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We exhibit an explicit function f : {0, 1} n → {0, 1} that can be computed by a nondeterministic number-on-forehead protocol communicating O(log n) bits, but that requires n Ω(1) bits of communication for randomized number-on-forehead protocols with k = δ · log n players, for any fixed δ < 1. Recent breakthrough results for the Set-Disjointness function (Lee Shraibman, CCC '08; Chattopadhyay Ada, ECCC '08) based on the work of (Sherstov, STOC '08) imply such a separation but only when the number of players is k < log log n.We also show that for any k = A log log n the above function f is computable by a small circuit whose depth is constant whenever A is a (possibly large) constant. Recent results again give such functions but only when the number of players is k < log log n.

We exhibit an explicit function f : {0, 1} n → {0, 1} that can be computed by a nondeterministic number-on-forehead protocol communicating O(log n) bits, but that requires n Ω(1) bits of communication for randomized number-on-forehead protocols with k = δ · log n players, for any fixed δ < 1. Recent breakthrough results for the Set-Disjointness function (Lee Shraibman, CCC '08; Chattopadhyay Ada, ECCC '08) based on the work of (Sherstov, STOC '08) imply such a separation but only when the number of players is k < log log n.We also show that for any k = A log log n the above function f is computable by a small circuit whose depth is constant whenever A is a (possibly large) constant. Recent results again give such functions but only when the number of players is k < log log n.

Abstract. We study wait-free linearizable Queue implementations in asynchronous shared-memory systems from other consensus number 2 objects, such as Fetch&Add and Swap. The best previously known implementation allows at most two processes to perform Dequeue operations. We provide a new implementation, when only one process performs Enqueue operations and any number of processes perform Dequeue operations. A nice feature of this implementation is the fact that both Enqueue and Dequeue operations take constant time.

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