Almost Time-Optimal Loosely-Stabilizing Leader Election on Arbitrary Graphs Without Identifiers in Population Protocols

2024-11-06

International Conference on Principles of Distributed Systems

AI Summary

This paper investigates the loosely stable leader election problem in arbitrary communication graphs where nodes lack unique identifiers. We propose two distributed protocols—randomized and deterministic—relying solely on upper bounds on the number of nodes $N$ and the maximum degree $Delta$, without requiring node IDs, global clocks, or centralized coordination. Our protocols employ state encoding, local graph traversal, and degree-aware counting to realize hierarchical competition and conflict resolution, achieving memory complexity $O(Delta log N)$. The expected and high-probability convergence time is $O(mN log N)$, approaching the theoretical lower bound $Omega(mN)$; the leader’s tenure is $Omega(e^{2N})$. Compared to prior solutions with $O(mN^2 log N)$ convergence time, our approach significantly improves efficiency, making it suitable for resource-constrained mobile agent networks.

Technology Category

• Multiagent Systems: Agent Communication
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Security and Privacy: Large-scale security measurements

Abstract

The population protocol model is a computational model for passive mobile agents. We address the leader election problem, which determines a unique leader on arbitrary communication graphs starting from any configuration. Unfortunately, self-stabilizing leader election is impossible to be solved without knowing the exact number of agents; thus, we consider loosely-stabilizing leader election, which converges to safe configurations in a relatively short time, and holds the specification (maintains a unique leader) for a relatively long time. When agents have unique identifiers, Sudo et al.(2019) proposed a protocol that, given an upper bound $N$ for the number of agents $n$, converges in $O(mNlog n)$ expected steps, where $m$ is the number of edges. When unique identifiers are not required, they also proposed a protocol that, using random numbers and given $N$, converges in $O(mN^2log{N})$ expected steps. Both protocols have a holding time of $Omega(e^{2N})$ expected steps and use $O(log{N})$ bits of memory. They also showed that the lower bound of the convergence time is $Omega(mN)$ expected steps for protocols with a holding time of $Omega(e^N)$ expected steps given $N$. In this paper, we propose protocols that do not require unique identifiers. These protocols achieve convergence times close to the lower bound with increasing memory usage. Specifically, given $N$ and an upper bound $Delta$ for the maximum degree, we propose two protocols whose convergence times are $O(mNlog n)$ and $O(mNlog N)$ both in expectation and with high probability. The former protocol uses random numbers, while the latter does not require them. Both protocols utilize $O(Delta log N)$ bits of memory and hold the specification for $Omega(e^{2N})$ expected steps.

Problem

• Solving leader election on arbitrary graphs without unique identifiers
• Achieving near-optimal convergence time with limited memory usage
• Maintaining a unique leader through loosely-stabilizing population protocols

Innovation

• Uses random numbers for fast convergence without identifiers
• Employs memory-efficient O(Δlog N) bits per agent
• Achieves near-optimal O(mN log n) convergence time