Lower Bounds for Randomized Consensus under a Weak Adversary

  • Authors:

  • Hagit Attiya;Keren Censor-Hillel

  • Affiliations:

  • hagit@cs.technion.ac.il;ckeren@csail.mit.edu

  • Venue:
  • SIAM Journal on Computing
  • Year:
  • 2010

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Abstract

This paper studies the inherent trade-off between termination probability and total step complexity of randomized consensus algorithms. It shows that for every integer $k$, the probability that an $f$-resilient randomized consensus algorithm of $n$ processes does not terminate with agreement within $k(n-f)$ steps is at least $\frac{1}{c^k}$, for some constant $c$. A corresponding result is proved for Monte-Carlo algorithms that may terminate in disagreement. The lower bound holds for asynchronous systems, where processes communicate either by message passing or through shared memory, under a very weak adversary that determines the schedule in advance, without observing the algorithm's actions. This complements algorithms of Kapron et al. [Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), ACM, New York, SIAM, Philadelphia, 2008, pp. 1038-1047] for message-passing systems, and of Aumann [Proceedings of the 16th Annual ACM Symposium on Principles of Distributed Computing (PODC), ACM, New York, 1997, pp. 209-218] and Aumann and Bender [Distrib. Comput., 17 (2005), pp. 191-207] for shared-memory systems.