Updates on Voronoi Diagrams

Abstract

The sweep line technique has been recently adapted to the sphere in order to build Voronoi diagrams of points on its surface. The resulting algorithm has proved to be simple and efficient, outperforming the freely available alternatives, which compute convex hulls of point sets in 3D. In this paper, we introduce two sweep algorithms for updating Voronoi diagrams, one for deleting and another for inserting a site, which are applicable to points on the sphere surface or on the plane. The algorithms operate directly on the doubly connected edge lists that implement the Voronoi diagram. This makes them preferable when the intended data is the Voronoi diagram, which happens, for instance, when natural neighbour interpolation is performed. Both algorithms require linear space. Besides, insertion runs in linear time, which is worst-case optimal, whereas deletion runs in super-linear time. Although the deletion running time is not asymptotically optimal, both algorithms cope very well with degenerated cases, are efficient, and are practical to implement. Experimental results in both domains reveal that their performances are better than or similar to those of the CGAL library, which work on Delaunay triangulations.