Nastya Is Transposing Matrices

1000ms

262144K

Description:

Nastya came to her informatics lesson, and her teacher who is, by the way, a little bit famous here gave her the following task.

Two matrices $$$A$$$ and $$$B$$$ are given, each of them has size $$$n \times m$$$. Nastya can perform the following operation to matrix $$$A$$$ unlimited number of times:

take any square square submatrix of $$$A$$$ and transpose it (i.e. the element of the submatrix which was in the $$$i$$$-th row and $$$j$$$-th column of the submatrix will be in the $$$j$$$-th row and $$$i$$$-th column after transposing, and the transposed submatrix itself will keep its place in the matrix $$$A$$$).

Nastya's task is to check whether it is possible to transform the matrix $$$A$$$ to the matrix $$$B$$$.

$\text{[math]}$ Example of the operation

As it may require a lot of operations, you are asked to answer this question for Nastya.

A square submatrix of matrix $$$M$$$ is a matrix which consist of all elements which comes from one of the rows with indeces $$$x, x+1, \dots, x+k-1$$$ of matrix $$$M$$$ and comes from one of the columns with indeces $$$y, y+1, \dots, y+k-1$$$ of matrix $$$M$$$. $$$k$$$ is the size of square submatrix. In other words, square submatrix is the set of elements of source matrix which form a solid square (i.e. without holes).

Input:

The first line contains two integers $$$n$$$ and $$$m$$$ separated by space ($$$1 \leq n, m \leq 500$$$) — the numbers of rows and columns in $$$A$$$ and $$$B$$$ respectively.

Each of the next $$$n$$$ lines contains $$$m$$$ integers, the $$$j$$$-th number in the $$$i$$$-th of these lines denotes the $$$j$$$-th element of the $$$i$$$-th row of the matrix $$$A$$$ ($$$1 \leq A_{ij} \leq 10^{9}$$$).

Output:

Print "YES" (without quotes) if it is possible to transform $$$A$$$ to $$$B$$$ and "NO" (without quotes) otherwise.

You can print each letter in any case (upper or lower).

Sample Input:

Sample Output:

YES

Sample Input:

Sample Output:

NO

Sample Input:

Sample Output:

YES

Note:

Consider the third example. The matrix $$$A$$$ initially looks as follows.

$$$$$$ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} $$$$$$

Then we choose the whole matrix as transposed submatrix and it becomes

$$$$$$ \begin{bmatrix} 1 & 4 & 7\\ 2 & 5 & 8\\ 3 & 6 & 9 \end{bmatrix} $$$$$$

Then we transpose the submatrix with corners in cells $$$(2, 2)$$$ and $$$(3, 3)$$$.

$$$$$$ \begin{bmatrix} 1 & 4 & 7\\ 2 & \textbf{5} & \textbf{8}\\ 3 & \textbf{6} & \textbf{9} \end{bmatrix} $$$$$$

So matrix becomes

$$$$$$ \begin{bmatrix} 1 & 4 & 7\\ 2 & 5 & 6\\ 3 & 8 & 9 \end{bmatrix} $$$$$$

and it is $$$B$$$.

Informação

Provedor Codeforces

Código CF1136C