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oai:drops-oai.dagstuhl.de:1343 2024-03-06T10:32:58Z ddc:004 open_access

Quantifying Homology Classes Chen, Chao Freedman, Daniel Computational Topology Computational Geometry Homology Persistent Homology Localization Optimization 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Chao Chen and Daniel Freedman 2008 Is Part Of LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2008.1343 urn:nbn:de:0030-drops-13434 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1343 eng https://creativecommons.org/licenses/by-nd/3.0/legalcode