A15468. Hero to Zero

There are no heroes in this problem. I guess we should have named it "To Zero".

You are given two arrays $a$ and $b$ , each of these arrays contains $n$ non-negative integers.

Let $c$ be a matrix of size $n \times n$ such that $c_{i,j} = |a_i - b_j|$ for every $i \in [1, n]$ and every $j \in [1, n]$ .

Your goal is to transform the matrix $c$ so that it becomes the zero matrix, i. e. a matrix where every element is exactly $0$ . In order to do so, you may perform the following operations any number of times, in any order:

- choose an integer $i$ , then decrease $c_{i,j}$ by $1$ for every $j \in [1, n]$ (i. e. decrease all elements in the $i$ -th row by $1$ ). In order to perform this operation, you pay $1$ coin;
- choose an integer $j$ , then decrease $c_{i,j}$ by $1$ for every $i \in [1, n]$ (i. e. decrease all elements in the $j$ -th column by $1$ ). In order to perform this operation, you pay $1$ coin;
- choose two integers $i$ and $j$ , then decrease $c_{i,j}$ by $1$ . In order to perform this operation, you pay $1$ coin;
- choose an integer $i$ , then increase $c_{i,j}$ by $1$ for every $j \in [1, n]$ (i. e. increase all elements in the $i$ -th row by $1$ ). When you perform this operation, you receive $1$ coin;
- choose an integer $j$ , then increase $c_{i,j}$ by $1$ for every $i \in [1, n]$ (i. e. increase all elements in the $j$ -th column by $1$ ). When you perform this operation, you receive $1$ coin.

You have to calculate the minimum number of coins required to transform the matrix $c$ into the zero matrix. Note that all elements of $c$ should be equal to $0$ simultaneously after the operations.

The first line contains one integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $0 \le a_i \le 10^8$ ).

The third line contains $n$ integers $b_1, b_2, \dots, b_n$ ( $0 \le b_j \le 10^8$ ).

Print one integer — the minimum number of coins required to transform the matrix $c$ into the zero matrix.

In the first example, the matrix looks as follows:

111000111You can turn it into a zero matrix using $2$ coins as follows:

- subtract $1$ from the first row, paying $1$ coin;
- subtract $1$ from the third row, paying $1$ coin.

In the second example, the matrix looks as follows:

221001221You can turn it into a zero matrix using $5$ coins as follows:

- subtract $1$ from the first row, paying $1$ coin;
- subtract $1$ from the third row, paying $1$ coin;
- subtract $1$ from the third row, paying $1$ coin;
- subtract $1$ from $a_{2,3}$ , paying $1$ coin;
- add $1$ to the third column, receiving $1$ coin;
- subtract $1$ from the first row, paying $1$ coin;
- subtract $1$ from $a_{2,3}$ , paying $1$ coin.