Some New Results With k-set agreement

In this article, we investigate the solvability of $k$-set agreement among $n$ processes in distributed systems prone to different types of process failures. Specifically, we explore two scenarios: synchronous message-passing systems prone to up to $t$ Byzantine failures of processes. And asynchronous shared memory systems prone to up to $t$ crash failures of processes. Our goal is to address the gaps left by previous works\cite{SSS,AsynchKset} in these areas. For Byzantine failures case we consider systems with authentication where processes have unforgeable signatures. For synchronous message-passing systems, we present an authenticated algorithm that achieves $k$-set agreement in only two rounds, with no constraints on the number of faults $t$, with $k$ determined as $k \geq \lfloor \frac{n}{n-t} \rfloor + 1$. In fact the lower bound for $k$ is $k \geq \lfloor \frac{n}{n-t} \rfloor$ that is obtained by an algorithm based on traditional consensus with $t+1$ rounds. In asynchronous shared memory systems, we introduce an algorithm that accomplishes $k$-set agreement for values of $k$ greater than $\lfloor \frac{n-t}{n-2t} \rfloor +1$. This algorithm uses a snapshot primitive to handle crash failures and enable effective set agreement.