Weighted Pseudorandom Generators for Read-Once Branching Programs via Weighted Pseudorandom Reductions

TL;DR

This paper introduces weighted pseudorandom generators for read-once branching programs, providing explicit seed lengths and derandomization techniques that improve efficiency and space complexity for various classes of ROBPs.

Contribution

The paper develops new weighted pseudorandom generators and derandomization methods for read-once branching programs, advancing the efficiency and scope of pseudorandomness in this domain.

Findings

Explicit epsilon-WPRG for standard ROBPs with optimized seed length

Seed length bounds for permutation ROBPs with unbounded widths

New derandomization approach achieving optimal space complexity for regular ROBPs

Abstract

We study weighted pseudorandom generators (WPRGs) and derandomizations for read-once branching programs (ROBPs). Denote $n$ and $w$ as the length and the width of a ROBP. We have the following results. For standard ROBPs, we give an explicit $\varepsilon$-WPRG with seed length $$O\left(\frac{\log n\log (nw)}{\max\left\{1,\log\log w-\log\log n\right\}}+\log w \left(\log\log\log w-\log\log\max\left\{2,\frac{\log w}{\log \frac{n}{\varepsilon}}\right\}\right)+\log\frac{1}{\varepsilon}\right).$$ For permutation ROBPs with unbounded widths and single accept nodes, we give an explicit $\varepsilon$-WPRG with seed length $$O\left( \log n\left( \log\log n + \sqrt{\log(1/\varepsilon)} \right)+\log(1/\varepsilon)\right). $$ We also give a new Nisan-Zuckerman style derandomization for regular ROBPs with width $w$, length $n = 2^{O(\sqrt{\log w})}$, and multiple accept nodes. We attain optimal space complexity $O(\log w)$ for arbitrary approximation error $\varepsilon = 1/\text{poly} (w)$. All our results are based on iterative weighted pseudorandom reductions, which can iteratively reduce fooling long ROBPs to fooling short ones.