Relative Neighborhood Graph

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## Algorithms

## Generalizations

## Related graphs

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Relative Neighborhood Graph

In computational geometry, the **relative neighborhood graph (RNG**) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points *p* and *q* by an edge whenever there does not exist a third point *r* that is closer to both *p* and *q* than they are to each other. This graph was proposed by Godfried Toussaint in 1980 as a way of defining a structure from a set of points that would match human perceptions of the shape of the set.^{[1]}^{[2]}

Supowit (1983) showed how to construct the relative neighborhood graph efficiently in O(*n* log *n*) time.^{[3]} It can be computed in O (*n*) expected time, for random set of points distributed uniformly in the unit square.^{[4]} The relative neighborhood graph can be computed in linear time from the Delaunay triangulation of the point set.^{[5]}^{[6]}

Because it is defined only in terms of the distances between points, the relative neighborhood graph can be defined for point sets in any and for non-Euclidean metrics.^{[1]}^{[5]}^{[9]}^{[10]}

The relative neighborhood graph is an example of a lens-based beta skeleton. It is a subgraph of the Delaunay triangulation. In turn, the Euclidean minimum spanning tree is a subgraph of it, from which it follows that it is a connected graph.

The Urquhart graph, the graph formed by removing the longest edge from every triangle in the Delaunay triangulation, was originally proposed as a fast method to compute the relative neighborhood graph.^{[11]} Although the Urquhart graph sometimes differs from the relative neighborhood graph^{[12]} it can be used as an approximation to the relative neighborhood graph.^{[13]}

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^{a}^{b}^{c}Toussaint, G. T. (1980), "The relative neighborhood graph of a finite planar set",*Pattern Recognition*,**12**(4): 261-268, doi:10.1016/0031-3203(80)90066-7. **^**Jaromczyk, J.W.; Toussaint, G.T. (1992), "Relative neighborhood graphs and their relatives",*Proceedings of the IEEE*,**80**(9): 1502-1517, doi:10.1109/5.163414.**^**Supowit, K. J. (1983), "The relative neighborhood graph, with an application to minimum spanning trees",*J. ACM*,**30**(3): 428-448, doi:10.1145/2402.322386.**^**Katajainen, Jyrki; Nevalainen, Olli; Teuhola, Jukka (1987), "A linear expected-time algorithm for computing planar relative neighbourhood graphs",*Information Processing Letters*,**25**(2): 77-86, doi:10.1016/0020-0190(87)90225-0.- ^
^{a}^{b}Jaromczyk, J. W.; Kowaluk, M. (1987), "A note on relative neighborhood graphs",*Proc. 3rd Symp. Computational Geometry*, New York, NY, USA: ACM, pp. 233-241, doi:10.1145/41958.41983. **^**Lingas, A. (1994), "A linear-time construction of the relative neighborhood graph from the Delaunay triangulation",*Computational Geometry*,**4**(4): 199-208, doi:10.1016/0925-7721(94)90018-3.**^**Jaromczyk, J. W.; Kowaluk, M. (1991), "Constructing the relative neighborhood graph in 3-dimensional Euclidean space",*Discrete Appl. Math.*,**31**(2): 181-191, doi:10.1016/0166-218X(91)90069-9.**^**Agarwal, Pankaj K.; Matau?ek, Ji?í (1992), "Relative neighborhood graphs in three dimensions",*Proc. 3rd ACM-SIAM Symp. Discrete Algorithms*, pp. 58-65.**^**O'Rourke, J. (1982), "Computing the relative neighborhood graph in the*L*_{1}and*L*_{?}metrics",*Pattern Recognition*,**15**(3): 189-192, doi:10.1016/0031-3203(82)90070-X.**^**Lee, D. T. (1985), "Relative neighborhood graphs in the*L*_{1}-metric",*Pattern Recognition*,**18**(5): 327-332, doi:10.1016/0031-3203(85)90023-8.**^**Urquhart, R. B. (1980), "Algorithms for computation of relative neighborhood graph",*Electronics Letters*,**16**(14): 556-557, doi:10.1049/el:19800386.**^**Toussaint, G. T. (1980), "Comment: Algorithms for computing relative neighborhood graph",*Electronics Letters*,**16**(22): 860, doi:10.1049/el:19800611. Reply by Urquhart, pp. 860-861.**^**Andrade, Diogo Vieira; de Figueiredo, Luiz Henrique (2001), "Good approximations for the relative neighbourhood graph",*Proc. 13th Canadian Conference on Computational Geometry*(PDF).

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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