AI 中文总结
研究针对置换分支程序构造ε-误差PRG，核心方法是利用经典INW PRG，通过依赖程序的半范数分析误差传播，贡献是得到种子长度\[O((\log w+\log(1/\varepsilon))\cdot \log n)\]，改进对w和n的依赖。
AI 中文摘要
我们构造了一个针对长度为n、宽度为w的置换一次读分支程序的ε-误差伪随机生成器（PRG），其种子长度为\[O((\log w+\log(1/\varepsilon))\cdot \log n)\]。与De（CCC 2011）和Steinke（ECCC 2012）的构造相比，这在对w的依赖上有指数级改进。与更一般适用于正则分支程序且已实现对w最优依赖的Braverman等人的工作相比，我们的结果改进了对长度n的依赖，达到最优对数依赖。生成器本身是Impagliazzo、Nisan和Wigderson（STOC 1994）的经典INW PRG。我们表明，对于置换分支程序，INW生成器可以用度为w和1/ε的多项式且与n无关的扩展器实例化。为证明这一点，我们使用针对手头分支程序定制的依赖程序的半范数分析误差传播。这些半范数基于Braverman等人引入的权重函数。关键在于，在这些适配的半范数下测量时，误差在整个递归过程中不会累积。由于我们的分析仅依赖于底层扩展器的谱扩展，我们的种子长度与Hoza、Pyne和Vadhan（Algorithmica 2024）最近对INW生成器的谱分析下界紧密匹配。
英文摘要
We construct an $\varepsilon$-error PRG for permutation read-once branching programs of length $n$ and width $w$ with seed length \[ O\left((\log w+\log(1/\varepsilon))\cdot \log n\right). \] This gives an exponential improvement in the dependence on $w$ compared with the constructions of De (CCC 2011) and Steinke (ECCC 2012). Compared with the work of Braverman, Rao, Raz, and Yehudayoff (FOCS 2010; SICOMP 2014), which applies more generally to regular branching programs and already achieves the optimal dependence on $w$, our result improves the dependence on the length $n$, attaining the optimal logarithmic dependence. The generator itself is the classical INW PRG of Impagliazzo, Nisan, and Wigderson (STOC 1994). We show that, for permutation branching programs, the INW generator can be instantiated with expanders whose degrees are polynomial in $w$ and $1/\varepsilon$ and, crucially, independent of $n$. To prove this, we analyze error propagation using program-dependent seminorms tailored to the branching program at hand. These seminorms build on the weight function introduced by Braverman et al. The key point is that, when measured in these adapted seminorms, the error does not accumulate throughout the recursion. Since our analysis relies only on the spectral expansion of the underlying expanders, our seed length tightly matches the recent lower bound for spectral analyses of the INW generator due to Hoza, Pyne, and Vadhan (Algorithmica 2024).
