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 point set 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 with respect to those using Delaunay triangulation. 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 that in 2-D the NNE-graph can be constructed in optimal Θ (n2) time in the worst case. We also present an O(n log n + nd) time algorithm, where d is the Ω(log n)-th largest degree in the utput 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 any value g with 1 ≤ g ≤ n\log n. We finally propose an O(n log n + nd log d*) time algorithm for the problem in 3-D, where d and d* are the Omega(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).