A more efficient algorithm for the min-plus multiplication

Dobosiewicz, W.

doi:10.1080/00207169008803814

Cited by 44 publications

(26 citation statements)

References 8 publications

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“…It seems that combinatorial algorithms for APSP (not using fast matrix multiplication) are much better suited to be implemented in a distributed way. Combinatorial APSP algorithms were studied first in [21] and then [9,10,15,19,23,42,43,50] but only yield polylogarithmic improvements over O(n 3 ).…”

Section: All Pairs Shortest Pathsmentioning

confidence: 99%

Optimal distributed all pairs shortest paths and applications

Holzer

Wattenhofer

2012

Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing

141

134

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We present an algorithm to compute All Pairs Shortest Paths (APSP) of a network in a distributed way. The model of distributed computation we consider is the message passing model: in each synchronous round, every node can transmit a different (but short) message to each of its neighbors. We provide an algorithm that computes APSP in O(n) communication rounds, where n denotes the number of nodes in the network. This implies a linear time algorithm for computing the diameter of a network. Due to a lower bound these two algorithms are optimal up to a logarithmic factor. Furthermore, we present a new lower bound for approximating the diameter D of a graph: Being allowed to answer D + 1 or D can speed up the computation by at most a factor D. On the positive side, we provide an algorithm that achieves such a speedup of D and computes an 1 + ε multiplicative approximation of the diameter. We extend these algorithms to compute or approximate other problems, such as girth, radius, center and peripheral vertices. At the heart of these approximation algorithms is the S-Shortest Paths problem which we solve in O(|S| + D) time.

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Section: All Pairs Shortest Pathsmentioning

confidence: 99%

Optimal distributed all pairs shortest paths and applications

Holzer

Wattenhofer

2012

Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing

141

134

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“…year O(n 3 ) Dijkstra/Floyd-Warshall 1959/1962 O(n 3 log 1/3 log n/ log 1/3 n) Fredman [17] 1976 O(n 3 log 1/2 log n/ log 1/2 n) Takaoka [34] 1991 O(n 3 / log 1/2 n) Dobosiewicz [15] 1990 O(n 3 log 5/7 log n/ log 5/7 n) Han [18] 2004 O(n 3 log 2 log n/ log n) Takaoka [35] 2004 O(n 3 log log n/ log n)…”

Section: Introductionmentioning

confidence: 99%

More Algorithms for All-Pairs Shortest Paths in Weighted Graphs

Chan¹

2010

SIAM J. Comput.

150

167

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In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n/ log 2 n), which improves all known algorithms for general real-weighted dense graphs.In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of "geometrically weighted" graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4) ), where ω < 2.376; in two dimensions, this is O(n 2.922 ). Our framework greatly extends the previously considered case of small-integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4 ) = O(n 2.844 ) time) for APSP in real-vertex-weighted graphs, as well as an improved result (near O(n (3+ω)/2 ) = O(n 2.688 ) time) for the all-pairs lightest shortest path problem for small-integer-weighted graphs.

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“…[7] discovered a slightly more efficient explicit implementation of Fredman's "algorithm" for rectangular min-plus products. Instead of using table-lookup, as done by Fredman [9] and Takaoka [25], the algorithm of Dobosiewicz simply uses bit-level parallelism.…”

Section: The Algorithm Of Fredmanmentioning

confidence: 99%