Minimum-weight spanning tree construction in O(log log n) communication rounds

We consider a simple model for overlay networks, where all n processes are connected to all other processes, and each message contains at most O(log n) bits. For this model, we present a distributed algorithm which constructs a minimum-weight spanning tree in O( log log n) communication rounds, where in each round any process can send a message to every other process. If message size is Theta(n(epsilon)) for some epsilon > 0, then the number of communication rounds is O(log 1/epsilon).