State-optimal snap-stabilizing PIF in tree networks

Abstract

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 (P F C ). We present two algorithms. The first one is a basic P F C scheme which is inherently snapstabilizing. However, it can be delayed Oh 2 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 P F C algorithm from Oh 2 to 1 step. The P F C scheme can be used to implement the distributed reset, the distributed infimum computation, and the global synchronizer in O1 waves (or PIF cycles). Moreover, assuming that a (local) checking mechanism exists to detect transient failures or topological changes, the P F C scheme allows processors to (locally) "detect" if the system is stabilized, in O1 waves without using any global metric (such as the diameter or size of the network).Finally, we show that the state requirement for both P F C 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.

Snap-Stabilization in Message-Passing Systems

“…Definition 1 (Snap-Stabilization [11]) Let SP T be a specification. An protocol P is snap-stabilizing for SP T if and only if starting from any configuration, any execution of P satisfies SP T .…”

“…That is, a self-stabilizing system may behave incorrectly as it recovers. Bui et al [11] introduce a related concept of snap-stabilization. Given a problem specification, a system is guaranteed to perform according to this specification regardless of the initial state.…”

“…In particular propagation of information with feedback (PIF) is a popular problem to address [11,10,12,20,14,9,19]. Several studies present snap-stabilizing token circulation protocols [30,16,18].…”

From Self- to Snap- Stabilization

“…A self-stabilizing system, regardless of the initial states of the processors and messages initialy in the links, is guaranteed to converge into the intended behavior in finite time. Recently, a new paradigm called snap-stabilization has been introduced in [2]. A snap-stabilizing protocol guarantees that, starting from any configuration, it always behaves according to its specification.…”

Snap-Stabilizing Detection of Cutsets

“…A self-stabilizing system, regardless of the initial states of the processors and messages initialy in the links, is guaranteed to converge to the intended behavior in a finite time. Snap-stabilization was introduced in [2]. A snap-stabilizing protocol guaranteed that it always behaves according to its specification.…”