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)]
        self.capacities = [dict() for _ in range(N)]

    def add_edge(self, v, w, cap):
        self.graph[v].append(w)
        self.graph[w].append(v)
        self.capacities[v][w] = cap
        self.capacities[w][v] = 0

    def bfs(self, s, t):
        self.level = [-1] * self.N
        q = deque([s])
        self.level[s] = 0
        while q:
            v = q.popleft()
            lw = self.level[v]+1
            for w, cap in self.capacities[v].items():
                if cap and self.level[w] == -1:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] != -1

    def dfs(self, s, t, up):
        st = [t]
        while st:
            v = st[-1]
            if v == s:
                st.pop()
                flow = up
                for w in st:
                    v = self.graph[w][self.it[w]]
                    flow = min(flow, self.capacities[v][w])
                for w in st:
                    v = self.graph[w][self.it[w]]
                    self.capacities[w][v] += flow
                    self.capacities[v][w] -= flow
                return flow
            lv = self.level[v]
            while self.it[v] < len(self.graph[v]):
                w = self.graph[v][self.it[v]]
                cap = self.capacities[w][v]
                if cap and self.level[w] != -1:
                    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.capacities[v].items():
                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.capacities[u].items():
        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] = ">"

print('\n'.join(("".join(res) for res in S)))

Solution (Python):



from collections import deque


class mf_graph:

    

    def __init__(self, N):
        self.N = N
        self.graph = [[] for _ in range(N)]
        self.capacities = [dict() for _ in range(N)]

    def add_edge(self, v, w, cap):
        self.graph[v].append(w)
        self.graph[w].append(v)
        self.capacities[v][w] = cap
        self.capacities[w][v] = 0

    def bfs(self, s, t):
        self.level = [-1] * self.N
        q = deque([s])
        self.level[s] = 0
        while q:
            v = q.popleft()
            lw = self.level[v]+1
            for w, cap in self.capacities[v].items():
                if cap and self.level[w] == -1:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] != -1

    def dfs(self, s, t, up):
        st = [t]
        while st:
            v = st[-1]
            if v == s:
                st.pop()
                flow = up
                for w in st:
                    v = self.graph[w][self.it[w]]
                    flow = min(flow, self.capacities[v][w])
                for w in st:
                    v = self.graph[w][self.it[w]]
                    self.capacities[w][v] += flow
                    self.capacities[v][w] -= flow
                return flow
            lv = self.level[v]
            while self.it[v] < len(self.graph[v]):
                w = self.graph[v][self.it[v]]
                cap = self.capacities[w][v]
                if cap and self.level[w] != -1:
                    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.capacities[v].items():
                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.capacities[u].items():
        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()
            nlevel = self.level[v]+1
            for w,cap,_ in self.graph[v]:
                if cap and self.level[w] is None:
                    self.level[w] = nlevel
                    q.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
        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)]
        self.capacities = [dict() for _ in range(N)]

    def add_edge(self, v, w, cap):
        self.graph[v].append(w)
        self.graph[w].append(v)
        self.capacities[v][w] = cap
        self.capacities[w][v] = 0

    def bfs(self, s, t):
        self.level = [-1] * self.N
        q = deque([s])
        self.level[s] = 0
        while q:
            v = q.popleft()
            lw = self.level[v]+1
            for w, cap in self.capacities[v].items():
                if cap and self.level[w] == -1:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] != -1

    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:
                    v = self.graph[w][self.it[w]]
                    flow = min(flow, self.capacities[v][w])
                for w in st:
                    v = self.graph[w][self.it[w]]
                    self.capacities[w][v] += flow
                    self.capacities[v][w] -= flow
                return flow
            lv = self.level[v]
            while self.it[v] < len(self.graph[v]):
                w = self.graph[v][self.it[v]]
                cap = self.capacities[w][v]
                if cap and self.level[w] != -1:
                    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.capacities[v].items():
                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.capacities[u].items():
        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:

    __slots__ = ["N", "graph", "level", "it"]

    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 = [-1] * 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] == -1:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] != -1

    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] != -1:
                    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 = [-1] * 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] == -1:
                    self.level[w] = lw
                    q.append(w)
        return self.level[t] != -1

    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] != -1:
                    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:

    __slots__ = ["N", "graph", "level", "it"]

    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 Dinic:
    def __init__(self, N):
        self.N = N
        self.G = [[] for i in range(N)]

    def add_edge(self, fr, to, cap):
        forward = [to, cap, None]
        forward[2] = backward = [fr, 0, forward]
        self.G[fr].append(forward)
        self.G[to].append(backward)

    def add_multi_edge(self, v1, v2, cap1, cap2):
        edge1 = [v2, cap1, None]
        edge1[2] = edge2 = [v1, cap2, edge1]
        self.G[v1].append(edge1)
        self.G[v2].append(edge2)

    def bfs(self, s, t):
        self.level = level = [None]*self.N
        deq = deque([s])
        level[s] = 0
        G = self.G
        while deq:
            v = deq.popleft()
            lv = level[v] + 1
            for w, cap, _ in G[v]:
                if cap and level[w] is None:
                    level[w] = lv
                    deq.append(w)
        return level[t] is not None

    def dfs(self, v, t, f):
        if v == t:
            return f
        level = self.level
        for e in self.it[v]:
            w, cap, rev = e
            if cap and level[v] < level[w]:
                d = self.dfs(w, t, min(f, cap))
                if d:
                    e[1] -= d
                    rev[1] += d
                    return d
        return 0

    def flow(self, s, t):
        flow = 0
        INF = 10**9 + 7
        G = self.G
        while self.bfs(s, t):
            *self.it, = map(iter, self.G)
            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.flow(s, t))
for u in range(N*M+2):
    for v, cap, _ in dc.G[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))