On Proximity Problems In Euclidean Spaces

Abstract

In this work, we focus on two kinds of problems involving the proximity of geometric objects. The first part revolves around intersection detection problems. In this setting, we are given two (or more) geometric objects and we are allowed to preprocess them. Then, the objects are translated and rotated within a geometric space, and we need to efficiently test if they intersect in these new positions. We develop representations of convex polytopes in any (constant) dimension that allow us to perform this intersection test in logarithmic time. In the second part of this work, we turn our attention to facility location problems. In this setting, we are given a set of sites in a geometric space and we want to place a facility at a specific place in such a way that the distance between the facility and its farthest site is minimized. We study first the constrained version of the problem, in which the facility can only be place within a given geometric domain. We then study the facility location problem under the geodesic metric. In this setting, we consider a different way to measure distances: Given a simple polygon, we say that the distance between two points is the length of the shortest path that connects them while staying within the given polygon. In both cases, we present algorithms to find the optimal location of the facility. In the process of solving facility location problems, we rely heavily on geometric structures called Voronoi diagrams. These structures summarize the proximity information of a set of simple'' geometric objects in the plane and encode it as a decomposition of the plane into interior disjoint regions whose boundaries define a plane graph. We study the problem of constructing Voronoi diagrams incrementally by analyzing the number of edge insertions and deletions needed to maintain its combinatorial structure as new sites are added.