The Massively Parallel Computation (MPC) model is an emerging model which distills core aspects of distributed and parallel computation. The aim is to solve (typically graph) problems in systems where the input is distributed over many machines with limited space. Recent work has focused on the regime in which machines have sublinear memory, with randomized algorithms presented for the tasks of Maximal Matching, Maximal Independent Set (MIS) and $(\Delta+1)$-Coloring. However, there have been no prior corresponding deterministic algorithms. We introduce a new graph sparsification technique that deterministically computes a low-degree subgraph with additional desired properties. The degree of the nodes in this subgraph is small in the sense that the edges of each node can be now stored on a single machine. Depending on the specific local problem, the low-degree subgraph also has the property that solving the problem on this subgraph provides a large global progress. We obtain the following deterministic MPC algorithms with $O(n^{\epsilon})$ space on each machine for any constant $\epsilon >0$: - We show an $O(\log \Delta+\log\log n)$ communication round algorithms for solving maximal matching and MIS, by derandomizing the well-known randomized algorithm of Luby [SICOMP'86]. Based on the recent work of Ghaffari et al. [FOCS'18], this additive $O(\log\log n)$ factor is conditionally essential. These algorithms can also be shown to run in $O(\log \Delta)$ rounds in the closely related model of Congested Clique. This improves up on the state-of-the-art bound of $O(\log^2 \Delta)$ rounds by Censor-Hillel et al. [DISC'17]. - By employing our graph sparsification technique accompanied with a palette sparsification, we give a deterministic (deg+1)-list coloring (and thus also a $(\Delta+1)$-coloring) algorithm in $O(\log \Delta+\log\log n)$ rounds.
翻译:质量平行计算( MPC) 模型是一个新兴模型, 它可以蒸馏分布和平行计算的核心方面。 我们引入一个新的图形解析技术, 可以用更多想要的特性来解析( 典型的图形) 输入在很多机器上分布的系统的问题。 最近的工作侧重于机器具有亚线性内存的系统, 并且根据特定本地问题, 低度的算法也具有解决本子图上问题的属性。 但是, 我们没有之前对应的确定性算法。 我们引入一个新的图形解析技术, 以较低水平解算法进行低水平的亚化亚化亚化亚化 。 在每家机器上使用一个低度的C- 亚化亚化亚化亚化亚化亚化亚化 。