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Bi- Lipschitz Bijection between the Boolean Cube and the Hamming Ball

Bi- Lipschitz Bijection between the Boolean Cube and the Hamming Ball. Gil Cohen. Joint work with Itai Benjamini and Igor Shinkar. Cube vs. Ball. Majority. Dictator. n. Strings with Hamming weight k. k. 0. The Problem.

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Bi- Lipschitz Bijection between the Boolean Cube and the Hamming Ball

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  1. Bi-LipschitzBijection between the Boolean Cube and the Hamming Ball Gil Cohen Joint work with ItaiBenjamini and Igor Shinkar

  2. Cube vs. Ball Majority Dictator n Strings with Hamming weight k k 0

  3. The Problem Open Problem [LovettViola11]. Prove that for any bijection, . Motivation. Related to proving lower bounds for sampling a natural distribution by circuits. The “Naïve” Upper Bound. Average stretch

  4. Main Theorem Theorem. such that 1) of of

  5. Main Theorem Theorem. such that 1) 2) is computable in DLOGTIME-uniform (Majority is -reducible to ) 3) is very local

  6. The BTK Partition [DeBruijnTengbergenKruyswijk51] • Partition of to symmetric chains. • A symmetric chain is a path . 1 0 0 1 1 0 0 1 1 0 _ 1 1 0 0 _ ^ ^ 1 0 0 1 1 0 0 1 1 1 1 0 _ 1 1 0 0 _ ^ ^ ^ ^ 1 0 0 1 1 0 0 1 0 1 0 _ 1 1 0 0 _ 1 ^ ^ ^ ^ ^ ^ 1 0 0 1 1 0 0 1 0 1 0 _ 1 1 0 0 _ 0

  7. The BTK Partition

  8. The Metric Properties • 1) • y • x

  9. The Metric Properties • 1) • 2) • y • x

  10. The Metric Properties • 1) • 2) • y • x

  11. The Metric Properties – Proof • .. • .. • ^ • ^ • ^ • ^

  12. 1 • 0 • 1 • 0 • 0 • 1 • 1

  13. The Hamming Ball isbi-Lipschitz Transitive Defintion. A metric space M is called k bi-Lipschitz transitive if for any there is a bijection such that , and Example. is 1 bi-Lipschitz transitive.

  14. The Hamming Ball isbi-Lipschitz Transitive Corollary. is 20 bi-Lipschitz transitive. • is convex in

  15. Open Problems

  16. The Constants Are the 4,5 optimal ? We know how to improve 4 to 3 at the expense of unbounded inverse. • Does the 20 in the corollary optimal ?

  17. General Balanced Halfspaces Switching to notation • Does the result hold for general balanced halfspaces ? • One possible approach: generalize BTK chains. • Applications to FPTAS for counting solutions to 0-1 knapsack problem [MorrisSinclair04].

  18. Lower Bounds for Average Stretch Exhibit a density half subset such that any bijection has super constant average stretch. Even average stretch 2.001 ! (for 2 take XOR). • Conjecture: monotone noise-sensitive functions like Recursive-Majority-of-Three (highly fractal) should work. • We believe a random subset of density half has a constant average stretch.

  19. Goldreich’s Question Is it true that for any with density, say, half there exist and , both with density half, with bi-Lipschitzbijection between them ?

  20. Thank you for your attention!

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