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# (Mono 4.6.2.0) (judge verdict: Accepted)

using Yorukatsu034.Questions;
using Yorukatsu034.Extensions;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace Yorukatsu034.Questions
{
    /// <summary>
    /// ABC073 D
    /// </summary>
    public class QuestionE : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var nmr = inputStream.ReadIntArray();
            var towns = nmr[0];
            var roads = nmr[1];
            var toVisit = nmr[2];

            var toVisits = inputStream.ReadIntArray().Select(i => i - 1).OrderBy(i => i).ToArray();
            var distances = new int[towns, towns];
            for (int i = 0; i < towns; i++)
            {
                for (int j = 0; j < towns; j++)
                {
                    distances[i, j] = 1 << 28;
                }
            }

            for (int i = 0; i < roads; i++)
            {
                var abc = inputStream.ReadIntArray();
                var a = abc[0] - 1;
                var b = abc[1] - 1;
                var c = abc[2];
                distances[a, b] = c;
                distances[b, a] = c;
            }

            WarshallFloyd(distances);

            yield return GetPermutations(toVisits).Min(visitOrder =>
            {
                long distance = 0;
                for (int i = 0; i < visitOrder.Length - 1; i++)
                {
                    distance += distances[visitOrder[i], visitOrder[i + 1]];
                }
                return distance;
            });
        }

        void WarshallFloyd(int[,] d)
        {
            var n = d.GetLength(0);
            for (int k = 0; k < n; k++)
            {
                for (int i = 0; i < n; i++)
                {
                    for (int j = 0; j < n; j++)
                    {
                        d[i, j] = Math.Min(d[i, j], d[i, k] + d[k, j]);
                    }
                }
            }
        }

        public static IEnumerable<T[]> GetPermutations<T>(IEnumerable<T> collection) where T : IComparable<T>
        {
            var a = collection.ToArray();

            yield return a; // ソート済み初期配列

            if (a.Length <= 2)
            {
                if (a.Length == 2 && a[0].CompareTo(a[1]) != 0)
                {
                    Swap(ref a[1], ref a[0]);
                    yield return a;
                    yield break;
                }

                yield break;
            }

            bool flag = true;
            while (flag)
            {
                flag = false;
                for (int i = a.Length - 2; i >= 0; i--)
                {
                    // iよりi+1の方が大きい（昇順）なら
                    if (a[i].CompareTo(a[i + 1]) < 0)
                    {
                        // 後ろから見ていってi<jとなるところを探して
                        int j;
                        for (j = a.Length - 1; a[i].CompareTo(a[j]) >= 0; j--) { }

                        // iとjを入れ替えて
                        Swap(ref a[j], ref a[i]);

                        // i+1以降を反転
                        if (i < a.Length - 2)
                        {
                            Array.Reverse(a, i + 1, a.Length - (i + 1));
                        }

                        flag = true;
                        yield return a;
                        break;
                    }
                }
            }
        }

        static void Swap<T>(ref T a, ref T b)
        {
            var temp = a;
            a = b;
            b = temp;
        }
    }
}

namespace Yorukatsu034
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionE();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace Yorukatsu034.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }

}

#endregion

#region Extensions

namespace Yorukatsu034.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion