Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space

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.