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Typically-Correct Derandomization for Small Time and Space

Author William M. Hoza

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Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Probabilistic computation
  • Theory of computation → Complexity classes
Keywords
  • Derandomization
  • pseudorandomness
  • space complexity

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Abstract

Suppose a language L can be decided by a bounded-error randomized algorithm that runs in space S and time n * poly(S). We give a randomized algorithm for L that still runs in space O(S) and time n * poly(S) that uses only O(S) random bits; our algorithm has a low failure probability on all but a negligible fraction of inputs of each length. As an immediate corollary, there is a deterministic algorithm for L that runs in space O(S) and succeeds on all but a negligible fraction of inputs of each length. We also give several other complexity-theoretic applications of our technique.

Cite As Get BibTex

William M. Hoza. Typically-Correct Derandomization for Small Time and Space. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 9:1-9:39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CCC.2019.9

Author Details

William M. Hoza
  • Department of Computer Science, University of Texas at Austin, USA

Funding

  • Hoza, William M.: Supported by the NSF GRFP under Grant DGE-1610403 and by a Harrington Fellowship from UT Austin.

Acknowledgements

We thank Michael Forbes, Scott Aaronson, David Zuckerman, Adam Klivans, and Anna Gál for helpful comments on an early draft of this paper. We thank Amnon Ta-Shma, Lijie Chen, Chris Umans, David Zuckerman, Adam Klivans, Anna Gál, Gil Cohen, Shachar Lovett, Oded Goldreich, and Avi Wigderson for helpful discussions.

References

  1. Leonard Adleman. Two theorems on random polynomial time. In Proceedings of the 19th Annual Symposium on Foundations of Computer Science (FOCS '78), pages 75-83. IEEE, 1978. Google Scholar
  2. Eric Allender, Klaus Reinhardt, and Shiyu Zhou. Isolation, matching, and counting uniform and nonuniform upper bounds. Journal of Computer and System Sciences, 59(2):164-181, 1999. URL: https://doi.org/10.1006/jcss.1999.1646.
  3. Josh Alman. An illuminating algorithm for the light bulb problem. In 2nd Symposium on Simplicity in Algorithms, volume 69 of OASIcs OpenAccess Ser. Inform., pages Art. No. 2, 11. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019. Google Scholar
  4. Roy Armoni. On the derandomization of space-bounded computations. In Proceedings of the 2nd International Workshop on Randomization and Computation (RANDOM '98), volume 1518 of Lecture Notes in Computer Science, pages 47-59. Springer, Berlin, 1998. URL: https://doi.org/10.1007/3-540-49543-6_5.
  5. Vikraman Arvind and Jacobo Toran. Solvable group isomorphism is (almost) in NP ∩ coNP. In Proceedings of the 19th Annual Conference on Computational Complexity (CCC '04), pages 91-103. IEEE, 2004. Google Scholar
  6. László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3(4):307-318, 1993. URL: https://doi.org/10.1007/BF01275486.
  7. László Babai, Noam Nisan, and Márió Szegedy. Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. Journal of Computer and System Sciences, 45(2):204-232, 1992. URL: https://doi.org/10.1016/0022-0000(92)90047-M.
  8. Paul Beame, Michael Saks, Xiaodong Sun, and Erik Vee. Time-space trade-off lower bounds for randomized computation of decision problems. Journal of the ACM, 50(2):154-195, 2003. URL: https://doi.org/10.1145/636865.636867.
  9. Manuel Blum and Silvio Micali. How to generate cryptographically strong sequences of pseudorandom bits. SIAM Journal on Computing, 13(4):850-864, 1984. URL: https://doi.org/10.1137/0213053.
  10. Jin-Yi Cai, Venkatesan T. Chakaravarthy, and Dieter van Melkebeek. Time-space tradeoff in derandomizing probabilistic logspace. Theory of Computing Systems, 39(1):189-208, 2006. URL: https://doi.org/10.1007/s00224-005-1264-9.
  11. Scott Diehl and Dieter van Melkebeek. Time-space lower bounds for the polynomial-time hierarchy on randomized machines. SIAM Journal on Computing, 36(3):563-594, 2006. URL: https://doi.org/10.1137/050642228.
  12. Lance Fortnow and Adam R. Klivans. Linear advice for randomized logarithmic space. In Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS '06), volume 3884 of Lecture Notes in Computer Science, pages 469-476. Springer, Berlin, 2006. URL: https://doi.org/10.1007/11672142_38.
  13. Lance Fortnow and Dieter van Melkebeek. Time-space tradeoffs for nondeterministic computation. In Proceedings of the 15th Annual Conference on Computational Complexity (CCC '00), pages 2-13. IEEE, 2000. URL: https://doi.org/10.1109/CCC.2000.856730.
  14. David Gillman. A Chernoff bound for random walks on expander graphs. SIAM Journal on Computing, 27(4):1203-1220, 1998. URL: https://doi.org/10.1137/S0097539794268765.
  15. Oded Goldreich and Avi Wigderson. Derandomization that is rarely wrong from short advice that is typically good. In Randomization and approximation techniques in computer science (RANDOM '02), volume 2483 of Lecture Notes in Computer Science, pages 209-223. Springer, Berlin, 2002. URL: https://doi.org/10.1007/3-540-45726-7_17.
  16. Venkatesan Guruswami, Christopher Umans, and Salil Vadhan. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. Journal of the ACM, 56(4):Art. 20, 34, 2009. URL: https://doi.org/10.1145/1538902.1538904.
  17. Dan Gutfreund and Emanuele Viola. Fooling parity tests with parity gates. In Proceedings of the 8th International Workshop on Randomization and Computation (RANDOM '04), volume 3122 of Lecture Notes in Computer Science, pages 381-392. Springer, 2004. Google Scholar
  18. Tzvika Hartman and Ran Raz. On the distribution of the number of roots of polynomials and explicit weak designs. Random Structures & Algorithms, 23(3):235-263, 2003. URL: https://doi.org/10.1002/rsa.10095.
  19. Lane A Hemaspaandra and Ryan Williams. SIGACT news complexity theory column 76: an atypical survey of typical-case heuristic algorithms. ACM SIGACT News, 43(4):70-89, 2012. Google Scholar
  20. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proceedings of the 29th Annual Symposium on Theory of Computing (STOC '97), pages 220-229, New York, NY, USA, 1997. ACM. URL: https://doi.org/10.1145/258533.258590.
  21. Neil D. Jones. Space-bounded reducibility among combinatorial problems. Journal of Computer and System Sciences, 11(1):68-85, 1975. URL: https://doi.org/10.1016/S0022-0000(75)80050-X.
  22. Daniel M Kane, Jelani Nelson, and David P Woodruff. Revisiting norm estimation in data streams. arXiv preprint, 2008. URL: http://arxiv.org/abs/0811.3648.
  23. Neeraj Kayal and Nitin Saxena. On the ring isomorphism & automorphism problems. In Proceedings of the 20th Annual Conference on Computational Complexity (CCC '05), pages 2-12. IEEE, 2005. Google Scholar
  24. Jeff Kinne, Dieter van Melkebeek, and Ronen Shaltiel. Pseudorandom generators, typically-correct derandomization, and circuit lower bounds. Computational Complexity, 21(1):3-61, 2012. URL: https://doi.org/10.1007/s00037-011-0019-z.
  25. Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing, 31(5):1501-1526, 2002. URL: https://doi.org/10.1137/S0097539700389652.
  26. Noam Nisan. Pseudorandom generators for space-bounded computation. Combinatorica, 12(4):449-461, 1992. URL: https://doi.org/10.1007/BF01305237.
  27. Noam Nisan. On read-once vs. multiple access to randomness in logspace. Theoretical Computer Science, 107(1):135-144, 1993. URL: https://doi.org/10.1016/0304-3975(93)90258-U.
  28. Noam Nisan. RL ⊆ SC. Computational Complexity, 4(1):1-11, 1994. URL: https://doi.org/10.1007/BF01205052.
  29. Noam Nisan and Avi Wigderson. Hardness vs. randomness. Journal of Computer and System Sciences, 49(2):149-167, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80043-1.
  30. Noam Nisan and David Zuckerman. Randomness is linear in space. Journal of Computer and System Sciences, 52(1):43-52, 1996. URL: https://doi.org/10.1006/jcss.1996.0004.
  31. Klaus Reinhardt and Eric Allender. Making nondeterminism unambiguous. SIAM Journal on Computing, 29(4):1118-1131, 2000. URL: https://doi.org/10.1137/S0097539798339041.
  32. Michael Saks and Shiyu Zhou. BP_H SPACE(S) ⊆ DSPACE(S^3/2). Journal of Computer and System Sciences, 58(2):376-403, 1999. Google Scholar
  33. Rahul Santhanam and Ryan Williams. On uniformity and circuit lower bounds. Computational Complexity, 23(2):177-205, 2014. URL: https://doi.org/10.1007/s00037-014-0087-y.
  34. Ronen Shaltiel. Typically-correct derandomization. ACM SIGACT News, 41(2):57-72, 2010. Google Scholar
  35. Ronen Shaltiel. Weak derandomization of weak algorithms: explicit versions of Yao’s lemma. Computational Complexity, 20(1):87-143, 2011. URL: https://doi.org/10.1007/s00037-011-0006-4.
  36. Ronen Shaltiel and Christopher Umans. Simple extractors for all min-entropies and a new pseudorandom generator. Journal of the ACM, 52(2):172-216, 2005. URL: https://doi.org/10.1145/1059513.1059516.
  37. Madhu Sudan, Luca Trevisan, and Salil Vadhan. Pseudorandom generators without the XOR lemma. Journal of Computer and System Sciences, 62(2):236-266, 2001. Special issue on the 14th Annual Conference on Computational Complexity (CCC '99). URL: https://doi.org/10.1006/jcss.2000.1730.
  38. Salil P. Vadhan. Pseudorandomness. Foundations and Trends® in Theoretical Computer Science, 7(1–3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
  39. J. H. van Lint. Introduction to coding theory, volume 86 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, third edition, 1999. URL: https://doi.org/10.1007/978-3-642-58575-3.
  40. Dieter van Melkebeek and Gautam Prakriya. Derandomizing Isolation in Space-Bounded Settings. In 32nd Annual Conference on Computational Complexity (CCC '17), volume 79 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1-5:32, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.5.
  41. Dieter van Melkebeek and Rahul Santhanam. Holographic proofs and derandomization. SIAM Journal on Computing, 35(1):59-90, 2005. URL: https://doi.org/10.1137/S0097539703438629.
  42. Andrew C. Yao. Theory and applications of trapdoor functions. In 23rd Annual Symposium on Foundations of Computer Science (FOCS '82), pages 80-91. IEEE, New York, 1982. Google Scholar
  43. Marius Zimand. Exposure-resilient extractors and the derandomization of probabilistic sublinear time. Computational Complexity, 17(2):220-253, 2008. URL: https://doi.org/10.1007/s00037-008-0243-3.
  44. David Zuckerman. Randomness-optimal oblivious sampling. Random Structures & 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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