Summary: Lower Bounds for Randomized Consensus
under a Weak Adversary
Keren Censor Hillel
April 26, 2010
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 1
ck , 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 ad-
vance, without observing the algorithm's actions. This complements algorithms of Kapron et al. ,
for message-passing systems, and of Aumann et al. [7,8], for shared-memory systems.
A preliminary version of this paper appeared in Proceedings of the 27th Annual ACM Symposium on Principles of Distributed
Computing, pages 315-324, 2008. This research is supported in part by the Israel Science Foundation (grant number 953/06).