Deterministic Massively Parallel Symmetry Breaking for Sparse Graphs

TL;DR

This paper introduces new deterministic algorithms for symmetry breaking problems in sparse graphs within the MPC model, achieving faster round complexities and extending to linear memory settings, thus advancing parallel graph algorithms.

Contribution

The paper develops deterministic MPC algorithms for MIS, MM, and vertex coloring that improve round complexity and memory efficiency, especially for graphs with low arboricity.

Findings

Deterministic degree reduction to poly(λ) in O(log log n) rounds.

Improved deterministic round complexity of O(log λ + log log n) for MIS and MM.

Single-step O(1)-round vertex coloring algorithm matching randomized results.

Abstract

We consider the problem of designing deterministic graph algorithms for the model of Massively Parallel Computation (MPC) that improve with the sparsity of the input graph, as measured by the notion of arboricity. For the problems of maximal independent set (MIS), maximal matching (MM), and vertex coloring, we improve the state of the art as follows. Let $\lambda$ denote the arboricity of the $n$-node input graph with maximum degree $\Delta$. MIS and MM: We develop a deterministic low-space MPC algorithm that reduces the maximum degree to $poly(\lambda)$ in $O(\log \log n)$ rounds, improving and simplifying the randomized $O(\log \log n)$-round $poly(\max(\lambda, \log n))$-degree reduction of Ghaffari, Grunau, Jin [DISC'20]. Our approach when combined with the state-of-the-art $O(\log \Delta + \log \log n)$-round algorithm by Czumaj, Davies, Parter [SPAA'20, TALG'21] leads to an improved deterministic round complexity of $O(\log \lambda + \log \log n)$ for MIS and MM in low-space MPC. We also extend above MIS and MM algorithms to work with linear global memory. Specifically, we show that both problems can be solved in deterministic time $O(\min(\log n, \log \lambda \cdot \log \log n))$, and even in $O(\log \log n)$ time for graphs with arboricity at most $\log^{O(1)} \log n$. In this setting, only a $O(\log^2 \log n)$-running time bound for trees was known due to Latypov and Uitto [ArXiv'21]. Vertex Coloring: We present a $O(1)$-round deterministic algorithm for the problem of $O(\lambda)$-coloring in linear-memory MPC with relaxed global memory of $n \cdot poly(\lambda)$ that solves the problem after just one single graph partitioning step. This matches the state-of-the-art randomized round complexity by Ghaffari and Sayyadi [ICALP'19] and improves upon the deterministic $O(\lambda^{\epsilon})$-round algorithm by Barenboim and Khazanov [CSR'18].