Papers › Algorithmic canonical stratifications of simplicial complexes
Algorithmic canonical stratifications of simplicial complexes
Ryo Asai, Jay Shah
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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 π of the poset P of simplices of a simplicial complex K, such that for each strata P_(π=i) ⊂P, P_(π=i) is maximal among all open subposets U ⊂P_(π=i) in its closure such that the restriction of the local ℤ-homology sheaf of P_(π=i) to U is locally constant. Passage to the localization of P dictated by π then attaches a canonical stratified homotopy type to K. Using ∞-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 provides experimental confirmation of this linear running time.
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