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 Ruby (2.7.1) (judge verdict: Accepted)

# frozen_string_literal: true

# MaxFlowGraph
class MaxFlow
  def initialize(n = 0)
    @n   = n
    @pos = []
    @g   = Array.new(n) { [] }
  end

  def add_edge(from, to, cap)
    edge_number = @pos.size

    @pos << [from, @g[from].size]

    from_id = @g[from].size
    to_id   = @g[to].size
    to_id += 1 if from == to
    @g[from] << Edge.new(to, to_id, cap)
    @g[to]   << Edge.new(from, from_id, 0)

    edge_number
  end

  def push(edge)
    add_edge(*edge)
  end
  alias << push

  # return edge = [from, to, cap, flow]
  def [](i)
    _e  = @g[@pos[i][0]][@pos[i][1]]
    _re = @g[_e.to][_e.rev]
    [@pos[i][0], _e.to, _e.cap + _re.cap, _re.cap]
  end
  alias get_edge []
  alias edge []

  def edges
    @pos.map do |(to, rev)|
      _e  = @g[to][rev]
      _re = @g[_e.to][_e.rev]
      [to, _e.to, _e.cap + _re.cap, _re.cap]
    end
  end

  def change_edge(i, new_cap, new_flow)
    _e  = @g[@pos[i][0]][@pos[i][1]]
    _re = @g[_e[0]][_e[1]]
    _e.cap  = new_cap - new_flow
    _re.cap = new_flow
  end

  def flow(s, t, flow_limit = 1 << 64)
    level = Array.new(@n)
    iter  = Array.new(@n)
    que   = []

    flow = 0
    while flow < flow_limit
      bfs(level, que, s, t)
      break if level[t] == -1

      iter.fill(0)
      while flow < flow_limit
        f = dfs(t, flow_limit - flow, s, level, iter)
        break if f == 0

        flow += f
      end
    end

    flow
  end
  alias max_flow flow

  def min_cut(s)
    visited = Array.new(@n) { false }
    que = [s]
    while (q = que.shift)
      visited[q] = true
      @g[q].each do |e|
        if e.cap > 0 && !visited[e.to]
          visited[e.to] = true
          que << e.to
        end
      end
    end
    visited
  end

  private

  class Edge
    def initialize(to, rev, cap)
      @to = to
      @rev = rev
      @cap = cap
    end

    attr_accessor :to, :rev, :cap
  end

  def bfs(level, que, s, t)
    level.fill(-1)
    level[s] = 0
    que.clear
    que << s

    while (v = que.shift)
      @g[v].each do |e|
        next if e.cap == 0 || level[e.to] >= 0

        level[e.to] = level[v] + 1
        return nil if e.to == t

        que << e.to
      end
    end
  end

  def dfs(v, up, s, level, iter)
    return up if v == s

    res = 0
    level_v = level[v]

    while iter[v] < @g[v].size
      i = iter[v]
      e = @g[v][i]
      re = @g[e.to][e.rev]
      if level_v > level[e.to] && re.cap > 0 then
        d = dfs(e.to, up-res<re.cap ? up-res : re.cap, s, level, iter)
        if d > 0 then
          e.cap += d
          re.cap -= d
          res += d
          break if res == up
        end
      end
      iter[v] += 1
    end

    res
  end
end

MaxFlowGraph = MaxFlow
MFGraph      = MaxFlow

n, m = gets.split.map &:to_i
str = (0 ... n).map{ gets.chomp }

g = MaxFlow.new(n*m+2)
s, t = n*m, n*m+1

n.times{ |i|
  m.times{ |j|
    next if str[i][j] == '#'
    (i+j).even? ? g.add_edge(s, i*m+j, 1) : g.add_edge(i*m+j, t, 1)
    [[i+1, j], [i, j+1]].each{ |x, y|
      if x<n && y<m && str[x][y]=='.' then
        (i+j).even? ? g.add_edge(i*m+j, x*m+y, 1) : g.add_edge(x*m+y, i*m+j, 1)
      end
    }
  }
}

puts g.max_flow(s, t)
es = g.edges

g.edges.each{ |from, to, _, f|
  if from != s && to != t && f == 1 then
    from, to = to, from if from > to
    a, b, c, d = from/m, from%m, to/m, to%m
    if a == c then
      str[a][b] = '>'
      str[c][d] = '<'
    else
      str[a][b] = 'v'
      str[c][d] = '^'
    end
  end
}

str.each{ |s| puts s }