Almost logarithmic-time space optimal leader election in population protocols

TL;DR

This paper introduces the first leader election protocol in population protocols that operates in expected time close to logarithmic, using optimal space per agent, by combining synthetic coins and phase clocks.

Contribution

It presents the fastest known leader election algorithm with near-logarithmic time complexity and optimal space, utilizing innovative techniques like synthetic coins and phase clocks.

Findings

Expected parallel time of O(log n log log n)

Each agent uses O(log log n) states asymptotically optimally

Achieves space and time efficiency in population protocols

Abstract

The model of population protocols refers to a large collection of simple indistinguishable entities, frequently called {\em agents}. The agents communicate and perform computation through pairwise interactions. We study fast and space efficient leader election in population of cardinality $n$ governed by a random scheduler, where during each time step the scheduler uniformly at random selects for interaction exactly one pair of agents. We propose the first $o(\log^2 n)$-time leader election protocol. Our solution operates in expected parallel time $O(\log n\log\log n)$ which is equivalent to $O(n \log n\log\log n)$ pairwise interactions. This is the fastest currently known leader election algorithm in which each agent utilises asymptotically optimal number of $O(\log\log n)$ states. The new protocol incorporates and amalgamates successfully the power of assorted {\em synthetic coins} with variable rate {\em phase clocks}.