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

2. 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)

3. 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

4. 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

5. 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]

6. 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

7. 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

8. 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

9. 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

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

11. 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

12. 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}

13. 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

14. 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

15. 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

16. 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

17. 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)

18. 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

19. 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)

20. 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!