License
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2008.1343
URN: urn:nbn:de:0030-drops-13434
URL: http://drops.dagstuhl.de/opus/volltexte/2008/1343/
Go to the corresponding Portal


Chen, Chao ; Freedman, Daniel

Quantifying Homology Classes

pdf-format:
Document 1.pdf (324 KB)


Abstract

We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in $O(\beta^4 n^3 log^2 n)$ time, where $n$ is the size of the simplicial complex and $\beta$ is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results.

BibTeX - Entry

@InProceedings{chen_et_al:LIPIcs:2008:1343,
  author =	{Chao Chen and Daniel Freedman},
  title =	{{Quantifying Homology Classes}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{169--180},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Susanne Albers and Pascal Weil},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2008/1343},
  URN =		{urn:nbn:de:0030-drops-13434},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2008.1343},
  annote =	{Keywords: Computational Topology, Computational Geometry, Homology, Persistent Homology, Localization, Optimization}
}

Keywords: Computational Topology, Computational Geometry, Homology, Persistent Homology, Localization, Optimization
Seminar: 25th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2008
Date of publication: 06.02.2008


DROPS-Home | Fulltext Search | Imprint Published by LZI