The Euclidean bottleneck Steiner tree and Steiner tree with minimum number of Steiner points

We study variations of Steiner tree problem. Let P = {p(1), p(2), .... p(n)} be a set of n terminals in the Euclidean plane. For a positive integer k, the bottleneck Steiner tree problem (BSTP for short) is to find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. For a positive constant R, the Steiner tree problem with minimum number of Steiner points (STP - MSP for short) asks for a Steiner tree such that each edge in the tree has length at most R and the number of Steiner points is minimized. In this paper, we give (1) a ratio-root3 + epsilon approximation algorithm for BSTP, where c is an arbitrary positive number; (2) a ratio-3 approximation algorithm for STP-MSP with running time O(n(3)); (3) a ratio-E approximation algorithm for STP-MSP.