Problem:
Score : 100 points

Problem Statement

You are given a grid of N rows and M columns. The square at the i-th row and j-th column will be denoted as (i,j).
Some of the squares contain an object. All the remaining squares are empty.
The state of the grid is represented by strings S_1,S_2,\cdots,S_N. The square (i,j) contains an object if S_{i,j}= # and is empty if S_{i,j}= ..

Consider placing 1 \times 2 tiles on the grid. Tiles can be placed vertically or horizontally to cover two adjacent empty squares.
Tiles must not stick out of the grid, and no two different tiles may intersect. Tiles cannot occupy the square with an object.

Calculate the maximum number of tiles that can be placed and any configulation that acheives the maximum.

Constraints

1 \leq N \leq 100

1 \leq M \leq 100

S_i is a string with length M consists of # and ..

Input

Input is given from Standard Input in the following format:

N M
S_1
S_2
\vdots
S_N

Output

On the first line, print the maximum number of tiles that can be placed.

On the next N lines, print a configulation that achieves the maximum.
Precisely, output the strings t_1,t_2,\cdots,t_N constructed by the following way.

t_i is initialized to S_i.

For each (i,j), if there is a tile that occupies (i,j) and (i+1,j), change t_{i,j}:=v, t_{i+1,j}:=^.

For each (i,j), if there is a tile that occupies (i,j) and (i,j+1), change t_{i,j}:=>, t_{i,j+1}:=<.

See samples for further information.

You may print any configulation that maximizes the number of tiles.

Sample Input 1

3 3
#..
..#
...

Sample Output 1

3
#><
vv#
^^.

The following output is also treated as a correct answer.

3
#><
v.#
^><

Solution in Rust (1.42.0) (judge verdict: Accepted)

use max_flow::MfGraph;
use std::cmp::*;
use std::io::*;
use utils::*;

pub fn main() {
    let i = stdin();
    let mut o = Vec::new();
    run(i.lock(), &mut o).unwrap();
    stdout().write_all(&o).unwrap();
}

pub fn run(i: impl BufRead, o: &mut impl Write) -> std::io::Result<()> {
    let mut i = CpReader::new(i);
    let (n, m) = i.read::<(usize, usize)>();
    let mut s = Vec::new();
    for _ in 0..n {
        s.push(i.read_line().to_vec());
    }
    let mut g = MfGraph::new(n * m + 2);
    let start = n * m;
    let goal = n * m + 1;
    for y in 0..n {
        for x in 0..m {
            if s[y][x] == b'.' {
                let v = y * m + x;
                if (y + x) % 2 == 0 {
                    g.add_edge(start, v, 1);
                } else {
                    g.add_edge(v, goal, 1);
                }
            }
        }
    }
    for y in 0..n {
        for x in 0..m {
            let v = y * m + x;
            if (y + x) % 2 == 0 && s[y][x] == b'.' {
                if y > 0 && s[y - 1][x] == b'.' {
                    g.add_edge(v, v - m, 1);
                }
                if y + 1 < n && s[y + 1][x] == b'.' {
                    g.add_edge(v, v + m, 1);
                }
                if x > 0 && s[y][x - 1] == b'.' {
                    g.add_edge(v, v - 1, 1);
                }
                if x + 1 < m && s[y][x + 1] == b'.' {
                    g.add_edge(v, v + 1, 1);
                }
            }
        }
    }
    let f = g.flow(start, goal);
    for edge in g.edges() {
        if edge.from == start || edge.to == goal || edge.flow == 0 {
            continue;
        }
        let from_x = edge.from % m;
        let from_y = edge.from / m;
        let to_x = edge.to % m;
        let to_y = edge.to / m;

        let min_x = min(from_x, to_x);
        let max_x = max(from_x, to_x);
        let min_y = min(from_y, to_y);
        let max_y = max(from_y, to_y);

        if min_y == max_y {
            s[min_y][min_x] = b'>';
            s[max_y][max_x] = b'<';
        }
        if min_x == max_x {
            s[min_y][min_x] = b'v';
            s[max_y][max_x] = b'^';
        }
    }
    writeln!(o, "{}", f)?;
    for s in &s {
        writeln!(o, "{}", std::str::from_utf8(s).unwrap())?;
    }
    Ok(())
}

pub mod utils {
    use super::*;

    pub struct CpReader<R: BufRead> {
        r: R,
        b: Vec<u8>,
    }
    impl<R: BufRead> CpReader<R> {
        pub fn new(r: R) -> Self {
            CpReader {
                r: r,
                b: Vec::new(),
            }
        }
        pub fn read_word(&mut self) -> &[u8] {
            self.b.clear();
            loop {
                let b = self.r.fill_buf().unwrap();
                assert!(!b.is_empty());
                if let Some(p) = b.iter().position(|&x| x == b' ' || x == b'\n') {
                    self.b.extend_from_slice(&b[..p]);
                    self.r.consume(p + 1);
                    return &self.b;
                }
                self.b.extend_from_slice(b);
                let b_len = b.len();
                self.r.consume(b_len);
            }
        }
        pub fn read_word_str(&mut self) -> &str {
            unsafe { std::str::from_utf8_unchecked(self.read_word()) }
        }

        pub fn read_line(&mut self) -> &[u8] {
            self.b.clear();
            self.r.read_until(b'\n', &mut self.b).unwrap();
            if self.b.last() == Some(&b'\n') {
                &self.b[..self.b.len() - 1]
            } else {
                &self.b
            }
        }
        pub fn read_line_str(&mut self) -> &str {
            unsafe { std::str::from_utf8_unchecked(self.read_line()) }
        }

        pub fn read<T: CpIn>(&mut self) -> T {
            T::read_from(self)
        }

        pub fn read_vec<T: CpIn>(&mut self, n: usize) -> Vec<T> {
            (0..n).map(|_| self.read()).collect()
        }
        pub fn read_iter<'a, T: CpIn>(&'a mut self, n: usize) -> CpIter<'a, R, T> {
            CpIter {
                r: self,
                n: n,
                _pd: Default::default(),
            }
        }
    }
    pub struct CpIter<'a, R: BufRead + 'a, T> {
        r: &'a mut CpReader<R>,
        n: usize,
        _pd: std::marker::PhantomData<fn() -> T>,
    }
    impl<'a, R: BufRead, T: CpIn> Iterator for CpIter<'a, R, T> {
        type Item = T;
        fn next(&mut self) -> Option<T> {
            if self.n == 0 {
                None
            } else {
                self.n -= 1;
                Some(self.r.read())
            }
        }
        fn size_hint(&self) -> (usize, Option<usize>) {
            (self.n, Some(self.n))
        }
    }
    impl<'a, R: BufRead, T: CpIn> ExactSizeIterator for CpIter<'a, R, T> {}

    pub trait CpIn {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self;
    }

    impl CpIn for u64 {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            read_u64_fast(&mut r.r)
        }
    }
    impl CpIn for i64 {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            read_i64_fast(&mut r.r)
        }
    }
    impl CpIn for char {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            let b = r.r.fill_buf().unwrap()[0] as char;
            r.r.consume(1);
            let s = r.r.fill_buf().unwrap()[0];
            assert!(s == b' ' || s == b'\n');
            r.r.consume(1);
            b
        }
    }

    macro_rules! cpin_tuple {
        ($($t:ident),*) => {
            impl<$($t: CpIn),*> CpIn for ($($t),*) {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    ($($t::read_from(r)),*)
                }
            }
        };
    }

    macro_rules! cpin_cast {
        ($t_self:ty, $t_read:ty) => {
            impl CpIn for $t_self {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    <$t_read>::read_from(r) as $t_self
                }
            }
        };
    }
    macro_rules! cpin_parse {
        ($t:ty) => {
            impl CpIn for $t {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    r.read_word_str().parse().unwrap()
                }
            }
        };
    }
    cpin_cast!(usize, u64);
    cpin_cast!(u32, u64);
    cpin_cast!(u16, u64);
    cpin_cast!(i32, i64);
    cpin_cast!(i16, i64);
    cpin_cast!(i8, i64);

    cpin_parse!(f64);

    cpin_tuple!(T1, T2);
    cpin_tuple!(T1, T2, T3);
    cpin_tuple!(T1, T2, T3, T4);
    cpin_tuple!(T1, T2, T3, T4, T5);

    fn read_u64_fast<R: BufRead>(r: &mut R) -> u64 {
        let mut value = 0;
        loop {
            let buf = r.fill_buf().unwrap();
            assert!(!buf.is_empty());
            let mut idx = 0;
            while let Some(&c) = buf.get(idx) {
                if b'0' <= c && c <= b'9' {
                    value = value * 10 + (c - b'0') as u64;
                    idx += 1;
                } else {
                    r.consume(idx + 1);
                    return value;
                }
            }
            r.consume(idx);
        }
    }
    fn read_i64_fast<R: BufRead>(r: &mut R) -> i64 {
        let sign = match r.fill_buf().unwrap()[0] {
            b'+' => {
                r.consume(1);
                1
            }
            b'-' => {
                r.consume(1);
                -1
            }
            _ => 1,
        };
        read_u64_fast(r) as i64 * sign
    }
}
mod max_flow {
    #![allow(dead_code)]

    use std::{
        cmp::min,
        collections::VecDeque,
        fmt::Debug,
        ops::AddAssign,
        ops::SubAssign,
        ops::{Add, Sub},
    };

    use num::{zero, Bounded, Zero};

    pub struct MfGraph<Cap> {
        n: usize,
        pos: Vec<(usize, usize)>,
        g: Vec<Vec<EdgeData<Cap>>>,
    }
    pub struct Edge<Cap> {
        pub from: usize,
        pub to: usize,
        pub cap: Cap,
        pub flow: Cap,
    }

    #[derive(Clone, Copy, Debug, Default)]
    struct EdgeData<Cap> {
        to: usize,
        rev: usize,
        cap: Cap,
    }

    impl<
            Cap: Copy
                + Zero
                + Add<Cap>
                + AddAssign
                + Sub<Cap, Output = Cap>
                + SubAssign
                + Ord
                + Bounded
                + Debug,
        > MfGraph<Cap>
    {
        pub fn new(n: usize) -> Self {
            Self {
                n,
                pos: Vec::new(),
                g: vec![Vec::new(); n],
            }
        }

        pub fn add_edge(&mut self, from: usize, to: usize, cap: Cap) -> usize {
            debug_assert!(from < self.n);
            debug_assert!(to < self.n);
            let m = self.pos.len();
            self.pos.push((from, self.g[from].len()));

            let rev = self.g[to].len();
            self.g[from].push(EdgeData { to, rev, cap });

            let rev = self.g[from].len() - 1;
            self.g[to].push(EdgeData {
                to: from,
                rev,
                cap: zero(),
            });
            m
        }
        pub fn get_edge(&self, i: usize) -> Edge<Cap> {
            let m = self.pos.len();
            debug_assert!(i < m);
            let e = &self.g[self.pos[i].0][self.pos[i].1];
            let re = &self.g[e.to][e.rev];
            Edge {
                from: self.pos[i].0,
                to: e.to,
                cap: e.cap + re.cap,
                flow: re.cap,
            }
        }
        pub fn edges(&self) -> Vec<Edge<Cap>> {
            let m = self.pos.len();
            let mut result = Vec::new();
            for i in 0..m {
                result.push(self.get_edge(i));
            }
            result
        }
        pub fn change_edge(&mut self, i: usize, new_cap: Cap, new_flow: Cap) {
            let m = self.pos.len();
            debug_assert!(i < m);
            debug_assert!(new_flow <= new_cap);
            let e = &mut self.g[self.pos[i].0][self.pos[i].1];
            e.cap = new_cap - new_flow;

            let (e_to, e_rev) = (e.to, e.rev);
            let re = &mut self.g[e_to][e_rev];
            re.cap = new_flow;
        }
        pub fn flow(&mut self, s: usize, t: usize) -> Cap {
            self.flow_with_limit(s, t, Cap::max_value())
        }
        pub fn flow_with_limit(&mut self, s: usize, t: usize, flow_limit: Cap) -> Cap {
            debug_assert!(s < self.n);
            debug_assert!(t < self.n);

            let mut level = vec![0; self.n];
            let mut iter = vec![0; self.n];
            let mut que = VecDeque::new();

            fn dfs<
                Cap: Copy
                    + Zero
                    + Add<Cap>
                    + AddAssign
                    + Sub<Cap, Output = Cap>
                    + SubAssign
                    + Ord
                    + Bounded
                    + Debug,
            >(
                v: usize,
                up: Cap,
                s: usize,
                level: &[i32],
                iter: &[usize],
                this: &mut MfGraph<Cap>,
            ) -> Cap {
                if v == s {
                    return up;
                }
                let mut res = zero();
                let level_v = level[v];
                for i in iter[v]..this.g[v].len() {
                    let e = &this.g[v][i];
                    let e_to = e.to;
                    let e_rev = e.rev;
                    if level_v <= level[e.to] || this.g[e.to][e.rev].cap == zero() {
                        continue;
                    }
                    let d = dfs(
                        e_to,
                        min(up - res, this.g[e_to][e_rev].cap),
                        s,
                        level,
                        iter,
                        this,
                    );
                    if d <= zero() {
                        continue;
                    }
                    this.g[v][i].cap += d;
                    this.g[e_to][e_rev].cap -= d;
                    res += d;
                    if res == up {
                        break;
                    }
                }
                res
            };
            let mut flow = zero();
            while flow < flow_limit {
                let mut bfs = || {
                    for x in &mut level {
                        *x = -1;
                    }
                    level[s] = 0;
                    que.clear();
                    que.push_back(s);
                    while let Some(v) = que.pop_front() {
                        for e in &self.g[v] {
                            if e.cap == zero() || level[e.to] >= 0 {
                                continue;
                            }
                            level[e.to] = level[v] + 1;
                            if e.to == t {
                                return;
                            }
                            que.push_back(e.to);
                        }
                    }
                };
                bfs();
                if level[t] == -1 {
                    break;
                }
                for x in &mut iter {
                    *x = 0;
                }
                while flow < flow_limit {
                    let f = dfs(t, flow_limit - flow, s, &level, &iter, self);
                    if f == zero() {
                        break;
                    }
                    flow += f;
                }
            }
            flow
        }
        pub fn min_cut(&self, s: usize) -> Vec<bool> {
            let mut visited = vec![false; self.n];
            let mut que = VecDeque::new();
            que.push_back(s);
            while let Some(p) = que.pop_front() {
                visited[p] = true;
                for &e in &self.g[p] {
                    if e.cap != zero() && !visited[e.to] {
                        visited[e.to] = true;
                        que.push_back(e.to);
                    }
                }
            }
            visited
        }
    }
}