Optimal Algorithm for the Planar Two-Center Problem
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.
two-center
r-coverage
disk coverage
circular hulls
Theory of computation~Computational geometry
Theory of computation~Design and analysis of algorithms
40:1-40:15
Regular Paper
https://arxiv.org/abs/2007.08784
Kyungjin
Cho
Kyungjin Cho
Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
https://orcid.org/0000-0003-2223-4273
Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2023-00209069).
Eunjin
Oh
Eunjin Oh
Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
https://orcid.org/0000-0003-0798-2580
Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2023-00209069).
Haitao
Wang
Haitao Wang
Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
https://orcid.org/0000-0001-8134-7409
Supported in part by NSF under Grant CCF-2300356.
Jie
Xue
Jie Xue
Department of Computer Science, New York University Shanghai, China
https://orcid.org/0000-0001-7015-1988
10.4230/LIPIcs.SoCG.2024.40
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Kyungjin Cho, Eunjin Oh, Haitao Wang, and Jie Xue
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