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

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 algorithms: an algorithm which uses randomized permutations can be analyzed assuming perfect k-wise independence, and then applied to an almost k-wise independent family. In pa...

• Type: Journal
• Year: 2011
• Where: ECCC
• Authors: Noga Alon, Shachar Lovett
• Keywords: ECCC 2011, ECommerce, K-wise Independent Distribution, K-wise Independent Families, Permutations