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: Time Limit Exceeded)

{-# LANGUAGE BangPatterns        #-}
{-# LANGUAGE FlexibleContexts    #-}
{-# LANGUAGE MultiWayIf          #-}
{-# LANGUAGE RankNTypes          #-}
{-# LANGUAGE ScopedTypeVariables #-}
import           Control.Monad
import           Control.Monad.ST
import           Control.Monad.State
import           Data.Array.IArray
import           Data.Array.ST
import           Data.Array.Unboxed
import qualified Data.ByteString.Char8 as BS
import           Data.Char
import           Data.List

main :: IO ()
main = do
  [h,w] <- readLnAsListWith unconsInt
  [cx,cy] <- readLnAsListWith unconsInt
  [dx,dy] <- readLnAsListWith unconsInt
  sgrid <- readLnAs2DArrayWith unconsChar h w :: IO (UArray (Int,Int) Char)

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

  print $ ggrid ! (dx,dy)

  return ()


type Grid a = UArray (Int,Int) a
type MGrid s = STUArray s (Int,Int)


gridBfs :: (Int,Int) -> (Int,Int) -> (Grid a -> (Int,Int) -> Bool) -> Grid a -> UArray (Int,Int) Int
gridBfs (h',w') (sh,sw) f g = runSTUArray $ do
  dist <- newArray ((1,1),(h',w')) (-1)
  let queue = pushBack empty (sh,sw)
  go dist 0 queue empty

  return dist

  where go :: MGrid s Int -> Int -> Queue (Int,Int) -> Queue (Int,Int) -> ST s ()
        go dist cost q qq =
          if isEmpty q
          then do if isEmpty qq
                    then do prev <- map fst . filter (\(_,e) -> e == cost) <$> getAssocs dist
                            candidate <- filterM (readArray dist >=> (return . (==(-1)))) (concatMap neighbour25 prev)
                            if null candidate then return () else do
                              let (newVal, newQueue) = popFront $ foldl' pushBack empty $ nub candidate
                              go dist (cost+1) (pushBack empty newVal) newQueue
                    else do let (q', qq') = popFront qq
                            s <- readArray dist q'
                            if s /= (-1)
                              then go dist cost empty qq'
                              else go dist cost (pushBack empty q') qq'
          else do
            let (curr, q') = popFront q
            writeArray dist curr cost
            candidate <- filterM (readArray dist >=> (return . (==(-1)))) (neighbour4 curr)
            go dist cost (foldl' pushBack q' candidate) qq



        neighbour4 :: (Int,Int) -> [(Int,Int)]
        neighbour4 (h,w) = filter inGrid [(h+x,w+y) | (x,y) <- [(-1,0),(1,0),(0,-1),(0,1)]]

        neighbour25 :: (Int,Int) -> [(Int,Int)]
        neighbour25 (h,w) = filter inGrid [(h+x,w+y) | x <- [-2..2], y <- [-2..2], (x,y) /= (0,0)]

        inGrid :: (Int,Int) -> Bool
        inGrid (h,w) = h >= 1 && h <= h' && w >= 1 && w <= w' && not (g `f` (h,w))


data Queue a = Q { front :: [a]
                 , rear  :: [a]
                 , thunk :: [a]
                 } deriving (Eq)

isEmpty :: Eq a => Queue a -> Bool
isEmpty = (==empty)

empty :: Queue a
empty = Q [] [] []

pushBack :: Queue a -> a -> Queue a
pushBack (Q f r t) e = check $! Q f (e:r) t

popFront :: Queue a -> (a, Queue a)
popFront (Q (e:f) r t) = (e, check $! Q f r t)
popFront (Q [] _ _)    = error "empty queue"

check :: Queue a -> Queue a
check (Q f r (_:t)) = Q f r t
check (Q f r []) = let ff = rotate f r []
                   in Q ff [] ff
  where rotate :: [a] -> [a] -> [a] -> [a]
        rotate [] (y:_) zs      = y:zs
        rotate (x:xs) (y:ys) zs = x : rotate xs ys (y:zs)

instance Show a => Show (Queue a) where
  show (Q f r t) = "Q { " <> show f <> "\n" <>
                   "    " <> show r <> "\n" <>
                   "    " <> show t <> "\n" <>
                   "  }"



-- gridDfs :: (Int,Int) -> (Int,Int) -> (Grid a -> (Int,Int) -> Bool) -> Grid a -> UArray (Int,Int) Int
-- gridDfs (h',w') (sh,sw) f g = runSTUArray $ do
--   seen <- newArray ((1,1),(h',w')) (-1) :: ST s (STUArray s (Int,Int) Int)
--   go seen (sh,sw) 0
--   return seen

--   where go :: MGrid s Int -> (Int,Int) -> Int -> ST s ()
--         go seen (h,w) color = do
--           c <- readArray seen (h,w)
--           if c /= (-1) then return () else do
--             writeArray seen (h,w) color
--             let nexts = neighbour4 (h,w)
--             forM_ nexts $ \(nh,nw) ->
--               if g `f` (nh,nw)
--               then do return ()
--               else do go seen (nh,nw) color

--             let nexts2 = neighbour25 (h,w)
--             forM_ nexts2 $ \(nh,nw) ->
--               if g `f` (nh,nw)
--               then do return ()
--               else do go seen (nh,nw) (color+1)


--         neighbour4 :: (Int,Int) -> [(Int,Int)]
--         neighbour4 (h,w) = filter inGrid [(h+x,w+y) | (x,y) <- [(-1,0),(1,0),(0,-1),(0,1)]]

--         neighbour25 :: (Int,Int) -> [(Int,Int)]
--         neighbour25 (h,w) = filter inGrid [(h+x,w+y) | x <- [-2..2], y <- [-2..2], (x,y) /= (0,0)]

--         inGrid :: (Int,Int) -> Bool
--         inGrid (h,w) = h >= 1 && h <= h' && w >= 1 && w <= 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

readLnAs2DArrayWith :: IArray a e => StateT BS.ByteString Maybe e -> Int -> Int -> IO (a (Int,Int) e)
readLnAs2DArrayWith !st !n !m = listArray ((1,1),(n,m)) . unfoldr (runStateT st) . BS.concat <$> replicateM n BS.getLine