SIAM Journal on Computing, Volume 49, Issue 4, Page 811-820, January 2020.
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 $O(\log^{3/2} n + \sqrt{\log n} \log(1/\epsilon))$.

中文翻译：

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小成功RL的简单最佳命中集

SIAM计算杂志，第49卷，第4期，第811-820页，2020年1月。
我们给出了一个简单的显式命中集生成器，用于宽度为$ w $且长度为$ r $的一次读取分支程序，其已知变量顺序和接受概率至少为$ \ epsilon $。当$ r = w $时，我们的生成器的种子长度为$ O（\ log ^ 2 r + \ log（1 / \ epsilon））$。当$ r = \ text {polylog} w $时，我们的生成器具有最佳种子长度$ O（\ log w + \ log（1 / \ epsilon））$。对于$ r $的中间值，我们的生成器的种子长度在这两个极端之间平滑地插值。Braverman，Cohen和Garg [SIAM J. Comput。，（2020年），doi：10.1137 / 18M1197734]最近的工作改进了我们发电机的种子长度。此外，我们的生成器及其分析比Braverman等人的工作要简单得多。当$ \ epsilon $较小时，我们的生成器的种子长度会在所有经典生成器上得到改进，从而进行有限的计算[N. 尼山 Combinatorica，12（1992），第449--461页; R.Impagliazzo，N.Nisan和A.Wigderson，在第26届ACM计算理论年度研讨会论文集中，ACM，1994年，第356--364页; N. Nisan和D. Zuckerman，J。Comput。System Sci。，52（1996），第43--52页]。但是，所有这些其他作品都比我们构造了更通用的对象。作为我们的推论，我们证明了使用$ r $随机位的每个$ {RL} $算法都可以由仅使用$ O（r / \ log ^ cn）$非确定性的$ {NL} $算法进行模拟位，其中$ c $是任意大的常量。最后，我们表明，可以在空间$ O（\ log ^ {3/2} n + \ log n \ log \ log（1 / \中确定性地模拟具有成功概率$ \ epsilon $的任何$ {RL} $算法epsilon））。这个空间范围改善了Saks和Zhou的工作[J. 计算 系统科学，58（1999），第376--403页]，