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Clustering with Few Disks to Minimize the Sum of Radii

Authors Mikkel Abrahamsen, Sarita de Berg, Lucas Meijer, André Nusser, Leonidas Theocharous

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Author Details

Mikkel Abrahamsen
  • University of Copenhagen, Denmark
Sarita de Berg
  • Utrecht University, The Netherlands
Lucas Meijer
  • Utrecht University, The Netherlands
André Nusser
  • University of Copenhagen, Denmark
Leonidas Theocharous
  • Eindhoven University of Technology, The Netherlands

Funding

  • Abrahamsen, Mikkel: is supported by Starting Grant 1054-00032B from the Independent Research Fund Denmark under the Sapere Aude research career programme and is part of Basic Algorithms Research Copenhagen (BARC), supported by the VILLUM Foundation grant 16582.
  • Meijer, Lucas: is supported by the Netherlands Organisation for Scientific Research (NWO) under project no. VI.Vidi.213.150.
  • Nusser, André: is part of Basic Algorithms Research Copenhagen (BARC), supported by the VILLUM Foundation grant 16582.
  • Theocharous, Leonidas: is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.

Acknowledgements

This work was initiated at the Workshop on New Directions in Geometric Algorithms, May 14-19 2023, Utrecht, The Netherlands.

Cite As Get BibTex

Mikkel Abrahamsen, Sarita de Berg, Lucas Meijer, André Nusser, and Leonidas Theocharous. Clustering with Few Disks to Minimize the Sum of Radii. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.2

Abstract

Given a set of n points in the Euclidean plane, the k-MinSumRadius problem asks to cover this point set using k disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV '12]; however, the running time of this algorithm is 𝒪(n^881), and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the k-MinSumRadius problem is that of small k. For the 2-MinSumRadius problem, a near-quadratic time algorithm with expected running time 𝒪(n² log² n log² log n) was given over 30 years ago [Eppstein '92].
We present the first improvement of this result, namely, a near-linear time algorithm to compute the 2-MinSumRadius that runs in expected 𝒪(n log² n log² log n) time. We generalize this result to any constant dimension d, for which we give an 𝒪(n^{2-1/(⌈d/2⌉ + 1) + ε}) time algorithm. Additionally, we give a near-quadratic time algorithm for 3-MinSumRadius in the plane that runs in expected 𝒪(n² log² n log² log n) time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • geometric clustering
  • minimize sum of radii
  • covering points with disks

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