#P2003E2. Turtle and Inversions (Hard Version)
Turtle and Inversions (Hard Version)
Description
This is a hard version of this problem. The differences between the versions are the constraint on $m$ and $r_i < l_{i + 1}$ holds for each $i$ from $1$ to $m - 1$ in the easy version. You can make hacks only if both versions of the problem are solved.
Turtle gives you $m$ intervals $[l_1, r_1], [l_2, r_2], \ldots, [l_m, r_m]$. He thinks that a permutation $p$ is interesting if there exists an integer $k_i$ for every interval ($l_i \le k_i < r_i$), and if he lets $a_i = \max\limits_{j = l_i}^{k_i} p_j, b_i = \min\limits_{j = k_i + 1}^{r_i} p_j$ for every integer $i$ from $1$ to $m$, the following condition holds:
$$\max\limits_{i = 1}^m a_i < \min\limits_{i = 1}^m b_i$$
Turtle wants you to calculate the maximum number of inversions of all interesting permutations of length $n$, or tell him if there is no interesting permutation.
An inversion of a permutation $p$ is a pair of integers $(i, j)$ ($1 \le i < j \le n$) such that $p_i > p_j$.
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.
The first line of each test case contains two integers $n, m$ ($2 \le n \le 5 \cdot 10^3, 0 \le m \le 5 \cdot 10^3$) — the length of the permutation and the number of intervals.
The $i$-th of the following $m$ lines contains two integers $l_i, r_i$ ($1 \le l_i < r_i \le n$) — the $i$-th interval. Note that there may exist identical intervals (i.e., there may exist two different indices $i, j$ such that $l_i = l_j$ and $r_i = r_j$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^3$ and the sum of $m$ over all test cases does not exceed $5 \cdot 10^3$.
For each test case, if there is no interesting permutation, output a single integer $-1$.
Otherwise, output a single integer — the maximum number of inversions.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.
The first line of each test case contains two integers $n, m$ ($2 \le n \le 5 \cdot 10^3, 0 \le m \le 5 \cdot 10^3$) — the length of the permutation and the number of intervals.
The $i$-th of the following $m$ lines contains two integers $l_i, r_i$ ($1 \le l_i < r_i \le n$) — the $i$-th interval. Note that there may exist identical intervals (i.e., there may exist two different indices $i, j$ such that $l_i = l_j$ and $r_i = r_j$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^3$ and the sum of $m$ over all test cases does not exceed $5 \cdot 10^3$.
Output
For each test case, if there is no interesting permutation, output a single integer $-1$.
Otherwise, output a single integer — the maximum number of inversions.
8
2 0
2 1
1 2
5 1
2 4
8 3
1 4
2 5
7 8
7 2
1 4
4 7
7 3
1 2
1 7
3 7
7 4
1 3
4 7
1 3
4 7
7 3
1 2
3 4
5 6
1
0
8
18
-1
-1
15
15
Note
In the third test case, the interesting permutation with the maximum number of inversions is $[5, 2, 4, 3, 1]$.
In the fourth test case, the interesting permutation with the maximum number of inversions is $[4, 3, 8, 7, 6, 2, 1, 5]$. In this case, we can let $[k_1, k_2, k_3] = [2, 2, 7]$.
In the fifth and the sixth test case, it can be proven that there is no interesting permutation.
In the seventh test case, the interesting permutation with the maximum number of inversions is $[4, 7, 6, 3, 2, 1, 5]$. In this case, we can let $[k_1, k_2, k_3, k_4] = [1, 6, 1, 6]$.
In the eighth test case, the interesting permutation with the maximum number of inversions is $[4, 7, 3, 6, 2, 5, 1]$.