Families of consensus algorithms

Abstract

Three main parameters characterize the efficiency of algorithms that solve the Consensus Problem. The ratio between the total number of processors and the maximum number of faulty processors (n and t, respectively), the number of rounds, and the size of any single message. Lower bounds exist for each one of the three. In this paper we present two families of algorithms, each achieving the lower bound for one parameter and a trade-off between the other two. The first family includes algorithms where, given an integer k, the algorithm always requires the minimal possible number of rounds (t+1), with n=k(3t+1) processors and messages of size at most t ^O(t/k). To the second family belong algorithms in which all messages are of one bit size, the number of processors is t ^O((k+1)/k) and the number of rounds is t+t ^O((k−1)/k). These two families are based on a very simple algorithm with (2t+1)(t+1) processors using the minimal number of rounds and the minimal message size (one bit).

Part of the work was done while the author was in the Hebrew University of Jerusalem. Supported in part by a Weizmann fellowship, and by contract ONR N0014-85-C-0731.