Optimality and Self-Stabilization Over Acyclic Distributed Systems

Self-stabilization was rst introduced by Dijkstra Dij74]: it is the property for a system to eventually recover itself a legitimate state after any perturbation modifying the memory state. In its pioneering paper, Dijkstra proposed three algorithms for token ring systems. Such systems are very useful to solve distributed mutual exclusion. Using the token deenition introduced by Dijkstra, i.e. the token is held by the processor enabled to make a move, Tchuente Tch81] showed the expected state number lower bound to solve mutual exclusion over tree networks, i.e. 2 n Q n i=1 i , n is the number of processors, i is the number of neighbors of each processor p i. In this paper, we use a weaker token formulation introduced by Villain Vil97]: a processor holds a token if it holds a particular state. This new light allows us to propose a self-stabilizing depth-rst token circulation for tree networks requiring fewer states than the Tchuente's lower bound. Futhermore, we show that ((1 + 1) Q n i=2 ((i + 2) is the minimal number of global states required for such a token circulation. This lower bound holds whether the system being self-stabilizing or not. Since the proposed algorithm exactly needs the same number of states, we conclude that it is optimal.