A Space Saving Trick for Directed Dynamic Transitive Closure and Shortest Path Algorithms
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ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
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SAGA'09 Proceedings of the 5th international conference on Stochastic algorithms: foundations and applications
ACM Transactions on Algorithms (TALG)
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PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
CPM'07 Proceedings of the 18th annual conference on Combinatorial Pattern Matching
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ICA3PP'12 Proceedings of the 12th international conference on Algorithms and Architectures for Parallel Processing - Volume Part I
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The subcubic (O(n/sup w/) for w(3) algorithms to multiply Boolean matrices do not provide the witnesses; namely, they compute C=A.B but if C/sub ij/=1 they do not find an index k (a witness) such that A/sub ik/=B/sub kj/=1. The authors design a deterministic algorithm for computing the matrix of witnesses that runs in O(n/sup w/) time, where here O(n/sup w/) denotes O(n/sup w/(log n)/sup O(1)/). The subcubic methods to compute the shortest distances between all pairs of vertices also do not provide for witnesses; namely they compute the shortest distances but do not generate information for computing quickly the paths themselves. A witness for a shortest path from v/sub i/ to v/sub j/ is an index k such that v/sub k/ is the first vertex on such a path. They describe subcubic methods to compute such witnesses for several versions of the all pairs shortest paths problem. As a result, they derive shortest paths algorithms that provide characterization of the shortest paths in addition to the shortest distances in the same time (up to a polylogarithmic factor) needed for computing the distances; namely O(n/sup (3+w)/2/) time in the directed case and O(n/sup w/) time in the undirected case. They also design an algorithm that computes witnesses for the transitive closure in the same time needed to compute witnesses for Boolean matrix multiplication.