Near-optimal population protocols on bounded-degree trees

2026-02-18

AI Summary

This work investigates the space-time trade-offs of population protocols on bounded-degree trees for leader election and exact majority problems. Addressing this sparse interaction topology, the paper introduces two novel self-stabilizing constant-space protocols: fast 2-hop coloring and optimal-time tree orientation, enabling efficient execution of simple protocols on directed trees. The study establishes, for the first time, that no significant asymptotic space-time trade-off exists for bounded-degree trees, achieving a near-optimal worst-case expected stabilization time of $O(n^2 \log n)$—yielding a linear speedup over prior methods. Key technical contributions include self-stabilizing population protocols, analysis of random drift processes, and annihilation dynamics.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We investigate space-time trade-offs for population protocols in sparse interaction graphs. In complete interaction graphs, optimal space-time trade-offs are known for the leader election and exact majority problems. However, it has remained open if other graph families exhibit similar space-time complexity trade-offs, as existing lower bound techniques do not extend beyond highly dense graphs. In this work, we show that -- unlike in complete graphs -- population protocols on bounded-degree trees do not exhibit significant asymptotic space-time trade-offs for leader election and exact majority. For these problems, we give constant-space protocols that have near-optimal worst-case expected stabilisation time. These new protocols achieve a linear speed-up compared to the state-of-the-art. Our results are based on two novel protocols, which we believe are of independent interest. First, we give a new fast self-stabilising 2-hop colouring protocol for general interaction graphs, whose stabilisation time we bound using a stochastic drift argument. Second, we give a self-stabilising tree orientation algorithm that builds a rooted tree in optimal time on any tree. As a consequence, we can use simple constant-state protocols designed for directed trees to solve leader election and exact majority fast. For example, we show that ``directed'' annihilation dynamics solve exact majority in $O(n^2 \log n)$ steps on directed trees.

Problem

• population protocols
• space-time trade-offs
• bounded-degree trees
• leader election
• exact majority

Innovation

• population protocols
• bounded-degree trees
• space-time trade-offs
• self-stabilising
• tree orientation