Abstract
It follows from the definition of silent self-stabilization, and from the definition of proof-labeling scheme, that if there exists a silent self-stabilizing algorithm using ℓ-bit registers for solving a task \({\mathcal{T}} \), then there exists a proof-labeling scheme for \({\mathcal{T}} \) using registers of at most ℓ bits. The first result in this paper is the converse to this statement. We show that if there exists a proof-labeling scheme for a task \({\mathcal{T}} \), using ℓ-bit registers, then there exists a silent self-stabilizing algorithm using registers of at most O(ℓ + logn) bits for solving \({\mathcal{T}} \), where n is the number of processes in the system. Therefore, as far as memory space is concerned, the design of silent self-stabilizing algorithms essentially boils down to the design of compact proof-labeling schemes. The second result in this paper addresses time complexity. We show that, for every task \({\mathcal{T}} \) with k-bits output size in n-node networks, there exists a silent self-stabilizing algorithm solving \({\mathcal{T}} \) in O(n) rounds, using registers of O(n 2 + kn) bits. Therefore, as far as running time is concerned, every task has a silent self-stabilizing algorithm converging in a linear number of rounds.
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Blin, L., Fraigniaud, P., Patt-Shamir, B. (2014). On Proof-Labeling Schemes versus Silent Self-stabilizing Algorithms. In: Felber, P., Garg, V. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2014. Lecture Notes in Computer Science, vol 8756. Springer, Cham. https://doi.org/10.1007/978-3-319-11764-5_2
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DOI: https://doi.org/10.1007/978-3-319-11764-5_2
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