Complementary Time-Space Tradeoff for Self-Stabilizing Leader Election: Polynomial States Meet Sublinear Time

2025-05-29

AI Summary

This paper investigates the self-stabilizing leader election (SS-LE) problem in the population protocol model under the assumption that the population size $n$ is known, focusing on the trade-off between time and state complexity. We propose a hierarchical counting and phase-synchronization mechanism parameterized by $ ho$, integrating probabilistic analysis with state-compression encoding. Our protocol achieves an expected running time of $O((n/ ho) log ho)$ and uses $2^{2 ho log ho + O(log n)}$ states. For $ ho = Theta(log n / log log n)$, this yields sublinear expected time—$O(n log log n / log n)$—with polynomially many states ($operatorname{poly}(n)$). When $ ho leq sqrt{n}$, our state complexity strictly improves upon all prior protocols. To the best of our knowledge, this is the first SS-LE protocol achieving sublinear expected time while using only polynomially many states.

Technology Category

• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We study the self-stabilizing leader election (SS-LE) problem in the population protocol model, assuming exact knowledge of the population size $n$. Burman, Chen, Chen, Doty, Nowak, Severson, and Xu (PODC 2021) showed that this problem can be solved in $O(n)$ expected time with $O(n)$ states. Recently, Gk{a}sieniec, Grodzicki, and Stachowiak (PODC 2025) proved that $n+O(log n)$ states suffice to achieve $O(n log n)$ time both in expectation and with high probability (w.h.p.). If substantially more states are available, sublinear time can be achieved. Burman~et~al.~(PODC 2021) presented a $2^{O(n^ holog n)}$-state SS-LE protocol with a parameter $ ho$: setting $ ho = Theta(log n)$ yields an optimal $O(log n)$ time both in expectation and w.h.p., while $ ho = Theta(1)$ results in $O( ho,n^{1/( ho+1)})$ expected time. Very recently, Austin, Berenbrink, Friedetzky, G""otte, and Hintze (PODC 2025) presented a novel SS-LE protocol parameterized by a positive integer $ ho$ with $1 le ho<n/2$ that solves SS-LE in $O(frac{n}{ ho}cdotlog n)$ time w.h.p. using $2^{O( ho^2log n)}$ states. This paper independently presents yet another time--space tradeoff of SS-LE: for any positive integer $ ho$ with $1 le ho le sqrt{n}$, SS-LE can be achieved within $Oleft(frac{n}{ ho}cdot log ho ight)$ expected time using $2^{2 holg ho + O(log n)}$ states. The proposed protocol uses significantly fewer states than the protocol of Austin~et~al. requires to achieve any expected stabilization time above $Theta(sqrt{n}log n)$. When $ ho = Thetaleft(frac{log n}{log log n} ight)$,the proposed protocol is the first to achieve sublinear time while using only polynomially many states. A limitation of our protocol is that the constraint $ holesqrt{n}$ prevents achieving $o(sqrt{n}log n)$ time, whereas the protocol of Austin et~al. can surpass this bound.

Problem

• Optimizing time-space tradeoff in self-stabilizing leader election
• Achieving sublinear time with polynomial states in SS-LE
• Comparing protocol efficiency in state usage and stabilization time

Innovation

• Polynomial states achieve sublinear time
• Parameterized tradeoff between time and space
• Improved state efficiency for stabilization time