By prior work, we have many results related to distributed graph algorithms for problems that can be defined with local constraints; the formal framework used in prior work is locally checkable labeling problems (LCLs), introduced by Naor and Stockmeyer in the 1990s. It is known, for example, that if we have a deterministic algorithm that solves an LCL in $o(\log n)$ rounds, we can speed it up to $O(\log^*n)$ rounds, and if we have a randomized $O(\log^*n)$ rounds algorithm, we can derandomize it for free. It is also known that randomness helps with some LCL problems: there are LCL problems with randomized complexity $\Theta(\log\log n)$ and deterministic complexity $\Theta(\log n)$. However, so far there have not been any LCL problems in which the use of shared randomness has been necessary; in all prior algorithms it has been enough that the nodes have access to their own private sources of randomness. Could it be the case that shared randomness never helps with LCLs? Could we have a general technique that takes any distributed graph algorithm for any LCL that uses shared randomness, and turns it into an equally fast algorithm where private randomness is enough? In this work we show that the answer is no. We present an LCL problem $\Pi$ such that the round complexity of $\Pi$ is $\Omega(\sqrt n)$ in the usual randomized \local model with private randomness, but if the nodes have access to a source of shared randomness, then the complexity drops to $O(\log n)$. As corollaries, we also resolve several other open questions related to the landscape of distributed computing in the context of LCL problems. In particular, problem $\Pi$ demonstrates that distributed quantum algorithms for LCL problems strictly benefit from a shared quantum state. Problem $\Pi$ also gives a separation between finitely dependent distributions and non-signaling distributions.

翻译：根据先前的研究，我们已获得许多关于分布式图算法的结果，这些算法针对可由局部约束定义的问题；先前工作中使用的形式化框架是由Naor和Stockmeyer在20世纪90年代提出的局部可检查标记问题（LCLs）。例如，已知若我们拥有在$o(\log n)$轮内求解LCL的确定性算法，则可将其加速至$O(\log^*n)$轮；若我们拥有随机化的$O(\log^*n)$轮算法，则可免费地将其去随机化。研究还表明随机性有助于解决某些LCL问题：存在随机复杂度为$\Theta(\log\log n)$而确定复杂度为$\Theta(\log n)$的LCL问题。然而，迄今为止尚未出现必须使用共享随机性的LCL问题；在所有现有算法中，节点仅需访问各自的私有随机源即可。是否存在共享随机性对LCL问题毫无帮助的情况？我们能否开发一种通用技术，将任何使用共享随机性的LCL分布式图算法转化为仅需私有随机性且保持同等速度的算法？本研究表明答案是否定的。我们提出了一个LCL问题$\Pi$，该问题在具有私有随机性的标准随机化\local模型中的轮复杂度为$\Omega(\sqrt n)$，但若节点能够访问共享随机源，则复杂度可降至$O(\log n)$。作为推论，我们还解决了与LCL问题背景下分布式计算格局相关的若干其他开放性问题。特别地，问题$\Pi$证明了LCL问题的分布式量子算法严格受益于共享量子态。问题$\Pi$还给出了有限依赖分布与非信号分布之间的分离性。