{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/algorithmic-canonical-stratifications-of","title":"Algorithmic canonical stratifications of simplicial complexes","arxiv_id":"1808.06568","date":"2018-08-20","proceeding":null,"authors":["Ryo Asai","Jay Shah"],"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.","url_abs":"https://arxiv.org/abs/1808.06568v3","url_pdf":"https://arxiv.org/pdf/1808.06568v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"algorithmic-canonical-stratifications-of","repo_url":"https://github.com/ColfaxResearch/CanonicalStratification","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}