Condition-Based Protocols for Set Agreement Problems

In this paper, we present condition based solutions to k-set agreement problems. A condition C is a set of input vectors to the problem. We identify a class of conditions that allow to solve k-set agreement in spite of f crashes in an asynchronous system. Moreover, we design a protocol that is always safe: it is guaranteed to terminate when the input vector belongs to C or there are less than k crashes, but it always decides on at most k diierent values, even if the input vector does not belong to C. While there are simple solutions when f < k, it has been shown that the k-set agreement problem has no solution when f k. Thus we identify classes of conditions that allow to solve this problem even when f k. We also describe two natural concrete conditions that belong to such a class. Finally, we present a more eecient k-set agreement protocol with only linear complexity (does not use snapshots), for any C that allows to solve consensus, when k f=(n ? f) + 1. This shows how to trade fault-tolerance for agreement precision using the condition based approach, and improves on the protocol we presented in STOC'01. Une solution au probl eme de l'accord sur un ensemble de valeurs fond ee sur les conditions R esum e : Ce rapport pr esente une solution au probl eme de l'accord sur un ensemble de k valeurs fond ee sur les conditions. Une condition C est un sous-ensemble des vecteurs d'entr ee possibles. Dans un syst eme asynchrone compos e de n processus et o u au plus f d'entre eux peuvent ^ etre d efaillants, le probl eme de l'accord sur un ensemble de k valeurs peut ^ etre facilement r esolu lorsque f < k alors qu'il n'a pas de solution d eterministe lorsque f k. Ce rapport d eenit des classes de conditions qui permettent de r esoudre ce probl eme m^ eme lorsque f k. Le protocole propos e poss ede la propri et e d'^ etre toujours s^ ur (\safe"): lorsqu'il termine, il ne d ecide pas plus de k dii erentes valeurs m^ eme lorsque le vecteur d'entr ee n'appartient pas a la condition avec laquelle le protocole a et e instanci e. De plus, un second protocole, de complexit e lin eaire, est propos e. Il r esout le probl eme …