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On mappings on the hypercube with small average stretch

Part of: Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  18 October 2022

Lucas Boczkowski  and
Igor Shinkar
Lucas Boczkowski
Affiliation:
CNRS, IRIF Université Paris 7, France
Igor Shinkar*
Affiliation:
Simon Fraser University, Burnaby, BC, V5A 1S6, Canada
*
*Corresponding author. Email: ishinkar@sfu.ca
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Abstract

Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$ , and let $\phi \,:\, \{0,1\}^{n-1} \to A$ be a bijection. We define the average stretch of $\phi$ as

\begin{equation*} {\sf avgStretch}(\phi ) = {\mathbb E}[{{\sf dist}}(\phi (x),\phi (x'))], \end{equation*}
where the expectation is taken over uniformly random $x,x' \in \{0,1\}^{n-1}$ that differ in exactly one coordinate.

In this paper, we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results.

  • For any set $A \subseteq \{0,1\}^n$ of density $1/2$ there exists a bijection $\phi _A \,:\, \{0,1\}^{n-1} \to A$ such that ${\sf avgStretch}(\phi _A) = O\left(\sqrt{n}\right)$ .

  • For $n = 3^k$ let ${A_{\textsf{rec-maj}}} = \{x \in \{0,1\}^n \,:\,{\textsf{rec-maj}}(x) = 1\}$ , where ${\textsf{rec-maj}} \,:\, \{0,1\}^n \to \{0,1\}$ is the function recursive majority of 3’s. There exists a bijection $\phi _{{\textsf{rec-maj}}} \,:\, \{0,1\}^{n-1} \to{A_{\textsf{rec-maj}}}$ such that ${\sf avgStretch}(\phi _{{\textsf{rec-maj}}}) = O(1)$ .

  • Let ${A_{{\sf tribes}}} = \{x \in \{0,1\}^n \,:\,{\sf tribes}(x) = 1\}$ . There exists a bijection $\phi _{{\sf tribes}} \,:\, \{0,1\}^{n-1} \to{A_{{\sf tribes}}}$ such that ${\sf avgStretch}(\phi _{{\sf tribes}}) = O(\!\log (n))$ .

These results answer the questions raised by Benjamini, Cohen, and Shinkar (Isr. J. Math 2016).

Keywords

boolean hypercubelipschitz mappings

MSC classification

Primary: 68R05: Combinatorics
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 2 , March 2023 , pp. 334 - 348
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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