Result: A Self-Stabilizing Distributed Algorithm for the Generalized Dominating Set Problem With Safe Convergence

A self-stabilizing distributed algorithm is guaranteed eventually to reach and stay at a legitimate configuration regardless of the initial configuration of a distributed system. In this paper, we propose the generalized dominating set problem, which is a generalization of the dominating set and $k$-redundant dominating set problems. In the generalized dominating set we propose in this paper, each node $P_{i}$ is given its set of domination wish sets, and a generalized dominating set is a set of nodes such that each node is contained in the set or has a wish set in which all its members are in the set. We propose a self-stabilizing distributed algorithm for finding a minimal generalized dominating set in an arbitrary network under the unfair distributed daemon. The proposed algorithm converges in $O(n^{3}m)$ steps and $O(n)$ rounds, where $n$ (resp., $m$) is the number of nodes (resp., edges). Furthermore, it has the safe convergence property with safe convergence time in $O(1)$ rounds. The space complexity of the proposed algorithm is $O(\Delta \log n)$ bits per node, where $\Delta $ is the maximum degree of nodes.