Shotgun identification on groups

TL;DR

This paper studies the problem of identifying global patterns on groups from local subpatterns, providing conditions for when such patterns can or cannot be reliably reconstructed as the pattern size grows.

Contribution

It extends shotgun identification theory to group structures, offering general conditions for pattern identifiability and non-identifiability in an asymptotic setting.

Findings

Established conditions for pattern identifiability with high probability.

Established conditions for pattern non-identifiability with high probability.

Applied results to specific examples of groups and patterns.

Abstract

We consider the problem of shotgun identification of patterns on groups, which extends previous work on shotgun identification of DNA sequences and labeled graphs. A shotgun identification problem on a group $G$ is specified by two finite subsets $C \subset G$ and $K \subset G$ and a finite alphabet $\mathcal{A}$. In such problems, there is a ``global" pattern $w \in \mathcal{A}^{CK}$, and one would like to be able to identify this pattern (up to translation) based only on observation of the ``local" $K$-shaped subpatterns of $w$, called reads, centered at the elements of $C$. We consider an asymptotic regime in which the size of $w$ tends to infinity and the symbols of $w$ are drawn in an i.i.d. fashion. Our first general result establishes sufficient conditions under which the random pattern $w$ is identifiable from its reads with probability tending to one, and our second general result establishes sufficient conditions under which the random pattern $w$ is non-identifiable with probability tending to one. Additionally, we illustrate our main results by applying them to several families of examples.