No Double Discount: Condition-Based Simultaneity Yields Limited Gain

Abstract

Assuming that each process proposes a value, the consensus problem requires the non-faulty processes to agree on the same value that, moreover, must be one of the proposed values. Solutions to the consensus problem in synchronous systems are based on the round-based model, namely, the processes progress in synchronous rounds. The well-known worst-case lower bound for consensus in the synchronous setting is t + 1 rounds, where t is an a priori bound on the number of failures. Simultaneous consensus is a variant of consensus in which the non-faulty processes are required to decide in the exact same round, in addition to the deciding on the same value. Dwork and Moses showed that, in a synchronous system prone to t process crashes, the earliest round at which a common decision can be simultaneously obtained is (t + 1) − W where t is a bound on the number of faulty processes and W is a non-negative integer determined by the actual failure pattern F. In the condition-based approach consensus the consensus requirement is relaxed by assuming that the input vectors (consisting of the proposed initial values) are restricted to belong to a predetermined set C. Initially considered as a means to achieve solvability for consensus in the asynchronous setting, condition-based consensus was shown by Mostéfaoui, Rajsbaum and Raynal to allow solutions with better worst-case behavior. They defined a hierarchy of sets of conditions (where the set contains the condition made up of all possible input vectors). It has been shown that t + 1 − d is a tight lower bound on the minimal number of rounds for synchronous condition-based consensus with a condition in .

This paper considers condition-based simultaneous consensus in the synchronous model. It first presents a simple algorithm in which processes decide simultaneously at the end of the round RS _t,d,F = (t + 1) −  max (W, d). The paper then shows that RS _t,d,F is a lower bound for simultaneous condition-based consensus. A consequence of the analysis is that the algorithm presented is optimal in each and every run, and not just in the worst case: For every choice of failure pattern by the adversary (and every input configuration), the algorithm reaches simultaneous agreement as fast as any correct algorithm could do under the same conditions. This shows that, contrary to what could be hoped, when considering condition-based consensus with simultaneous decision, we can benefit from the best of both actual worlds (either the failure world when RS _t,d,F= (t + 1) − W, or the condition world when RS _t,d,F= d + 1), but we cannot benefit from the sum of savings offered by both. Only the best discount applies.