An Efficient Representation for Filtrations of Simplicial Complexes

TL;DR

This paper introduces the Critical Simplex Diagram, a compact data structure for efficiently representing filtrations of simplicial complexes, supporting fast operations and optimal storage, with applications to Flag and Delaunay complexes.

Contribution

The paper presents the Critical Simplex Diagram, a novel, space-efficient data structure for filtrations, improving on existing methods and supporting fast queries and construction algorithms.

Findings

CSD is space-efficient and supports fast operations.

CSD is optimal in storage space.

Fast construction algorithms for specific complexes.

Abstract

A filtration over a simplicial complex $K$ is an ordering of the simplices of $K$ such that all prefixes in the ordering are subcomplexes of $K$. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica '14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest. In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica '17]. Our data structure allows one to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Finally, we show that the CSD representation admits fast construction algorithms for Flag complexes and relaxed Delaunay complexes.