Near-optimal population protocols on bounded-degree trees

TL;DR

This work investigates space-time trade-offs for population protocols on sparse interaction graphs, focusing on bounded-degree trees. It introduces two key technical ingredients: a fast self-stabilising 2-hop colouring protocol and a time-optimal self-stabilising tree orientation algorithm; together, they enable fast, constant-state protocols for leader election and exact majority on trees. The results show that, unlike in cliques, constant-state protocols on bounded-degree trees can achieve near-optimal stabilisation times, yielding linear speed-ups and, in some cases, optimal $\\Theta(Dn)$ time for leader election. The paper also establishes strong lower bounds on orientation and leader election times on trees and discusses broader applications and open problems in the space-time trade-off landscape for sparse graphs.

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.

Figures (7)

• Figure 1: (a) An undirected input tree. (b) A 2-hop coloured tree. (b) The orientation protocol orients a 2-hop coloured tree. The grey node may be a "hallucinated" parent of the root. (d) An example configuration of the leader election protocol. The leader tokens ($\mathsf{L}$) are pushed towards the root. When two leader tokens meet, the child's token is removed. (e) An example configuration of the majority protocol. The $\mathsf{A}$- and $\mathsf{B}$-tokens are pushed towards the root; when $\mathsf{A}$ and $\mathsf{B}$ tokens meet, both turn into a $\mathsf{C}$-token. In contrast to $\mathsf{A}$- and $\mathsf{B}$-tokens, the $\mathsf{C}$-tokens move away from the root.
• Figure 2: Different types of edge conflicts. The array indexed by colours next to each node represents the $\textup{stamp}$ variable of the corresponding node. In this example, the only edge with a stamp conflict is $\{u_2,u_4\}$. In addition, $\{u_1,u_2\}$ and $\{u_2,u_3\}$ both have a colour conflict, and $(u_1,u_2,u_3)$ is a "red"-conflict path. Finally, $\{u_3,u_5\}$ and $\{u_4,u_5\}$ do not have any conflict.
• Figure 3: In this example, $(u,v_1, v_2)$ is yellow-risky; $(u,v_2, v_1)$ is blue-risky; and $(u,v_2, v_3)$ is red-risky.
• Figure 4: The three orientation statuses of an edge $\{u,v\}$ induced by the $\textup{children}$ and $\textup{parent}$ variables of nodes $u$ and $v$. There is an outcoming blue arrow from $u$ if $\textup{parent}(u) = \textup{colour}(v)$, and an incoming red arrow into $v$ if $\textup{children}(v,\textup{colour}(u)) = 1$. (a) Properly and weakly oriented edges. (b) Examples of disoriented edges (i.e., edges that are neither properly or weakly oriented); here the list of configurations where the edge $\{u,v\}$ is disoriented is not exhaustive.
• Figure 5: The potential $\Phi_t(\cdot)$. Here $\mathcal{M} = \{x_1, \ldots, x_7\}$ are the edge-markers. We adopt the same convention as in \ref{['fig:definition']} to represent the $\textup{parent}$ and $\textup{children}$ variables of every node involved. This tree has 3 properly oriented edges, and 4 weakly oriented edges. The potential of the three markers $x_5,x_6,x_7$ on the properly oriented edges is zero to $0$ and not drawn on the figure.

Theorems & Definitions (72)

• Theorem 1
• Theorem 2
• Corollary 2
• Theorem 3
• Corollary 3
• Theorem 4
• Corollary 5
• Theorem 6
• Theorem 6
• Lemma 7