A self-stabilizing distributed algorithm for the local (1,|Ni|)-critical section problem

We consider the local (1,|N-i|)-critical section (CS) problem where N-i is the set of neighboring processes for each process P-i. It dynamically maintains two disjoint dominating sets and is one of the generalizations of the mutual exclusion problem. The problem is one of controlling the system in such a way that, for each process, among its neighbors and itself, at least one process must be in the CS and at least one process must be out of the CS at each time. That is, in the system G=(V,E), there are always two disjoint dominating sets A(1)(subset of V) and A(2)(=V\A(1)) and each process alternates between its rule A(1) and A(2) infinitely. It is useful for sleep scheduling or cluster head scheduling in sensor networks. In this paper, first, we show the necessary and sufficient conditions to solve the problem without any deadlock detection. To discuss the conditions, we consider an inefficient (costly) self-stabilizing algorithm for the local (1,|N-i|)-CS problem. After that, an efficient self-stabilizing algorithm for the local (1,|N-i|)-CS problem is proposed under an additional assumption that the graph does not have a special matching, which we call unpreventable colorable maximal matching. The convergence time of the proposed algorithm is O(n) rounds under the weakly fair distributed daemon.