On the Planar Two-Center Problem and Circular Hulls

Abstract

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 $$O(n\log ^2\!n)$$ expected time and Chan (CGTA’99) presented a deterministic algorithm of $$O(n\log ^2\!n\log ^2\log n)$$ time. In this paper, we propose an $$O(n\log ^2\!n)$$ 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 $$O(n\log n\log \log n)$$ deterministic time. Our results rely on new techniques for dynamically maintaining circular hulls under point insertions and deletions, which are of independent interest.