Construction of the nearest neighbor embracing graph of a point set

Abstract

This paper gives optimal algorithms for the construction of the Nearest Neighbor Embracing Graph (NNE-graph) of a given set of points V of size n in the k-dimensional space (k-D) for k = (2, 3). The NNE-graph provides another way of connecting points in a communication network, which has lower expected degree at each point and shorter total length of connections than Delaunay graph. In fact, the NNE-graph can also be used as a tool to test whether a point set is randomly generated or has some particular properties. We show in 2-D that the NNE-graph can be constructed in optimal O(n2) time in the worst case. We also present an O(n log n + nd) algorithm, where d is the Ω(log n)th largest degree in the output NNE-graph. The algorithm is optimal when d = O(log n). The algorithm is also sensitive to the structure of the NNE-graph, for instance when d = g · (log n), the number of edges in NNE-graph is bounded by O(gn log n) for 1 ≤ g ≤ n/log n. We finally propose an O(n log n + nd log d*) algorithm for the problem in 3-D, where d and d* are the Ω(log n/log log n)th largest vertex degree and the largest vertex degree in the NNE-graph, respectively. The algorithm is optimal when the largest vertex degree of the NNE-graph d* is O(log n/log log n). © Springer-Verlag Berlin Heidelberg 2004.