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Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams

Authors Kevin Buchin , Mark Joachim Krallmann , Frank Staals

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LIPIcs.ESA.2026.124.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Data structures design and analysis
Keywords
  • Range searching
  • smallest enclosing disk
  • farthest point Voronoi diagram

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Abstract

Let S be a set of n points in R². Our goal is to preprocess S to efficiently compute the smallest enclosing disk of the points in S that lie inside an axis-aligned query rectangle. Previous data structures for this problem achieve a query time of O(log⁶ n) with O(n log² n) preprocessing time and space by lifting the points to 3D, dualizing them into polyhedra, and searching through their intersections. We present a significantly simpler approach, solely based on 2D geometric structures, specifically 2D farthest-point Voronoi diagrams. Our approach achieves a deterministic query time of O(log⁴ n) and, via randomization, an expected query time of O(log^{5/2} n log log n) with the same preprocessing bounds.

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Kevin Buchin, Mark Joachim Krallmann, and Frank Staals. Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 124:1-124:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.124

Author Details

Kevin Buchin
  • Department of Computer Science, TU Dortmund University, Germany
Mark Joachim Krallmann
  • Department of Computer Science, TU Dortmund University, Germany
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Krallmann, Mark Joachim: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 550797858.

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