Complementary Time-Space Tradeoff for Self-Stabilizing Leader Election: Polynomial States Meet Sublinear Time
Yuichi Sudo
TL;DR
This work studies self-stabilizing leader election in population protocols with exact population size $n$, introducing a time–space tradeoff that achieves sublinear stabilization time using polynomial-to-subexponential state counts. The core construction SSRK(\rho) combines three components—FindTarget, Detect, and Rank—coordinated by a phase clock and aided by a loosely-stabilizing leader, an epidemic broadcast, and careful collision-detection to tolerate outdated information. The main result shows that for any $1\le\rho\le\sqrt{n}$, SS-RK (and thus SS-LE) can be solved in $O\left(\frac{n}{\rho}\log \rho\right)$ expected time using $2^{2\rho\lg\rho+O(\log n)}$ states; taking $\rho=\Theta\left(\frac{\log n}{\log\log n}\right)$ yields sublinear time with polynomially many states. This complements recent sublinear-time protocols that require super-exponential state sizes, offering a distinct regime with markedly reduced state complexity while achieving sublinear stabilization time. The results advance understanding of practical SS-LE tradeoffs and raise questions about achieving sublinear time with purely polynomial-state protocols and the status of the SS-RK conjecture in broader interaction graphs.
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, Gąsieniec, 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 $2^{O(n^ρ\log n)}$-state SS-LE protocol with a parameter $ρ$: setting $ρ= Θ(\log n)$ yields an optimal $O(\log n)$ time both in expectation and w.h.p., while $ρ= Θ(1)$ results in $O(ρ\,n^{1/(ρ+1)})$ expected time. Very recently, Austin, Berenbrink, Friedetzky, Götte, and Hintze (PODC 2025) presented a novel SS-LE protocol parameterized by a positive integer $ρ$ with $1 \le ρ< n/2$ that solves SS-LE in $O(\frac{n}ρ\cdot\log n)$ time w.h.p.\ using $2^{O(ρ^2\log n)}$ states. This paper independently presents yet another time--space tradeoff of SS-LE: for any positive integer $ρ$ with $1 \le ρ\le \sqrt{n}$, SS-LE can be achieved within $O\left(\frac{n}ρ\cdot \logρ\right)$ expected time using $2^{2ρ\lgρ+ O(\log n)}$ states. The proposed protocol uses significantly fewer states than the protocol of Austin~et~al.\ requires to achieve any expected stabilization time above $Θ(\sqrt{n}\log n)$. When $ρ= Θ\left(\frac{\log n}{\log \log n}\right)$,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 $ρ\le\sqrt{n}$ prevents achieving $o(\sqrt{n}\log n)$ time, whereas the protocol of Austin et~al.\ can surpass this bound.