# Optimal Boolean Locality-Sensitive Hashing

## TL;DR

This paper characterizes the optimal distribution over Boolean functions for locality-sensitive hashing, showing it assigns nonzero probability only to dictator functions to minimize a specific correlation ratio.

## Contribution

It provides a theoretical characterization of the optimal Boolean LSH scheme, identifying dictator functions as the only functions with nonzero probability in the optimal distribution.

## Key findings

- Optimal distribution over Boolean functions is supported only on dictator functions.
- The ratio 
ho_{\u03b1, } is minimized by dictator functions.
- Theoretical foundation for the design of optimal Boolean LSH schemes.

## Abstract

For $0 \leq \beta < \alpha < 1$ the distribution $\mathcal{H}$ over Boolean functions $h \colon \{-1, 1\}^d \to \{-1, 1\}$ that minimizes the expression \begin{equation*} \rho_{\alpha, \beta} = \frac{\log(1/\Pr_{\substack{h \sim \mathcal{H} \\ (x, y) \text{ $\alpha$-corr.}}}[h(x) = h(y)])}{\log(1/\Pr_{\substack{h \sim \mathcal{H} \\ (x, y) \text{ $\beta$-corr.}}}[h(x) = h(y)])} \end{equation*} assigns nonzero probability only to members of the set of dictator functions $h(x) = \pm x_i$.