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 (4.02.3) (judge verdict: Accepted)

module WeightedDirectedGraph
  (Weight : sig
    type t
    val inf : t (* オーバーフローの恐れはないので，max_intとか突っ込んでも良い *)
    val zero : t
    val neg_inf : t (* オーバーフローの恐れはないので，min_intとか突っ込んでも良い *)
    val ( + ) : t -> t -> t
    val compare : t -> t -> int
  end) :
sig
  type 'a church_list = { fold : 'b. ('a -> 'b -> 'b) -> 'b -> 'b }

  val warshall_floyd :
    (* 頂点の数n *)
    int ->
    (* 辺のリスト *)
    (* 頂点は0からn-1までの整数でなくてはならない *)
    (int * int * Weight.t) church_list ->
    (* 辿り着けなければinf，負閉路を含む経路があればneg_infを返す関数 *)
    (int -> int -> Weight.t)
end =
struct
  type 'a church_list = { fold : 'b. ('a -> 'b -> 'b) -> 'b -> 'b }

  let warshall_floyd n es =
    let d = Array.make_matrix n n Weight.inf in
    for v = 0 to n - 1 do
      d.(v).(v) <- Weight.zero
    done;
    es.fold (fun (u, v, c) () ->
      (* c < d.(u).(v) *)
      if 0 < Weight.compare d.(u).(v) c
      then d.(u).(v) <- c) ();
    for i = 0 to n - 1 do
      for j = 0 to n - 1 do
        for k = 0 to n - 1 do
          let open Weight in
          if 
            (* 経路がない場合は更新しない *)
            0 < Weight.compare inf d.(j).(i)
            && 0 < Weight.compare inf d.(i).(k)
            (* d.(j).(i) + d.(i).(k) < d.(j).(k) *)
            && 0 < Weight.compare d.(j).(k) (d.(j).(i) + d.(i).(k))
          then d.(j).(k) <- d.(j).(i) + d.(i).(k)
        done
      done
    done;
    for i = 0 to n - 1 do
      (* 頂点iを通る負閉路がある *)
      if 0 < Weight.compare Weight.zero d.(i).(i) then
        for j = 0 to n - 1 do
          for k = 0 to n - 1 do
            (* 負閉路を通る経路があれば，最短距離を負の無限大で更新 *)
            let open Weight in
            if 0 < Weight.compare inf d.(j).(i) && 0 < Weight.compare inf d.(i).(k)
            then d.(j).(k) <- Weight.neg_inf
          done
        done
    done;
    fun u v -> try d.(u).(v) with _ -> Weight.inf
end

module G = WeightedDirectedGraph (struct
  type t = int
  let zero = 0
  let ( + ) = ( + )
  let inf = max_int
  let neg_inf = min_int
  let compare = compare
end)

let perm k xs =
  let rec perm_aux trace n k i xs ys acc =
    match k, i, xs with
    | 0, _, _ -> trace :: acc
    | _, 0, _ -> acc
    | _, _, [] -> perm_aux trace n k i (List.rev ys) [] acc
    | _, _, x :: xs ->
        perm_aux trace n k (i - 1) xs (x :: ys) @@
        perm_aux (x :: trace) (n - 1) (k - 1) (n - 1) xs ys acc in
  let n = List.length xs in
  perm_aux [] n k n xs [] []

let () = Scanf.scanf "%d %d %d\n" @@ fun n m r ->
  let rs = Array.init r @@ fun _ -> Scanf.scanf "%d " pred in
  let abcs = Array.init m @@ fun _ -> Scanf.scanf "%d %d %d\n" @@ fun a b c -> a - 1, b - 1, c in
  let d = G.warshall_floyd n { G.fold = fun f -> Array.fold_right (fun ((a, b, c) as e) acc -> f (b, a, c) @@ f e acc) abcs } in
  Printf.printf "%d\n" @@
  List.fold_left (fun acc (r :: rs) ->
    min acc @@
    snd @@
    List.fold_left (fun (r, a) s -> (s, d r s + a)) (r, 0) rs) max_int @@
  perm r @@
  Array.to_list rs