Self-stabilizing Minimal Global Offensive Alliance Algorithm with Safe Convergence in an Arbitrary Graph

Abstract

In a graph or a network G = (V, E), a set ${\mathcal{S}} \subseteq V$ is a global offensive alliance if each node $i \in \{V-{\mathcal{S}}\}$ has $|N[i] \cap{\mathcal{S}}| \ge |N[i]-{\mathcal{S}}|$. A global offensive alliance ${\mathcal{S}}$ is called minimal when there does not exist a node $i \in{\mathcal{S}}$ such that the set ${\mathcal{S}}-\{i\}$ is a global offensive alliance. In this paper, we propose a new self-stabilizing algorithm for minimal global offensive alliance. It has safe convergence property under synchronous daemon in the sense that starting from an arbitrary state, it quickly converges to a global offensive alliance (a safe state) in two rounds, and then stabilizes in a minimal global offensive alliance (the legitimate state) in O(n) rounds without breaking safety during the convergence interval, where n is the number of nodes. Space requirement at each node is O(logn) bits.