Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system tha...

Abstract Voronoi diagrams were introduced by R. Klein (1988) as an axiomatic basis of Voronoi diagrams. We show how to construct abstract Voronoi diagrams in time O(n log n) by a randomized algorithm, which is based on Clarkson and Shor’s randomized incremental construction technique (1989). The new algorithm has the following advantages over previous algorithms: l It can handle a much wider cl...

We consider the Euclidean Voronoi diagram for a set of n parallel halflines in R. A relation of this diagram to planar power diagrams is shown, and is used to analyze its geometric and topological properties. Moreover, a simple plane-sweep algorithm is given that computes the Voronoi diagram for parallel halflines at logarithmic cost per face.

For a set P of n points in Rd, we define a new type of space decomposition. The new diagram provides an ε-approximation to the distance function associated with the Voronoi diagram of P , while being of near linear size, for d ≥ 2. This contrasts with the standard Voronoi diagram that has Ω ( ndd/2e ) complexity in the worst case.

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. Algorith...

Euclidean Voronoi diagram of spheres in 3-dimensional space has not been explored as much as it deserves even though it has significant potential impacts on diverse applications in both science and engineering. In addition, studies on the data structure for its topology have not been reported yet. Presented in this paper is the topological representation for Euclidean Voronoi diagram of spheres...