A fast parallel algorithm for the maximal independent set problem
Journal of the ACM (JACM)
A fast and simple randomized parallel algorithm for the maximal independent set problem
Journal of Algorithms
A simple parallel algorithm for the maximal independent set problem
SIAM Journal on Computing
Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
Expected complexity of graph partitioning problems
Discrete Applied Mathematics - Special issue: Combinatorial Optimization 1992 (CO92)
Algorithmic theory of random graphs
Random Structures & Algorithms - Special issue: average-case analysis of algorithms
Finding a large hidden clique in a random graph
proceedings of the eighth international conference on Random structures and algorithms
Finding and certifying a large hidden clique in a semirandom graph
Random Structures & Algorithms
Hiding Cliques for Cryptographic Security
Designs, Codes and Cryptography
Testing that distributions are close
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Testing Random Variables for Independence and Identity
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Almost k-wise independence versus k-wise independence
Information Processing Letters
Sublinear algorithms for testing monotone and unimodal distributions
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Testing monotone high-dimensional distributions
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On the fourier tails of bounded functions over the discrete cube
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Declaring independence via the sketching of sketches
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Testing symmetric properties of distributions
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Small Sample Spaces Cannot Fool Low Degree Polynomials
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
How hard is it to approximate the best Nash equilibrium?
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Property Testing: A Learning Theory Perspective
Foundations and Trends® in Machine Learning
Perturbed identity matrices have high rank: Proof and applications
Combinatorics, Probability and Computing
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
Measuring independence of datasets
Proceedings of the forty-second ACM symposium on Theory of computing
Testing monotone continuous distributions on high-dimensional real cubes
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Testing non-uniform k-wise independent distributions over product spaces
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Invariance in property testing
Property testing
Invariance in property testing
Property testing
How Hard Is It to Approximate the Best Nash Equilibrium?
SIAM Journal on Computing
Approximating and testing k-histogram distributions in sub-linear time
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Testing Symmetric Properties of Distributions
SIAM Journal on Computing
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Testing Closeness of Discrete Distributions
Journal of the ACM (JACM)
Statistical algorithms and a lower bound for detecting planted cliques
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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.