Given some binary matrix $M$, suppose we are presented with the collection of its rows and columns in independent arbitrary orderings. From this information, are we able to recover the unique original orderings and matrix? We present an algorithm that identifies whether there is a unique ordering associated with a set of rows and columns, and outputs either the unique correct orderings for the rows and columns or the full collection of all valid orderings and valid matrices. We show that there is a constant $c > 0$ such that the algorithm terminates in $O(n^2)$ time with high probability and in expectation for random $n \times n$ binary matrices with i.i.d.\ Bernoulli $(p)$ entries $(m_{ij})_{ij=1}^n$ such that $\frac{c\log^2(n)}{n(\log\log(n))^2} \leq p \leq \frac{1}{2}$.

翻译：给定某个二进制矩阵$M$，假设我们已知其行和列各自独立地以任意顺序排列的集合。我们能否从这些信息中恢复出唯一的原始顺序和矩阵？本文提出了一种算法，用于判断是否存在与行列集合相关的唯一顺序，并输出唯一的正确行、列顺序，或所有有效顺序和有效矩阵的完整集合。我们证明存在常数$c > 0$，使得对于元素$(m_{ij})_{ij=1}^n$服从独立同分布伯努利$(p)$分布、且满足$\frac{c\log^2(n)}{n(\log\log(n))^2} \leq p \leq \frac{1}{2}$的随机$n \times n$二进制矩阵，该算法高概率地以及期望意义下在$O(n^2)$时间内终止。