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

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 [BCC+21] (PODC) showed that this problem can be solved in $O(n)$ expected time with $O(n)$ states. Recently, G\k{a}sieniec, Grodzicki, and Stachowiak [GGS25] (PODC) 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. The authors of [BCC+21] presented a $2^{O(n^\rho\log n)}$-state SS-LE protocol with a parameter $\rho$: setting $\rho = \Theta(\log n)$ yields an optimal $O(\log n)$ time both in expectation and w.h.p., while $\rho = \Theta(1)$ results in $O(\rho\,n^{1/(\rho+1)})$ expected time. Recently, Austin, Berenbrink, Friedetzky, G\"otte, and Hintze [ABF+25] (PODC) presented a novel SS-LE protocol parameterized by a positive integer $\rho$ with $1 \le \rho < n/2$ that solves SS-LE in $O(\frac{n}{\rho}\cdot\log n)$ time w.h.p.\ using $2^{O(\rho^2\log n)}$ states. This paper independently presents yet another time--space tradeoff of SS-LE: for any positive integer $\rho$ with $2 \le \rho \le \sqrt{n}$, SS-LE can be achieved within $O\left(\frac{n}{\rho}\cdot \log\rho\right)$ expected time using $2^{2\rho\lg^2\rho + O(\log n)}$ states. The proposed protocol uses significantly fewer states than [ABF+25] for any expected stabilization time above $\Theta(\sqrt{n}\log n)$. When $\rho = \Theta\left(\frac{\log n}{\log^2 \log n}\right)$, 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 $\rho\le\sqrt{n}$ prevents achieving $o(\sqrt{n}\log n)$ time, whereas the protocol of [ABF+25] can surpass this bound.