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标题：All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication
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作者：Uri Zwick
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2000
卷号：2000
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：We present two new algorithms for solving the {\em All Pairs Shortest Paths\/} (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in \Ot(n2+) time, where satisfies the equation (11)=1+2 and (11) is the exponent of the multiplication of an nn matrix by an nn matrix. Currently, the best available bounds on (11) , obtained by Coppersmith, imply that 0575 . The running time of our algorithm is therefore O(n2575). Our algorithm improves on the \Ot(n(3+)2) time algorithm, where =(111)2376 is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be O(n2688). The second algorithm solves the APSP problem {\em almost\/} exactly for directed graphs with {\em arbitrary\/} non-negative real weights. The algorithm runs in \Ot((n\eps)log(W\eps)) time, where \eps0 is an error parameter and~W is the largest edge weight in the graph, after the edge weights are scaled so that the smallest non-zero edge weight in the graph is~1. It returns estimates of all the distances in the graph with a stretch of at most 1+\eps. Corresponding paths can also be found efficiently.
关键词：Graph Algorithms, matrix multiplication, Probabilistic Algorithms, Shortest Paths