License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ESA.2016.72
URN: urn:nbn:de:0030-drops-64207
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6420/
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Porat, Ely ; Shahbazian, Eduard ; Tov, Roei

New Parameterized Algorithms for APSP in Directed Graphs

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Abstract

All Pairs Shortest Path (APSP) is a classic problem in graph theory. While for general weighted graphs there is no algorithm that computes APSP in O(n^{3-epsilon}) time (epsilon > 0), by using fast matrix multiplication algorithms, we can compute APSP in O(n^{omega}*log(n)) time (omega < 2.373) for undirected unweighted graphs, and in O(n^{2.5302}) time for directed unweighted graphs. In the current state of matters, there is a substantial gap between the upper bounds of the problem for undirected and directed graphs, and for a long time, it is remained an important open question whether it is possible to close this gap.

In this paper we introduce a new parameter that measures the symmetry of directed graphs (i.e. their closeness to undirected graphs), and obtain a new parameterized APSP algorithm for directed unweighted graphs, that generalizes Seidel's O(n^{omega}*log(n)) time algorithm for undirected unweighted graphs. Given a directed unweighted graph G, unless it is highly asymmetric, our algorithms can compute APSP in o(n^{2.5}) time for G, providing for such graphs a faster APSP algorithm than the state-of-the-art algorithms for the problem.

BibTeX - Entry

@InProceedings{porat_et_al:LIPIcs:2016:6420,
  author =	{Ely Porat and Eduard Shahbazian and Roei Tov},
  title =	{{New Parameterized Algorithms for APSP in Directed Graphs}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{72:1--72:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Piotr Sankowski and Christos Zaroliagis},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6420},
  URN =		{urn:nbn:de:0030-drops-64207},
  doi =		{10.4230/LIPIcs.ESA.2016.72},
  annote =	{Keywords: Graphs, distances, APSP, fast matrix multiplication}
}

Keywords: Graphs, distances, APSP, fast matrix multiplication
Collection: 24th Annual European Symposium on Algorithms (ESA 2016)
Issue Date: 2016
Date of publication: 18.08.2016

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