Shotgun assembly threshold for lattice labeling model

TL;DR

This paper determines the precise threshold for the recoverability of lattice labels from local observations, sharpening previous bounds and solving an open problem in the field.

Contribution

It establishes the exact threshold for the shotgun assembly problem in lattice models, improving upon prior work and confirming the feasibility of efficient recovery.

Findings

Identified the sharp transition threshold for label recovery

Improved the constant factor in the threshold compared to previous work

Confirmed the possibility of efficient recovery above the threshold

Abstract

We study the shotgun assembly problem for the lattice labeling model, where i.i.d. uniform labels are assigned to each vertex in a $d$-dimensional box of side length $n$. We wish to recover the labeling configuration on the whole box given empirical profile of labeling configurations on all boxes of side length $r$. We determine the threshold around which there is a sharp transition from impossible to recover with probability tending to 1, to possible to recover with an efficient algorithm with probability tending to 1. Our result sharpens a constant factor in a previous work of Mossel and Ross (2019) and thus solves a question therein.