Jakob Solnerzik | Searcharxiv

arXiv2026

Near-optimal population protocols on bounded-degree trees

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.

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arXiv2025

Space-efficient population protocols for exact majority on general graphs

We study exact majority consensus in the population protocol model. In this model, the system is described by a graph $G = (V,E)$ with $n$ nodes, and in each time step, a scheduler samples uniformly at random a pair of adjacent nodes to interact. In the exact majority consensus task, each node is given a binary input, and the goal is to design a protocol that almost surely reaches a stable configuration, where all nodes output the majority input value. We give improved upper and lower bounds for exact majority in general graphs. First, we give asymptotically tight time lower bounds for general (unbounded space) protocols. Second, we obtain new upper bounds parameterized by the relaxation time $\tau_{\mathsf{rel}}$ of the random walk on $G$ induced by the scheduler and the degree imbalance $\Delta/\delta$ of $G$. Specifically, we give a protocol that stabilizes in $O\left( \tfrac{\Delta}{\delta} \tau_{\mathsf{rel}} \log^2 n \right)$ steps in expectation and with high probability and uses $O\left( \log n \cdot \left( \log\left(\tfrac{\Delta}{\delta}\right) + \log \left(\tfrac{\tau_{\mathsf{rel}}}{n}\right) \right) \right)$ states in any graph with minimum degree at least $\delta$ and maximum degree at most $\Delta$. For regular expander graphs, this matches the optimal space complexity of $\Theta(\log n)$ for fast protocols in complete graphs [Alistarh et al., SODA 2016 and Doty et al., FOCS 2022] with a nearly optimal stabilization time of $O(n \log^2 n)$ steps. Finally, we give a new upper bound of $O(\tau_{\mathsf{rel}} \cdot n \log n)$ for the stabilization time of a constant-state protocol.

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