Time-optimal Loosely-stabilizing Leader Election in Population Protocols

TL;DR

This paper presents a time-optimal loosely-stabilizing leader election protocol in population protocols, achieving faster convergence and longer stable periods than previous methods, with proven optimality in expectation.

Contribution

It introduces a new protocol that attains the fastest known convergence time while maintaining a long leader stability period, both in expectation, under loose-stabilization.

Findings

Achieves $O( au ext{log} n)$ convergence time in expectation.

Maintains a leader for $ ext{Omega}(n^{ au})$ time in expectation.

Proves the protocol's optimality in both convergence and holding times.

Abstract

We consider the leader election problem in population protocol models. In pragmatic settings of population protocols, self-stabilization is a highly desired feature owing to its fault resilience and the benefit of initialization freedom. However, the design of self-stabilizing leader election is possible only under a strong assumption (i.e. the knowledge of the \emph{exact} size of a network) and rich computational resources (i.e. the number of states). Loose-stabilization, introduced by Sudo et al [Theoretical Computer Science, 2012], is a promising relaxed concept of self-stabilization to address the aforementioned issue. Loose-stabilization guarantees that starting from any configuration, the network will reach a safe configuration where a single leader exists within a short time, and thereafter it will maintain the single leader for a long time, but not forever. The main contribution of the paper is a time-optimal loosely-stabilizing leader election protocol. While the shortest convergence time achieved so far in loosely-stabilizing leader election is $O(\log^3 n)$ parallel time, the proposed protocol with design parameter $\tau \ge 1$ attains $O(\tau \log n)$ parallel convergence time and $\Omega(n^{\tau})$ parallel holding time (i.e. the length of the period keeping the unique leader), both in expectation. This protocol is time-optimal in the sense of both the convergence and holding times in expectation because any loosely-stabilizing leader election protocol with the same length of the holding time is known to require $\Omega(\tau \log n)$ parallel time.