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Deterministic Massively Parallel Symmetry Breaking for Sparse Graphs

26 January 2023
Manuela Fischer
Jeff Giliberti
Christoph Grunau
ArXiv (abs)PDFHTML
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 λ\lambdaλ denote the arboricity of the nnn-node input graph with maximum degree Δ\DeltaΔ. MIS and MM: We develop a deterministic low-space MPC algorithm that reduces the maximum degree to poly(λ)poly(\lambda)poly(λ) in O(log⁡log⁡n)O(\log \log n)O(loglogn) rounds, improving and simplifying the randomized O(log⁡log⁡n)O(\log \log n)O(loglogn)-round poly(max⁡(λ,log⁡n))poly(\max(\lambda, \log n))poly(max(λ,logn))-degree reduction of Ghaffari, Grunau, Jin [DISC'20]. Our approach when combined with the state-of-the-art O(log⁡Δ+log⁡log⁡n)O(\log \Delta + \log \log n)O(logΔ+loglogn)-round algorithm by Czumaj, Davies, Parter [SPAA'20, TALG'21] leads to an improved deterministic round complexity of O(log⁡λ+log⁡log⁡n)O(\log \lambda + \log \log n)O(logλ+loglogn) 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⁡λ⋅log⁡log⁡n))O(\min(\log n, \log \lambda \cdot \log \log n))O(min(logn,logλ⋅loglogn)), and even in O(log⁡log⁡n)O(\log \log n)O(loglogn) time for graphs with arboricity at most log⁡O(1)log⁡n\log^{O(1)} \log nlogO(1)logn. In this setting, only a O(log⁡2log⁡n)O(\log^2 \log n)O(log2logn)-running time bound for trees was known due to Latypov and Uitto [ArXiv'21]. Vertex Coloring: We present a O(1)O(1)O(1)-round deterministic algorithm for the problem of O(λ)O(\lambda)O(λ)-coloring in linear-memory MPC with relaxed global memory of n⋅poly(λ)n \cdot poly(\lambda)n⋅poly(λ) 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(λϵ)O(\lambda^{\epsilon})O(λϵ)-round algorithm by Barenboim and Khazanov [CSR'18].

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