Combinatorica

, Volume 17, Issue 2, pp 215–234 | Cite as

Efficient construction of a small hitting set for combinatorial rectangles in high dimension

  • Nathan Linial
  • Michael Luby
  • Michael Saks
  • David Zuckerman
Article

Abstract

We describe a deterministic algorithm which, on input integersd, m and real number ∈∃ (0,1), produces a subset S of [m]d={1,2,3,...,m}d that hits every combinatorial rectangle in [m]d of volume at least ∈, i.e., every subset of [m]d the formR1×R2×...×Rd of size at least ∈md. The cardinality of S is polynomial inm(logd)/∈, and the time to construct it is polynomial inmd/∈. The construction of such sets has applications in derandomization methods based on small sample spaces for general multivalued random variables.

Mathematics Subject Classification (1991)

068Q25 05B40 68R05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Ajtai, J. Komlós, andE. Szemerédi: Deterministic Simulation in LOGSPACE, in:Proc. of 19th ACM Symposium on Theory of Computing, 1987, 132–140.Google Scholar
  2. [2]
    R. Armoni, M. Saks, A. Wigderson, andS. Zhou: Discrepancy sets and pseudorandom generators for combinatorial rectangles,Proc. 37th IEEE Symposium on Foundations of Computer Science, 1996.Google Scholar
  3. [3]
    J. L. Carter, M. N. Wegman: Universal classes of hash functions,J. Comput. System Sci.,18 (1979), 143–154.Google Scholar
  4. [4]
    B. Chor, O. Goldreich: On the Power of Two-Point Based Sampling,Journal of Complexity,5 (1989), 96–106.Google Scholar
  5. [5]
    G. Even, O. Goldreich, M. Luby, N. Nisan, andB. Boban Veliĉković: Approximations of General Independent Distributions. in:Proc. of the 24th ACM Symposium on Theory of Computing, 1992.Google Scholar
  6. [6]
    J. Friedman: ConstructingO(n log(n)) size monotone formulae for thekth threshold function ofn boolean variables.SIAM J. on Computing,15 (1986).Google Scholar
  7. [7]
    O. Gabber, Z. Galil: Explicit constructions of linear-sized superconcentrators,Journal of Computer System Science,22 (1981), 407–420.Google Scholar
  8. [8]
    J. Kahn, N. Linial, andA. Samorodnitsky: Inclusion-Exclusion: Exact and approximate,Combinatorica,16 (1996), 465–477.Google Scholar
  9. [9]
    R. Karp, M. Pippenger, andM. Sipser:Time-Randomness Tradeoff, presented at the AMS conference on probabilistic computational complexity, Durham, New Hampshire, 1982.Google Scholar
  10. [10]
    M. Karpinski, M. Luby: Approximating the Number of Solutions to aGF[2] Formula,Journal of Algorithms,14 (1993), 280–287.Google Scholar
  11. [11]
    M. Luby, B. Veliĉković: On Deterministic Approximation of DNF, in:Proceedings of 23rd ACM Symposium on Theory of Computing, 1991, 430–438,Algorithmica (special issue devoted to randomized algorithms), Vol. 16, No. 4/5, October/November 1996, 415–433.Google Scholar
  12. [12]
    M. Luby, A. Wigderson, andB. Veliĉković: Deterministic Approximate Counting of Depth-2 Circuits, in:Proceedings of the Second Israeli Symposium on Theory of Computing and Systems, 1993, 18–24.Google Scholar
  13. [13]
    G. A. Margulis: Explicit construction of concentrators,Problemy Peredaĉi Informacii 9, (1973) 71–80. (English translation inProblems Inform. Transmission, 1975).Google Scholar
  14. [14]
    N. Nisan: Pseudorandom Generators for Space-Bounded Computation,Combinatorica,12 (1992), 449–461.Google Scholar
  15. [15]
    J. Schmidt, A. Siegel: The Spatial Complexity of obliviousk-probe hash functions,SIAM Journal on Computing,19 (1990), 775–786.Google Scholar
  16. [16]
    M. Sipser: Expanders, Randomness, or Time vs. Space,Journal of Computer and System Sciences,36 (1988), 379–383.Google Scholar
  17. [17]
    D. Zuckerman: Simulating BPP Using a General Weak Random Source,Algorithmica,16 (1996), 367–391.Google Scholar

Copyright information

© János Bolyai Mathematical Society 1997

Authors and Affiliations

  • Nathan Linial
    • 1
  • Michael Luby
    • 2
  • Michael Saks
    • 3
  • David Zuckerman
    • 4
  1. 1.Computer Science DepartmentHebrew UniversityJerusalemIsrael
  2. 2.Palo Alto
  3. 3.Department of MathematicsRutgers UniversityNew BrunswickUSA
  4. 4.Department of Computer SciencesThe University of Texas at AustinAustinUSA

Personalised recommendations