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 Ruby (2.3.3) (judge verdict: Wrong Answer)

class Atcoder
  MAX_N = 8

  def initialize
    _n, m, _r = gets.chomp.split.map(&:to_i)
    r = gets.chomp.split.map {|i| i.to_i - 1 }
    @rsize = r.size
    @dp = Array.new(1 << MAX_N) { Array.new(MAX_N, -1) }

    d = Hash.new { |h, k| h[k] = {} }
    dist = Hash.new { |h, k| h[k] = {} }
    @table = Array.new(@rsize) { Array.new(@rsize, 0) }

    m.times do
      a, b, c = gets.chomp.split.map(&:to_i)
      a -= 1
      b -= 1
      d[a][b] = c
      d[b][a] = c
    end

    r.each do |s|
      queue = []
      queue << [s, 0]
      check_list = Hash.new

      until queue.empty?
        node = queue.shift
        next if check_list[node.first]
        check_list[node.first] = true

        dist[s][node.first] = node.last
        dist[node.first][s] = node.last

        d[node.first].each_key do |n|
          queue << [n, node.last + d[node.first][n]]
          queue.sort_by! { |q| q.last }
        end
      end
    end

    r.each_with_index do |a, i|
      r.each_with_index do |b, j|
        next if i == j
        @table[i][j] = dist[a][b]
      end
    end

=begin
    puts r.permutation.map { |list|
      cost = 0

      list.each_cons(2) do |a, b|
        cost += dist[a][b]
      end

      cost
    }.min
=end

    puts rec(0, 0)
  end

  def rec(s, v)
    if @dp[s][v] >= 0
      return @dp[s][v]
    end

    if s == (1 << @rsize) - 1
      return @dp[s][v] = 0
    end

    res = Float::INFINITY

    @rsize.times do |u|
      if s[u] == 0
        res = [res, rec(s | 1 << u, u) + @table[v][u]].min
      end
    end

    @dp[s][v] = res
  end
end

Atcoder.new