A self-stabilizing distributed algorithm for the 1-MIS problem under the distance-3 model

Fault-tolerance and self-organization are critical properties in modern distributed systems. Self-stabilization is a class of fault-tolerant distributed algorithms which has the ability to recover from any kind and any finite number of transient faults and topology changes. In this article, we propose a self-stabilizing distributed algorithm for the 1-MIS problem under the unfair central daemon assuming the distance-3 model. Here, in the distance-3 model, each process can refer to the values of local variables of processes within three hops. Intuitively speaking, the 1-MIS problem is a variant of the maximal independent set (MIS) problem with improved local optimizations. The time complexity (convergence time) of our algorithm is $O\left(n\right)$ steps and the space complexity is $O\left(\log n\right)$ bits, where $n$ is the number of processes. Finally, we extend the notion of 1-MIS to $p$-MIS for each nonnegative integer $p$, and compare the set sizes of $p$-MIS ($p=0,1,2,\dots$) and the maximum independent set.