Abstract
We consider the problem of consensus in the challenging classic model. In this model, the adversary is adaptive; it can choose which processors crash at any point during the course of the algorithm. Further, communication is via asynchronous message passing: there is no known upper bound on the time to send a message from one processor to another, and all messages and coin flips are seen by the adversary.
We describe a new randomized consensus protocol with expected message complexity O(n 2 log2 n) when fewer than n/2 processes may fail by crashing. This is an almost-linear improvement over the best previously known protocol, and within logarithmic factors of a known Ω(n 2) message lower bound. The protocol further ensures that no process sends more than O(nlog3 n) messages in expectation, which is again within logarithmic factors of optimal.We also present a generalization of the algorithm to an arbitrary number of failures t, which uses expected O(nt + t 2 log2 t) total messages. Our protocol uses messages of size O(log n), and can therefore scale to large networks.
Our approach is to build a message-efficient, resilient mechanism for aggregating individual processor votes, implementing the message-passing equivalent of a weak shared coin. Roughly, in our protocol, a processor first announces its votes to small groups, then propagates them to increasingly larger groups as it generates more and more votes. To bound the number of messages that an individual process might have to send or receive, the protocol progressively increases the weight of generated votes. The main technical challenge is bounding the impact of votes that are still “in flight” (generated, but not fully propagated) on the final outcome of the shared coin, especially since such votes might have different weights. We achieve this by leveraging the structure of the algorithm, and a technical argument based on martingale concentration bounds. Overall, we show that it is possible to build an efficient message-passing implementation of a shared coin, and in the process (almost-optimally) solve the classic consensus problem in the asynchronous message-passing model.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abrahamson, K.: On achieving consensus using a shared memory. In: Proceedings of the Seventh Annual ACM Symposium on Principles of Distributed Computing, PODC 1988, pp. 291–302. ACM, New York (1988)
Aguilera, M.K., Toueg, S.: The correctness proof of Ben-Or’s randomized consensus algorithm. Distributed Computing 25(5), 371–381 (2012)
Alistarh, D., Aspnes, J., King, V., Saia, J.: Communication-efficient randomized consensus (2014), Full version available at http://www.cs.yale.edu/homes/aspnes/papers/disc2014-submission.pdf
Aspnes, J.: Lower bounds for distributed coin-flipping and randomized consensus. J. ACM 45(3), 415–450 (1998)
Aspnes, J., Attiya, H., Censor, K.: Randomized consensus in expected O(n logn) individual work. In: PODC 2008: Proceedings of the Twenty-Seventh ACM Symposium on Principles of Distributed Computing, pp. 325–334 (August 2008)
Aspnes, J., Attiya, H., Censor-Hillel, K.: Polylogarithmic concurrent data structures from monotone circuits. J. ACM 59(1), 2 (2012)
Aspnes, J., Censor, K.: Approximate shared-memory counting despite a strong adversary. ACM Transactions on Algorithms 6(2), 1–23 (2010)
Aspnes, J., Censor-Hillel, K.: Atomic snapshots in O(log3 n) steps using randomized helping. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 254–268. Springer, Heidelberg (2013)
Aspnes, J., Herlihy, M.: Fast randomized consensus using shared memory. Journal of Algorithms 11(3), 441–461 (1990)
Aspnes, J., Waarts, O.: Randomized consensus in expected O(n log2 n) operations per processor. SIAM J. Comput. 25(5), 1024–1044 (1996)
Attiya, H., Bar-Noy, A., Dolev, D.: Sharing memory robustly in message-passing systems. J. ACM 42(1), 124–142 (1995)
Attiya, H., Censor, K.: Tight bounds for asynchronous randomized consensus. J. ACM 55(5), 20:1–20:26 (2008)
Ben-Or, M.: Another advantage of free choice (extended abstract): Completely asynchronous agreement protocols. In: Proceedings of the Second Annual ACM Symposium on Principles of Distributed Computing, PODC 1983, pp. 27–30. ACM, New York (1983)
Bracha, G.: An asynchronous [(n - 1)/3]-resilient consensus protocol. In: PODC 1984: Proceedings of the Third Annual ACM Symposium on Principles of Distributed Computing, pp. 154–162. ACM, New York (1984)
Bracha, G., Rachman, O.: Randomized consensus in expected O(n2log n) operations. In: Toueg, S., Spirakis, P.G., Kirousis, L.M. (eds.) WDAG 1991. LNCS, vol. 579, pp. 143–150. Springer, Heidelberg (1992)
Chandra, T.D.: Polylog randomized wait-free consensus. In: Proceedings of the Fifteenth Annual ACM Symposium on Principles of Distributed Computing, Philadelphia, Pennsylvania, USA, May 23-26, pp. 166-175 (1996)
Chandra, T.D., Toueg, S.: Unreliable failure detectors for reliable distributed systems. J. ACM 43(2), 225–267 (1996)
Chor, B., Israeli, A., Li, M.: On processor coordination using asynchronous hardware. In: Proceedings of the Sixth Annual ACM Symposium on Principles of Distributed Computing, PODC 1987, pp. 86–97. ACM, New York (1987)
Dwork, C., Lynch, N., Stockmeyer, L.: Consensus in the presence of partial synchrony. J. ACM 35(2), 288–323 (1988)
Fich, F., Herlihy, M., Shavit, N.: On the space complexity of randomized synchronization. J. ACM 45(5), 843–862 (1998)
Fischer, M.J., Lynch, N.A., Paterson, M.S.: Impossibility of distributed consensus with one faulty process. J. ACM 32(2), 374–382 (1985)
Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes. Oxford University Press (2001)
Hall, P., Heyde, C.: Martingale Limit Theory and Its Application. Academic Press (1980)
Karlin, A., Yao, A.: Probabilistic lower bounds for byzantine agreement and clock synchronization. Unpublished manuscript
Keralapura, R., Cormode, G., Ramamirtham, J.: Communication-efficient distributed monitoring of thresholded counts. In: Proceedings of the 2006 ACM SIGMOD International Conference on Management of Data, SIGMOD 2006, pp. 289–300. ACM, New York (2006)
King, V., Saia, J.: Byzantine agreement in polynomial expected time. In: Proceedings of the ACM Symposium on Theory of Computing, STOC (2013)
King, V., Saia, J.: Faster agreement via a spectral method for detecting malicious behavior. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA) (2014)
Lamport, L., Shostak, R., Pease, M.: The byzantine generals problem. ACM Trans. Program. Lang. Syst. 4(3), 382–401 (1982)
Pease, M., Shostak, R., Lamport, L.: Reaching agreement in the presence of faults. J. ACM 27(2), 228–234 (1980)
Saks, M., Shavit, N., Woll, H.: Optimal time randomized consensus - making resilient algorithms fast in practice. In: Proc. of the 2nd ACM Symposium on Discrete Algorithms (SODA), pp. 351–362 (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alistarh, D., Aspnes, J., King, V., Saia, J. (2014). Communication-Efficient Randomized Consensus. In: Kuhn, F. (eds) Distributed Computing. DISC 2014. Lecture Notes in Computer Science, vol 8784. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45174-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-45174-8_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-45173-1
Online ISBN: 978-3-662-45174-8
eBook Packages: Computer ScienceComputer Science (R0)