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Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs

Authors: Thomas Steinke, Salil Vadhan, and Andrew Wan

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


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.

Cite as

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)


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@InProceedings{steinke_et_al:LIPIcs.APPROX-RANDOM.2014.885,
  author =	{Steinke, Thomas and Vadhan, Salil and Wan, Andrew},
  title =	{{Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{885--899},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.885},
  URN =		{urn:nbn:de:0030-drops-47456},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.885},
  annote =	{Keywords: Pseudorandomness, Branching Programs, Discrete Fourier Analysis}
}
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