Ranking Opinions with Few States in Population Protocols

2026-05-18

AI Summary

This work addresses the problem of efficiently solving plurality consensus and opinion ranking in population protocols with low state complexity. The authors propose the CIRCLES protocol, which innovatively organizes agents into a hierarchy of decreasing-size circular linked lists, ensuring that agents of the same color never reside in the same ring. This approach achieves the first known state complexity of $O(k^3)$ for the plurality problem and is further extended to support arbitrary color orderings, multiple tie-breaking mechanisms, and full sorting tasks—surmounting the limitations of traditional pairwise-comparison paradigms. The resulting deterministic distributed algorithms guarantee correctness under weak fairness scheduling, requiring only $k^3$ states for plurality consensus and $2k^4$ states for complete sorting.

Abstract

Population protocols are a model of distributed computing where $n$ agents, each a simple finite-state machine, interact in pairs to solve a common task against a (adversarial) interaction scheduler. This model was intensively studied in recent years; in particular, the problem of relative majority received much attention: Each agent starts with an input opinion (or color) out of $k$ possibilities, and the goal is for each agent to eventually output the color with the largest support in the population. Before our work, the state complexity (the minimum number of states required per agent) was only known to be between $Ω(k^2)$ and $O(k^{7})$. Our main contribution is a population protocol that solves the relative majority problem with $k^3$ states. We achieve this result with a new protocol called CIRCLES. While prior approaches in the literature relied on duels of agents to find the majority color -- an approach that proved effective for the case with two colors -- CIRCLES partitions the agents into circular linked lists of decreasing sizes, with the property that no two agents with the same initial color lie in the same circle. We show that CIRCLES always correctly computes the desired structure against the most adversarial of schedulers (weakly fair). We then show that a trivial extension of CIRCLES solves the relative majority problem. We extend our protocol to handle various tie-breaking mechanisms or to support the case where the agents do not share a prior ordering of the colors. Finally, we show that a modification of CIRCLES solves the ranking problem with $2 \cdot k^4$ states, where each agent must output the rank of its initial color in the population.

Problem

• population protocols
• relative majority
• state complexity
• ranking problem
• distributed computing

Innovation

• population protocols
• relative majority
• state complexity
• CIRCLES protocol
• ranking problem