On the road to the weakest failure detector for k -set agreement in message-passing systems

• Published in: Theoretical computer science Vol. 412; no. 33; pp. 4273 - 4284
• Main Authors: Bonnet, François, Raynal, Michel
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Oxford Elsevier B.V 29.07.2011; Elsevier
• Subjects: [formula omitted]-set agreement; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Asynchronous systems; Computer Science; Computer science; control theory; systems; Crashes; Detectors; Distributed, Parallel, and Cluster Computing; Eventual leaders; Exact sciences and technology; Failure; Failure detectors; Message-passing systems; Miscellaneous; Quorums; Reduction; Roads; Theoretical computing; Wait-freedom; Message-passing systems; Failure detectors; Reduction; Quorums; k -set agreement; Asynchronous systems; Eventual leaders; Wait-freedom; Computer theory; Algorithmics; Algorithm; Failures; Consensus; Message passing; k-set agreement; Shared memory systems; Failure
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2010.11.007

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 . For 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 presents four contributions whose aim is to help pave the way to discover the weakest failure detector class for k -set agreement in message-passing systems. These contributions are the following. (a) The first is a new failure detector class, denoted Π k , that is such that Π 1 = Σ × Ω (the weakest class for k = 1 ), and Π n − 1 = L (the weakest class for k = n − 1 ). (b) The second is an investigation of the structure of Π k that shows that Π k is the combination of two failure detector classes Σ k (that is new) and Ω k (they generalize the previous “quorums” and “eventual leaders” failure detector classes, respectively). (c) The third contribution concerns Σ k that is shown to be a necessary requirement (as far as information on failure is concerned) to solve the k -set agreement problem in message-passing systems. (d) Finally, the last contribution is a Π n − 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 agreement problem can be solved in asynchronous message-passing systems. It is hoped that these contributions will help discover the weakest failure detector class for k -set agreement in message-passing systems.