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：给定某个二元矩阵 $M$，假设我们获得其各行与各列独立随机排列后的集合。能否从这些信息中恢复出唯一正确的原始排列顺序与矩阵？本文提出一种算法，可判定与给定行、列集合关联的唯一性存在性，并输出两种结果之一：各行与各列的唯一正确排列顺序，或所有有效排列顺序与有效矩阵的完整集合。我们证明存在常数 $c > 0$，使得对于独立同分布伯努利 $(p)$ 随机元 $(m_{ij})_{ij=1}^n$ 构成的 $n \times n$ 二元矩阵，当 $\frac{c\log^2(n)}{n(\log\log(n))^2} \leq p \leq \frac{1}{2}$ 时，该算法以高概率（期望意义下亦然）在 $O(n^2)$ 时间内终止。