ABSTRACT
In this work, we consider the problems of testing whether adistribution over (0,1n) is k-wise (resp. (ε,k)-wise) independentusing samples drawn from that distribution.
For the problem of distinguishing k-wise independent distributions from those that are δ-far from k-wise independence in statistical distance, we upper bound the number ofrequired samples by Õ(nk/δ2) and lower bound it by Ω(nk-1/2/δ) (these bounds hold for constantk, and essentially the same bounds hold for general k). Toachieve these bounds, we use Fourier analysis to relate adistribution's distance from k-wise independence to its biases, a measure of the parity imbalance it induces on a setof variables. The relationships we derive are tighter than previouslyknown, and may be of independent interest.
To distinguish (ε,k)-wise independent distributions from thosethat are δ-far from (ε,k)-wise independence in statistical distance, we upper bound thenumber of required samples by O(k log n / δ2ε2) and lower bound it by Ω(√ k log n / 2k(ε+δ)√ log 1/2k(ε+δ)). Although these bounds are anexponential improvement (in terms of n and k) over thecorresponding bounds for testing k-wise independence, we give evidence thatthe time complexity of testing (ε,k)-wise independence isunlikely to be poly(n,1/ε,1/δ) for k=Θ(log n),since this would disprove a plausible conjecture concerning the hardness offinding hidden cliques in random graphs. Under the conjecture, ourresult implies that for, say, k = log n and ε = 1 / n0.99,there is a set of (ε,k)-wise independent distributions, and a set of distributions at distance δ=1/n0.51 from (ε,k)-wiseindependence, which are indistinguishable by polynomial time algorithms.
- N. Alon. Problems and results in extremal combinatorics (Part I). Discrete Math., 273:31--53, 2003.Google ScholarDigital Library
- N. Alon, L. Babai, and A. Itai. A fast and simple randomized algorithm for the maximal independent set problem. J. of Algorithms, 7:567--583, 1986. Google ScholarDigital Library
- N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3(3):289--304, 1992. Earlier version in FOCS'90.Google ScholarCross Ref
- N. Alon, O. Goldreich, and Y. Mansour. Almost k-wise independence versus k-wise independence. Inform. Process. Lett., 88:107--110, 2003. Google ScholarDigital Library
- N. Alon, M. Krivelevich, and B. Sudakov. Finding a large hidden clique in a random graph. Random Structures and Algorithms, 13:457--466, 1998. Google ScholarDigital Library
- N. Alon and J. Spencer. The Probabilistic Method. John Wiley and Sons, second edition, 2000.Google Scholar
- T. Batu, E. Fisher, L. Fortnow, R. Kumar, R. Rubinfeld, and P. White. Testing random variables for independence and identity. In Proc. 42nd Annual IEEE Symposium on Foundations of Computer Science, pages 442--451, 2001. Google ScholarDigital Library
- T. Batu, L. Fortnow, R. Rubinfeld, W.D. Smith, and P. White. Testing that distributions are close. In Proc. 41st Annual IEEE Symposium on Foundations of Computer Science, pages 189--197, 2000. Google ScholarDigital Library
- T. Batu, R. Kumar, and R. Rubinfeld. Sublinear algorithms for testing monotone and unimodal distributions. In Proc. 36th Annual ACM Symposium on the Theory of Computing, pages 381--390, New York, NY, USA, 2004. ACM Press. Google ScholarDigital Library
- W. Beckner. Inequalities in fourier analysis. Annals of Mathematics, 102:159--182, 1975.Google ScholarCross Ref
- S. Bernstein. The Theory of Probabilities. Gostehizdat Publishing House, Moscow, 1946.Google Scholar
- A. Bonami. Etude des coefficients fourier des fonctiones de lp(g). Ann. Inst. Fourier (Grenoble), 20(2):335--402, 1970.Google ScholarCross Ref
- B. Chor, J. Friedman, O. Goldreich, J. Hastad, S. Rudich, and R. Smolensky. The bit extraction problem and t-resilient functions. In Proc. 26th Annual IEEE Symposium on Foundations of Computer Science, pages 396--407, 1985.Google ScholarDigital Library
- I. Dinur, E. Friedgut, G. Kindler, and R. O'Donnell. On the Fourier tails of bounded functions over the discrete cube. In Proc. 38th Annual ACM Symposium on the Theory of Computing, pages 437--446, New York, NY, USA, 2006. ACM Press. Google ScholarDigital Library
- U. Feige and R. Krauthgamer. Finding and certifying a large hidden clique in a semirandom graph. Random Structures and Algorithms, 16:195--208, 2000. Google ScholarDigital Library
- A. Frieze and C. McDiarmid. Algorithmic theory of random graphs. Random Structures and Algorithms, 10(1-2):5--42, 1997. Google ScholarDigital Library
- O. Goldreich and D. Ron. On testing expansion in bounded-degree graphs. Technical Report TR00-020, Electronic Colloquium on Computational Complexity, 2000.Google Scholar
- M. Jerrum. Large cliques elude the metropolis process. Random Structures and Algorithms, 3(4):347--359, 1992.Google ScholarCross Ref
- A. Joffe. On a set of almost deterministic k-independent random variables. Annals of Probability, 2:161--162, 1974.Google ScholarCross Ref
- A. Juels and M. Peinado. Hiding cliques for cryptographic security. Des. Codes Cryptography, 20(3):269--280, 2000. Google ScholarDigital Library
- R. Karp and A. Wigderson. A fast parallel algorithm for the maximal independent set problem. Journal of the ACM, 32(4):762--773, 1985. Google ScholarDigital Library
- R.M. Karp. The probabilistic analysis of some combinatorial search algorithms. In JF. Traub, editor, Algorithms and Complexity: New directions and Recent Results, pages 1--19, New York, 1976. Academic Press.Google Scholar
- L. Kucera. Expected complexity of graph partitioning problems. Disc. Applied Math., 57(2-3):193--212, 1995. Google ScholarDigital Library
- M. Luby. A simple parallel algorithm for the maximal independent set problem. SIAM J. on Comput., 15(4):1036--1053, 1986. Earlier version in STOC'85. Google ScholarDigital Library
- J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. on Comput., 22(4):838--856, 1993. Earlier version in STOC'90. Google ScholarDigital Library
- R. Rubinfeld and R.A. Servedio. Testing monotone high-dimensional distributions. In Proc. 37th Annual ACM Symposium on the Theory of Computing, pages 147--156, New York, NY, USA, 2005. ACM Press. Google ScholarDigital Library
Index Terms
- Testing k-wise and almost k-wise independence
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