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 10.0.0) (judge verdict: Accepted)

#include<stdio.h>
#include<stdlib.h>
#include<string.h>
int digg(int here, int N, int *map, int *been);

int digg(int here, int N, int *map, int *been) {
	int i;
	int sum;
	int paths;

	been[here] = 1;
	//if \sum been = N, vertex here is the end of path.
	sum = 0;
	for(i = 0; i < N; i += 1) {
		sum += been[i];
	}
	if(sum == N) {
		//printf("exit %d:endofpath\n", here);
		been[here] = 0;
		return 1;
	}

	paths= 0;
	for(i = 0; i < N; i += 1) {
		//Check if path vertex here to vertex i be open,
		//and also never been to vertex i.
		//If true, repeat digging from vertex i
		//else, check vertex i+1.
		//printf("now %d, check %d\n", here, i);
		if(map[here + N * i] == 0) {
			//printf("  %d closed\n", i);
			continue;
		}
		if(been[i] == 1) {
			//printf("  %d visited\n", i);
			continue;
		}
		//printf("  %d open\n", i);
		paths += digg(i, N, map, been);
	}
	been[here] = 0;
	//printf("exit %d\n", here);
	//paths is the number of paths whose end vertex is deeper than vertex 'here'
	return paths;
}

int main(void) {
	int N, M;//vertices, bonds
	int a, b;
	int *map;//map[i+n*j)]=0:i to j closed, =1:i to j open
	int *been;
	int ans;
	int i, j;
	char buf[128], *ch;


	fgets(buf, sizeof(buf), stdin);
	ch = strtok(buf, " ");
	N = atoi(ch);
	ch = strtok(NULL, " ");
	M = atoi(ch);

	been = (int *)calloc(N, sizeof(int));
	map = (int *)calloc(N * N, sizeof(int));

	//make map
	for(i = 0; i < M; i += 1) {
		fgets(buf, sizeof(buf), stdin);
		ch = strtok(buf, " ");
		a = atoi(ch);
		ch = strtok(NULL, " ");
		b = atoi(ch);

		map[(a - 1) + (b - 1) * N] = 1;
		map[(b - 1) + (a - 1) * N] = 1;
	}

	//search path to the end
	ans = digg(0, N, map, been);
	printf("%d\n", ans);

	free(been);
	free(map);
	return 0;
}