Problem:
Score : 400 points

Problem Statement

A maze is composed of a grid of H \times W squares - H vertical, W horizontal.

The square at the i-th row from the top and the j-th column from the left - (i,j) - is a wall if S_{ij} is # and a road if S_{ij} is ..

There is a magician in (C_h,C_w). He can do the following two kinds of moves:

Move A: Walk to a road square that is vertically or horizontally adjacent to the square he is currently in.

Move B: Use magic to warp himself to a road square in the 5\times 5 area centered at the square he is currently in.

In either case, he cannot go out of the maze.

At least how many times does he need to use the magic to reach (D_h, D_w)?

Constraints

1 \leq H,W \leq 10^3

1 \leq C_h,D_h \leq H

1 \leq C_w,D_w \leq W

S_{ij} is # or ..

S_{C_h C_w} and S_{D_h D_w} are ..

(C_h,C_w) \neq (D_h,D_w)

Input

Input is given from Standard Input in the following format:

H W
C_h C_w
D_h D_w
S_{11}\ldots S_{1W}
\vdots
S_{H1}\ldots S_{HW}

Output

Print the minimum number of times the magician needs to use the magic. If he cannot reach (D_h,D_w), print -1 instead.

Sample Input 1

4 4
1 1
4 4
..#.
..#.
.#..
.#..

Sample Output 1

1

For example, by walking to (2,2) and then using the magic to travel to (4,4), just one use of magic is enough.

Note that he cannot walk diagonally.

Sample Input 2

4 4
1 4
4 1
.##.
####
####
.##.

Sample Output 2

-1

He cannot move from there.

Sample Input 3

4 4
2 2
3 3
....
....
....
....

Sample Output 3

0

No use of magic is needed.

Sample Input 4

4 5
1 2
2 5
#.###
####.
#..##
#..##

Sample Output 4

2

Solution in Haskell (GHC 8.8.3) (judge verdict: Accepted)

{-# LANGUAGE BangPatterns        #-}
{-# LANGUAGE FlexibleContexts    #-}
{-# LANGUAGE MultiWayIf          #-}
{-# LANGUAGE RankNTypes          #-}
{-# LANGUAGE ScopedTypeVariables #-}
import           Control.Monad
import           Control.Monad.ST
import           Control.Monad.State
import qualified Data.ByteString.Char8       as BS
import           Data.Char
import           Data.List
import           Data.Sequence               (Seq ((:<|), (:|>)), ViewL ((:<)),
                                              ViewR ((:>)), viewl, viewr, (<|),
                                              (><), (|>))
import qualified Data.Sequence               as Seq
import qualified Data.Vector.Unboxed         as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import           Debug.Trace

main :: IO ()
main = do
  [h,w] <- readLnAsListWith unconsInt
  [cx,cy] <- readLnAsListWith unconsInt
  [dx,dy] <- readLnAsListWith unconsInt
  sgrid <- readLnAs2DUVecWith unconsChar h w :: IO (VU.Vector Char)

  let ggrid = gridBfs (h,w) (cx-1,cy-1) (\g (i,j) -> g VU.! gridIndex w i j == '.') sgrid

  print $ ggrid VU.! gridIndex w (dx-1) (dy-1)

  return ()


type Grid a = VU.Vector a
type MGrid s = VUM.MVector s

gridBfs :: (Int,Int) -> (Int,Int) -> (Grid a -> (Int,Int) -> Bool) -> Grid a -> VU.Vector Int
gridBfs (h',w') (sh,sw) f g = VU.create $ do
  seen <- VUM.replicate (h'*w') False
  dist <- VUM.replicate (h'*w') (-1)

  VUM.write dist (ix (sh,sw)) 0
  let deque = Seq.empty |> ix (sh,sw)
  go dist seen deque

  return dist

  where go :: MGrid s Int -> MGrid s Bool -> Seq Int -> ST s ()
        go dist seen dq =
          case dq of
            Seq.Empty -> return ()
            _         -> do
              let (curr :< dq') = viewl dq
              s <- VUM.read seen curr
              if s then go dist seen dq' else do
                VUM.write seen curr True
                cost <- VUM.read dist curr

                candCost0 <- filterM (VUM.read dist >=> return . ((||) <$> (==(-1)) <*> (>cost))) $ neighbour4 curr
                forM_ candCost0 $ \next -> VUM.write dist next cost

                candCost1 <- filterM (VUM.read dist >=> return . (==(-1))) $ neighbour25 curr
                forM_ candCost1 $ \next -> VUM.write dist next (cost+1)

                let nextDq = foldr (<|) (foldl' (|>) dq' candCost1) candCost0
                go dist seen nextDq

        neighbour4 :: Int -> [Int]
        neighbour4 hw = let (h,w) = xi hw
                        in map ix $ filter inGrid [(h+x,w+y) | (x,y) <- [(-1,0),(1,0),(0,-1),(0,1)]]

        neighbour25 :: Int -> [Int]
        neighbour25 hw = let (h,w) = xi hw
                         in map ix $ filter inGrid [(h+x,w+y) | x <- [-2..2], y <- [-2..2], (x,y) /= (0,0)]

        inGrid :: (Int,Int) -> Bool
        inGrid (h,w) = h>=0 && h<h' && w>=0 && w<w' && g `f` (h,w)

        ix :: (Int,Int) -> Int
        ix (i,j) = gridIndex w' i j

        xi :: Int -> (Int,Int)
        xi ij = (ij`div`w',ij`mod`w')



-- converter
unconsChar :: StateT BS.ByteString Maybe Char
unconsChar = StateT BS.uncons

unconsInt :: StateT BS.ByteString Maybe Int
unconsInt = StateT $ BS.readInt . BS.dropWhile isSpace

readLnAsListWith :: StateT BS.ByteString Maybe a -> IO [a]
readLnAsListWith !st = unfoldr (runStateT st) <$> BS.getLine

-- for unboxed vector
readLnAs2DUVecWith :: VU.Unbox a => StateT BS.ByteString Maybe a -> Int -> Int -> IO (VU.Vector a)
readLnAs2DUVecWith !st !n !m = VU.unfoldrN (n * m) (runStateT st) . BS.concat <$> replicateM n BS.getLine

-- indexing
-- i: height (row)
-- j: width (column)
gridIndex :: Int -> Int -> Int -> Int
gridIndex w i j = i * w + j