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: Time Limit Exceeded)

class PriorityQueue
  attr_reader :size

  def initialize(order: :asc)
    @size = 0
    @nodes = []
    @operator = (order == :asc)? [:>, :<] : [:<, :>]
  end

  def empty?
    @size.zero?
  end

  def push(node)
    if @nodes[0].nil?
      @nodes[0] = node
    else
      index = size
      @nodes[index] = node

      loop {
        parent = (index-1)/2

        if (@nodes[parent].last <=> node.last).public_send(@operator.first, 0)
          @nodes[index], @nodes[parent] = @nodes[parent], @nodes[index]
          index = parent

          break if index.zero?
        else
          break
        end
      }
    end

    @size += 1
  end
  alias_method :<<, :push

  def pop
    node = @nodes.first
    @nodes[0], @nodes[size - 1] = @nodes[size - 1], nil
    index = 0

    loop {
      left = 2*index+1
      right = 2*index+2

      if @nodes[left].nil? && @nodes[right].nil?
        break
      elsif @nodes[right].nil? && @nodes[left]
        @nodes[index], @nodes[left] = @nodes[left], @nodes[index]
        index = left
      elsif @nodes[left].nil? && @nodes[right]
        @nodes[index], @nodes[right] = @nodes[right], @nodes[index]
        index = right
      elsif (@nodes[left].last <=> @nodes[right].last).public_send(@operator.last, 0)
        @nodes[index], @nodes[left] = @nodes[left], @nodes[index]
        index = left
      else
        @nodes[index], @nodes[right] = @nodes[right], @nodes[index]
        index = right
      end
    }

    @size -= 1
    node
  end

  def top
    @nodes.first
  end
end

class Atcoder
  MAX_N = 9

  def initialize
    _n, m, _r = gets.chomp.split.map(&:to_i)
    r = [0] + gets.chomp.split.map(&:to_i)
    @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)
      d[a][b] = c
      d[b][a] = c
    end

    r[1..-1].each do |s|
      pque = PriorityQueue.new
      pque << [s, 0]
      check_list = Hash.new { |h, k| h[k] = Hash.new(Float::INFINITY) }

      until pque.empty?
        id, cost = pque.pop
        next if check_list[s][id] <= cost
        check_list[s][id] = cost

        dist[s][id] = cost
        dist[id][s] = cost

        d[id].each_key do |n|
          pque << [n, cost + d[id][n]]
        end
      end
    end

    r.each_with_index do |a, i|
      r.each_with_index do |b, j|
        next if i == j
        if i == 0 || j == 0
          @table[i][j] = 0
        else
          @table[i][j] = dist[a][b]
        end
      end
    end

    puts r[1..-1].permutation.map { |list|
      cost = 0

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

      cost
    }.min

    #puts rec(0, 0)
  end

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

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

    res = Float::INFINITY

    @rsize.times do |u|
      if (s >> u & 1) == 0
        val = rec(s | 1 << u, u)

        res = [res, val + @table[v][u]].min
      end
    end

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

Atcoder.new