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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.io.*;
import java.util.*;

class Solver {
  final int n, m;
  final char[][] cm;

  private MaxFlowSolver maxFlowSolver;

  Solver(int n, int m, char[][] cm) {
    this.n = n;
    this.m = m;
    this.cm = cm;
  }

  private int toNodeIndex(int i, int j) {
    return m * i + j;
  }

  private int getI(int nodeIndex) {
    return nodeIndex / m;
  }

  private int getJ(int nodeIndex) {
    return nodeIndex % m;
  }

  private void addEdge(int i1, int j1, int i2, int j2) {
    int nodeIndex1 = toNodeIndex(i1, j1);
    int nodeIndex2 = toNodeIndex(i2, j2);
    if ((i1 + j1) % 2 == 0) {
      maxFlowSolver.addEdge(nodeIndex1, nodeIndex2, 1);
    } else {
      maxFlowSolver.addEdge(nodeIndex2, nodeIndex1, 1);
    }
  }

  List<String> solve() {
    maxFlowSolver = new MaxFlowSolver(n * m + 2);
    int source = n * m;
    int destination = n * m + 1;

    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        if (cm[i][j] == '#') {
          continue;
        }
        int nodeIndex = toNodeIndex(i, j);
        if ((i + j) % 2 == 0) {
          maxFlowSolver.addEdge(source, nodeIndex, 1);
        } else {
          maxFlowSolver.addEdge(nodeIndex, destination, 1);
        }
      }
    }

    for (int i = 0; i < n; i++) {
      for (int j = 1; j < m; j++) {
        if (cm[i][j - 1] == '#' || cm[i][j] == '#') {
          continue;
        }
        addEdge(i, j, i, j - 1);
      }
    }

    for (int i = 1; i < n; i++) {
      for (int j = 0; j < m; j++) {
        if (cm[i - 1][j] == '#' || cm[i][j] == '#') {
          continue;
        }
        addEdge(i, j, i - 1, j);
      }
    }

    int f = maxFlowSolver.flow(n * m, n * m + 1);
//    System.err.println(f);
    MaxFlowSolver.Edge[] edges = maxFlowSolver.getEdges();
    for (MaxFlowSolver.Edge edge : edges) {
//      System.err.printf("%d -> %d: %d / %d\n", edge.fromNodeIndex, edge.toNodeIndex, edge.flow, edge.capacity);
      if (edge.fromNodeIndex == n * m || edge.toNodeIndex == n * m + 1 || edge.flow == 0) {
        continue;
      }
      int minNodeIndex = Math.min(edge.fromNodeIndex, edge.toNodeIndex);
      int maxNodeIndex = Math.max(edge.fromNodeIndex, edge.toNodeIndex);
      int minI = getI(minNodeIndex);
      int maxI = getI(maxNodeIndex);
      int minJ = getJ(minNodeIndex);
      int maxJ = getJ(maxNodeIndex);
      if (minI == maxI) {
        cm[minI][minJ] = '>';
        cm[maxI][maxJ] = '<';
      } else {
        cm[minI][minJ] = 'v';
        cm[maxI][maxJ] = '^';
      }
    }

    List<String> answers = new ArrayList<>();
    answers.add("" + f);
    for (char[] cs : cm) {
      answers.add(new String(cs));
    }
    return answers;
  }
}

public class Main {
  private static void execute(ContestReader reader, ContestWriter out) {
    int n = reader.nextInt();
    int m = reader.nextInt();
    char[][] cm = reader.nextCharArray(n);
    out.printList(new Solver(n, m, cm).solve());
  }
  
  public static void main(String[] args) {
    ContestReader reader = new ContestReader(System.in);
    ContestWriter out = new ContestWriter(System.out);
    execute(reader, out);
    out.flush();
  }
}

class MaxFlowSolver {
  public class Edge {
    final int fromNodeIndex, toNodeIndex;
    final int capacity, flow;

    Edge(int fromNodeIndex, int toNodeIndex, int capacity, int flow) {
      this.fromNodeIndex = fromNodeIndex;
      this.toNodeIndex = toNodeIndex;
      this.capacity = capacity;
      this.flow = flow;
    }
  }

  private class InternalEdge {
    int id;
    int fromNodeIndex;
    int toNodeIndex;
    InternalEdge reverseEdge;
    int capacity;

    void set(int id, int fromNodeIndex, int toNodeIndex, InternalEdge reverseEdge, int capacity) {
      this.fromNodeIndex = fromNodeIndex;
      this.toNodeIndex = toNodeIndex;
      this.reverseEdge = reverseEdge;
      this.capacity = capacity;
    }
  }

  private final int n;
  private final List<InternalEdge> edges;
//  private final ArrayList[] graph;
  private final List<List<InternalEdge>> graph;

  private final int[] bfsLevel;
  private final Deque<Integer> bfsQueue;

  private final int[] dfsIterators;

  public MaxFlowSolver(int n) {
    this.n = n;
    this.edges = new ArrayList<>();
    this.graph = new ArrayList<>();
    for (int i = 0; i < n; i++) {
      this.graph.add(new ArrayList<>());
    }

    this.bfsLevel = new int[n];
    this.bfsQueue = new ArrayDeque<>();
    this.dfsIterators = new int[n];
  }

  public int addEdge(int fromNodeIndex, int toNodeIndex, int capacity) {
    InternalEdge fromEdge = new InternalEdge();
    InternalEdge toEdge = new InternalEdge();
    int m = edges.size();
    fromEdge.set(m, fromNodeIndex, toNodeIndex, toEdge, capacity);
    toEdge.set(-1, toNodeIndex, fromNodeIndex, fromEdge, 0);
    graph.get(fromNodeIndex).add(fromEdge);
    graph.get(toNodeIndex).add(toEdge);
    edges.add(fromEdge);
    return m;
  }

  Edge getEdge(int edgeIndex) {
    InternalEdge edge = edges.get(edgeIndex);
    InternalEdge reverseEdge = edge.reverseEdge;
    return new Edge(edge.fromNodeIndex, edge.toNodeIndex, edge.capacity + reverseEdge.capacity, reverseEdge.capacity);
  }

  Edge[] getEdges() {
    Edge[] returnEdges = new Edge[edges.size()];
    for (int i = 0; i < edges.size(); i++) {
      returnEdges[i] = getEdge(i);
    }
    return returnEdges;
  }

  void changeEdge(int edgeIndex, int new_capacity, int new_flow) {
    InternalEdge edge = edges.get(edgeIndex);
    InternalEdge reverseEdge = edge.reverseEdge;
    edge.capacity = new_capacity - new_flow;
    reverseEdge.capacity = new_flow;
  }

  private void bfsForFindingPath(int srcNodeIndex, int dstNodeIndex) {
    Arrays.fill(bfsLevel, -1);
    bfsLevel[srcNodeIndex] = 0;
    bfsQueue.clear();
    bfsQueue.add(srcNodeIndex);
    while (!bfsQueue.isEmpty()) {
      int nodeIndex = bfsQueue.poll();
      for (InternalEdge edge : graph.get(nodeIndex)) {
        if (edge.capacity == 0 || bfsLevel[edge.toNodeIndex] >= 0) {
          continue;
        }
        bfsLevel[edge.toNodeIndex] = bfsLevel[nodeIndex] + 1;
        if (edge.toNodeIndex == dstNodeIndex) {
          return;
        }
        bfsQueue.addLast(edge.toNodeIndex);
      }
    }
  }

  private int dfsCalculateFlow(int srcNodeIndex, int nodeIndex, int flowLimit) {
//    System.err.printf("dfs: 0, srcNodeIndex: %d, nodeIndex: %d, flowLimit: %d\n", srcNodeIndex, nodeIndex, flowLimit);
    if (nodeIndex == srcNodeIndex) {
//      System.err.printf("dfs: 1, srcNodeIndex: %d, nodeIndex: %d, flowLimit: %d\n", srcNodeIndex, nodeIndex, flowLimit);
      return flowLimit;
    }
    int flow = 0;
    int level = bfsLevel[nodeIndex];
    for (; dfsIterators[nodeIndex] < graph.get(nodeIndex).size(); dfsIterators[nodeIndex]++) {
      int i = dfsIterators[nodeIndex];
      InternalEdge edge = graph.get(nodeIndex).get(i);
      if (level <= bfsLevel[edge.toNodeIndex] || edge.reverseEdge.capacity <= 0) {
        continue;
      }
      int d = dfsCalculateFlow(srcNodeIndex, edge.toNodeIndex, Math.min(flowLimit - flow, edge.reverseEdge.capacity));
      if (d <= 0) {
        continue;
      }
      edge.capacity += d;
      edge.reverseEdge.capacity -= d;
      flow += d;
      if (flow == flowLimit) {
        break;
      }
    }
    return flow;
  }

  public int flow(int srcNodeIndex, int dstNodeIndex) {
    return flow(srcNodeIndex, dstNodeIndex, Integer.MAX_VALUE);
  }

  public int flow(int srcNodeIndex, int dstNodeIndex, int flowLimit) {
//    System.err.printf("flow: 0: \n");
    int flow = 0;
    while (flow < flowLimit) {
//      System.err.printf("flow: 1: \n");
      bfsForFindingPath(srcNodeIndex, dstNodeIndex);
      if (bfsLevel[dstNodeIndex] == -1) {
//        System.err.printf("flow: 2: \n");
        break;
      }
      Arrays.fill(dfsIterators, 0);
//      System.err.printf("flow: 3: \n");
/*
      for (int i = 0; i < n; i++) {
        System.err.printf("flow: 3: Level[%2d] = %d\n", i, bfsLevel[i]);
      }
      */
      while (flow < flowLimit) {
//        System.err.printf("flow: 4: \n");
        int f = dfsCalculateFlow(srcNodeIndex, dstNodeIndex, flowLimit - flow);
        if (f <= 0) {
//          System.err.printf("flow: 5: \n");
          break;
        }
        flow += f;
      }
    }
//    System.err.printf("flow: 6: \n");
    return flow;
  }

  private List<Boolean> minCut(int srcNodeIndex) {
    return null;
  }
}


/*
class MaxFlowSolver {
  private class InternalEdge {
    int to, rev;
    int cap;

    InternalEdge(int to, int rev, int cap) {
      this.to = to;
      this.rev = rev;
      this.cap = cap;
    }
  }

  public class Edge {
    int from,  to;
    int cap, flow;

    Edge(int from, int to, int cap, int flow) {
      this.from = from;
      this.to = to;
      this.cap = cap;
      this.flow = flow;
    }
  }

  private final int n;
  private List<InternalEdge>[] g;
  private List<Pair> pos;

  @SuppressWarnings("unchecked")
  MaxFlowSolver(int n) {
    this.n = n;
    this.g = (ArrayList<InternalEdge>[])new Object[n];
    for (int i = 0; i < n; i++) {
      g[i] = new ArrayList<>();
    }
    this.pos = new ArrayList<>();
  }

  int addEdge(int from, int to, int cap) {
    assert(0 <= from && from < n);
    assert(0 <= to && to < n);
    assert(0 <= cap);
    int m = pos.size();
    pos.add(new Pair(from, g[from].size()));
    g[from].add(new InternalEdge(to, g[to].size(), cap));
    g[to].add(new InternalEdge(from, g[from].size() - 1, 0));
    return m;
  }

  Edge get(int i) {
    int m = pos.size();
    assert(0 <= i && i < m);
    InternalEdge edge = g[pos.get(i).first].get(pos.get(i).second);
    InternalEdge reverseEdge = g[edge.to].get(edge.rev);
    return new Edge(pos.get(i).first, edge.to, edge.cap + reverseEdge.cap, reverseEdge.cap);
  }






}
*/


class ContestWriter extends PrintWriter {
  ContestWriter(PrintStream printStream) {
    super(printStream);
  }

  public void printList(List<? extends Object> list) {
    for (Object object : list) {
      println(object);
    }
  }

  public void printListOneLine(List<? extends Object> list) {
    List<String> stringList = new ArrayList<>();
    for (Object object : list) {
      stringList.add(object.toString());
    }
    println(String.join(" ", stringList));
  }
}

class ContestReader {
  private static final int BUFFER_SIZE = 1024;
  
  private final InputStream stream;
  private final byte[] buffer;
  private int pointer;
  private int bufferLength;
  
  ContestReader(InputStream stream) {
    this.stream = stream;
    this.buffer = new byte[BUFFER_SIZE];
    this.pointer = 0;
    this.bufferLength = 0;
  }
  
  private boolean hasNextByte() {
    if (pointer < bufferLength) {
      return true;
    }
    
    pointer = 0;
    try {
      bufferLength = stream.read(buffer);
    } catch (IOException e) {
      throw new RuntimeException(e);
    }
    return bufferLength > 0;
  }
  
  private int readByte() {
    if (hasNextByte()) {
      return buffer[pointer++];
    } else {
      return -1;
    }
  }
  
  private static boolean isPrintableChar(int c) {
    return 33 <= c && c <= 126;
  }
  
  public boolean hasNext() {
    while (hasNextByte() && !isPrintableChar(buffer[pointer])) {
      pointer++;
    }
    return hasNextByte();
  }
  
  public String next() {
    if (!hasNext()) {
      throw new NoSuchElementException();
    }
    StringBuilder sb = new StringBuilder();
    while(true) {
      int b = readByte();
      if (!isPrintableChar(b)) {
        break;
      }
      sb.appendCodePoint(b);
    }
    return sb.toString();
  }
  
  public String nextLine() {
    if (!hasNext()) {
      throw new NoSuchElementException();
    }
    StringBuilder sb = new StringBuilder();
    while(true) {
      int b = readByte();
      if (!isPrintableChar(b) && b != 0x20) {
        break;
      }
      sb.appendCodePoint(b);
    }
    return sb.toString();
  }
  
  public char nextChar() {
    return next().charAt(0);
  }
  
  public int nextInt() {
    if (!hasNext()) {
      throw new NoSuchElementException();
    }
    
    int n = 0;
    boolean minus = false;
    
    {
      int b = readByte();
      if (b == '-') {
        minus = true;
      } else if ('0' <= b && b <= '9') {
        n = b - '0';
      } else {
        throw new NumberFormatException();
      }
    }
    
    while(true){
      int b = readByte();
      if ('0' <= b && b <= '9') {
        n *= 10;
        n += b - '0';
      } else if (b == -1 || !isPrintableChar(b)) {
        return minus ? -n : n;
      } else {
        throw new NumberFormatException();
      }
    }
  }
  
  public long nextLong() {
    if (!hasNext()) {
      throw new NoSuchElementException();
    }
    
    long n = 0;
    boolean minus = false;
    
    {
      int b = readByte();
      if (b == '-') {
        minus = true;
      } else if ('0' <= b && b <= '9') {
        n = b - '0';
      } else {
        throw new NumberFormatException();
      }
    }
    
    while(true){
      int b = readByte();
      if ('0' <= b && b <= '9') {
        n *= 10;
        n += b - '0';
      } else if (b == -1 || !isPrintableChar(b)) {
        return minus ? -n : n;
      } else {
        throw new NumberFormatException();
      }
    }
  }
  
  public double nextDouble() {
    return Double.parseDouble(next());
  }
  
  public String[] next(int n) {
    String[] array = new String[n];
    for (int i = 0; i < n; i++) {
      array[i] = next();
    }
    return array;
  }
  
  public String[] nextLine(int n) {
    String[] array = new String[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextLine();
    }
    return array;
  }
  
  public char[] nextChar(int n) {
    char[] array = new char[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextChar();
    }
    return array;
  }
  
  public int[] nextInt(int n) {
    int[] array = new int[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextInt();
    }
    return array;
  }
  
  public long[] nextLong(int n) {
    long[] array = new long[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextLong();
    }
    return array;
  }
  
  public double[] nextDouble(int n) {
    double[] array = new double[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextDouble();
    }
    return array;
  }
  
  public char[] nextCharArray() {
    return next().toCharArray();
  }
  
  public String[][] next(int n, int m) {
    String[][] matrix = new String[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = next();
      }
    }
    return matrix;
  }
  
  public int[][] nextInt(int n, int m) {
    int[][] matrix = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextInt();
      }
    }
    return matrix;
  }
  
  public char[][] nextChar(int n, int m) {
    char[][] matrix = new char[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextChar();
      }
    }
    return matrix;
  }
  
  public long[][] nextLong(int n, int m) {
    long[][] matrix = new long[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextLong();
      }
    }
    return matrix;
  }
  
  public double[][] nextDouble(int n, int m) {
    double[][] matrix = new double[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextDouble();
      }
    }
    return matrix;
  }
  
  public char[][] nextCharArray(int n) {
    char[][] matrix = new char[n][];
    for (int i = 0; i < n; i++) {
      matrix[i] = next().toCharArray();
    }
    return matrix;
  }
}

class MyAssert {
  public static void myAssert(boolean flag, String message) {
    if (!flag) {
      throw new RuntimeException(message);
    }
  }
 
  public static void myAssert(boolean flag) {
    myAssert(flag, "");
  }
}