The region approach for computing relative neighborhood graphs in the L p metric

The following geometrical proximity concepts are discussed: relative closeness and geographic closeness. Consider a setV={v 1,v 2, ...,v n } of distinct points in atwo-dimensional space. The pointv j is said to be arelative neighbour ofv i ifd p (v i ,v j )≤max{d p (v j ,v k ),d p (v j ,v k )} for allv k ∈V, whered p denotes the distance in theL p metric, 1≤p≤∞. After dividing the space around the pointv i into eight sectors (regions) of equal size, a closest point tov i in some region is called anoctant (region, orgeographic) neighbour ofv i. For anyL p metric, a relative neighbour ofv i is always an octant neighbour in some region atv i. This gives a direct method for computing all relative neighbours, i.e. for establishing therelative neighbourhood graph ofV. For every pointv i ofV, first search for the octant neighbours ofv i in each region, and then for each octant neighbourv j found check whether the pointv j is also a relative neighbour ofv i. In theL p metric, 1<p<∞, the total number of octant neighbours is shown to be θ(n) for any set ofn points; hence, even a straightforward implementation of the above method runs in θn 2) time. In theL 1 andL ∞ metrics the method can be refined to a θ(n logn+m) algorithm, wherem is the number of relative neighbours in the output,n-1≤m≤n(n-1). TheL 1 (L ∞) algorithm is optimal within a constant factor.