Self-Stabilizing Diffusion

This paper deals with a peculiar instance of the consensus problem. In our scenario, the system can be assimilated to an irregular network composed of clusters of processing units. Each unit holds a ”copy” of a value, that changes continuously according with a known function. The basic problem we address is to keep all the copies sufficiently close to the real value, given that most units cannot keep the local copy of the value exactly updated, with a minimal computational load. With respect to the ”precision” of the local copy of the global value, we distinguish two kinds of units: a restricted number of units that continuously keep the correct value, upon which any other unit should agree, and the rest of the units, that can only estimate locally the current global value. We do not investigate the way in which the former units keep their copies exactly updated: since we count a few units of this kind in the system, they may use an expensive strategy. On the contrary, we are interested in the behavior of the rest of the system. The precision of the local estimate of the other units progressively degrades, and they need to periodically refresh their local estimate in order to keep it acceptably near to the global value. Our problem is the design of a protocol that performs this task. The relevance of the problem should become evident if we recognize that the above description is a paraphrase of the clock synchronization problem: a few units in the system hold accurate (and expensive) clocks, while the others periodically synchronize their in-accurate (and cheap) clocks with the accurate ones. The interested reader can find a technical discussion of the problem and its practical solution in Internet in [7]: the protocol suite is known as Network Time Protocol (NTP).