K-wise vs almost K-wise permutations, and general group actions Noga Alon, Tel-Aviv University Shachar Lovett, IAS / UCSD

Limited indepdence Distributions with limited independence are a powerful derandomization tool K-wise bits: well understood K-wise permutations: not so much… This work: Simplify analysis of algorithms (using existing constructions)

K-wise bits A distribution D over {0,1}n is k-wise if Explicit, efficient constructions (based on error-correcting codes) Sample x using O(k log n) random bits

K-wise permutations Distribution D over permutations on n elements is k-wise if Explicit constructions: k=1,2,3 only One solution: allow errors

Almost K-wise permutations Distribution D over permutations on n elements is almost k-wise with error if Explicit, efficient constructions known [...,Kaplan-Naor-Reingold’05, Kassabov’07]

K-wise vs almost K-wise permutations No errors No constructions… Almost K-wise permutations: Allow errors Explicit efficient constructions This work: bridge the gap

Main results (1) Thm 1: Any almost k-wise distribution over permutations with good enough error is close in statistical distance to a k-wise distribution over permutations Extends [Alon,Goldreich,Mansour’03] who showed a similar result for k-wise bits

Main results (1) Thm 1: Any almost k-wise distribution over permutations with good enough error is close in statistical distance to a k-wise distribution over permutations What does it mean? To derandomize a decision algorithm: Analyze assuming k-wise permutations Actually use almost k-wise permutations

Main results (2) Thm 2: Any almost 2k-wise distribution over permutations with good enough error supports a k-wise distribution over permutations What does it mean? To derandomize a search algorithm: Analyze assuming k-wise permutations Actually use almost 2k-wise permutations

General group actions It turns out that k-wise permutations is an instance of a more general framework General setup: group actions

General setup: group actions Group G acts on a set X (e.g. permutations on k-tuples) Distribution D on G acts on X uniformly if It acts almost uniformly with error if

Examples K-wise permutations: K-wise bits: Group G=Sn acts on disjoint k-tuples X={(i1,…,ik): i1,…,ik[n]} K-wise bits: Group G=SnZ2n acts of indices & values X={(i1,…,ik,v): i1,…,ik[n], vZ2n}

Main results (1) G acts on X Thm 1: any almost X-uniform distribution with good enough error is close in statistical distance to an X-uniform distribution

Main results (2) G acts on X G naturally acts on X2 Thm 2: Any almost X2-uniform distribution with good enough error supports an X-uniform distribution

Proof idea Main tool: basic representation theory Focus on thm 1 in this talk Thm 1: any almost X-uniform distribution with good enough error is close in statistical distance to an X-uniform distribution

Alon, Goldreich, Mansour Thm [AGM]: any distribution on {0,1}n which is almost k-wise with good enough error is close in statistical distance to a k-wise distribution Proof idea: Correct all Fourier coefficients of size ≤k to be zero

Extension to group action Thm 1: Any almost X-uniform distribution with good enough error is close in statistical distance to an X-uniform distribution What is the analog of “Fourier coefs of size at most k” used in [AGM]? Answer: irreducible representations occurring in the action of G on X (X is small - not too many of them)

Proof idea Proof uses only elementary properties of irreducible representations (e.g. nothing specific for permutations) This is why the results extend to general group actions

Summary Main message: simplify analysis of algorithms Analyze assuming you have k-wise permutations (which we currently don’t know how to construct) Actually use only almost k-wise permutations (which we know how to construct)

Thank you! Further research Combinatorial problem: construct efficient sample spaces of k-wise independent permutations [Kuperberg-L.-Peled’12]: they exist Now we need to find them… Thank you!