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：我们研究计算几何中的一个基本问题——平面双中心问题。在该问题中，输入为平面上的一个包含 $n$ 个点的集合 $S$，目标是找到两个最小的全等圆盘，使得它们的并集覆盖 $S$ 中的所有点。一个长期未解决的开放问题是：是否存在一个时间复杂度为 $O(n\log n)$ 的平面双中心算法，以匹配 Eppstein [SODA'97] 给出的 $\Omega(n\log n)$ 下界。为此，研究人员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 $O(n\log^2 n)$。本文给出一种确定性的 $O(n\log n)$ 时间平面双中心算法，彻底解决了这一开放问题。