Nets, useful topological tools, used to generalize certainconcepts that may only be general enough in the context of metricspaces. In this work we introduce this concept in an $S$-poset, aposet with an action of a posemigroup $S$ on it whichis a very useful structure in computer sciences and interestingfor mathematicians, and give the the concept of $S$-net. Using $S$-nets and itsconvergency we...

nets, useful topological tools, used to generalize certainconcepts that may only be general enough in the context of metricspaces. in this work we introduce this concept in an $s$-poset, aposet with an action of a posemigroup $s$ on it whichis a very useful structure in computer sciences and interestingfor mathematicians, and give the the concept of $s$-net. using $s$-nets and itsconvergency we...

[3] P. B. Callahan, " Optimal parallel all-nearest-neighbors using the well-separated pair decomposition, " in

The monoid $Cb$ of name substitutions and the notion of finitely supported $Cb$-sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported $Cb$-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported $Cb$-set is an injective map. Also, for every two distinct elements of a simple finitely supported $Cb$-set, there exists...

Let P be a set of n points in R d. For any parameter s ≥ 1, A, B ⊆ P are said to be s-separated or well separated if A and B can be enclosed by congruent d-spheres S A and S B of radius r such that the minimum distance between S A and S B is at least sr. There is a stronger notion of well-separated point sets. For any parameter s ≥ 1, A, B ⊆ P are said to be s-separated if the bounding boxes of...

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 ...