On the space complexity of randomized synchronization

Abstract

The “waite-free hierarchy” provides a classification of multiprocessor synchronization primitives based on the values of n for which there are deterministic wait-free implementations of n -process consensus using instances of these objects and read-write registers. In a randomized wait-free setting, this classification is degenerate, since n -process consensus can be solved using only O(n) read-write registers. In this paper, we propose a classification of synchronization primitives based on the space complexity of randomized solutions to n -process consensus. A historyless object, such as a read-write register, a swap register, or a test&set register, is an object whose state depends only on the lost nontrivial operation thate was applied to it. We show that, using historyless objects, Ω(√n) object instances are necessary to solve n -process consensus. This lower bound holds even if the objects have unbounded size and the termination requirement is nondeterministic solo termination , a property strictly weaker than randomized wait-freedom. We then use this result to related the randomized space complexity of basic multiprocessor synchronization primitives such as shared counters, fetch&add registers, and compare&swap registers. Viewed collectively, our results imply that there is a separation based on space complexity for synchronization primitives in randomized computation, and that this separation differs from that implied by the deterministic “wait-free hierarchy.”