# Graph sparsification for derandomizing massively parallel computation with low space

A. Czumaj, P. Davies, M. Parter, in:, Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020), Association for Computing Machinery, 2020, pp. 175–185.

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Abstract

The Massively Parallel Computation (MPC) model is an emerging model which distills core aspects of distributed and parallel computation. It has been developed as a tool 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 (in $n$, the number of nodes in the input graph) space, with randomized algorithms presented for fundamental graph 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 the edges incident to a single node. This poses a considerable obstacle compared to the 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 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. This low-degree subgraph also has the property that solving the problem on this subgraph provides \emph{significant} global progress, i.e., progress towards solving the problem for the original input graph.
Using this framework to derandomize the well-known randomized algorithm of Luby [SICOMP'86], we obtain $O(\log \Delta+\log\log n)$-round deterministic MPC algorithms for solving the fundamental problems of Maximal Matching and Maximal Independent Set with $O(n^{\epsilon})$ space on each machine for any constant $\epsilon > 0$. 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, improving upon the state-of-the-art bound of $O(\log^2 \Delta)$ rounds by Censor-Hillel et al. [DISC'17].

Publishing Year

Date Published

2020-07-01

Proceedings Title

Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020)

Issue

7

Page

175-185

Conference

SPAA: Symposium on Parallelism in Algorithms and Architectures

Conference Location

Virtual Event, United States

Conference Date

2020-07-15 – 2020-07-17

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### Cite this

Czumaj A, Davies P, Parter M. Graph sparsification for derandomizing massively parallel computation with low space. In:

*Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020)*. Association for Computing Machinery; 2020:175-185. doi:10.1145/3350755.3400282Czumaj, A., Davies, P., & Parter, M. (2020). Graph sparsification for derandomizing massively parallel computation with low space. In

*Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020)*(pp. 175–185). Virtual Event, United States: Association for Computing Machinery. https://doi.org/10.1145/3350755.3400282Czumaj, Artur, Peter Davies, and Merav Parter. “Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space.” In

*Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020)*, 175–85. Association for Computing Machinery, 2020. https://doi.org/10.1145/3350755.3400282.A. Czumaj, P. Davies, and M. Parter, “Graph sparsification for derandomizing massively parallel computation with low space,” in

*Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020)*, Virtual Event, United States, 2020, no. 7, pp. 175–185.Czumaj A, Davies P, Parter M. 2020. Graph sparsification for derandomizing massively parallel computation with low space. Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020). SPAA: Symposium on Parallelism in Algorithms and Architectures, 175–185.

Czumaj, Artur, et al. “Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space.”

*Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020)*, no. 7, Association for Computing Machinery, 2020, pp. 175–85, doi:10.1145/3350755.3400282.**All files available under the following license(s):**

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arXiv 1912.05390