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Near-Optimal Population Protocols on Bounded-Degree Trees

Authors Joel Rybicki , Jakob Solnerzik , Robin Vacus

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ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • population protocols
  • distributed graph algorithms

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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 whether 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(n² log n) steps on directed trees.

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Joel Rybicki, Jakob Solnerzik, and Robin Vacus. Near-Optimal Population Protocols on Bounded-Degree Trees. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.30

Author Details

Joel Rybicki
  • Humboldt-Universität zu Berlin, Germany
Jakob Solnerzik
  • Humboldt-Universität zu Berlin, Germany
Robin Vacus
  • Sorbonne Université, CNRS, LIP6, France

Funding

This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number 539576251.

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