Tight Bounds for k-Set Agreement with Limited-Scope Failure Detectors

Abstract

A system with limited-scope failure detectors ensures that there are q subsets X _i of x _i processes, 0 ≤ i ≤ q-1, such that some correct process in X _i is never suspected by any process in X _i . Let x be the sum of x _i and X be the union of X _i . The failure detector class S _{x, q} satisfies this property all the time, while ⋄S _{x, 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 S _{x, q} or ⋄S _{x, q} classes.

For S _{x, q} , we show that any k-set agreement protocol that tolerates f failures must satisfy f < k + x – q. This result establishes for the first time that the protocol of Mostéfaoui and Raynal for the S _x = S _x,1 failure detector is optimal.

For ⋄S _{x, q} , our lower bound is $f < min (\frac{n+1}{2}, k+x-q)$. We give a novel protocol that matches our lower bound, disproving a conjecture of Mostéfaoui and Raynal for the ⋄S _x = ⋄S _x,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.