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 OCaml (4.10.0) (judge verdict: Accepted)

Scanf.scanf "%d %d %d" (fun n m r ->
    let next_permutation list =
        let rec phase3 taddr = function
            | _  :: tl when tl == taddr -> []
            | hd :: tl                  -> hd :: phase3 taddr tl
            | []                        -> [] in

        let rec phase2 h2 taddr = function
            | h1 :: tl when h1 > h2 -> let lis = phase3 taddr tl in
                                       h1, h2 :: lis
            | h1 :: tl              -> let hh, lis = phase2 h2 taddr tl in
                                       hh, h1 :: lis
            | _                     -> failwith "??" in

        let rec phase1 list = function
            | h1 :: h2 :: tl when h1 > h2  -> let hh, lis = phase2 h2   tl list in
                                              List.rev_append tl (hh :: lis)
            | _  :: tl                     -> phase1 list tl
            | _                            -> list in

        let rev_list = List.rev list in
        phase1 rev_list rev_list
    in
    let ri = Array.init r (fun _ -> Scanf.scanf " %d" (fun r -> r - 1)) in
    let nei = Array.make n [] in
    for i = 1 to m do
        Scanf.scanf " %d %d %d" (fun a b c ->
            let a = a - 1 in
            let b = b - 1 in
            nei.(a) <- (c, b) :: nei.(a);
            nei.(b) <- (c, a) :: nei.(b);
        )
    done;
    let module S = Set.Make (struct type t = int * int let compare = compare end) in
    let dijkstra s g =
        let d = Array.make n max_int in
        d.(s) <- 0;
        let rec loop set =
            if S.is_empty set then () else
                let ((dd, cur) as m) = S.min_elt set in
                let set = S.remove m set in
                let set = List.fold_left (fun set (nc, nv) ->
                    if d.(nv) <= nc + dd then set else (
                        d.(nv) <- nc + dd;
                        S.add (nc + dd, nv) set
                    )
                ) set nei.(cur)
                in
                loop set
        in
        loop (S.singleton (0, s));
        d.(g)
    in
    let dist = Array.make_matrix r r max_int in
    for i = 0 to r - 1 do
        dist.(i).(i) <- 0;
        for j = i + 1 to r - 1 do
            let c = dijkstra ri.(i) ri.(j) in
            dist.(i).(j) <- c;
            dist.(j).(i) <- c;
        done
    done;
    let rec iota a = if a < 0 then [] else a :: iota (a - 1) in
    let rec fact a = if a = 0 then 1  else a *  fact (a - 1) in

    let calc l =
        match l with
        | [] -> 0
        | hd :: tl ->
                let rec loop prev sum = function
                    | [] -> sum
                    | x :: xs -> loop x (sum + dist.(prev).(x)) xs
                in
                loop hd 0 tl
    in

    let rec loop i lis acc =
        if i = 0 then acc else
            loop (i - 1) (next_permutation lis) (min acc (calc lis))
    in
    loop (fact r) (iota (r - 1)) max_int |> Printf.printf "%d\n"
)