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 $O(\log{1\over\epsilon})$.