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 $\Theta(n^2)$ time in the worst case. We also present an $O(n \log n + nd)$ time algorithm, where d is the $\Omega(\log n)$-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 $d=g \cdot(\log n)$, the number of edges in NNE-graph is bounded by $O(gn \log n)$ for any value g with $1 \leq g \leq \frac{n}{\log n}$. 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 $\Omega(\frac{\log n}{\log \log n})$-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 $O(\frac{\log n}{\log \log n})$.