A linear expected-time algorithm for computing planar relative neighbourhood graphs

Abstract

A new algorithm for computing the relative neighbourhood graph (RNG) of a planar point set is given. The expected running time of the algorithm is linear for a point set in a unit square when the points have been generated by a homogeneous planar Poisson point process. The worst-case running time is quadratic on the number of the points. The algorithm proceeds in two steps. First, a supergraph of the RNG is constructed with the aid of a cell organization of the points. Here, a point is connected by an edge to some of its nearest neighbours in eight regions around the point. The nearest region neighbours are chosen in a special way to minimize the costs. Second, extra edges are pruned from the graph by a simple scan.