Synchronous vs. asynchronous unison

This paper considers the self-stabilizing unison problem in uniform distributed systems. The contribution of this paper is threefold. First, we establish that when any self-stabilizing asynchronous unison protocol runs in synchronous systems, it converges to synchronous unison if the size of the clock K is greater than C (G) , C (G) being the length of the maximal cycle of the shortest maximal cycle basis if the graph contains cycles, 2 otherwise (tree networks). The second result demonstrates that the asynchronous unison in Boulinier et al. (In PODC '04: Proceedings of the twenty-third annual ACM symposium on principles of distributed computing, pp. 150-159, 2004) provides a general self-stabilizing synchronous unison for trees which is optimal in memory space, i.e., it works with any K >= 3, without any extra state, and stabilizes within 2D rounds, where D is the diameter of the network. This protocol gives a positive answer to the question whether there exists or not a general self-stabilizing synchronous unison for tree networks with a state requirement independent of local or global information of the tree. If K=3, then the stabilization time of this protocol is equal to D only, i.e., it reaches the optimal performance of Herman and Ghosh (Inf. Process. Lett. 54:259-265, 1995). The third result of this paper is a self-stabilizing unison for general synchronous systems. It requires K >= 2 only, at least K+D states per process, and its stabilization time is 2D only. This is the best solution for general synchronous systems, both for the state requirement and the stabilization time.