All Pairs Shortest Paths in Undirected Graphs with Integer Weights (1999)
| Venue: | In IEEE Symposium on Foundations of Computer Science |
| Citations: | 37 - 6 self |
BibTeX
@INPROCEEDINGS{Shoshan99allpairs,
author = {Avi Shoshan and Uri Zwick},
title = {All Pairs Shortest Paths in Undirected Graphs with Integer Weights},
booktitle = {In IEEE Symposium on Foundations of Computer Science},
year = {1999},
pages = {605--614}
}
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Abstract
We show that the All Pairs Shortest Paths (APSP) problem for undirected graphs with integer edge weights taken from the range f1; 2; : : : ; Mg can be solved using only a logarithmic number of distance products of matrices with elements in the range f1; 2; : : : ; Mg. As a result, we get an algorithm for the APSP problem in such graphs that runs in ~ O(Mn ! ) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and ! ! 2:376 is the exponent of matrix multiplication. This improves, and also simplifies, an ~ O(M (!+1)=2 n ! ) time algorithm of Galil and Margalit. 1. Introduction The All Pairs Shortest Paths (APSP) problem is one of the most fundamental algorithmic graph problems. The APSP problem for directed or undirected graphs with real weights can be solved using classical methods, in O(mn + n 2 log n) time (Dijkstra [4], Johnson [10], Fredman and Tarjan [7]), or in O(n 3 ((log log n)= log n) 1=2 ) time (Fredman [6], ...
