A Space-Time Trade-off for Fast Self-Stabilizing Leader Election in Population Protocols

TL;DR

The paper tackles self-stabilizing leader election in population protocols, where $n$ anonymous agents with fixed state spaces interact in random pairs. It introduces ElectLeader$_r$, a parameterized protocol that trades space for time by partitioning ranks into groups of size $\Theta(r)$ and employing a novel message-based collision-detection mechanism, followed by a soft-reset framework to handle initialization errors. The main results show that for $1 \le r \le n/2$, the protocol stabilizes in $O\left(\frac{n^2}{r} \log n\right)$ interactions with $2^{O(r^2 \log n)}$ states w.h.p., and for $r = \Theta(n)$ achieves the time-optimal $O(n \log n)$ with $2^{O(n^2 \log n)}$ states; additionally, with $r = \log^2 n$, sublinear-time stabilization is obtained using $2^{O((\log n)^3)}$ states. The approach advances the state of the art by reducing state-space blow-up, answering open questions about sublinear-time stabilization with sub-exponential state complexity, and providing a framework that can derandomize via scheduler-based techniques. The results have broad implications for robust, scalable leader election in large, asynchronous populations and related ranking problems.

Abstract

We consider the problem of self-stabilizing leader election in the population model by Angluin, Aspnes, Diamadi, Fischer, and Peralta (JDistComp '06). The population model is a well-established and powerful model for asynchronous, distributed computation with a large number of applications. For self-stabilizing leader election, the population of $n$ anonymous agents, interacting in uniformly random pairs, must stabilize with a single leader from any possible initial configuration. The focus of this paper is to develop time-efficient self-stabilizing protocols whilst minimizing the number of states. We present a parametrized protocol, which, for a suitable setting, achieves the asymptotically optimal time $O(\log n)$ using $2^{O(n^2\log n)}$ states (throughout the paper, ``time'' refers to ``parallel time'', i.e., the number of pairwise interactions divided by $n$). This is a significant improvement over the previously best protocol Sublinear-Time-SSR due to Burman, Chen, Chen, Doty, Nowak, Severson, and Xu (PODC '21), which requires $2^{O(n^{\log n}\log n)}$ states for the same time bound. In general, for $1\le r\le n/2$, our protocol requires $2^{O(r^2\log{n})}$ states and stabilizes in time $O((n\log{n})/r)$, w.h.p.; the above result is achieved for $r=Θ(n)$. For $r=\log^2n$ our protocol requires only sub-linear time using only $2^{O(\log^3 n)}$ states, resolving an open problem stated in that paper. Sublinear-Time-SSR requires $O(\log n\cdot n^{1/(H+1)})$ time using $2^{Θ(n^H) \cdot \log n}$ states for all $1\le H\leΘ(\log n)$. Similar to previous works, it solves leader election by assigning a unique rank from $1$ through $n$ to each agent. The principal bottleneck for self-stabilizing ranking usually is to detect if there exist agents with the same rank. One of our main conceptual contributions is a novel technique for collision detection.

Figures (4)

• Figure 1: An overview of $\textsc{ElectLeader}_r$'s state space $Q$. The subspaces $Q_{PR}, Q_{AR}$, and $Q_{SV}$ and described in \ref{['apx:PropagateReset']}, \ref{['apx:ranking']}, and \ref{['sec:verify']} repectively.
• Figure 2: An overview of $\textsc{StableVerify}_r$'s state space. The field $\textup{qDC} \in Q_{DC}$ is a state from $\textsc{DetectCollision}_r$ described in \ref{['sec:error_detection']}.
• Figure 3: An overview of $\textsc{DetectCollision}_r$'s state space. The overall state complexity is $2^{{\operatorname{O}}(r^2\log{r})}$.
• Figure 4: An overview of $\textsc{FastLeaderElect}$'s state space. The overall state complexity is $2^{{\operatorname{O}}(\log{n})}$.

Theorems & Definitions (59)

• Theorem 1.1: Main Theorem
• proof : Proof of \ref{['thm:framework']}
• Lemma 6.1: Safety
• proof : Proof Sketch
• Lemma 6.2: Correctness after a full reset
• proof : Proof Sketch
• Lemma 6.3: Recovery
• proof
• Lemma A.1: Technical Lemma for counting interactions in the population as a whole
• proof