A linear-time self-stabilizing distributed algorithm for the minimal minus ($L, K, Z$) -domination problem under the distance-2 model.

The domination problem is one of the fundamental graph problems and there are many variations. The problem has practical applications in a distributed setting, and studied well in the distributed computing community. In this paper, we propose a new problem called the minus ( $L, K, Z$ ) -domination problem where $L, K$ , and $Z$ are integers such that $L\leq-1,K\geq 1$ , and $Z\geq 1$ . The minus ( $L, K, Z$ ) -domination problem is a problem to assign a value $L, L+1, \ldots, 0, \ldots, K-1, K$ for each vertex in a graph such that the local summation of values is greater than or equal to $Z$ . Because it is the same as the minus domination problem when $L=-1, K=1$ and $Z=1$ , it is an extension of the minus domination problem. Then, we propose a self-stabilizing distributed algorithm for the minus ( $L, K, Z$ ) -domination problem, where self-stabilization is a class of fault-tolerant distributed algorithms that tolerate arbitrary finite number of transient faults. The proposed algorithm is designed under the distance-2 model and the unfair central daemon, and its convergence time is $O(n)$ , that is, linear to $n$ , where $n$ is the number of processes. If it is converted into the ordinary distance-1 model with a transformer, we obtain an algorithm whose convergence time is $O(nm)$ .