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Optimal Algorithm for the Planar Two-Center Problem

Authors Kyungjin Cho , Eunjin Oh , Haitao Wang , Jie Xue

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Author Details

Kyungjin Cho
  • Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
Eunjin Oh
  • Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Jie Xue
  • Department of Computer Science, New York University Shanghai, China

Funding

  • Cho, Kyungjin: Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2023-00209069).
  • Oh, Eunjin: Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2023-00209069).
  • Wang, Haitao: Supported in part by NSF under Grant CCF-2300356.

Cite AsGet BibTex

Kyungjin Cho, Eunjin Oh, Haitao Wang, and Jie Xue. Optimal Algorithm for the Planar Two-Center Problem. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.40

Abstract

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set S of n points in the plane and the goal is to find two smallest congruent disks whose union contains all points of S. A longstanding open problem has been to obtain an O(nlog n)-time algorithm for planar two-center, matching the Ω(nlog n) lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in O(nlog² n) time. In this paper, we present an O(nlog n)-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • two-center
  • r-coverage
  • disk coverage
  • circular hulls

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