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Quantifying Homology Classes

Authors Chao Chen, Daniel Freedman

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LIPIcs.STACS.2008.1343.pdf
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Keywords
  • Computational Topology
  • Computational Geometry
  • Homology
  • Persistent Homology
  • Localization
  • Optimization

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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.

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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

Author Details

Chao Chen
Daniel Freedman
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