length of the shortest path from i to j and is infinity if there is no directed path from i to j.</span></li>
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<div class="single"><h2>Description</h2>
<div><div>This function uses the Floyd-Warshall algorithm to compute the lengths of shortest paths between all pairs of vertices in a directed graph.</div>
<table class="examples"><tr><td><pre>i1 : G = directedGraph( {a,b,c,d,e}, {(a,b),(b,c),(a,c),(a,d),(d,e)});</pre>
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<tr><td><pre>i2 : allPairsShortestPath(G)
o2 = | 0 1 1 1 2 |
| infinity 0 1 infinity infinity |
| infinity infinity 0 infinity infinity |
| infinity infinity infinity 0 1 |
| infinity infinity infinity infinity 0 |
5 5
o2 : Matrix RR <--- RR
53 53</pre>
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<tr><td><pre>i3 : allPairsShortestPath(adjacencyMatrix(G))
o3 = | 0 1 1 1 2 |
| infinity 0 1 infinity infinity |
| infinity infinity 0 infinity infinity |
| infinity infinity infinity 0 1 |
| infinity infinity infinity infinity 0 |
5 5
o3 : Matrix RR <--- RR
53 53</pre>
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<tr><td><pre>i4 : A = matrix({{0,1,3,5},{1/0.,0,1,3},{1/0.,1/0.,0,1},{2,1/0.,1/0.,0}})
o4 = | 0 1 3 5 |
| infinity 0 1 3 |
| infinity infinity 0 1 |
| 2 infinity infinity 0 |
4 4
o4 : Matrix RR <--- RR
53 53</pre>
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<tr><td><pre>i5 : allPairsShortestPath(A)
o5 = | 0 1 2 3 |
| 4 0 1 2 |
| 3 4 0 1 |
| 2 3 4 0 |
4 4
o5 : Matrix RR <--- RR
53 53</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>Assume there is no negative cycles. Output matrix is over RR.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Directed__Graph.html" title="a class for directed graphs">DirectedGraph</a> -- a class for directed graphs</span></li>
<li><span><a href="_adjacency__Matrix.html" title="returns adjacency matrix of a directed graph">adjacencyMatrix</a> -- returns adjacency matrix of a directed graph</span></li>
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<div class="waystouse"><h2>Ways to use <tt>allPairsShortestPath</tt> :</h2>
<ul><li>allPairsShortestPath(DirectedGraph)</li>
<li>allPairsShortestPath(Matrix)</li>
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