23 research outputs found
Ball hulls, ball intersections, and 2-center problems for gauges
The notions of ball hull and ball intersection of nite sets, important in Banach space theory, are extended from normed planes to generalized normed planes, i.e., to (asymmetric) convex distance functions which are also called gauges. In this more general setting we derive various new results about these notions and their relations to each other. Further on, we extend the known 2-center problem and a modified version of it from the Euclidean situation to norms and gauges or, in other words, from Euclidean circles to arbitrary closed convex curves. We derive algorithmical results on the construction of ball hulls and ball intersections, and computational approaches to the 2-center problem with constrained circles and, in case of strictly convex norms and gauges, for the fixed 2-center problem are also given
Minimum Perimeter-Sum Partitions in the Plane
Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time
On the Planar Two-Center Problem and Circular Hulls
Given a set of points in the Euclidean plane, the two-center problem
is to find two congruent disks of smallest radius whose union covers all points
of . Previously, Eppstein [SODA'97] gave a randomized algorithm of
expected time and Chan [CGTA'99] presented a deterministic
algorithm of time. In this paper, we propose an
time deterministic algorithm, which improves Chan's
deterministic algorithm and matches the randomized bound of Eppstein. If is
in convex position, then we solve the problem in
deterministic time. Our results rely on new techniques for dynamically
maintaining circular hulls under point insertions and deletions, which are of
independent interest.Comment: A preliminary version to appear in SoCG 202