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+21a] (PODC) showed that this problem can be solved in $O(n)$ expected time with $O(n)$ states. Recently, Gąsieniec, 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 $2^{O(n^ρ\log n)}$-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. Recently, Austin, Berenbrink, Friedetzky, Götte, and Hintze [ABF+25] (PODC) 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 $2 \le ρ\le \sqrt{n}$, SS-LE can be achieved within $O\left(\frac{n}ρ\cdot \logρ\right)$ expected time using $2^{2ρ\lg^2ρ+ O(\log n)}$ states. The proposed protocol uses significantly fewer states than [ABF+25] for any expected stabilization time above $Θ(\sqrt{n}\log n)$. When $ρ= Θ\left(\frac{\log n}{\log^2 \log n}\right)$, 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 $ρ\le\sqrt{n}$ prevents achieving $o(\sqrt{n}\log n)$ time, whereas the protocol of [ABF+25] can surpass this bound.

翻译：本文研究种群协议模型中的自稳定领导者选举（SS-LE）问题，假设已知种群规模 $n$ 的精确值。Burman、Chen、Chen、Doty、Nowak、Severson 和 Xu [BCC+21a]（PODC）表明，该问题可以在 $O(n)$ 期望时间内以 $O(n)$ 状态解决。近期，Gąsieniec、Grodzicki 和 Stachowiak [GGS25]（PODC）证明了 $n+O(\log n)$ 状态足以在期望时间和高概率（w.h.p.）情况下均实现 $O(n \log n)$ 时间。若可用状态显著增加，则可实现亚线性时间。[BCC+21] 的作者提出了一种参数化为 $ρ$ 的 $2^{O(n^ρ\log n)}$ 状态 SS-LE 协议：当 $ρ= Θ(\log n)$ 时，在期望时间和 w.h.p. 情况下均能实现最优的 $O(\log n)$ 时间；当 $ρ= Θ(1)$ 时，则产生 $O(ρ\,n^{1/(ρ+1)})$ 期望时间。近期，Austin、Berenbrink、Friedetzky、Götte 和 Hintze [ABF+25]（PODC）提出了一种新颖的 SS-LE 协议，该协议以正整数 $ρ$（$1 \le ρ< n/2$）为参数，使用 $2^{O(ρ^2\log n)}$ 状态在 w.h.p. 情况下以 $O(\frac{n}ρ\cdot\log n)$ 时间解决 SS-LE 问题。本文独立提出另一种 SS-LE 的时空权衡：对于任意满足 $2 \le ρ\le \sqrt{n}$ 的正整数 $ρ$，SS-LE 可以使用 $2^{2ρ\lg^2ρ+ O(\log n)}$ 状态在 $O\left(\frac{n}ρ\cdot \logρ\right)$ 期望时间内实现。对于任意高于 $Θ(\sqrt{n}\log n)$ 的期望稳定时间，所提协议使用的状态显著少于 [ABF+25]。当 $ρ= Θ\left(\frac{\log n}{\log^2 \log n}\right)$ 时，所提协议是首个在使用多项式状态的同时实现亚线性时间的方案。本协议的一个局限性在于约束 $ρ\le\sqrt{n}$ 阻止了实现 $o(\sqrt{n}\log n)$ 时间，而 [ABF+25] 的协议能够超越这一界限。