One main genre of algorithmic derandomization comes from the construction of probability distributions with small support that fool a randomized algorithm. This is especially well-suited to parallelization, i.e. NC algorithms. A significant abstraction of these methods can be formulated in terms of fooling polynomial-space statistical tests computed via finite automata (Sivakumar 2002); this encompasses $k$-wise independence, sums of random variables, and many other properties. We describe new parallel algorithms to fool general finite-state automata with significantly reduced processor complexity. The analysis is also simplified because we can cleanly separate the problem-specific optimizations from the general lattice discrepancy problems at the core of the automaton-fooling construction. We illustrate with improved applications to the Gale-Berlekamp Switching Game and to approximate MAX-CUT via SDP rounding.

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