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E - Count Simple Paths

Score : $500$ points

Problem Statement

You are given a simple undirected graph with $N$ vertices numbered $1$ to $N$ and $M$ edges numbered $1$ to $M$. Edge $i$ connects vertex $u_i$ and vertex $v_i$. The degree of each vertex is at most $10$.
Let $K$ be the number of simple paths (paths without repeated vertices) starting from vertex $1$. Print $\min(K, 10^6)$.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq \min \left(2 \times 10^5, \frac{N(N-1)}{2}\right)$
• $1 \leq u_i, v_i \leq N$
• The given graph is simple.
• The degree of each vertex in the given graph is at most $10$.
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_M$ $v_M$

Output

Print the answer.

4 2
1 2
2 3

3

We have the following three paths that count. (Note that a path of length $0$ also counts.)

• Vertex $1$;
• vertex $1$, vertex $2$;
• vertex $1$, vertex $2$, vertex $3$.

4 6
1 2
1 3
1 4
2 3
2 4
3 4

16

8 21
2 6
1 3
5 6
3 8
3 6
4 7
4 6
3 4
1 5
2 4
1 2
2 7
1 4
3 5
2 5
2 3
4 5
3 7
6 7
5 7
2 8

2023