A Topological Treatment of Early-Deciding Set-Agreement

This paper considers the k-set-agreement problem in a synchronous message passing distributed system where up to t processes can fail by crashing. We determine the number of communication rounds needed for all correct processes to reach a decision in a given run, as a function of k, the degree of coordination, and f ≤ t the number of processes that actually fail in the run. We prove a lower bound of min(bf/kc+2, bt/kc+1) rounds. Our proof uses simple topological tools to reason about runs of a full information set-agreement protocol. In particular, we introduce a topological operator, which we call the early deciding operator, to capture rounds where k processes fail but correct processes see only k − 1 failures.