Computer Science > Computational Geometry
[Submitted on 17 Jul 2020 (v1), last revised 29 Apr 2024 (this version, v4)]
Title:Optimal Algorithm for the Planar Two-Center Problem
View PDF HTML (experimental)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 $O(n\log n)$-time algorithm for planar two-center, matching the $\Omega(n\log n)$ 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 $O(n\log^2 n)$ time. In this paper, we present an $O(n\log n)$-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.
Submission history
From: Haitao Wang [view email][v1] Fri, 17 Jul 2020 07:20:27 UTC (394 KB)
[v2] Tue, 21 Nov 2023 16:14:52 UTC (525 KB)
[v3] Tue, 12 Dec 2023 17:44:50 UTC (527 KB)
[v4] Mon, 29 Apr 2024 18:50:10 UTC (469 KB)
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