Geometric optimization and querying: exact & approximate

This thesis has two main parts. The first part deals with the stage illumination problem. Given a stage represented by a line segment L and a set of lightsources represented by a set of points S in the plane, assign powers to the lightsources such that every point on the stage receives a sufficient amount, e.g. one unit, of light while minimizing the overall power consumption. By assuming that the amount of light arriving from a fixed lightsource decreases rapidly with the distance from the lightsource, this becomes an interesting geometric optimization problem. We present different solutions, based on convex optimization, discretization and linear programming, as well as a purely combinatorial approximation algorithm. Some experimental results are also provided. In the second part of this thesis, we are concerned with two different geometric problems whose solutions are based on the construction of a data structure that would allow for efficient queries. The central idea of our data structures is the well-separated pair decomposition. The first problem we address is the k-hop restricted shortest path under the power-euclidean distance function. Given a set P of n points in the plane and the distance function |pq|δ +Cp for some constant δ > 1, nonnegative offset cost Cp and p,q ∈ P, where |pq| denotes the Euclidean distance between p and q, we consider the problem of finding paths between any pair of points that minimize the lenght of the path and do not use more than some constant number k of hops. Known exact algorithms for this problem required Ω(n logn) per query pair (p,q). We relax the exactness requirement and only require approximate (1+ε) solutions which allows us to derive schemes which guarantee constant query time using linear space and O(n logn) preprocessing time. The dependence on ε is polynomial in 1/ε. We also develop a tool that might be of independent interest: For any pair of points p,q ∈ P report in constant time the cluster pair (A,B) representing (p,q) in a well-separated pair decomposition of P. The second problem in this part is so-called cone-restricted nearest neighbor. For a given point set in Euclidean space we consider the problem of finding (approximate) nearest neighbors of a query point but restricting only to points that lie within a fixed cone with apex at the query point. We investigate the structure of the Voronoi diagram induced by this notion of proximity and present approximate and exact data structures for answering cone-restricted nearest neighbor queries. In particular, we develop an approximate Voronoi diagram of size O((n/εd) log(1/ε)) that can be used to answer cone-restricted nearest neighbor queries in O(log(n/ε)) time.