We study the self-stabilizing leader election problem in anonymous $n$-nodes networks. Achieving self-stabilization with low space memory complexity is particularly challenging, and designing space-optimal leader election algorithms remains an open problem for general graphs. In deterministic settings, it is known that $Ω(\log \log n)$ bits of memory per node are necessary [Blin et al., Disc. Math. \& Theor. Comput. Sci., 2023], while in probabilistic settings the same lower bound holds for some values of $n$, but only for an unfair scheduler [Beauquier et al., PODC 1999]. Several deterministic and probabilistic protocols have been proposed in models ranging from the state model to the population protocols. However, to the best of our knowledge, existing solutions either require $Ω(\log n)$ bits of memory per node for general worst case graphs, or achieve low state complexity only under restricted network topologies such as rings, trees, or bounded-degree graphs. In this paper, we present a probabilistic self-stabilizing leader election algorithm for arbitrary anonymous networks that uses $O(\log \log n)$ bits of memory per node. Our algorithm operates in the state model under a synchronous scheduler and assumes knowledge of a global parameter $N = Θ(\log n)$. We show that, under our protocol, the system converges almost surely to a stable configuration with a unique leader and stabilizes within $O(\mathrm{poly}(n))$ rounds with high probability. To achieve $O(\log \log n)$ bits of memory, our algorithm keeps transmitting information after convergence, i.e. it does not verify the silence property. Moreover, like most works in the field, our algorithm does not provide explicit termination detection (i.e., nodes do not detect when the algorithm has converged).

翻译：我们研究匿名$n$节点网络中的自稳定领导者选举问题。以低空间内存复杂度实现自稳定尤为困难，设计空间最优的领导者选举算法对于一般图而言仍是一个开放问题。在确定性设置中，已知每个节点需要$Ω(\log \log n)$位内存[Blin等人，Disc. Math. & Theor. Comput. Sci., 2023]，而在概率性设置中，相同的下界对某些$n$值成立，但仅适用于非公平调度器[Beauquier等人，PODC 1999]。已有若干确定性和概率性协议被提出，模型涵盖从状态模型到群体协议。然而，据我们所知，现有解决方案要么在一般最坏情况图下要求每个节点$Ω(\log n)$位内存，要么仅在受限网络拓扑（如环、树或有界度图）下实现低状态复杂度。本文提出一种适用于任意匿名网络的概率性自稳定领导者选举算法，该算法每个节点使用$O(\log \log n)$位内存。我们的算法在同步调度器下的状态模型中运行，并假设已知全局参数$N = Θ(\log n)$。我们证明，在我们的协议下，系统几乎必然收敛至具有唯一领导者的稳定配置，并以高概率在$O(\mathrm{poly}(n))$轮内稳定。为实现$O(\log \log n)$位内存，我们的算法在收敛后持续传输信息，即不满足静默性。此外，与该领域大多数工作类似，我们的算法不提供显式终止检测（即节点无法检测算法何时收敛）。