Conditions for Set Agreement with an Application to Synchronous Systems

The $k$-set agreement problem is a generalization of the consensus problem: considering a system made up of $n$ processes where each process proposes a value, each non-faulty process has to decide a value such that a decided value is a proposed value, and no more than $k$ different values are decided. While this problem cannot be solved in an asynchronous system prone to $t$ process crashes when $t \geq k$, it can always be solved in a synchronous system; $\lfloor \frac{t}{k} \rfloor +1$ is then a lower bound on the number of rounds (consecutive communication steps) for the non-faulty processes to decide. The {\it condition-based} approach has been introduced in the consensus context. Its aim was to both circumvent the consensus impossibility in asynchronous systems, and allow for more efficient consensus algorithms in synchronous systems. This paper addresses the condition-based approach in the context of the $k$-set agreement problem. It has two main contributions. The first is the definition of a framework that allows defining conditions suited to the $\ell$-set agreement problem. More precisely, a condition is defined as a set of input vectors such that each of its input vectors can be seen as ``encoding'' $\ell$ values, namely, the values that can be decided from that vector. A condition is characterized by the parameters $t$, $\ell$, and a parameter denoted $d$ such that the greater $d+\ell$, the least constraining the condition (i.e., it includes more and more input vectors when $d+\ell$ increases, and there is a condition that includes all the input vectors when $d+\ell>t$). The conditions characterized by the triple of parameters $t$, $d$ and $\ell$ define the class of conditions denoted ${\cal S}_t^{d,\ell}$, $0\leq d\leq t$, $1\leq \ell \leq n-1$. The properties of the sets ${\cal S}_t^{d,\ell}$are investigated, and it is shown that they have a lattice structure. The second contribution is a generic synchronous $k$-set agreement algorithm based on a condition $C\in {\cal S}_t^{d,\ell}$, i.e., a condition suitedto the $\ell$-set agreement problem, for $\ell \leq k$. This algorithm requires at most $\left\lfloor \frac{d-1+\ell}{k} \right\rfloor +1$ rounds when the input vector belongs to $C$, and $\left\lfloor \frac{t}{k} \right\rfloor +1$ rounds otherwise. (Interestingly, this algorithm includes as particular cases the classical synchronous $k$-set agreement algorithm that requires $\left\lfloor \frac{t}{k} \right\rfloor+1$ rounds (case $d=t$ and $\ell=1$), and the synchronous consensus condition-based algorithm that terminates in $d+1$ rounds when the input vector belongs to the condition, and in $t+1$ rounds otherwise (case $k=\ell=1$).