State-optimal snap-stabilizing PIF in tree networks

In this paper, we introduce the notion of snapstabilization. A snap-stabilizing algorithm protocol guarantees that, starting from an arbitrary system configuration, the protocol always behaves according to its specification. So, a snap-stabilizing protocol is a self-stabilizing protocol which stabilizes in 0 steps. We propose a snap-stabilizing Propagation of Information with Feedback (PIF) scheme on a rooted tree network. We call this scheme Propagation of information with Feedback and Cleaning (PFC). We present two algorithms. The first one is a basic PFC scheme which is inherently snapstabilizing. However, it can be delayed O(h2) steps (where h is the height of the tree) due to some undesirable local states. The second algorithm improves the worst delay of the basic PFC algorithm from O(h2) to 1 step. The PFC scheme can be used to implement the distributed reset, the distributed infimum computation, and the global synchronizer in O(1) waves (or PIF cycles). Moreover, assuming that a (local) checking mechanism exists to detect transient failures or topological changes, the PFC scheme allows processors to (locally) “detect” if the system is stabilized, in O(1) waves without using any global metric (such as the diameter or size of the network). Finally, we show that the state requirement for both PFC algorithms matches the exact lower bound of the PIF algorithms on tree networks—3 states per processor, except for the root and leaf processors which use only 2 states. Thus, the proposed algorithms are optimal PIF schemes in terms of the number of states.