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 Scala (2.11.7) (judge verdict: Accepted)

import java.util.Scanner

import scala.collection.mutable

object Main extends App {
  val sc = new Scanner(System.in)
  val Vector(n, m) = Vector.fill(2)(sc.nextInt)
  val map: mutable.Map[Int, Set[Int]] = mutable.Map.empty
  (1 to n).foreach { i => map.update(i, Set.empty) }

  val es = Vector.fill(m)(Edge(sc.nextInt, sc.nextInt))
  es.foreach { e =>
    map.update(e.v1, map(e.v1) + e.v2)
    map.update(e.v2, map(e.v2) + e.v1)
  }

  println(solve(1, Set(1), "1"))

  def solve(current: Int, already: Set[Int], route: String): Int = {
    val ints: Set[Int] = map(current) -- already
    if (already.size == n) 1 else {
      if (ints.isEmpty) 0
      else ints.toStream.map(n => solve(n, already + n, route + s"->$n")).sum
    }
  }
}

case class Edge(v1: Int, v2: Int)

case class Vertex(index: Int, nexts: Set[Int])