Simple Optimal Hitting Sets for Small-Success RL

Abstract

We give a simple explicit hitting set generator for read-once branching programs of width $w$ and length $r$ with known variable order and acceptance probability at least $\epsilon$. When $r = w$, our generator has seed length $O(\log^2 r + \log(1/\epsilon))$. When $r = \text{polylog } w$, our generator has optimal seed length $O(\log w + \log(1/\epsilon))$. For intermediate values of $r$, our generator's seed length smoothly interpolates between these two extremes. Our generator's seed length improves on recent work by Braverman, Cohen, and Garg [SIAM J. Comput., (2020), doi:10.1137/18M1197734]. In addition, our generator and its analysis are dramatically simpler than the work by Braverman et al. When $\epsilon$ is small, our generator's seed length improves on all the classic generators for space-bounded computation [N. Nisan, Combinatorica, 12 (1992), pp. 449--461; R. Impagliazzo, N. Nisan, and A. Wigderson, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, ACM, 1994, pp. 356--364; N. Nisan and D. Zuckerman, J. Comput. System Sci., 52 (1996), pp. 43--52]. However, all of these other works construct more general objects than we do. As a corollary of our construction, we show that every ${RL}$ algorithm that uses $r$ random bits can be simulated by an ${NL}$ algorithm that uses only $O(r/\log^c n)$ nondeterministic bits, where $c$ is an arbitrarily large constant. Finally, we show that any ${RL}$ algorithm with small success probability $\epsilon$ can be simulated deterministically in space $O(\log^{3/2} n + \log n \log \log(1/\epsilon))$. This space bound improves on work by Saks and Zhou [J. Comput. System Sci., 58 (1999), pp. 376--403], who gave an algorithm for the more general “two-sided” problem that runs in space $O(\log^{3/2} n + \sqrt{\log n} \log(1/\epsilon))$.