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Almost k-wise vs. k-wise independent permutations, and uniformity for general group actions
 

Summary: Almost k-wise vs. k-wise independent permutations,
and uniformity for general group actions
Noga Alon
Tel-Aviv University and IAS, Princeton
nogaa@tau.ac.il
Shachar Lovett
IAS, Princeton
slovett@math.ias.edu
June 24, 2011
Abstract
A family of permutations in Sn is k-wise independent if a uniform permutation
chosen from the family maps any distinct k elements to any distinct k elements equally
likely. Efficient constructions of k-wise independent permutations are known for k = 2
and k = 3, but are unknown for k 4. In fact, it is known that there are no nontrivial
subgroups of Sn for n 25 which are 4-wise independent. Faced with this adversity,
research has turned towards constructing almost k-wise independent families, where
small errors are allowed. Optimal constructions of almost k-wise independent families
of permutations were achieved by several authors.
Our first result is that any such family with small enough error is statistically close
to a distribution which is perfectly k-wise. This allows for a simplified analysis of algo-

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics