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DOI: 10.4230/LIPIcs.WADS.2025.38

Clustering Point Sets Revisited

Authors: Md. Billal Hossain and Benjamin Raichel

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract

In the sets clustering problem one is given a collection of point sets 𝒫 = {P_1,… P_m} in ℝ^d, where for any set of k centers in ℝ^d, each P_i is assigned to its nearest center as determine by some local cost functions. The goal is then to select a set of k centers to minimize some global cost function of the corresponding local assignment costs. Specifically, we consider either summing or taking the maximum cost over all P_i, where for each P_i the cost of assigning it to a center c is either max_{p ∈ P_i} ‖c-p‖, ∑_{p ∈ P_i} ‖c-p‖, or ∑_{p ∈ P_i} ‖c-p‖². Different combinations of the global and local cost functions naturally generalize the k-center, k-median, and k-means clustering problems. In this paper, we improve the prior results for the natural generalization of k-center, give the first result for the natural generalization of k-means, and give results for generalizations of k-median and k-center which differ from those previously studied.

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Md. Billal Hossain and Benjamin Raichel. Clustering Point Sets Revisited. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hossain_et_al:LIPIcs.WADS.2025.38,
  author =	{Hossain, Md. Billal and Raichel, Benjamin},
  title =	{{Clustering Point Sets Revisited}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{38:1--38:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.38},
  URN =		{urn:nbn:de:0030-drops-242693},
  doi =		{10.4230/LIPIcs.WADS.2025.38},
  annote =	{Keywords: Clustering, k-center, k-median, k-means}
}

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Clustering Point Sets Revisited

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