Looking for the Weakest Failure Detector for k-Set Agreement in Message-passing Systems: Is Πk the End of the Road?

In the k-set agreement problem, each process (in a set of n processes) proposes a value and has to decide a proposed value in such a way that at most k different values are decided. While this problem can easily be solved in asynchronous systems prone to t process crashes when k > t, it cannot be solved when k ≤ t. Since several years, the failure detector-based approach has been investigated to circumvent this impossibility. While the weakest failure detector class to solve the k-set agreement problem in read/write shared-memory systems has recently been discovered (PODC 2009), the situation is different in message-passing systems where the weakest failure detector classes are known only for the extreme cases k = 1 (consensus) and k = n − 1 (set agreement). This paper introduces a candidate for the general case. It presents a new failure detector class, denoted Πk, and shows Π1 = Σ × Ω (the weakest class for k = 1), and Πn−1 = L (the weakest class for k = n− 1). Then, the paper investigates the structure of Πk and shows it is the combination of two failures detector classes denoted Σk and Ωk (that generalize the previous “quorums” and “eventual leaders” failure detectors classes). Finally, the paper proves that Σk is a necessary requirement (as far as information on failure is concerned) to solve the k-set agreement problem in message-passing systems. The paper presents also a Πn−1-based algorithm that solves the (n − 1)-set agreement problem. This algorithm provides us with a new algorithmic insight on the way the (n − 1)-set agreeement problem can be solved in asynchronous message-passing systems (insight from the point of view of the non-partitioning constraint defined by Σn−1). Key-words: Asynchronous systems, Eventual leaders, Failure detectors, Message passing system, Quorums, Reduction, k-Set agreement, Wait-freedom. A la recherche du plus faible détecteur de fautes pour l’accord k-ensembliste dans les systèmes à passage de messages Résumé : Ce rapport étudie le problème du plus faible détecteur de fautes pour l’accord k-ensembliste dans les systèmes asynchrones à passage de messages sujets aux défaillances des processeurs. Mots clés : Accord ensembliste, Asynchronisme, Défaillances des processus, Détecteur de fautes. * Projet ASAP: équipe commune avec l’INRIA, le CNRS, l’universit Rennes 1 et l’INSA de Rennes ** Projet ASAP: équipe commune avec l’INRIA, le CNRS, l’universit Rennes 1 et l’INSA de Rennes c ©IRISA – Campus de Beaulieu – 35042 Rennes Cedex – France – +33 2 99 84 71 00 – www.irisa.fr in ria -0 03 84 99 3, v er si on 1 18 M ay 2 00 9 2 F. Bonnet & M. Raynal