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Israel Journal of Mathematics

, Volume 212, Issue 2, pp 677–703 | Cite as

Bi-Lipschitz bijection between the Boolean cube and the Hamming ball

  • Itai BenjaminiEmail author
  • Gil CohenEmail author
  • Igor Shinkar
Article

Abstract

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n ∈ N there exists an explicit bijection ψ: {0, 1} n → {x ∈ {0, 1} n+1 : |x| > n/2} such that for every xy ∈ {0, 1} n , \(\frac{1}{5} \leqslant \frac{{dis\tan ce\left( {\psi \left( x \right),\psi \left( y \right)} \right)}}{{dis\tan ce\left( {x,y} \right)}} \leqslant 4,\) where distance(·, ·) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive.

This result gives a strong negative answer to an open problem of Lovett and Viola (2012), who raised the question in the context of sampling distributions in low-level complexity classes. The conceptual implication is that the problem of proving lower bounds in the context of sampling distributions requires ideas beyond the sensitivity-based structural results of Boppana (1997).

We study the mapping ψ further and show that it (and its inverse) are computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that ψ is “approximately local” in the sense that all but the last output bit of ψ are essentially determined by a single input bit.

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Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Department of MathematicsWeizmann Institute of ScienceRehovotIsrael
  2. 2.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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