Optimal Steiner hull algorithm

Abstract

Given a set Z of n points in the plane, we consider the problem of finding the Steiner hull for Z which is a non-trivial polygon containing every Euclidean Steiner minimal tree for Z. We give an optimal Θ( nlog n) time and Θ( n) space algorithm exploiting a Delaunay triangulation of Z. If the Delaunay triangulation is given, the algorithm requires linear time and space. Furthermore, we argue that the uniqueness argument for the O( n 3) time Steiner hull algorithm given in [4] is incorrect, and we give a correct uniqueness proof.