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].

翻译：摘要：我们研究针对大规模并行计算（MPC）模型设计确定性图算法的问题，这些算法可随输入图稀疏性（通过树度概念衡量）的提升而优化。针对极大独立集（MIS）、极大匹配（MM）和顶点着色问题，我们改进了当前最优结果。设$\lambda$表示最大度为$\Delta$的$n$节点输入图的树度。MIS与MM：我们提出一种确定性低空间MPC算法，可在$O(\log \log n)$轮内将最大度降至$poly(\lambda)$，改进并简化了Ghaffari、Grunau、Jin [DISC'20] 的随机$O(\log \log n)$轮$poly(\max(\lambda, \log n))$度降阶方法。将该方法与Czumaj、Davies、Parter [SPAA'20, TALG'21] 的当前最优$O(\log \Delta + \log \log n)$轮算法结合，可实现对低空间MPC中MIS与MM的确定性轮复杂度改进至$O(\log \lambda + \log \log n)$。我们还将上述MIS与MM算法扩展至适用于线性全局内存场景。具体而言，我们证明两个问题均可在确定性时间$O(\min(\log n, \log \lambda \cdot \log \log n))$内解决，对于树度不超过$\log^{O(1)} \log n$的图甚至可在$O(\log \log n)$时间内解决。此前该场景仅对树结构存在Latypov和Uitto [ArXiv'21] 提出的$O(\log^2 \log n)$运行时间复杂度。顶点着色：我们提出一种$O(1)$轮确定算法，用于在具有$n \cdot poly(\lambda)$宽松全局内存的线性内存MPC中实现$O(\lambda)$着色问题，该算法仅通过单步图分区即可求解问题。该结果匹配Ghaffari和Sayyadi [ICALP'19] 的当前最优随机轮复杂度，并改进了Barenboim和Khazanov [CSR'18] 的确定性$O(\lambda^{\epsilon})$轮算法。