Stability of voronoi neighborship under perturbations of the sites

This paper considers the eeect of site perturbations on Voronoi diagrams, where the sites are points in the plane. Given a bound on the distance that any site may move, we ask which pairs of Voronoi neighbors may become non-neighbors and which are guaranteed to remain neighbors. A pair of the second kind is called stable. The paper shows necessary and suucient conditions for stability. Algorithms are proposed for deciding stability with regard to a given perturbation bound and for determining the supremal bound up to which a particular pair of Voronoi neighbors remains stable. 1. Introduction The Voronoi diagram of a set of points is a powerful tool for proximity-related computations. It is used by many algorithms in computational geometry and related elds. When dealing with real-world data, errors of measurement can have a non-negligible impact on the result. In the case of Voronoi diagrams, even very slight perturbations of the sites may change the diagram's topology. If an algorithm makes decisions based on a Voronoi diagram, it may beneet from the knowledge how reliable the diagram as a whole or certain parts of it are. The stability which we investigate here can serve as a measure of reliability. This paper is concerned with the eeect that site perturbations have on the topology of a Voronoi diagram in the plane. In particular, we want to know which pairs of strong Voronoi neighbors arèsepa-rated' by the perturbation and which are not. It is assumed that the reader is familiar with planar