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 fully-scalable 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].

翻译：我们考虑为大规模并行计算（MPC）模型设计确定性图算法的问题，这些算法的性能随输入图稀疏度的提升而改进，稀疏度由树度（arboricity）概念衡量。针对最大独立集（MIS）、最大匹配（MM）和顶点着色问题，我们改进了现有技术如下。设$\lambda$表示具有最大度$\Delta$的$n$节点输入图的树度。MIS和MM：我们开发了一种确定性完全可扩展算法，可在$O(\log \log n)$轮内将最大度降低至$poly(\lambda)$，改进并简化了Ghaffari、Grunau、Jin [DISC'20]的随机化$O(\log \log n)$轮$poly(\max(\lambda, \log n))$度降低方法。当我们的方法与Czumaj、Davies、Parter [SPAA'20, TALG'21]的最优$O(\log \Delta + \log \log n)$轮算法结合时，在低空间MPC中，MIS和MM的确定性轮复杂度改进为$O(\log \lambda + \log \log n)$。我们还将上述MIS和MM算法扩展到线性全局内存场景。具体而言，我们证明这两个问题可在确定性时间$O(\min(\log n, \log \lambda \cdot \log \log n))$内解决，甚至对于树度不超过$\log^{O(1)} \log n$的图可在$O(\log \log n)$时间内解决。在此设定下，此前仅Latypov和Uitto [ArXiv'21]对树实现了$O(\log^2 \log n)$运行时间界限。顶点着色：我们提出了一种$O(1)$轮确定性算法，用于在线性内存MPC中解决$O(\lambda)$着色问题，该算法具有$n \cdot poly(\lambda)$的松弛全局内存，仅需一次图划分步骤即可解决问题。这与Ghaffari和Sayyadi [ICALP'19]的最优随机化轮复杂度相匹配，并改进了Barenboim和Khazanov [CSR'18]的确定性$O(\lambda^{\epsilon})$轮算法。