Problem:
Score : 300 points

Problem Statement

You are given an undirected unweighted graph with N vertices and M edges that contains neither self-loops nor double edges.

Here, a self-loop is an edge where a_i = b_i (1≤i≤M), and double edges are two edges where (a_i,b_i)=(a_j,b_j) or (a_i,b_i)=(b_j,a_j) (1≤i<j≤M).

How many different paths start from vertex 1 and visit all the vertices exactly once?

Here, the endpoints of a path are considered visited.

For example, let us assume that the following undirected graph shown in Figure 1 is given.

Figure 1: an example of an undirected graph

The following path shown in Figure 2 satisfies the condition.

Figure 2: an example of a path that satisfies the condition

However, the following path shown in Figure 3 does not satisfy the condition, because it does not visit all the vertices.

Figure 3: an example of a path that does not satisfy the condition

Neither the following path shown in Figure 4, because it does not start from vertex 1.

Figure 4: another example of a path that does not satisfy the condition

Constraints

2≦N≦8

0≦M≦N(N-1)/2

1≦a_i<b_i≦N

The given graph contains neither self-loops nor double edges.

Input

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

N M
a_1 b_1
a_2 b_2
:
a_M b_M

Output

Print the number of the different paths that start from vertex 1 and visit all the vertices exactly once.

Sample Input 1

3 3
1 2
1 3
2 3

Sample Output 1

2

The given graph is shown in the following figure:

The following two paths satisfy the condition:

Sample Input 2

7 7
1 3
2 7
3 4
4 5
4 6
5 6
6 7

Sample Output 2

1

This test case is the same as the one described in the problem statement.

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::vec::Vec;

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>().split_whitespace().map(|e| e.parse().ok().unwrap()).collect()
}

fn read_touple<T: std::str::FromStr + Copy>() -> (T, T) {
  let v: Vec<T> = read_vec();
  assert_eq!(v.len(), 2);
  (v[0], v[1])
}


fn main() {
  solve();
}

fn solve() {
    let (n, m) = read_touple::<usize>();
    let mut g = vec![vec![]; n];
    for _ in 0..m {
      let (a, b) = read_touple::<usize>();
      g[a-1].push(b-1);
      g[b-1].push(a-1);
    }

    let mut visited = vec![false; n];
    visited[0] = true;

    let ans = dfs_rec(0, &g, &mut visited);
    //let ans = dfs_loop(n, &g);

    println!("{}", ans);
}

fn dfs_rec(v: usize, g: &Vec<Vec<usize>>,  visited: &mut Vec<bool>) -> usize {
  let mut all_visited = visited
                          .iter()
                          .fold(true, |acc, &x| acc && x);

  if all_visited {
     return 1;
  }

  let mut res = 0;

  for &e in &g[v] {
    if visited[e] {
       continue
    }

    visited[e] = true;
    res += dfs_rec(e, g, visited);
    visited[e] = false;
  }

  return res
}


// stack に訪問状況の情報も付随しないと解けない
fn dfs_loop(n: usize, g: &Vec<Vec<usize>>) -> usize {
  let mut res = 0;
  let mut stack = Vec::new();

  // init
  let now = 0;
  let visited = vec![false; n];
  stack.push((now, visited));


  while let Some(t) = stack.pop() {
    let (now, mut visited) = t;
    visited[now] = true;
    let all_visited = visited
                        .iter()
                        .fold(true, |acc, &x| acc && x);

    if all_visited {
       res += 1;
       continue
    }

    for &e in &g[now] {
      if visited[e] {
         continue
      }
      stack.push((e, visited.clone()));
    }
  }

  res
}