Problem:
Score : 400 points

Problem Statement

There are N towns in the State of Atcoder, connected by M bidirectional roads.

The i-th road connects Town A_i and B_i and has a length of C_i.

Joisino is visiting R towns in the state, r_1,r_2,..,r_R (not necessarily in this order).

She will fly to the first town she visits, and fly back from the last town she visits, but for the rest of the trip she will have to travel by road.

If she visits the towns in the order that minimizes the distance traveled by road, what will that distance be?

Constraints

2≤N≤200

1≤M≤N×(N-1)/2

2≤R≤min(8,N) (min(8,N) is the smaller of 8 and N.)

r_i≠r_j (i≠j)

1≤A_i,B_i≤N, A_i≠B_i

(A_i,B_i)≠(A_j,B_j),(A_i,B_i)≠(B_j,A_j) (i≠j)

1≤C_i≤100000

Every town can be reached from every town by road.

All input values are integers.

Input

Input is given from Standard Input in the following format:

N M R
r_1 ... r_R
A_1 B_1 C_1
:
A_M B_M C_M

Output

Print the distance traveled by road if Joisino visits the towns in the order that minimizes it.

Sample Input 1

3 3 3
1 2 3
1 2 1
2 3 1
3 1 4

Sample Output 1

2

For example, if she visits the towns in the order of 1, 2, 3, the distance traveled will be 2, which is the minimum possible.

Sample Input 2

3 3 2
1 3
2 3 2
1 3 6
1 2 2

Sample Output 2

4

The shortest distance between Towns 1 and 3 is 4. Thus, whether she visits Town 1 or 3 first, the distance traveled will be 4.

Sample Input 3

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

Sample Output 3

3

Solution in C++ (Clang 10.0.0) (judge verdict: Wrong Answer)

#define _USE_MATH_DEFINES
#include <stdio.h>
#include <iostream>
#include <string>
#include <vector>
#include <utility>
#include <algorithm>
#include <queue>
#include <list>
#include <stack>
#include <math.h>
#include <map>
#include <unordered_map>
#include <set>
using namespace std;

#define ll long long
#define ull unsigned long long

#define	max	1000000007

ll gcd(ll a, ll b) {
	if (a < b)
		swap(a, b);
	while (ll r = a % b)
		a = b, b = r;
	return b;
}

class mod
{
public:
	ll val;

	mod(ll v)
	{
		val = v % max;
	}

	mod()
	{
		val = 0;
	}

	void operator -=(const mod &v)
	{
		val -= v.val;

		if (val < 0)
			val += max;
	}

	void operator +=(const mod &v)
	{
		if (val + v.val >= max)
			val -= max;
		val += v.val;
	}

	void operator +=(ll &v)
	{
		if (val + v >= max)
			val -= max;
		val += v;
	}

	void operator *=(const mod &v)
	{
		val *= v.val;
		val %= max;
	}

	void operator *=(ll &v)
	{
		val *= v;
		val %= max;
	}

	ll modinv(ll a)
	{
		ll b = max, u = 1, v = 0;
		while (b)
		{
			ll t = a / b;
			a -= t * b; swap(a, b);
			u -= t * v; swap(u, v);
		}
		u %= max;
		if (u < 0)
			u += max;
		return u;
	}

	void operator /=(const mod &v)
	{
		val *= modinv(v.val);
		val %= max;
	}

	void operator /=(ll &v)
	{
		val *= modinv(v);
		val %= max;
	}
};

struct UnionFind
{
	vector<int> par; // par[i]:iの親の番号　(例) par[3] = 2 : 3の親が2
	vector<int> childN;
	ll r = 0;

	UnionFind(int N) : par(N), childN(N)
	{ 
		//最初は全てが根であるとして初期化
		for (int i = 0; i < N; i++) 
			par[i] = i, childN[i] = 1;
	}

	int root(int x) 
	{ 
		// データxが属する木の根を再帰で得る：root(x) = {xの木の根}
		if (par[x] == x) return x;
		return par[x] = root(par[x]);
	}

	void unite(int x, int y) 
	{
		// xとyの木を併合
		int rx = root(x); //xの根をrx
		int ry = root(y); //yの根をry
		if (rx == ry) return; //xとyの根が同じ(=同じ木にある)時はそのまま
		par[rx] = ry; //xとyの根が同じでない(=同じ木にない)時：xの根rxをyの根ryにつける
		r -= childN[rx] * (childN[rx] - 1) / 2;
		r -= childN[ry] * (childN[ry] - 1) / 2;
		childN[ry] += childN[rx];
		childN[rx] = 0;
		r += childN[ry] * (childN[ry] - 1) / 2;
	}

	bool same(int x, int y)
	{
		// 2つのデータx, yが属する木が同じならtrueを返す
		int rx = root(x);
		int ry = root(y);
		return rx == ry;
	}
};

//	for priority queue
//	priority_queue<ipair, vector<ipair>, greater<ipair>>
using ipair = pair<int, pair<int, int>>;
bool operator>(const ipair a, const ipair b) { return a.first > b.first; };
bool operator<(const ipair a, const ipair b) { return a.first < b.first; };

using xpair = pair<int, int>;
bool operator>(const xpair &a, const xpair &b) { return a.second > b.second || (a.second == b.second && a.first > b.first); };
bool operator<(const xpair &a, const xpair &b) { return a.second < b.second || (a.second == b.second && a.first > b.first); };
bool operator==(const xpair &a, const xpair &b) { return a.first== b.first; };

int main()
{
	int n, m, R;
	cin >> n >> m >> R;
	vector<int> r(R);
	for (int i = 0; i < R; i++)
	{
		cin >> r[i];
		r[i]--;
	}
	vector<vector<int>> a(n, vector<int>(n, max));
	for (int i = 0; i < n; i++)
		a[i][i] = 0;
	for (int i = 0; i < m; i++)
	{
		int x, y, z;
		cin >> x >> y >> z;
		x--;
		y--;
		a[x][y] = a[y][x] = z;
	}

	for (int k = 0; k < n; k++)
		for (int i = 0; i < n; i++)
			for (int j = 0; j < n; j++)
			{
				int d = a[i][k] + a[k][j];
				if (d < a[i][j])
					a[i][j] = a[j][i] = d;
			}
	vector<vector<int>> b(R, vector<int>(R, max));
	for (int i = 0; i < R; i++)
	{
		for (int j = 0; j < R; j++)
			b[j][i] = b[i][j] = a[r[i]][r[j]];
		b[i][i] = max;
	}

	int result = max;
	for (int head = 0; head < R; head++)
	{
		int now = head;
		vector<bool> done(R, false);
		done[head] = true;
		int rt = 0;
		for (int i = 1; i < R; i++)
		{
			int mi = max, next = -1;
			for (int j = 0; j < R; j++)
				if (!done[j] && j != now && mi > b[now][j])
					next = j, mi = b[now][j];
			now = next;
			rt += mi;
		}
		if (rt < result)
			result = rt;
	}

	printf("%d", result);
	return 0;
}