An optimal algorithm for the Euclidean bottleneck full Steiner tree problem

Let P and S be two disjoint sets of n and m points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to S, in which each point of P   is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in O((n+m)logm) time, improving the previously best result by a logarithmic factor. We also prove a matching lower bound in the algebraic computation tree model.