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An output-sensitive algorithm for persistent homology

Published: 13 June 2011 Publication History

Abstract

In this paper, we present the first output-sensitive algorithm to compute the persistence diagram of a filtered simplicial complex. For any Γ>0, it returns only those homology classes with persistence at least Γ. Instead of the classical reduction via column operations, our algorithm performs rank computations on submatrices of the boundary matrix. For an arbitrary constant δ ∈ (0,1), the running time is O(C(1-δ)ΓR(n)log n), where C(1-δ)Γ is the number of homology classes with persistence at least (1-δ)Γ, n is the total number of simplices, and R(n) is the complexity of computing the rank of an n x n matrix with O(n) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O(C(1-δ)Γn2.376) algorithm, a O(C(1-δ)Γn2.28) Las-Vegas algorithm, or a O(C(1-δ)Γn2+ε) Monte-Carlo algorithm for an arbitrary ε>0.

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  • (2014)Clear and Compress: Computing Persistent Homology in ChunksTopological Methods in Data Analysis and Visualization III10.1007/978-3-319-04099-8_7(103-117)Online publication date: 19-Mar-2014
  • (2012)On persistent homotopy, knotted complexes and the Alexander moduleProceedings of the 3rd Innovations in Theoretical Computer Science Conference10.1145/2090236.2090270(428-441)Online publication date: 8-Jan-2012
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cover image ACM Conferences
SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry
June 2011
532 pages
ISBN:9781450306829
DOI:10.1145/1998196
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 13 June 2011

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Author Tags

  1. computational topology
  2. persistent homology
  3. randomized algorithms
  4. rank computation

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SoCG '11
SoCG '11: Symposium on Computational Geometry
June 13 - 15, 2011
Paris, France

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

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  • (2016)Multiscale mapperProceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms10.5555/2884435.2884506(997-1013)Online publication date: 10-Jan-2016
  • (2014)Clear and Compress: Computing Persistent Homology in ChunksTopological Methods in Data Analysis and Visualization III10.1007/978-3-319-04099-8_7(103-117)Online publication date: 19-Mar-2014
  • (2012)On persistent homotopy, knotted complexes and the Alexander moduleProceedings of the 3rd Innovations in Theoretical Computer Science Conference10.1145/2090236.2090270(428-441)Online publication date: 8-Jan-2012
  • (2012)Efficient computation of 3D Morse–Smale complexes and persistent homology using discrete Morse theoryThe Visual Computer10.1007/s00371-012-0726-828:10(959-969)Online publication date: 26-May-2012
  • (2011)Memory-Efficient Computation of Persistent Homology for 3D Images Using Discrete Morse TheoryProceedings of the 2011 24th SIBGRAPI Conference on Graphics, Patterns and Images10.1109/SIBGRAPI.2011.24(25-32)Online publication date: 28-Aug-2011
  • (2011)Efficient Computation of Persistent Homology for Cubical DataTopological Methods in Data Analysis and Visualization II10.1007/978-3-642-23175-9_7(91-106)Online publication date: 14-Nov-2011

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