An Efficient Solution to the k-Set Agreement Problem

In this paper, we present a ◊Sx-based solution to solve the k-Set agreement problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f < \frac{{n + k - 1}} {2} $$\end{document} where f is the maximum number of crashes that can occur and n is the total number of processes. Just as the k-Set agreement problem is a generalization of the consensus problem (at most k different values can be decided), the class of ◊Sx failure detectors is a generalization of ◊Sx failure detectors, where x, the scope of the accuracy property, is the number of processes that do not have to suspect a correct process. We propose a simple protocol based on the following idea: k - 1 “privileged” processes directly decide their initial value, while the others (n - k + 1 processes) run a 1-Set agreement protocol (i.e., a consensus protocol) to decide on one value. This simple idea enables to tolerate up to (n + k - 1)/2 crash failures, and may lead the k-set agreement problem to be solved in only one broadcast. The protocol is decomposed into three modules. Each process executes either 1, 2 or 3 modules. To reduce the scope of accuracy, more processes have to execute the three modules. But, in that case, more messages have to be exchanged. Finally, the proposed solution considers the degree of repetition of the proposed values and takes advantage of a possible high degree of redundancy of one of them.