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On the Planar Two-Center Problem and Circular Hulls

Author Haitao Wang

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  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational geometry
Keywords
  • two-center
  • disk coverage
  • circular hulls
  • dynamic data structures

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Abstract

Given a set S of n points in the Euclidean plane, the two-center problem is to find two congruent disks of smallest radius whose union covers all points of S. Previously, Eppstein [SODA'97] gave a randomized algorithm of O(nlog²n) expected time and Chan [CGTA'99] presented a deterministic algorithm of O(nlog² nlog²log n) time. In this paper, we propose an O(nlog² n) time deterministic algorithm, which improves Chan’s deterministic algorithm and matches the randomized bound of Eppstein. If S is in convex position, we solve the problem in O(nlog nlog log n) deterministic time. Our results rely on new techniques for dynamically maintaining circular hulls under point insertions and deletions, which are of independent interest.

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Haitao Wang. On the Planar Two-Center Problem and Circular Hulls. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.68

Author Details

Haitao Wang
  • Department of Computer Science, Utah State University, Logan, UT 84322, USA

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