2026-07-21T14:16:24Z https://drops.dagstuhl.de/oai
oai:drops-oai.dagstuhl.de:12269 2024-03-06T10:49:29Z ddc:004 open_access
Clustering Moving Entities in Euclidean Space Durocher, Stephane Hassan, Md Yeakub trajectories clustering moving entities k-CENTER algorithms Clustering is a fundamental problem of spatio-temporal data analysis. Given a set 𝒳 of n moving entities, each of which corresponds to a sequence of Ο„ time-stamped points in ℝ^d, a k-clustering of 𝒳 is a partition of 𝒳 into k disjoint subsets that optimizes a given objective function. In this paper, we consider two clustering problems, k-Center and k-MM, where the goal is to minimize the maximum value of the objective function over the duration of motion for the worst-case input 𝒳. We show that both problems are NP-hard when k is an arbitrary input parameter, even when the motion is restricted to ℝ. We provide an exact algorithm for the 2-MM clustering problem in ℝ^d that runs in O(Ο„ d nΒ²) time. The running time can be improved to O(Ο„ n log{n}) when the motion is restricted to ℝ. We show that the 2-Center clustering problem is NP-hard in ℝ². Our 2-MM clustering algorithm provides a 1.15-approximate solution to the 2-Center clustering problem in ℝ². Moreover, finding a (1.15-Ξ΅)-approximate solution remains NP-hard for any Ξ΅ >0. For both the k-MM and k-Center clustering problems in ℝ^d, we provide a 2-approximation algorithm that runs in O(Ο„ d n k) time. Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik Stephane Durocher and Md Yeakub Hassan 2020 Is Part Of LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SWAT.2020.22 urn:nbn:de:0030-drops-122698 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.22 eng https://creativecommons.org/licenses/by/3.0/legalcode