A general approach to dominance in the plane

Abstract

Given two points p and q and a (finite) set of points O in the plane, p is said to dominate q with respect to O if p dominates q and there is no o ϵ O such that p dominates o and o dominates q. In other words, O is a set of obstacles that might block the “rectangular view” from p to q. Given sets P and O we are interested in determining all pairs ( p, q) ϵ P × P such that p dominates q with respect to O. This generalizes notions of direct dominance and rectangular visibility that have been studied before. An algorithm is presented that solves the problem in optimal time O( n log n + k), where n is the size of P ∪ O and k is the number of answers. We also study query versions of the problem in which we ask for all points that are dominated with respect to O by a given query point. Both static and dynamic data structures are presented. Finally, the notion of dominance with respect to obstacles is extended to obstacle sets that may contain arbitrary objects.