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)

// The main code is at the very bottom.

#[allow(unused_imports)]
use {
  lib::byte::ByteChar,
  std::cell::{Cell, RefCell},
  std::cmp::{
    self,
    Ordering::{self, *},
    Reverse,
  },
  std::collections::*,
  std::convert::identity,
  std::fmt::{self, Debug, Display, Formatter},
  std::io::prelude::*,
  std::iter::{self, FromIterator},
  std::marker::PhantomData,
  std::mem,
  std::num::Wrapping,
  std::ops::{Range, RangeFrom, RangeInclusive, RangeTo, RangeToInclusive},
  std::process,
  std::rc::Rc,
  std::thread,
  std::time::{Duration, Instant},
  std::{char, f32, f64, i128, i16, i32, i64, i8, isize, str, u128, u16, u32, u64, u8, usize},
};

#[allow(unused_imports)]
#[macro_use]
pub mod lib {
  pub mod byte {
    pub use self::byte_char::*;

    mod byte_char {
      use std::error::Error;
      use std::fmt::{self, Debug, Display, Formatter};
      use std::str::FromStr;

      #[derive(Clone, Copy, Default, PartialEq, Eq, PartialOrd, Ord, Hash)]
      #[repr(transparent)]
      pub struct ByteChar(pub u8);

      impl Debug for ByteChar {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          write!(f, "b'{}'", self.0 as char)
        }
      }

      impl Display for ByteChar {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          write!(f, "{}", self.0 as char)
        }
      }

      impl FromStr for ByteChar {
        type Err = ParseByteCharError;

        fn from_str(s: &str) -> Result<ByteChar, ParseByteCharError> {
          match s.as_bytes().len() {
            1 => Ok(ByteChar(s.as_bytes()[0])),
            0 => Err(ParseByteCharErrorKind::EmptyStr.into()),
            _ => Err(ParseByteCharErrorKind::TooManyBytes.into()),
          }
        }
      }

      #[derive(Clone, Copy, PartialEq, Eq, Hash, Debug)]
      pub struct ParseByteCharError {
        kind: ParseByteCharErrorKind,
      }

      impl Display for ParseByteCharError {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          f.write_str(match self.kind {
            ParseByteCharErrorKind::EmptyStr => "empty string",
            ParseByteCharErrorKind::TooManyBytes => "too many bytes",
          })
        }
      }

      impl Error for ParseByteCharError {}

      #[derive(Clone, Copy, PartialEq, Eq, Hash, Debug)]
      enum ParseByteCharErrorKind {
        EmptyStr,
        TooManyBytes,
      }

      impl From<ParseByteCharErrorKind> for ParseByteCharError {
        fn from(kind: ParseByteCharErrorKind) -> ParseByteCharError {
          ParseByteCharError { kind }
        }
      }
    }
  }

  pub mod io {
    pub use self::scanner::*;

    mod scanner {
      use std::io::{self, BufRead};
      use std::iter;
      use std::str::FromStr;

      #[derive(Debug)]
      pub struct Scanner<R> {
        reader: R,
        buf: String,
        pos: usize,
      }

      impl<R: BufRead> Scanner<R> {
        pub fn new(reader: R) -> Self {
          Scanner {
            reader,
            buf: String::new(),
            pos: 0,
          }
        }

        pub fn next(&mut self) -> io::Result<&str> {
          let start = loop {
            match self.rest().find(|c| c != ' ') {
              Some(i) => break i,
              None => self.fill_buf()?,
            }
          };
          self.pos += start;
          let len = self.rest().find(' ').unwrap_or(self.rest().len());
          let s = &self.buf[self.pos..][..len]; // self.rest()[..len]
          self.pos += len;
          Ok(s)
        }

        pub fn parse_next<T>(&mut self) -> io::Result<Result<T, T::Err>>
        where
          T: FromStr,
        {
          Ok(self.next()?.parse())
        }

        pub fn parse_next_n<T>(&mut self, n: usize) -> io::Result<Result<Vec<T>, T::Err>>
        where
          T: FromStr,
        {
          iter::repeat_with(|| self.parse_next()).take(n).collect()
        }

        pub fn map_next_bytes<T, F>(&mut self, mut f: F) -> io::Result<Vec<T>>
        where
          F: FnMut(u8) -> T,
        {
          Ok(self.next()?.bytes().map(&mut f).collect())
        }

        pub fn map_next_bytes_n<T, F>(&mut self, n: usize, mut f: F) -> io::Result<Vec<Vec<T>>>
        where
          F: FnMut(u8) -> T,
        {
          iter::repeat_with(|| self.map_next_bytes(&mut f))
            .take(n)
            .collect()
        }

        fn rest(&self) -> &str {
          &self.buf[self.pos..]
        }

        fn fill_buf(&mut self) -> io::Result<()> {
          self.buf.clear();
          self.pos = 0;
          let read = self.reader.read_line(&mut self.buf)?;
          if read == 0 {
            return Err(io::ErrorKind::UnexpectedEof.into());
          }
          if *self.buf.as_bytes().last().unwrap() == b'\n' {
            self.buf.pop();
          }
          Ok(())
        }
      }
    }
  }
}

#[allow(unused_imports)]
pub mod quick_find {
  use std::mem;

  #[derive(Clone, Debug)]
  pub struct QuickFind {
    elems: Vec<Vec<usize>>,
    reprs: Vec<usize>,
    sets_len: usize,
  }

  impl QuickFind {
    pub fn new(len: usize) -> Self {
      Self {
        elems: (0..len).map(|i| vec![i]).collect(),
        reprs: (0..len).collect(),
        sets_len: len,
      }
    }

    pub fn len(&self) -> usize {
      self.reprs.len()
    }

    pub fn sets_len(&self) -> usize {
      self.sets_len
    }

    pub fn repr(&self, i: usize) -> usize {
      self.reprs[i]
    }

    pub fn unite_sets(&mut self, i: usize, j: usize) -> bool {
      let mut i = self.repr(i);
      let mut j = self.repr(j);

      if i == j {
        return false;
      }
      if self.elems[i].len() < self.elems[j].len() {
        mem::swap(&mut i, &mut j);
      }

      debug_assert!(self.elems[i].len() >= self.elems[j].len());

      for &k in &self.elems[j] {
        self.reprs[k] = i;
      }
      let mut elems_j = mem::replace(&mut self.elems[j], vec![]);
      self.elems[i].append(&mut elems_j);
      self.sets_len -= 1;
      true
    }

    pub fn belong_to_same_set(&self, i: usize, j: usize) -> bool {
      self.repr(i) == self.repr(j)
    }

    pub fn set(&self, i: usize) -> &[usize] {
      &self.elems[self.repr(i)]
    }
  }
}

#[allow(unused_macros)]
macro_rules! eprint {
  ($($arg:tt)*) => {
    if cfg!(debug_assertions) {
      std::eprint!($($arg)*)
    }
  };
}
#[allow(unused_macros)]
macro_rules! eprintln {
  ($($arg:tt)*) => {
    if cfg!(debug_assertions) {
      std::eprintln!($($arg)*)
    }
  };
}
#[allow(unused_macros)]
macro_rules! dbg {
  ($($arg:tt)*) => {
    if cfg!(debug_assertions) {
      std::dbg!($($arg)*)
    } else {
      ($($arg)*)
    }
  };
}

const CUSTOM_STACK_SIZE_MIB: Option<usize> = Some(1024);
const INTERACTIVE: bool = false;

fn main() -> std::io::Result<()> {
  match CUSTOM_STACK_SIZE_MIB {
    Some(stack_size_mib) => std::thread::Builder::new()
      .name("run_solver".to_owned())
      .stack_size(stack_size_mib * 1024 * 1024)
      .spawn(run_solver)?
      .join()
      .unwrap(),
    None => run_solver(),
  }
}

fn run_solver() -> std::io::Result<()> {
  let stdin = std::io::stdin();
  let reader = stdin.lock();
  let stdout = std::io::stdout();
  let writer = stdout.lock();
  macro_rules! with_wrapper {
    ($($wrapper:expr)?) => {{
      let mut writer = $($wrapper)?(writer);
      solve(reader, &mut writer)?;
      writer.flush()
    }};
  }
  if cfg!(debug_assertions) || INTERACTIVE {
    with_wrapper!()
  } else {
    with_wrapper!(std::io::BufWriter::new)
  }
}

fn solve<R, W>(reader: R, mut writer: W) -> std::io::Result<()>
where
  R: BufRead,
  W: Write,
{
  let mut _scanner = lib::io::Scanner::new(reader);
  #[allow(unused_macros)]
  macro_rules! scan {
    ($T:ty) => {
      _scanner.parse_next::<$T>()?.unwrap()
    };
    ($($T:ty),+) => {
      ($(scan!($T)),+)
    };
    ($T:ty; $n:expr) => {
      _scanner.parse_next_n::<$T>($n)?.unwrap()
    };
    ($($T:ty),+; $n:expr) => {
      iter::repeat_with(|| -> std::io::Result<_> { Ok(($(scan!($T)),+)) })
        .take($n)
        .collect::<std::io::Result<Vec<_>>>()?
    };
  }
  #[allow(unused_macros)]
  macro_rules! scan_bytes_map {
    ($f:expr) => {
      _scanner.map_next_bytes($f)?
    };
    ($f:expr; $n:expr) => {
      _scanner.map_next_bytes_n($n, $f)?
    };
  }
  #[allow(unused_macros)]
  macro_rules! print {
    ($($arg:tt)*) => {
      write!(writer, $($arg)*)?
    };
  }
  #[allow(unused_macros)]
  macro_rules! println {
    ($($arg:tt)*) => {
      writeln!(writer, $($arg)*)?
    };
  }
  #[allow(unused_macros)]
  macro_rules! answer {
    ($($arg:tt)*) => {{
      println!($($arg)*);
      return Ok(());
    }};
  }
  {
    let (h, w) = scan!(usize, usize);
    let (c_h, c_w) = scan!(usize, usize);
    let (d_h, d_w) = scan!(usize, usize);
    let s = scan_bytes_map!(ByteChar; h);

    let (c_h, c_w) = (c_h - 1, c_w - 1);
    let (d_h, d_w) = (d_h - 1, d_w - 1);

    use quick_find::QuickFind;

    let idx = move |i: usize, j: usize| w * i + j;
    let pos = move |idx: usize| (idx / w, idx % w);

    let mut uf = QuickFind::new(h * w);
    for i in 0..h {
      for j in 0..w {
        if s[i][j].0 == b'#' {
          continue;
        }
        for k in 0..4 {
          let dy = [!0, 1, 0, 0];
          let dx = [0, 0, !0, 1];
          let i2 = i.wrapping_add(dy[k]);
          let j2 = j.wrapping_add(dx[k]);
          if i2 < h && j2 < w && s[i2][j2].0 == b'.' {
            uf.unite_sets(idx(i, j), idx(i2, j2));
          }
        }
      }
    }

    // debug
    {
      let mut set = HashSet::new();
      for id in 0..h * w {
        if set.insert(uf.repr(id)) {
          let (y, x) = pos(id);
          eprintln!("({}, {}) (repr = {}): {:?}", y, x, uf.repr(id), uf.set(id));
        }
      }
    }

    let mut g = vec![HashSet::new(); h * w];
    let mut visited = vec![false; h * w];
    for i in 0..h {
      for j in 0..w {
        if s[i][j].0 == b'#' {
          continue;
        }
        let repr = uf.repr(idx(i, j));
        if visited[repr] {
          continue;
        }
        visited[repr] = true;
        for &id_inner in uf.set(idx(i, j)) {
          let (i, j) = pos(id_inner);
          eprintln!("({}, {}) ->", i, j);
          for &dy in &[!0 - 1, !0, 0, 1, 2] {
            for &dx in &[!0 - 1, !0, 0, 1, 2] {
              let i2 = i.wrapping_add(dy);
              let j2 = j.wrapping_add(dx);
              if i2 >= h || j2 >= w || s[i2][j2].0 == b'#' {
                continue;
              }
              eprintln!("  ({}, {})", i2, j2);
              let repr2 = uf.repr(idx(i2, j2));
              g[repr].insert(repr2);
              g[repr2].insert(repr);
            }
          }
        }
      }
    }
    eprintln!("{:?}", g);

    let mut ans = None;
    let mut queue = VecDeque::new();
    let mut visited = vec![false; h * w];
    let d_repr = uf.repr(idx(d_h, d_w));
    {
      let c_repr = uf.repr(idx(c_h, c_w));
      if c_repr == d_repr {
        answer!("{}", 0);
      }
      queue.push_back((c_repr, 0u64));
      visited[c_repr] = true;
    }
    'outer: while let Some((repr, dist)) = queue.pop_front() {
      for &repr2 in &g[repr] {
        if repr2 == d_repr {
          ans = Some(dist + 1);
          break 'outer;
        }
        if visited[repr2] {
          continue;
        }
        visited[repr2] = true;
        queue.push_back((repr2, dist + 1));
      }
    }
    if let Some(ans) = ans {
      println!("{}", ans);
    } else {
      println!("-1");
    }
  }
  #[allow(unreachable_code)]
  Ok(())
}