A13226. Obtain a Permutation
编程题
普及/提高-
知识点
题目描述
You are given a rectangular matrix of size $n \times m$ consisting of integers from $1$ to $2 \cdot 10^5$ .
In one move, you can:
- choose any element of the matrix and change its value to any integer between $1$ and $n \cdot m$ , inclusive;
- take any column and shift it one cell up cyclically (see the example of such cyclic shift below).
A cyclic shift is an operation such that you choose some $j$ ( $1 \le j \le m$ ) and set $a_{1, j} := a_{2, j}, a_{2, j} := a_{3, j}, \dots, a_{n, j} := a_{1, j}$ simultaneously.
 Example of cyclic shift of the first column You want to perform the minimum number of moves to make this matrix look like this:
In other words, the goal is to obtain the matrix, where $a_{1, 1} = 1, a_{1, 2} = 2, \dots, a_{1, m} = m, a_{2, 1} = m + 1, a_{2, 2} = m + 2, \dots, a_{n, m} = n \cdot m$ (i.e. $a_{i, j} = (i - 1) \cdot m + j$ ) with the minimum number of moves performed.
In one move, you can:
- choose any element of the matrix and change its value to any integer between $1$ and $n \cdot m$ , inclusive;
- take any column and shift it one cell up cyclically (see the example of such cyclic shift below).
A cyclic shift is an operation such that you choose some $j$ ( $1 \le j \le m$ ) and set $a_{1, j} := a_{2, j}, a_{2, j} := a_{3, j}, \dots, a_{n, j} := a_{1, j}$ simultaneously.
 Example of cyclic shift of the first column You want to perform the minimum number of moves to make this matrix look like this:
In other words, the goal is to obtain the matrix, where $a_{1, 1} = 1, a_{1, 2} = 2, \dots, a_{1, m} = m, a_{2, 1} = m + 1, a_{2, 2} = m + 2, \dots, a_{n, m} = n \cdot m$ (i.e. $a_{i, j} = (i - 1) \cdot m + j$ ) with the minimum number of moves performed.
输入格式
The first line of the input contains two integers $n$ and $m$ ( $1 \le n, m \le 2 \cdot 10^5, n \cdot m \le 2 \cdot 10^5$ ) — the size of the matrix.
The next $n$ lines contain $m$ integers each. The number at the line $i$ and position $j$ is $a_{i, j}$ ( $1 \le a_{i, j} \le 2 \cdot 10^5$ ).
The next $n$ lines contain $m$ integers each. The number at the line $i$ and position $j$ is $a_{i, j}$ ( $1 \le a_{i, j} \le 2 \cdot 10^5$ ).
输出格式
Print one integer — the minimum number of moves required to obtain the matrix, where $a_{1, 1} = 1, a_{1, 2} = 2, \dots, a_{1, m} = m, a_{2, 1} = m + 1, a_{2, 2} = m + 2, \dots, a_{n, m} = n \cdot m$ ( $a_{i, j} = (i - 1)m + j$ ).
输入输出样例
输入 #1
3 3 3 2 1 1 2 3 4 5 6
输出 #1
6
输入 #2
4 3 1 2 3 4 5 6 7 8 9 10 11 12
输出 #2
0
输入 #3
3 4 1 6 3 4 5 10 7 8 9 2 11 12
输出 #3
2
说明/提示
In the first example, you can set $a_{1, 1} := 7, a_{1, 2} := 8$ and $a_{1, 3} := 9$ then shift the first, the second and the third columns cyclically, so the answer is $6$ . It can be shown that you cannot achieve a better answer.
In the second example, the matrix is already good so the answer is $0$ .
In the third example, it is enough to shift the second column cyclically twice to obtain a good matrix, so the answer is $2$ .
In the second example, the matrix is already good so the answer is $0$ .
In the third example, it is enough to shift the second column cyclically twice to obtain a good matrix, so the answer is $2$ .