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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.Arrays;
import java.util.Scanner;
import java.util.stream.IntStream;

public class Main {

	public static void main(String[] args) {
		try (Scanner scanner = new Scanner(System.in)) {
			int n = scanner.nextInt(), m = scanner.nextInt(), q = scanner.nextInt();
			// 町と町の間の移動最短距離
			long[][] d = new long[n][n];
			IntStream.range(0, n).forEach(i -> {
				Arrays.fill(d[i], Integer.MAX_VALUE);
				d[i][i] = 0;
			});
			int[] r = new int[q];
			IntStream.range(0, q).forEach(i -> r[i] = scanner.nextInt() - 1);
			IntStream.range(0, m).forEach(i -> {
				int a = scanner.nextInt() - 1;
				int b = scanner.nextInt() - 1;
				int c = scanner.nextInt();
				d[a][b] = d[b][a] = c;
			});
			warshallFloyd(d, n);
			System.out.println(getResult(d, r, -1, new boolean[q], 1));
		}
	}

	/**
	 * 町と町の間の移動最短距離をワーシャル–フロイド法で計算する
	 * 
	 * @param d 町と町の間の移動最短距離の配列
	 * @param n 町の数
	 */
	private static void warshallFloyd(long[][] d, int n) {
		IntStream.range(0, n).forEach(k -> IntStream.range(0, n)
				.forEach(j -> IntStream.range(0, n).forEach(i -> d[i][j] = Math.min(d[i][j], d[i][k] + d[k][j]))));
	}

	/**
	 * 解説通りに実装したメソッド
	 * 
	 * @param d       町と町の最短距離の配列
	 * @param r       訪ねる町の配列
	 * @param prev    直前に訪ねた町（存在しないときは-1）
	 * @param visited 町が訪ねたかどうかの配列
	 * @param depth   訪ねた町の数
	 * @return 訪ねる町をすべて訪ねたときの移動距離
	 */
	private static long getResult(final long[][] d, final int[] r, int prev, boolean[] visited, int depth) {
		if (depth > r.length) {
			return 0L;
		}
		long result = Integer.MAX_VALUE;
		for (int i = 0; i < r.length; i++) {
			if (!visited[i]) {
				visited[i] = true;
				if (-1 == prev) {
					result = Math.min(result, getResult(d, r, i, visited, depth + 1));
				} else {
					result = Math.min(result, getResult(d, r, i, visited, depth + 1) + d[r[prev]][r[i]]);
				}
				visited[i] = false;
			}
		}
		return result;
	}
}