{"title":"Merge Trees of Periodic Filtrations","authors":"Herbert Edelsbrunner, Teresa Heiss","doi":"arxiv-2408.16575","DOIUrl":null,"url":null,"abstract":"Motivated by applications to crystalline materials, we generalize the merge\ntree and the related barcode of a filtered complex to the periodic setting in\nEuclidean space. They are invariant under isometries, changing bases, and\nindeed changing lattices. In addition, we prove stability under perturbations\nand provide an algorithm that under mild geometric conditions typically\nsatisfied by crystalline materials takes $\\mathcal{O}({(n+m) \\log n})$ time, in\nwhich $n$ and $m$ are the numbers of vertices and edges in the quotient\ncomplex, respectively.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16575","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Motivated by applications to crystalline materials, we generalize the merge
tree and the related barcode of a filtered complex to the periodic setting in
Euclidean space. They are invariant under isometries, changing bases, and
indeed changing lattices. In addition, we prove stability under perturbations
and provide an algorithm that under mild geometric conditions typically
satisfied by crystalline materials takes $\mathcal{O}({(n+m) \log n})$ time, in
which $n$ and $m$ are the numbers of vertices and edges in the quotient
complex, respectively.