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Optimal Error Pseudodistributions for Read-Once Branching Programs

Authors Eshan Chattopadhyay, Jyun-Jie Liao

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Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Derandomization
  • explicit constructions
  • space-bounded computation

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Abstract

In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length n and width w read-once branching programs with seed length O(log n⋅ log(nw)+log n⋅log(1/ε)) and error ε. It remains a central question to reduce the seed length to O(log (nw/ε)), which would prove that 𝐁𝐏𝐋 = 𝐋. However, there has been no improvement on Nisan’s construction for the case n = w, which is most relevant to space-bounded derandomization.
Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit construction of a PRPD with seed length Õ(log n⋅ log(nw)+log(1/ε)). A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing 𝐁𝐏𝐋 and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length O(log n⋅ log(nw)+log(1/ε)), but their techniques are restricted to hitting sets.
In this work, we construct a PRPD with seed length O(log n⋅ log (nw)⋅ log log(nw)+log(1/ε)). This improves upon the construction by Braverman, Cogen and Garg by a O(log log(1/ε)) factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg.

Cite As Get BibTex

Eshan Chattopadhyay and Jyun-Jie Liao. Optimal Error Pseudodistributions for Read-Once Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 25:1-25:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CCC.2020.25

Author Details

Eshan Chattopadhyay
  • Department of Computer Science, Cornell University, Ithaca, NY, USA
Jyun-Jie Liao
  • Department of Computer Science, Cornell University, Ithaca, NY, USA

Funding

  • Chattopadhyay, Eshan: Supported by NSF grant CCF-1849899.
  • Liao, Jyun-Jie: Supported by NSF grant CCF-1849899.

Acknowledgements

We thank anonymous reviewers for helpful comments.

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