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.

翻译：我们研究了计算几何中的一个基本问题——平面双中心问题。该问题的输入是平面上包含$n$个点的集合$S$，目标是找到两个最小的全等圆盘，使得它们的并集包含$S$中的所有点。长期以来，一个悬而未决的问题是为平面双中心问题设计一个$O(n\log n)$时间复杂度的算法，以匹配Eppstein [SODA'97]给出的$\Omega(n\log n)$下界。为此，研究人员在过去数十年间付出了大量努力。此前的最佳算法由Wang [SoCG'20]提出，其时间复杂度为$O(n\log^2 n)$。在本文中，我们提出了一种$O(n\log n)$时间（确定性）算法来解决平面双中心问题，从而彻底解决了这一开放性问题。