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.SoCG.2020.64
URN: urn:nbn:de:0030-drops-122223
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12222/
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Rathod, Abhishek

Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis

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LIPIcs-SoCG-2020-64.pdf (0.5 MB)

Abstract

We study the problem of finding a minimum homology basis, that is, a shortest set of cycles that generates the 1-dimensional homology classes with ℤ₂ coefficients in a given simplicial complex K. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al. [Dey et al., 2018], runs in O(N^ω + N² g) time, where N denotes the number of simplices in K, g denotes the rank of the 1-homology group of K, and ω denotes the exponent of matrix multiplication. In this paper, we present two conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex K. The first algorithm runs in Õ(m^ω) time, where m denotes the number of edges in K, whereas the second algorithm runs in O(m^ω + N m^{ω-1}) time.
We also study the problem of finding a minimum cycle basis in an undirected graph G with n vertices and m edges. The best known algorithm for this problem runs in O(m^ω) time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in Õ(m^ω) time.

BibTeX - Entry

@InProceedings{rathod:LIPIcs:2020:12222,
  author =	{Abhishek Rathod},
  title =	{{Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{64:1--64:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Sergio Cabello and Danny Z. Chen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12222},
  URN =		{urn:nbn:de:0030-drops-122223},
  doi =		{10.4230/LIPIcs.SoCG.2020.64},
  annote =	{Keywords: Computational topology, Minimum homology basis, Minimum cycle basis, Simplicial complexes, Matrix computations}
}

Keywords: Computational topology, Minimum homology basis, Minimum cycle basis, Simplicial complexes, Matrix computations
Collection: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue Date: 2020
Date of publication: 08.06.2020

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