CONSTRUCTING THE RELATIVE NEIGHBORHOOD GRAPH IN 3-DIMENSIONAL EUCLIDEAN-SPACE

The relative neighborhood graph for a finite set S = {p1,...,p(N)} of points, briefly RNG(S), is defined by the following formation rule: p(i)p(j)BAR is an edge in RNG(S) if and only if for all p(k) is-an-element-of S - {p(i),p(j)}, dist(p(i),p(j) less-than-or-equal-to max(dist(p(i),p(k)), dist(j),p(k)). We show that RNG for point sets in R3 can be constructed in optimal space and O(N2logN) time. Also, combinatorial estimates on the size of RNG in R3 are given.