A note on relative neighborhood graphs

Authors:
J. W. Jaromczyk;M. Kowaluk
Affiliations:
University of Kentucky, Dep. of Comp. Sci., Lexington, KY, USA and Warsaw University, Inst. of Informatics, 00-901 Warsaw, PKiN 8 p., Poland;Warsaw University, Inst. of Informatics, 00-901 Warsaw, PKiN 8 p., Poland
Venue:
SCG '87 Proceedings of the third annual symposium on Computational geometry
Year:
1987

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Abstract

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 (R2,Lp) 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 (Rd,Lp), 1 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 (Rd,Lp).