CF1765J.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.

本题中没有英雄。我们本应将其命名为“归零”。

给定两个数组 $a$ 和 $b$，每个数组均包含 $n$ 个非负整数。

定义一个 $n \times n$ 的矩阵 $c$，其中对任意 $i \in [1, n]$ 和任意 $j \in [1, n]$，均有 $c_{i,j} = |a_i - b_j|$。

你的目标是将矩阵 $c$ 变为零矩阵（即所有元素均为 $0$ 的矩阵）。为此，你可以以任意顺序、任意次数执行以下操作：

• 选择一个整数 $i$，将第 $i$ 行所有元素 $c_{i,j}$（$j \in [1, n]$）减 $1$（即整行减 $1$）。执行该操作需花费 $1$ 枚硬币；
• 选择一个整数 $j$，将第 $j$ 列所有元素 $c_{i,j}$（$i \in [1, n]$）减 $1$（即整列减 $1$）。执行该操作需花费 $1$ 枚硬币；
• 选择两个整数 $i$ 和 $j$，将元素 $c_{i,j}$ 减 $1$。执行该操作需花费 $1$ 枚硬币；
• 选择一个整数 $i$，将第 $i$ 行所有元素 $c_{i,j}$（$j \in [1, n]$）加 $1$（即整行加 $1$）。执行该操作可获得 $1$ 枚硬币；
• 选择一个整数 $j$，将第 $j$ 列所有元素 $c_{i,j}$（$i \in [1, n]$）加 $1$（即整列加 $1$）。执行该操作可获得 $1$ 枚硬币。

你需要计算将矩阵 $c$ 变为零矩阵所需的最少硬币数量。注意：所有操作完成后，矩阵 $c$ 的每个元素必须同时等于 $0$。

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 $b_1, b_2, \dots, b_n$ ($0 \le b_j \le 10^8$).

第一行包含一个整数 $n$（$2 \le n \le 2 \cdot 10^5$）。

第二行包含 $n$ 个整数 $a_1, a_2, \dots, a_n$（$0 \le a_i \le 10^8$）。

第三行包含 $n$ 个整数 $b_1, b_2, \dots, b_n$（$0 \le b_j \le 10^8$）。

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

输出一个整数——将矩阵 $c$ 变为零矩阵所需的最少硬币数量。

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.

在第一个例子中，矩阵如下所示：

1

1

1

0

0

0

1

1

1

你可以花费 $2$ 枚硬币将其变为零矩阵，具体操作如下：

• 从第一行减去 $1$，花费 $1$ 枚硬币；
• 从第三行减去 $1$，花费 $1$ 枚硬币。

在第二个例子中，矩阵如下所示：

2

2

1

0

0

1

2

2

1

你可以花费 $5$ 枚硬币将其变为零矩阵，具体操作如下：

• 从第一行减去 $1$，花费 $1$ 枚硬币；
• 从第三行减去 $1$，花费 $1$ 枚硬币；
• 从第三行减去 $1$，花费 $1$ 枚硬币；
• 从元素 $a_{2,3}$ 减去 $1$，花费 $1$ 枚硬币；
• 向第三列加 $1$，获得 $1$ 枚硬币；
• 从第一行减去 $1$，花费 $1$ 枚硬币；
• 从元素 $a_{2,3}$ 减去 $1$，花费 $1$ 枚硬币。