Simple Optimal Hitting Sets For Small-Success Rl

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 = polylog w, our generator has optimal seed length 0(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 T. Comput., (2020), doi:10.1137/18M1197734]. In addition, our generator and its analysis are dramatically simpler than the work by Braverman et al. When E 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, T. 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 + root log n log (1/epsilon)).