Looking for the weakest failure detector for k-set agreement in message-passing systems: Is Πk the end of the road?

Abstract

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 , and shows (the weakest class for k∈=∈1), and (the weakest class for k∈=∈n∈-∈1). Then, the paper investigates the structure of and shows it is the combination of two failures detector classes denoted and (that generalize the previous quorums and eventual leaders failure detectors classes). Finally, the paper proves that 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 -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 ). © 2009 Springer-Verlag Berlin Heidelberg.