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

Lélia Blin, Gabriel Le Bouder Laurent Feuilloley

DMTCS: Discrete Mathematics & Theoretical Computer Science. 2023.
doi:10.46298/dmtcs.9335

Given a boolean predicate $\Pi$ on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for $\Pi$ 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 $\Pi$. 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 $\Omega(\log\log n)$-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.

This paper first appeared as a brief announcement at DISC 2019, then as a full paper at OPODIS, and now as a final extended and polished journal version. It is also part of Gabriel's PhD thesis.

Gabriel presented the paper as a BA at DISC 2019, and as a full paper at OPODIS 2021.

The paper received the best student paper at OPODIS 2021.