Informative Trains: A Memory-Efficient Journey to a Self-Stabilizing Leader Election Algorithm in Anonymous Graphs

2026-02-19

AI Summary

This work addresses the long-standing challenge of achieving self-stabilizing leader election in anonymous graph networks with minimal memory overhead, particularly under the space complexity constraints inherent to general graph topologies. The paper proposes a probabilistic self-stabilizing leader election algorithm that operates under a synchronous scheduler and state model, applicable to arbitrary anonymous networks. By leveraging a global parameter \(N = \Theta(\log n)\) and a non-silent, continuous information propagation mechanism, each node requires only \(O(\log \log n)\) bits of memory—breaking prior barriers that restricted low-memory solutions to specific topologies or incurred high space costs. With high probability, the algorithm converges within \(O(\text{poly}(n))\) rounds to a unique, stable leader configuration, achieving memory efficiency approaching the theoretical lower bound.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We study the self-stabilizing leader election problem in anonymous $n$-nodes networks. Achieving self-stabilization with low space memory complexity is particularly challenging, and designing space-optimal leader election algorithms remains an open problem for general graphs. In deterministic settings, it is known that $Ω(\log \log n)$ bits of memory per node are necessary [Blin et al., Disc. Math. \& Theor. Comput. Sci., 2023], while in probabilistic settings the same lower bound holds for some values of $n$, but only for an unfair scheduler [Beauquier et al., PODC 1999]. Several deterministic and probabilistic protocols have been proposed in models ranging from the state model to the population protocols. However, to the best of our knowledge, existing solutions either require $Ω(\log n)$ bits of memory per node for general worst case graphs, or achieve low state complexity only under restricted network topologies such as rings, trees, or bounded-degree graphs. In this paper, we present a probabilistic self-stabilizing leader election algorithm for arbitrary anonymous networks that uses $O(\log \log n)$ bits of memory per node. Our algorithm operates in the state model under a synchronous scheduler and assumes knowledge of a global parameter $N = Θ(\log n)$. We show that, under our protocol, the system converges almost surely to a stable configuration with a unique leader and stabilizes within $O(\mathrm{poly}(n))$ rounds with high probability. To achieve $O(\log \log n)$ bits of memory, our algorithm keeps transmitting information after convergence, i.e. it does not verify the silence property. Moreover, like most works in the field, our algorithm does not provide explicit termination detection (i.e., nodes do not detect when the algorithm has converged).

Problem

• self-stabilizing
• leader election
• anonymous graphs
• memory-efficient
• space complexity

Innovation

• self-stabilizing
• leader election
• anonymous graphs
• memory-efficient
• probabilistic algorithm