On the Planar Two-Center Problem and Circular Hulls

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