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 C (Clang 10.0.0) (judge verdict: Wrong Answer)

#include <stdio.h>
#include <stdlib.h>

typedef struct vector2 {
  int x;
  int y;
}vector2;

int qcount;
vector2 *queue;
int qfirst = 0, qlast = 0;

void makeQueue(int count) {
  qcount = count;
  qfirst = qlast = count / 2;
  queue = malloc(count * sizeof(vector2));
}

void push_back(int x, int y) {
  vector2 v = { x, y };
  qlast %= qcount;
  queue[qlast++] = v;
}

void push_front(int x, int y) {
  vector2 v = { x, y };
  qfirst = (qfirst + qcount) % qcount;
  queue[--qfirst] = v;
}

vector2 dequeue(void) {
  qfirst %= qcount;
  vector2 v = queue[qfirst++];
  return v;
}

int main(void){
  int h, w;
  scanf("%d%d", &h, &w);
  int ch, cw;
  scanf("%d%d", &ch, &cw);
  ch++, cw++;
  int dh, dw;
  scanf("%d%d", &dh, &dw);
  dh++, dw++;
  makeQueue(h * w);
  char s[h+4][w+4];
  
  // clear map
  for(int i = 0;i < h + 4;i++) {
    for(int j = 0;j < w + 4;j++) {
      s[i][j] = -2;
    }
  }

  // read map
  for(int i = 2;i < h + 2;i++) {
    scanf("%s", &s[i][2]);
    for(int j = 2;j < w + 2;j++) {
      if(s[i][j] == '.') s[i][j] = -1;
      else if(s[i][j] == '#') s[i][j] = -2;
    }
    s[i][w + 2] = -2;
  }

  push_back(cw, ch);
  s[ch][cw] = 0;

  while(qfirst != qlast) {

    vector2 point = dequeue();
    int x = point.x, y = point.y;
    if(x == dw && y == dh) break;
    int data = s[y][x];

    // work
    int dx[] = { -1, 1, 0, 0 };
    int dy[] = { 0, 0, -1, 1 };
    for(int i = 0;i < 4;i++) {
      int ss = s[y + dy[i]][x + dx[i]];
      // wall
      if(ss == -2) continue;
      //already visit
      if(ss > -1) { 
        if(ss > data) {
          s[y + dy[i]][x + dx[i]] = data;
          push_front(x + dx[i], y + dy[i]);
        }
      } else if(ss == -1) {
        s[y + dy[i]][x + dx[i]] = data;
        push_front(x + dx[i], y + dy[i]);
      }
    }

    // warp
    for(int i = -2;i <= 2;i++) {
      for(int j = -2;j <= 2;j++) {
        int ss = s[y + i][x + j];
        // cannot visit
        if(ss == -2) { continue; }
        // already visit
        if(ss > -1) {
          if(ss > data + 1) {
            s[y + i][x + j] = data + 1;
            push_back(x + j, y + i);
          }
        }else if(ss == -1) {
          s[y+i][x+j] = data + 1;
          push_back(x + j, y + i);
        }
      }
    }

  }
  if(s[dh][dw] == -2) {
    puts("-1");
  } else {
    printf("%d\n", s[dh][dw]);
  }
  return 0;
}