2026-07-25T03:15:51Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20020 2024-06-06T06:21:17Z ddc:004 open_access

Morse Theory for the k-NN Distance Function Reani, Yohai Bobrowski, Omer Applied topology Morse theory Distance function k-nearest neighbor We study the k-th nearest neighbor distance function from a finite point-set in ℝ^d. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-k Delaunay mosaics, and random k-fold coverage. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Yohai Reani and Omer Bobrowski 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.75 urn:nbn:de:0030-drops-200202 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.75 eng https://creativecommons.org/licenses/by/4.0/legalcode