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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][151][152][153][154][155][156][157][158][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.

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We introduce a simple tool called the wavelet (or, ̺-wavelet) scheme. Wavelets deals with coordination among processes which are at most ̺ hops away of each other. We present a selfstabilizing solution for this scheme. Our solution requires no underlying structure and works in arbritrary anonymous networks, i.e., no process identifier is required. Moreover, our solution works under any (even unfair) daemon.Next, we use the wavelet scheme to design self-stabilizing layer clocks. We show that they provide an efficient device in the design of local coordination problems at distance ̺, i.e., ̺-barrier synchronization and ̺-local resource allocation (LRA) such as ̺-local mutual exclusion (LME), ̺-group mutual exclusion (GME), and ̺-Reader/Writers. Some solutions to the ̺-LRA problem (e.g., ̺-LME) also provide transformers to transform algorithms written assuming any ̺-central daemon into algorithms working with any distributed daemon.̺-local computations [NS93], i.e., running in constant time independent of any global parameter like the size of the network or the diameter. Computation in constant time ̺ can be achieved if the processes can collect informations from processes located within radius of ̺ from them. In [NS93], the authors mainly address Local Checkable Labeling problems. Local computation is also considered in [GMM04] by considering the recognition problem. The computing model is a relabelling system.Wireless networks bring new trends in distributed systems which also motivate research the local control of concurrency at distance ̺. In [DNT06], the authors propose a generalization of the well-known dining philosophers problem [Dij68]. They extends the conflict processes beyond the immediate neighbors of the processes. As an application, their solution provide a solution to the interfering transmitter problem in wireless networks.Another motivation consists in assuming that the knowledge of the processes goes beyond their immediate neighbors could help in the design of non-trivial tasks [GGH + 04, GHJT06]. An efficient self-stabilizing solution is given to the maximal 2-packing problem assuming the knowledge at distance 2 [GGH + 04]. (The maximal 2-packing problem consists to find a maximal set of nodes S, such that no two nodes in S are adjacent and no two nodes in S have a common neighbor.) The solution in [GGH + 04] requires process ID's and works under a central daemon. In [GHJT06], the authors propose a ̺-distance knowledge transformer to construct self-stabilizing algorithms which use a ̺ distance knowledge. Again, their solution works only if the daemon is central and with process ID's.Note that various kinds of transformers have been proposed in the area of self-stabilization to refine self-stabilizing algorithms which use tight scheduling constraints like the central daemon into the corresponding self-stabilizing algorithm working assuming weaker daemons,e.g., [MN98, GH99, NA02, CDP03]. A popular technique consists in composing the algorithm with a self-stabilizing local mutual exclusion (LME) alg...