A self-stabilizing algorithm to maximal 2-packing with improved complexity

In self-stabilization, each node has a local view of the distributed network system, in a finite amount of time the system converges to a global setup with desired property, in this case establishing a 2-packing set. Using a graph G=(V,E)$G=(V,E)$ to represent the network, a subset S⊆V$S\subseteq V$ is a 2-packing if ∀i∈V:|N[i]∩S|?1$\forall i\in V:|N[i]\cap S|?1$. In this paper, we first propose an ID-based, constant space, self-stabilizing algorithm that stabilizes to a maximal 2-packing in an arbitrary graph. We show that the algorithm stabilizes in O(mn)$O(mn)$ moves under any scheduler (such as a distributed daemon). Secondly, we show that the algorithm stabilizes in O(n²)$O(n^2)$ rounds under a synchronous daemon where every privileged node moves at each round.