Applications of Parametric Searching in Geometric Optimization

Abstract

We present several applications in computational geometry of Megiddo′s parametric searching technique. These applications include: (1) Finding the minimum Hausdorff distance in the Euclidean metric between two polygonal regions under translation; (2) Computing the biggest line segment that can be placed inside a simple polygon; (3) Computing the smallest width annulus that contains a given set of given points in the plane; (4) Given a set of n points in 3-space, finding the largest radius r such that if we place a ball of radius r around each point, no segment connecting a pair of points is intersected by a third ball. Besides obtaining efficient solutions to all these problems (which, in every case, either improve considerably previous solutions or are the first nontrivial solutions to these problems), our goal is to demonstrate the versatility of the parametric searching technique.