A13613. Rearrange

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.

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.

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.

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 $S(A') = (X, Y) = (\{ 9, 8, 6 \}, \{ 9, 8, 4 \})$