Shotgun Assembly of the Non-Uniform Labeled Lattice Model

Abstract

In this paper, we study the graph shotgun assembly problem in the labeled lattice model, as introduced by Mossel and Ross (2019). We start with a $d$-dimensional lattice of length $n$, where the color of each vertex is sampled i.i.d. from a distribution $\pi$. Then, we ask whether it is possible to reconstruct the coloring of the lattice given the coloring of each subgrid of length $k$. While previous work by Ding and Liu (2023) and Demidovich, Panichkin, and Zhukovskii (2023) identified the reconstruction threshold when $\pi$ is uniform, we examine the non-uniform case. This paper establishes a sufficient condition for the impossibility of reconstruction for $d\ge 2$ using information-theoretic arguments. Our negative result matches the known threshold in the uniform case.

arXiv comment13 pages