Chasing the Weakest Failure Detector for k-Set Agreement in Message-Passing Systems

This paper continues our quest for the weakest failure detector which allows the k-set agreement problem to be solved in asynchronous message-passing systems prone to process failures. It has two main contributions which will be instrumental to complete this quest. The first contribution is a new failure detector (denoted PiSigma(x, y)) that has several noteworthy properties. (a) It is stronger than Sigma(k) which has been shown to be necessary. (b) It is equivalent to the pair (Sigma, Omega) when x=y=1 (optimal to solve consensus). (c) It is equivalent to the pair (Sigma(n-1), Omega(n-1)) when x=y=n-1 (optimal for (n-1)-set agreement). (d) It is strictly weaker than the pair (Sigma(x), anti-Omega(y)) (which has been investigated in previous works). (e) It is operational: the paper presents a PiSigma(x, y)-based algorithm that solves k-set agreement for k greater or equal to xy (intuitively, x refers to the maximum number of isolated groups of processes and y to the number of leaders in each of these groups). The second contribution of the paper is a proof that, for k strictly between 1 and n-1, the eventual leaders failure detector Omega(k) (which eventually provides each process with the same set of k process identities, this set including at least one correct process) is not necessary to solve k-set agreement problem.