The purpose of this paper is to discuss some recent generalizations of the basic covering radius problem in coding theory. In particular we discuss multiple coverings, multiple coverings of the farthest-o points and weighted coverings.

We present an algorithm that computes the diameter of a set of n points in the cylinder in optimal time O(n log n); this algorithm uses as a fundamental tool the farthest point Voronoi diagram.

In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log2 n) time. Our data structure needs O(n logn) space and preprocessing time. To the best of our kno...

Optical Multi-Trees with Shuffle Exchange (OMTSE) architecture is an efficient model of an optoelectronic computer. The network has a total of 3n/2 nodes. The diameter and bisection width of the network are 6 logn − 1 and n/4 respectively. In this note, we present synchronous SIMD algorithms on an OMTSE optoelectronic computer for the following problems in computational geometry: Convex Hull, S...

We consider approximation of diameter of a set S of n points in dimension m. Eg̃eciog̃lu and Kalantari [8] have shown that given any p ∈ S, by computing its farthest in S, say q, and in turn the farthest point of q, say q′, we have diam(S) ≤ √ 3 d(q, q′). Furthermore, iteratively replacing p with an appropriately selected point on the line segment pq, in at most t ≤ n additional iterations, the c...

Finding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(n logn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum)...

Given a set S of n static points and a free point p in the Euclidean plane, we study a new variation of the minimum enclosing circle problem, in which a dynamic weight that equals to the reciprocal of the distance from the free point p to the undetermined circle center is included. In this work, we prove the optimal solution of the new problem is unique and lies on the boundary of the farthest-...

A problem in distance geometry is to find the location of an unknown point in a given convex set in Rk such that its farthest distance to n fixed points is minimum. In this paper we present an algorithm based on subgradient method and convex hull computation for solving this problem. A recent improvement of Quickhull algorithm for computing the convex hull of a finite set of planar points is ap...

An earlier result states that a point of the surface of a convex polyhedron with n vertices, endowed with its intrinsic metric, cannot have more than n antipodes (farthest points). In this paper we produce examples of polyhedra with n vertices, on which some suitable point admits exactly n antipodes. MSC (2000): 52B10, 53C45.