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 C (Clang 3.8.0) (judge verdict: Accepted)

#include <stdio.h>
#include <string.h>
#include <stdlib.h>

void get_int2(char *line, int size, int *a1, int *a2) {
  if(!fgets(line, size, stdin)) return;
  sscanf(line, "%d %d", a1, a2);
}

#define BUF_SIZE 10
#define VERTEX_MAX 8

static int mat[VERTEX_MAX+1][VERTEX_MAX+1];
int cnt = 0;

void swap(int *a, int *b) {
  int tmp = *a;
  *a = *b;
  *b = tmp;
}

void permute(int *arr, int size, int start, int end) {
  int i;
  if(start+1 >= end) {
    int prev = arr[1];
    int flag = 1;
    for(i = 2; i < size; i++) {
      int cur = arr[i];
      if(!mat[prev][cur]) {
        flag = 0; break;
      }
      prev = cur;
    }
    if (flag) cnt++;
    return;
  }
  for(i = start; i < end; i++) {
    swap(&arr[start], &arr[i]);
    permute(arr, size, start+1, end);
    swap(&arr[start], &arr[i]);
  }
}

int main(void) {
  int vs, es;
  char line[BUF_SIZE];
  get_int2(line, BUF_SIZE, &vs, &es);
  int i, j;
  for(i = 0; i < es; i++) {
    int v1, v2;
    get_int2(line, BUF_SIZE, &v1, &v2);
    mat[v1][v2] = mat[v2][v1] = 1;
  }
  int *arr = malloc(sizeof(int)*(vs+1));
  for(i = 1; i <= vs; i++) {
    arr[i] = i;
  }

  // start: 1 fixed
  permute(arr, vs+1, 2, vs+1);
  printf("%d\n", cnt);



  return 0;
}