Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For 1-D persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most n-D persistence modules, n>1, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called 2-D interval decomposable modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the bottleneck distance for these modules from indecomposables, which bounds the interleaving distance from above, and give another algorithm to compute a new distance called dimension distance that bounds it from below.
@InProceedings{dey_et_al:LIPIcs.SoCG.2018.32, author = {Dey, Tamal K. and Xin, Cheng}, title = {{Computing Bottleneck Distance for 2-D Interval Decomposable Modules}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {32:1--32:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.32}, URN = {urn:nbn:de:0030-drops-87453}, doi = {10.4230/LIPIcs.SoCG.2018.32}, annote = {Keywords: Persistence modules, bottleneck distance, interleaving distance} }
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