We consider the planar Euclidean two-center problem in which given n points in the plane we are to find two congruent disks of the smallest radius covering the points. We present a deterministic $O(n\log n)$-time algorithm for the case that the centers of the two optimal disks are close to each other, that is, the overlap of the two optimal disks is a constant fraction of the disk area. We also present a deterministic $O(n\log n)$-time algorithm for the case that the input points are in convex position. Both results improve the previous best $O(n\log n\log \log n)$ bound on the problems.

中文翻译：

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高效的平面两中心算法

我们考虑平面欧几里德两中心问题，在该问题中，给定平面中的n个点，我们将找到两个半径最小的全等圆盘覆盖这些点。我们提出确定性$Ø（ñ\mathrm{日志}ñ）$两个最佳磁盘的中心彼此靠近的情况下的实时算法，即两个最佳磁盘的重叠是磁盘区域的恒定分数。我们还提出了确定性$Ø（ñ\mathrm{日志}ñ）$输入点处于凸位置的情况下的时间算法。两项结果均提高了以往的最佳水平$Ø（ñ\mathrm{日志}ñ\mathrm{日志}\mathrm{日志}ñ）$ 束手无策。