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

## TL;DR

This paper establishes the optimal space complexity for deterministic self-stabilizing leader election algorithms, proving that they require at least logarithmic double-logarithmic bits per node, matching existing upper bounds.

## Contribution

It provides a tight lower bound of (  n) bits per node for leader election, confirming the optimality of known algorithms and extending to other problems without anonymous solutions.

## Key findings

- Deterministic self-stabilizing leader election requires (  n) bits per node.
- Existing algorithms using O(      n) bits are optimal.
- Lower bounds apply to all problems unsolvable by anonymous algorithms.

## Abstract

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.