The Hausdorff Voronoi diagram of a set of clusters of points in the plane is a generalization of the classic Voronoi diagram, where distance between a point t and a cluster P is measured as the maximum distance, or equivalently the Hausdorff distance between t and P . The size of the diagram for non-crossing clusters is O(n ), where n is the total number of points in all clusters. In this paper...

In pattern recognition the problem of input variable selection has been traditionally focused on technological issues, e.g., performance enhancement, lowering computational requirements, and reduction of data acquisition costs. However, in the last few years, it has found many applications in basic science as a model selection and discovery technique, as shown by a rich literature on this subje...

We tackle the problem of computing the Voronoi diagram of a 3-D polyhedron whose faces are planar. The main difficulty with the computation is that the diagram’s edges and vertices are of relatively high algebraic degrees. As a result, previous approaches to the problem have been non-robust, difficult to implement, or not provenly correct. We introduce three new proximity skeletons related to t...

This paper studies local properties of Voronoi diagrams of sets of disjoint compact convex sites in R. It is established that bisectors are C surfaces and trisectors are C curves, and that as a point moves along a trisector its clearance sphere develops monotonically (Lemma 2.4). This monotonicity property is useful in establishing the existence of Voronoi vertices bounding edges in certain sit...

Given a protein, it is often necessary to study its geometric and physicochemical properties for studying its structure and predicting funtions of a protein. In this case, a connolly surface of a protein plays important roles for these purpose. A protein consists of a set of amino acids and a set of atoms comprise an amino acide. Since an atom can be represented by a hard 3D sphere in van der W...

A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it is an affine image of a Voronoi polytope.

We study the combinatorial complexity of Voronoi diagram of point sites on a general triangulated 2-manifold surface, based on the geodesic metric. Given a triangulated 2-manifold T of n faces and a set of m point sites S = {s1, s2, · · · , sm} ∈ T , we prove that the complexity of Voronoi diagram VT (S) of S on T is O(mn) if the genus of T is zero. For a genus-g manifold T in which the samples...

Let S be a set of n + m sites, of which n are red and have weight wR, and m are blue and weigh wB. The objective of this paper is to calculate the minimum value of the red sites’ weight such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected region. This problem is solved for the multiplicativelyweighted Voronoi diagram in O((n+m)2 log(nm)) time and for ...

We present a systematic study of the expected complexity of the intersection of geometric objects. We first study the expected size of the intersection between a random Voronoi diagram and a generic geometric object that consists of a finite collection of line segments in the plane. Using this result, we explore the intersection complexity of a random Voronoi diagram with the following target o...

We analyze structural properties of the order-k Voronoi diagram of line segments, which surprisingly has not received any attention in the computational geometry literature. We show that order-k Voronoi regions of line segments may be disconnected; in fact a single orderk Voronoi region may consist of Ω(n) disjoint faces. Nevertheless, the structural complexity of the order-k Voronoi diagram of...