Applications of a semi-dynamic convex hull algorithm

Abstract

We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for three problems in geometry and scheduling. The new convex hull results ate logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. We then apply these results to solve the following problems: (1) [matching] given n red points and n blue points in the plane, find a matching of red and blue points (by line segments) in which no two edges cross, (2) [scheduling] given n jobs with due dates, linear penalties for late completion, and a single machine on which to process them, find a schedule of jobs that minimizes the maximum penalty, and (3) [covering] given n points in the plane and two real numbers r1 and r2 determine if one can cover all the points with two disks of radii r1 and r2. Our time bounds are O(n log n) for problems (1) and (2) and 0(n2 log n) for problem (3), an improvement by a factor of log n over the previous bounds.