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Journal Articles Discrete Mathematics and Theoretical Computer Science Year : 2023

Optimal Space Lower Bound for Deterministic Self-Stabilizing Leader Election Algorithms

Lélia Blin
Gabriel Le Bouder

Abstract

Given a boolean predicate Π on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for Π is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying Π. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size n of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of O(log log n) bits per node in any n-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use Ω(log log n)-bit per node registers in some n-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms.
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Dates and versions

hal-03536828 , version 1 (20-01-2022)
hal-03536828 , version 2 (21-05-2023)

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Lélia Blin, Laurent Feuilloley, Gabriel Le Bouder. Optimal Space Lower Bound for Deterministic Self-Stabilizing Leader Election Algorithms. Discrete Mathematics and Theoretical Computer Science, 2023, LIPIcs, 25 (1), pp.5. ⟨10.46298/dmtcs.9335⟩. ⟨hal-03536828v2⟩
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