(anti-Ωx x Σz)-Based k-Set Agreement Algorithms

This paper considers the k-set agreement problem in a crash-prone asynchronous message passing system enriched with failure detectors. Two classes of failure detectors have been previously identified as necessary to solve asynchronous k-set agreement: the class anti-leader anti-Omega(k) and the weak-quorum class Sigma(k). The paper investigates the families of failure detector (anti-Omega(x))1 <= x <= n and (Sigma(z))1 <= z<n. It characterizes in an n processes system equipped with failure detectors anti-Omega(x) and Sigma(z) for which values of k, x and z k-set-agreement can be solved. While doing so, the paper (1) disproves previous conjunctures about the weakest failure detector to solve k-set-agreement in the asynchronous message passing model and, (2) introduces the first indulgent algorithm that tolerates a majority of processes failures.