On Lipschitz Bijections between Boolean Functions

TL;DR

This paper investigates Lipschitz bijections between Boolean functions, constructing specific mappings, proving limitations, analyzing average stretch, and showing probabilistic results for random functions.

Contribution

It provides new constructions and bounds for Lipschitz mappings between Boolean functions, answering open problems and exploring average stretch properties.

Findings

Constructed a C-Lipschitz mapping from Majority to Dictator

Proved no n/2-Lipschitz mapping from Dictator to Majority exists

Established a lower bound of Ω(√n) for average stretch from XOR to Majority

Abstract

For two functions $f,g:\{0,1\}^n\to\{0,1\}$ a mapping $\psi:\{0,1\}^n\to\{0,1\}^n$ is said to be a $\textit{mapping from $f$ to $g$}$ if it is a bijection and $f(z)=g(\psi(z))$ for every $z\in\{0,1\}^n$. In this paper we study Lipschitz mappings between boolean functions. Our first result gives a construction of a $C$-Lipschitz mapping from the ${\sf Majority}$ function to the ${\sf Dictator}$ function for some universal constant $C$. On the other hand, there is no $n/2$-Lipschitz mapping in the other direction, namely from the ${\sf Dictator}$ function to the ${\sf Majority}$ function. This answers an open problem posed by Daniel Varga in the paper of Benjamini et al. (FOCS 2014). We also show a mapping from ${\sf Dictator}$ to ${\sf XOR}$ that is 3-local, 2-Lipschitz, and its inverse is $O(\log(n))$-Lipschitz, where by $L$-local mapping we mean that each of its output bits depends on at most $L$ input bits. Next, we consider the problem of finding functions such that any mapping between them must have large \emph{average stretch}, where the average stretch of a mapping $\phi$ is defined as ${\sf avgStretch}(\phi) = {\mathbb E}_{x,i}[dist(\phi(x),\phi(x+e_i)]$. We show that any mapping $\phi$ from ${\sf XOR}$ to ${\sf Majority}$ must satisfy ${\sf avgStretch}(\phi) \geq \Omega(\sqrt{n})$. In some sense, this gives a "function analogue" to the question of Benjamini et al. (FOCS 2014), who asked whether there exists a set $A \subset \{0,1\}^n$ of density 0.5 such that any bijection from $\{0,1\}^{n-1}$ to $A$ has large average stretch. Finally, we show that for a random balanced function $f:\{0,1\}^n\to\{0,1\}^n$ with high probability there is a mapping $\phi$ from ${\sf Dictator}$ to $f$ such that both $\phi$ and $\phi^{-1}$ have constant average stretch. In particular, this implies that one cannot obtain lower bounds on average stretch by taking uniformly random functions.