Let G = (V, E) be a graph with n nodes in which each pair of nodes is joined by an edge. There is a positive weight wij on each edge (i, j); and we will assume these weights satisfy the triangle inequality wik wij + wjk. For a subset V V, we will use G[V] to denote the subgraph (with edge weights) induced on the nodes in V. We are given a set X V of k terminals that must be connected by edges. We say that a Steiner tree on X is a set Z so that X Z V, together with a spanning subtree T of G[Z]. The weight of the Steiner tree is the weight of the tree T.

Show that the problem of finding a minimum-weight Steiner tree on X can be solved in time O(n^O(k)).