# Abstract voronoi diagrams and their applications

## Abstract

Given a set *S* of *n* points in the plane, and for every two of them a separating Jordan curve, the abstract Voronoi diagram *V*(*S*) can be defined, provided that the regions obtained as the intersections of all the “halfplanes” containing a fixed point of *S* are path-connected sets and together form an exhaustive partition of the plane. This definition does not involve any notion of distance. The underlying planar graph, \(\hat V\)(*S*), turns out to have *O*(*n*) edges and vertices. If *S*=*L* ∪ *R* is such that the set of edges separating *L*-faces from *R*-faces in \(\hat V\)(*S*) does not contain loops then \(\hat V\)(*L*) and \(\hat V\)(*R*) can be merged within *O*(*n*) steps giving \(\hat V\)(*S*). This result implies that for a large class of metrics *d* in the plane the *d*-Voronoi diagram of *n* points can be computed within optimal *O*(*n* log *n*) time. Among these metrics are, for example, the symmetric convex distance functions as well as the metric defined by the city layout of Moscow or Karlsruhe.

## Keywords

Voronoi diagram metric computational geometry## Preview

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