Language: C++

using namespace std;
using namespace atcoder;

namespace internal {

template <class T> struct simple_queue {
    std::vector<T> payload;
    int pos = 0;
    void reserve(int n) { payload.reserve(n); }
    int size() const { return int(payload.size()) - pos; }
    bool empty() const { return pos == int(payload.size()); }
    void push(const T& t) { payload.push_back(t); }
    T& front() { return payload[pos]; }
    void clear() {
        payload.clear();
        pos = 0;
    }
    void pop() { pos++; }
};

}  // namespace internal

template <class Cap> struct mf_graph {
  public:
    mf_graph() : _n(0) {}
    mf_graph(int n) : _n(n), g(n) {}

    int add_edge(int from, int to, Cap cap) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        assert(0 <= cap);
        int m = int(pos.size());
        pos.push_back({from, int(g[from].size())});
        g[from].push_back(_edge{to, int(g[to].size()), cap});
        g[to].push_back(_edge{from, int(g[from].size()) - 1, 0});
        return m;
    }

    struct edge {
        int from, to;
        Cap cap, flow;
    };

    edge get_edge(int i) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        auto _e = g[pos[i].first][pos[i].second];
        auto _re = g[_e.to][_e.rev];
        return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
    }
    std::vector<edge> edges() {
        int m = int(pos.size());
        std::vector<edge> result;
        for (int i = 0; i < m; i++) {
            result.push_back(get_edge(i));
        }
        return result;
    }
    void change_edge(int i, Cap new_cap, Cap new_flow) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        assert(0 <= new_flow && new_flow <= new_cap);
        auto& _e = g[pos[i].first][pos[i].second];
        auto& _re = g[_e.to][_e.rev];
        _e.cap = new_cap - new_flow;
        _re.cap = new_flow;
    }

    Cap flow(int s, int t) {
        return flow(s, t, std::numeric_limits<Cap>::max());
    }
    Cap flow(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);

        std::vector<int> level(_n), iter(_n);
        internal::simple_queue<int> que;

        auto bfs = [&]() {
            std::fill(level.begin(), level.end(), -1);
            level[s] = 0;
            que.clear();
            que.push(s);
            while (!que.empty()) {
                int v = que.front();
                que.pop();
                for (auto e : g[v]) {
                    if (e.cap == 0 || level[e.to] >= 0) continue;
                    level[e.to] = level[v] + 1;
                    if (e.to == t) return;
                    que.push(e.to);
                }
            }
        };
        auto dfs = [&](auto self, int v, Cap up) {
            if (v == s) return up;
            Cap res = 0;
            int level_v = level[v];
            for (int& i = iter[v]; i < int(g[v].size()); i++) {
                _edge& e = g[v][i];
                if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
                Cap d =
                    self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
                if (d <= 0) continue;
                g[v][i].cap += d;
                g[e.to][e.rev].cap -= d;
                res += d;
                if (res == up) break;
            }
            return res;
        };

        Cap flow = 0;
        while (flow < flow_limit) {
            bfs();
            if (level[t] == -1) break;
            std::fill(iter.begin(), iter.end(), 0);
            while (flow < flow_limit) {
                Cap f = dfs(dfs, t, flow_limit - flow);
                if (!f) break;
                flow += f;
            }
        }
        return flow;
    }

    std::vector<bool> min_cut(int s) {
        std::vector<bool> visited(_n);
        internal::simple_queue<int> que;
        que.push(s);
        while (!que.empty()) {
            int p = que.front();
            que.pop();
            visited[p] = true;
            for (auto e : g[p]) {
                if (e.cap && !visited[e.to]) {
                    visited[e.to] = true;
                    que.push(e.to);
                }
            }
        }
        return visited;
    }

  private:
    int _n;
    struct _edge {
        int to, rev;
        Cap cap;
    };
    std::vector<std::pair<int, int>> pos;
    std::vector<std::vector<_edge>> g;
};

int main() {
    int n, m;
    cin >> n >> m;

    vector<string> grid(n);
    for (int i = 0; i < n; i++) {
        cin >> grid[i];
    }

    /**
     * generate (s -> even grid -> odd grid -> t) graph
     * grid(i, j) correspond to vertex (i * m + j)
     **/
    mf_graph<int> g(n * m + 2);
    int s = n * m, t = n * m + 1;

    // s -> even / odd -> t
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < m; j++) {
            if (grid[i][j] == '#') continue;
            int v = i * m + j;
            if ((i + j) % 2 == 0) {
                g.add_edge(s, v, 1);
            } else {
                g.add_edge(v, t, 1);
            }
        }
    }

    // even -> odd
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < m; j++) {
            if ((i + j) % 2 || grid[i][j] == '#') continue;
            int v0 = i * m + j;
            if (i && grid[i - 1][j] == '.') {
                int v1 = (i - 1) * m + j;
                g.add_edge(v0, v1, 1);
            }
            if (j && grid[i][j - 1] == '.') {
                int v1 = i * m + (j - 1);
                g.add_edge(v0, v1, 1);
            }
            if (i + 1 < n && grid[i + 1][j] == '.') {
                int v1 = (i + 1) * m + j;
                g.add_edge(v0, v1, 1);
            }
            if (j + 1 < m && grid[i][j + 1] == '.') {
                int v1 = i * m + (j + 1);
                g.add_edge(v0, v1, 1);
            }
        }
    }

    cout << g.flow(s, t) << endl;

    auto edges = g.edges();
    for (auto e : edges) {
        if (e.from == s || e.to == t || e.flow == 0) continue;
        int i0 = e.from / m, j0 = e.from % m;
        int i1 = e.to / m, j1 = e.to % m;

        if (i0 == i1 + 1) {
            grid[i1][j1] = 'v';
            grid[i0][j0] = '^';
        } else if (j0 == j1 + 1) {
            grid[i1][j1] = '>'; grid[i0][j0] = '<';
        } else if (i0 == i1 - 1) {
            grid[i0][j0] = 'v';
            grid[i1][j1] = '^';
        } else {
            grid[i0][j0] = '>'; grid[i1][j1] = '<';
        }
    }

    for (int i = 0; i < n; i++) {
        cout << grid[i] << endl;
    }

    return 0;
}

Compiler output:
a.cc:2:17: error: 'atcoder' is not a namespace-name
    2 | using namespace atcoder;
      |                 ^~~~~~~
a.cc:7:10: error: 'vector' in namespace 'std' does not name a template type
    7 |     std::vector<T> payload;
      |          ^~~~~~
a.cc:1:1: note: 'std::vector' is defined in header '<vector>'; this is probably fixable by adding '#include <vector>'
  +++ |+#include <vector>
    1 | using namespace std;
a.cc: In member function 'void internal::simple_queue<T>::reserve(int)':
a.cc:9:27: error: 'payload' was not declared in this scope
    9 |     void reserve(int n) { payload.reserve(n); }
      |                           ^~~~~~~
a.cc: In member function 'int internal::simple_queue<T>::size() const':
a.cc:10:35: error: 'payload' was not declared in this scope
   10 |     int size() const { return int(payload.size()) - pos; }
      |                                   ^~~~~~~
a.cc: In member function 'bool internal::simple_queue<T>::empty() const':
a.cc:11:44: error: 'payload' was not declared in this scope
   11 |     bool empty() const { return pos == int(payload.size()); }
      |                                            ^~~~~~~
a.cc: In member function 'void internal::simple_queue<T>::push(const T&)':
a.cc:12:29: error: 'payload' was not declared in this scope
   12 |     void push(const T& t) { payload.push_back(t); }
      |                             ^~~~~~~
a.cc: In member function 'T& internal::simple_queue<T>::front()':
a.cc:13:25: error: 'payload' was not declared in this scope
   13 |     T& front() { return payload[pos]; }
      |                         ^~~~~~~
a.cc: In member function 'void internal::simple_queue<T>::clear()':
a.cc:15:9: error: 'payload' was not declared in this scope
   15 |         payload.clear();
      |         ^~~~~~~
a.cc: At global scope:
a.cc:51:10: error: 'vector' in namespace 'std' does not name a template type
   51 |     std::vector<edge> edges() {
      |          ^~~~~~
a.cc:51:5: note: 'std::vector' is defined in header '<vector>'; this is probably fixable by adding '#include <vector>'
   51 |     std::vector<edge> edges() {
      |     ^~~
a.cc:127:10: error: 'vector' in namespace 'std' does not name a template type
  127 |     std::vector<bool> min_cut(int s) {
      |          ^~~~~~
a.cc:127:5: note: 'std::vector' is defined in header '<vector>'; this is probably fixable by adding '#include <vector>'
  127 |     std::vector<bool> min_cut(int s) {
      |     ^~~
a.cc:151:10: error: 'vector' in namespace 'std' does not name a template type
  151 |     std::vector<std::pair<int, int>> pos;
      |          ^~~~~~
a.cc:151:5: note: 'std::vector' is defined in header '<vector>'; this is probably fixable by adding '#include <vector>'
  151 |     std::vector<std::pair<int, int>> pos;
      |     ^~~
a.cc:152:10: error: 'vector' in namespace 'std' does not name a template type
  152 |     std::vector<std::vector<_edge>> g;
      |          ^~~~~~
a.cc:152:5: note: 'std::vector' is defined in header '<vector>'; this is probably fixable by adding '#include <vector>'
  152 |     std::vector<std::vector<_edge>> g;
      |     ^~~
a.cc: In constructor 'mf_graph<Cap>::mf_graph(int)':
a.cc:26:30: error: class 'mf_graph<Cap>' does not have any field named 'g'
   26 |     mf_graph(int n) : _n(n), g(n) {}
      |                              ^
a.cc: In member function 'int mf_graph<Cap>::add_edge(int, int, Cap)':
a.cc:29:9: error: there are no arguments to 'assert' that depend on a template parameter, so a declaration of 'assert' must be available [-fpermissive]
   29 |         assert(0 <= from && from < _n);
      |         ^~~~~~
a.cc:29:9: note: (if you use '-fpermissive', G++ will accept your code, but allowing the use of an undeclared name is deprecated)
a.cc:30:9: error: there are no arguments to 'assert' that depend on a template parameter, so a declaration of 'assert' must be available [-fpermissive]
   30 |         assert(0 <= to && to < _n);
      |         ^~~~~~
a.cc:32:21: error: 'pos' was not declared in this scope
   32 |         int m = int(pos.size());
      |                     ^~~
a.cc:33:34: error: 'g' was not declared in this scope
   33 |         pos.push_back({from, int(g[from].size())});
      |                                  ^
a.cc: In member function 'mf_graph<Cap>::edge mf_graph<Cap>::get_edge(int)':
a.cc:45:21: error: 'pos' was not declared in this scope
   45 |         int m = int(pos.size());
      |                     ^~~
a.cc:46:9: error: there are no arguments to 'assert' that depend on a template parameter, so a declaration of 'assert' must be available [-fpermissive]
   46 |         assert(0 <= i && i < m);
      |         ^~~~~~
a.cc:47:19: error: 'g' was not declared in this scope
   47 |         auto _e = g[pos[i].first][pos[i].second];
      |                   ^
a.cc: In member function 'void mf_graph<Cap>::change_edge(int, Cap, Cap)':
a.cc:60:21: error: 'pos' was not declared in this scope
   60 |         int m = int(pos.size());
      |                     ^~~
a.cc:61:9: error: there are no arguments to 'assert' that depend on a template parameter, so a declaration of 'assert' must be available [-fpermissive]
   61 |         assert(0 <= i && i < m);
      |         ^~~~~~
a.cc:63:20: error: 'g' was not declared in this scope
   63 |         auto& _e = g[pos[i].first][pos[i].second];
      |                    ^
a.cc: In member function 'Cap mf_graph<Cap>::flow(int, int)':
a.cc:70:32: error: 'numeric_limits' is not a member of 'std'
   70 |         return flow(s, t, std::numeric_limits<Cap>::max());
      |                                ^~~~~~~~~~~~~~
a.cc:70:50: error: expected primary-expression before '>' token
   70 |         return flow(s, t, std::numeric_limits<Cap>::max());
      |                                                  ^
a.cc:70:53: error: '::max' has not been declared
   70 |         return flow(s, t, std::numeric_limits<Cap>::max());
      |                                                     ^~~
a.cc: In member function 'Cap mf_graph<Cap>::flow(int, int, Cap)':
a.cc:73:9: error: there are no arguments to 'assert' that depend on a template parameter, so a declaration of 'assert' must be available [-fpermissive]
   73 |         assert(0 <= s && s < _n);
      |         ^~~~~~
a.cc:74:9: error: there are no arguments to 'assert' that depend on a template parameter, so a declaration of 'assert' must be available [-fpermissive]
   74 |         assert(0 <= t && t < _n);
      |         ^~~~~~
a.cc:76:14: error: 'vector' is not a member of 'std'
   76 |         std::vector<int> level(_n), iter(_n);
      |              ^~~~~~
a.cc:76:14: note: 'std::vector' is defined in header '<vector>'; this is probably fixable by adding '#include <vector>'
a.cc:76:21: error: expected primary-expression before 'int'
   76 |         std::vector<int> level(_n), iter(_n);
      |                     ^~~
a.cc: In lambda function:
a.cc:80:18: error: 'fill' is not a member of 'std'
   80 |             std::fill(level.begin(), level.end(), -1);
      |                  ^~~~
a.cc:80:23: error: 'level' was not declared in this scope
   80 |             std::fill(level.begin(), level.end(), -1);
      |                       ^~~~~
a.cc:87:31: error: 'g' was not declared in this scope
   87 |                 for (auto e : g[v]) {
      |                               ^
a.cc: In lambda function:
a.cc:98:27: error: 'level' was not declared in this scope; did you mean 'level_v'?
   98 |             int level_v = level[v];
      |                           ^~~~~
      |                           level_v
a.cc:99:27: error: 'iter' was not declared in this scope
   99 |             for (int& i = iter[v]; i < int(g[v].size()); i++) {
      |                           ^~~~
a.cc:99:44: error: 'g' was not declared in this scope
   99 |             for (int& i = iter[v]; i < int(g[v].size()); i++) {
      |                                            ^
a.cc:103:43: error: 'min' is not a member of 'std'
  103 |                     self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
      |                                           ^~~
a.cc: In member function 'Cap mf_graph<Cap>::flow(int, int, Cap)':
a.cc:116:17: error: 'level' was not declared in this scope
  116 |             if (level[t] == -1) break;
      |                 ^~~~~
a.cc:117:18: error: 'fill' is not a member of 'std'
  117 |             std::fill(iter.begin(), iter.end(), 0);
      |                  ^~~~
a.cc:117:23: error: 'iter' was not declared in this scope
  117 |             std::fill(iter.begin(), iter.end(), 0);
      |                       ^~~~
a.cc: In function 'int main()':
a.cc:157:5: error: 'cin' was not declared in this scope
  157 |     cin >> n >> m;
      |     ^~~
a.cc:1:1: note: 'std::cin' is defined in header '<iostream>'; this is probably fixable by adding '#include <iostream>'
  +++ |+#include <iostream>
    1 | using namespace std;
a.cc:159:5: error: 'vector' was not declared in this scope
  159 |     vector<string> grid(n);
      |     ^~~~~~
a.cc:159:5: note: 'std::vector' is defined in header '<vector>'; this is probably fixable by adding '#include <vector>'
a.cc:159:12: error: 'string' was not declared in this scope
  159 |     vector<string> grid(n);
      |            ^~~~~~
a.cc:1:1: note: 'std::string' is defined in header '<string>'; this is probably fixable by adding '#include <string>'
  +++ |+#include <string>
    1 | using namespace std;
a.cc:159:20: error: 'grid' was not declared in this scope
  159 |     vector<string> grid(n);
      |                    ^~~~
a.cc:208:5: error: 'cout' was not declared in this scope
  208 |     cout << g.flow(s, t) << endl;
      |     ^~~~
a.cc:208:5: note: 'std::cout' is defined in header '<iostream>'; this is probably fixable by adding '#include <iostream>'
a.cc:208:29: error: 'endl' was not declared in this scope
  208 |     cout << g.flow(s, t) << endl;
      |                             ^~~~
a.cc:1:1: note: 'std::endl' is defined in header '<ostream>'; this is probably fixable by adding '#include <ostream>'
  +++ |+#include <ostream>
    1 | using namespace std;
a.cc:210:20: error: 'struct mf_graph<int>' has no member n