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 Java (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.util.*;
import java.io.*;
import java.nio.charset.StandardCharsets;

public class Main {

	public static void main(String[] args) throws IOException {
		InputStreamReader reader = new InputStreamReader(System.in, StandardCharsets.UTF_8);
		BufferedReader in = new BufferedReader(reader);
		Main ins = new Main(in);
		ins.calc();
		ins.showResult();
	}

	int H, W;
	int cH, cW, dH, dW;
	List<String> lines;
	boolean[][] map;
	Tuple[] nodes;
	boolean enable = false;

	static class Tuple {
		int id, h, w, hop;
		Map<Integer, Integer> conns;

		Tuple(int id, int h, int w, int hop) {
			this.id = id;
			this.h = h;
			this.w = w;
			this.hop = hop;
			this.conns = new HashMap<>();
		}
	}

	static class Next {
		int h, w, dist;

		Next(int h, int w, int dist) {
			this.h = h;
			this.w = w;
			this.dist = dist;
		}
	}

	Main(BufferedReader in) throws IOException {
		String[] tokens = in.readLine().split(" ");
		H = Integer.parseInt(tokens[0]);
		W = Integer.parseInt(tokens[1]);
		tokens = in.readLine().split(" ");
		cH = Integer.parseInt(tokens[0]) - 1;
		cW = Integer.parseInt(tokens[1]) - 1;
		tokens = in.readLine().split(" ");
		dH = Integer.parseInt(tokens[0]) - 1;
		dW = Integer.parseInt(tokens[1]) - 1;
		lines = new ArrayList<>(H);
		for (int i = 0; i < H; ++i) {
			lines.add(in.readLine());
		}
		map = new boolean[H][W];
		for (int i = 0; i < H; ++i) {
			String line = lines.get(i);
			for (int j = 0; j < W; ++j) {
				map[i][j] = (line.charAt(j) == '.') ? true : false;
			}
		}

		nodes = new Tuple[H * W];
		for (int h = 0; h < H; ++h) {
			for (int w = 0; w < W; ++w) {
				nodes[h * W + w] = new Tuple(h * W + w, h, w, Integer.MAX_VALUE / 2);
			}
		}
		for (int h = 0; h < H; ++h) {
			for (int w = 0; w < W; ++w) {
				if (map[h][w]) {
					int i = h * W + w;
					for (int y = Math.max(0, h - 2); y <= Math.min(h + 2, H - 1); ++y) {
						for (int x = Math.max(0, w - 2); x <= Math.min(w + 2, W - 1); ++x) {
							if (x != w || y != h) {
								if (map[y][x]) {
									int conn = y * W + x;
									nodes[i].conns.put(conn, 1);
									nodes[conn].conns.put(i, 1);
								}
							}
						}
					}
					if (w + 1 < W) {
						if (map[h][w + 1]) {
							int conn = h * W + (w + 1);
							nodes[i].conns.put(conn, 0);
							nodes[conn].conns.put(i, 0);
						}
					}
					if (w - 1 >= 0) {
						if (map[h][w - 1]) {
							int conn = h * W + (w - 1);
							nodes[i].conns.put(conn, 0);
							nodes[conn].conns.put(i, 0);
						}
					}
					if (h + 1 < H) {
						if (map[h + 1][w]) {
							int conn = (h + 1) * W + w;
							nodes[i].conns.put(conn, 0);
							nodes[conn].conns.put(i, 0);
						}
					}
					if (h - 1 >= 0) {
						if (map[h - 1][w]) {
							int conn = (h - 1) * W + w;
							nodes[i].conns.put(conn, 0);
							nodes[conn].conns.put(i, 0);
						}
					}
				}
			}
		}
		nodes[cH * W + cW].hop = 0;
	}

	Set<Next> getNexts(int h, int w, int cdist) {
		Set<Next> set = new HashSet<>();
		for (Integer next : nodes[h * W + w].conns.keySet()) {
			int n = next;
			int nh = n / W;
			int nw = n % W;
			int dist = nodes[h * W + w].conns.get(next);
			set.add(new Next(nh, nw, cdist + dist));
		}
		Set<Integer> nextKeys = new HashSet<>();
		nextKeys.addAll(nodes[h * W + w].conns.keySet());
		int key = h * W + w;
		for (Integer nextKey : nextKeys) {
			nodes[nextKey].conns.remove(key);
		}
		nodes[key].conns.clear();

		return set;
	}

	void calc() {
		Queue<Next> que = new PriorityQueue<>(new Comparator<Next>() {
			@Override
			public int compare(Next o1, Next o2) {
				return Integer.compare(o1.dist, o2.dist);
			}
		});
		que.addAll(getNexts(cH, cW, 0));
		while (!que.isEmpty()) {
			Next next = que.poll();
			int nextIndex = next.h * W + next.w;
			// System.out.println(" " + (next.h + 1) + " " + (next.w + 1) + " " + next.dist);
			nodes[nextIndex].hop = next.dist;
			if (next.h == dH && next.w == dW) {
				enable = true;
				break;
			}
			que.addAll(getNexts(next.h, next.w, next.dist));
		}
	}

	void showResult() {
		if (enable) {
			System.out.println(nodes[dH * W + dW].hop);
		} else {
			System.out.println("-1");
		}
	}
}