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(n ²) time in the worst case. We also present an O(n log n + nd) algorithm, where d is the Ω(logn)^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 $\leq g \leq \frac{n}{{\rm log} n}$. We finally propose an O(n log n + nd log d ^*) algorithm for the problem in 3-D, where d and d ^* are the $\Omega(\frac{{\rm log} n}{{\rm log log} n})^{th}$ 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 $O(\frac{{\rm log} n}{{\rm log log} n})$.