Generating Cofaces in Vietoris--Rips Filtration Order

TL;DR

This paper introduces efficient algorithms for generating cofaces in Vietoris-Rips filtrations directly in filtration order, improving the process of computing persistent cohomology and simplifying the construction of simplicial complexes.

Contribution

It presents novel algorithms that generate cofaces in filtration order directly, reducing the need for post-processing sorting steps in Vietoris-Rips filtrations.

Findings

Algorithms generate cofaces with the same diameter efficiently.

Methods produce simplices in filtration order by construction.

Facilitates direct coboundary matrix computation in persistent cohomology.

Abstract

Cofaces -- simplices that contain a given simplex -- have multiple important uses in generating and using a Vietoris-Rips filtration: both in creating the coboundary matrix for computing persistent cohomology, and for generating the ordered sequence of simplices in the first place. Traditionally, most methods have generated simplices first, and then sorted them in filtration order after the generation step. In this paper, we propose fast algorithms for generating the sequence of simplices by generating cofaces of a given simplex with the same diameter, which by construction produces simplices in filtration order, and for generating additional cofaces in filtration order using sorted neighborhood lists in order to generate coboundaries directly in filtration order.