Testing of random matrices

TL;DR

This paper investigates algorithms to determine if a random matrix with independent uniform entries is 'good', meaning each row and column forms a permutation of numbers 1 to n, and analyzes their effectiveness.

Contribution

It introduces and analyzes four algorithms for testing whether a matrix's rows and columns are permutations, providing insights into their performance.

Findings

Algorithms effectively identify 'good' matrices

Performance varies with matrix size and algorithm type

Provides theoretical analysis of algorithm accuracy

Abstract

Let $n$ be a positive integer and $X = [x_{ij}]_{1 \leq i, j \leq n}$ be an $n \times n$\linebreak \noindent sized matrix of independent random variables having joint uniform distribution $$\hbox{Pr} {x_{ij} = k \hbox{for} 1 \leq k \leq n} = \frac{1}{n} \quad (1 \leq i, j \leq n) \koz. $$ A realization $\mathcal{M} = [m_{ij}]$ of $X$ is called \textit{good}, if its each row and each column contains a permutation of the numbers $1, 2,..., n$. We present and analyse four typical algorithms which decide whether a given realization is good.