Discrete descriptions of geometric objects

Abstract

The core of this thesis is the wireless localization or internet cafe problem, a relatively new problem which evolved within the field of art gallery problems: An internet cafe wants to provide wireless internet access for its customers. How can they prevent customers in the competitor’s cafe next door from using their internet connection? The customers should have the possibility to prove that they are inside while anybody outside should not be able to provide such a proof. We place point stations (usually called guards) that broadcast a unique signal within an angular range. The signals are not blocked by the walls of the cafe. The goal is to place the guards and adjust their angular range in such a way that a customer in the cafe can prove to be inside by naming the keys it receives whereas anyone outside of the cafe cannot provide such a proof. This can be phrased as a constructive solid geometry (CSG) problem: Describe a simple polygon by a formula over primitive objects—in our case the cones representing the area of broadcast of the guards—using the set operators union and intersection. Instead of asking for a description using cones one can try to use other primitives such as triangles. More generally, we are interested in simple and compact descriptions of polygons and polyhedra using different classes of primitives. Decomposing polygons into primitive objects is a important problem in computational geometry. Besides direct applications, for example in pattern recognition or VLSI design, decomposition is often the first step of geometric algorithms: A polygon is first decomposed into simple pieces, then the problem is solved for each piece individually, and then the partial solutions are combined to give a solution for the original polygon. How can a complex geometric object be described by simpler objects? We provide a general framework in order to give this question a mathematical meaning. Depending on which objects and which primitives are used, and what operations are allowed to combine the primitives, we get a rich variety of problems that are precise variants of the informal quest for “nice descriptions”. Then, we turn our attention to specific variants. It is well-known that n/3 guards are sufficient and sometimes necessary to watch an art gallery with n walls. Similarly, we try to bound the number of guards needed for wireless localization. On one hand, 3n/4 guards are sufficient to describe any simple polygon with n non-parallel edges. On the other hand, there is a class of simple