Let S be a set of n points in < d. We present an algorithm that uses the well-separated pair decomposition and computes the minimum spanning tree of S under any Lp or polyhedral metric. It has an expected running time of O(n logn) for uniform distributions. Experimentalresults show that this approachis practical. Under a variety of input distributions, the resultingimplementation is robust and ...

We present an optimal parallel algorithm to construct the well-separated pair decomposition of a point set P in < d. We show how this leads to a deterministic optimal O(logn) time parallel algorithm for nding the k-nearest-neighbors of each point in P, where k is a constant. We discuss several additional applications of the well-separated pair decomposition for which we can derive faster parall...

Let S be a set of points in R. Given a geometric spanner graph, G = (S, E), with constant dilation t, and a positive constant ε, we show how to construct a (1 + ε)-spanner of G with O(|S|) edges in time O(|E|+ |S| log |S|). Previous algorithms require a preliminary step in which the edges are sorted in non-decreasing order of their lengths and, thus, have running time Ω(|E| log |S|). We obtain ...

Given the positions of n sites in a radio network we consider the problem of finding routes between any pair of sites that minimize energy consumption and do not use more than some constant number k of hops. Known exact algorithms for this problem required Ω(n logn) per query pair (p, q). In this paper we relax the exactness requirement and only compute approximate (1+2) solutions which allows ...

Given a real number t > 1, a geometric t-spanner is a geometric graph for a point set in R with straight lines between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t. An imprecise point set is modeled by a set R of regions in R. If one chooses a point in each reg...