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Simulating BPP using a general weak random source

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Abstract

We show how to simulate BPP and approximation algorithms in polynomial time using the output from a δ-source. A δ-source is a weak random source that is asked only once forR bits, and must output anR-bit string according to some distribution that places probability no more than 2−δR on any particular string. We also give an application to the unapproximability of MAX CLIQUE.

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References

  1. N. Alon and V. D. Milman, λ1, Isoperimetric Inequalities for Graphs and Superconcentrators,J. Combin. Theory Ser. B, 38:73–88, 1985.

    Google Scholar 

  2. S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy, Proof Verification and Intractibility of Approximation Problems,Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, 1992, pp. 14–23.

  3. S. Arora and S. Safra, Approximating Clique is NP-Complete,Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, 1992, pp. 2–13.

  4. L. Babai, L. Fortnow, and C. Lund, Non-Deterministic Exponential Time has Two-Prover Interactive Protocols,Comput. Complexity, 1:16–25, 1991.

    Google Scholar 

  5. M. Bellare and M. Sudan, Improved Non-Approximability Results,Proceedings of the 26th ACM Symposium on Theory of Computing, 1994, pp. 184–193.

  6. C. H. Bennett, G. Brassard, and J.-M. Robert, Privacy Amplification by Public Discussion,SIAM J. Comput., 17(2):210–229, 1988.

    Google Scholar 

  7. M. Ben-Or and N. Linial, Collective Coin Flipping Robust Voting Schemes and Minimal Banzhaf Values, inAdvances in Computing Research 5: Randomness and Computation, S. Micali, ed., JAI Press, Greenwich, CT, 1989, pp. 91–115.

    Google Scholar 

  8. M. Blum, Independent Unbiased Coin Flips from a Correlated Biased Source: a Finite Markov Chain,Combinatorica, 6(2):97–108, 1986.

    Google Scholar 

  9. L. Carter and M. Wegman, Universal Classes of Hash Functions,J. Comp. Syst. Sci., 18(2): 143–154, 1979.

    Google Scholar 

  10. B. Chor and O. Goldreich, Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity,SIAM J. Comput., 17(2):230–261, 1988.

    Google Scholar 

  11. B. Chor and O. Goldreich, On the Power of Two-Point Based Sampling,J. Complexity, 5:96–106, 1989.

    Google Scholar 

  12. B. Chor, O. Goldreich, J. Hastad, J. Friedman, S. Rudich, and R. Smolensky, The Bit Extraction Problem ort-Resilient Functions,Proceedings of the 26th Annual Symposium on Foundations of Computer Science, 1985, pp. 396–407.

  13. A. Cohen and A. Wigderson, Dispersers, Deterministic Amplification, and Weak Random Sources,Proceedings of the 30th Annual Symposium on Foundations of Computer Science, 1989, pp. 14–19.

  14. M. Dyer, A. Frieze, and R. Kannan, A Random Polynomial Time Algorithm for Approximating the Volume of a Convex Body,J. Assoc. Comput. Mach., 38:1–17, 1991.

    Google Scholar 

  15. P. Elias, The Efficient Construction of an Unbiased Random Sequence,Ann. Math. Statist., 43(3):865–870, 1972.

    Google Scholar 

  16. U. Feige, S. Goldwasser, L. Lovasz, S. Safra, and M. Szegedy, Approximating Clique is Almost NP-Complete,Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, 1991, pp. 2–12.

  17. L. Fortnow, J. Rompel, and M. Sipser, On the Power of Multi-Prover Interactive Protocols,Theoret. Comput. Sci. A, 134:545–557, 1994.

    Google Scholar 

  18. J. Friedman and A. Wigderson, On the Second Eigenvalue of Hypergraphs, Technical Report CS-TR-232-89, Princeton University, 1989.

  19. O. Gabber and Z. Galil, Explicit Constructions of Linear-Sized Supercomputers,J. Comput. System Sci., 22:407–420, 1981.

    Google Scholar 

  20. R. Impagliazzo, L. Levin, and M. Luby, Pseudo-Random Generation from One-Way Functions,Proceedings of the 21st ACM Symposium on Theory of Computing, 1989, pp. 12–24.

  21. R. Impagliazzo and D. Zuckerman, How To Recycle Random Bits,Proceedings of the 30th Annual Symposium on Foundations of Computer Science, 1989, pp. 248–253.

  22. R. M. Karp, Reducibility Among Combinatorial Problems, inComplexity of Computer Computations, R. E. Miller and J. W. Thatcher, eds., Plenum, New York, 1972, pp. 85–103.

    Google Scholar 

  23. D. Lichtenstein, N. Linial, and M. Saks, Some Extremal Problems Arising from Discrete Control Processes,Combinatorica, 9:269–287, 1989.

    Google Scholar 

  24. N. Linial, M. Luby, M. Saks, and D. Zuckerman, Efficient Construction of a Small Hitting Set for Combinatorial Rectangles in High Dimension,Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 258–267.

  25. N. Nisan and D. Zuckerman, Randomness Is Linear in Space,J. Comput. System Sci., 52(1):43–52, 1996.

    Google Scholar 

  26. M. O. Rabin, Probabilistic Algorithm for Testing Primality,J. Number Theory, 12:128–138, 1980.

    Google Scholar 

  27. A. Sahay and M. Sudan, personal communication.

  28. M. Saks, A. Srinivasan, and S. Zhou, Explicit Dispersers with Polylog Degree,Proceedings of the 27th ACM Symposium on Theory of Computing, 1995, to appear.

  29. M. Santha, On Using Deterministic Functions in Probabilistic Algorithms,Inform. and Comput., 74(3):241–249, 1987.

    Google Scholar 

  30. M. Santha and U. Vazirani, Generating Quasi-Random Sequences from Slightly Random Sources,J. Comput. System Sci., 33:75–87, 1986.

    Google Scholar 

  31. M. Sipser, Expanders, Randomness, or Time Versus Space,J. Comput. System Sci., 36:379–383, 1988.

    Google Scholar 

  32. A. Srinivasan and D. Zuckerman, Computing with Very Weak Random Sources,Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 264–275.

  33. R. M. Tanner, Explicit Construction of Concentrators from GeneralizedN-gons,SIAM J. Algebraic Discrete Methods, 5:287–293, 1984.

    Google Scholar 

  34. U. Vazirani, Randomness, Adversaries and Computation, Ph.D. Thesis, University of California, Berkeley, 1986.

    Google Scholar 

  35. U. Vazirani, Efficiency Considerations in Using Semi-Random Sources,Proceedings of the 19th ACM Symposium on Theory of Computing, 1987, pp. 160–168.

  36. U. Vazirani, Strong Communication Complexity or Generating Quasi-Random Sequences from Two Communicating Semi-Random Sources,Combinatorica, 7(4):375–392, 1987.

    Google Scholar 

  37. U. Vazirani and V. Vazirani, Random Polynomial Time is Equal to Semi-Random Polynomial Time,Proceedings of the 26th Annual Symposium on Foundations of Computer Science, 1985, pp. 417–428.

  38. J. von Neumann,Various Techniques Used in Connection with Random Digits, Notes by G. E. Forsythe, National Bureau of Standards, Applied Math Series, Vol. 12, Pitman, Boston, MA, 1951, pp. 36–38. Reprinted inVon Neumann's Collected Works, Vol. 5, Pergamon, New York, 1963, pp. 768–770.

    Google Scholar 

  39. A. Wigderson and D. Zuckerman, Expanders that Beat the Eigenvalue Bound: Explicit Construction and Applications,Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 245–251.

  40. D. Zuckerman, General Weak Random Sources,Proceedings of the 31st Annual Symposium on Foundations of Computer Science, 1990, pp. 534–543.

  41. D. Zuckerman, Simulating BPP Using a General Weak Random Source,Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, 1991, pp. 79–89.

  42. D. Zuckerman, NP-Complete Problems Have a Version That's Hard To Approximate,Proceedings of the 8th IEEE Conference on Structure in Complexity Theory, 1993, pp. 305–312.

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Authors and Affiliations

  1. Department of Computer Sciences, The University of Texas at Austin, 78712, Austin, TX, USA

    D. Zuckerman

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Communicated by M. Luby.

This paper appeared in preliminary form in theProceedings of the 32nd Annual Symposium on Foundations of Computer Science, 1991, pp. 79–89. Most of this research was done while the author was at U.C. Berkeley, and supported by an AT&T Graduate Fellowship, NSF PYI Grant No. CCR-8896202, and NSF Grant No. IRI-8902813. Part of this research was done while the author was at MIT, supported by an NSF Postdoctoral Fellowship, NSF Grant No. 92-12184 CCR, and DARPA Grant No. N00014-92-J-1799. Part of this research was done at UT Austin, where the author was supported by NSF NYI Grant No. CCR-9457799.

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Zuckerman, D. Simulating BPP using a general weak random source. Algorithmica 16, 367–391 (1996). https://doi.org/10.1007/BF01940870

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