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

TL;DR

This work tackles leader election in population protocols over arbitrary graphs without unique identifiers, where true self-stabilization is impossible. It introduces a loosely-stabilizing framework that converges quickly to a safe configuration and sustains a single leader for a long period, driven by a two-hop coloring backbone and a Same Speed Timer mechanism. The key contributions are two protocols, one randomized and one deterministic, that realize near-optimal convergence times: $O(mN\log n)$ and $O(mN\log N)$ respectively, while maintaining a memory footprint of $O(\Delta\log N)$ and a holding time of $\Omega(Ne^{2N})$; these bounds improve upon prior work that required more stringent assumptions. The approach integrates self-stabilizing two-hop coloring (via ${\mathcal{P}_{\mathrm{LRU}}}$ or ${\mathcal{P'}_{\mathrm{LRU}}}$) with a cohesive leader-election protocol ${\mathcal{P}_{\mathrm{BC}}}$ that uses epidemic/broadcast techniques and identifiers generated through interactions, achieving robust leader election in a fully distributed, identifier-free setting with provable guarantees.

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 $O(mN\log n)$ 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 $O(mN^2\log{N})$ expected steps. Both protocols have a holding time of $Ω(e^{2N})$ expected steps and use $O(\log{N})$ bits of memory. They also showed that the lower bound of the convergence time is $Ω(mN)$ expected steps for protocols with a holding time of $Ω(e^N)$ 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 $Δ$ for the maximum degree, we propose two protocols whose convergence times are $O(mN\log n)$ and $O(mN\log N)$ both in expectation and with high probability. The former protocol uses random numbers, while the latter does not require them. Both protocols utilize $O(Δ\log N)$ bits of memory and hold the specification for $Ω(e^{2N})$ expected steps.

Figures (1)

• Figure 1: An example flow of ${\mathcal{P}_{\mathrm{BC}}}$.

Theorems & Definitions (57)

• Definition 1: Loosely-stabilizing leader electionSUDOLSLE
• Definition 2
• Definition 3
• Theorem 4
• Theorem 5
• Lemma 5
• Lemma 6
• Lemma 6
• Lemma 6
• Lemma 7