Reconstruction of shredded random matrices

A matrix is given in ``shredded'' form if we are presented with the multiset of rows and the multiset of columns, but not told which row is which or which column is which. The matrix is reconstructible if it is uniquely determined by this information. Let $M$ be a random binary $n\times n$...

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Bibliographic Details
Main Authors Balister, Paul, Kronenberg, Gal, Scott, Alex, Tamitegama, Youri
Format Journal Article
LanguageEnglish
Published 10.01.2024
Subjects
Mathematics - Combinatorics
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Summary:A matrix is given in ``shredded'' form if we are presented with the multiset of rows and the multiset of columns, but not told which row is which or which column is which. The matrix is reconstructible if it is uniquely determined by this information. Let $M$ be a random binary $n\times n$ matrix, where each entry independently is $1$ with probability $p=p(n)\le\frac12$. Atamanchuk, Devroye and Vicenzo introduced the problem and showed that $M$ is reconstructible with high probability for $p\ge (2+\varepsilon)\frac{1}{n}\log n$. Here we find that the sharp threshold for reconstructibility is at $p\sim\frac{1}{2n}\log n$.
DOI:10.48550/arxiv.2401.05058