In this paper we investigate the Voronoi diagram that is induced by a set of sites in the plane, where each site’s precise location is uncertain but is known to be within a particular region, and the cells of this diagram contain those points guaranteed to be closest to a particular site. We then examine the diagram for sites with disc-shaped regions of uncertainty, prove that it has linear com...

The 3D tree visualization faces multiple challenges: the election of an appropriate layout, the use of the interactions that make the data exploration easier and a metaphor that helps in the process of information understanding. A good combination of these elements will result in a visualization that effectively conveys the key features of a complex structure or system to a wide range of users ...

We discuss in this paper a method of finding skyline or non-dominated points in a set P of nP points with respect to a set S of nS sites. A point pi ∈ P is non-dominated if and only if for each pj ∈ P , j 6= i, there exists at least one point s ∈ S that is closer to pi than pj . We reduce this problem of determining non-dominated points to the problem of finding sites that have non-empty cells ...

Recently, simple conditions for well-behaved-ness of anisotropic Voronoi diagrams have been proposed. While these conditions ensure well-behaved-ness of two types of practical anisotropic Voronoi diagrams, as well as the geodesic-distance one, in any dimension, they are both prohibitively expensive to evaluate, and not well-suited for typical problems in approximation or optimization. We propos...

In this paper, we propose a postclustering process that robustly computes cluster regions at different levels of granularity through the complete Order-k Voronoi diagrams. The robustness and flexibility of the proposed method overcome the application-dependency and rigidity of traditional approaches. The proposed cluster tessellation method robustly models monotonic and nonmonotonic cluster gro...

In this paper we develop the concept of a polygon-ooset distance function and show how to compute the respective nearest-and furthest-site Voronoi diagrams of point sites in the plane. We provide optimal deterministic O(n(log n + log m) + m)-time algorithms, where n is the number of points and m is the complexity of the underlying polygon, for computing compact representations of both diagrams.

Given two point sets in the plane, we study the minimization of the bottleneck distance between a point set B and an equallysized subset of a point set A under translations. We relate this problem to a Voronoi-type diagram and derive polynomial bounds for its complexity that are optimal in the size of A. We devise efficient algorithms for the construction of such a diagram and its lexicographic...

We present a graphics hardware implementation of the tangent-plane algorithm for computing the kth-order Voronoi diagram of a set of point sites in image space. Correct and efficient implementation of this algorithm using graphics hardware is possible only with the use of an appropriate shader program on the GPU. This is achieved by rendering in k passes the parallel projection of the top k lev...

Evolutionary algorithms have been applied to function identification problems with great success. This paper presents an approach in which the individuals represent a partition of the input space in Voronoi regions together with a set of local functions associated to each one of these regions. In this way, the solution corresponds to a combination of local functions over a spatial structure top...

The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the (n2) lower bound and the O(n3+ε) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Mi...