Loosely-Stabilizing Leader Election in Population Protocol Model
A self-stabilizing protocol guarantees that starting from an arbitrary initial configuration, a system eventually comes to satisfy its specification and keeps the specification forever. Although self-stabilizing protocols show excellent fault-tolerance against any transient faults (e.g. memory crash), designing self-stabilizing protocols is difficult and, what is worse, might be impossible due to the severe requirements. To circumvent the difficulty and impossibility, we introduce a novel notion of loose-stabilization, that relaxes the closure requirement of self-stabilization; starting from an arbitrary configuration, a system comes to satisfy its specification in a relatively short time, and it keeps the specification for a long time, though not forever. To show effectiveness and feasibility of this new concept, we present a probabilistic loosely-stabilizing leader election protocol in the Probabilistic Population Protocol (PPP) model of complete networks. Starting from any configuration, the protocol elects a unique leader within O(nNlogn) expected steps and keeps the unique leader for Ω(NeN) expected steps, where n is the network size (not known to the protocol) and N is a known upper bound of n. This result proves that introduction of the loose-stabilization circumvents the already-known impossibility result; the self-stabilizing leader election problem in the PPP model of complete networks cannot be solved without the knowledge of the exact network size.
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