From a static impossibility to an adaptive lower bound: the complexity of early deciding set agreement

Set agreement, where processors decisions constitute a set of outputs, is notoriously harder to analyze than consensus where the decisions are restricted to a single output. This is because the topological questions that underly set agreement are not about simple connectivity as in consensus. Analyzing set agreement inspired the discovery of the relation between topology and distributed algorithms, and consequently the impossibility of asynchronous set agreement.Yet, the application of topological reasoning has been to the static case, that of asynchronous and synchronous tasks. It is not known yet for example, how to characterize starvation-free solvability of non-terminating tasks. Non-terminating tasks are dynamic entities with no defined end. In a similar vain, early deciding synchronous set agreement, in which the number of rounds it takes a processor to decide adapts to the actual number of failures, falls in this category of dynamic entities.This paper develops a simulation technique that brings to bear topological results to deal with the dynamic situation that arises with early decisions. The novelty of the new simulation is the ability of simulators to look back at the transcript of past rounds of the simulation to influence their current behavior.Using our new technique, we not only re-derive past results, but we propose and prove a lower bound to synchronous early stopping set agreement. We then provide an algorithm to match the lower bound. Our technique uses the BG simulation, in the most creative way it was used to-date, to obtain a rather simple reduction from a static asynchronous impossibility. This reduction is a simple alternative to yet unknown topological argument, and in fact may suggest the way of finding such an argument.