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 Haskell (GHC 7.10.3) (judge verdict: Wrong Answer)

import qualified Data.ByteString.Char8 as BC
import Data.Maybe (fromJust)
import Data.List
import Data.Array.IO
import Data.Bits
import Control.Monad

type Memo = IOUArray (Int, Int) Int

main = do
  [n, m, r] <- map read . words <$> getLine :: IO [Int]
  rr <- map read . words <$> getLine :: IO [Int]
  input <- (\[x, y, z] -> zip3 x y z) . transpose <$> getInt2DListBC
  edge <- newArray ((1, 1), (n, n)) (10^5 + 1) :: IO Memo
  pass <- newArray ((1, 1), (n, n)) 0 :: IO Memo
  foldM (memoize pass) edge input
  let index = [(i, j, k) | i <- [1..n], j <- [1..n], k <- [1..n], i /= j]
      init = [(i, i, 0) | i <- [1..n]]
      purpose = foldl (\x y -> x .|. shift 1 (y - 1)) 0 rr
  foldM memoize' edge init
  foldM (warshallFloyd pass) edge index
  ans <- mapM (readMemo purpose 0 pass edge) $ permutations rr
  putStrLn . show . minimum $ ans

bsToInt :: BC.ByteString -> Int
bsToInt = fst . fromJust . BC.readInt

getInt2DListBC :: IO [[Int]]
getInt2DListBC = map (map bsToInt . BC.words) . BC.lines <$> BC.getContents

memoize :: Memo -> Memo -> (Int, Int, Int) -> IO Memo
memoize pass memo (x, y, d) = do
  writeArray pass (x, y) $ shift 1 (x - 1)
  writeArray pass (y, x) $ shift 1 (y - 1)
  writeArray memo (x, y) d
  writeArray memo (y, x) d
  return memo

memoize' :: Memo -> (Int, Int, Int) -> IO Memo
memoize' memo (x, y, d) = do
  writeArray memo (x, y) d
  return memo

warshallFloyd :: Memo -> Memo -> (Int, Int, Int) -> IO Memo
warshallFloyd pass memo (i, j, k) = do
  p1 <- readArray pass (i, j)
  p2 <- readArray pass (i, k)
  p3 <- readArray pass (k, j)
  x <- readArray memo (i, j)
  y <- readArray memo (i, k)
  z <- readArray memo (k, j)
  let minWeight = min x $ y + z
      pp = if x < y + z then p1 else p2 .|. p3
  writeArray memo (i, j) minWeight
  writeArray pass (i, j) pp
  return memo

readMemo :: Int -> Int -> Memo -> Memo -> [Int] -> IO Int
readMemo purpose points pass memo (_:[]) = return 0
readMemo purpose points pass memo (s:t:ts)
  | purpose == purpose .&. (points .|. shift 1 (s - 1)) = do
    return 0
  | otherwise = do
    d <- readArray memo (s, t)
    pp <- readArray pass (s, t)
    return =<< (+d) <$> readMemo purpose (points .|. pp) pass memo (t:ts)