An Efficient Representation for Filtrations of Simplicial 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. 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. This direction has been recently pursued for the case of maintaining simplicial complexes. For instance, Boissonnat et al. [Algorithmica’17] considered storing the simplices that are maximal with respect to inclusion and Attali et al. [IJCGA’12] considered storing the simplices that block the expansion of the complex. Nevertheless, so far there has been no data structure that compactly stores the filtration of a simplicial complex, while also allowing the efficient implementation of basic operations on the complex. In this article, we propose a new data structure called the Critical Simplex Diagram (CSD), which is a variant of the Simplex Array List [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.