A Self-stabilizing 1-Maximal Independent Set Algorithm

Abstract

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.

This work was supported by JSPS KAKENHI Grant Numbers 17K19977, 18K18000, 19H04085, and 19K11826 and JST SICORP Grant Number JPMJSC1606.

A. K. Datta passed away on May 26, 2019. Rest in Peace, Ajoy.