One-Step Consensus Solvability

While any fault-tolerant asynchronous consensus algorithm requires two communication steps even in failure-free executions, it is known that we can construct an algorithm terminating in one step for some good inputs (e.g. all processes propose a same value). In this paper, we present the necessary and sufficient constraint for the set of inputs for which we can construct an asynchronous consensus algorithm terminating in one step. Our investigation is based on the notion of the condition-based approach: it introduces conditions on input vectors to specify subsets of all possible input vectors and condition-based algorithms can circumvent some impossibility if the actual input vector satisfy a particular condition. More interestingly, conditions treated in this paper are adaptive. That is, we consider hierarchical sequences of conditions whose k-th condition is the set of input vectors for which the consensus can be solved in one step if at most k processes crash. The necessary and sufficient constraint we propose in this paper is one for such condition sequences. In addition, we present an instance of the sufficient condition sequences. Compared with existing constraints for inputs this instance is more relaxed.