Sample spaces with small bias on neighborhoods and error-correcting communication protocols

  • Michael Saks
  • Shiyu Zhou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1269)


We give a deterministic algorithm which, on input an integer n, collection f of subsets of {1,2,⋯, n} and ∃ (0,1) runs in time polynomial in nf‖/∈ and produces a ±1-matrix M with n columns and m=O(log ‖f‖/∈2) rows with the following property: for any subset Ff, the fraction of 1's in the n-vector obtained by coordinate-wise multiplication of the column vectors indexed by F is between (1−∈)/2 and (1+)/2. In the case that f is the set of all subsets of size at most k, k constant, this gives a polynomial time construction for a k-wise -biased sample space of size O(log n/∈2), as compared to the best previous constructions of [NN90] and [AGHP91] which were, respectively, O(log n/∈4) and O((log n)2/2). The number of rows in the construction matches the upper bound given by the probabilistic existence argument. Such constructions are of interest for derandomizing algorithms. As an application, we present a family of essentially optimal deterministic communication protocols for the problem of checking the consistency of two files.


Sample Space Explicit Construction Deterministic Algorithm Support Size Efficient Construction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [Alo86]
    N. Alon, Explicit constructions of exponential sized families of k-independent sets, Discrete Math, 58, pp 191–193, 1986.Google Scholar
  2. [Alo91]
    N. Alon, A parallel algorithm version of the Local Lemma, Random Structures and Algorithms 2(4), pp 367–378, 1991.Google Scholar
  3. [ABI86]
    N. Alon, L. Babai, A. Itai, A fast and simple randomized parallel algorithm for the Maximal Independent Set Problem, J. Algorithms 7, pp 567–583, 1986.Google Scholar
  4. [ABNNR92]
    N. Alon, J. Bruck, J. Naor, M. Naor, R. Roth, Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs, IEEE Trans. on Info. Theory, Vol 38(2), pp 509–515, 1992.Google Scholar
  5. [AGHP91]
    N. Alon, O. Goldreich, J. Hastad, R. Peralta, Simple constructions of almost k-wise independent random variables, Random Structures and Algorithms 3(3), pp 289–303, 1992.Google Scholar
  6. [AS91]
    N. Alon, J. Spencer, The Probabilistic Method, Wiley, 1991.Google Scholar
  7. [BR89]
    B. Berger, J. Rompel, Simulating (log c n)-wise independence in NC, J. Assoc. Comput. Mach. 38(4), pp 1026–1046, 1991.Google Scholar
  8. [CGHFRS85]
    B. Chor, O. Goldreich, J. Hastad, J. Friedman, S. Rudich, R. Smolensky, The bit extraction problem or t-resilient functions, 26th IEEE FOCS, pp 396–407, 1985.Google Scholar
  9. [CRS94]
    S. Chari, P. Rohatgi, A. Srinivasan, Improved algorithms via approximations of probability distributions, 26th ACM STOC, pp 584–592, 1994.Google Scholar
  10. [EGLNV92]
    G. Even, O. Goldreich, M. Luby, N. Nisan, B. Velickovic, Approximations of general independent distributions, 24th ACM STOC, pp 10–16, 1992.Google Scholar
  11. [ES73]
    P. Erdös, J. Selfridge, On a combinatorial game, Journal of Combinatorial Theory, Series A 14, pp 298–301, 1973.Google Scholar
  12. [Fun94]
    A. Fundia, Derandomizing Chebyshev's inequality to find independent sets in uncrowded hypergraghs, Random Structures and Algorithms, to appear.Google Scholar
  13. [KK94]
    D. Karger and D. Koller, (De)randomized construction of small sample spaces in NC, 35th IEEE FOCS, pp 252–263, 1994.Google Scholar
  14. [KM94]
    H. Karloff and Y. Mansour, On construction of k-wise independent random variables, Proceedings of the 26th Annual ACM Symposium on Theory of Computing, 1994.Google Scholar
  15. [KW84]
    R. Karp and A. Wigderson, A fast parallel algorithm for the maximal independent set problem, Proceedings of the 16th Annual ACM Symposium on Theory of Computing, 1984.Google Scholar
  16. [KM93]
    D. Koller and N. Megiddo, Finding small sample spaces satisfying given constraints, Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pp 268–277, 1993.Google Scholar
  17. [KN95]
    Kuselevitz and N. Nisan, Communication Complexity.Google Scholar
  18. [LLSZ95]
    N. Linial, M. Luby, M. Saks, D. Zuckerman, Efficient construction of a small hitting set for combinatorial rectangles in high dimension, 25th ACM STOC, pp 258–267, 1993.Google Scholar
  19. [Lub85]
    M. Luby, A simple parallel algorithm for the maximal independent set problem, SIAM J. Comput. 15(4), pp 1036–1053, 1986.Google Scholar
  20. [MNN89]
    R. Motwani, J. Naor, M. Naor, The probabilistic method yields deterministic parallel algorithms, 30th IEEE FOCS, pp 8–13, 1989.Google Scholar
  21. [NN90]
    J. Naor, M. Naor, Small-bias probability spaces: efficient constructions and applications, SIAM J. Comput. 22(4), pp 838–856, 1993.Google Scholar
  22. [NSS95]
    M. Naor, L. Schulman, A. Srinivasan, Splitters and near-optimal deran-domization, 36th IEEE FOCS, pp 182–191, 1995.Google Scholar
  23. [Rag88]
    P. Raghavan, Probabilistic construction of deterministic algorithms: approximating packing integer programs, JCSS 37, pp 130–143, 1988.Google Scholar
  24. [Sch92]
    L. Schulman, Sample spaces uniform, on neighborhoods, 24th ACM STOC, pp 17–25, 1992.Google Scholar
  25. [SB88]
    G. Seroussi, N. Bshouty, Vector sets for exhaustive testing of logic circuits, IEEE Trans. on Info. Theory, Vol 34(3), pp 513–522, 1988.Google Scholar
  26. [Spe87]
    J. Spencer, Ten Lectures on the Probabilistic Method, SIAM(Philadelphia), 1987.Google Scholar
  27. [Vaz86]
    U. Vazirani, Randomness, Adversaries and Computation, Ph.D. Thesis, University of California, Berkeley, 1986.Google Scholar
  28. [Yao79]
    A. C-C Yao, Some complexity questions related to distributive computing, 11th ACM STOC, pp 209–213, 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Michael Saks
    • 1
  • Shiyu Zhou
    • 2
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Bell LaboratoriesMurray HillUSA

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