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

TL;DR

This paper introduces almost time-optimal loosely-stabilizing leader election protocols for arbitrary graphs without identifiers, achieving near lower-bound convergence times with increased memory.

Contribution

The authors develop new protocols that do not require unique identifiers and approach the theoretical lower bound for convergence time, improving upon previous methods.

Findings

Protocols achieve $O(mN ext{log}n)$ and $O(mN ext{log}N)$ convergence times.

Protocols use $O( ext{max degree} imes ext{log}N)$ bits of memory.

Hold the leader election specification for exponential expected steps.

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 $\Omega(e^{2N})$ 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 $O(mN\log n)$ 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 $O(\Delta \log N)$ bits of memory and hold the specification for $\Omega(e^{2N})$ expected steps.