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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Diagnostics.CodeAnalysis;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Reflection.Metadata;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading;

namespace AtCoder
{
    public class Program
    {
        static void Main()
        {
            using var cin = new Scanner();
            var (n, m) = cin.Int2();
            var s = new string[n];
            for (int i = 0; i < n; i++)
            {
                s[i] = cin.Next();
            }

            int num = n * m + 2;
            int si = n * m;
            int ti = n * m + 1;
            var flow = MFGraph.Long(num);
            for (int i = 0; i < n; i++)
            {
                for (int j = 0; j < m; j++)
                {
                    if (s[i][j] == '#')
                    {
                        continue;
                    }

                    int k = i * m + j;
                    if ((i + j) % 2 == 0)
                    {
                        flow.AddEdge(si, k, 1);
                        Helper.DoAt4(i, j, n, m, (ii, jj) =>
                        {
                            if (s[ii][jj] == '.')
                            {
                                int kk = ii * m + jj;
                                flow.AddEdge(k, kk, 1);
                            }
                        });
                    }
                    else
                    {
                        flow.AddEdge(k, ti, 1);
                    }
                }
            }

            long maxFlow = flow.Flow(si, ti);
            Console.WriteLine(maxFlow);

            var ans = new char[n][];
            for (int i = 0; i < n; i++)
            {
                ans[i] = new char[m];
            }

            for (int i = 0; i < n; i++)
            {
                for (int j = 0; j < m; j++)
                {
                    ans[i][j] = s[i][j];
                }
            }

            var edges = flow.Edges();
            foreach (var e in edges)
            {
                if (e.From == si || e.To == ti || e.Flow == 0)
                {
                    continue;
                }

                int i0 = e.From / m;
                int j0 = e.From % m;
                int i1 = e.To / m;
                int j1 = e.To % m;

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

            for (int i = 0; i < n; i++)
            {
                Console.WriteLine(new string(ans[i]));
            }
        }
    }
    public interface INumOperater<T> : IComparer<T> where T : struct
    {
        public T Zero { get; }
        public bool IsZero(T v);
        public T MaxValue { get; }
        T Add(T v1, T v2);
        T Sub(T v1, T v2);
    }
    public readonly struct IntOperater : INumOperater<int>
    {
        public int Zero => 0;
        public bool IsZero(int v) => v == 0;
        public int MaxValue => int.MaxValue;
        public int Add(int v1, int v2) => v1 + v2;
        public int Sub(int v1, int v2) => v1 - v2;
        public int Compare(int x, int y) => x.CompareTo(y);
    }
    public readonly struct LongOperater : INumOperater<long>
    {
        public long Zero => 0;
        public bool IsZero(long v) => v == 0;
        public long MaxValue => long.MaxValue;
        public long Add(long v1, long v2) => v1 + v2;
        public long Sub(long v1, long v2) => v1 - v2;
        public int Compare(long x, long y) => x.CompareTo(y);
    }
    public static class MFGraph
    {
        public static MFGraph<long, LongOperater> Long(int n) => new MFGraph<long, LongOperater>(n);
    }
    public class MFGraph<TValue, TOp>
        where TValue : struct
        where TOp : struct, INumOperater<TValue>
    {
        static TOp op = default;
        public MFGraph(int n)
        {
            _n = n;
            _g = new List<EdgeInternal>[n];
            for (int i = 0; i < n; i++)
            {
                _g[i] = new List<EdgeInternal>();
            }
            _pos = new List<(int first, int second)>();
        }

        public int AddEdge(int from, int to, TValue cap)
        {
            int m = _pos.Count;
            Debug.Assert(0 <= from && from < _n);
            Debug.Assert(0 <= to && to < _n);
            //Debug.Assert(0 <= cap);
            _pos.Add((from, _g[from].Count));
            _g[from].Add(new EdgeInternal(to, _g[to].Count, cap));
            _g[to].Add(new EdgeInternal(from, _g[from].Count - 1, default));
            return m;
        }

        public Edge GetEdge(int i)
        {
            int m = _pos.Count;
            Debug.Assert(0 <= i && i < m);
            var _e = _g[_pos[i].first][_pos[i].second];
            var _re = _g[_e.To][_e.Rev];
            return new Edge(_pos[i].first, _e.To, op.Add(_e.Cap, _re.Cap), _re.Cap);
        }

        public List<Edge> Edges()
        {
            int m = _pos.Count;
            var result = new List<Edge>();
            for (int i = 0; i < m; i++)
            {
                result.Add(GetEdge(i));
            }
            return result;
        }

        public void ChangeEdge(int i, TValue newCap, TValue newFlow)
        {
            int m = _pos.Count;
            Debug.Assert(0 <= i && i < m);
            //Debug.Assert(0 <= new_flow && new_flow <= new_cap);
            var _e = _g[_pos[i].first][_pos[i].second];
            var _re = _g[_e.To][_e.Rev];
            _e.Cap = op.Sub(newCap, newFlow);
            _re.Cap = newFlow;
        }

        public TValue Flow(int s, int t)
        {
            return Flow(s, t, op.MaxValue);
        }

        public TValue Flow(int s, int t, TValue flowLimit)
        {
            Debug.Assert(0 <= s && s < _n);
            Debug.Assert(0 <= t && t < _n);

            var level = new int[_n];
            var iter = new int[_n];
            var que = new Queue<int>();

            void Bfs()
            {
                for (int i = 0; i < _n; i++)
                {
                    level[i] = -1;
                }

                level[s] = 0;
                que.Clear();
                que.Enqueue(s);
                while (que.Count > 0)
                {
                    int v = que.Dequeue();
                    foreach (var e in _g[v])
                    {
                        if (op.IsZero(e.Cap) || level[e.To] >= 0) continue;
                        level[e.To] = level[v] + 1;
                        if (e.To == t) return;
                        que.Enqueue(e.To);
                    }
                }
            }

            TValue Dfs(int v, TValue up)
            {
                if (v == s) return up;
                var res = default(TValue);
                int level_v = level[v];
                for (int i = iter[v]; i < _g[v].Count; i++)
                {
                    EdgeInternal e = _g[v][i];
                    if (level_v <= level[e.To] || op.IsZero(_g[e.To][e.Rev].Cap)) continue;
                    var up1 = op.Sub(up, res);
                    var up2 = _g[e.To][e.Rev].Cap;
                    var d = Dfs(e.To, op.Compare(up1, up2) < 0 ? up1 : up2);
                    if (op.Compare(d, op.Zero) <= 0) continue;
                    _g[v][i].Cap = op.Add(_g[v][i].Cap, d);
                    _g[e.To][e.Rev].Cap = op.Sub(_g[e.To][e.Rev].Cap, d);
                    res = op.Add(res, d);
                    if (res.Equals(up)) break;
                }

                return res;
            }

            TValue flow = default;
            while (op.Compare(flow, flowLimit) < 0)
            {
                Bfs();
                if (level[t] == -1) break;
                for (int i = 0; i < _n; i++)
                {
                    iter[i] = 0;
                }
                while (op.Compare(flow, flowLimit) < 0)
                {
                    var f = Dfs(t, op.Sub(flowLimit, flow));
                    if (op.IsZero(f)) break;
                    flow = op.Add(flow, f);
                }
            }
            return flow;
        }

        public bool[] MinCut(int s)
        {
            var visited = new bool[_n];
            var que = new Queue<int>();
            que.Enqueue(s);
            while (que.Count > 0)
            {
                int p = que.Dequeue();
                visited[p] = true;
                foreach (var e in _g[p])
                {
                    if (!op.IsZero(e.Cap) && !visited[e.To])
                    {
                        visited[e.To] = true;
                        que.Enqueue(e.To);
                    }
                }
            }

            return visited;
        }

        public struct Edge
        {
            public int From { get; set; }
            public int To { get; set; }
            public TValue Cap { get; set; }
            public TValue Flow { get; set; }
            public Edge(int from, int to, TValue cap, TValue flow)
            {
                From = from;
                To = to;
                Cap = cap;
                Flow = flow;
            }
        };

        private class EdgeInternal
        {
            public int To { get; set; }
            public int Rev { get; set; }
            public TValue Cap { get; set; }
            public EdgeInternal(int to, int rev, TValue cap)
            {
                To = to;
                Rev = rev;
                Cap = cap;
            }
        };

        private readonly int _n;
        private readonly List<(int first, int second)> _pos;
        private readonly List<EdgeInternal>[] _g;
    }

    public class HashMap<TKey, TValue> : Dictionary<TKey, TValue>
    {
        private readonly Func<TKey, TValue> initialzier_;
        public HashMap(Func<TKey, TValue> initialzier)
            : base()
        {
            initialzier_ = initialzier;
        }

        public HashMap(Func<TKey, TValue> initialzier, int capacity)
            : base(capacity)
        {
            initialzier_ = initialzier;
        }

        new public TValue this[TKey key]
        {
            get
            {
                if (ContainsKey(key) == false)
                {
                    base[key] = initialzier_(key);
                }

                return base[key];
            }

            set { base[key] = value; }
        }
    }

    public struct ModInt
    {
        public const long P = 1000000007;
        //public const long P = 998244353;

        public static ModInt Inverse(ModInt value) => Pow(value, P - 2);
        public static ModInt Pow(ModInt value, long k) => Pow(value.value_, k);
        public static ModInt Pow(long value, long k)
        {
            long ret = 1;
            for (k %= P - 1; k > 0; k >>= 1, value = value * value % P)
            {
                if ((k & 1) == 1)
                {
                    ret = ret * value % P;
                }
            }
            return new ModInt(ret);
        }

        private long value_;

        public ModInt(long value)
            => value_ = value;
        public ModInt(long value, bool mods)
        {
            if (mods)
            {
                value %= P;
                if (value < 0)
                {
                    value += P;
                }
            }

            value_ = value;
        }

        public static ModInt operator +(ModInt lhs, ModInt rhs)
        {
            lhs.value_ = (lhs.value_ + rhs.value_) % P;
            return lhs;
        }
        public static ModInt operator +(long lhs, ModInt rhs)
        {
            rhs.value_ = (lhs + rhs.value_) % P;
            return rhs;
        }
        public static ModInt operator +(ModInt lhs, long rhs)
        {
            lhs.value_ = (lhs.value_ + rhs) % P;
            return lhs;
        }

        public static ModInt operator -(ModInt lhs, ModInt rhs)
        {
            lhs.value_ = (P + lhs.value_ - rhs.value_) % P;
            return lhs;
        }
        public static ModInt operator -(long lhs, ModInt rhs)
        {
            rhs.value_ = (P + lhs - rhs.value_) % P;
            return rhs;
        }
        public static ModInt operator -(ModInt lhs, long rhs)
        {
            lhs.value_ = (P + lhs.value_ - rhs) % P;
            return lhs;
        }

        public static ModInt operator *(ModInt lhs, ModInt rhs)
        {
            lhs.value_ = lhs.value_ * rhs.value_ % P;
            return lhs;
        }
        public static ModInt operator *(long lhs, ModInt rhs)
        {
            rhs.value_ = lhs * rhs.value_ % P;
            return rhs;
        }
        public static ModInt operator *(ModInt lhs, long rhs)
        {
            lhs.value_ = lhs.value_ * rhs % P;
            return lhs;
        }

        public static ModInt operator /(ModInt lhs, ModInt rhs)
        {
            long exp = P - 2;
            while (exp > 0)
            {
                if (exp % 2 > 0)
                {
                    lhs *= rhs;
                }

                rhs *= rhs;
                exp /= 2;
            }

            return lhs;
        }

        public static implicit operator ModInt(long n) => new ModInt(n, true);
        public long ToLong() => value_;
        public override string ToString() => value_.ToString();
    }

    public static class Helper
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void UpdateMin<T>(ref T target, T value) where T : IComparable<T>
            => target = target.CompareTo(value) > 0 ? value : target;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void UpdateMin<T>(ref T target, T value, Action<T> onUpdated) where T : IComparable<T>
        {
            if (target.CompareTo(value) > 0)
            {
                target = value;
                onUpdated(value);
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void UpdateMax<T>(ref T target, T value) where T : IComparable<T>
            => target = target.CompareTo(value) < 0 ? value : target;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void UpdateMax<T>(ref T target, T value, Action<T> onUpdated) where T : IComparable<T>
        {
            if (target.CompareTo(value) < 0)
            {
                target = value;
                onUpdated(value);
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static T[] Array<T>(int n, Func<int, T> init)
            => Enumerable.Range(0, n).Select(x => init(x)).ToArray();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static List<T> List<T>(int n, Func<int, T> init)
            => Enumerable.Range(0, n).Select(x => init(x)).ToList();

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static T[,] Array2<T>(int n, int m, T init)
            where T : struct
        {
            var array = new T[n, m];
            for (int i = 0; i < n; i++)
            {
                for (int j = 0; j < m; j++)
                {
                    array[i, j] = init;
                }
            }

            return array;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static T[,] Array2<T>(int n, int m, Func<int, int, T> initializer)
        {
            var array = new T[n, m];
            for (int i = 0; i < n; i++)
            {
                for (int j = 0; j < m; j++)
                {
                    array[i, j] = initializer(i, j);
                }
            }

            return array;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static T[,,] Array3<T>(int n1, int n2, int n3, T init)
            where T : struct
        {
            var array = new T[n1, n2, n3];
            for (int i1 = 0; i1 < n1; i1++)
            {
                for (int i2 = 0; i2 < n2; i2++)
                {
                    for (int i3 = 0; i3 < n3; i3++)
                    {
                        array[i1, i2, i3] = init;
                    }
                }
            }

            return array;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static T[,,,] Array4<T>(int n1, int n2, int n3, int n4, T init)
            where T : struct
        {
            var array = new T[n1, n2, n3, n4];
            for (int i1 = 0; i1 < n1; i1++)
            {
                for (int i2 = 0; i2 < n2; i2++)
                {
                    for (int i3 = 0; i3 < n3; i3++)
                    {
                        for (int i4 = 0; i4 < n4; i4++)
                        {
                            array[i1, i2, i3, i4] = init;
                        }
                    }
                }
            }

            return array;
        }

        private static readonly int[] delta4_ = { 1, 0, -1, 0, 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void DoAt4(int i, int j, int imax, int jmax, Action<int, int> action)
        {
            for (int dn = 0; dn < 4; dn++)
            {
                int d4i = i + delta4_[dn];
                int d4j = j + delta4_[dn + 1];
                if ((uint)d4i < (uint)imax && (uint)d4j < (uint)jmax)
                {
                    action(d4i, d4j);
                }
            }
        }

        private static readonly int[] delta8_ = { 1, 0, -1, 0, 1, 1, -1, -1, 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void DoAt8(int i, int j, int imax, int jmax, Action<int, int> action)
        {
            for (int dn = 0; dn < 8; dn++)
            {
                int d8i = i + delta8_[dn];
                int d8j = j + delta8_[dn + 1];
                if ((uint)d8i < (uint)imax && (uint)d8j < (uint)jmax)
                {
                    action(d8i, d8j);
                }
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void ForEachSubBit(int bit, Action<int> action)
        {
            for (int sub = bit; sub >= 0; sub--)
            {
                sub &= bit;
                action(sub);
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static string Reverse(string src)
        {
            var chars = src.ToCharArray();
            for (int i = 0, j = chars.Length - 1; i < j; i++, j--)
            {
                var tmp = chars[i];
                chars[i] = chars[j];
                chars[j] = tmp;
            }
            return new string(chars);
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static string Join<T>(this IEnumerable<T> values, string separator = "")
            => string.Join(separator, values);
    }

    public class Scanner : IDisposable
    {
        private const int BUFFER_SIZE = 1024;
        private const int ASCII_CHAR_BEGIN = 33;
        private const int ASCII_CHAR_END = 126;
        private readonly string filePath_;
        private readonly Stream stream_;
        private readonly byte[] buf_ = new byte[BUFFER_SIZE];
        private int length_ = 0;
        private int index_ = 0;
        private bool isEof_ = false;

        public Scanner(string file = "")
        {
            if (string.IsNullOrWhiteSpace(file))
            {
                stream_ = Console.OpenStandardInput();
            }
            else
            {
                filePath_ = file;
                stream_ = new FileStream(file, FileMode.Open);
            }

            Console.SetOut(new StreamWriter(Console.OpenStandardOutput())
            {
                AutoFlush = false
            });
        }

        public void Dispose()
        {
            Console.Out.Flush();
            stream_.Dispose();
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public char Char()
        {
            byte b;
            do
            {
                b = Read();
            } while (b < ASCII_CHAR_BEGIN || ASCII_CHAR_END < b);

            return (char)b;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public string Next()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= ASCII_CHAR_BEGIN && b <= ASCII_CHAR_END; b = (char)Read())
            {
                sb.Append(b);
            }

            return sb.ToString();
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public int Int() => (int)Long();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public int Int(int offset) => Int() + offset;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (int, int) Int2(int offset = 0)
            => (Int(offset), Int(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (int, int, int) Int3(int offset = 0)
            => (Int(offset), Int(offset), Int(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (int, int, int, int) Int4(int offset = 0)
            => (Int(offset), Int(offset), Int(offset), Int(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public int[] ArrayInt(int length, int offset = 0)
        {
            int[] Array = new int[length];
            for (int i = 0; i < length; i++)
            {
                Array[i] = Int(offset);
            }
            return Array;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public long Long()
        {
            long ret = 0;
            byte b;
            bool ng = false;
            do
            {
                b = Read();
            } while (b != '-' && (b < '0' || '9' < b));

            if (b == '-')
            {
                ng = true;
                b = Read();
            }

            for (; true; b = Read())
            {
                if (b < '0' || '9' < b)
                {
                    return ng ? -ret : ret;
                }
                else
                {
                    ret = ret * 10 + b - '0';
                }
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public long Long(long offset) => Long() + offset;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (long, long) Long2(long offset = 0)
            => (Long(offset), Long(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (long, long, long) Long3(long offset = 0)
            => (Long(offset), Long(offset), Long(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (long, long, long, long) Long4(long offset = 0)
            => (Long(offset), Long(offset), Long(offset), Long(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public long[] ArrayLong(int length, long offset = 0)
        {
            long[] Array = new long[length];
            for (int i = 0; i < length; i++)
            {
                Array[i] = Long(offset);
            }
            return Array;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public double Double() => double.Parse(Next(), CultureInfo.InvariantCulture);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public double Double(double offset) => Double() + offset;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (double, double) Double2(double offset = 0)
            => (Double(offset), Double(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (double, double, double) Double3(double offset = 0)
            => (Double(offset), Double(offset), Double(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (double, double, double, double) Double4(double offset = 0)
            => (Double(offset), Double(offset), Double(offset), Double(offset));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public double[] ArrayDouble(int length, double offset = 0)
        {
            double[] Array = new double[length];
            for (int i = 0; i < length; i++)
            {
                Array[i] = Double(offset);
            }
            return Array;
        }

        private byte Read()
        {
            if (isEof_)
            {
                throw new EndOfStreamException();
            }

            if (index_ >= length_)
            {
                index_ = 0;
                if ((length_ = stream_.Read(buf_, 0, BUFFER_SIZE)) <= 0)
                {
                    isEof_ = true;
                    return 0;
                }
            }

            return buf_[index_++];
        }

        public void Save(string text)
        {
            if (string.IsNullOrWhiteSpace(filePath_))
            {
                return;
            }

            File.WriteAllText(filePath_ + "_output.txt", text);
        }
    }
}