On the performance of Dijkstra's third self-stabilizing algorithm for mutual exclusion and related algorithms

摘要

In Dijkstra (Commun ACM 17(11):643-644, 1974) introduced the notion of self-stabilizing algorithms and presented three such algorithms for the problem of mutual exclusion on a ring of n processors. The third algorithm is the most interesting of these three but is rather non intuitive. In Dijkstra (Distrib Comput 1:5-6, 1986) a proof of its correctness was presented, but the question of determining its worst case complexity-that is, providing an upper bound on the number of moves of this algorithm until it stabilizes-remained open. In this paper we solve this question and prove an upper bound of 3 13/18n2 +O (n) for the complexity of this algorithm. We also show a lower bound of 1 5/6n2 - O(n) for the worst case complexity. For computing the upper bound, we use two techniques: potential functions and amortized analysis. We also present a new-three state self-stabilizing algorithm for mutual exclusion and show a tight bound of 5/6n2 + O(n) for the worst case complexity of this algorithm. In Beauquier and Debas (Proceedings of the second workshop on self-stabilizing systems, pp 17.1-17.13, 1995) presented a similar three-state algorithm, with an upper bound of 5 3/4n~2 + O (n) and a lower bound of 1/8n~2 - O (n) for its stabilization time. For this algorithm we prove an upper bound of 1 1/2n~2 + O (n) and show a lower bound of n~2 - O (n). As far as the worst case performance is considered, the algorithm in Beauquier and Debas (Proceedings of the second workshop on self-stabilizing systems, pp 17.1-17.13, 1995) is better than the one in Dijkstra (Commun ACM 17(11):643-644, 1974) and our algorithm is better than both.