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
        q = deque([s])
        self.level[s] = 0
        while q:
            v = q.popleft()
            lw = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,t,up):
        st = deque([t])
        while st:
            v = st[-1]
            if v == s:
                st.pop()
                flow = up
                for w in st:
                    e = self.graph[w][self.it[w]][2]
                    flow = min(flow,e[1])
                for w in st:
                    e = self.graph[w][self.it[w]]
                    e[1] += flow
                    e[2][1] -= flow
                return flow
            lv = self.level[v]
            while self.it[v] < len(self.graph[v]):
                w,_,e = self.graph[v][self.it[v]]
                cap = e[1]
                if cap and self.level[w] is not None:
                    if lv > self.level[w]:
                        st.append(w)
                        break
                self.it[v] += 1
            if self.it[v] == len(self.graph[v]):
                st.pop()
                self.level[v] = self.N
        return 0
    
    def flow(self,s,t,flow_limit=float('inf')):
        flow = 0
        while flow < flow_limit and self.bfs(s,t):
            self.it = [0]*self.N
            while flow < flow_limit:
                f = self.dfs(s,t,flow_limit-flow)
                if not f:
                    break
                flow += f
        return flow

    def min_cut(self,s):
        visited = [False]*self.N
        q = deque([s])
        while q:
            v = q.pop()
            visited[v] = True
            for w,cap,_ in self.graph[v]:
                if cap and not visited[w]:
                    q.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
        q = deque([s])
        self.level[s] = 0
        while q:
            v = q.popleft()
            lw = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,t,up):
        st = deque([t])
        while st:
            v = st[-1]
            if v == s:
                st.pop()
                flow = up
                for w in st:
                    e = self.graph[w][self.it[w]][2]
                    flow = min(flow,e[1])
                for w in st:
                    e = self.graph[w][self.it[w]]
                    e[1] += flow
                    e[2][1] -= flow
                return flow
            lv = self.level[v]
            while self.it[v] < len(self.graph[v]):
                w,_,e = self.graph[v][self.it[v]]
                cap = e[1]
                if cap and self.level[w] is not None:
                    if lv > self.level[w]:
                        st.append(w)
                        break
                self.it[v] += 1
            if self.it[v] == len(self.graph[v]):
                st.pop()
                self.level[v] = self.N
        return 0
    
    def flow(self,s,t,flow_limit=float('inf')):
        flow = 0
        while flow < flow_limit and self.bfs(s,t):
            self.it = [0]*self.N
            while flow < flow_limit:
                f = self.dfs(s,t,flow_limit-flow)
                if not f:
                    break
                flow += f
        return flow

    def min_cut(self,s):
        visited = [False]*self.N
        q = deque([s])
        while q:
            v = q.pop()
            visited[v] = True
            for w,cap,_ in self.graph[v]:
                if cap and not visited[w]:
                    q.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
        q = deque([s])
        self.level[s] = 0
        while q:
            v = q.popleft()
            lw = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,t,up):
        st = deque([t])
        while st:
            v = st[-1]
            if v == s:
                st.pop()
                flow = up
                for w in st:
                    e = self.graph[w][self.it[w]][2]
                    flow = min(flow,e[1])
                for w in st:
                    e = self.graph[w][self.it[w]]
                    e[1] += flow
                    e[2][1] -= flow
                return flow
            lv = self.level[v]
            while self.it[v] < len(self.graph[v]):
                w,_,e = self.graph[v][self.it[v]]
                cap = e[1]
                if cap and self.level[w] is not None:
                    if lv > self.level[w]:
                        st.append(w)
                        break
                self.it[v] += 1
            if self.it[v] == len(self.graph[v]):
                st.pop()
                self.level[v] = self.N
        return 0
    
    def flow(self,s,t,flow_limit=float('inf')):
        flow = 0
        while flow < flow_limit and self.bfs(s,t):
            self.it = [0]*self.N
            while flow < flow_limit:
                f = self.dfs(s,t,flow_limit-flow)
                if not f:
                    break
                flow += f
        return flow

    def min_cut(self,s):
        visited = [False]*self.N
        q = deque([s])
        while q:
            v = q.pop()
            visited[v] = True
            for w,cap,_ in self.graph[v]:
                if cap and not visited[w]:
                    q.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):
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
        q = deque([s])
        self.level[s] = 0
        while q:
            v = q.popleft()
            lw = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,t,up):
        st = deque([t])
        while st:
            v = st[-1]
            if v == s:
                st.pop()
                flow = up
                for w in st:
                    e = self.graph[w][self.it[w]][2]
                    flow = min(flow,e[1])
                for w in st:
                    e = self.graph[w][self.it[w]]
                    e[1] += flow
                    e[2][1] -= flow
                return flow
            lv = self.level[v]
            while self.it[v] < len(self.graph[v]):
                w,_,e = self.graph[v][self.it[v]]
                cap = e[1]
                if cap and self.level[w] is not None:
                    if lv > self.level[w]:
                        st.append(w)
                        break
                self.it[v] += 1
            if self.it[v] == len(self.graph[v]):
                st.pop()
                self.level[v] = self.N
        return 0
    
    def flow(self,s,t,flow_limit=float('inf')):
        flow = 0
        while flow < flow_limit and self.bfs(s,t):
            self.it = [0]*self.N
            while flow < flow_limit:
                f = self.dfs(s,t,flow_limit-flow)
                if not f:
                    break
                flow += f
        return flow

    def min_cut(self,s):
        visited = [False]*self.N
        q = deque([s])
        while q:
            v = q.pop()
            visited[v] = True
            for w,cap,_ in self.graph[v]:
                if cap and not visited[w]:
                    q.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):
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
        q = deque([s])
        self.level[s] = 0
        while q:
            v = q.popleft()
            lw = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] is not None
    
    def dfs(self,s,t,up):
        st = deque([t])
        while st:
            v = st[-1]
            if v == s:
                st.pop()
                flow = up
                for w in st:
                    e = self.graph[w][self.it[w]][2]
                    flow = min(flow,e[1])
                for w in st:
                    e = self.graph[w][self.it[w]]
                    e[1] += flow
                    e[2][1] -= flow
                return flow
            lv = self.level[v]
            while self.it[v] < len(self.graph[v]):
                w,_,e = self.graph[v][self.it[v]]
                cap = e[1]
                if cap and self.level[w] is not None:
                    if lv > self.level[w]:
                        st.append(w)
                        break
                self.it[v] += 1
            if self.it[v] == len(self.graph[v]):
                st.pop()
                self.level[v] = self.N
        return 0
    
    def flow(self,s,t,flow_limit=float('inf')):
        flow = 0
        while flow < flow_limit and self.bfs(s,t):
            self.it = [0]*self.N
            while flow < flow_limit:
                f = self.dfs(s,t,flow_limit-flow)
                if not f:
                    break
                flow += f
        return flow

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

"""
sから奇数へ，奇数から偶数へ，偶数からtへ
"""
def main():
    
    import collections
    class Dinic:
        def __init__(self, vnum):
            self.edge = [[] for i in range(vnum)]
            self.n = vnum
            self.inf = 10**10
        def add_edge(self, start, end, c):
            self.edge[start].append([end, c, len(self.edge[end])])
            self.edge[end].append([start, 0, len(self.edge[start])-1])
        def bfs(self, vst):
            dist = [-1]*self.n
            dist[vst] = 0
            Q = collections.deque([vst])
            while Q:
                nv = Q.popleft()
                for vt, c, r in self.edge[nv]:
                    if dist[vt] == -1 and c > 0:
                        dist[vt] = dist[nv] + 1
                        Q.append(vt)
            self.dist = dist
        def dfs(self, nv, en, nf):
            nextv = self.nextv
            if nv == en:
                return nf
            dist = self.dist
            ist = nextv[nv]
            for i, (vt, c, r) in enumerate(self.edge[nv][ist:], ist):
                if dist[nv] < dist[vt] and c > 0:
                    df = self.dfs(vt, en, min(nf, c))
                    if df > 0:
                        self.edge[nv][i][1] -= df
                        self.edge[vt][r][1] += df
                        return df
                nextv[nv] += 1
            return 0
        def get_max_flow(self, st, en):
            mf = 0
            while True:
                self.bfs(st)
                if self.dist[en] == -1:
                    break
                self.nextv = [0]*self.n
                while True:
                    fl = self.dfs(st, en, self.inf)
                    if fl > 0:
                        mf += fl
                    else:
                        break
            return mf
        
    mod=10**9+7
    N,M=map(int, input().split())
    S=[]
    for _ in range(N):
        s=list(input())
        S.append(s)
        
    din=Dinic(N*M+2)
    s=N*M
    t=N*M+1
    
    
    for i in range(N):
        for j in range(M):
            if S[i][j]=="
                continue
            v=M*i + j
            if (i+j)%2==1:
                din.add_edge(s,v,1)
            else:
                din.add_edge(v,t,1)
                
    
    dx=[0,0,1,-1]
    dy=[1,-1,0,0]
    
    for i in range(N):
        for j in range(M):
            if S[i][j]=="
                continue
            v=M*i + j
            if (i+j)%2==1:
                for k in range(4):
                    ni=i+dx[k]
                    nj=j+dy[k]
                    
                    if 0<=ni<N and 0<=nj<M:
                        if S[ni][nj]== "
                            continue
                        nv=M*ni + nj
                        din.add_edge(v,nv,1)
                        
    
    ans=din.get_max_flow(s,t)
    
    T=[s[:] for s in S]
            
    
    for v in range(N*M):
        i,j = divmod(v,M)
        if (i+j)%2==1:
            if S[i][j]==".":
                for nv,cap,_ in din.edge[v]:
                    if cap==0 and nv!=t and nv!=s:
                        ni,nj=divmod(nv,M)
                        
                        if i==ni:
                            if j<nj:
                                T[i][j]=">"
                                T[ni][nj]="<"
                            else:
                                T[i][j]="<"
                                T[ni][nj]=">"
                        else:
                            if i<ni:
                                T[i][j]="v"
                                T[ni][nj]="^"
                            else:
                                T[i][j]="^"
                                T[ni][nj]="v"
                                
                                
    print(ans)
    for i in range(N):
        print(''.join(map(str, T[i])))
                                
                                
                                
            
    
                    
        
    
    

main()

Solution (Python):
from collections import defaultdict
import sys
sys.setrecursionlimit(10**7)
def dfs(v,t,f,used):
    if v==t:
        return f
    used[v]=True
    for nv,cap in G[v].items():
        if not used[nv] and cap>0:
            d=dfs(nv,t,min(f,cap),used)
            if d>0:
                G[v][nv]-=d
                G[nv][v]+=d
                return d
    return 0

def Max_Flow(s,t):
    flow=0
    while(True):
        used=[False]*X
        f=dfs(s,t,INF,used)
        if f==0:
            return flow
        flow+=f

INF=float('inf')
N,M=map(int,input().split())
X=N*M+2

Num={}
def ID(i,j):return Num[(i,j)]

G=[defaultdict(int) for i in range(X)]
T=[list(input()) for i in range(N)]
s,t=0,N*M+1
cnt=1
for i in range(N):
    for j in range(M):
        if T[i][j]=='
            continue
        Num[(i,j)]=cnt
        cnt+=1
        if (i+j)%2==0:
            G[s][ID(i,j)]=1
            G[ID(i,j)][s]=0
        else:
            G[ID(i,j)][t]=1
            G[t][ID(i,j)]=0
DX,DY=[0,1,0,-1],[1,0,-1,0]
for i in range(N):
    for j in range(M):
        if T[i][j]=='
            continue
        for dx,dy in zip(DX,DY):
            if i+dx<0 or i+dx>=N or j+dy<0 or j+dy>=M:
                continue
            if T[i+dx][j+dy]=='
                continue
            G[ID(i,j)][ID(i+dx,j+dy)]=1
            G[ID(i+dx,j+dy)][ID(i,j)]=0

res=Max_Flow(s,t)
for i in range(N):
    for j in range(M):
        if T[i][j]=='
            continue
        for dx,dy in zip(DX,DY):
            if i+dx<0 or i+dx>=N or j+dy<0 or j+dy>=M:
                continue
            if T[i+dx][j+dy]=='
                continue
            if G[ID(i,j)][ID(i+dx,j+dy)]==0:
                if dx==1:
                    T[i][j]='v'
                    T[i+dx][j+dy]='^'
                if dx==-1:
                    T[i][j]='^'
                    T[i+dx][j+dy]='v'
                if dy==1:
                    T[i][j]='>'
                    T[i+dx][j+dy]='<'
                if dy==-1:
                    T[i][j]='<'
                    T[i+dx][j+dy]='>'
                    
print(res)
for i in range(N):
    print(''.join(T[i]))

Solution (Python):
import networkx as nx
from networkx.algorithms.bipartite import hopcroft_karp_matching

n, m = map(int, input().split())

def inside(x, y):
	if 0 <= x < n and 0 <= y < m:
		return True
	else:
		return False

DI = [1, 0, -1, 0]
DJ = [0, 1, 0, -1]

a = [list(input()) for _ in range(n)]
g = nx.Graph()
g.add_nodes_from(range(n*m))
zero = True
top = set()
for i in range(n):
	for j in range(m):
		if a[i][j] == "
			continue
		if (i+j)%2 == 0:
			for k in range(4):
				di, dj = i+DI[k], j+DJ[k]
				if inside(di, dj) and a[di][dj] == ".":
					zero = False
					g.add_edge(i*m+j, di*m+dj)
					top.add(i*m+j)
if zero:
	print(0)
else:
	h = hopcroft_karp_matching(g, top_nodes=top)
	print(len(h)//2)
	for fr, to in h.items():
		i0, j0 = fr//m, fr%m
		i1, j1 = to//m, to%m
		if a[i0][j0] != ".":
			continue
		if i1 == i0+1:
			a[i0][j0] = "v"
			a[i1][j1] = "^"
		elif i1 == i0-1:
			a[i0][j0] = "^"
			a[i1][j1] = "v"
		elif j1 == j0+1:
			a[i0][j0] = ">"
			a[i1][j1] = "<"
		else:
			a[i0][j0] = "<"
			a[i1][j1] = ">"
print(*["".join(x) for x in a], sep="\n")

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))