Rearrange

1000ms

262144K

Description:

Koa the Koala has a matrix $$$A$$$ of $$$n$$$ rows and $$$m$$$ columns. Elements of this matrix are distinct integers from $$$1$$$ to $$$n \cdot m$$$ (each number from $$$1$$$ to $$$n \cdot m$$$ appears exactly once in the matrix).

For any matrix $$$M$$$ of $$$n$$$ rows and $$$m$$$ columns let's define the following:

The $$$i$$$-th row of $$$M$$$ is defined as $$$R_i(M) = [ M_{i1}, M_{i2}, \ldots, M_{im} ]$$$ for all $$$i$$$ ($$$1 \le i \le n$$$).
The $$$j$$$-th column of $$$M$$$ is defined as $$$C_j(M) = [ M_{1j}, M_{2j}, \ldots, M_{nj} ]$$$ for all $$$j$$$ ($$$1 \le j \le m$$$).

Koa defines $$$S(A) = (X, Y)$$$ as the spectrum of $$$A$$$, where $$$X$$$ is the set of the maximum values in rows of $$$A$$$ and $$$Y$$$ is the set of the maximum values in columns of $$$A$$$.

More formally:

$$$X = \{ \max(R_1(A)), \max(R_2(A)), \ldots, \max(R_n(A)) \}$$$
$$$Y = \{ \max(C_1(A)), \max(C_2(A)), \ldots, \max(C_m(A)) \}$$$

Koa asks you to find some matrix $$$A'$$$ of $$$n$$$ rows and $$$m$$$ columns, such that each number from $$$1$$$ to $$$n \cdot m$$$ appears exactly once in the matrix, and the following conditions hold:

$$$S(A') = S(A)$$$
$$$R_i(A')$$$ is bitonic for all $$$i$$$ ($$$1 \le i \le n$$$)
$$$C_j(A')$$$ is bitonic for all $$$j$$$ ($$$1 \le j \le m$$$)

An array $$$t$$$ ($$$t_1, t_2, \ldots, t_k$$$) is called bitonic if it first increases and then decreases.

More formally: $$$t$$$ is bitonic if there exists some position $$$p$$$ ($$$1 \le p \le k$$$) such that: $$$t_1 < t_2 < \ldots < t_p > t_{p+1} > \ldots > t_k$$$.

Help Koa to find such matrix or to determine that it doesn't exist.

Input:

The first line of the input contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 250$$$) — the number of rows and columns of $$$A$$$.

Each of the ollowing $$$n$$$ lines contains $$$m$$$ integers. The $$$j$$$-th integer in the $$$i$$$-th line denotes element $$$A_{ij}$$$ ($$$1 \le A_{ij} \le n \cdot m$$$) of matrix $$$A$$$. It is guaranteed that every number from $$$1$$$ to $$$n \cdot m$$$ appears exactly once among elements of the matrix.

Output:

If such matrix doesn't exist, print $$$-1$$$ on a single line.

Otherwise, the output must consist of $$$n$$$ lines, each one consisting of $$$m$$$ space separated integers — a description of $$$A'$$$.

The $$$j$$$-th number in the $$$i$$$-th line represents the element $$$A'_{ij}$$$.

Every integer from $$$1$$$ to $$$n \cdot m$$$ should appear exactly once in $$$A'$$$, every row and column in $$$A'$$$ must be bitonic and $$$S(A) = S(A')$$$ must hold.

If there are many answers print any.

Sample Input:

Sample Output:

9 5 1
7 8 2
3 6 4

Sample Input:

2 2
4 1
3 2

Sample Output:

4 1
3 2

Sample Input:

Sample Output:

12 8 6 1
10 11 9 2
3 4 5 7

Note:

Let's analyze the first sample:

For matrix $$$A$$$ we have:

Rows:
- $$$R_1(A) = [3, 5, 6]; \max(R_1(A)) = 6$$$
- $$$R_2(A) = [1, 7, 9]; \max(R_2(A)) = 9$$$
- $$$R_3(A) = [4, 8, 2]; \max(R_3(A)) = 8$$$
Columns:
- $$$C_1(A) = [3, 1, 4]; \max(C_1(A)) = 4$$$
- $$$C_2(A) = [5, 7, 8]; \max(C_2(A)) = 8$$$
- $$$C_3(A) = [6, 9, 2]; \max(C_3(A)) = 9$$$

$$$X = \{ \max(R_1(A)), \max(R_2(A)), \max(R_3(A)) \} = \{ 6, 9, 8 \}$$$
$$$Y = \{ \max(C_1(A)), \max(C_2(A)), \max(C_3(A)) \} = \{ 4, 8, 9 \}$$$
So $$$S(A) = (X, Y) = (\{ 6, 9, 8 \}, \{ 4, 8, 9 \})$$$

For matrix $$$A'$$$ we have:

Rows:
- $$$R_1(A') = [9, 5, 1]; \max(R_1(A')) = 9$$$
- $$$R_2(A') = [7, 8, 2]; \max(R_2(A')) = 8$$$
- $$$R_3(A') = [3, 6, 4]; \max(R_3(A')) = 6$$$
Columns:
- $$$C_1(A') = [9, 7, 3]; \max(C_1(A')) = 9$$$
- $$$C_2(A') = [5, 8, 6]; \max(C_2(A')) = 8$$$
- $$$C_3(A') = [1, 2, 4]; \max(C_3(A')) = 4$$$

Note that each of this arrays are bitonic.
$$$X = \{ \max(R_1(A')), \max(R_2(A')), \max(R_3(A')) \} = \{ 9, 8, 6 \}$$$
$$$Y = \{ \max(C_1(A')), \max(C_2(A')), \max(C_3(A')) \} = \{ 9, 8, 4 \}$$$
So $$$S(A') = (X, Y) = (\{ 9, 8, 6 \}, \{ 9, 8, 4 \})$$$

Informação

Provedor Codeforces

Código CF1383D