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DOI: 10.4230/LIPIcs.STACS.2008.1343
URN: urn:nbn:de:0030-drops-13434
URL: https://drops.dagstuhl.de/opus/volltexte/2008/1343/
Chen, Chao ; Freedman, Daniel
Quantifying Homology Classes
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Abstract
We develop a method for measuring homology classes. This involvesthree 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 = {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 |
