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Optimal Error Pseudodistributions for Read-Once Branching Programs

Authors Eshan Chattopadhyay, Jyun-Jie Liao



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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

Acknowledgements

We thank anonymous reviewers for helpful comments.

Cite AsGet 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

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.

Subject Classification

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

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References

  1. Roy Armoni. On the derandomization of space-bounded computations. In Michael Luby, José D. P. Rolim, and Maria J. Serna, editors, Randomization and Approximation Techniques in Computer Science, Second International Workshop, RANDOM'98, Barcelona, Spain, October 8-10, 1998, Proceedings, volume 1518 of Lecture Notes in Computer Science, pages 47-59. Springer, 1998. URL: https://doi.org/10.1007/3-540-49543-6_5.
  2. Andrej Bogdanov, Zeev Dvir, Elad Verbin, and Amir Yehudayoff. Pseudorandomness for width-2 branching programs. Theory of Computing, 9:283-293, 2013. URL: https://doi.org/10.4086/toc.2013.v009a007.
  3. Allan Borodin, Stephen A. Cook, and Nicholas Pippenger. Parallel computation for well-endowed rings and space-bounded probabilistic machines. Information and Control, 58(1-3):113-136, 1983. URL: https://doi.org/10.1016/S0019-9958(83)80060-6.
  4. Mark Braverman, Gil Cohen, and Sumegha Garg. Hitting sets with near-optimal error for read-once branching programs. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 353-362. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188780.
  5. Mark Braverman, Anup Rao, Ran Raz, and Amir Yehudayoff. Pseudorandom generators for regular branching programs. SIAM J. Comput., 43(3):973-986, 2014. URL: https://doi.org/10.1137/120875673.
  6. Joshua Brody and Elad Verbin. The coin problem and pseudorandomness for branching programs. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 30-39. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/FOCS.2010.10.
  7. Kuan Cheng and William Hoza. Hitting sets give two-sided derandomization of small space. Electronic Colloquium on Computational Complexity (ECCC), 2020. URL: https://eccc.weizmann.ac.il/report/2020/016/.
  8. Benny Chor and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput., 17(2):230-261, 1988. URL: https://doi.org/10.1137/0217015.
  9. Anindya De. Pseudorandomness for permutation and regular branching programs. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC 2011, San Jose, California, USA, June 8-10, 2011, pages 221-231. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/CCC.2011.23.
  10. Oded Goldreich. A sample of samplers: A computational perspective on sampling. In Oded Goldreich, editor, Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation - In Collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman, volume 6650 of Lecture Notes in Computer Science, pages 302-332. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22670-0_24.
  11. Oded Goldreich and Avi Wigderson. Tiny families of functions with random properties: A quality-size trade-off for hashing. Random Struct. Algorithms, 11(4):315-343, 1997. URL: https://doi.org/10.1002/(SICI)1098-2418(199712)11:4<315::AID-RSA3>3.0.CO;2-1.
  12. Venkatesan Guruswami, Christopher Umans, and Salil P. Vadhan. Unbalanced expanders and randomness extractors from parvaresh-vardy codes. J. ACM, 56(4):20:1-20:34, 2009. URL: https://doi.org/10.1145/1538902.1538904.
  13. William Hoza and David Zuckerman. Simple optimal hitting sets for small-success RL. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 59-64. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00015.
  14. William M. Hoza and Chris Umans. Targeted pseudorandom generators, simulation advice generators, and derandomizing logspace. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 629-640. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055414.
  15. Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In Frank Thomson Leighton and Michael T. Goodrich, editors, Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23-25 May 1994, Montréal, Québec, Canada, pages 356-364. ACM, 1994. URL: https://doi.org/10.1145/195058.195190.
  16. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Frank Thomson Leighton and Peter W. Shor, editors, Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 220-229. ACM, 1997. URL: https://doi.org/10.1145/258533.258590.
  17. H. Jung. Relationships between probabilistic and deterministic tape complexity. In Jozef Gruska and Michal Chytil, editors, Mathematical Foundations of Computer Science 1981, Strbske Pleso, Czechoslovakia, August 31 - September 4, 1981, Proceedings, volume 118 of Lecture Notes in Computer Science, pages 339-346. Springer, 1981. URL: https://doi.org/10.1007/3-540-10856-4_101.
  18. Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004. URL: https://doi.org/10.1007/s00037-004-0182-6.
  19. Daniel M. Kane, Jelani Nelson, and David P. Woodruff. Revisiting norm estimation in data streams. CoRR, abs/0811.3648, 2008. URL: http://arxiv.org/abs/0811.3648.
  20. Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31(5):1501-1526, 2002. URL: https://doi.org/10.1137/S0097539700389652.
  21. Michal Koucký, Prajakta Nimbhorkar, and Pavel Pudlák. Pseudorandom generators for group products: extended abstract. In Lance Fortnow and Salil P. Vadhan, editors, Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 263-272. ACM, 2011. URL: https://doi.org/10.1145/1993636.1993672.
  22. Raghu Meka, Omer Reingold, and Avishay Tal. Pseudorandom generators for width-3 branching programs. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 626-637. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316319.
  23. Noam Nisan. Pseudorandom generators for space-bounded computation. Combinatorica, 12(4):449-461, 1992. URL: https://doi.org/10.1007/BF01305237.
  24. Noam Nisan. RL less= SC. Computational Complexity, 4:1-11, 1994. URL: https://doi.org/10.1007/BF01205052.
  25. Noam Nisan and David Zuckerman. Randomness is linear in space. J. Comput. Syst. Sci., 52(1):43-52, 1996. URL: https://doi.org/10.1006/jcss.1996.0004.
  26. Ran Raz and Omer Reingold. On recycling the randomness of states in space bounded computation. In Jeffrey Scott Vitter, Lawrence L. Larmore, and Frank Thomson Leighton, editors, Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1-4, 1999, Atlanta, Georgia, USA, pages 159-168. ACM, 1999. URL: https://doi.org/10.1145/301250.301294.
  27. Omer Reingold, Salil P. Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. Electronic Colloquium on Computational Complexity (ECCC), 8(18), 2001. URL: http://eccc.hpi-web.de/eccc-reports/2001/TR01-018/index.html.
  28. Michael Saks. Randomization and derandomization in space-bounded computation. In Proceedings of Computational Complexity (Formerly Structure in Complexity Theory), pages 128-149. IEEE, 1996. Google Scholar
  29. Michael E. Saks and Shiyu Zhou. BP _hspace(s) subseteq dspace(s^3/2). J. Comput. Syst. Sci., 58(2):376-403, 1999. URL: https://doi.org/10.1006/jcss.1998.1616.
  30. Walter J. Savitch. Relationships between nondeterministic and deterministic tape complexities. J. Comput. Syst. Sci., 4(2):177-192, 1970. URL: https://doi.org/10.1016/S0022-0000(70)80006-X.
  31. Thomas Steinke. Pseudorandomness for permutation branching programs without the group theory. Electronic Colloquium on Computational Complexity (ECCC), 19:83, 2012. URL: http://eccc.hpi-web.de/report/2012/083.
  32. Salil P. Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
  33. David Zuckerman. Randomness-optimal oblivious sampling. Random Struct. Algorithms, 11(4):345-367, 1997. URL: https://doi.org/10.1002/(SICI)1098-2418(199712)11:4<345::AID-RSA4>3.0.CO;2-Z.
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