Feb22010

# N use an n x n matrix in which to compute the lengths

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Unformatted text preview: –  Nodes are pathnodes in your map –  Labels are the distances between connecDng nodes (or other speciﬁc cost informaDon) •  G=(V, E) in which each arc v → w has a non ­ negaDve cost C[v,w]. •  Find, for each ordered pair (v,w) the smallest length of any path from v to w. UC Santa Cruz Computer Science – Game Design All Pairs Shortest Path (APSP) Floyd’s Algorithm •  Assume the verDces in V are numbered 1,...,n. •  Use an n X n matrix in which to compute the lengths of the shortest paths. UC Santa Cruz Computer Science – Game Design All Pairs Shortest Path (APSP) Floyd’s Algorithm •  Set A[i,j] = C[i,j] for all i not equal to j. •  If there is no arc from i to j, then assume C[i,j] = ∞. •  Each diagonal element is set to 0. UC Santa Cruz Computer Science – Game Design All Pairs Shortest Path (APSP) Floyd’s Algorithm •  Make n iteraDons over the matrix •  On the kth iteraDon, A[i,j] will have for its value the smallest length of any path from i to j that doesn’t pass through a vertex numbered higher than k. UC Santa Cruz Computer Science – Game Design All Pairs Shortest Path (APSP) Floyd’s Algorithm •  On the kth iteraDon, use the following formula to compute A Ak ­1[i,j] •  A[i,j] = min { Ak ­1[i,k] + Ak ­1[k,j] UC Santa Cruz Computer Science – Game Design All Pairs Shortest Path (APSP) Floyd’s Algorithm •  On the kth iteraDon, use the following formula to compute A The cost of going from i Ak ­1[i,j] to j without going through •  A[i,j] = min { k or any higher node Ak ­1[i,k] + Ak ­1[k,j] UC Santa Cruz Computer Science – Game Design All Pairs Shortest Path (APSP) Floyd’s Algorithm •  On the kth iteraDon, use the following formula to compute A The cost of going from i Ak ­1[i,j] to k, then k to j, without •  A[i,j] = min { going through any node higher than k....
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## This note was uploaded on 11/10/2010 for the course CMPS 101 taught by Professor Tantalo,p during the Fall '08 term at UCSC.

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