A15317. Misha and Paintings

Misha has a square $n \times n$ matrix, where the number in row $i$ and column $j$ is equal to $a_{i, j}$ . Misha wants to modify the matrix to contain exactly $k$ distinct integers. To achieve this goal, Misha can perform the following operation zero or more times:

1. choose any square submatrix of the matrix (you choose $(x_1,y_1)$ , $(x_2,y_2)$ , such that $x_1 \leq x_2$ , $y_1 \leq y_2$ , $x_2 - x_1 = y_2 - y_1$ , then submatrix is a set of cells with coordinates $(x, y)$ , such that $x_1 \leq x \leq x_2$ , $y_1 \leq y \leq y_2$ ),
2. choose an integer $k$ , where $1 \leq k \leq n^2$ ,
3. replace all integers in the submatrix with $k$ .

Please find the minimum number of operations that Misha needs to achieve his goal.

The first input line contains two integers $n$ and $k$ ( $1 \leq n \leq 500, 1 \leq k \leq n^2$ ) — the size of the matrix and the desired amount of distinct elements in the matrix.

Then $n$ lines follows. The $i$ -th of them contains $n$ integers $a_{i, 1}, a_{i, 2}, \ldots, a_{i, n}$ ( $1 \leq a_{i,j} \leq n^2$ ) — the elements of the $i$ -th row of the matrix.

Output one integer — the minimum number of operations required.

In the first test case the answer is $1$ , because one can change the value in the bottom right corner of the matrix to $1$ . The resulting matrix can be found below:

111112341In the second test case the answer is $2$ . First, one can change the entire matrix to contain only $1$ s, and the change the value of any single cell to $2$ . One of the possible resulting matrices is displayed below:

111111112