Space-efficient population protocols for exact majority on general graphs

TL;DR

This work advances exact majority computation in population protocols on general graphs by deriving asymptotically tight time lower bounds for unbounded-space settings and introducing fast, space-efficient protocols whose performance scales with the graph’s relaxation time $\tau_{\mathsf{rel}}$ and degree imbalance $\Delta/\delta$. Central to the approach are two innovations: (i) a detailed analysis of two-species annihilation–diffusion dynamics that bounds extinction and clearing times, and (ii) a space-efficient graphical phase clock that synchronizes cancellation–doubling across arbitrary graphs. The resulting fast protocol achieves $S = O\bigl( \log n \cdot (\log(\Delta/\delta) + \log(\tau_{\mathsf{rel}}/n)) \bigr)$ states and $T = O\bigl( (\Delta/\delta) \tau_{\mathsf{rel}} \log n \cdot \log(1/\gamma) \bigr)$ stabilization steps (in expectation and w.h.p.), with near-optimal performance on regular expanders, where $\tau_{\mathsf{rel}} \in O(n)$ and space is $O(\log n)$. A constant-state protocol is shown to stabilize in $O\bigl( \tau_{\mathsf{rel}} \log n / \gamma \bigr)$ steps for graphs with bias $\gamma>0$, linking to classical spectral bounds. Collectively, these results provide new tools for population protocols on graphs and pave the way for broader space-time trade-offs in distributed, asynchronous settings.

Abstract

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 $τ_{\mathsf{rel}}$ of the random walk on $G$ induced by the scheduler and the degree imbalance $Δ/δ$ of $G$. Specifically, we give a protocol that stabilizes in $O\left( \tfracΔδ τ_{\mathsf{rel}} \log^2 n \right)$ steps in expectation and with high probability and uses $O\left( \log n \cdot \left( \log\left(\tfracΔδ\right) + \log \left(\tfrac{τ_{\mathsf{rel}}}{n}\right) \right) \right)$ states in any graph with minimum degree at least $δ$ and maximum degree at most $Δ$. For regular expander graphs, this matches the optimal space complexity of $Θ(\log n)$ 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 $O(τ_{\mathsf{rel}} \cdot n \log n)$ for the stabilization time of a constant-state protocol.

Figures (3)

• Figure 1: Illustration of the matrix $R_S$. All coefficients in colorless areas are equal to $0$. The submatrix $Q[V \setminus S]$, corresponding to the restriction of $Q$ to $V\setminus S$, is block diagonal since the sets $U_1, \ldots, U_k$ are the connected components of the subgraph induced by $V \setminus S$.
• Figure 2: An implementation of the internal clock using $H(2K-1)$ states. Each column implements a $p$-coin with $p=2^{-K}$. Tossing a $p$-coin takes exactly $K$ local interactions. The coin flip is successful if the token remains on the right part of the column (depicted in gray) for all $K$ consecutive local interactions, which happens with probability $p$. In that case, the token moves to the next column; otherwise, it goes back to the start of the same column. Each transition requires one random bit. The token generates a clock tick after getting $H$ (not necessarily consecutive) successful $p$-coin flips.
• Figure 3: Illustration of a phase clock execution with $\Phi=4$. The time $t^*_i$ indicates the first time when some clock token makes its $i^{\text{th}}$ internal tick and $r_i$ denotes the $i^{\text{th}}$ synchronization step. (a) The synchronization, agreement and bounded delay properties. Agreement property ensures that during an interval $[r_i, r_{i+1})$ all values are within distance one from one another (w.h.p.). Bounded delay property ensures that the consecutive synchronization steps are (w.h.p.) at controlled distance $R \le r_{i+1}-{r_i} \le \eta R$. (b) The time steps used in the analysis for a single interval $[r_i, r_{i+1})$.

Theorems & Definitions (62)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Corollary 4
• Lemma 5
• proof : Proof of (c)
• Lemma 5
• Lemma 5
• Theorem 5