The concept of convex polygon-offset distance function was introduced in 2001 by Barequet, Dickerson, and Goodrich. Using this notion of point-to-point distance, they showed how to compute the corresponding nearestand farthest-site Voronoi diagram for a set of points. In this paper we generalize the polygon-offset distance function to be from a point to any convex object with respect to an m-si...

Given n point sites in the plane each painted in one of k colors, the region of a c-colored site p in the Farthest Color Voronoi Diagram (FCVD) contains all points of the plane for which c is the farthest color and p the nearest c-colored site. This novel structure generalizes both the standard Voronoi diagram (k = 1) and the farthest site Voronoi diagram (k = n). We show that the FCVD has stru...

Codes with minimum distance at least d and covering radius at most d− 1 are considered. The minimal cardinality of such codes is investigated. Herewith, their connection to covering problems is applied and a new construction theorem is given. Additionally, a new lower bound for the covering problem is proved. A necessary condition on an existence problem is presented by using a multiple coverin...

For a finite set of points X on the unit hypersphere in R we consider the iteration ui+1 = ui +χi, where χi is the point of X farthest from ui. Restricting to the case where the origin is contained in the convex hull of X we study the maximal length of ui. We give sharp upper bounds for the length of ui independently of X . Precisely, this upper bound is infinity for d ≥ 3 and √ 2 for d = 2.

The k-modes algorithm has become a popular technique in solving categorical data clustering problems in different application domains. However, the algorithm requires random selection of initial points for the clusters. Different initial points often lead to considerable distinct clustering results. In this paper we present an experimental study on applying a farthest-point heuristic based init...

Let X be a complete CAT(0) space with the geodesic extension property and Alexandrov curvature bounded below. It is shown that if C is a closed subset of X , then the set of points of X which have a unique nearest point in C is Gδ and of the second Baire category inX. If, in addition,C is bounded, then the set of points ofX which have a unique farthest point in C is dense in X. A proximity resu...

The eccentricity transform associates to each point of a shape the geodesic distance to the points farthest away. This can be used to decompose shapes into parts based on the connectivity of the isoheight lines. The adjacency graph of the obtained decomposition is similar to a Reeb graph and characterizes the topology of the object. Graph pyramids are used to efficiently compute the decompositi...

We address the problem of computing the fitting line of a set of circles in the Laguerre metric, that minimizes the distance to the farthest circle. To solve the fitting line problem we introduce a generalization of the concept of the width of a set of points using the Laguerre metric. We also present an efficient algorithm for finding the fitting line of a set of circles using minimization dia...