On the space complexity of randomized synchronization

Abstract

The “waite-free hierarchy” provides a classification of multiprocessor synchronization primitives based on the values ofnfor which there are deterministic wait-free implementations ofn-process consensus using instances of these objects andread-writeregisters. In a randomized wait-free setting, this classification is degenerate, sincen-process consensus can be solved using onlyO(n) read-writeregisters.In this paper, we propose a classification of synchronization primitives based on thespace complexityof randomized solutions ton-process consensus. Ahistoryless object,such as aread-writeregister, aswapregister, or atest&setregister, is an object whose state depends only on the lost nontrivial operation thate was applied to it. We show that, usinghistorylessobjects, Ω(√n) object instances are necessary to solven-process consensus. This lower bound holds even if the objects have unbounded size and the termination requirement isnondeterministic 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 asshared counters, fetch&addregisters, andcompare&swapregisters. 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.”