LIPIcs.APPROX-RANDOM.2014.885.pdf
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Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Randomization and Computation (RANDOM)
Conference: International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) - License:
Creative Commons Attribution 3.0 Unported license
- Publication date: 2014-09-04
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Thomas Steinke, Salil Vadhan, and Andrew Wan. Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 885-899, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.885
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.885
Abstract
We present an explicit pseudorandom generator for oblivious, read-once, width-3 branching programs, which can read their input bits in any order. The generator has seed length O~( log^3 n ).
The previously best known seed length for this model is n^{1/2+o(1)} due to Impagliazzo, Meka, and Zuckerman (FOCS'12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM'13) for permutation branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any f : {0,1}^n -> {0,1} computed by such a branching program, and k in [n], sum_{|s|=k} |hat{f}(s)| < n^2 * (O(\log n))^k,
where f(x) = sum_s hat{f}(s) (-1)^<s,x> is the standard Fourier transform over Z_2^n. The base O(log n) of the Fourier growth is tight up to a factor of log log n.
Keywords
- Pseudorandomness
- Branching Programs
- Discrete Fourier Analysis
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