Tight bounds for k-set agreement with limited-scope failure detectors

Abstract

In a system with limited-scope failure detectors, there are q disjoint clusters of processes such that some correct process in each cluster is never suspected by any process in that cluster. The failure detector class Sx,q satisfies this property all the time, while ♦Sx,q satisfies it eventually. This paper gives the first tight bounds for the k-set agreement task in asynchronous message-passing models augmented with failure detectors from either the Sx,q or ♦Sx,q classes. For Sx,q, we show that any k-set agreement protocol that tolerates f failures must satisfy f < k+x-q if q < k and f < x otherwise, where x is the combined size of the q disjoint clusters if q < k (or the k largest, otherwise). This result establishes for the first time that the protocol of Mostefaoui and Raynal for the Sx = Sx,1 failure detector is optimal.For ♦Sx,q, we show that any k-set agreement protocol that tolerates f failures must satisfy f < min(n+1/2, k+x-q) if q < k and f < min(n+1/2,x) otherwise, where n + 1 is the total number of processes. We give a novel protocol that matches our lower bound, disproving a conjecture of Mostefaoui and Raynal for the ♦Sx = ♦Sx,1 failure detector.Our lower bounds exploit techniques borrowed from Combinatorial Topology, demonstrating for the first time that this approach is applicable to models that encompass failure detectors.