The well-separated pair decomposition (WSPD) of the complete Euclidean graph defined on points in R, introduced by Callahan and Kosaraju [JACM, 42 (1): 67-90, 1995], is a technique for partitioning the edges of the complete graph based on length into a linear number of sets. Among the many different applications of WSPDs, Callahan and Kosaraju proved that the sparse subgraph that results by sel...

A spanner graph on a set of points in Rd provides shortest paths between any pair of points with lengths at most a constant factor of their Euclidean distance. A spanner with a sparse set of edges is thus a good candidate for network backbones, as desired in many practical scenarios such as the transportation network and peer-to-peer network overlays. In this paper we investigate new models and...

Given a cone C and a set S of n points in R, we want to preprocess S into a data structure so that we can find fast an approximate nearest neighbor to a query point q with respect to the points of S contained in the translation of C with apex at q. We develop an approximate conic Voronoi diagram of Õ(n/ε) size that supports conic nearest neighbor queries in O(log(n/ε)) time. Our preprocessing u...

Let P be a set of n points in the plane. The geometric minimum-diameter spanning tree (MDST) of P is a tree that spans P and minimizes the Euclidian length of the longest path. It is known that there is always a monoor a dipolar MDST, i.e. a MDST whose longest path consists of two or three edges, respectively. The more difficult dipolar case can so far only be solved in O(n) time. This paper ha...