Relative neighborhood graph explained

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]}

Algorithms

showed how to construct the relative neighborhood graph of

n points in the plane 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]}

Generalizations

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] [7] [8]} Computing the relative neighborhood graph, for higher-dimensional point sets, can be done in time

O(n ^{2)} .

Related graphs

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.^[9] Although the Urquhart graph sometimes differs from the relative neighborhood graph^[10] it can be used as an approximation to the relative neighborhood graph.^[11]

Notes and References

1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .
9. .
10. . Reply by Urquhart, pp. 860–861.
11. .