A Self-stabilizing 1-maximal Independent Set Algorithm

We consider the 1-maximal independent set (1-MIS) problem: given a graph \(G=(V,E)\), our goal is to find an 1-maximal independent set (1-MIS) of a given network G, that is, a maximal independent set (MIS) \(S \subset V\) of G such that \(S \cup \{v,w\} \setminus \{u\}\) is not an independent set for any nodes \(u \in S\), and \(v,w \notin S\) (\(v \ne w\)). We give a silent, self-stabilizing, and asynchronous distributed algorithm to construct 1-MIS on a network of any topology. We assume the processes have unique identifiers and the scheduler is unfair and distributed. The time complexity, i.e., the number of rounds to reach a legitimate configuration in the worst case, of the proposed algorithm is O(nD), where n is the number of processes in the network and D is the diameter of the network. We use a composition technique called loop composition [Datta et al. 2017] to iterate the same procedure consistently, which results in a small space complexity, \(O(\log n)\) bits per process.