A central approach to algorithmic derandomization is to construct probability distributions with small support that "fool" randomized algorithms, often enabling efficient parallel (NC) implementations. An abstraction of this idea is fooling polynomial-space statistical tests computed via finite automata (Sivakumar 2002); this encompasses a wide range of properties including $k$-wise independence and sums of random variables. We present new parallel algorithms to fool automata, with significantly reduced processor complexity. Briefly, our approach is to iteratively sparsify distributions via work-efficient lattice discrepancy rounding, while tracking an aggregate weighted error that is determined by the Lipschitz value of the statistical tests. We illustrate with applications to the Gale-Berlekamp Switching Game and approximate MAX-CUT via SDP rounding. These involve several optimizations, including truncating the state space of the automata and using FFT-based convolutions to compute transition probabilities efficiently.

翻译：算法去随机化的核心方法之一是构造具有小支撑集的概率分布来“欺骗”随机化算法，这通常能实现高效的并行（NC）实现。该思想的一个抽象形式是欺骗通过有限自动机计算的多项式空间统计检验（Sivakumar 2002）；这涵盖了包括$k$阶独立性和随机变量和在内的广泛性质。我们提出了欺骗自动机的新并行算法，其处理器复杂度显著降低。简而言之，我们的方法是通过工作高效的格偏差舍入迭代地稀疏化分布，同时跟踪由统计检验的Lipschitz值确定的聚合加权误差。我们通过Gale-Berlekamp开关博弈和基于SDP舍入的近似MAX-CUT问题来说明其应用。这些应用涉及多项优化技术，包括截断自动机的状态空间以及使用基于FFT的卷积来高效计算转移概率。