Relations Linking Failure Detectors Associated with k-Set Agreement in Message-Passing Systems

Abstract

AbstractThe k-set agreement problem is a coordination problem where each process is assumed to propose a value and each process that does not crash has to decide a value such that each decided value is a proposed value and at most k different values are decided. While it can always be solved in synchronous systems, k-set agreement has no solution in asynchronous send/receive message-passing systems where up to t ≥ k processes may crash.A failure detector is a distributed oracle that provides processes with additional information related to failed processes and can consequently be used to enrich the computability power of asynchronous send/receive message-passing systems. Several failure detectors have been proposed to circumvent the impossibility of k-set agreement in pure asynchronous send/receive message-passing systems. Considering three of them (namely, the generalized quorum failure detector Σ k , the generalized loneliness failure detector \({\cal L}_k\) and the generalized eventual leader failure detector Ω k ) the paper investigates their computability power and the relations that link them. It has three mains contributions: (a) it shows that the failure detector Ω k and the eventual version of \({\cal L}_k\) have the same computational power; (b) it shows that \({\cal L}_k\) is realistic if and only if k ≥ n/2; and (c) it gives an exact characterization of the difference between \({\cal L}_k\) (that is too strong for k-set agreement) and Σ k (that is too weak for k-set agreement).KeywordsAsynchronous message-passing systemDistributed computabilityEquivalenceEventual leaderFailure detectorFault-toleranceImpossibilityQuorumRealistic failure detectorReduction k-Set agreementTheory