NDLI: All pairs shortest paths using bridging sets and rectangular matrix multiplication

Content Provider	ACM Digital Library
Author	Zwick, Uri
Copyright Year	2002
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.
Starting Page	289
Ending Page	317
Page Count	29
File Format	PDF
ISSN	00045411
e-ISSN	1557735X
DOI	10.1145/567112.567114
Journal	Journal of the ACM (JACM)
Volume Number	49
Issue Number	3
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2002-05-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Matrix multiplication Shortest paths
Content Type	Text
Resource Type	Article
Subject	Hardware and Architecture Information Systems Control and Systems Engineering Artificial Intelligence Software

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
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9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

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