Hero to Zero

4000ms

524288K

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.

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.

Sample Input:

3
1 2 3
2 2 2

Sample Output:

Sample Input:

3
3 1 3
1 1 2

Sample Output:

Sample Input:

2
1 0
2 1

Sample Output:

Sample Input:

2
1 4
2 3

Sample Output:

Sample Input:

4
1 3 3 7
6 9 4 2

Sample Output:

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.

Informação

Provedor Codeforces

Código CF1765J