2026-07-24T00:34:41Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:18681 2024-03-06T11:02:26Z ddc:004 open_access

On k-Means for Segments and Polylines Cabello, Sergio Giannopoulos, Panos k-means clustering segments polylines Hausdorff distance Fréchet mean We study the problem of k-means clustering in the space of straight-line segments in ℝ² under the Hausdorff distance. For this problem, we give a (1+ε)-approximation algorithm that, for an input of n segments, for any fixed k, and with constant success probability, runs in time O(n + ε^{-O(k)} + ε^{-O(k)} ⋅ log^O(k) (ε^{-1})). The algorithm has two main ingredients. Firstly, we express the k-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron [Antoine Vigneron, 2014] to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg [Dan Feldman and Michael Langberg, 2011; Feldman et al., 2020]. Our results can be extended to polylines of constant complexity with a running time of O(n + ε^{-O(k)}). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Sergio Cabello and Panos Giannopoulos 2023 Is Part Of LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2023.28 urn:nbn:de:0030-drops-186812 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.28 eng https://creativecommons.org/licenses/by/4.0/legalcode