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 $O((n+m)\log m)$ time, improving the previously best result by a logarithmic factor. We also prove a matching lower bound in the algebraic computation tree model.