310743: CF1879B. Chips on the Board

You are given a board of size $n \times n$ ($n$ rows and $n$ colums) and two arrays of positive integers $a$ and $b$ of size $n$.

Your task is to place the chips on this board so that the following condition is satisfied for every cell $(i, j)$:

• there exists at least one chip in the same column or in the same row as the cell $(i, j)$. I. e. there exists a cell $(x, y)$ such that there is a chip in that cell, and either $x = i$ or $y = j$ (or both).

The cost of putting a chip in the cell $(i, j)$ is equal to $a_i + b_j$.

For example, for $n=3$, $a=[1, 4, 1]$ and $b=[3, 2, 2]$. One of the possible chip placements is as follows:

White squares are empty

The total cost of that placement is $(1+3) + (1+2) + (1+2) = 10$.

Calculate the minimum possible total cost of putting chips according to the rules above.

The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 3 \cdot 10^5$).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$).

The third line contains $n$ integers $b_1, b_2, \dots, b_n$ ($1 \le b_i \le 10^9$).

The sum of $n$ over all test cases doesn't exceed $3 \cdot 10^5$.

For each test case, print a single integer — the minimum possible total cost of putting chips according to the rules.

4
3
1 4 1
3 2 2
1
4
5
2
4 5
2 3
5
5 2 4 5 3
3 4 2 1 5

10
9
13
24

The first test case of the example is described in the statement.