We show that the planar dual to the Euclidean farthest point Voronoi diagram for the set of vertices of a convex polygon has the lexicographic minimum possible sequence of triangle angles, sorted from sharpest to least sharp. As a consequence, the sharpest angle determined by three vertices of a convex polygon can be found in linear time.

A method is presented to partition a given set of data entries embedded in Euclidean space by recursively bisecting clusters into smaller ones. The initial set is subdivided into two subsets whose centroids are farthest from each other, and the process is repeated recursively on each subset. The bisection task can be formulated as an integer programming problem, which is NP-hard. Instead, an ap...

In contrast to classical offline k-k routing, the online packet routing problem allows for an arbitrary number of packets with arbitrary end points and release times. We study this problem on linear array and ring networks. We generalize an earlier result for the offline problem by showing that Farthest First (FF) scheduling is optimal with respect to makespan on linear arrays. We also show tha...

We will describe an O(n) time reduction from the set-disjointness problem to the farthest pairs problem. This will imply the Ω(n log n) lower bound. Consider an instance of the set-disjointness problem A = {a1, a2, . . . , an} and B = {b1, b2, . . . , bn}. We need to find out whether there exist i and j such that ai = bj. We map the sets A and B to the first and third quadrant of the unit circl...

We maintain the maximum spanning tree of a planar point set, as points are inserted or deleted, in O(log n) time per update in Mulmuley’s expected-case model of dynamic geometric computation. We use as subroutines dynamic algorithms for two other geometric graphs: the farthest neighbor forest and the rotating caliper graph related to an algorithm for static computation of point set widths and d...

In this paper, we present a Voronoi based algorithm for closed curve reconstruction and medial axis approximation from planar points. In principle, the algorithm estimates one of the poles (farthest Voronoi vertices of a Voronoi cell) and hence the normals at each sample point by drawing an analogy between a residential water distribution system and Voronoi diagram of input samples. The algorit...

We propose a new image compression method based on geodesic Delaunay triangulations. Triangulations are generated by a progressive geodesic meshing algorithm which exploits the anisotropy of images through a farthest point sampling strategy. This seeding is performed according to anisotropic geodesic distances which force the anisotropic Delaunay triangles to follow the geometry of the image. G...

A new method of farthest point strategy (FPS) for progressive image acquisition-an acquisition process that enables an approximation of the whole image at each sampling stage-is presented. Its main advantage is in retaining its uniformity with the increased density, providing efficient means for sparse image sampling and display. In contrast to previously presented stochastic approaches, the FP...

Quorum systems are a mechanism for obtaining fault-tolerance and efficient distributed systems. We consider geographic quorum systems; a geographic quorum system is a partitioning of a set X of points in the plane (representing servers) into quorums (i.e. clusters) of size k. The distance between a point p and a cluster C is the Euclidean distance between p and the farthest point in C. We prese...