Construct a minimum spanning tree covering a specific subset of the vertices

Question

I have an undirected, positive-edge-weight graph (V,E) for which I want a minimum spanning tree covering a subset k of vertices V (the Steiner tre

Answer 1

Run Prim's algorithm on the restricted graph (k, E') where E' = {(x, y) ∈ V : x ∈ k and y ∈ k}). Constructing that graph takes O(|E|).