Problem:
Score : 500 points

Problem Statement

We have N locked treasure boxes, numbered 1 to N.

A shop sells M keys. The i-th key is sold for a_i yen (the currency of Japan), and it can unlock b_i of the boxes: Box c_{i1}, c_{i2}, ..., c_{i{b_i}}. Each key purchased can be used any number of times.

Find the minimum cost required to unlock all the treasure boxes. If it is impossible to unlock all of them, print -1.

Constraints

All values in input are integers.

1 \leq N \leq 12

1 \leq M \leq 10^3

1 \leq a_i \leq 10^5

1 \leq b_i \leq N

1 \leq c_{i1} < c_{i2} < ... < c_{i{b_i}} \leq N

Input

Input is given from Standard Input in the following format:

N M
a_1 b_1
c_{11} c_{12} ... c_{1{b_1}}
:
a_M b_M
c_{M1} c_{M2} ... c_{M{b_M}}

Output

Print the minimum cost required to unlock all the treasure boxes.
If it is impossible to unlock all of them, print -1.

Sample Input 1

2 3
10 1
1
15 1
2
30 2
1 2

Sample Output 1

25

We can unlock all the boxes by purchasing the first and second keys, at the cost of 25 yen, which is the minimum cost required.

Sample Input 2

12 1
100000 1
2

Sample Output 2

-1

We cannot unlock all the boxes.

Sample Input 3

4 6
67786 3
1 3 4
3497 1
2
44908 3
2 3 4
2156 3
2 3 4
26230 1
2
86918 1
3

Sample Output 3

69942

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

{-# OPTIONS_GHC -O2 #-}
{-# LANGUAGE BangPatterns          #-}
{-# LANGUAGE CPP                   #-}
{-# LANGUAGE LambdaCase            #-}
{-# LANGUAGE MagicHash             #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE MultiWayIf            #-}
{-# LANGUAGE OverloadedStrings     #-}
{-# LANGUAGE TupleSections         #-}
{-# LANGUAGE TypeFamilies          #-}
{-# LANGUAGE ViewPatterns          #-}
{-# LANGUAGE RecordWildCards       #-}

import           Control.Applicative
import           Control.Exception
import           Control.Monad
import           Control.Monad.Primitive
import           Control.Monad.ST
import           Control.Monad.State.Strict
import           Data.Bool
import qualified Data.ByteString             as B
import qualified Data.ByteString.Builder     as B
import qualified Data.ByteString.Char8       as C
import qualified Data.ByteString.Internal    as B
import qualified Data.ByteString.Unsafe      as B
import           Data.Char
import qualified Data.Foldable               as F
import           Data.Function
import qualified Data.IntMap.Strict          as IM
import qualified Data.IntSet                 as IS
import qualified Data.List                   as L
import qualified Data.Map.Strict             as M
import           Data.Monoid
import           Data.Ord
import           Data.Primitive.MutVar
import qualified Data.Set                    as S
import           Data.Tuple
import qualified Data.Vector                 as V
import qualified Data.Vector.Mutable         as VM
import qualified Data.Vector.Generic         as G
import qualified Data.Vector.Generic.Mutable as GM
import qualified Data.Vector.Unboxed         as U
import qualified Data.Vector.Unboxed.Mutable as UM
import           Debug.Trace
import           Foreign                     hiding (void)
import           GHC.Exts
import qualified System.IO                   as IO
import           Unsafe.Coerce

main :: IO ()
main = do
    [n, m] <- map read.words <$> getLine :: IO [Int]
    edges <- V.replicateM m $ do
        [!a, !b] <- map read.words <$> getLine :: IO [Int]
        xs <- U.unfoldrN b parseInt <$> C.getLine
        return (a, b, xs)
    print . maybe (-1) id $ solve n m edges

solve :: Int -> Int -> V.Vector (Int, Int, U.Vector Int) -> Maybe Int
solve n m keys
    | dist U.! dst == inf = Nothing
    | otherwise = Just $ dist U.! dst
  where
    src = 0
    dst = shiftL 1 n - 1
    dist = dijkstraDense src gr
    inf = 0x3f3f3f3f
    gr = runST $ do
        let ix i j = unsafeShiftL i n .|. j
        m <- UM.replicate (shiftL 1 $ 2 * n) inf
        rep (shiftL 1 n) $ \i -> do
            UM.unsafeWrite m (ix i i) 0
        V.forM_ keys $ \(cost, _, boxes) -> do
            let !key = U.foldl' (\acc i -> setBit acc (i - 1)) 0 boxes
            rep (shiftL 1 n) $ \s -> do
                let s' = s .|. key
                c <- UM.unsafeRead m (ix s s')
                when (cost < c) $ do
                    UM.unsafeWrite m (ix s s') cost
        DenseGraph (shiftL 1 n) <$> U.unsafeFreeze m

type Vertex = Int
type Cost = Int
data DenseGraph a = DenseGraph
    { numVerticesDG :: !Int
    , adjDG :: U.Vector a
    }
instance (U.Unbox a, Show a) => Show (DenseGraph a) where
    show DenseGraph{..} =
        let n = numVerticesDG
        in F.foldMap (<> "\n")
            $ V.generate n $ \i ->
                show . U.toList $ U.unsafeSlice (i * n) n adjDG

dijkstraDense :: Vertex -> DenseGraph Cost -> U.Vector Cost
dijkstraDense src DenseGraph{..} = U.create $ do
    let n = numVerticesDG
    let ix i j = i * n + j
    dist <- UM.replicate (n + 1) maxBound
    used <- UM.replicate n False
    UM.write dist src 0
    let nothing = n
    vars <- UM.replicate 2 0
    let [_v, _dv] = [0..1]
    fix $ \loop -> do
        UM.unsafeWrite vars _v nothing
        UM.unsafeWrite vars _dv maxBound
        rep n $ \i -> do
            UM.unsafeRead used i >>= \case
                False -> do
                    di <- UM.unsafeRead dist i
                    dv <- UM.unsafeRead vars _dv
                    when (di < dv) $ do
                        UM.unsafeWrite vars _v i
                        UM.unsafeWrite vars _dv di
                True -> return ()
        v <- UM.unsafeRead vars _v
        when (v /= nothing) $ do
            UM.unsafeWrite used v True
            dv <- UM.unsafeRead vars _dv
            rep n $ \i -> do
                let di' = dv + U.unsafeIndex adjDG (ix v i)
                UM.unsafeModify dist (min di') i
            loop
    return dist


-------------------------------------------------------------------------------

rep :: Monad m => Int -> (Int -> m ()) -> m ()
rep !n = U.forM_ $ U.generate n id
{-# INLINE rep #-}

rev :: Monad m => Int -> (Int -> m ()) -> m ()
rev !n = U.forM_ $ U.iterateN n (subtract 1) (n - 1)
{-# INLINE rev #-}


type Parser a = C.ByteString -> Maybe (a, C.ByteString)

parseInt :: Parser Int
parseInt = C.readInt . C.dropWhile isSpace

parseInt2 :: Parser (Int, Int)
parseInt2 = runStateT $
    (,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)

parseInt3 :: Parser (Int, Int, Int)
parseInt3 = runStateT $
    (,,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)

parseInt4 :: Parser (Int, Int, Int, Int)
parseInt4 = runStateT $
    (,,,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)