Constructing the relative neighborhood graph in 3-dimensional Euclidean space

Abstract

The relative neighborhood graph for a finite set S={ p 1,…, p N } of points, briefly RNG( S), is defined by the following formation rule: p ip j is an edge in RNG( S) if and only if for all p k ϵ S−{ p i , p j }, dist( p i , p j )≤max(dist( p i , p k ), dist( p j , p k )). We show that RNG for point sets in R 3 can be constructed in optimal space and O( N 2log N) time. Also, combinatorial estimates on the size of RNG in R 3 are given.