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-csc 3.5.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;

namespace Contest
{
  class Program
  {
    static void Main(string[] args)
    {
      ABC072.D();
    }
    class ABC072
    {
      public static void A()
      {
        var N = Read.Str();
        Console.WriteLine(N.Contains('9') ? "Yes" : "No");
      }
      public static void B()
      {
        var N = Read.Int();
        var LR = Read.TupleInt(N);
        int sum = 0;
        foreach(var lr in LR)
        {
          sum += lr.Item2 - lr.Item1 + 1;
        }
        Console.WriteLine(sum);
      }
      public static void C()
      {
        var N = Read.Int();
        var A = Enumerable.Range(0, N).Select(i => Read.Int()).ToArray();
        var freq = Util.Frequency(A);
        Console.WriteLine(freq.Values.Count(x => x % 2 == 1));
      }
      public static void D()
      {
        var N = Read.Int();
        var M = Read.Int();
        var Rs = Read.Int();
        var R = Read.IntN();
        var Adj = Read.AdjacencyListWithWeight(N, M);
        var dist = Util.DistFromAdj(N, Adj);
        Util.WarshallFloyd(dist);
        var dist2 = new long[Rs, Rs];
        for(int i = 0; i < Rs; ++i)
        {
          for(int j = 0; j < Rs; ++j)
          {
            dist2[i, j] = i == j ? 0 : dist[R[i] - 1, R[j] - 1];
          }
        }
        var dp = new long[Rs, 1 << Rs];
        for (int i = 0; i < Rs; ++i)
        {
          for (int j = 0; j < (1 << Rs) - 1; ++j)
          {
            dp[i, j] = (long)1e18;
          }
        }
        for (int s = (1 << Rs) - 2; s >= 0; --s)
        {
          for (int v = 0; v < Rs; ++v)
          {
            for (int u = 0; u < Rs; ++u)
            {
              dp[v, s] = Math.Min(dp[v, s], dp[u, s | 1 << u] + dist2[v, u]);
            }
          }
        }
        long min = (long)1e18;
        for (int i = 0; i < Rs; ++i)
        {
          min = Math.Min(min, dp[i, 0]);
        }
        Console.WriteLine(min);
      }
    }

    class Util
    {
      public static T[,] Gen<T>(int r, int c, T v)
      {
        var matrix = new T[r, c];
        for (int i = 0; i < r; ++i)
        {
          for (int j = 0; j < c; ++j)
          {
            matrix[i, j] = v;
          }
        }
        return matrix;
      }
      public static long[] ASum(long[] array)
      {
        var res = new long[array.Length + 1];
        for (int i = 0; i < array.Length; ++i)
        {
          res[i + 1] = res[i] + array[i];
        }
        return res;
      }
      public static long[,] ASum(long[,] matrix)
      {
        var N = matrix.GetLength(0);
        var M = matrix.GetLength(1);
        var res = new long[N + 1, M + 1];
        for(int i = 0; i < N; ++i)
        {
          for(int j = 0; j < M; ++j)
          {
            res[i + 1, j + 1] = res[i, j + 1] + res[i + 1, j] - res[i, j] + matrix[i, j];
          }
        }
        return res;
      }
      public static Dictionary<T, long> Frequency<T>(T[] array)
      {
        var dict = new Dictionary<T, long>();
        foreach (var a in array)
        {
          if (!dict.ContainsKey(a)) { dict.Add(a, 0); }
          dict[a]++;
        }
        return dict;
      }

      public static void WarshallFloyd(long[,] dist)
      {
        var N = dist.GetLength(0);
        for (int k = 0; k < N; ++k)
        {
          for (int i = 0; i < N; ++i)
          {
            for (int j = 0; j < N; ++j)
            {
              dist[i, j] = Math.Min(dist[i, j], dist[i, k] + dist[k, j]);
            }
          }
        }
      }
      public static long[,] DistFromAdj(int N, Dictionary<int, Dictionary<int, long>> adj)
      {
        var dist = new long[N, N];
        for (int i = 0; i < N; ++i)
        {
          for (int j = 0; j < N; ++j)
          {
            dist[i, j] = adj.ContainsKey(i + 1) && adj[i + 1].ContainsKey(j + 1) ? adj[i + 1][j + 1] : (long)1e18;
          }
        }
        return dist;
      }
    }

    static class Read
{
static Queue<string> queue = new Queue<string>();
static void Enqueue() { var ss = Console.ReadLine().Split(' '); foreach (var s in ss) { queue.Enqueue(s); } }
public static int Int() { if (queue.Count == 0) { Enqueue(); } return Convert.ToInt32(queue.Dequeue()); }
public static long Long() { if (queue.Count == 0) { Enqueue(); } return Convert.ToInt64(queue.Dequeue()); }
public static string Str() { if (queue.Count == 0) { Enqueue(); } return queue.Dequeue(); }
public static int[] IntN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToInt32(s)).ToArray(); }
public static long[] LongN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToInt64(s)).ToArray(); }
public static ulong[] ULongN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToUInt64(s)).ToArray(); }
public static string[] StrN() { return Console.ReadLine().Split(' '); }
public static Tuple<int, int>[] TupleInt(int n) { List<Tuple<int, int>> list = new List<Tuple<int, int>>(); for (int i = 0; i < n; ++i) { var ab = Read.IntN(); list.Add(new Tuple<int, int>(ab[0], ab[1])); } return list.ToArray(); }
public static Tuple<long, long>[] TupleLong(int n) { List<Tuple<long, long>> list = new List<Tuple<long, long>>(); for (int i = 0; i < n; ++i) { var ab = Read.LongN(); list.Add(new Tuple<long, long>(ab[0], ab[1])); } return list.ToArray(); }
public static Tuple<string, string>[] TupleStr(int n) { List<Tuple<string, string>> list = new List<Tuple<string, string>>(); for (int i = 0; i < n; ++i) { var ab = Read.StrN(); list.Add(new Tuple<string, string>(ab[0], ab[1])); } return list.ToArray(); }
public static Tuple<double, double>[] TupleDouble(int n) { List<Tuple<double, double>> list = new List<Tuple<double, double>>(); for (int i = 0; i < n; ++i) { var ab = Read.LongN(); list.Add(new Tuple<double, double>(ab[0], ab[1])); } return list.ToArray(); }
public static long[,] LongMatrix(int r, int c) { var mat = new long[r, c]; for (int i = 0; i < r; ++i) { var x = Read.LongN(); for (int j = 0; j < c; ++j) { mat[i, j] = x[j]; } } return mat; }
public static Dictionary<int, HashSet<int>> AdjacencyList(int N, int M, bool isDirectional = false)
{
Dictionary<int, HashSet<int>> dict = new Dictionary<int, HashSet<int>>();
for (int i = 0; i <= N; ++i) { dict.Add(i, new HashSet<int>()); }
        for (int i = 0; i < M; ++i) { var ab = Read.IntN(); if (!dict[ab[0]].Contains(ab[1])) { dict[ab[0]].Add(ab[1]); } if (!isDirectional && !dict[ab[1]].Contains(ab[0])) { dict[ab[1]].Add(ab[0]); } }
        return dict;
}
public static Dictionary<int, Dictionary<int, long>> AdjacencyListWithWeight(int N, int M)
{
Dictionary<int, Dictionary<int, long>> dict = new Dictionary<int, Dictionary<int, long>>(); for (int i = 0; i <= N; ++i) { dict.Add(i, new Dictionary<int, long>()); }
for (int i = 0; i < M; ++i) { var ab = Read.IntN(); dict[ab[0]].Add(ab[1], ab[2]); dict[ab[1]].Add(ab[0], ab[2]); }
return dict;
}
}
    }
}