# Shotgun reconstruction in the hypercube

## TL;DR

This paper demonstrates that most random 2-colourings of an n-dimensional hypercube can be reconstructed from small local information, significantly reducing the complexity compared to worst-case scenarios.

## Contribution

It establishes that almost all random colourings of the hypercube are reconstructible from local ball colourings of small radius, contrasting with worst-case requirements.

## Key findings

- Almost every 2-colouring can be reconstructed from 2-radius ball colourings.
- For q ≥ n^{2+ε}, almost all q-colourings are reconstructible from 1-radius ball colourings.
- Reconstruction is dramatically easier for typical random colourings than in worst-case scenarios.

## Abstract

Mossel and Ross raised the question of when a random colouring of a graph can be reconstructed from local information, namely the colourings (with multiplicity) of balls of given radius. In this paper, we are concerned with random $2$-colourings of the vertices of the $n$-dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter $\Omega(n)$ are required to reconstruct. However, the situation for random colourings is dramatically different: we show that almost every $2$-colouring can be reconstructed from the multiset of colourings of balls of radius $2$. Furthermore, we show that for $q \ge n^{2+\epsilon}$, almost every $q$-colouring can be reconstructed from the multiset of colourings of $1$-balls.