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 Rust (1.42.0) (judge verdict: Time Limit Exceeded)

//use itertools::Itertools;
use std::cmp;
use std::collections::BTreeMap;
use std::collections::BTreeSet;
use std::collections::BinaryHeap;
use std::collections::HashMap;
use std::collections::HashSet;
use std::collections::VecDeque;
use std::io::Read;
use std::usize::MAX;

macro_rules! input {(source = $s:expr, $($r:tt)*) => {let mut iter = $s.split_whitespace();let mut next = || { iter.next().unwrap() };input_inner!{next, $($r)*}};($($r:tt)*) => {let stdin = std::io::stdin();let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));let mut next = move || -> String{bytes.by_ref().map(|r|r.unwrap() as char).skip_while(|c|c.is_whitespace()).take_while(|c|!c.is_whitespace()).collect()};input_inner!{next, $($r)*}};}
macro_rules! input_inner {($next:expr) => {};($next:expr, ) => {};($next:expr, $var:ident : $t:tt $($r:tt)*) => {let $var = read_value!($next, $t);input_inner!{$next $($r)*}};}
macro_rules! read_value {($next:expr, ( $($t:tt),* )) => {( $(read_value!($next, $t)),* )};($next:expr, [ $t:tt ; $len:expr ]) => {(0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()};($next:expr, chars) => {read_value!($next, String).chars().collect::<Vec<char>>()};($next:expr, usize1) => {read_value!($next, usize) - 1};($next:expr, [ $t:tt ]) => {{let len = read_value!($next, usize);(0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()}};($next:expr, $t:ty) => {$next().parse::<$t>().expect("Parse error")};}

fn solve() {
    input! {
        h: usize,
        w:usize,
        c : (usize1,usize1),
        d: (usize1,usize1),
        maze: [chars;h]
    }

    let mut visited = vec![vec![false; w]; h];
    let mut dist = vec![vec![MAX; w]; h];

    let mut q = BinaryHeap::new();
    q.push((MAX, c.1, c.0, 0));
    while !q.is_empty() {
        let now = q.pop().unwrap();
        let now_x = now.1;
        let now_y = now.2;
        let now_dist = now.3;

        visited[now_y][now_x] = true;
        dist[now_y][now_x] = cmp::min(dist[now_y][now_x], now_dist);
        for i in 0..5 {
            for j in 0..5 {
                if now_x + i >= 2 && now_y + j >= 2 {
                    let new_x = now_x + i - 2;
                    let new_y = now_y + j - 2;

                    if new_x < w && new_y < h && maze[new_y][new_x] == '.' && !visited[new_y][new_x]
                    {
                        let x_abs = cmp::max(now_x, new_x) - cmp::min(now_x, new_x);
                        let y_abs = cmp::max(now_y, new_y) - cmp::min(now_y, new_y);

                        if (x_abs == 0 && y_abs == 1) || (x_abs == 1 && y_abs == 0) {
                            q.push((MAX - now_dist, new_x, new_y, now_dist));
                        } else {
                            q.push((MAX - (now_dist + 1), new_x, new_y, now_dist + 1));
                        }
                    }
                }
            }
        }
    }

    //println!("{:?}", dist);

    let start_dist = dist[c.0][c.1];
    let goal_dist = dist[d.0][d.1];

    if goal_dist == MAX {
        println!("{}", -1);
    } else {
        println!("{}", goal_dist);
    }
}

fn main() {
    let thd = std::thread::Builder::new().stack_size(104_857_600);
    thd.spawn(|| solve()).unwrap().join().unwrap();

    /*
    // 入力を一括で読み込む場合
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();
    let mut input = buf.split_whitespace();

    // inputに対しnext()で読み込んでいく
    let q: usize = input.next().unwrap().parse().unwrap();
    */
}

const LARGE_PRIME: u64 = 1_000_000_007;

// @〜でsnippet
struct UnionFind {
    parent: Vec<usize>,
    rank: Vec<usize>,
    size: Vec<usize>,
}

impl UnionFind {
    fn new(n: usize) -> UnionFind {
        UnionFind {
            parent: (0..n).collect(),
            rank: vec![0; n],
            size: vec![1; n],
        }
    }

    fn root(&mut self, x: usize) -> usize {
        let parent = self.parent[x];
        if parent != x {
            self.parent[x] = self.root(parent);
            self.rank[x] = 1;
        }
        return self.parent[x];
    }

    fn unite(&mut self, x: usize, y: usize) {
        if self.same(x, y) {
            return;
        }
        let x_root = self.root(x);
        let y_root = self.root(y);
        if self.rank[x_root] >= self.rank[y_root] {
            self.size[x_root] += self.size(y);
            self.rank[y_root] = 1;
            self.parent[y_root] = x_root;
        } else {
            let x_root = self.root(x);
            let y_root = self.root(y);
            self.size[y_root] += self.size(x);
            self.rank[x_root] = 1;
            self.parent[x_root] = y_root;
        }
    }

    fn size(&mut self, x: usize) -> usize {
        let x_root = self.root(x);
        return self.size[x_root];
    }

    fn same(&mut self, x: usize, y: usize) -> bool {
        self.root(x) == self.root(y)
    }
}