In this paper, we consider the \emph{planar two-center problem}: Given a set $S$ of $n$ points in the plane, the goal is to find two smallest congruent disks whose union contains all points of $S$. We present an $O(n\log n)$-time algorithm for the planar two-center problem. This matches the best known lower bound of $\Omega(n\log n)$ as well as improving the previously best known algorithms which takes $O(n\log^2 n)$ time.

翻译：本文考虑平面双中心问题：给定平面上$n$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。我们提出了一种时间复杂度为$O(n\log n)$的平面双中心问题算法。该结果不仅匹配了已知最优下界$\Omega(n\log n)$，且改进了先前最优算法的$O(n\log^2 n)$时间复杂度。