Optimal Hitting Sets for Combinatorial Shapes

  • Aditya Bhaskara
  • Devendra Desai
  • Srikanth Srinivasan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)

Abstract

We consider the problem of constructing explicit Hitting sets for Combinatorial Shapes, a class of statistical tests first studied by Gopalan, Meka, Reingold, and Zuckerman (STOC 2011). These generalize many well-studied classes of tests, including symmetric functions and combinatorial rectangles. Generalizing results of Linial, Luby, Saks, and Zuckerman (Combinatorica 1997) and Rabani and Shpilka (SICOMP 2010), we construct hitting sets for Combinatorial Shapes of size polynomial in the alphabet, dimension, and the inverse of the error parameter. This is optimal up to polynomial factors. The best previous hitting sets came from the Pseudorandom Generator construction of Gopalan et al., and in particular had size that was quasipolynomial in the inverse of the error parameter.

Our construction builds on natural variants of the constructions of Linial et al. and Rabani and Shpilka. In the process, we construct fractional perfect hash families and hitting sets for combinatorial rectangles with stronger guarantees. These might be of independent interest.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: 20th Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, October 29-31, pp. 29–31. IEEE (1979)Google Scholar
  2. 2.
    Alon, N., Feige, U., Wigderson, A., Zuckerman, D.: Derandomized graph products. Computational Complexity 5, 60–75 (1995), doi:10.1007/BF01277956MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Armoni, R., Saks, M., Wigderson, A., Zhou, S.: Discrepancy sets and pseudorandom generators for combinatorial rectangles. In: 37th Annual Symposium on Foundations of Computer Science, Burlington, VT, pp. 412–421. IEEE Comput. Soc. Press, Los Alamitos (1996)Google Scholar
  5. 5.
    Blum, A., Kalai, A., Wasserman, H.: Noise-tolerant learning, the parity problem, and the statistical query model. J. ACM 50(4), 506–519 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Even, G., Goldreich, O., Luby, M., Nisan, N., Veličković, B.: Efficient approximation of product distributions. Random Structures Algorithms 13(1), 1–16 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2. Wiley (1971)Google Scholar
  8. 8.
    Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with 0(1) worst case access time. J. ACM 31(3), 538–544 (1984)MATHCrossRefGoogle Scholar
  9. 9.
    Gopalan, P., Meka, R., Reingold, O., Zuckerman, D.: Pseudorandom generators for combinatorial shapes. In: STOC, pp. 253–262 (2011)Google Scholar
  10. 10.
    Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bulletin of the AMS 43(4), 439–561 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, El Paso, Texas, May 4-6, pp. 220–229 (1997)Google Scholar
  12. 12.
    Linial, N., Luby, M., Saks, M., Zuckerman, D.: Efficient construction of a small hitting set for combinatorial rectangles in high dimension. Combinatorica 17, 215–234 (1997), doi:10.1007/BF01200907MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Lovett, S., Reingold, O., Trevisan, L., Vadhan, S.: Pseudorandom Bit Generators That Fool Modular Sums. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX and RANDOM 2009. LNCS, vol. 5687, pp. 615–630. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Lu, C.-J.: Hyper-encryption against space-bounded adversaries from on-line strong extractors, pp. 257–271Google Scholar
  15. 15.
    Meka, R., Zuckerman, D.: Small-bias spaces for group products. These proceedings (2009)Google Scholar
  16. 16.
    Moser, R.A., Tardos, G.: A constructive proof of the general lovász local lemma. J. ACM 57(2) (2010)Google Scholar
  17. 17.
    Naor, J., Naor, M.: Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing 22(4), 838–856 (1993)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Nisan, N.: Pseudorandom generators for space-bounded computation. Combinatorica 12(4), 449–461 (1992)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Nisan, N., Wigderson, A.: Hardness vs. randomness. J. Comput. Syst. Sci. 49(2), 149–167 (1994)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Nisan, N., Zuckerman, D.: Randomness is linear in space. Journal of Computer and System Sciences 52(1), 43–52 (1996)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Rabani, Y., Shpilka, A.: Explicit construction of a small epsilon-net for linear threshold functions. SIAM J. Comput. 39(8), 3501–3520 (2010)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Schmidt, J.P., Siegel, A.: The analysis of closed hashing under limited randomness (extended abstract). In: STOC, pp. 224–234 (1990)Google Scholar
  23. 23.
    Shaltiel, R., Umans, C.: Pseudorandomness for approximate counting and sampling. Computational Complexity 15(4), 298–341 (2006)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Aditya Bhaskara
    • 1
  • Devendra Desai
    • 2
  • Srikanth Srinivasan
    • 3
  1. 1.Department of Computer SciencePrinceton UniversityUSA
  2. 2.Department of Computer ScienceRutgers UniversityUSA
  3. 3.DIMACSRutgers UniversityUSA

Personalised recommendations