Testing k-wise and almost k-wise independence

Noga Alon, Alexandr Andoni, Tali Kaufman, Kevin Matulef, Ronitt Rubinfeld, Ning Xie

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

86 Scopus citations


In this work, we consider the problems of testing whether adistribution over (0,1 n) is k-wise (resp. (,k)-wise) independentusing samples drawn from that distribution. For the problem of distinguishing k-wise independent distributions from those that are -far from k-wise independence in statistical distance, we upper bound the number ofrequired samples by (n k/ 2) and lower bound it by (n k-1/2/) (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 (,k)-wise independent distributions from thosethat are -far from (,k)-wise independence in statistical distance, we upper bound thenumber of required samples by O(k log n / 22) and lower bound it by ( k log n / 2 k(+) log 1/2 k(+)). 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 (,k)-wise independence isunlikely to be poly(n,1/,1/) for k=(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 = 1 / n 0.99,there is a set of (,k)-wise independent distributions, and a set of distributions at distance =1/n 0.51 from (,k)-wiseindependence, which are indistinguishable by polynomial time algorithms.

Original languageEnglish
Title of host publicationSTOC'07
Subtitle of host publicationProceedings of the 39th Annual ACM Symposium on Theory of Computing
Number of pages10
StatePublished - 2007
Externally publishedYes
EventSTOC'07: 39th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: 11 Jun 200713 Jun 2007

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


ConferenceSTOC'07: 39th Annual ACM Symposium on Theory of Computing
Country/TerritoryUnited States
CitySan Diego, CA


  • Almost k-wise independence
  • Fourier analysis
  • Hidden clique
  • K-wise independence
  • Property testing


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