A16016. Tenzing and Random Real Numbers

There are $n$ uniform random real variables between 0 and 1, inclusive, which are denoted as $x_1, x_2, \ldots, x_n$ .

Tenzing has $m$ conditions. Each condition has the form of $x_i+x_j\le 1$ or $x_i+x_j\ge 1$ .

Tenzing wants to know the probability that all the conditions are satisfied, modulo $998~244~353$ .

Formally, let $M = 998~244~353$ . It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$ . Output the integer equal to $p \cdot q^{-1} \bmod M$ . In other words, output the integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$ .

The first line contains two integers $n$ and $m$ ( $1\le n\le 20$ , $0\le m\le n^2+n$ ).

Then following $m$ lines of input contains three integers $t$ , $i$ and $j$ ( $t \in \{0,1\}$ , $1\le i\le j\le n$ ).

- If $t=0$ , the condition is $x_i+x_j\le 1$ .
- If $t=1$ , the condition is $x_i+x_j\ge 1$ .

It is guaranteed that all conditions are pairwise distinct.

Output the probability that all the conditions are satisfied, modulo $M = 998~244~353$ .

In the first test case, the conditions are $x_1+x_2 \le 1$ and $x_3+x_3\ge 1$ , and the probability that each condition is satisfied is $\frac 12$ , so the probability that they are both satisfied is $\frac 12\cdot \frac 12=\frac 14$ , modulo $998~244~353$ is equal to $748683265$ .

In the second test case, the answer is $\frac 14$ .

In the third test case, the answer is $\frac 1{16}$ .

In the fourth test case, the sum of all variables must equal $2$ , so the probability is $0$ .