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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.Scanner;

public class Main {
	private static int[] encode = {0, 1, 2, 3, 5, 7, 11, 13, 17};
	private static int[] decode = {0, 1, 2, 3, 0, 4, 0, 5, 0, 0, 0, 6, 0, 7, 0, 0, 0, 8};
	private static int[][] graph = new int[9][9];
	private static int goal = 1;

	public static void main(String[] args) {
		Scanner scanner = new Scanner(System.in);
		String[] conf = scanner.nextLine().split(" ");
		int numNodes = Integer.parseInt(conf[0]);
		int numEdges = Integer.parseInt(conf[1]);

		for (int i = 1; i <= numNodes; i++) goal *= encode[i];
		for (int i = 0; i < numEdges; i++) {
			String[] input = scanner.nextLine().split(" ");
			int start = Integer.parseInt(input[0]);
			int end = Integer.parseInt(input[1]);
			graph[start][end] = 1;
			graph[end][start] = 1;
		}

		System.out.println(numPaths(1, 1));
	}

	private static int numPaths(int current, int passed) {
		if (passed == goal) return 1;

		int numPaths = 0;
		for (int dest = 1; dest <= 8; dest++) {
			if (graph[current][dest] == 1 && passed % encode[dest] != 0) {
				numPaths += numPaths(dest, passed * encode[dest]);
			}
		}
		return numPaths;
	}
}