One of the central models in distributed computing is Linial's LOCAL model [SIAM J. Comp. 1992]. Over time, researchers have studied distributed graph problems in the LOCAL model under slightly different assumptions, such as whether nodes know the exact network size $n$, only a polynomial upper bound on $n$, or nothing at all. We ask whether these differences are merely technical or fundamentally affect the theory of Locally Checkable Labelings (LCLs), one of the most studied problem classes. LCLs are graph problems whose valid solutions can be characterized by a finite set of allowed constant-radius neighborhoods. Since their introduction by Naor and Stockmeyer [FOCS 1995], they have become central in distributed computing, and the last decade has seen major progress in understanding their complexity. For example, Chang, Kopelowitz, and Pettie [FOCS 2016] showed that the randomized complexity of any LCL on $n$-node graphs is at least its deterministic complexity on $\sqrt{\log n}$-node graphs. Later, Chang and Pettie [FOCS 2017] showed that any randomized $n^{o(1)}$-round algorithm for LCLs on bounded-degree trees can be turned into a deterministic $O(\log n)$-round algorithm. Then, Balliu et al. [STOC 2018] showed that such automatic speedups are impossible for general bounded-degree graphs. However, these results fundamentally rely on nodes knowing $n$. How much does this assumption affect the theory of LCLs? Our work shows that if nodes are oblivious to $n$, or know only a polynomial upper bound on it, then even on trees, the theory of LCLs changes significantly. While the fundamental classification of problems remains the same, we show the landscape becomes much more complex: for example, for LCLs, randomness helps in more cases; some problems have very unnatural complexities; and some have a lower bound that depends on which definition of $Ω$ we use!

翻译：分布式计算的核心模型之一是Linial提出的LOCAL模型[SIAM J. Comp. 1992]。长期以来，研究者们在LOCAL模型下对分布式图问题开展了研究，但所采用的基本假设略有不同，例如节点是否知道确切的网络规模$n$、只知道$n$的一个多项式上界，或者完全不知道。我们提出疑问：这些差异仅仅是技术性的，还是从根本上影响了局部可检查标记（LCL）——这一研究最广泛的问题类别之一——的理论？LCL是一类图问题，其有效解可通过一个有限集合的允许的常数半径邻域来刻画。自Naor和Stockmeyer[FOCS 1995]引入以来，LCL已成为分布式计算的核心，过去十年中对其复杂度的理解取得了重大进展。例如，Chang、Kopelowitz和Pettie[FOCS 2016]证明，在$n$节点图上任何LCL的随机化复杂度至少是在$\sqrt{\log n}$节点图上其确定式复杂度。随后，Chang和Pettie[FOCS 2017]证明，在有限度树上任何用于LCL的随机化$n^{o(1)}$轮算法均可转化为确定式$O(\log n)$轮算法。接着，Balliu等人[STOC 2018]表明，对于一般有限度图而言，此类自动加速是不可能的。然而，这些结果从根本上依赖于节点知道$n$。这一假设对LCL理论的影响有多大？我们的工作表明，如果节点对$n$无认知，或只知道$n$的一个多项式上界，那么即使在树上，LCL理论也会发生显著变化。尽管问题的基本分类保持不变，但我们发现其景观变得更为复杂：例如，对于LCL而言，随机化在更多情况下是有帮助的；某些问题具有非常不自然的复杂度；并且有些问题的下界取决于我们采用哪一种$\Omega$定义！