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On k-Means for Segments and Polylines

Authors Sergio Cabello , Panos Giannopoulos

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Author Details

Sergio Cabello
  • Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
  • Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Panos Giannopoulos
  • Department of Computer Science, City, University of London, UK

Funding

  • Cabello, Sergio: Research partially supported by the Slovenian Research Agency (P1-0297, J1-2452, N1-0218, N1-0285).

Cite AsGet BibTex

Sergio Cabello and Panos Giannopoulos. On k-Means for Segments and Polylines. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 28:1-28:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.28

Abstract

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)}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • k-means clustering
  • segments
  • polylines
  • Hausdorff distance
  • Fréchet mean

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