LIPIcs.STACS.2008.1343.pdf
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Quantifying Homology Classes
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25th International Symposium on Theoretical Aspects of Computer Science
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Theoretical Aspects of Computer Science (STACS) - License:
Creative Commons Attribution-NoDerivs 3.0 Unported license
- Publication date: 2008-02-06
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Chao Chen and Daniel Freedman. Quantifying Homology Classes. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 169-180, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)
https://doi.org/10.4230/LIPIcs.STACS.2008.1343
https://doi.org/10.4230/LIPIcs.STACS.2008.1343
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.
Keywords
- Computational Topology
- Computational Geometry
- Homology
- Persistent Homology
- Localization
- Optimization
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