On Lipschitz Bijections between Boolean Functions

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.