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On Small Pair Decompositions for Point Sets

Authors Kevin Buchin , Jacobus Conradi , Sariel Har-Peled , Antonia Kalb , Abhiruk Lahiri , Lukas Plätz , Carolin Rehs , Sampson Wong

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LIPIcs.ESA.2026.102.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Well-separated pair decomposition
  • Semi-separated pair decomposition
  • Approximation algorithms
  • Doubling metrics
  • Computational geometry

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Abstract

We study the problem of computing a minimum-size Well-Separated Pair Decomposition (WSPD) of a given point set. We obtain the following results for the minimum-size WSPD: (1) a constant-factor approximation in doubling metrics, (2) a simple 3-approximation in ℝ, and (3) an NP-hardness proof in ℝ². We also provide an optimal output-sensitive runtime for the algorithm in doubling metrics and an implementation of the 3-approximation algorithm in ℝ. Furthermore, we introduce a new pair-decomposition for point sets in a metric space. It is defined using a relaxed requirement that, for all pairs {X,Y} in the decomposition, all the distances of pairs of points in X × Y are equal up to a factor in [1 ± ε]. Surprisingly, we show that in a general metric space, one can compute such a decomposition of size O(n/ε log n), which is dramatically smaller than the Θ(n²) bound for WSPDs. For a point set in ℝ^d, the bound improves to O(d n/ε log 1/ε).

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Kevin Buchin, Jacobus Conradi, Sariel Har-Peled, Antonia Kalb, Abhiruk Lahiri, Lukas Plätz, Carolin Rehs, and Sampson Wong. On Small Pair Decompositions for Point Sets. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 102:1-102:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.102

Author Details

Kevin Buchin
  • Department of Computer Science, TU Dortmund University, Germany
Jacobus Conradi
  • Institute of Computer Science, Universität Bonn, Germany
  • Basic Algorithms Research Copenhagen, University of Copenhagen, Denmark
Sariel Har-Peled
  • Department of Computer Science, University of Illinois, Urbana, IL, USA
Antonia Kalb
  • Department of Computer Science, TU Dortmund University, Germany
Abhiruk Lahiri
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
  • Heinrich Heine Universität Düsseldorf, Germany
Lukas Plätz
  • Institute of Computer Science, Ruhr University Bochum, Germany
Carolin Rehs
  • Department of Computer Science, TU Dortmund University, Germany
  • Department of Mathematics and Computing Science, TU Eindhoven, The Netherlands
Sampson Wong
  • Basic Algorithms Research Copenhagen, University of Copenhagen, Denmark

Funding

  • Lahiri, Abhiruk: Partially supported by the Strategic Research Fund, Heinrich Heine Universität, Düsseldorf.
  • Rehs, Carolin: Partly supported by the PRIME programme of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Research, Technology and Space (BMFTR).

Acknowledgements

This paper results from the Rhine-Ruhr Computational Geometry Workshop 2025. We thank the organizers from FernUniversität Hagen.

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