Voronoi Diagrams

Abstract

Let S be a finite point set in ℝⁿ. Since S is compact, for every point x∈ℝⁿ there exists a closest point in S (which is not necessarily unique) with respect to the Euclidean norm ∥⋅∥. The set of all points in ℝⁿ that have a fixed point s∈S as their nearest “neighbor” is a polyhedron. This mapping induces a decomposition of ℝⁿ into polyhedral “regions”, the Voronoi diagram of S. Numerous applications of computational geometry begin with the computation of a Voronoi diagram.

We will first study the geometry of single Voronoi regions. To be able to discuss the arrangement of all Voronoi regions, we will introduce the general concept of a polyhedral complex. The main result of this chapter is the relationship between Voronoi diagrams and the convex hull problem from the previous chapter. We conclude the chapter by discussing an algorithm for the computation of Voronoi diagrams in the plane and its application to the post-office problem from the introduction.