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

Abstract

a b s t r a c tIn 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 messagepassing 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 messagepassing systems. It is hoped that these contributions will help discover the weakest failure detector class for k-set agreement in message-passing systems.