Journal of Combinatorial Optimization

, Volume 11, Issue 4, pp 435–443

# Construction of the nearest neighbor embracing graph of a point set

• M. Y. Chan
• Danny Z. Chen
• Francis Y. L. Chin
• Cao An Wang
Article

## 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

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## References

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© Springer Science + Business Media, LLC 2006

## Authors and Affiliations

• M. Y. Chan
• 1
• Danny Z. Chen
• 2
• Francis Y. L. Chin
• 1
• Cao An Wang
• 3
1. 1.Department of Computer Science and Information SystemsThe University of Hong KongHong Kong
2. 2.Department of Computer Science and EngineeringUniversity of Notre DomeNotre DomeUSA
3. 3.Department of Computer ScienceMemorial University of NewfoundlandSt. John’sCanada