# An output sensitive algorithm for persistent homology

Chen C, Kerber M. 2013. An output sensitive algorithm for persistent homology. Computational Geometry: Theory and Applications. 46(4), 435–447.

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*English*

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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).

Publishing Year

Date Published

2013-05-01

Journal Title

Computational Geometry: Theory and Applications

Acknowledgement

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.

Volume

46

Issue

4

Page

435 - 447

IST-REx-ID

### Cite this

Chen C, Kerber M. An output sensitive algorithm for persistent homology.

*Computational Geometry: Theory and Applications*. 2013;46(4):435-447. doi:10.1016/j.comgeo.2012.02.010Chen, C., & Kerber, M. (2013). An output sensitive algorithm for persistent homology.

*Computational Geometry: Theory and Applications*. Elsevier. https://doi.org/10.1016/j.comgeo.2012.02.010Chen, Chao, and Michael Kerber. “An Output Sensitive Algorithm for Persistent Homology.”

*Computational Geometry: Theory and Applications*. Elsevier, 2013. https://doi.org/10.1016/j.comgeo.2012.02.010.C. Chen and M. Kerber, “An output sensitive algorithm for persistent homology,”

*Computational Geometry: Theory and Applications*, vol. 46, no. 4. Elsevier, pp. 435–447, 2013.Chen, Chao, and Michael Kerber. “An Output Sensitive Algorithm for Persistent Homology.”

*Computational Geometry: Theory and Applications*, vol. 46, no. 4, Elsevier, 2013, pp. 435–47, doi:10.1016/j.comgeo.2012.02.010.