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 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 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 Mostèfaoui and Raynal for the S _x = S _x,1 failure detector is optimal.

For ⋄S _x,q , we show that any k-set agreement protocol that tolerates f failures must satisfy $f \textless \hbox{min}(\frac{n+1}{2},k+x-q)$ if q < k and $f \textless \hbox{min}(\frac{n+1}{2},x)$ otherwise, where n + 1 is the total number of processes. We give a novel protocol that matches our lower bound, disproving a conjecture of Mostèfaoui 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.