Answering distance queries in directed graphs using fast matrix multiplication

Raphael Yuster, Uri Zwick

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Let G = (V, E, w) be a weighted directed graph, where w : E → {-M,..., 0,..., M}. We show that G can be preprocessed in Õ(Mn ω) time, where ω < 2.376 is the exponent of fast matrix multiplication, such that subsequently, each distance δ(u, v) in the graph, where u, v ∈ V, can be computed exactly in O(n) time. We also present a tradeoff between the processing time and the query answering time. As a very special case, we obtain an Õ(Mn ω) time algorithm for the Single Source Shortest Paths (SSSP) problem for directed graphs with integer weights of absolute value at most M. For sufficiently dense graphs, with small enough edge weights, this improves upon the O(m√n log M) time algorithm of Goldberg. We note that even the case M = 1, in which all the edge weights are in {-1, 0, +1}, is an interesting case for which no improvement over Goldberg's O(m√n) algorithm was known. Our new Õ(n ω) algorithm is faster whenever m > n ω-1/2 ≃ n 1.876.

Original languageEnglish
Title of host publicationProceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
Pages389-396
Number of pages8
DOIs
StatePublished - 2005
Event46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005 - Pittsburgh, PA, United States
Duration: 23 Oct 200525 Oct 2005

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2005
ISSN (Print)0272-5428

Conference

Conference46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
Country/TerritoryUnited States
CityPittsburgh, PA
Period23/10/0525/10/05

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