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 (GCC 5.4.1) (judge verdict: Accepted)

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

typedef struct node{
    int n;
    int flag;
    struct node* next[8];
}node;

void reset(node* target){
    for(int i=0; i<8; i++){
        target->next[i] = NULL;
    }
}

int t[8][8];

int func(node* target, int n){
    if((target->flag & n) == n){
        return 1;
    }
    int x = 0;
    for(int i=0; i<8; i++){
        if(t[target->n][i]){
            if((target->flag & (int)pow(2, i)) == 0){
                target->next[x] = (node*)malloc(sizeof(node));
                target->next[x]->n = i;
                target->next[x]->flag = target->flag + (int)pow(2,i);
                reset(target->next[x]);
                x++;
            }
        }
    }
    int sum = 0, y = 0;
    while(target->next[y] != NULL){
        sum += func(target->next[y], n);
        y++;
    }
    return sum;
}

int main(){
    int n, m;
    scanf("%d%d", &n, &m);
    for(int i=0; i<8; i++){
        for(int j=0; j<8; j++){
            t[i][j] = 0;
        }
    }
    for(int i=0; i<m; i++){
        int a, b;
        scanf("%d%d", &a, &b);
        a--;
        b--;
        t[a][b] = 1;
        t[b][a] = 1;
    }
    node root;
    root.n = 0;
    root.flag = 1;
    reset(&root);
    printf("%d", func(&root, (int)pow(2, n)-1));

    return 0;
}