The randomized online-LOCAL model captures a number of models of computing; it is at least as strong as all of these models: - the classical LOCAL model of distributed graph algorithms, - the quantum version of the LOCAL model, - finitely dependent distributions [e.g. Holroyd 2016], - any model that does not violate physical causality [Gavoille, Kosowski, Markiewicz, DISC 2009], - the SLOCAL model [Ghaffari, Kuhn, Maus, STOC 2017], and - the dynamic-LOCAL and online-LOCAL models [Akbari et al., ICALP 2023]. In general, the online-LOCAL model can be much stronger than the LOCAL model. For example, there are locally checkable labeling problems (LCLs) that can be solved with logarithmic locality in the online-LOCAL model but that require polynomial locality in the LOCAL model. However, in this work we show that in trees, many classes of LCL problems have the same locality in deterministic LOCAL and randomized online-LOCAL (and as a corollary across all the above-mentioned models). In particular, these classes of problems do not admit any distributed quantum advantage. We present a near-complete classification for the case of rooted regular trees. We also fully classify the super-logarithmic region in unrooted regular trees. Finally, we show that in general trees (rooted or unrooted, possibly irregular, possibly with input labels) problems that are global in deterministic LOCAL remain global also in the randomized online-LOCAL model.

翻译：随机化在线LOCAL模型涵盖了多种计算模型，其能力至少与以下所有模型相当：- 经典的分布式图算法LOCAL模型，- LOCAL模型的量子版本，- 有限依赖分布[例如Holroyd 2016]，- 任何不违反物理因果性的模型[Gavoille, Kosowski, Markiewicz, DISC 2009]，- SLOCAL模型[Ghaffari, Kuhn, Maus, STOC 2017]，以及- 动态LOCAL与在线LOCAL模型[Akbari等人, ICALP 2023]。一般而言，在线LOCAL模型可能比LOCAL模型强大得多。例如，存在某些局部可检查标记问题（LCLs），在在线LOCAL模型中仅需对数级局部性即可求解，而在LOCAL模型中却需要多项式级局部性。然而，本研究表明在树结构中，多类LCL问题在确定性LOCAL模型与随机化在线LOCAL模型（并作为推论在所有上述模型中）具有相同的局部性。特别地，这些类别的问题不存在任何分布式量子优势。我们针对有根正则树的情形给出了近乎完备的分类，并对无根正则树中的超对数区域进行了完整分类。最后，我们证明在一般树结构（有根或无根、可能非正则、可能带有输入标记）中，在确定性LOCAL模型中具有全局性质的问题，在随机化在线LOCAL模型中同样保持全局性。