310425: CF1831E. Hyperregular Bracket Strings

You are given an integer $n$ and $k$ intervals. The $i$-th interval is $[l_i,r_i]$ where $1 \leq l_i \leq r_i \leq n$.

Let us call a regular bracket sequence$^{\dagger,\ddagger}$ of length $n$ hyperregular if for each $i$ such that $1 \leq i \leq k$, the substring $\overline{s_{l_i} s_{l_{i}+1} \ldots s_{r_i}}$ is also a regular bracket sequence.

Your task is to count the number of hyperregular bracket sequences. Since this number can be really large, you are only required to find it modulo $998\,244\,353$.

$^\dagger$ A bracket sequence is a string containing only the characters "(" and ")".

$^\ddagger$ A bracket sequence is called regular if one can turn it into a valid math expression by adding characters + and 1. For example, sequences (())(), (), (()(())) and the empty string are regular, while )(, ((), and (()))( are not.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 10^5$) — the number of test cases. The description of test cases follows.

The first line of each test case contains two integers $n$ and $k$ ($1 \le n \le 3 \cdot 10^5$, $0 \le k \le 3 \cdot 10^5$) — the length of the hyperregular bracket sequences and the number of intervals respectively.

The following $k$ lines of each test case contains two integers $l_i$ and $r_i$ ($1 \le l \le r \le n$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $3 \cdot 10^5$ and the sum of $k$ across all test cases does not exceed $3 \cdot 10^5$.

For each test case, output the number of hyperregular bracket sequences modulo $998\,244\,353$.

7
6 0
5 0
8 1
1 3
10 2
3 4
6 9
1000 3
100 701
200 801
300 901
28 5
1 12
3 20
11 14
4 9
18 19
4 3
1 4
1 4
1 4

5
0
0
4
839415253
140
2

• For the first testcase, the $5$ hyperregular bracket strings of length $6$ are: ((())), (()()), (())(), ()(()) and ()()().
• For the second testcase, there are no regular bracket strings of length $5$, and consequently, there are no hyperregular bracket strings of length $5$.
• For the third testcase, there are no hyperregular bracket strings of length $8$ for which the substring $[1 \ldots 3]$ is a regular bracket string.
• For the fourth testcase, there $4$ hyperregular bracket strings are: ((())(())), ((())()()), ()()((())) and ()()(()())