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 System;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Extentions;

class IO
{
    int idx;
    string[] input = In.ReadToEnd().Split(new[] { " ", "\n", "\r" },
        StringSplitOptions.RemoveEmptyEntries);

    T Get<T>(Func<string, T> parser) => parser(input[idx++]);

    public string S => Get(s => s);
    public char C => Get(char.Parse);
    public int I => Get(int.Parse);
    public long L => Get(long.Parse);
    public double F => Get(double.Parse);
    public decimal D => Get(decimal.Parse);
    public BigInteger B => Get(BigInteger.Parse);

    T[] Gets<T>(int n, Func<string, T> parser)
        => input.Skip((idx += n) - n).Take(n).Select(parser).ToArray();

    public string[] Ss(int n) => Gets(n, s => s);
    public char[] Cs(int n) => Gets(n, char.Parse);
    public int[] Is(int n) => Gets(n, int.Parse);
    public long[] Ls(int n) => Gets(n, long.Parse);
    public double[] Fs(int n) => Gets(n, double.Parse);
    public decimal[] Ds(int n) => Gets(n, decimal.Parse);
    public BigInteger[] Bs(int n) => Gets(n, BigInteger.Parse);

    public void Write<T>(params T[] xs) => WriteLine(string.Join(" ", xs));
    public void Write(params object[] xs) => WriteLine(string.Join(" ", xs));
}

#region Library

public interface IMonoid<T>
{
    T Unit { get; }
    T Append(T left, T right);
}

public class Monoid<T> : IMonoid<T>
{
    public T Unit { get; private set; }
    Func<T, T, T> append;
    public T Append(T left, T right) => append(left, right);

    public Monoid(T unit, Func<T, T, T> append)
    {
        this.Unit = unit;
        this.append = append;
    }
}

public abstract class Graph<T> where T : IEquatable<T>
{
    public abstract T this[int source, int target] { get; set; }

    public int V { get; protected set; }
    public IMonoid<T> Monoid { get; protected set; }
    public IMonoid<T> Comparer { get; protected set; }

    public event Action GraphChanged;
    protected void OnGraphChanged() => GraphChanged?.Invoke();

    public bool HasEdge(int source, int target)
        => !this[source, target].Equals(this.Comparer.Unit);
}

public class AdjacencyMatrix<T> : Graph<T> where T : IEquatable<T>
{
    protected readonly T[,] graph;

    public override T this[int source, int target]
    {
        // O(1)
        get { return graph[source, target]; }

        // O(1)
        set
        {
            var a = this.Comparer.Append(graph[source, target], value);
            if (a.Equals(graph[source, target])) return;
            graph[source, target] = a;
            OnGraphChanged();
        }
    }

    // O(V^2)
    public AdjacencyMatrix(int v, IMonoid<T> monoid, IMonoid<T> comparer)
    {
        this.V = v;
        this.Monoid = monoid;
        this.Comparer = comparer;
        graph = new T[v, v];
        for (var i = 0; i < v; i++)
            for (var j = 0; j < v; j++)
                graph[i, j] = i == j ? monoid.Unit : comparer.Unit;
    }
}

public class WarshallFloyd<T> where T : IEquatable<T>
{
    AdjacencyMatrix<T> graph;
    T[,] d;
    bool calculated = false;

    // O(1)
    public WarshallFloyd(AdjacencyMatrix<T> graph)
    {
        this.graph = graph;
        d = new T[graph.V, graph.V];
        graph.GraphChanged += () => calculated = false;
    }

    // O(V^3)
    public void Calculate()
    {
        if (calculated) return;

        for (var i = 0; i < graph.V; i++)
            for (var j = 0; j < graph.V; j++) d[i, j] = graph[i, j];

        for (var k = 0; k < graph.V; k++)
            for (var i = 0; i < graph.V; i++)
                for (var j = 0; j < graph.V; j++)
                    d[i, j] = graph.Comparer.Append(d[i, j],
                        d[i, k].Equals(graph.Comparer.Unit)
                        || d[k, j].Equals(graph.Comparer.Unit)
                        ? graph.Comparer.Unit
                        : graph.Monoid.Append(d[i, k], d[k, j]));

        calculated = true;
    }

    // O(1)
    public T GetMinCost(int source, int target)
    {
        if (!calculated) throw new InvalidOperationException();
        return d[source, target];
    }
}

static class Extentions
{
}

#endregion

static class Program
{
    public static void Main()
    {
        var sw = new StreamWriter(OpenStandardOutput()) { NewLine = "\n" };
#if DEBUG
        sw.AutoFlush = true;
#else
        sw.AutoFlush = false;
#endif
        SetOut(sw);
        Solve(new IO());
        Out.Flush();
    }

    static void Solve(IO io)
    {
        var n = io.I;
        var m = io.I;
        var r = io.I;
        var rs = io.Is(r).Select(x => x - 1).ToArray();

        var gr = new AdjacencyMatrix<int>(n,
            new Monoid<int>(0, (a, b) => a + b),
            new Monoid<int>(int.MaxValue >> 1, Min));

        for (var i = 0; i < m; i++)
        {
            var a = io.I - 1;
            var b = io.I - 1;
            var c = io.I;
            gr[a, b] = gr[b, a] = c;
        }

        var wf = new WarshallFloyd<int>(gr);
        wf.Calculate();

        var visited = new bool[r];

        Func<int, int, int> Search = null; Search = (int i, int s) =>
        {
            if (visited.All(x => x)) return s;

            var t = int.MaxValue;

            for (var j = 0; j < r; j++)
            {
                if (!visited[j])
                {
                    visited[j] = true;
                    t = Min(t, Search(j, s + wf.GetMinCost(rs[i], rs[j])));
                    visited[j] = false;
                }
            }

            return t;
        };

        var ans = int.MaxValue;
        for (var i = 0; i < r; i++) ans = Min(ans, Search(i, 0));

        io.Write(ans);
    }
}