Hero to Zero
Description
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.
Input
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 $b_1, b_2, \dots, b_n$ ($0 \le b_j \le 10^8$).
Output
Print one integer — the minimum number of coins required to transform the matrix $c$ into the zero matrix.
3
1 2 3
2 2 2
3
3 1 3
1 1 2
2
1 0
2 1
2
1 4
2 3
4
1 3 3 7
6 9 4 2
2
5
2
4
29
Note
In the first example, the matrix looks as follows:
| 1 | 1 | 1 |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
You 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:
| 2 | 2 | 1 |
| 0 | 0 | 1 |
| 2 | 2 | 1 |
You 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.