Points Movement
Description
There are $n$ points and $m$ segments on the coordinate line. The initial coordinate of the $i$-th point is $a_i$. The endpoints of the $j$-th segment are $l_j$ and $r_j$ — left and right endpoints, respectively.
You can move the points. In one move you can move any point from its current coordinate $x$ to the coordinate $x - 1$ or the coordinate $x + 1$. The cost of this move is $1$.
You should move the points in such a way that each segment is visited by at least one point. A point visits the segment $[l, r]$ if there is a moment when its coordinate was on the segment $[l, r]$ (including endpoints).
You should find the minimal possible total cost of all moves such that all segments are visited.
The input consists of multiple test cases. The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Description of the test cases follows.
The first line of each test case contains two integers $n$ and $m$ ($1 \le n, m \le 2 \cdot 10^5$) — the number of points and segments respectively.
The next line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ($-10^9 \le a_i \le 10^9$) — the initial coordinates of the points.
Each of the next $m$ lines contains two integers $l_j$, $r_j$ ($-10^9 \le l_j \le r_j \le 10^9$) — the left and the right endpoints of the $j$-th segment.
It's guaranteed that the sum of $n$ and the sum of $m$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case print a single integer — the minimal total cost of all moves such that all segments are visited.
Input
The input consists of multiple test cases. The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Description of the test cases follows.
The first line of each test case contains two integers $n$ and $m$ ($1 \le n, m \le 2 \cdot 10^5$) — the number of points and segments respectively.
The next line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ($-10^9 \le a_i \le 10^9$) — the initial coordinates of the points.
Each of the next $m$ lines contains two integers $l_j$, $r_j$ ($-10^9 \le l_j \le r_j \le 10^9$) — the left and the right endpoints of the $j$-th segment.
It's guaranteed that the sum of $n$ and the sum of $m$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case print a single integer — the minimal total cost of all moves such that all segments are visited.
Samples
2
4 11
2 6 14 18
0 3
4 5
11 15
3 5
10 13
16 16
1 4
8 12
17 19
7 13
14 19
4 12
-9 -16 12 3
-20 -18
-14 -13
-10 -7
-3 -1
0 4
6 11
7 9
8 10
13 15
14 18
16 17
18 19
5
22
Note
In the first test case the points can be moved as follows:
- Move the second point from the coordinate $6$ to the coordinate $5$.
- Move the third point from the coordinate $14$ to the coordinate $13$.
- Move the fourth point from the coordinate $18$ to the coordinate $17$.
- Move the third point from the coordinate $13$ to the coordinate $12$.
- Move the fourth point from the coordinate $17$ to the coordinate $16$.
The total cost of moves is $5$. It is easy to see, that all segments are visited by these movements. For example, the tenth segment ($[7, 13]$) is visited after the second move by the third point.
Here is the image that describes the first test case:
