We show the first conditionally optimal deterministic algorithm for $3$-coloring forests in the low-space massively parallel computation (MPC) model. Our algorithm runs in $O(\log \log n)$ rounds and uses optimal global space. The best previous algorithm requires $4$ colors [Ghaffari, Grunau, Jin, DISC'20] and is randomized, while our algorithm are inherently deterministic. Our main technical contribution is an $O(\log \log n)$-round algorithm to compute a partition of the forest into $O(\log n)$ ordered layers such that every node has at most two neighbors in the same or higher layers. Similar decompositions are often used in the area and we believe that this result is of independent interest. Our results also immediately yield conditionally optimal deterministic algorithms for maximal independent set and maximal matching for forests, matching the state of the art [Giliberti, Fischer, Grunau, SPAA'23]. In contrast to their solution, our algorithms are not based on derandomization, and are arguably simpler.

翻译：我们在低空间大规模并行计算（MPC）模型中首次提出针对森林的3-着色问题的最优确定性算法。该算法在 $O(\log \log n)$ 轮内运行，并采用最优的全局空间。先前最优算法需使用4种颜色 [Ghaffari, Grunau, Jin, DISC'20] 且为随机算法，而我们的算法本质上是确定性的。主要技术贡献在于提出了一种 $O(\log \log n)$ 轮算法，将森林划分为 $O(\log n)$ 个有序层，使得每个节点在相同层或更高层中最多有两个邻居。类似分解常被用于该领域，我们相信该结果具有独立意义。我们的结果还直接推导出森林最大独立集与最大匹配的条件最优确定性算法，与当前最优成果 [Giliberti, Fischer, Grunau, SPAA'23] 相当。与他们的解决方案不同，我们的算法不基于去随机化，并且可以说更为简洁。