Problem:
Score : 900 points

Problem Statement

Mole decided to live in an abandoned mine. The structure of the mine is represented by a simple connected undirected graph which consists of N vertices numbered 1 through N and M edges.
The i-th edge connects Vertices a_i and b_i, and it costs c_i yen (the currency of Japan) to remove it.

Mole would like to remove some of the edges so that there is exactly one path from Vertex 1 to Vertex N that does not visit the same vertex more than once. Find the minimum budget needed to achieve this.

Constraints

2 \leq N \leq 15

N-1 \leq M \leq N(N-1)/2

1 \leq a_i, b_i \leq N

1 \leq c_i \leq 10^{6}

There are neither multiple edges nor self-loops in the given graph.

The given graph is connected.

Input

Input is given from Standard Input in the following format:

N M
a_1 b_1 c_1
:
a_M b_M c_M

Output

Print the answer.

Sample Input 1

4 6
1 2 100
3 1 100
2 4 100
4 3 100
1 4 100
3 2 100

Sample Output 1

200

By removing the two edges represented by the red dotted lines in the figure below, the objective can be achieved for a cost of 200 yen.

Sample Input 2

2 1
1 2 1

Sample Output 2

0

It is possible that there is already only one path from Vertex 1 to Vertex N in the beginning.

Sample Input 3

15 22
8 13 33418
14 15 55849
7 10 15207
4 6 64328
6 9 86902
15 7 46978
8 14 53526
1 2 8720
14 12 37748
8 3 61543
6 5 32425
4 11 20932
3 12 55123
8 2 45333
9 12 77796
3 9 71922
12 15 70793
2 4 25485
11 6 1436
2 7 81563
7 11 97843
3 1 40491

Sample Output 3

133677

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<bits/stdc++.h>
using namespace std;
#define chmax(a,b) a=max(a,b)
const int N=17;

int n,m;
int g[N][N];
int cost[1<<N];
int dp[N][1<<N];

int main()
{
	
	cin>>n>>m;
	while(m--)
	{
		int u,v,w;
		cin>>u>>v>>w;
		u--; v--;
		g[u][v]=g[v][u]=w;
	}
	for(int mask=0;mask<(1<<n);mask++)
	for(int u=0;u<n;u++)
	for(int v=u+1;v<n;v++)
		if(mask&(1<<u) && mask&(1<<v)) cost[mask]+=g[u][v];
	memset(dp,-1,sizeof(dp));
	for(int mask=1;mask<(1<<n);mask+=2)
		dp[0][mask]=cost[mask];
	for(int mask=0;mask<(1<<(n-1));mask++)
	for(int i=0;i<n-1;i++)
	{
		if(dp[i][mask]<0) continue;
		int _=(1<<n)-1-mask;
		for(int mask2=_;mask2>0;mask2=(mask2-1)&_)
		for(int j=0;j<n;j++)
		{
			if(!(mask2&(1<<j))) continue;
			if(!g[i][j]) continue;
			chmax(dp[j][mask|mask2],dp[i][mask]+g[i][j]+cost[mask2]);
		}
	}
	cout<<cost[(1<<n)-1]-dp[n-1][(1<<n)-1]<<'\n';
	
	return 0;
}