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.STACS.2017.56
URN: urn:nbn:de:0030-drops-69773
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/6977/
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Sankowski, Piotr ; Wegrzycki, Karol

Improved Distance Queries and Cycle Counting by Frobenius Normal Form

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LIPIcs-STACS-2017-56.pdf (0.5 MB)

Abstract

Consider an unweighted, directed graph G with the diameter D. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time O-tilde(n^omega). The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the following problems efficiently.

* All Nodes Shortest Cycles - for every node return the length of the shortest cycle containing it. We give an O-tilde(n^omega) algorithm that improves the previous O-tilde(n^((omega + 3)/2)) algorithm for unweighted digraphs.

* We show how to compute all D sets of vertices lying on cycles of length c in {1, ..., D} in randomized time O-tilde(n^omega). It improves upon an algorithm by Cygan where algorithm that computes a single set is presented.

* We present a functional improvement of distance queries for directed, unweighted graphs.

* All Pairs All Walks - we show almost optimal O-tilde(n^3) time algorithm for all walks counting problem. We improve upon the naive O(D n^omega) time algorithm.

BibTeX - Entry

@InProceedings{sankowski_et_al:LIPIcs:2017:6977,
  author =	{Piotr Sankowski and Karol Wegrzycki},
  title =	{{Improved Distance Queries and Cycle Counting by Frobenius Normal Form}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{56:1--56:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Heribert Vollmer and Brigitte ValleĢe},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/6977},
  URN =		{urn:nbn:de:0030-drops-69773},
  doi =		{10.4230/LIPIcs.STACS.2017.56},
  annote =	{Keywords: Frobenius Normal Form, Graph Algorithms, All Nodes Shortest Cycles}
}

Keywords: Frobenius Normal Form, Graph Algorithms, All Nodes Shortest Cycles
Collection: 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)
Issue Date: 2017
Date of publication: 06.03.2017

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