The Euclidean Bottleneck Full Steiner Tree Problem

Abstract

Given two sets in the plane, R of n (terminal) points and S of m (Steiner) points, a full Steiner tree is a Steiner tree in which all points of R are leaves. In the bottleneck full Steiner tree (BFST) problem, one has to find a full Steiner tree T (with any number of Steiner points from S), such that the length of the longest edge in T is minimized, and, in the k-BFST problem, has to find a full Steiner tree T with at most k≤m Steiner points from S such that the length of the longest edge in T is minimized. The problems are motivated by wireless network design. In this paper, we present an exact algorithm of ${{{\mathcal {O}}}}((n+m)\log^{2}{m})$ time to solve the BFST problem. Moreover, we show that the k-BFST problem is NP-hard and that there exists a polynomial-time approximation algorithm for the problem with performance ratio 4.