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 Fortran (gfortran v4.8.4) (judge verdict: Runtime Error)

program main
    integer :: N,M,R,a,b,c
    integer :: rr(8)
    integer :: g(200,200)
    logical :: used(8)
    integer, parameter :: INF = 100000000
    integer :: ans
    common g, R, rr, ans
    ans = INF
    do i=1,200
        do j=1,200
            g(i,j) = INF
        end do
    end do
    do i=1,8
        used(i) = .false.
    end do

    read *, N,M,R

    read *, (rr(i), i=1,R)

    do i = 1,M
        read *, a, b, c
        g(a,b) = c
        g(b,a) = c
    end do

    do i = 1,N
        do j=1,N
            do k=1,N
                if(g(j,k) > g(j,i)+g(i,k) ) then
                    g(j,k) = g(j,i) + g(i,k)
                end if
            end do
        end do
    end do

    call dfs(0, 1, -1)

    print *, ans

contains
    recursive subroutine dfs(dist, c, now_pos)
        integer :: dist, c, now_pos
        integer :: R, ans
        integer :: rr(8)
        logical :: used(8)
        integer :: g(200,200)
        common g, R, rr, ans, used
        if(now_pos == R) then
            if(ans > dist) then
                ans = dist
            end if
        else
            do i=1,8
                if(used(i) .eqv. .false.) then
                    used(i) = .true.
                    if(now_pos == -1) then
                        call dfs(0, c, rr(i))
                    else
                        call dfs(dist + g(now_pos, rr(i)), c+1, rr(i))
                    end if
                    used(i) = .false.
                end if
            end do
        end if
    end subroutine dfs
end program main