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Author Zwick, Uri
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2002
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Matrix multiplication ♦ Shortest paths
Abstract We present two new algorithms for solving the 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 $Õ(n^{2+μ})$ time, where μ satisfies the equation ω(1, μ, 1) = 1 + 2μ and ω(1, μ, 1) is the exponent of the multiplication of an $\textit{n}$ × $n^{μ}$ matrix by an $n^{μ}$ × $\textit{n}$ matrix. Currently, the best available bounds on ω(1, μ, 1), obtained by Coppersmith, imply that μ < 0.575. The running time of our algorithm is therefore $O(n^{2.575}).$ Our algorithm improves on the $&Otilede;(n^{(3c+ω)/2})$ time algorithm, where ω = ω(1, 1, 1) < 2.376 is the usual exponent of matrix multiplication, obtained by Alon et al., whose running time is only known to be $O(n^{2.688}).The$ second algorithm solves the APSP problem $\textit{almost}$ exactly for directed graphs with $\textit{arbitrary}$ nonnegative real weights. The algorithm runs in $Õ((n^{ω}/&epsis;)$ $log(\textit{W}/&epsis;))$ time, where &epsis; > 0 is an error parameter and $\textit{W}$ 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 + &epsis;. Corresponding paths can also be found efficiently.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2002-05-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 49
Issue Number 3
Page Count 29
Starting Page 289
Ending Page 317


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Source: ACM Digital Library