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Preprints, Working Papers, ... Year : 2023

Fast persistent homology computation for functions on ℝ

Marc Glisse

Abstract

0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the connected components and we get the persistence diagram in time $O(n\alpha(n))$. The running time is thus usually dominated by sorting the edges in $\Theta(n\log(n))$. A little-known fact is that, in the particularly simple case of studying the sublevel sets of a piecewise-linear function on $\mathbb{R}$ or $\mathbb{S}^1$, persistence can actually be computed in linear time. This note presents a simple algorithm that achieves this complexity and an extension to image persistence. An implementation is available in Gudhi.

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hal-04148137 , version 1 (02-07-2023)

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Marc Glisse. Fast persistent homology computation for functions on ℝ. 2023. ⟨hal-04148137⟩
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