Given a boolean predicate $\Pi$ on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for $\Pi$ is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying $\Pi$. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size $n$ of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of $O(\log \log n)$ bits per node in any $n$-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use $\Omega(\log \log n)$-bit per node registers in some $n$-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms.

翻译：给定标记网络上的布尔谓词$\Pi$（例如，正确着色、领导者选举等），针对$\Pi$的自稳定算法是一种分布式算法，它可以从网络的任意初始配置（即每个节点的每个变量都被赋予任意值）开始，并最终收敛到满足$\Pi$的配置。已知领导者选举问题不存在使用常数大小寄存器（每个节点）的确定性自稳定算法，即对于某些网络，部分节点必须使用其大小随网络规模$n$增长的寄存器。另一方面，也已知在任何$n$节点有界度数网络中，可以使用每个节点$O(\log \log n)$比特寄存器的确定性自稳定算法解决领导者选举问题。我们证明后者的空间复杂度是最优的。具体而言，我们证明每个解决领导者选举问题的确定性自稳定算法在某些$n$节点网络中必须使用$\Omega(\log \log n)$比特每节点寄存器。此外，我们证明我们的下界超越了领导者选举问题，并适用于所有无法通过匿名算法解决的问题。