License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2017.74
URN: urn:nbn:de:0030-drops-74510
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7451/
Georgiadis, Loukas ;
Graf, Daniel ;
Italiano, Giuseppe F. ;
Parotsidis, Nikos ;
Uznanski, Przemyslaw
All-Pairs 2-Reachability in O(n^w log n) Time
Abstract
In the 2-reachability problem we are given a directed graph G and we wish to determine if there are two (edge or vertex) disjoint paths from u to v, for given pair of vertices u and v. In this paper, we present an algorithm that computes 2-reachability information for all pairs of vertices in O(n^w log n) time, where n is the number of vertices and w is the matrix multiplication exponent. Hence, we show that the running time of all-pairs 2-reachability is only within a log factor of transitive closure. Moreover, our algorithm produces a witness (i.e., a separating edge or a separating vertex) for all pair of vertices where 2-reachability does not hold. By processing these witnesses, we can compute all the edge- and vertex-dominator trees of G in O(n^2) additional time, which in turn enables us to answer various connectivity queries in O(1) time. For instance, we can test in constant time if there is a path from u to v avoiding an edge e, for any pair of query vertices u and v, and any query edge e, or if there is a path from u to v avoiding a vertex w, for any query vertices u, v, and w.
BibTeX - Entry
@InProceedings{georgiadis_et_al:LIPIcs:2017:7451,
author = {Loukas Georgiadis and Daniel Graf and Giuseppe F. Italiano and Nikos Parotsidis and Przemyslaw Uznanski},
title = {{All-Pairs 2-Reachability in O(n^w log n) Time}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {74:1--74:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-041-5},
ISSN = {1868-8969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7451},
URN = {urn:nbn:de:0030-drops-74510},
doi = {10.4230/LIPIcs.ICALP.2017.74},
annote = {Keywords: 2-reachability, All Dominator Trees, Directed Graphs, Boolean Matrix Multiplication}
}
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Keywords: |
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2-reachability, All Dominator Trees, Directed Graphs, Boolean Matrix Multiplication |
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Collection: |
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44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) |
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Issue Date: |
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2017 |
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Date of publication: |
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07.07.2017 |
