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 (Python):
from collections import deque

class mf_graph:
    def __init__(self,N):
        self.N = N
        self.graph = [[] for _ in range(N)]
    
    def add_edge(self,v,w,cap):
        edge = [w,cap,None]
        edge[2] = rev_edge = [v,0,edge]
        self.graph[v].append(edge)
        self.graph[w].append(rev_edge)
    
    def bfs(self,s,t):
        self.level = [None]*self.N
        dq = deque([s])
        self.level[s] = 0
        while dq:
            v = dq.popleft()
            nlevel = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = nlevel
                    dq.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,v,f):
        if v == s:
            return f
        for rev in self.it[v]:
            w,_,e = rev
            cap = e[1]
            if cap and self.level[w] is not None:
                if self.level[v] > self.level[w]:
                    d = self.dfs(s,w,min(f,cap))
                    if d:
                        e[1] -= d
                        rev[1] += d
                        return d
        return 0
    
    def flow(self,s,t,flow_limit=float('inf')):
        flow = 0
        while self.bfs(s,t):
            *self.it, = map(iter,self.graph)
            f = flow_limit
            while f:
                f = self.dfs(s,t,flow_limit-flow)
                flow += f
        return flow

    def min_cut(self,s):
        visited = [False]*self.N
        dq = deque([s])
        while dq:
            v = dq.pop()
            visited[v] = True
            for w,cap,_ in self.graph[v]:
                if cap and not visited[w]:
                    dq.append(w)


N,M = map(int, input().split())
S = [list(input()) for i in range(N)]
dc = mf_graph(N*M+2)

s =N*M
t = N*M+1
for i in range(N):
    for j in range(M):
        if (i+j) % 2 == 0:
            dc.add_edge(s,M*i+j,1)
        else:
            dc.add_edge(M*i+j,t,1)

for i in range(N):
    for j in range(M):
        if j+1 < M and S[i][j] == "." and S[i][j+1] == ".":
            u,v = M*i+j,M*i+j+1
            if (i+j) % 2 == 1:
                u,v = v,u
            dc.add_edge(u,v,1)
        if i+1 < N and S[i][j] == "." and S[i+1][j] == ".":
            u,v = M*i+j, M*(i+1)+j
            if (i+j) % 2 == 1:
                u,v = v,u
            dc.add_edge(u,v,1)

print(dc.flow(s,t,10**9))
for u in range(N*M+2):
    for v,cap,_ in dc.graph[u]:
        ui,uj = divmod(u,M)
        vi,vj = divmod(v,M)
        if (ui+uj) % 2 == 0 and cap == 0 and u != s and u != t and v != s and v!= t:
            if ui+1 == vi:
                S[ui][uj] = "v"
                S[vi][vj] = "^"
            elif ui == vi+1:
                S[ui][uj] = "^"
                S[vi][vj] = "v"
            elif uj+1 == vj:
                S[ui][uj] = ">"
                S[vi][vj] = "<"
            elif uj == vj+1:
                S[ui][uj] = "<"
                S[vi][vj] = ">"

for res in S:
    print("".join(res))

Solution (Python):
from collections import deque

class mf_graph:
    def __init__(self,N):
        self.N = N
        self.graph = [[] for _ in range(N)]
    
    def add_edge(self,v,w,cap):
        edge = [w,cap,None]
        edge[2] = rev_edge = [v,0,edge]
        self.graph[v].append(edge)
        self.graph[w].append(rev_edge)
    
    def bfs(self,s,t):
        self.level = [None]*self.N
        dq = deque([s])
        self.level[s] = 0
        while dq:
            v = dq.popleft()
            nlevel = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = nlevel
                    dq.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,v,f):
        if v == s:
            return f
        for rev in self.it[v]:
            w,_,e = rev
            cap = e[1]
            if cap and self.level[w] is not None:
                if self.level[v] > self.level[w]:
                    d = self.dfs(s,w,min(f,cap))
                    if d:
                        e[1] -= d
                        rev[1] += d
                        return d
        return 0
    
    def flow(self,s,t,flow_limit=float('inf')):
        flow = 0
        while self.bfs(s,t):
            *self.it, = map(iter,self.graph)
            f = flow_limit
            while f:
                f = self.dfs(s,t,flow_limit-flow)
                flow += f
        return flow

    def min_cut(self,s):
        visited = [False]*self.N
        dq = deque([s])
        while dq:
            v = dq.pop()
            visited[v] = True
            for w,cap,_ in self.graph[v]:
                if cap and not visited[w]:
                    dq.append(w)


N,M = map(int, input().split())
S = [list(input()) for i in range(N)]
dc = mf_graph(N*M+2)

s =N*M
t = N*M+1
for i in range(N):
    for j in range(M):
        if (i + j) % 2 == 0:
            dc.add_edge(s,M*i+j,1)
        else:
            dc.add_edge(M*i+j,t,1)

for i in range(N):
    for j in range(M):
        if j+1 < M and S[i][j] == "." and S[i][j+1] == ".":
            u,v = M*i+j,M*i+j+1
            if (i+j) % 2 == 1:
                u,v = v,u
            dc.add_edge(u, v, 1)
        if i+1 < N and S[i][j] == "." and S[i + 1][j] == ".":
            u,v = M*i+j, M*(i+1)+j
            if (i+j) % 2 == 1:
                u,v = v,u
            dc.add_edge(u, v, 1)

print(dc.flow(s, t))
for u in range(N*M+2):
    for v, cap, _ in dc.graph[u]:
        ui, uj = divmod(u, M)
        vi, vj = divmod(v, M)
        if (ui+uj) % 2 == 0 and cap == 0 and u != s and u != t and v != s and v!= t:
            if ui+1 == vi:
                S[ui][uj] = "v"
                S[vi][vj] = "^"
            elif ui == vi+1:
                S[ui][uj] = "^"
                S[vi][vj] = "v"
            elif uj+1 == vj:
                S[ui][uj] = ">"
                S[vi][vj] = "<"
            elif uj == vj+1:
                S[ui][uj] = "<"
                S[vi][vj] = ">"

for res in S:
    print("".join(res))

Solution (Python):
import sys

sys.path.append("/home/contestant/.local/lib/python3.8/site-packages")


import networkx as nx

R, C = [int(x) for x in input().split()]
grid = [list(input()) for r in range(R)]


def pack(r, c):
    return r * C + c


graph = nx.Graph()
top_nodes = set()  
for r in range(R):
    for c in range(C):
        if r + 1 < R and grid[r][c] == grid[r + 1][c] == ".":
            u, v = pack(r, c), pack(r + 1, c)
            graph.add_edge(u, v)
            if (r + c) % 2 == 0:
                top_nodes.add(v)
            else:
                top_nodes.add(u)
        if c + 1 < C and grid[r][c] == grid[r][c + 1] == ".":
            u, v = pack(r, c), pack(r, c + 1)
            graph.add_edge(u, v)
            if (r + c) % 2 == 0:
                top_nodes.add(v)
            else:
                top_nodes.add(u)


matching = nx.bipartite.maximum_matching(graph, top_nodes=top_nodes)

count = 0
for u, v in matching.items():
    if u > v:
        assert matching[v] == u
        r1, c1 = divmod(u, C)
        r2, c2 = divmod(v, C)
        if r1 == r2:
            if c1 > c2:
                c1, c2 = c2, c1
            grid[r1][c1], grid[r2][c2] = ">", "<"
        elif c1 == c2:
            if r1 > r2:
                r1, r2 = r2, r1
            grid[r1][c1] = "v"
            grid[r2][c2] = "^"
        count += 1

print(count)
for row in grid:
    print("".join(row))

Solution (Python):
import networkx as nx

R, C = [int(x) for x in input().split()]
grid = [list(input()) for r in range(R)]


def pack(r, c):
    return r * C + c


graph = nx.Graph()
top_nodes = set()  
for r in range(R):
    for c in range(C):
        if r + 1 < R and grid[r][c] == grid[r + 1][c] == ".":
            u, v = pack(r, c), pack(r + 1, c)
            graph.add_edge(u, v)
            if (r + c) % 2 == 0:
                top_nodes.add(v)
            else:
                top_nodes.add(u)
        if c + 1 < C and grid[r][c] == grid[r][c + 1] == ".":
            u, v = pack(r, c), pack(r, c + 1)
            graph.add_edge(u, v)
            if (r + c) % 2 == 0:
                top_nodes.add(v)
            else:
                top_nodes.add(u)


matching = nx.bipartite.maximum_matching(graph, top_nodes=top_nodes)

count = 0
for u, v in matching.items():
    if u > v:
        assert matching[v] == u
        r1, c1 = divmod(u, C)
        r2, c2 = divmod(v, C)
        if r1 == r2:
            if c1 > c2:
                c1, c2 = c2, c1
            grid[r1][c1], grid[r2][c2] = ">", "<"
        elif c1 == c2:
            if r1 > r2:
                r1, r2 = r2, r1
            grid[r1][c1] = "v"
            grid[r2][c2] = "^"
        count += 1

print(count)
for row in grid:
    print("".join(row))

Solution (Python):
from collections import deque
class Dinic:
    def __init__(self,N):
        self.N = N
        self.graph = [[] for _ in range(N)]
    
    def add_edge(self,v,w,cap):
        edge = [w,cap,None]
        edge[2] = rev_edge = [v,0,edge]
        self.graph[v].append(edge)
        self.graph[w].append(rev_edge)
    
    def bfs(self,s,t):
        self.level = [None]*self.N
        d = deque([s])
        self.level[s] = 0
        while d:
            v = d.popleft()
            nlevel = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = nlevel
                    d.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,v,f):
        if v == s:
            return f
        for rev in self.it[v]:
            w,_,e = rev
            cap = e[1]
            if cap and self.level[w] is not None:
                if self.level[v] > self.level[w]:
                    d = self.dfs(s,w,min(f,cap))
                    if d:
                        e[1] -= d
                        rev[1] += d
                        return d
        return 0
    
    def max_flow(self,s,t):
        flow = 0
        INF = 10**9
        while self.bfs(s,t):
            *self.it, = map(iter,self.graph)
            f = INF
            while f:
                f = self.dfs(s,t,INF)
                flow += f
        return flow


N,M = map(int, input().split())
S = [list(input()) for i in range(N)]
dc = Dinic(N*M+2)

s =N*M
t = N*M+1
for i in range(N):
    for j in range(M):
        if (i + j) % 2 == 0:
            dc.add_edge(s,M*i+j,1)
        else:
            dc.add_edge(M*i+j,t,1)

for i in range(N):
    for j in range(M):
        if j+1 < M and S[i][j] == "." and S[i][j+1] == ".":
            u,v = M*i+j,M*i+j+1
            if (i+j) % 2 == 1:
                u,v = v,u
            dc.add_edge(u, v, 1)
        if i+1 < N and S[i][j] == "." and S[i + 1][j] == ".":
            u,v = M*i+j, M*(i+1)+j
            if (i+j) % 2 == 1:
                u,v = v,u
            dc.add_edge(u, v, 1)

print(dc.max_flow(s, t))
for u in range(N*M+2):
    for v, cap, _ in dc.graph[u]:
        ui, uj = divmod(u, M)
        vi, vj = divmod(v, M)
        if (ui+uj) % 2 == 0 and cap == 0 and u != s and u != t and v != s and v!= t:
            if ui+1 == vi:
                S[ui][uj] = "v"
                S[vi][vj] = "^"
            elif ui == vi+1:
                S[ui][uj] = "^"
                S[vi][vj] = "v"
            elif uj+1 == vj:
                S[ui][uj] = ">"
                S[vi][vj] = "<"
            elif uj == vj+1:
                S[ui][uj] = "<"
                S[vi][vj] = ">"

for res in S:
    print("".join(res))

Solution (Python):
from collections import deque
import sys
sys.setrecursionlimit(1<<30)
class Dinic:
    def __init__(self,N):
        self.N = N
        self.graph = [[] for _ in range(N)]
    
    def add_edge(self,v,w,cap):
        edge = [w,cap,None]
        edge[2] = rev_edge = [v,0,edge]
        self.graph[v].append(edge)
        self.graph[w].append(rev_edge)
    
    def bfs(self,s,t):
        self.level = [None]*self.N
        d = deque([s])
        self.level[s] = 0
        while d:
            v = d.popleft()
            nlevel = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = nlevel
                    d.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,v,f):
        if v == s:
            return f
        for rev in self.it[v]:
            w,_,e = rev
            cap = e[1]
            if cap and self.level[w] is not None:
                if self.level[v] > self.level[w]:
                    d = self.dfs(s,w,min(f,cap))
                    if d:
                        e[1] -= d
                        rev[1] += d
                        return d
        return 0
    
    def max_flow(self,s,t):
        flow = 0
        INF = 10**9
        while self.bfs(s,t):
            *self.it, = map(iter,self.graph)
            f = INF
            while f:
                f = self.dfs(s,t,INF)
                flow += f
        return flow


N,M = map(int, input().split())
S = [list(input()) for i in range(N)]
dc = Dinic(N*M+2)

s =N*M
t = N*M+1
for i in range(N):
    for j in range(M):
        if (i + j) % 2 == 0:
            dc.add_edge(s,M*i+j,1)
        else:
            dc.add_edge(M*i+j,t,1)

for i in range(N):
    for j in range(M):
        if j+1 < M and S[i][j] == "." and S[i][j+1] == ".":
            u,v = M*i+j,M*i+j+1
            if (i+j) % 2 == 1:
                u,v = v,u
            dc.add_edge(u, v, 1)
        if i+1 < N and S[i][j] == "." and S[i + 1][j] == ".":
            u,v = M*i+j, M*(i+1)+j
            if (i+j) % 2 == 1:
                u,v = v,u
            dc.add_edge(u, v, 1)

print(dc.max_flow(s, t))
for u in range(N*M+2):
    for v, cap, _ in dc.graph[u]:
        ui, uj = divmod(u, M)
        vi, vj = divmod(v, M)
        if (ui+uj) % 2 == 0 and cap == 0 and u != s and u != t and v != s and v!= t:
            if ui+1 == vi:
                S[ui][uj] = "v"
                S[vi][vj] = "^"
            elif ui == vi+1:
                S[ui][uj] = "^"
                S[vi][vj] = "v"
            elif uj+1 == vj:
                S[ui][uj] = ">"
                S[vi][vj] = "<"
            elif uj == vj+1:
                S[ui][uj] = "<"
                S[vi][vj] = ">"

for res in S:
    print("".join(res))

Solution (Python):
class mf_graph:
    def __init__(self, n=0):
        self._n = n
        self.g = [[] for _ in range(n)]
        self.pos = []
    
    def add_edge(self, frm, to, cap):
        m = len(self.pos)
        e1 = mf_graph._edge(to, cap)
        e2 = mf_graph._edge(frm, 0)
        e1.rev = e2
        e2.rev = e1
        self.pos.append(e1)
        self.g[frm].append(e1)
        self.g[to].append(e2)
        return m
    
    class edge:
        def __init__(self, frm, to, cap, flow):
            self.frm = frm
            self.to = to
            self.cap = cap
            self.flow = flow
        
        def __iter__(self):
            yield self.frm
            yield self.to
            yield self.cap
            yield self.flow
    
    def get_edge(self, i):
        e1 = self.pos[i]
        e2 = e1.rev
        return mf_graph.edge(e2.to, e1.to, e1.cap + e2.cap, e2.cap)
    
    def edges(self):
        return [self.get_edge(i) for i in range(len(self.pos))]
    
    def change_edge(self, i, new_cap, new_flow):
        e = self.pos[i]
        e.cap = new_cap - new_flow
        e.rev.cap = new_flow
    
    def flow(self, s, t, flow_limit=0XFFFFFFFFFFFFFFF):
        g = self.g
        flow = 0
        es = [[] for _ in range(self._n)]
        que = [None] * self._n
        while flow < flow_limit:
            level = [-1] * self._n
            level[s] = 0
            ql = 0
            qr = 1
            que[0] = s
            unreached = s != t
            lvl = 1
            for p in es:
                p.clear()
            while unreached and ql < qr:
                nqr = qr
                while ql < qr:
                    v = que[ql]
                    ql += 1
                    for e in g[v]:
                        to = e.to
                        if e.cap and not 0 <= level[to] < lvl:
                            es[to].append(e.rev)
                            if level[to] < 0:
                                level[to] = lvl
                                if to == t:
                                    unreached = False
                                que[nqr] = to
                                nqr += 1
                qr = nqr
                lvl += 1
            if unreached:
                return flow
            stack = []
            v = t
            up = flow_limit - flow
            res = 0
            while True:
                if v == s or not es[v]:
                    if v == s:
                        res = up
                    while stack:
                        tmp = res
                        e, up, res = stack.pop()
                        e.cap -= tmp
                        e.rev.cap += tmp
                        res += tmp
                        if res < up:
                            v = e.to
                            break
                    else:
                        flow += res
                        break
                else:
                    e = es[v].pop()
                    stack.append((e.rev, up, res))
                    v = e.to
                    up = min(up, e.rev.cap)
                    res = 0
        return flow
    
    def min_cut(self, s):
        visited = [False] * self._n
        que = [None] * self._n
        ql = 0
        qr = 1
        que[0] = s
        visited[s] = True
        while ql < qr:
            p = que[ql]
            ql += 1
            for e in self.g[p]:
                if e.cap and not visited[e.to]:
                    visited[e.to] = True
                    que[qr] = e.to
                    qr += 1
        return visited
    
    class _edge:
        def __init__(self, to, cap):
            self.to = to
            self.cap = cap


n, m = map(int, input().split())
s = [input() for _ in range(n)]
g = mf_graph(n * m + 2)
for i in range(n):
    for j in range(i % 2, m, 2):
        if s[i][j] == '
            continue
        for di, dj in (1, 0), (0, 1), (-1, 0), (0, -1):
            x, y = i + di, j + dj
            if 0 <= x < n and 0 <= y < m and s[x][y] == '.':
                g.add_edge(i * m + j, x * m + y, 1)
for i in range(n):
    for j in range(m):
        if s[i][j] == '
            continue
        if (i + j) % 2 == 0:
            g.add_edge(n * m, i * m + j, 1)
        else:
            g.add_edge(i * m + j, n * m + 1, 1)
print(g.flow(n * m, n * m + 1))
ans = [list(si) for si in s]
for frm, to, cap, flow in g.edges():
    if flow and frm < n * m and to < n * m:
        i, j = frm // m, frm % m
        x, y = to // m, to % m
        if j == y:
            if i > x:
                i, j, x, y = x, y, i, j
            ans[i][j], ans[x][y] = 'v^'
        else:
            if j > y:
                i, j, x, y = x, y, i, j
            ans[i][j], ans[x][y] = '><'

for a in ans:
    print(''.join(a))

Solution (Python):
class mf_graph:
    def __init__(self, n=0):
        self._n = n
        self.g = [[] for _ in range(n)]
        self.pos = []
    
    def add_edge(self, frm, to, cap):
        m = len(self.pos)
        e1 = mf_graph._edge(to, cap)
        e2 = mf_graph._edge(frm, 0)
        e1.rev = e2
        e2.rev = e1
        self.pos.append(e1)
        self.g[frm].append(e1)
        self.g[to].append(e2)
        return m
    
    class edge:
        def __init__(self, frm, to, cap, flow):
            self.frm = frm
            self.to = to
            self.cap = cap
            self.flow = flow
        
        def __iter__(self):
            yield self.frm
            yield self.to
            yield self.cap
            yield self.flow
    
    def get_edge(self, i):
        e1 = self.pos[i]
        e2 = e1.rev
        return mf_graph.edge(e2.to, e1.to, e1.cap + e2.cap, e2.cap)
    
    def edges(self):
        return [self.get_edge(i) for i in range(len(self.pos))]
    
    def change_edge(self, i, new_cap, new_flow):
        e = self.pos[i]
        e.cap = new_cap - new_flow
        e.rev.cap = new_flow
    
    def flow(self, s, t, flow_limit=0XFFFFFFFFFFFFFFF):
        g = self.g
        flow = 0
        que = [None] * self._n
        while flow < flow_limit:
            level = [-1] * self._n
            level[s] = 0
            ql = 0
            qr = 1
            que[0] = s
            unreached = s != t
            while unreached and ql < qr:
                v = que[ql]
                ql += 1
                for e in g[v]:
                    to = e.to
                    if e.cap and level[to] < 0:
                        level[to] = level[v] + 1
                        if to == t:
                            unreached = False
                            break
                        que[qr] = to
                        qr += 1
            if unreached:
                return flow
            ptr = [len(es) for es in g]
            stack = []
            v = t
            up = flow_limit - flow
            res = 0
            while True:
                if v == s or not ptr[v]:
                    if v == s:
                        res = up
                    while stack:
                        tmp = res
                        e, up, res = stack.pop()
                        e.cap -= tmp
                        e.rev.cap += tmp
                        res += tmp
                        if res < up:
                            v = e.to
                            break
                    else:
                        flow += res
                        break
                i = ptr[v]
                while i:
                    i -= 1
                    e = g[v][i]
                    if level[e.to] == level[v] - 1 and e.rev.cap:
                        ptr[v] = i
                        stack.append((e.rev, up, res))
                        v = e.to
                        up = min(up, e.rev.cap)
                        res = 0
                        break
                else:
                    ptr[v] = i
        return flow
    
    def min_cut(self, s):
        visited = [False] * self._n
        que = [None] * self._n
        ql = 0
        qr = 1
        que[0] = s
        visited[s] = True
        while ql < qr:
            p = que[ql]
            ql += 1
            for e in self.g[p]:
                if e.cap and not visited[e.to]:
                    visited[e.to] = True
                    que[qr] = e.to
                    qr += 1
        return visited
    
    class _edge:
        def __init__(self, to, cap):
            self.to = to
            self.cap = cap

n, m = map(int, input().split())
s = [input() for _ in range(n)]
g = mf_graph(n * m + 2)
for i in range(n):
    for j in range(i % 2, m, 2):
        if s[i][j] == '
            continue
        for di, dj in (1, 0), (0, 1), (-1, 0), (0, -1):
            x, y = i + di, j + dj
            if 0 <= x < n and 0 <= y < m and s[x][y] == '.':
                g.add_edge(i * m + j, x * m + y, 1)
for i in range(n):
    for j in range(m):
        if s[i][j] == '
            continue
        if (i + j) % 2 == 0:
            g.add_edge(n * m, i * m + j, 1)
        else:
            g.add_edge(i * m + j, n * m + 1, 1)
print(g.flow(n * m, n * m + 1))
ans = [list(si) for si in s]
for frm, to, cap, flow in g.edges():
    if flow and frm < n * m and to < n * m:
        i, j = frm // m, frm % m
        x, y = to // m, to % m
        if j == y:
            if i > x:
                i, j, x, y = x, y, i, j
            ans[i][j], ans[x][y] = 'v^'
        else:
            if j > y:
                i, j, x, y = x, y, i, j
            ans[i][j], ans[x][y] = '><'

for a in ans:
    print(''.join(a))

Solution (Python):
class mf_graph:
    def __init__(self, n=0):
        self._n = n
        self.g = [[] for _ in range(n)]
        self.pos = []
    
    def add_edge(self, frm, to, cap):
        m = len(self.pos)
        e1 = mf_graph._edge(to, cap)
        e2 = mf_graph._edge(frm, 0)
        e1.rev = e2
        e2.rev = e1
        self.pos.append(e1)
        self.g[frm].append(e1)
        self.g[to].append(e2)
        return m
    
    class edge:
        def __init__(self, frm, to, cap, flow):
            self.frm = frm
            self.to = to
            self.cap = cap
            self.flow = flow
        
        def __iter__(self):
            yield self.frm
            yield self.to
            yield self.cap
            yield self.flow
    
    def get_edge(self, i):
        e1 = self.pos[i]
        e2 = e1.rev
        return mf_graph.edge(e2.to, e1.to, e1.cap + e2.cap, e2.cap)
    
    def edges(self):
        return [self.get_edge(i) for i in range(len(self.pos))]
    
    def change_edge(self, i, new_cap, new_flow):
        e = self.pos[i]
        e.cap = new_cap - new_flow
        e.rev.cap = new_flow
    
    def flow(self, s, t, flow_limit=0XFFFFFFFFFFFFFFF):
        g = self.g
        flow = 0
        es = [[] for _ in range(self._n)]
        while flow < flow_limit:
            level = [-1] * self._n
            level[s] = 0
            que = [None] * self._n
            ql = 0
            qr = 1
            que[0] = s
            unreached = s != t
            lvl = 1
            for p in es:
                p.clear()
            while unreached and ql < qr:
                nqr = qr
                while ql < qr:
                    v = que[ql]
                    ql += 1
                    for e in g[v]:
                        to = e.to
                        if e.cap and not 0 <= level[to] < lvl:
                            es[to].append(e.rev)
                            if level[to] < 0:
                                level[to] = lvl
                                if to == t:
                                    unreached = False
                                que[nqr] = to
                                nqr += 1
                qr = nqr
                lvl += 1
            if unreached:
                return flow
            stack = []
            v = t
            up = flow_limit - flow
            res = 0
            while True:
                if v == s or not es[v]:
                    if v == s:
                        res = up
                    while stack:
                        tmp = res
                        e, up, res = stack.pop()
                        e.cap -= tmp
                        e.rev.cap += tmp
                        res += tmp
                        if res < up:
                            v = e.to
                            break
                    else:
                        flow += res
                        break
                else:
                    e = es[v].pop()
                    stack.append((e.rev, up, res))
                    v = e.to
                    up = min(up, e.rev.cap)
                    res = 0
        return flow
    
    def min_cut(self, s):
        visited = [False] * self._n
        que = [None] * self._n
        ql = 0
        qr = 1
        que[0] = s
        visited[s] = True
        while ql < qr:
            p = que[ql]
            ql += 1
            for e in self.g[p]:
                if e.cap and not visited[e.to]:
                    visited[e.to] = True
                    que[qr] = e.to
                    qr += 1
        return visited
    
    class _edge:
        def __init__(self, to, cap):
            self.to = to
            self.cap = cap


n, m = map(int, input().split())
s = [input() for _ in range(n)]
g = mf_graph(n * m + 2)
for i in range(n):
    for j in range(i % 2, m, 2):
        if s[i][j] == '
            continue
        for di, dj in (1, 0), (0, 1), (-1, 0), (0, -1):
            x, y = i + di, j + dj
            if 0 <= x < n and 0 <= y < m and s[x][y] == '.':
                g.add_edge(i * m + j, x * m + y, 1)
for i in range(n):
    for j in range(m):
        if s[i][j] == '
            continue
        if (i + j) % 2 == 0:
            g.add_edge(n * m, i * m + j, 1)
        else:
            g.add_edge(i * m + j, n * m + 1, 1)
print(g.flow(n * m, n * m + 1))
ans = [list(si) for si in s]
for frm, to, cap, flow in g.edges():
    if flow and frm < n * m and to < n * m:
        i, j = frm // m, frm % m
        x, y = to // m, to % m
        if j == y:
            if i > x:
                i, j, x, y = x, y, i, j
            ans[i][j], ans[x][y] = 'v^'
        else:
            if j > y:
                i, j, x, y = x, y, i, j
            ans[i][j], ans[x][y] = '><'

for a in ans:
    print(''.join(a))