a note on relative neighborhood graphs
Two new algorithms finding relative neighborhood graph RNG(V) for a set V of n points are presented. The first algorithm solves this problem for input points in (R 2 ,L p ) metric space in time &Ogr;(n &agr;(n,n)) if the Delaunay triangulation DT(V) is given. This time performance is achieved due to attractive and natural application of FIND-UNION data structure to represent so-called elimination forest of edges in DT(V). The second algorithm solves the relative neighborhood graph problem in (R d ,L p ), 1 <p<oo, metric space in time &Ogr;(n 2 ) when no three points in V form an isosceles triangle. The complexity analysis of this algorithm is based on some general facts pertaining to properties of equilateral triangles in the metric space (R d ,L p ).
graph algorithm point input metric spaces timing triangulation performance applications data structures complex analysis general relation