### All pairs shortest paths using bridging sets and rectangular matrix multiplicationAll pairs shortest paths using bridging sets and rectangular matrix multiplication

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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