The Massively Parallel Computation (MPC) model is an emerging model that distills core aspects of distributed and parallel computation, developed as a tool to solve combinatorial (typically graph) problems in systems of many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in n , the number of nodes in the input graph) space, with randomized algorithms presented for the fundamental problems of Maximal Matching and Maximal Independent Set. However, there have been no prior corresponding deterministic algorithms. A major challenge underlying the sublinear space setting is that the local space of each machine might be too small to store all edges incident to a single node. This poses a considerable obstacle compared to classical models in which each node is assumed to know and have easy access to its incident edges. To overcome this barrier, we introduce a new graph sparsification technique that deterministically computes a low-degree subgraph, with the additional property that solving the problem on this subgraph provides significant progress towards solving the problem for the original input graph. Using this framework to derandomize the well-known algorithm of Luby [SICOMP’86], we obtain O (log Δ + log log n )-round deterministic MPC algorithms for solving the problems of Maximal Matching and Maximal Independent Set with O ( n ɛ ) space on each machine for any constant ɛ > 0. These algorithms also run in O (log Δ) rounds in the closely related model of CONGESTED CLIQUE, improving upon the state-of-the-art bound of O (log 2 Δ) rounds by Censor-Hillel et al. [DISC’17].

中文翻译：

---

用于对低空间大规模并行计算进行去随机化的图稀疏化

大规模并行计算 (MPC) 模型是一种新兴模型，它提炼了分布式和并行计算的核心方面，被开发为解决空间有限的许多机器系统中的组合（通常是图形）问题的工具。最近的工作集中在机器具有次线性（在n，输入图中的节点数）空间，针对最大匹配和最大独立集的基本问题提出了随机算法。但此前并没有相应的确定性的算法。亚线性空间设置的一个主要挑战是，每台机器的本地空间可能太小而无法存储入射到单个节点的所有边。与假设每个节点都知道并且可以轻松访问其入射边缘的经典模型相比，这构成了相当大的障碍。为了克服这个障碍，我们引入了一个新的图稀疏化技术确定性地计算一个低度子图，其附加属性是解决该子图上的问题为解决原始输入图的问题提供了重大进展。使用这个框架去随机化 Luby [SICOMP'86] 的著名算法，我们得到○(log Δ + log logn）-圆形的确定性的解决问题的 MPC 算法最大匹配和最大独立集和○(n ε) 每台机器上的空间对于任何常数 ɛ > 0。这些算法也可以在○(log Δ) 在紧密相关的 CLIQUE 模型中循环，改进了最先进的边界○（日志2Δ) Censor-Hillel 等人的轮次。[光盘'17]。