In this paper, we study the power and limitations of component-stable algorithms in the low-space model of massively parallel computation (MPC). Recently Ghaffari, Kuhn and Uitto (FOCS 2019) introduced the class of component-stable low-space MPC algorithms, which are, informally, those algorithms for which the outputs reported by the nodes in different connected components are required to be independent. This very natural notion was introduced to capture most (if not all) of the known efficient MPC algorithms to date, and it was the first general class of MPC algorithms for which one can show non-trivial conditional lower bounds. In this paper we enhance the framework of component-stable algorithms and investigate its effect on the complexity of randomized and deterministic low-space MPC. Our key contributions include: 1. We revise and formalize the lifting approach of Ghaffari, Kuhn and Uitto. This requires a very delicate amendment of the notion of component stability, which allows us to fill in gaps in the earlier arguments. 2. We also extend the framework to obtain conditional lower bounds for deterministic algorithms and fine-grained lower bounds that depend on the maximum degree \(\Delta \). 3. We demonstrate a collection of natural graph problems for which deterministic component-unstable algorithms break the conditional lower bound obtained for component-stable algorithms. This implies that, in the context of deterministic algorithms, component-stable algorithms are conditionally weaker than the component-unstable ones. 4. We also show that the restriction to component-stable algorithms has an impact in the randomized setting. We present a natural problem which can be solved in O(1) rounds by a component-unstable MPC algorithm, but requires \(\Omega (\log \log ^* n)\) rounds for any component-stable algorithm, conditioned on the connectivity conjecture. Altogether our results imply that component-stability might limit the computational power of the low-space MPC model, at least in certain contexts, paving the way for improved upper bounds that escape the conditional lower bound setting of Ghaffari, Kuhn, and Uitto.

在本文中，我们研究了大规模并行计算（MPC ）低空间模型中组件稳定算法的能力和局限性。最近，Ghaffari、Kuhn 和 Uitto (FOCS 2019) 引入了一类组件稳定的低空间MPC算法，非正式地，这些算法要求不同连接组件中的节点报告的输出是独立的。引入这个非常自然的概念是为了捕获迄今为止大多数（如果不是全部）已知的高效MPC算法，并且它是第一类可以显示非平凡条件下界的通用MPC算法。在本文中，我们增强了组件稳定算法的框架，并研究了其对随机和确定性低空间MPC复杂性的影响。我们的主要贡献包括： 1. 我们修改并规范了 Ghaffari、Kuhn 和 Uitto 的提升方法。这需要对组件稳定性的概念进行非常微妙的修正，这使我们能够填补先前论点中的空白。 2. 我们还扩展了框架以获得确定性算法的条件下界和取决于最大度\(\Delta \)的细粒度下界。 3. 我们演示了一系列自然图问题，对于这些问题，确定性组件不稳定算法打破了组件稳定算法获得的条件下界。这意味着，在确定性算法的背景下，组件稳定算法在条件上弱于组件不稳定算法。 4. 我们还表明，对组件稳定算法的限制对随机设置有影响。我们提出了一个自然问题，可以通过组件不稳定的MPC算法在O (1) 轮中解决，但对于任何组件稳定的算法都需要\(\Omega (\log \log ^* n)\)轮，条件是连通性猜想。总而言之，我们的结果表明，至少在某些情况下，组件稳定性可能会限制低空间MPC模型的计算能力，从而为逃避 Ghaffari、Kuhn 和 Uitto 的条件下界设置的改进上限铺平道路。