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：给定一个关于标记网络的布尔谓词 $\Pi$（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，它可以从网络的任何初始配置（即，每个节点的每个变量被赋予任意值）开始运行，并最终收敛到满足 $\Pi$ 的配置。已知领导者选举不存在每个节点使用常数大小寄存器的确定性自稳定算法，即对于某些网络，它们的一些节点必须使用寄存器，其大小随网络规模 $n$ 增长。另一方面，同样已知的是，在任何 $n$ 节点有界度网络中，领导者选举可以通过每个节点使用 $O(\log \log n)$ 位寄存器的确定性自稳定算法来解决。我们证明了后一种空间复杂度是最优的。具体来说，我们证明了在求解领导者选举的每个确定性自稳定算法中，某些 $n$ 节点网络必须使用每个节点 $\Omega(\log \log n)$ 位的寄存器。此外，我们证明了我们的下界超越了领导者选举，并适用于所有无法通过匿名算法解决的问题。