An output-sensitive algorithm for persistent homology

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 ran...

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Vydáno v:Computational geometry : theory and applications Ročník 46; číslo 4; s. 435 - 447
Hlavní autoři: Chen, Chao, Kerber, Michael
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 01.05.2013
Témata:
Algorithms
Boundaries
Complexity
Computational geometry
Computational topology
Computing time
Homology
Monte Carlo methods
Persistent homology
Randomized algorithms
Rank computation
Running
Randomized algorithms
Rank computation
Persistent homology
Computational topology
ISSN:0925-7721
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Shrnutí: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)ΓRd(n)logn), 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 Rd(n) is the complexity of computing the rank of an n×n matrix with O(dn) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O(C(1−δ)Γn2.376) algorithm, an O(C(1−δ)Γn2.28) Las-Vegas algorithm, or an O(C(1−δ)Γn2+ϵ) Monte-Carlo algorithm for an arbitrary ϵ>0. The space complexity of the Monte-Carlo version is bounded by O(dn)=O(nlogn).
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ISSN:0925-7721
DOI:10.1016/j.comgeo.2012.02.010