An output sensitive algorithm for persistent homology Journal Article

Author(s): Chen, Chao; Kerber, Michael
Article Title: An output sensitive algorithm for persistent homology
Affiliation IST Austria
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) Γ R d (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 in the complex, d its dimension, and R d (n) is the complexity of computing the rank of an n × n matrix with O (d n) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O (C (1 - δ) Γ n 2.376) algorithm, an O (C (1 - δ) Γ n 2.28) Las-Vegas algorithm, or an O (C (1 - δ) Γ n 2 + ε{lunate}) Monte-Carlo algorithm for an arbitrary ε{lunate} > 0. The space complexity of the Monte-Carlo version is bounded by O (d n) = O (n log n).
Keywords: persistent homology; Computational topology; Randomized algorithms; Rank computation
Journal Title: Computational Geometry: Theory and Applications
Volume: 46
Issue 4
ISSN: 0925-7721
Publisher: Elsevier  
Date Published: 2013-05-01
Start Page: 435
End Page: 447
Copyright Statement: Copyright © 2012 Elsevier B.V. All rights reserved.
DOI: 10.1016/j.comgeo.2012.02.010
Notes: The authors thank Herbert Edelsbrunner for many helpful discussions and suggestions. Moreover, they are grateful for the careful reviews that helped to improve the quality of the paper.
Open access: no
IST Austria Authors
  1. Michael Kerber
    21 Kerber
  2. Chao Chen
    14 Chen
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