We study a fundamental problem in Computational Geometry, the \emph{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。该问题的输入为平面上包含$n$个点的集合$S$，目标是找到两个最小的全等圆盘，使其并集覆盖$S$中的所有点。一个长期未解决的开放问题是：是否存在时间复杂度为$O(n\log n)$的算法来解决平面双中心问题，从而匹配Eppstein [SODA'97]提出的$\Omega(n\log n)$下界。为此，研究人员在数十年间进行了大量努力。此前的最佳算法由Wang [SoCG'20]给出，其时间复杂度为$O(n\log^2 n)$。本文提出一种$O(n\log n)$时间的（确定性）算法，彻底解决了这一开放问题。