We consider the following Euclidean 2-center problem. Given n disks in the plane, find two smallest congruent disks such that every input disk intersects at least one of the two congruent disks. We present a deterministic algorithm for the problem that returns an optimal pair of congruent disks in $O(n^2{\log }^{3}n/\log \log n)$ time. We also present a randomized algorithm with $O(n^2{\log }^{2}n/\log \log n)$ expected time. These results improve upon the previously best deterministic and randomized algorithms, making a step closer to the optimal algorithms for the problem. We show that the same algorithms also work for two variants of the problem, the 2-piercing problem and the restricted 2-cover problem on disks. We also consider the 2-center problem and its two variants on n convex polygons, each with $O(1)$ vertices in the plane and present efficient algorithms for them.

我们考虑以下欧几里德 2 中心问题。给定平面中的n 个圆盘，找到两个最小的全等圆盘，使得每个输入圆盘至少与两个全等圆盘中的一个相交。我们提出了一种确定性算法，用于返回一对最佳全等圆盘的问题$○(n^{\mathrm{2个}}{\mathrm{日志}}^{\mathrm{3个}}n/\mathrm{日志}\mathrm{日志}n)$时间。我们还提出了一个随机算法$○(n^{\mathrm{2个}}{\mathrm{日志}}^{\mathrm{2个}}n/\mathrm{日志}\mathrm{日志}n)$预计时间。这些结果改进了以前最好的确定性和随机算法，使问题的最佳算法更近了一步。我们表明，相同的算法也适用于该问题的两个变体，即 2-piercing 问题和磁盘上的受限 2-cover 问题。我们还考虑了 2 中心问题及其在n 个凸多边形上的两个变体，每个变体都有$○(\mathrm{1个})$平面中的顶点并为它们提供有效的算法。