Gradual Stabilization under τ -Dynamics

We introduce the notion of gradually stabilizing algorithm as any self-stabilizing algorithm achieving the following additional feature. Assuming that at most τ dynamic steps occur starting from a legitimate configuration, a gradually stabilizing algorithm first quickly recover to a configuration from which a specification offering a minimum quality of service is satisfied. It then gradually converges to specifications offering stronger and stronger safety guarantees until fully recovering to its initial (strong) specification. We illustrate this new property by considering three variants of a synchronization problem respectively called strong, weak, and partial weak unison. We propose a self-stabilizing algorithm which is also gradually stabilizing in the sense that after one dynamic step from a legitimate configuration, it immediately satisfies the specification of partial weak unison, then converges to the specification of weak unison in at most one round, and finally retrieves the specification of strong unison after at most (μ+ 1)D1 + 1 additional rounds, where D1 is the diameter of the network after the dynamic step and μ is a parameter which should be greater than or equal to sum of n (the initial size of the network) and #J (an upper bound on the number of processes that join the system during the dynamic step).