k-Immediate Snapshot and x-Set Agreement - How Are They Related?

An immediate snapshot object is a high level communication object, built on top of a read/write distributed system in which all except one processes may crash. This object provides the processes with a single operation, denoted write _ snapshot ( ) , which allows the invoking process to write a value and obtain a set of pairs ⟨ process id, value ⟩ satisfying some set containment properties, that represent a snapshot of the values written to the object, occurring immediately after the write step. Considering an n -process model in which up to t processes may crash, this paper introduces first the k -resilient immediate snapshot object, which is a natural generalization of the basic immediate snapshot (which corresponds to the case k = t = n - 1 ). In addition to the set containment properties of the basic immediate snapshot, a k -resilient immediate snapshot object requires that each set returned to a process contains at least ( n - k ) pairs. The paper first shows that, for k , t < n - 1 , k -resilient immediate snapshot is impossible in asynchronous read/write systems. Then it investigates a model of computation where the processes communicate with each other by accessing k -immediate snapshot objects, and shows that this model is stronger than the t -crash model. Considering the space of x -set agreement problems (which are impossible to solve in systems such that x ≤ t ), the paper shows then that x -set agreement can be solved in read/write systems enriched with k -immediate snapshot objects for x = max ( 1 , t + k - ( n - 2 ) ) . It also shows that, in these systems, k -resilient immediate snapshot and consensus are equivalent when 1 ≤ t < n / 2 and t ≤ k ≤ ( n - 1 ) - t . Hence, the paper establishes strong relations linking fundamental distributed computing objects (one related to communication, the other to agreement), which are impossible to solve in pure read/write systems.