Relative neighborhood graphs in three dimensions

Abstract

The relative neighborhood graph (RNG) of a set S of n points in R d is a graph ( S, E), where ( p, q)∈ E if and only if there is no point z∈ S such that max{ d( p, z), d( q, z)}< d( p, q). We show that in R 3, RNG( S) has O(n 4 3 ) edges. We present a randomized algorithm that constructs RNG( S) in expected time O(n 3 2 +ε ) assuming that the points of S are in general position. If the points of S are arbitrary, the expected running time is O(n 7 4 +ε ) . These algorithms can be made deterministic without affecting their asymptotic running time.