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Pseudorandom Pseudo-distributions with Near-Optimal Error for Read-Once Branching Programs
SIAM Journal on Computing ( IF 1.6 ) Pub Date : 2020-02-18 , DOI: 10.1137/18m1197734
Mark Braverman , Gil Cohen , Sumegha Garg

SIAM Journal on Computing, Ahead of Print.
Nisan [Combinatorica, 12 (1992), pp. 449--461] constructed a pseudorandom generator for length $n$, width $n$ read-once branching programs (ROBPs) with error $\varepsilon$ and seed length $O(\log^2{n} + \log{n} \cdot \log(1/\varepsilon))$. A major goal in complexity theory is to reduce the seed length, hopefully, to the optimal $O(\log{n}+\log(1/\varepsilon))$, or to construct improved hitting sets, as these would yield stronger derandomization of ${BPL}$ and ${RL}$, respectively. In contrast to a successful line of work in restricted settings, no progress has been made for general, unrestricted, ROBPs. Indeed, Nisan's construction is the best pseudorandom generator and, prior to this work, also the best hitting set for unrestricted ROBPs. In this work, we make the first improvement for the general case by constructing a hitting set with seed length $\widetilde{O}(\log^2{n}+\log(1/\varepsilon))$. That is, we decouple $\varepsilon$ and $n$, and obtain near-optimal dependence on the former. The regime of parameters in which our construction strictly improves upon prior works, namely, $\log(1/\varepsilon) \gg \log{n}$, is also motivated by the work of Saks and Zhou [J. Comput. System Sci., 58 (1999), pp. 376--403], who use pseudorandom generators with error $\varepsilon$, for length $n$, width $w$ ROBPs, such that $w,1/\varepsilon = 2^{(\log{n})^{2}}$ in their proof for ${BPL} \subseteq \mathbf{L}^{3/2}$. In fact, we introduce and construct a new type of primitive we call pseudorandom pseudo-distributions. Informally, this is a generalization of pseudorandom generators in which one may assign negative and unbounded weights to paths, as opposed to working with probability distributions. We show that such a primitive yields hitting sets and, for derandomization purposes, can be used to derandomize two-sided error algorithms.


中文翻译:

一次分支程序的具有接近最佳错误的伪随机伪分布

《 SIAM计算杂志》,预印本。
Nisan [Combinatorica,12(1992),第449--461页]构建了一个伪随机生成器,其长度为$ n $,宽度为$ n $,一次读取分支程序(ROBP),错误为$ \ varepsilon $,种子长度为$ O( \ log ^ 2 {n} + \ log {n} \ cdot \ log(1 / \ varepsilon))$。复杂性理论的主要目标是希望将种子长度减少到最佳$ O(\ log {n} + \ log(1 / \ varepsilon))$,或构建改进的命中集,因为它们会产生更强的分别对$ {BPL} $和$ {RL} $进行非随机化。与在受限环境中成功开展工作相反,对于一般性非受限ROBP而言,没有取得任何进展。的确,Nisan的构造是最好的伪随机生成器,并且在进行这项工作之前,它也是不受限制的ROBP的最佳命中方案。在这项工作中 我们通过构造种子长度为$ \ widetilde {O}(\ log ^ 2 {n} + \ log(1 / \ varepsilon))$的命中集对一般情况进行了第一个改进。也就是说,我们将$ \ varepsilon $和$ n $解耦,并获得对前者的近似最佳依赖。Saks和Zhou的工作也激发了我们的构造严格改进先前工作的参数范围,即\\ log(1 / \ varepsilon)\ gg \ log {n} $。计算 System Sci。,58(1999),pp。376--403],他们使用错误$ \ varepsilon $,长度$ n $,宽度$ w $ ROBP的伪随机生成器,使得$ w,1 / \ varepsilon = 2 ^ {(\ log {n})^ {2}} $作为$ {BPL} \ subseteq \ mathbf {L} ^ {3/2} $的证明。实际上,我们引入并构造了一种新型的原语,我们称其为伪随机伪分布。非正式地,这是伪随机生成器的一种概括,其中可以为路径分配负权重和无界权重,这与使用概率分布相反。我们表明,这样的原始数据会产生命中集,并且出于非随机化的目的,可用于对两侧错误算法进行非随机化。
更新日期:2020-02-18
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