Improved and Partially-Tight Lower Bounds for Message-Passing Implementations of Multiplicity Queues

Abstract

A multiplicity queue is a concurrently-defined data type which relaxes the conditions of a linearizable FIFO queue to allow concurrent Dequeue instances to return the same value. It would seem that this should allow faster implementations, as processes should not need to wait as long to learn about concurrent operations at remote processes and previous work has shown that multiplicity queues are computationally less complex than the unrelaxed version. Intriguingly, recent work has shown that there is, in fact, not much speedup possible versus an unrelaxed queue implementation. Seeking to understand this difference between intuition and real behavior, we extend that work, increasing the lower bound for uniform algorithms. Further, we outline a path forward toward building proofs for even higher lower bounds, allowing us to hypothesize that the worst-case time to Dequeue approaches maximum message delay, which is similar to the time required for an unrelaxed Dequeue. We also give an upper bound for a special case to show that our bounds are tight at that point. To achieve our lower bounds, we use extended shifting arguments, which have been rarely used but allow larger lower bounds than traditional shifting arguments. We use these in series of inductive indistinguishability proofs which allow us to extend our proofs beyond the usual limitations of shifting arguments. This proof structure is an interesting contribution independently of the main result, as developing new lower bound proof techniques may have many uses in future work.