In this paper, a very simple model of parallel computation is considered, and the question of how restricting the flow of data to be one way compares with two-way flow is studied. It is shown that the one-way version is surprisingly very powerful in that it can solve problems that seemingly require two-way communication. Whether or not one-way communication is strictly weaker than two-way is an open problem, although the conjecture in this paper is in the positive. It is shown, however, that proving this conjecture is at least as hard as some well-known open problems in complexity theory.