Algorithmic canonical stratifications of simplicial complexes

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 C P, P pi=i is maximal among all open subposets U C P pi=i in its closure such that the restriction of the local Z-homology sheaf of 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 oo-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 < 3, and show that it runs in polynomial time. In particular, an n -dimensional simplicial complex with size s and n < 3 can be processed in O(s2) time or O(s) given one further assumption on the structure. Processing Delaunay triangulations of 2-spheres and 3-balls provide experimental confirmation of this linear running time.(c) 2022 Elsevier B.V. All rights reserved.