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: Accepted)

#[allow(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use num::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::collections::*;

#[derive(Clone)]
pub struct Graph {
    graph: WeightedGraph,
}
pub type Edge = (usize, usize);
impl Graph {
    pub fn new(edges: &[Edge], vn: usize) -> Self {
        let edges = Self::to_weighted_edges(&edges);
        let graph = WeightedGraph::new(&edges, vn);
        Graph { graph }
    }
    pub fn new_directed(edges: &[Edge], vn: usize) -> Self {
        let edges = Self::to_weighted_edges(&edges);
        let graph = WeightedGraph::new_directed(&edges, vn);
        Graph { graph }
    }
    pub fn add_directed_edge(&mut self, e: Edge) {
        self.graph.add_directed_edge((e.0, e.1, 1));
    }
    pub fn add_edge(&mut self, e: Edge) {
        self.graph.add_edge((e.0, e.1, 1));
    }
    pub fn shortest_path(&self, start: usize) -> Vec<Option<i64>> {
        self.graph.shortest_path_1(start)
    }
    pub fn to_weighted_graph(&self) -> WeightedGraph {
        self.graph.clone()
    }
    fn to_weighted_edges(edges: &[Edge]) -> Vec<WeightedEdge> {
        edges
            .into_iter()
            .map(|(u, v)| (*u, *v, 1))
            .collect::<Vec<_>>()
    }
}
#[derive(Clone)]
pub struct WeightedGraph {
    graph: Vec<Vec<(usize, i64)>>,
    vn: usize,
}
pub type WeightedEdge = (usize, usize, i64);
impl WeightedGraph {
    const INF: i64 = 1 << 60;
    pub fn new(edges: &[WeightedEdge], vn: usize) -> Self {
        let mut graph = vec![Vec::new(); vn];
        for &(u, v, w) in edges {
            graph[u].push((v, w));
            graph[v].push((u, w));
        }
        Self { graph, vn }
    }
    pub fn new_directed(edges: &[WeightedEdge], vn: usize) -> Self {
        let mut graph = vec![Vec::new(); vn];
        for &(u, v, w) in edges {
            graph[u].push((v, w));
        }
        Self { graph, vn }
    }
    pub fn add_directed_edge(&mut self, e: WeightedEdge) {
        self.graph[e.0].push((e.1, e.2));
    }
    pub fn add_edge(&mut self, e: WeightedEdge) {
        self.graph[e.0].push((e.1, e.2));
        self.graph[e.1].push((e.0, e.2));
    }
    pub fn bellman_ford(&self, s: usize) -> Option<Vec<Option<i64>>> {
        let vn = self.vn;
        let inf = Self::INF;
        let mut cost_list = vec![inf; vn];
        cost_list[s] = 0;
        for c in 0..vn {
            for u in 0..vn {
                for &(v, w) in &self.graph[u] {
                    if cost_list[u] == inf {
                        continue;
                    }
                    let new_cost = cost_list[u] + w;
                    if new_cost < cost_list[v] {
                        cost_list[v] = new_cost;
                        if c == vn - 1 {
                            return None;
                        }
                    }
                }
            }
        }
        Some(self.optionalize(cost_list))
    }
    pub fn prim(&self) -> i64 {
        let mut used = std::collections::HashSet::new();
        let mut heap = std::collections::BinaryHeap::new();
        let mut res = 0_i64;
        heap.push(std::cmp::Reverse((0_i64, 0)));
        while let Some(std::cmp::Reverse((weight, u))) = heap.pop() {
            if used.contains(&u) {
                continue;
            }
            used.insert(u);
            res += weight;
            for &(v, w) in &self.graph[u] {
                if used.contains(&v) {
                    continue;
                }
                heap.push(std::cmp::Reverse((w, v)));
            }
        }
        res
    }
    pub fn reachable_vertexes(&self, s: usize) -> std::collections::HashSet<usize> {
        let mut visited = std::collections::HashSet::new();
        let mut queue = std::collections::VecDeque::new();
        visited.insert(s);
        queue.push_back(s);
        while let Some(u) = queue.pop_front() {
            for &(v, _) in &self.graph[u] {
                if visited.contains(&v) {
                    continue;
                }
                visited.insert(v);
                queue.push_back(v);
            }
        }
        visited
    }
    pub fn rev(&self) -> WeightedGraph {
        let mut edges = Vec::new();
        for u in 0..self.vn {
            for &(v, w) in &self.graph[u] {
                edges.push((v, u, w));
            }
        }
        Self::new_directed(&edges, self.vn)
    }
    pub fn shortest_path(&self, start: usize) -> Vec<Option<i64>> {
        let mut cost_list = vec![Self::INF; self.vn];
        let mut heap = std::collections::BinaryHeap::new();
        cost_list[start] = 0;
        heap.push(std::cmp::Reverse((0_i64, start)));
        while let Some(std::cmp::Reverse((cost, u))) = heap.pop() {
            if cost > cost_list[u] {
                continue;
            }
            for &(v, w) in &self.graph[u] {
                let new_cost = cost + w;
                if new_cost < cost_list[v] {
                    heap.push(std::cmp::Reverse((new_cost, v)));
                    cost_list[v] = new_cost;
                }
            }
        }
        self.optionalize(cost_list)
    }
    pub fn shortest_path_1(&self, start: usize) -> Vec<Option<i64>> {
        let mut cost_list = vec![None; self.vn];
        let mut que = std::collections::VecDeque::new();
        cost_list[start] = Some(0);
        que.push_back(start);
        while let Some(u) = que.pop_front() {
            for &(v, w) in &self.graph[u] {
                if w != 1 {
                    panic!("weight is not 1");
                }
                if cost_list[v].is_some() {
                    continue;
                }
                let new_cost = cost_list[u].unwrap() + 1;
                cost_list[v] = Some(new_cost);
                que.push_back(v);
            }
        }
        cost_list
    }
    pub fn shortest_path_01(&self, start: usize) -> Vec<Option<i64>> {
        let mut cost_list = vec![Self::INF; self.vn];
        let mut que = std::collections::VecDeque::new();
        cost_list[start] = 0;
        que.push_front((start, 0_i64));
        while let Some((u, cost)) = que.pop_front() {
            if cost > cost_list[u] {
                continue;
            }
            for &(v, w) in &self.graph[u] {
                if w != 0 && w != 1 {
                    panic!("weight is not 01");
                }
                let new_cost = cost + w;
                if new_cost < cost_list[v] {
                    cost_list[v] = new_cost;
                    if w == 0 {
                        que.push_front((v, new_cost));
                    } else {
                        que.push_back((v, new_cost));
                    }
                }
            }
        }
        self.optionalize(cost_list)
    }
    pub fn warshall_floyd(&self) -> Vec<Vec<Option<i64>>> {
        let inf = Self::INF;
        let vn = self.vn;
        let mut cost_list = vec![vec![inf; vn]; vn];
        for u in 0..vn {
            for &(v, w) in &self.graph[u] {
                cost_list[u][v] = w;
            }
        }
        for i in 0..vn {
            cost_list[i][i] = 0;
        }
        for k in 0..vn {
            for i in 0..vn {
                for j in 0..vn {
                    cost_list[i][j] =
                        std::cmp::min(cost_list[i][j], cost_list[i][k] + cost_list[k][j]);
                }
            }
        }
        cost_list
            .into_iter()
            .map(|v| self.optionalize(v))
            .collect::<Vec<_>>()
    }
    fn optionalize(&self, v: Vec<i64>) -> Vec<Option<i64>> {
        v.into_iter()
            .map(|x| if x == Self::INF { None } else { Some(x) })
            .collect::<Vec<_>>()
    }
}
pub struct Grid {
    grid: Vec<Vec<char>>,
    h: usize,
    w: usize,
    ng_char: char,
}
impl Grid {
    pub fn new(grid: &[Vec<char>], ng_char: char) -> Self {
        assert!(grid.len() > 0);
        let grid = grid.into_iter().cloned().collect::<Vec<_>>();
        let h = grid.len();
        let w = grid[0].len();
        Self {
            grid,
            h,
            w,
            ng_char,
        }
    }
    pub fn to_graph<F>(&self, generator: F) -> (WeightedGraph, VertexTable)
    where
        F: Fn(usize, usize, &Grid) -> Vec<GridDestination>,
    {
        let mut edges = Vec::new();
        let mut vertex_table = std::collections::HashMap::new();
        let mut v = 0;
        for i in 0..self.h {
            for j in 0..self.w {
                if self.grid[i][j] == self.ng_char {
                    continue;
                }
                let from = Self::gen_vertex_if_needed((i, j), &mut vertex_table, &mut v);
                for (pos, w) in generator(i, j, &self) {
                    let to = Self::gen_vertex_if_needed(pos, &mut vertex_table, &mut v);
                    edges.push((from, to, w));
                }
            }
        }
        let graph = WeightedGraph::new_directed(&edges, vertex_table.len());
        (graph, vertex_table)
    }
    pub fn in_grid(&self, i: isize, j: isize) -> bool {
        i >= 0 && i < self.h as isize && j >= 0 && j < self.w as isize
    }
    pub fn cell(&self, i: usize, j: usize) -> char {
        self.grid[i][j]
    }
    pub fn ng_char(&self) -> char {
        self.ng_char
    }
    fn gen_vertex_if_needed(
        pos: (usize, usize),
        vertex_table: &mut VertexTable,
        cur_v: &mut usize,
    ) -> usize {
        if let Some(&v) = vertex_table.get(&pos) {
            return v;
        } else {
            let v = *cur_v;
            vertex_table.insert(pos, *cur_v);
            *cur_v += 1;
            return v;
        };
    }
}
pub type VertexTable = std::collections::HashMap<(usize, usize), usize>;
/// 上下左右 (i, j)
pub const UDLR_DIRS: [(isize, isize); 4] = [(-1, 0), (1, 0), (0, -1), (0, 1)];
/// 上下左右 + 斜め (i, j)
pub const ALL_DIRS: [(isize, isize); 8] = [
    (-1, 0),
    (-1, 1),
    (0, 1),
    (1, 1),
    (1, 0),
    (1, -1),
    (0, -1),
    (-1, -1),
];
pub type GridPos = (usize, usize);
pub type GridDestination = (GridPos, i64);
pub fn gen_grid_destinations(
    i: usize,
    j: usize,
    directions: &[(isize, isize)],
    grid: &Grid,
) -> Vec<GridDestination> {
    let mut edges = Vec::new();
    for &(di, dj) in directions {
        let new_i = i as isize + di;
        let new_j = j as isize + dj;
        if !grid.in_grid(new_i, new_j) {
            continue;
        }
        let new_i = new_i as usize;
        let new_j = new_j as usize;
        if grid.cell(new_i, new_j) == grid.ng_char() {
            continue;
        }
        edges.push(((new_i, new_j), 1));
    }
    edges
}

fn solve() {
    input! {
        h: usize, _: usize,
        ch: Usize1, cw: Usize1,
        dh: Usize1, dw: Usize1,
        grid: [Chars; h]
    };

    let (graph, table) = Grid::new(&grid, '#').to_graph(|i, j, grid| {
        let mut edges = Vec::new();
        for di in -2..=2 {
            for dj in -2..=2 {
                let new_i = i as isize + di;
                let new_j = j as isize + dj;
                if !grid.in_grid(new_i, new_j) {
                    continue;
                }
                let is_udlr = (new_i - i as isize).abs() + (new_j - j as isize).abs() == 1;
                let new_i = new_i as usize;
                let new_j = new_j as usize;
                if grid.cell(new_i, new_j) == grid.ng_char() {
                    continue;
                }
                let w = if is_udlr { 0 } else { 1 };
                edges.push(((new_i, new_j), w));
            }
        }
        edges
    });
    let start = *table.get(&(ch, cw)).unwrap();
    let goal = *table.get(&(dh, dw)).unwrap();
    let res = graph.shortest_path_01(start)[goal].unwrap_or(-1);
    println!("{}", res);
}

fn main() {
    std::thread::Builder::new()
        .name("big stack size".into())
        .stack_size(256 * 1024 * 1024)
        .spawn(|| {
            solve();
        })
        .unwrap()
        .join()
        .unwrap();
}