License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.28
URN: urn:nbn:de:0030-drops-186812
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18681/
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Cabello, Sergio ; Giannopoulos, Panos

On k-Means for Segments and Polylines

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LIPIcs-ESA-2023-28.pdf (0.7 MB)

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)}).

BibTeX - Entry

@InProceedings{cabello_et_al:LIPIcs.ESA.2023.28,
  author =	{Cabello, Sergio and Giannopoulos, Panos},
  title =	{{On k-Means for Segments and Polylines}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{28:1--28:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18681},
  URN =		{urn:nbn:de:0030-drops-186812},
  doi =		{10.4230/LIPIcs.ESA.2023.28},
  annote =	{Keywords: k-means clustering, segments, polylines, Hausdorff distance, Fr\'{e}chet mean}
}

Keywords: k-means clustering, segments, polylines, Hausdorff distance, Fréchet mean
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

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