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Worst-Case Examples for the Computation of Persistent Homology

Uzay Çetin, Ergun Yalcin

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.

Worst-Case Examples for the Computation of Persistent Homology

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.
Paper Structure (7 sections, 9 equations, 1 figure, 3 algorithms)

This paper contains 7 sections, 9 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Matrix Reduction Algorithms
  3. The Standard Algorithm for Reduction
  4. Reduction Algorithm with a Twist
  5. Reduction Algorithm with Look-ahead
  6. The runtime of reduction algorithms
  7. An Alternative Worst-case Example: Strip

Figures (1)

  • Figure 1:

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7