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Weak Derandomization of Weak Algorithms: Explicit Versions of Yao’s Lemma

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Abstract

A simple averaging argument shows that given a randomized algorithm A and a function f such that for every input x, Pr[A(x) = f(x)] ≥ 1 − ρ (where the probability is over the coin tosses of A), there exists a non-uniform deterministic algorithm B “of roughly the same complexity” such that Pr[B(x) = f(x)] ≥ 1 − ρ (where the probability is over a uniformly chosen input x). This implication is often referred to as “the easy direction of Yao’s lemma” and can be thought of as “weak derandomization” in the sense that B is deterministic but only succeeds on most inputs. The implication follows as there exists a fixed value r′ for the random coins of A such that “hardwiring r′ into A” produces a deterministic algorithm B. However, this argument does not give a way to explicitly construct B.

In this paper, we consider the task of proving uniform versions of the implication above. That is, how to explicitly construct a deterministic algorithm B when given a randomized algorithm A. We prove such derandomization results for several classes of randomized algorithms. These include randomized communication protocols, randomized decision trees (here we improve a previous result by Zimand), randomized streaming algorithms, and randomized algorithms computed by polynomial-size constant-depth circuits.

Our proof uses an approach suggested by Goldreich and Wigderson and “extracts randomness from the input”. We introduce a new type of (seedless) extractors that extract randomness from distributions that are “recognizable” by the given randomized algorithm. We show that such extractors produce randomness that is in some sense not correlated with the input.

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References

  • L. Adleman (1978). Two theorems on random polynomial time. In Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science.

  • M. Ajtai (1990). Approximate counting with uniform constant depth circuits. Advances in computational complexity theory 1–20.

  • Alon N., Matias Y., Szegedy M. (1999) The Space Complexity of Approximating the Frequency Moments. J. Comput. Syst. Sci. 58(1): 137–147

    Article  MathSciNet  MATH  Google Scholar 

  • Babai L., Fortnow L., Nisan N., Wigderson A. (1993) BPP Has Subexponential Time Simulations Unless EXPTIME has Publishable Proofs. Computational Complexity 3(4): 307–318

    Article  MathSciNet  MATH  Google Scholar 

  • Ben-Or M., Linial N. (1989) Collective Coin Flipping. ADVCR: Advances in Computing Research 5: 91–115

    Google Scholar 

  • Blum M., Micali S. (1984) How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits. SIAM Journal on Computing 13(4): 850–864

    Article  MathSciNet  MATH  Google Scholar 

  • Bourgain J. (2005) More on the sum-product phenomenon in prime fields and its applications. International Journal of Number Theory 1: 1–32

    Article  MathSciNet  MATH  Google Scholar 

  • Buhrman H., de Wolf R. (2002) Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci. 288(1): 21–43

    Article  MATH  Google Scholar 

  • B. Chor & O. Goldreich (1988). Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing 17(2), 230–261. ISSN 0097-5397 (print), 1095-7111 (electronic). Special issue on cryptography.

    Google Scholar 

  • B. Chor, O. Goldreich, J. Hastad, J. Friedman, S. Rudich & R. Smolensky (1985). The bit extraction problem or t-resilient functions. In Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, 396–407.

  • A. Cohen & A. Wigderson (1989). Dispersers, deterministic amplification and weak random sources. In Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, 14–25.

  • Y. Dodis, A. Elbaz, R. Oliveira & R. Raz (2004). Improved Randomness Extraction from Two Independent Sources. In RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science, 334–344.

  • Z. Dvir (2009). Extractors for Varieties. In 24th Annual IEEE Conference on Computational Complexity, 102–113. IEEE Computer Society, Washington, DC, USA. ISBN 978-0-7695-3717-7.

  • Dvir Z., Gabizon A., Wigderson A. (2009) Extractors And Rank Extractors For Polynomial Sources. Computational Complexity 18(1): 1–58

    Article  MathSciNet  MATH  Google Scholar 

  • J. Feigenbaum, S. Kannan, M. Strauss & M. Viswanathan (2000). Testing and spot-checking of data streams (extended abstract). In SODA, 165–174.

  • Gabizon A., Raz R., Shaltiel R. (2006) Deterministic Extractors for Bit-Fixing Sources by Obtaining an Independent Seed. SICOMP: SIAM Journal on Computing 36(4): 1072–1094

    Article  MathSciNet  MATH  Google Scholar 

  • O. Goldreich & A. Wigderson (2002). Derandomization That Is Rarely Wrong from Short Advice That Is Typically Good. In Randomization and Approximation Techniques, 6th International Workshop, RANDOM 2002, Cambridge, MA, USA, 209–223.

  • J. Håstad (1986). Almost Optimal Lower Bounds for Small Depth Circuits. In STOC, 6–20. ACM.

  • R. Impagliazzo (2006). Can every randomized algorithm be derandomized? In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, 373–374.

  • R. Impagliazzo, N. Nisan & A. Wigderson (1994). Pseudorandomness for network algorithms. In Proceedings of the 26th ACM Symposium on Theory of Computing.

  • R. Impagliazzo & A. Wigderson (1997). P = BPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, 220–229. El Paso, Texas.

  • R. Impagliazzo & A. Wigderson (1998). Randomness vs. Time: De-randomization under a uniform assumption. In 39th Annual Symposium on Foundations of Computer Science. IEEE.

  • V. Kabanets (2002). Derandomization: A Brief Overview. In Electronic Colloquium on Computational Complexity, technical reports, TR 02-008.

  • Kabanets V., Impagliazzo R. (2004) Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds. Computational Complexity 13(1–2): 1–46

    Article  MathSciNet  MATH  Google Scholar 

  • J. Kamp, A. Rao, S. Vadhan & D. Zuckerman (2006). Deterministic extractors for small-space sources. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 691–700.

  • Kamp J., Zuckerman D. (2007) Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography. SIAM J. Comput 36(5): 1231–1247

    Article  MathSciNet  Google Scholar 

  • J. Kinne, D. van Melkebeek & R. Shaltiel (2009). Pseudorandom Generators and Typically-Correct Derandomization. In APPROX-RANDOM, volume 5687 of Lecture Notes in Computer Science, 574–587. Springer. ISBN 978-3-642-03684-2.

  • A. Klivans (2001). On the Derandomization of Constant Depth Circuits. In RANDOM-APPROX, Michel~X. Goemans, Klaus Jansen, José D. P. Rolim & Luca Trevisan, editors, volume 2129 of Lecture Notes in Computer Science, 249–260. Springer. ISBN 3-540-42470-9.

  • R. König & U. M. Maurer (2005). Generalized Strong Extractors and Deterministic Privacy Amplification. In IMA Int. Conf., 322–339.

  • E. Kushilevitz & N. Nisan (1997). Communication Complexity. Cambridge University Press.

  • P. Bro Miltersen (2001). Derandomizing complexity classes. In Handbook of Randomized Computing, Kluwer, 843–941.

  • E. Mossel & C. Umans (2001). On the complexity of approximating the vc dimension. In Sixteenth Annual IEEE Conference on Computational Complexity, 220–225.

  • Newman I. (1991) Private vs. Common Random Bits in Communication Complexity. Inf. Process. Lett. 39(2): 67–71

    Article  MATH  Google Scholar 

  • Nisan N. (1991) CREW PRAMs and Decision Trees. SIAM J. Comput. 20(6): 999–1007

    Article  MathSciNet  MATH  Google Scholar 

  • Nisan N. (1992) Pseudorandom generators for space bounded computation. Combinatorica 12(4): 449–461

    Article  MathSciNet  MATH  Google Scholar 

  • N. Nisan & A. Ta-Shma (1999). Extracting Randomness: A Survey and New Constructions. JCSS: Journal of Computer and System Sciences 58.

  • Nisan N., Wigderson A. (1994) Hardness vs Randomness. Journal of Computer and System Sciences 49(2): 149–167

    Article  MathSciNet  MATH  Google Scholar 

  • Nisan N., Zuckerman D. (1996) Randomness is Linear in Space. Journal of Computer and System Sciences 52(1): 43–52

    Article  MathSciNet  MATH  Google Scholar 

  • Radhakrishnan J., Ta-Shma A. (2000) Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators. SIAM Journal on Discrete Mathematics 13(1): 2–24

    Article  MathSciNet  MATH  Google Scholar 

  • A. Rao (2009). Extractors for Low-Weight Affine Sources. In IEEE Conference on Computational Complexity, 95–101. IEEE Computer Society. ISBN 978-0-7695-3717-7.

  • R. Raz & O. Reingold (1999). On Recycling the Randomness of States in Space Bounded Computation. In Proceedings of the 31st ACM Symposium on Theory of Computing, 159–168.

  • O. Reingold (2008). Undirected connectivity in log-space. J. ACM 55(4).

  • Santha M., Vazirani U.V. (1986) Generating Quasi-Random Sequences from Semi-Random Sources. Journal of Computer and System Sciences 33: 75–87

    Article  MathSciNet  MATH  Google Scholar 

  • R. Shaltiel (2002). Bulletin of the EATCS 77, 67–95.

    Google Scholar 

  • R. Shaltiel (2006). How to Get More Mileage from Randomness Extractors. In CCC ’06: Proceedings of the 21st Annual IEEE Conference on Computational Complexity, 46–60.

  • R. Shaltiel & C. Umans (2001). Simple extractors for all min-entropies and a new pseudo-random generator. In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science.

  • M. Sudan, L. Trevisan & S. Vadhan (1999). Pseudorandom generators without the XOR Lemma. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing.

  • L. Trevisan & S. Vadhan (2000). Extracting Randomness from samplable distributions. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science.

  • Trevisan L., Vadhan S.P. (2007) Pseudorandomness and Average-Case Complexity Via Uniform Reductions. Computational Complexity 16(4): 331–364

    Article  MathSciNet  MATH  Google Scholar 

  • C. Umans (2002). Pseudo-random generators for all hardnesses. In Proceedings of the Thirty-fourth Annual ACM Symposium on the Theory of Computing.

  • S. Vadhan (2002). Randomness Extractors and their Many Guises. In Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 9–12.

  • Vazirani U. (1987) Strong communication complexity or generating quasi-random sequences from two communicating semi-random sources. Combinatorica 7: 375–392

    Article  MathSciNet  MATH  Google Scholar 

  • A. C. Yao (1979). Some Complexity Questions Related to Distributive Computing (Preliminary Report). In Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing, 30 April–2 May, 1979, Atlanta, Georgia, USA, 209–213.

  • A. C. Yao (1982). Theory and Applications of Trapdoor Functions (Extended Abstract). In 23rd Annual Symposium on Foundations of Computer Science, 80–91. IEEE, Chicago, Illinois.

  • A. C. Yao (1983). Lower Bounds by Probabilistic Arguments (Extended Abstract). In 24th Annual Symposium on Foundations of Computer Science, 7-9 November 1983, Tucson, Arizona, USA, 420–428.

  • M. Zimand (2006). Exposure-Resilient Extractors. In IEEE Conference on Computational Complexity, 61–72.

  • M. Zimand (2007). On Derandomizing Probabilistic Sublinear-Time Algorithms. In IEEE Conference on Computational Complexity, 1–9.

  • Zuckerman D. (1997) Randomness-Optimal Oblivious Sampling. Random Structures and Algorithms 11: 345–367

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Computer Science, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Ronen Shaltiel

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  1. Ronen Shaltiel
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Shaltiel, R. Weak Derandomization of Weak Algorithms: Explicit Versions of Yao’s Lemma. comput. complex. 20, 87–143 (2011). https://doi.org/10.1007/s00037-011-0006-4

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