Synchronous vs. Asynchronous Unison

This paper considers the self-stabilizing unison problem. 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 CG, CG 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 [3] provides a universal self-stabilizing synchronous unison for trees which is optimal in memory space. 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 universal self-stabilizing synchronous unison for tree networks with a state requirement independant 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 [8]. 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.