We present a novel technique for work-efficient parallel derandomization, for algorithms that rely on the concentration of measure bounds such as Chernoff, Hoeffding, and Bernstein inequalities. Our method increases the algorithm's computational work and depth by only polylogarithmic factors. Before our work, the only known method to obtain parallel derandomization with such strong concentrations was by the results of [Motwani, Naor, and Naor FOCS'89; Berger and Rompel FOCS'89], which perform a binary search in a $k$-wise independent space for $k=poly(\log n)$. However, that method blows up the computational work by a high $poly(n)$ factor and does not yield work-efficient parallel algorithms. Their method was an extension of the approach of [Luby FOCS'88], which gave a work-efficient derandomization but was limited to algorithms analyzed with only pairwise independence. Pushing the method from pairwise to the higher $k$-wise analysis resulted in the $poly(n)$ factor computational work blow-up. Our work can be viewed as an alternative extension from the pairwise case, which yields the desired strong concentrations while retaining work efficiency up to logarithmic factors. Our approach works by casting the problem of determining the random variables as an iterative process with $poly(\log n)$ iterations, where different iterations have independent randomness. This is done so that for the desired concentrations, we need only pairwise independence inside each iteration. In particular, we model each binary random variable as a result of a gradual random walk, and our method shows that the desired Chernoff-like concentrations about the endpoints of these walks can be boiled down to some pairwise analysis on the steps of these random walks in each iteration (while having independence across iterations).

翻译：我们提出了一种新颖的工作高效并行去随机化技术，适用于依赖切尔诺夫、霍夫丁和伯恩斯坦不等式等度量集中界限的算法。该方法仅需对数多项式因子即可提升算法的计算工作量和深度。在我们的工作之前，已知的能实现此类强集中度的并行去随机化方法仅源于[Motwani, Naor, and Naor FOCS'89; Berger and Rompel FOCS'89]的研究结果，该方法在$k$次独立空间（$k=poly(\log n)$）中执行二分搜索。然而，该方法的计算工作量会放大为高次多项式$poly(n)$因子，且无法产生工作高效的并行算法。该方法是对[Luby FOCS'88]方法的扩展——Luby的方法虽能实现工作高效的去随机化，但仅限于仅依赖对偶独立性分析的算法。将方法从对偶情形推广到更高阶$k$次分析时，导致了计算工作量的多项式因子膨胀。我们的工作可被视为对偶情形的另一种推广，在保留对数因子内的工作效率的同时，实现了所需的强集中度。我们的方法将随机变量的确定问题建模为拥有$poly(\log n)$轮迭代的迭代过程，其中不同轮次使用独立随机性。通过这种设计，我们仅需每轮迭代内部的对偶独立性即可获得所需的集中度。具体而言，我们将每个二元随机变量建模为渐进随机游走的结果，该方法表明这些游走终点的类切尔诺夫集中度可简化为每轮迭代中随机游走步长上的某种对偶分析（同时保持轮次间的独立性）。