Problem:
Score : 500 points

Problem Statement

N players will participate in a tennis tournament. We will call them Player 1, Player 2, \ldots, Player N.

The tournament is round-robin format, and there will be N(N-1)/2 matches in total.
Is it possible to schedule these matches so that all of the following conditions are satisfied? If the answer is yes, also find the minimum number of days required.

Each player plays at most one matches in a day.

Each player i (1 \leq i \leq N) plays one match against Player A_{i, 1}, A_{i, 2}, \ldots, A_{i, N-1} in this order.

Constraints

3 \leq N \leq 1000

1 \leq A_{i, j} \leq N

A_{i, j} \neq i

A_{i, 1}, A_{i, 2}, \ldots, A_{i, N-1} are all different.

Input

Input is given from Standard Input in the following format:

N
A_{1, 1} A_{1, 2} \ldots A_{1, N-1}
A_{2, 1} A_{2, 2} \ldots A_{2, N-1}
:
A_{N, 1} A_{N, 2} \ldots A_{N, N-1}

Output

If it is possible to schedule all the matches so that all of the conditions are satisfied, print the minimum number of days required; if it is impossible, print -1.

Sample Input 1

3
2 3
1 3
1 2

Sample Output 1

3

All the conditions can be satisfied if the matches are scheduled for three days as follows:

Day 1: Player 1 vs Player 2

Day 2: Player 1 vs Player 3

Day 3: Player 2 vs Player 3

This is the minimum number of days required.

Sample Input 2

4
2 3 4
1 3 4
4 1 2
3 1 2

Sample Output 2

4

All the conditions can be satisfied if the matches are scheduled for four days as follows:

Day 1: Player 1 vs Player 2, Player 3 vs Player 4

Day 2: Player 1 vs Player 3

Day 3: Player 1 vs Player 4, Player 2 vs Player 3

Day 4: Player 2 vs Player 4

This is the minimum number of days required.

Sample Input 3

3
2 3
3 1
1 2

Sample Output 3

-1

Any scheduling of the matches violates some condition.

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          #-}

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 <- readLn :: IO Int
    mat <- U.unfoldrN (n*(n-1)) parseInt <$> C.getContents
    print . maybe (-1) id $ solve n $ V.generate n $ \i -> U.slice (i * (n - 1)) (n - 1) mat

type TaskId = Int

taskId :: Int -> Int -> TaskId
taskId x y = min x y * 1024 + max x y

solve :: Int -> V.Vector (U.Vector Int) -> Maybe Int
solve n (V.imap (U.map . taskId . (+1)) -> priority0) = runST $ do
    next <- VM.replicate (1024 * 1024) []
    inDeg <- UM.replicate (1024 * 1024) (0 :: Int)
    V.forM_ priority0 $ \row -> do
        U.forM_ (U.zip row $ U.tail row) $ \(v, nv) -> do
            VM.unsafeModify next (nv:) v
            UM.unsafeModify inDeg (+1) nv
    queue <- newQueueM
    tasks0 <- V.filterM (\p -> (== 0) <$> UM.unsafeRead inDeg p)
        (V.map U.unsafeHead priority0)
    forM_ (IS.toList $ V.foldl' (flip IS.insert) IS.empty tasks0) $ \p -> do
        d <- UM.unsafeRead inDeg p
        when (d == 0) $ do
            enqueueM p queue
    flip fix 0 $ \loop !i -> do
        tasks <- runVecQueue queue
        -- traceShowM ("out", tasks)
        if U.null tasks
        then do
            count <- UM.read (queueInfo queue) 1
            -- traceShowM ("count", count)
            if count == n * (n - 1) `div` 2
            then return $ Just i
            else return Nothing
        else do
            U.forM_ tasks $ \task -> do
                childlen <- VM.unsafeRead next task
                forM_ childlen $ \child -> do
                    d <- UM.unsafeRead inDeg child
                    UM.unsafeWrite inDeg child (d - 1)
                    when (d == 1) $ do
                        enqueueM child queue
                        -- traceShowM ("in", child)
            loop (i + 1)

runVecQueue :: (PrimMonad m, UM.Unbox a) => (VecQueue (PrimState m) a) -> m (U.Vector a)
runVecQueue (VecQueue info q) = do
    h <- UM.unsafeRead info 0
    t <- UM.unsafeRead info 1
    UM.unsafeWrite info 0 t
    U.freeze (UM.slice h (t - h) q)

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

data VecQueue m a = VecQueue
    { queueInfo :: !(UM.MVector m Int)
    , queueData :: !(UM.MVector m a)
    }

newQueueM :: (PrimMonad m, UM.Unbox a) => m (VecQueue (PrimState m) a)
newQueueM = VecQueue <$> UM.replicate 2 0 <*> UM.unsafeNew (1024 * 1024)

dequeueM :: (PrimMonad m, UM.Unbox a) => VecQueue (PrimState m) a -> m (Maybe a)
dequeueM (VecQueue info q) = do
    h <- UM.unsafeRead info 0
    t <- UM.unsafeRead info 1
    if h < t
    then do
        UM.unsafeWrite info 0 (h + 1)
        pure <$> UM.unsafeRead q h
    else return Nothing
{-# INLINE dequeueM #-}

enqueueM :: (PrimMonad m, UM.Unbox a) => a -> VecQueue (PrimState m) a -> m ()
enqueueM x (VecQueue info q) = do
    t <- UM.unsafeRead info 1
    UM.unsafeWrite q t x
    UM.unsafeWrite info 1 (t + 1)
{-# INLINE enqueueM #-}


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)