On Mappings on the Hypercube with Small Average Stretch

TL;DR

This paper constructs mappings on the hypercube with small average stretch, showing that for various subsets, one can find bijections with average stretch bounds from constant to square root of n.

Contribution

It provides new bounds on the average stretch of bijections on the hypercube for specific subsets, advancing understanding of geometric mappings in high-dimensional discrete spaces.

Findings

Existence of bijections with average stretch O(√n) for half-density subsets.

Bijections with constant average stretch for recursive majority subsets.

Bijections with O(log n) average stretch for tribes function subsets.

Abstract

Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$, and let $\phi \colon \{0,1\}^{n-1} \to A$ be a bijection. We define the average stretch of $\phi$ as ${\sf avgStretch}(\phi) = {\mathbb E}[{\sf dist}(\phi(x),\phi(x'))]$, 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. (1) For any set $A \subseteq \{0,1\}^n$ of density $1/2$ there exists a bijection $\phi_A \colon \{0,1\}^{n-1} \to A$ such that ${\sf avgstretch}(\phi_A) = O(\sqrt{n})$. (2) For $n = 3^k$ let $A_{{\sf rec\text{-}maj}} = \{x \in \{0,1\}^n : {\sf rec\text{-}maj}(x) = 1\}$, where ${\sf rec\text{-}maj} : \{0,1\}^n \to \{0,1\}$ is the function recursive majority of 3's. There exists a bijection $\phi_{{\sf rec\text{-}maj}} \colon \{0,1\}^{n-1} \to A_{\sf rec\text{-}maj}$ such that ${\sf avgstretch}(\phi_{\sf rec\text{-}maj}) = O(1)$. (3) Let $A_{\sf tribes} = \{x \in \{0,1\}^n : {\sf tribes}(x) = 1\}$. There exists a bijection $\phi_{{\sf tribes}} \colon \{0,1\}^{n-1} \to A_{\sf tribes}$ such that ${\sf avgstretch}(\phi_{{\sf tribes}}) = O(\log(n))$. These results answer the questions raised by Benjamini et al.\ (FOCS 2014).