On the Space Complexity of Set Agreement

TL;DR

This paper investigates the space complexity of the $k$-set agreement problem under various progress conditions, providing bounds on the number of registers needed for solutions in different settings.

Contribution

It establishes tight upper and lower bounds on register requirements for $m$-obstruction-free $k$-set agreement, including both one-shot and repeated cases.

Findings

Repeated $k$-set agreement can be solved with $n+2m-k$ registers.

Lower bound of $n+m-k$ registers for solving the problem.

Results unify and extend understanding of space complexity in set agreement.

Abstract

The $k$-set agreement problem is a generalization of the classical consensus problem in which processes are permitted to output up to $k$ different input values. In a system of $n$ processes, an $m$-obstruction-free solution to the problem requires termination only in executions where the number of processes taking steps is eventually bounded by $m$. This family of progress conditions generalizes wait-freedom ($m=n$) and obstruction-freedom ($m=1$). In this paper, we prove upper and lower bounds on the number of registers required to solve $m$-obstruction-free $k$-set agreement, considering both one-shot and repeated formulations. In particular, we show that repeated $k$ set agreement can be solved using $n+2m-k$ registers and establish a nearly matching lower bound of $n+m-k$.