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)

public class Main {
    public static void main(String[] args) throws Exception {
        FastScanner sc = new FastScanner(
            // new java.io.FileInputStream(new java.io.File("MaxFlow/in"))
        );
        FastPrintStream pw = new FastPrintStream(
            // new java.io.File("MaxFlow/out")
        );
        solve(sc, pw);
        sc.close();
        pw.flush();
        pw.close();
    }

    public static void solve(FastScanner sc, FastPrintStream pw) {
        int n = sc.nextInt();
        int m = sc.nextInt();
        var g = new char[n][];
        java.util.Arrays.setAll(g, i -> sc.nextChars(m));
        MaxFlow mf = new MaxFlow(n * m + 2);
        int s = n * m;
        int t = s + 1;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                if (g[i][j] == '#') continue;
                if (((i ^ j) & 1) == 0) {
                    mf.addEdge(s, i * m + j, 1);
                } else {
                    mf.addEdge(i * m + j, t, 1);
                }
            }
        }
        for (int i = 0; i < n; i++) {
            for (int j = i & 1; j < m; j += 2) {
                if (g[i][j] == '#') continue;
                if (j - 1 >= 0 && g[i][j - 1] == '.') {
                    mf.addEdge(i * m + j, i * m + (j - 1), 1);
                }
                if (j + 1 < m && g[i][j + 1] == '.') {
                    mf.addEdge(i * m + j, i * m + (j + 1), 1);
                }
                if (i - 1 >= 0 && g[i - 1][j] == '.') {
                    mf.addEdge(i * m + j, (i - 1) * m + j, 1);
                }
                if (i + 1 < n && g[i + 1][j] == '.') {
                    mf.addEdge(i * m + j, (i + 1) * m + j, 1);
                }
            }
        }
        int ans = (int) mf.maxFlow(s, t);
        int cnt = 0;
        for (var e : mf.getEdges()) {
            if (e.getFlow() == 0) continue;
            int u = e.getFrom();
            int v = e.getTo();
            if (u == s || v == t) continue;
            int ui = u / m, uj = u % m;
            int vi = v / m, vj = v % m;
            if (g[ui][uj] != '.' || g[vi][vj] != '.') {
                throw new AssertionError();
            }
            if (ui == vi) {
                if (uj + 1 == vj) {
                    g[ui][uj] = '>';
                    g[vi][vj] = '<';
                } else if (uj == vj + 1) {
                    g[ui][uj] = '<';
                    g[vi][vj] = '>';
                } else {
                    throw new AssertionError();
                }
            } else if (uj == vj) {
                if (ui + 1 == vi) {
                    g[ui][uj] = 'v';
                    g[vi][vj] = '^';
                } else if (ui == vi + 1) {
                    g[ui][uj] = '^';
                    g[vi][vj] = 'v';
                } else {
                    throw new AssertionError();
                }
            } else {
                throw new AssertionError();
            }
            cnt++;
        }
        if (ans != cnt) throw new AssertionError();
        pw.println(ans);
        for (var gi : g) {
            pw.println(gi);
        }
    }

    static class MaxFlow {
        public class CapEdge {
            private final int from, to;
            private long cap;
            private final int rev;
            CapEdge(int from, int to, long cap, int rev) {
                this.from = from;
                this.to = to;
                this.cap = cap;
                this.rev = rev;
            }
            public int getFrom()  {return from;}
            public int getTo()    {return to;}
            public long getCap()  {return cap;}
            public long getFlow() {return g[to][rev].cap;}
        }
    
        private static final long INF = Long.MAX_VALUE;
    
        private final int n;
        private int m;
        private final java.util.ArrayList<CapEdge> edges;
        private final int[] count;
        private final CapEdge[][] g;

        private final int[] label;
        private final long[] excess;
        private final int[] actVerStack;
        private int stPtr = 0;
    
        public MaxFlow(int n) {
            this.n = n;
            this.edges = new java.util.ArrayList<>();
            this.count = new int[n];
            this.g = new CapEdge[n][];
            this.label = new int[n];
            this.excess = new long[n];
            this.actVerStack = new int[n];
        }
    
        public int addEdge(int from, int to, long cap) {
            rangeCheck(from, 0, n);
            rangeCheck(to, 0, n);
            nonNegativeCheck(cap, "Capacity");
            CapEdge e = new CapEdge(from, to, cap, count[to]);
            count[from]++; count[to]++;
            edges.add(e);
            return m++;
        }
    
        private void buildGraph() {
            for (int i = 0; i < n; i++) {
                g[i] = new CapEdge[count[i]];
            }
            int[] idx = new int[n];
            for (CapEdge e : edges) {
                g[e.to][idx[e.to]++] = new CapEdge(e.to, e.from, 0, idx[e.from]);
                g[e.from][idx[e.from]++] = e;
            }
        }

        public long maxFlow(int s, int t) {
            return flow(s, t, INF);
        }
        
        public long flow(int s, int t, long flowLimit) {
            rangeCheck(s, 0, n);
            rangeCheck(t, 0, n);
            buildGraph();
            excess[s] = INF;
            for (CapEdge e : g[s]) { // push
                int v = e.to;
                long d = e.cap;
                if (d == 0) continue;
                excess[s] -= d;
                excess[v] += d;
                g[v][e.rev].cap += d;
                e.cap -= d;
                if (v != t) actVerStack[stPtr++] = v;
            }
            globalLabeling(s);
            while (true) {
                if (stPtr == 0) break;
                int u = actVerStack[--stPtr];
                CapEdge e = getActiveEdge(u, label);
                if (e == null) { // relabel
                    actVerStack[stPtr++] = u;
                    for (CapEdge eu : g[u]) {
                        int v = eu.to;
                        if (eu.cap <= 0) continue;
                        label[u] = Math.min(label[u], label[v]);
                    }
                    label[u]++;
                } else { // push
                    int v = e.to;
                    long d = Math.min(e.cap, excess[u]);
                    if (d == 0) continue;
                    excess[u] -= d;
                    excess[v] += d;
                    g[v][e.rev].cap += d;
                    e.cap -= d;
                    if (v != t && 0 < excess[v] && excess[v] <= d) {
                        actVerStack[stPtr++] = v;
                    }
                }
            }
            long flow = 0;
            for (CapEdge e : g[s]) {
                flow += e.getFlow();
            }
            return flow;
        }

        private void globalLabeling(int s) {
            int[] que = new int[n];
            int hd = 0, tl = 0;
            que[tl++] = s;
            while (tl > hd) {
                int u = que[hd++];
                for (CapEdge e : g[u]) {
                    int v = e.to;
                    if (e.cap <= 0 || label[v] > 0) continue;
                    label[v] = label[u] + 1;
                    que[tl++] = v;
                }
            }
            label[s] = n;
        }

        private CapEdge getActiveEdge(int u, int[] label) {
            for (CapEdge e : g[u]) {
                int v = e.to;
                if (label[u] == label[v] + 1 && e.cap > 0) {
                    return e;
                }
            }
            return null;
        }

        public CapEdge getEdge(int i) {
            rangeCheck(i, 0, m);
            return edges.get(i);
        }

        public void changeEdge(int i, long newCap, long newFlow) {
            rangeCheck(i, 0, m);
            nonNegativeCheck(newCap, "Capacity");
            if (newFlow > newCap) {
                throw new IllegalArgumentException(
                    String.format("Flow %d is greater than capacity %d.", newCap, newFlow)
                );
            }
            CapEdge e = edges.get(i);
            CapEdge er = g[e.to][e.rev];
            e.cap = newCap - newFlow;
            er.cap = newFlow;
        }
    
        public java.util.ArrayList<CapEdge> getEdges() {
            return edges;
        }
    
        public boolean[] minCut(int s) {
            rangeCheck(s, 0, n);
            boolean[] reachable = new boolean[n];
            int[] stack = new int[n];
            int ptr = 0;
            stack[ptr++] = s;
            reachable[s] = true;
            while (ptr > 0) {
                int u = stack[--ptr];
                for (CapEdge e : g[u]) {
                    int v = e.to;
                    if (reachable[v] || e.cap <= 0) continue;
                    reachable[v] = true;
                    stack[ptr++] = v;
                }
            }
            return reachable;
        }
    
        private void rangeCheck(int i, int minInlusive, int maxExclusive) {
            if (i < 0 || i >= maxExclusive) {
                throw new IndexOutOfBoundsException(
                    String.format("Index %d out of bounds for length %d", i, maxExclusive)
                );
            }
        }
    
        private void nonNegativeCheck(long cap, java.lang.String attribute) {
            if (cap < 0) {
                throw new IllegalArgumentException(
                    String.format("%s %d is negative.", attribute, cap)
                );
            }
        }
    }
}

class FastPrintStream implements AutoCloseable {
    private static final int BUF_SIZE = 1 << 15;
    private final byte[] buf = new byte[BUF_SIZE];
    private int ptr = 0;
    private final java.lang.reflect.Field strField;
    private final java.nio.charset.CharsetEncoder encoder;

    private java.io.OutputStream out;

    public FastPrintStream(java.io.OutputStream out) {
        this.out = out;
        java.lang.reflect.Field f;
        try {
            f = java.lang.String.class.getDeclaredField("value");
            f.setAccessible(true);
        } catch (NoSuchFieldException | SecurityException e) {
            f = null;
        }
        this.strField = f;
        this.encoder = java.nio.charset.StandardCharsets.US_ASCII.newEncoder();
    }

    public FastPrintStream(java.io.File file) throws java.io.IOException {
        this(new java.io.FileOutputStream(file));
    }

    public FastPrintStream(java.lang.String filename) throws java.io.IOException {
        this(new java.io.File(filename));
    }

    public FastPrintStream() {
        this(System.out);
        try {
            java.lang.reflect.Field f = java.io.PrintStream.class.getDeclaredField("autoFlush");
            f.setAccessible(true);
            f.set(System.out, false);
        } catch (IllegalAccessException | IllegalArgumentException | NoSuchFieldException e) {
            // ignore
        }
    }

    public FastPrintStream println() {
        if (ptr == BUF_SIZE) internalFlush();
        buf[ptr++] = (byte) '\n';
        return this;
    }

    public FastPrintStream println(java.lang.Object o) {
        return print(o).println();
    }

    public FastPrintStream println(java.lang.String s) {
        return print(s).println();
    }

    public FastPrintStream println(char[] s) {
        return print(s).println();
    }

    public FastPrintStream println(char c) {
        return print(c).println();
    }

    public FastPrintStream println(int x) {
        return print(x).println();
    }

    public FastPrintStream println(long x) {
        return print(x).println();
    }

    public FastPrintStream println(double d, int precision) {
        return print(d, precision).println();
    }

    private FastPrintStream print(byte[] bytes) {
        int n = bytes.length;
        if (ptr + n > BUF_SIZE) {
            internalFlush();
            try {
                out.write(bytes);
            } catch (java.io.IOException e) {
                throw new RuntimeException();
            }
        } else {
            System.arraycopy(bytes, 0, buf, ptr, n);
            ptr += n;
        }
        return this;
    }

    public FastPrintStream print(java.lang.Object o) {
        return print(o.toString());
    }

    public FastPrintStream print(java.lang.String s) {
        if (strField == null) {
            return print(s.getBytes());
        } else {
            try {
                return print((byte[]) strField.get(s));
            } catch (IllegalAccessException e) {
                return print(s.getBytes());
            }
        }
    }

    public FastPrintStream print(char[] s) {
        try {
            return print(encoder.encode(java.nio.CharBuffer.wrap(s)).array());
        } catch (java.nio.charset.CharacterCodingException e) {
            byte[] bytes = new byte[s.length];
            for (int i = 0; i < s.length; i++) {
                bytes[i] = (byte) s[i];
            }
            return print(bytes);
        }
    }

    public FastPrintStream print(char c) {
        if (ptr == BUF_SIZE) internalFlush();
        buf[ptr++] = (byte) c;
        return this;
    }

    public FastPrintStream print(int x) {
        if (x == 0) {
            if (ptr == BUF_SIZE) internalFlush();
            buf[ptr++] = '0';
            return this;
        }
        int d = len(x);
        if (ptr + d > BUF_SIZE) internalFlush();
        if (x < 0) {
            buf[ptr++] = '-';
            x = -x;
            d--;
        }
        int j = ptr += d; 
        while (x > 0) {
            buf[--j] = (byte) ('0' + (x % 10));
            x /= 10;
        }
        return this;
    }

    public FastPrintStream print(long x) {
        if (x == 0) {
            if (ptr == BUF_SIZE) internalFlush();
            buf[ptr++] = '0';
            return this;
        }
        int d = len(x);
        if (ptr + d > BUF_SIZE) internalFlush();
        if (x < 0) {
            buf[ptr++] = '-';
            x = -x;
            d--;
        }
        int j = ptr += d; 
        while (x > 0) {
            buf[--j] = (byte) ('0' + (x % 10));
            x /= 10;
        }
        return this;
    }

    public FastPrintStream print(double d, int precision) {
        if (d < 0) {
            print('-');
            d = -d;
        }
        d += Math.pow(10, -d) / 2;
        print((long) d).print('.');
        d -= (long) d;
        for(int i = 0; i < precision; i++){
            d *= 10;
            print((int) d);
            d -= (int) d;
        }
        return this;
    }

    private void internalFlush() {
        try {
            out.write(buf, 0, ptr);
            ptr = 0;
        } catch (java.io.IOException e) {
            throw new RuntimeException(e);
        }
    }

    public void flush() {
        try {
            out.write(buf, 0, ptr);
            out.flush();
            ptr = 0;
        } catch (java.io.IOException e) {
            throw new RuntimeException(e);
        }
    }

    public void close() {
        try {
            out.close();
        } catch (java.io.IOException e) {
            throw new RuntimeException(e);
        }
    }

    private static int len(int x) {
        int d = 1;
        if (x >= 0) {d = 0; x = -x;}
        int p = -10;
        for (int i = 1; i < 10; i++, p *= 10) if (x > p) return i + d;
        return 10 + d;
    }

    private static int len(long x) {
        int d = 1;
        if (x >= 0) {d = 0; x = -x;}
        long p = -10;
        for (int i = 1; i < 19; i++, p *= 10) if (x > p) return i + d;
        return 19 + d;
    }
}

class FastScanner implements AutoCloseable {
    private final java.io.InputStream in;
    private final byte[] buf = new byte[2048];
    private int ptr = 0;
    private int buflen = 0;

    public FastScanner(java.io.InputStream in) {
        this.in = in;
    }

    public FastScanner() {
        this(System.in);
    }

    private boolean hasNextByte() {
        if (ptr < buflen) return true;
        ptr = 0;
        try {
            buflen = in.read(buf);
        } catch (java.io.IOException e) {
            throw new RuntimeException(e);
        }
        return buflen > 0;
    }

    private int readByte() {
        return hasNextByte() ? buf[ptr++] : -1;
    }

    public boolean hasNext() {
        while (hasNextByte() && !(32 < buf[ptr] && buf[ptr] < 127)) ptr++;
        return hasNextByte();
    }

    private StringBuilder nextSequence() {
        if (!hasNext()) throw new java.util.NoSuchElementException();
        StringBuilder sb = new StringBuilder();
        for (int b = readByte(); 32 < b && b < 127; b = readByte()) {
            sb.appendCodePoint(b);
        }
        return sb;
    }

    public String next() {
        return nextSequence().toString();
    }

    public String next(int len) {
        return new String(nextChars(len));
    }

    public char nextChar() {
        if (!hasNextByte()) throw new java.util.NoSuchElementException();
        return (char) readByte();
    }

    public char[] nextChars() {
        StringBuilder sb = nextSequence();
        int l = sb.length();
        char[] dst = new char[l];
        sb.getChars(0, l, dst, 0);
        return dst;
    }
    public char[] nextChars(int len) {
        if (!hasNext()) throw new java.util.NoSuchElementException();
        char[] s = new char[len];
        int i = 0;
        int b = readByte();
        while (32 < b && b < 127 && i < len) {
            s[i++] = (char) b; b = readByte();
        }
        if (i != len) {
            throw new java.util.NoSuchElementException(
                String.format("Next token has smaller length than expected.", len)
            );
        }
        return s;
    }
    public long nextLong() {
        if (!hasNext()) throw new java.util.NoSuchElementException();
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {
            minus = true;
            b = readByte();
        }
        if (b < '0' || '9' < b) throw new NumberFormatException();
        while (true) {
            if ('0' <= b && b <= '9') {
                n = n * 10 + b - '0';
            } else if (b == -1 || !(32 < b && b < 127)) {
                return minus ? -n : n;
            } else throw new NumberFormatException();
            b = readByte();
        }
    }
    public int nextInt() {
        return Math.toIntExact(nextLong());
    }
    public double nextDouble() {
        return Double.parseDouble(next());
    }
    public void close() {
        try {
            in.close();
        } catch (java.io.IOException e) {
            throw new RuntimeException(e);
        }
    }
}