On the Planar Two-Center Problem and Circular Hulls

02/19/2020

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by   Haitao Wang, et al.

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Given a set S of n points in the Euclidean plane, the two-center problem is to find two congruent disks of smallest radius whose union covers all points of S. Previously, Eppstein [SODA'97] gave a randomized algorithm of O(nlog^2n) expected time and Chan [CGTA'99] presented a deterministic algorithm of O(nlog^2 nlog^2log n) time. In this paper, we propose an O(nlog^2 n) time deterministic algorithm, which improves Chan's deterministic algorithm and matches the randomized bound of Eppstein. If S is in convex position, then we solve the problem in O(nlog nloglog n) deterministic time. Our results rely on new techniques for dynamically maintaining circular hulls under point insertions and deletions, which are of independent interest.