We study the graph constructed on a Poisson point process in @math dimensions by connecting each point to the @math points nearest to it. This graph a.s. has an infinite cluster if @math where @math , known as the critical value, depends only on the dimension @math . This paper presents an improved upper bound of 188 on the value of @math . We also show that if @math the infinite cluster of @math has an infinite subset of points with the property that the distance along the edges of the graphs between these points is at most a constant multiplicative factor larger than their Euclidean distance. Finally we discuss in detail the relevance of our results to the study of multi-hop wireless sensor networks.
Let @math be a set of elements and d a distance function defined among them. Let NNk(u) be the k elements in @math having the smallest distance to u. The k-nearest neighbor graph (knng) is a weighted directed graph @math such that E= (u,v), v∈NNk(u) . Several knng construction algorithms are known, but they are not suitable to general metric spaces. We present a general methodology to construct knngs that exploits several features of metric spaces. Experiments suggest that it yields costs of the form c1n1.27 distance computations for low and medium dimensional spaces, and c2n1.90 for high dimensional ones. Given a set S of points in a metric space with distance function D, the nearest-neighbor searching problem is to build a data structure for S so that for an input query point q, the point s 2 S that minimizes D(s,q) can be found quickly. We survey approaches to this problem, and its relation to concepts of metric space dimension. Several measures of dimension can be estimated using nearest-neighbor searching, while others can be used to estimate the cost of that searching. In recent years, several data structures have been proposed that are provably good for low-dimensional spaces, for some particular measures of dimension. These and other data structures for nearest-neighbor searching are surveyed. Given a setV ofn points ink-dimensional space, and anLq-metric (Minkowski metric), the all-nearest-neighbors problem is defined as follows: for each pointp inV, find all those points inV? p that are closest top under the distance metricLq. We give anO(n logn) algorithm for the all-nearest-neighbors problem, for fixed dimensionk and fixed metricLq. Since there is an ?(n logn) lower bound, in the algebraic decision-tree model of computation, on the time complexity of any algorithm that solves the all-nearest-neighbors problem (fork=1), the running time of our algorithm is optimal up to a constant factor.