Efficient Stable Population Protocols for Parity and Beyond

Abstract

The population protocol model studies pairwise interactions among simple, indistinguishable agents, each storing a single state from a fixed space. In the probabilistic variant, a random scheduler selects an interacting pair uniformly at random at each step. Besides state utilisation, we consider stabilisation time, defined as the number of interactions until a final configuration, divided by the population size n. A protocol is time-efficient if it stabilises in polylogarithmic time, space-efficient if it uses polylogarithmic number of states, stable if it converges to the correct answer with probability 1, and silent if no states change after stabilisation.
We present a systematic methodology that enables the design of protocols that are efficient, stable and silent for computing congruence predicates. We first address the parity problem, where agents compute the parity of a designated sub‑population size, and then generalise the approach to congruence modulo an arbitrary integer m. While prior work focused on problem‑specific designs, our approach provides a systematic framework for efficient multi‑stage protocols. Central to this approach is the weight system, an efficient counting mechanism based on the novel concept of population weights (or simply weights), combined with majority computation and operating via implicit conversion between unary and binary representations. The weight system is integrated within the proposed modular MC+ computing paradigm, which leverages extensive prior work on clocking mechanisms, efficient anomaly signalling, and a switching mechanism that guarantees fallback to a slow yet always‑correct stable protocol. The new approach yields the first efficient parity and congruence protocols using O(log³ n) states and achieving silent stabilisation in O(log³ n) time, providing significant progress towards the research directions identified by Czerner et al. (2024). We also discuss the impact of the implicit conversion on other problems, including the computation and alternative representation of (sub-)population sizes, a postulate raised by Doty and Eftekhari (2021).
All algorithmic solutions presented in this paper guarantee silent stabilisation in time both expected and with high probability (whp) defined as 1-n^{-η}, for any fixed constant η > 0.