Faith Fich
Maurice Herlihy
Nir Shavit
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.”
- ABRAHAMSON, K. 1988. On achieving consensus using a shared memory. In Proceedings of the 7th Annual ACM Symposium on Principles of Distributed Computing (Toronto, Ont., Canada, Aug. 15-17). ACM, New York, pp. 291-302. Google Scholar
- AFEK, Y., AND STUPP, G. 1993. Synchronization power depends on the register size. In Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science (Nov.). IEEE Computer Science Press, Los Alamitos, Calif., pp. 196-205.Google Scholar
- AFEK, Y., ATTIYA, H., DOLEV, D., GAFNI, E., MERRITT, M., AND SHAVIT, N. 1993. Atomic snapshots of shared memory. J. ACM 40, 4 (Sept.), 873-890. Google Scholar
- AFEK, f., GREENBERG, D., MERRITT, M., AND TAUBENFELD, G. 1995. Computing with faulty shared objects. J. ACM 42, 6 (Nov.), 1231-1274. Google Scholar
- ALEMANY, J., AND FELTEN, E.W. 1992. Performance issues in non-blocking synchronization on shared-memory multiprocessors. In Proceedings of the llth Annual ACM Symposium on Principles of Distributed Computing (Vancouver, B.C., Canada, Aug. 10-12). ACM, New York, pp. 125-134. Google Scholar
- ASPNES, J. 1990. Time- and-space efficient randomized consensus. In Proceedings of the 9th Annual ACM Symposium on Principles of Distributed Computing (Quebec City, Que., Canada, Aug. 22-24). ACM, New York, pp. 325-331. Google Scholar
- ASPNES, J. 1997. Lower bounds for distributed coin-flipping and randomized consensus. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (El Paso, Tex., May 4-6). ACM, New York, pp. 559-568. Google Scholar
- ASPNES, J., AND HERLIHY, M. 1990. Fast, randomized consensus using shared memory. J. Algorithms 11 (Sept.), 441-461. Google Scholar
- ASPNES, J., AND WAARTS, O. 1992. Randomized Consensus in Expected O(n log2 n) operations per processor. In Proceedings of the 33rd Annual IEEE Symposium on the Foundation of Computer Science (Oct.). IEEE Computer Science Press, Los Alamitos, Calif., pp. 137-146.Google Scholar
- ATTIYA, H., DOLEV, D., AND SHAVIT, N. 1989. Bounded polynomial randomized consensus. In Proceedings of the 8th Annual ACM Symposium on Principles of Distributed Computing (Edmonton, Alb., Canada, Aug. 14-16). ACM, New York, pp. 281-293. Google Scholar
- BERSHAD, B. 1993. Practical considerations for non-blocking concurrent objects. In Proceedings of the 13th International Conference on Distributed Computing Systems (May). IEEE Computer Society Press, Los Alamitos, Calif., pp. 264-274.Google Scholar
- BRACHA, G., AND RACHMAN, O. 1991. Ransomized consensus in expected O(n2 log n) operations. In Proceedings of the 5th International Workshop on Distributed Algorithms (Delphi, Greece, Oct.). Lecture Notes in Computer Science, vol. 579, Springer-Verlag, New York, pp. 143-150. Google Scholar
- BURNS, J., AND LYNCH, N. 1989. Mutual exclusion using indivisible reads and writes. In Proceedings of the 18th Annual Allerton Conference on Communication Control, and Computing. Monticello, Ill. pp. 833-842.Google Scholar
- CHOR, B., ISRAELI, A., AND LI, M. 1987. On processor coordination using asynchronous hardware. In Proceedings of the 6th ACM Symposium on Principles of Distributed Computing (Vancouver, B.C., Canada, Aug. 10-12). ACM, New York, pp. 86-97. Google Scholar
- DOLEV, D., DWORK, C., AND STOCKMEYER, L. 1987. On the minimal synchronism needed for distributed consensus. J. ACM 34, 1 (Jan.), 77-97. Google Scholar
- DWORK, C., HERLIHY, M., PLOTKIN, S. A., AND WAARTS, 0. 1992. Time-lapse snapshots. Tech. Rep. STAN//CS-TR-92-1423. Dept. Computer Science, Stanford Univ., Stanford, Calif. Google Scholar
- GRAHAM, R. L., AND YAO, A.C. 1989. On the improbability of reaching Byzantine agreements. In Proceedings of the 21st Annual ACM Symposium on the Theory of Computing (Seattle, Wash., May 15-17). ACM, New York, pp. 467-478. Google Scholar
- HERLIHY, M. P. 1991a. Randomized wait-free concurrent objects. In Proceedings of the lOth Annual ACM Symposium on Principles of Distributed Computing (Montreal, Que., Canada, Aug. 19-21). ACM, New York, pp. 11-21. Google Scholar
- HERLIHY, M. P. 1991b. Wait-free synchronization. ACM Trans. Prog. Lang. Syst. 13, 1 (Jan.), 124-149. Google Scholar
- HERLIHY, M. P., AND WING, J.M. 1990. Linearizability: A correctness condition for concurrent objects. ACM Trans. Prog. Lang. Syst. 12, 3 (July), 463-492. Google Scholar
- JAYANTI, P., TAN, K., AND TOUEG, S. 1996. Time and space lower bounds for non-blocking implementations (preliminary version). In Proceedings of the 15th Annual ACM Symposium on Principles of Distributed Computing (Philadelphia, Pa., May 23-26). ACM, New York, pp. 257-266. Google Scholar
- KUSHILEVITZ, E., MANSOUR, Y., RABIN, M., AND ZUCKERMAN, D. 1993. Lower bounds for randomized mutual exclusion. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, pp. 154-163. Google Scholar
- LAMPORT, L. 1986. On interprocess communication. Part II: Algorithms. Dist. Comput. 1, 2, 86-101.Google Scholar
- LouI, M., AND ABU-AMARA, H. 1987. Memory requirements for agreement among unreliable asynchronous processes. Adv. Comput. Res. 4, 163-183.Google Scholar
- LYNCH, N.A. 1996. Distributed Algorithms. Morgan-Kaufmann, San Francisco, Calif. Google Scholar
- LYNCH, N. A., AND TUTTLE, M. R. 1987. Hierarchical correctness proofs for distributed algorithms. In Proceedings of the 6th Annual A CM Symposium on Principles of Distributed Computing (Vancouver, B.C., Canada, Aug. 10-12). ACM, New York, pp. 137-151. (Full version available as HIT Tech Rep. MIT/LCS/TR-387.) Google Scholar
- LYNCH, N. A., AND TUTTLE, M.R. 1988. An introduction to input/output automata. HIT Tech. Rep. MIT/LCS/TR-373.Google Scholar
- MORAN, S., TAUBENFELD, G., AND YADIN, I. 1992. Concurrent counting. In Proceedings of the 11th Annual ACM Symposium on Principles of Distributed Computing (Vancouver, B.C., Canada, Aug. 10-12). ACM, New York, pp. 59-70. Google Scholar
- POGOSYANTS, A., SEGALA, R., AND LYNCH, N. 1996. Verification of the randomized consensus algorithm of Aspnes and Herlihy: A case study. Unpublished manuscript. HIT, Cambridge, Mass. Google Scholar
- SAKS, M., SHAVIT, N., AND WOLL, U. 1991. Optimal time randomized consensus--Making resilient algorithms fast in practice. In Proceedings of the 2nd Annual ACM Symposium on Discrete Algorithms. ACM, New York, pp. 351-362. Google Scholar
Index Terms
- On the space complexity of randomized synchronization
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