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.
@InProceedings{chen_et_al:LIPIcs.STACS.2008.1343, author = {Chen, Chao and Freedman, Daniel}, 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 = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.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} }
Feedback for Dagstuhl Publishing