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 PyPy3 (7.3.0) (judge verdict: Accepted)

# D - Wizard in Maze
import sys
from collections import deque
from typing import List, Tuple


class WarpableMaze:
    __slots__ = ["height", "width", "is_unsearched", "is_unreachable", "dist"]

    def __init__(
        self, height: int, width: int, grid: List[bytes], road: str = ".",
    ) -> None:
        self.height = height + 4
        self.width = width + 4
        self.is_unsearched = 1 << 30
        self.is_unreachable = -1
        self.dist = self._convert_grid_to_dist(grid, road)

    def _convert_grid_to_dist(self, grid: List[bytes], road: str) -> List[int]:
        """Convert a 2D grid to a 1D dist."""
        dist = [self.is_unreachable] * self.height * self.width
        i = self.width * 2 + 1
        for row in grid:
            for c in row.decode():
                i += 1
                if c == road:
                    dist[i] = self.is_unsearched
            i += 4
        return dist

    def _flatten_coordinate(self, x: int, y: int) -> int:
        """Flatten 2D coordinate values.
        Values of a coordinate must be 1-origin.
        """
        return self.width * (x + 1) + y + 1

    def bfs(self, start_2d: Tuple[int, int], goal_2d: Tuple[int, int]) -> int:
        """BFS to compute the distance from start to goal.
        Values of start_2d and goal_2d must be 1-origin.
        """
        start = self._flatten_coordinate(*start_2d)
        goal = self._flatten_coordinate(*goal_2d)

        w = self.width
        walk_to_warp = {
            -w: (-2 * w - 2, -2 * w - 1, -2 * w, -2 * w + 1, -w - 1),
            w: (w + 1, 2 * w - 1, 2 * w, 2 * w + 1, 2 * w + 2),
            -1: (-w - 2, -2, w - 2, w - 1, 2 * w - 2),
            1: (-w + 1, -2 * w + 2, -w + 2, 2, w + 2),
        }

        dist = self.dist[:]
        dist[start] = 0
        queue = deque([start])
        while queue:
            x = queue.popleft()
            cur_dist = dist[x]
            if x == goal:
                break

            for walk, warp in walk_to_warp.items():
                nx = x + walk
                if dist[nx] > cur_dist:
                    dist[nx] = cur_dist
                    queue.appendleft(nx)
                    continue
                if dist[nx] >= 0:
                    continue

                for dx in warp:
                    nx = x + dx
                    if dist[nx] > cur_dist + 1:
                        dist[nx] = cur_dist + 1
                        queue.append(nx)

        return dist[goal] if dist[goal] != self.is_unsearched else self.is_unreachable


def main():
    read = sys.stdin.buffer.read
    readline = sys.stdin.buffer.readline

    H, W = map(int, readline().split())
    (*start,) = map(int, readline().split())
    (*goal,) = map(int, readline().split())
    (*S,) = read().split()

    maze = WarpableMaze(H, W, S)
    print(maze.bfs(start, goal))


if __name__ == "__main__":
    main()