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: Wrong Answer)

//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 group = UnionFind::new(w * 10000 + h);

    for i in 0..w {
        for j in 0..h {
            if maze[j][i] == '.' {
                if i != 0 {
                    if maze[j][i - 1] == '.' {
                        group.unite(i * 10000 + j, (i - 1) * 10000 + j);
                    }
                }

                if j != 0 {
                    if maze[j - 1][i] == '.' {
                        group.unite(i * 10000 + j, i * 10000 + (j - 1));
                    }
                }
            }
        }
    }

    let mut map: HashMap<usize, HashSet<usize>> = HashMap::new();

    for i in 0..w {
        for j in 0..h {
            if maze[j][i] == '.' {
                let mut mini_set = map
                    .entry(group.root(i * 10000 + j))
                    .or_insert(HashSet::new());
                for ii in 0..4 {
                    for jj in 0..4 {
                        let worp_x: i64 = (j + jj) as i64 - 2;
                        let worp_y: i64 = (i + ii) as i64 - 2;
                        if worp_x < h as i64 && worp_y < w as i64 && worp_x >= 0 && worp_y >= 0 {
                            if maze[j + jj - 2][i + ii - 2] == '.' {
                                if !group.same(i * 10000 + j, (i + ii - 2) * 10000 + (j + jj - 2)) {
                                    mini_set
                                        .insert(group.root((i + ii - 2) * 10000 + (j + jj - 2)));
                                }
                            }
                        }
                    }
                }
            }
        }
    }

    let start = group.root(c.1 * 10000 + c.0);
    let goal = group.root(d.1 * 10000 + d.0);

    let mut map_dist: HashMap<usize, usize> = HashMap::new();

    for (&index, set) in &map {
        map_dist.entry(index).or_insert(MAX - 1);
    }

    let len = map_dist.len();

    map_dist.insert(start, 0);

    for i in 0..len {
        for (from, set) in &map {
            let dist = map_dist.get(from).unwrap().clone();
            for to in set {
                let to_dist = map_dist.get(to).unwrap().clone();
                if to_dist > dist + 1 {
                    map_dist.insert(*to, dist + 1);
                }
            }
        }
    }

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

    let mut ans = map_dist.get(&goal).unwrap();

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

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)
    }
}