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Optimal Algorithm for the Planar Two-Center Problem (2007.08784v5)

Published 17 Jul 2020 in cs.CG

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 $O(n\log2 n)$ time. In this paper, we present an $O(n\log n)$-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

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