Worst-Case Examples for the Computation of Persistent Homology

TL;DR

This paper constructs explicit worst-case examples for the standard persistent homology reduction algorithm, demonstrating their robustness under subdivisions and their realizability as Vietoris-Rips complexes, aiding understanding of algorithmic limits.

Contribution

It introduces a new class of worst-case examples for persistent homology computation, with explicit construction methods and experimental comparisons.

Findings

Strip examples remain worst-case after subdivisions.

Examples can be realized as clique complexes of filtered graphs.

Experiments compare different reduction algorithm versions.

Abstract

We construct worst-case examples for the standard reduction algorithm for computing persistent homology. Our constructions are similar to the worst-case examples introduced by Morozov, but we replace the single-triangle arrangement with a strip of base and fin triangles. This structure allows us to give an explicit algorithm for their construction and to perform experiments comparing the runtime of different versions of the reduction algorithm. We further show that, after suitable edge and triangle subdivisions, these strip examples remain worst-case and can be realized as clique complexes of filtered graphs, and hence as Vietoris--Rips complexes of finite point clouds for a sequence of scale parameters.