I'm working on the following problem:
Suppose that we're given a connected, undirected graph $G = (V, E)$ with edge weights $w_e$ and a subset of vertices $U \subset V$. We want to find the lightest spanning tree in which the nodes of $U$ are leaves (there may be other leaves as well). We want to do so in $O(|E|\log(|V|))$ time.
Here's my thinking: since every node $v \in U$ must be a leaf, there must exist a vertex $u \in V \setminus U$ that is the source (i.e. each leaf in $U$ is connected to $u$). However, I'm having trouble find a way to do this that doesn't involve running a polynomial time algorithm. Can anyone help?