On mappings on the hypercube with small average stretch

Abstract Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$ , and let $\phi \,:\, \{0,1\}^{n-1} \to A$ bijection. We define the average stretch $\phi$ as \begin{equation*} {\sf avgStretch}(\phi ) = {\mathbb E}[{{\sf dist}}(\phi (x),\phi (x'))], \end{equation*} where expectation is taken over uniformly random $x,x' \in \{0,1\}^{n-1}$ that differ in exactly one coordinate. In this paper, we continue line research studying mappings on discrete hypercube with small stretch. prove following results. For any density $1/2$ there exists bijection _A such ${\sf _A) O\left(\sqrt{n}\right)$ . $n 3^k$ ${A_{\textsf{rec-maj}}} \{x \{0,1\}^n \,:\,{\textsf{rec-maj}}(x) 1\}$ ${\textsf{rec-maj}} \{0,1\}$ function recursive majority 3’s There _{{\textsf{rec-maj}}} \to{A_{\textsf{rec-maj}}}$ _{{\textsf{rec-maj}}}) O(1)$ ${A_{{\sf tribes}}} \,:\,{\sf tribes}(x) _{{\sf tribes}} \to{A_{{\sf tribes}}}$ tribes}}) O(\!\log (n))$ These results answer questions raised by Benjamini, Cohen, Shinkar (Isr. J. Math 2016).