Algorithmic canonical stratifications of simplicial complexes

TL;DR

This paper presents a new algorithm for stratifying finite simplicial complexes based on local homology, enabling canonical stratified homotopy type computation with proven correctness and polynomial runtime for low dimensions.

Contribution

The paper introduces a novel algorithm for canonical stratification of simplicial complexes using local homology, with theoretical validation and efficient implementation for dimensions up to three.

Findings

Algorithm correctly computes canonical stratification.

Runs in polynomial time for dimensions ≤ 3.

Experimental results confirm linear runtime on Delaunay triangulations.

Abstract

We introduce a new algorithm for the structural analysis of finite abstract simplicial complexes based on local homology. Through an iterative and top-down procedure, our algorithm computes a stratification $\pi$ of the poset $P$ of simplices of a simplicial complex $K$, such that for each strata $P_{\pi=i} \subset P$, $P_{\pi=i}$ is maximal among all open subposets $U \subset \overline{P_{\pi=i}}$ in its closure such that the restriction of the local $\mathbb{Z}$-homology sheaf of $\overline{P_{\pi=i}}$ to $U$ is locally constant. Passage to the localization of $P$ dictated by $\pi$ then attaches a canonical stratified homotopy type to $K$. Using $\infty$-categorical methods, we first prove that the proposed algorithm correctly computes the canonical stratification of a simplicial complex; along the way, we prove a few general results about sheaves on posets and the homotopy types of links that may be of independent interest. We then present a pseudocode implementation of the algorithm, with special focus given to the case of dimension $\leq 3$, and show that it runs in polynomial time. In particular, an $n$-dimensional simplicial complex with size $s$ and $n\leq3$ can be processed in O($s^2$) time or O($s$) given one further assumption on the structure. Processing Delaunay triangulations of $2$-spheres and $3$-balls provides experimental confirmation of this linear running time.