# 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 \(O(\frac{\log n}{\log \log n})\).

## Keywords

Computational geometry Nearest neighbors Network connections## Preview

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