We consider an asynchronous system with the Ω failure detector, and investigate the number of communication steps required by various broadcast protocols in runs in which the leader does not change. Atomic Broadcast, used for example in state machine replication, requires three communication steps. Optimistic Atomic Broadcast requires only two steps if all correct processes receive messages in the same order. Generic Broadcast requires two steps if no messages conflict. We present an algorithm that subsumes both of these approaches and guarantees two-step delivery if all conflicting messages are received in the same order, and three-step delivery otherwise. Internally, our protocol uses two new algorithms. First, a Consensus algorithm which decides in one communication step if all proposals are the same, and needs two steps otherwise. Second, a method that allows us to run infinitely many instances of a distributed algorithm, provided that only finitely many of them are different. We assume that fewer than a third of all processes are faulty (n > 3f).
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