Near-State and State-Optimal Self-Stabilising Leader Election Population Protocols

2025-02-03

AI Summary

This paper investigates leader election in self-stabilizing population protocols: given $n$ agents with arbitrary initial states, the goal is to converge silently and deterministically to a unique leader via pairwise interactions. We introduce a novel “agent-trap” mechanism, achieving— for the first time—state-optimal protocols: exactly $n$ rank states and zero auxiliary states, with stabilization time asymptotically better than $O(n^2)$. Further, adding only one auxiliary state yields $O(n^{7/4} log^2 n)$ convergence time; with $O(log n)$ auxiliary states, we attain the current best $O(n log n)$. All three protocols are strictly self-stabilizing, silent, and achieve either state-optimality or near-optimality. They succeed with high probability ($1 - n^{-eta}$ for any constant $eta > 0$) from *any* initial configuration. Collectively, these results significantly advance the state of the art in both time and space complexity for self-stabilizing leader election in population protocols.

Technology Category

• Multiagent Systems: Mechanism Design
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

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

Abstract

We investigate leader election problem via ranking within self-stabilising population protocols. In this scenario, the agent's state space comprises $n$ rank states and $x$ extra states. The initial configuration of $n$ agents consists of arbitrary arrangements of rank and extra states, with the objective of self-ranking. Specifically, each agent is tasked with stabilising in a unique rank state silently, implying that after stabilisation, each agent remains in its designated state indefinitely. In this paper, we present several new self-stabilising ranking protocols, greatly enriching our comprehension of these intricate problems. All protocols ensure self-stabilisation time with high probability (whp), defined as $1-n^{-eta},$ for a constant $eta>0.$ We delve into three scenarios, from which we derive stable (always correct), either state-optimal or almost state-optimal, silent ranking protocols that self-stabilise within a time frame of $o(n^2)$ whp, including: - Utilising a novel concept of an agent trap, we derive a state-optimal ranking protocol that achieves self-stabilisation in time $O(min(kn^{3/2},n^2log^2 n)),$ for any $k$-distant starting configuration. - Furthermore, we show that the incorporation of a single extra state ($x=1$) ensures a ranking protocol that self-stabilises in time $O(n^{7/4}log^2 n)=o(n^2)$, regardless of the initial configuration. - Lastly, we show that extra $x=O(log n)$ states admit self-stabilising ranking with the best currently known stabilisation time $O(nlog n)$, when whp and $x=O(log n)$ guarantees are imposed.

Problem

• Swarm Robotics
• Hierarchical Status
• Self-repairing Algorithms

Innovation

• Self-repairing leadership
• Logarithmic special states
• Optimized ranking method