The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle versus two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., \(\textsf {P}\ne \textsf {NP}\)), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by \(\textsf {MPC}(o(\log N))\), and the standard space complexity classes \(\textsf {L}\) and \(\textsf {NL}\), and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model. Specifically, our main results are as follows.

• Lower bounds conditioned on the one cycle versus two cycles conjecture can be instead argued under the \(\textsf {L}\nsubseteq \textsf {MPC}(o(\log N))\) conjecture: these two assumptions are equivalent, and refuting either of them would lead to \(o(\log N)\)-round MPC algorithms for a large number of challenging problems, including list ranking, minimum cut, and planarity testing. In fact, we show that these problems and many others require asymptotically the same number of rounds as the seemingly much easier problem of distinguishing between a graph being one cycle or two cycles.

• Many lower bounds previously argued under the one cycle versus two cycles conjecture can be argued under an even more robust (thus harder to refute) conjecture, namely \(\textsf {NL}\nsubseteq \textsf {MPC}(o(\log N))\). Refuting this conjecture would lead to \(o(\log N)\)-round MPC algorithms for an even larger set of problems, including all-pairs shortest paths, betweenness centrality, and all aforementioned ones. Lower bounds under this conjecture hold for problems such as perfect matching and network flow.

大规模并行计算(MPC) 模型作为许多现代大规模数据处理框架的通用抽象，在过去几年中受到越来越多的关注，尤其是在经典图问题的背景下。到目前为止，争论该模型下界的唯一方法是对一些特定问题的难度的猜想进行条件化，例如承诺图上的图连通性是一个循环或两个循环，通常称为一个循环与两个循环问题。这与基于复杂性类猜想的传统论证不同（例如，\(\textsf {P}\ne \textsf {NP}\))，从某种意义上说，它们通常更健壮，因为反驳它们会导致针对一大堆问题的开创性算法。在本文中，我们提出了允许后一种论证的问题和问题类别之间的联系。这些连接涉及在 MPC 模型中以次对数轮数可解决的问题类别，用\(\textsf {MPC}(o(\log N))\)表示，以及标准空间复杂度类\(\textsf { L}\)和\(\textsf {NL}\)，并提出在某种意义上是稳健的猜想，即反驳它们会导致 MPC 模型中出现许多速度惊人的新算法。我们还获得了新的条件下界，并证明了 MPC 模型中问题之间的新约简和等价性。具体来说，我们的主要结果如下。

• 一个循环与两个循环猜想的下界可以在\(\textsf {L}\nsubseteq \textsf {MPC}(o(\log N))\)猜想下进行论证：这两个假设是等价的，并且驳斥其中任何一个都会导致\(o(\log N)\) -round MPC 算法解决大量具有挑战性的问题，包括列表排序、最小切割和平面性测试。事实上，我们证明了这些问题和许多其他问题都需要渐近相同的轮数，这与区分图是一个循环还是两个循环的看似简单得多的问题相同。

• 以前在一个循环与两个循环猜想下争论的许多下界可以在一个更稳健（因此更难反驳）的猜想下争论，即\(\textsf {NL}\nsubseteq \textsf {MPC}(o(\log N ))\)。反驳这个猜想将导致\(o(\log N)\) -round MPC 算法解决更大的问题，包括所有对最短路径、中介中心性以及所有上述问题。该猜想下的下界适用于完美匹配和网络流等问题。