Accord $k$-ensembliste asynchrone et anonyme avec $(n-k+1)$ registres atomiques

The $k$-set agreement problem is a generalization of the consensus problem.Namely, assuming each process proposes a value, each non-faulty process has to decide a value such that each decided value was proposed, and no more than $k$ different values are decided. This is a hard problem in the sense thatit cannot be solved in asynchronous systems as soon as $k$ or more processes may crash. One way to circumvent this impossibility consists in weakening its termination property, requiring that a process terminates (decides) only if it executes alone during a long enough period. This is the well-known obstruction-freedom progress condition. Considering a system of $n$ {\it anonymous asynchronous} processes, which communicate through atomic {\it read/write registers only}, and where {\it any number of processes may crash}, this paper addresses and solves the challenging open problem of designing an obstruction-free $k$-set agreement algorithm with $(n-k+1)$ atomic registers only. From a shared memory cost point of view, this algorithm is the best algorithm known so far, thereby establishing a new upper bound on the numberof registers needed to solve the problem (its gain is $(n-k)$ with respect to the previous upper bound). The algorithm is then extended to address the repeated version of $(n,k)$-set agreement. As it is optimal in the number of atomic read/write registers, this algorithm closes the gap on previously established lower/upper boundsfor both the anonymous and non-anonymous versions of the repeated $(n,k)$-set agreement problem. Finally, for $1 \leq x\leq k < n$, a generalization suited to $x$-obstruction-freedom is also described, which requires $(n-k+x)$ atomic registers only.Cet article préente un algorithme asynchrone qui résoud l'accord k-ensenbliste dans un système de n processus asynchrones et anonymes communiquant via $(n-k+1)$ registres atomiques du type lire/écrire, et dans lequel un nombre quelconque d'entre eux peut s'arrêterde façon inopinée (crash failure). La propriété de vivacité garantie par l'algorithme est appelée "obstruction-freedom"