We study the deterministic complexity of the $2$-Ruling Set problem in the model of Massively Parallel Computation (MPC) with linear and strongly sublinear local memory. Linear MPC: We present a constant-round deterministic algorithm for the $2$-Ruling Set problem that matches the randomized round complexity recently settled by Cambus, Kuhn, Pai, and Uitto [DISC'23], and improves upon the deterministic $O(\log \log n)$-round algorithm by Pai and Pemmaraju [PODC'22]. Our main ingredient is a simpler analysis of CKPU's algorithm based solely on bounded independence, which makes its efficient derandomization possible. Sublinear MPC: We present a deterministic algorithm that computes a $2$-Ruling Set in $\tilde O(\sqrt{\log n})$ rounds deterministically. Notably, this is the first deterministic ruling set algorithm with sublogarithmic round complexity, improving on the $O(\log \Delta + \log \log^* n)$-round complexity that stems from the deterministic MIS algorithm of Czumaj, Davies, and Parter [TALG'21]. Our result is based on a simple and fast randomness-efficient construction that achieves the same sparsification as that of the randomized $\tilde O(\sqrt{\log n})$-round LOCAL algorithm by Kothapalli and Pemmaraju [FSTTCS'12].

翻译：本研究探讨了在具有线性和强次线性本地内存的大规模并行计算（MPC）模型中，2-支配集问题的确定性计算复杂度。线性内存MPC：我们提出了一种常数轮确定性算法求解2-支配集问题，该算法与Cambus、Kuhn、Pai和Uitto近期确立的随机算法轮复杂度相匹配[DISC'23]，并改进了Pai与Pemmaraju提出的确定性$O(\log \log n)$轮算法[PODC'22]。我们的核心贡献在于基于有限独立性对CKPU算法进行了更简洁的分析，这使其高效去随机化成为可能。次线性内存MPC：我们提出了一种确定性算法，可在$\tilde O(\sqrt{\log n})$轮内计算2-支配集。值得注意的是，这是首个具有亚对数轮复杂度的确定性支配集算法，改进了源自Czumaj、Davies和Parter确定性最大独立集算法的$O(\log \Delta + \log \log^* n)$轮复杂度[TALG'21]。我们的成果基于一种简单高效的随机性节约构造，该构造实现了与Kothapalli和Pemmaraju随机化$\tilde O(\sqrt{\log n})$轮LOCAL算法相同的稀疏化效果[FSTTCS'12]。