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.#
^><



Sample input:
3 3
#..
..#
...


Sample output:
3
#><
vv#
^^.

Submission status: Accepted

Solution code:
from collections import deque

class MaxFlow():
    def __init__(self, n):
        self.n = n
        self.graph = [[] for _ in range(n)]
        self.pos = []

    def add_edge(self, fr, to, cap):
        #assert 0 <= fr < self.n
        #assert 0 <= to < self.n
        #assert 0 <= cap
        m = len(self.pos)
        self.pos.append((fr, len(self.graph[fr])))
        self.graph[fr].append([to, len(self.graph[to]), cap])
        self.graph[to].append([fr, len(self.graph[fr]) - 1, 0])
        return m

    def get_edge(self, idx):
        #assert 0 <= idx < len(self.pos)
        to, rev, cap = self.graph[self.pos[idx][0]][self.pos[idx][1]]
        rev_to, rev_rev, rev_cap = self.graph[to][rev]
        return rev_to, to, cap + rev_cap, rev_cap

    def edges(self):
        m = len(self.pos)
        for i in range(m):
            yield self.get_edge(i)

    def change_edge(self, idx, new_cap, new_flow):
        #assert 0 <= idx < len(self.pos)
        #assert 0 <= new_flow <= new_cap
        to, rev, cap = self.graph[self.pos[idx][0]][self.pos[idx][1]]
        self.graph[self.pos[idx][0]][self.pos[idx][1]][2] = new_cap - new_flow
        self.graph[to][rev][2] = new_flow

    def dfs(self, s, v, up):
        if v == s: return up
        res = 0
        lv = self.level[v]
        for i in range(self.iter[v], len(self.graph[v])):
            to, rev, cap = self.graph[v][i]
            if lv <= self.level[to] or self.graph[to][rev][2] == 0: continue
            d = self.dfs(s, to, min(up - res, self.graph[to][rev][2]))
            if d <= 0: continue
            self.graph[v][i][2] += d
            self.graph[to][rev][2] -= d
            res += d
            if res == up: break
        return res

    def max_flow(self, s, t):
        return self.max_flow_with_limit(s, t, 2**63 - 1)

    def max_flow_with_limit(self, s, t, limit):
        #assert 0 <= s < self.n
        #assert 0 <= t < self.n
        flow = 0
        while flow < limit:
            self.level = [-1] * self.n
            self.level[s] = 0
            queue = deque()
            queue.append(s)
            while queue:
                v = queue.popleft()
                for to, rev, cap in self.graph[v]:
                    if cap == 0 or self.level[to] >= 0: continue
                    self.level[to] = self.level[v] + 1
                    if to == t: break
                    queue.append(to)
            if self.level[t] == -1: break
            self.iter = [0] * self.n
            while flow < limit:
                f = self.dfs(s, t, limit - flow)
                if not f: break
                flow += f
        return flow

    def min_cut(self, s):
        visited = [0] * self.n
        queue = deque()
        queue.append(s)
        while queue:
            p = queue.popleft()
            visited[p] = True
            for to, rev, cap in self.graph[p]:
                if cap and not visited[to]:
                    visited[to] = True
                    queue.append(to)
        return visited

import sys
input = sys.stdin.readline

N, M = map(int, input().split())
S = [input() for _ in range(N)]

mf = MaxFlow(N * M + 2)

res = [['.' for j in range(M)] for i in range(N)]

for i in range(N):
    for j in range(M):
        if S[i][j] == '#':
            res[i][j] = '#'
            continue
        if (i + j) % 2 == 0:
            mf.add_edge(N * M, i * M + j, 1)
            for di, dj in (1, 0), (0, 1), (-1, 0), (0, -1):
                ni = i + di
                nj = j + dj
                if 0 <= ni < N and 0 <= nj < M and S[ni][nj] == '.':
                    mf.add_edge(i * M + j, ni * M + nj, 1)
        else:
            mf.add_edge(i * M + j, N * M + 1, 1)

print(mf.max_flow(M * N, M * N + 1))

for fr, to, cap, flow in mf.edges():
    if flow == 0 or fr == N * M or to == N * M + 1:
        continue
    si, sj = divmod(fr, M)
    ti, tj = divmod(to, M)
    if si + 1 == ti:
        res[si][sj] = 'v'
        res[ti][tj] = '^'
    elif sj + 1 == tj:
        res[si][sj] = '>'
        res[ti][tj] = '<'
    elif si == ti + 1:
        res[si][sj] = '^'
        res[ti][tj] = 'v'
    else:
        res[si][sj] = '<'
        res[ti][tj] = '>'

for r in res:
    print(''.join(r))

Submission status: Accepted

Solution code:
from collections import deque

class MaxFlow():
    def __init__(self, n):
        self.n = n
        self.graph = [[] for _ in range(n)]
        self.pos = []

    def add_edge(self, fr, to, cap):
        #assert 0 <= fr < self.n
        #assert 0 <= to < self.n
        #assert 0 <= cap
        m = len(self.pos)
        self.pos.append((fr, len(self.graph[fr])))
        self.graph[fr].append([to, len(self.graph[to]), cap])
        self.graph[to].append([fr, len(self.graph[fr]) - 1, 0])
        return m

    def get_edge(self, idx):
        #m = len(self.pos)
        #assert 0 <= idx < m
        to, rev, cap = self.graph[self.pos[idx][0]][self.pos[idx][1]]
        rev_to, rev_rev, rev_cap = self.graph[to][rev]
        return rev_to, to, cap + rev_cap, rev_cap

    def edges(self):
        m = len(self.pos)
        for i in range(m):
            yield self.get_edge(i)

    def change_edge(self, idx, new_cap, new_flow):
        #m = len(self.pos)
        #assert 0 <= idx < m
        #assert 0 <= new_flow <= new_cap
        to, rev, cap = self.graph[self.pos[idx][0]][self.pos[idx][1]]
        self.graph[self.pos[idx][0]][self.pos[idx][1]][2] = new_cap - new_flow
        self.graph[to][rev][2] = new_flow

    def dfs(self, s, v, up):
        if v == s: return up
        res = 0
        lv = self.level[v]
        for i in range(len(self.graph[v])):
            to, rev, cap = self.graph[v][i]
            if lv <= self.level[to] or self.graph[to][rev][2] == 0: continue
            d = self.dfs(s, to, min(up - res, self.graph[to][rev][2]))
            if d <= 0: continue
            self.graph[v][i][2] += d
            self.graph[to][rev][2] -= d
            res += d
            if res == up: break
        return res

    def max_flow(self, s, t):
        return self.max_flow_with_limit(s, t, 2**63 - 1)

    def max_flow_with_limit(self, s, t, limit):
        #assert 0 <= s < self.n
        #assert 0 <= t < self.n
        flow = 0
        while flow < limit:
            self.level = [-1] * self.n
            self.level[s] = 0
            queue = deque()
            queue.append(s)
            while queue:
                v = queue.popleft()
                for to, rev, cap in self.graph[v]:
                    if cap == 0 or self.level[to] >= 0: continue
                    self.level[to] = self.level[v] + 1
                    if to == t: break
                    queue.append(to)
            if self.level[t] == -1: break
            #self.iter = [0] * self.n
            while flow < limit:
                f = self.dfs(s, t, limit - flow)
                if not f: break
                flow += f
        return flow

    def min_cut(self, s):
        visited = [0] * self.n
        queue = deque()
        queue.append(s)
        while queue:
            p = queue.popleft()
            visited[p] = True
            for to, rev, cap in self.graph[p]:
                if cap and not visited[to]:
                    visited[to] = True
                    queue.append(to)
        return visited

import sys
input = sys.stdin.readline

N, M = map(int, input().split())
S = [input() for _ in range(N)]

mf = MaxFlow(N * M + 2)

dx = [1, 0, -1, 0]
dy = [0, 1, 0, -1]

res = [['.' for j in range(M)] for i in range(N)]

for i in range(N):
    for j in range(M):
        if S[i][j] == '#':
            res[i][j] = '#'
            continue
        if (i + j) % 2 == 0:
            mf.add_edge(N * M, i * M + j, 1)
            for k in range(4):
                ni = i + dx[k]
                nj = j + dy[k]
                if 0 <= ni < N and 0 <= nj < M and S[ni][nj] == '.':
                    mf.add_edge(i * M + j, ni * M + nj, 1)
        else:
            mf.add_edge(i * M + j, N * M + 1, 1)

print(mf.max_flow(M * N, M * N + 1))

for fr, to, cap, flow in mf.edges():
    if fr == N * M or to == N * M + 1 or flow == 0:
        continue
    si = fr // M
    sj = fr % M
    ti = to // M
    tj = to % M
    if si + 1 == ti:
        res[si][sj] = 'v'
        res[ti][tj] = '^'
    elif sj + 1 == tj:
        res[si][sj] = '>'
        res[ti][tj] = '<'
    elif si == ti + 1:
        res[si][sj] = '^'
        res[ti][tj] = 'v'
    else:
        res[si][sj] = '<'
        res[ti][tj] = '>'

for s in res:
    print(''.join(s))

Submission status: Accepted

Solution code:
from collections import deque

class MaxFlow():
    def __init__(self, n):
        self.n = n
        self.graph = [[] for _ in range(n)]
        self.pos = []

    def add_edge(self, fr, to, cap):
        #assert 0 <= fr < self.n
        #assert 0 <= to < self.n
        #assert 0 <= cap
        m = len(self.pos)
        self.pos.append((fr, len(self.graph[fr])))
        self.graph[fr].append([to, len(self.graph[to]), cap])
        self.graph[to].append([fr, len(self.graph[fr]) - 1, 0])
        return m

    def get_edge(self, idx):
        m = len(self.pos)
        #assert 0 <= idx < m
        to, rev, cap = self.graph[self.pos[idx][0]][self.pos[idx][1]]
        rev_to, rev_rev, rev_cap = self.graph[to][rev]
        return rev_to, to, cap + rev_cap, rev_cap

    def edges(self):
        m = len(self.pos)
        res = []
        for i in range(m):
            res.append(self.get_edge(i))
        return res

    def change_edge(self, idx, new_cap, new_flow):
        m = len(self.pos)
        #assert 0 <= idx < m
        #assert 0 <= new_flow <= new_cap
        to, rev, cap = self.graph[self.pos[idx][0]][self.pos[idx][1]]
        self.graph[self.pos[idx][0]][self.pos[idx][1]][2] = new_cap - new_flow
        self.graph[to][rev][2] = new_flow

    def dfs(self, s, v, up):
        if v == s: return up
        res = 0
        lv = self.level[v]
        for i in range(self.iter[v], len(self.graph[v])):
            to, rev, cap = self.graph[v][i]
            if lv <= self.level[to] or self.graph[to][rev][2] == 0: continue
            d = self.dfs(s, to, min(up - res, self.graph[to][rev][2]))
            if d <= 0: continue
            self.graph[v][i][2] += d
            self.graph[to][rev][2] -= d
            res += d
            if res == up: break
        return res

    def max_flow(self, s, t):
        return self.max_flow_with_limit(s, t, 2**63 - 1)

    def max_flow_with_limit(self, s, t, limit):
        #assert 0 <= s < self.n
        #assert 0 <= t < self.n
        flow = 0
        while flow < limit:
            self.level = [-1] * self.n
            self.level[s] = 0
            queue = deque()
            queue.append(s)
            while queue:
                v = queue.popleft()
                for to, rev, cap in self.graph[v]:
                    if cap == 0 or self.level[to] >= 0: continue
                    self.level[to] = self.level[v] + 1
                    if to == t: break
                    queue.append(to)
            if self.level[t] == -1: break
            self.iter = [0] * self.n
            while flow < limit:
                f = self.dfs(s, t, limit - flow)
                if not f: break
                flow += f
        return flow

    def min_cut(self, s):
        visited = [0] * self.n
        queue = deque()
        queue.append(s)
        while queue:
            p = queue.popleft()
            visited[p] = True
            for to, rev, cap in self.graph[p]:
                if cap and not visited[to]:
                    visited[to] = True
                    queue.append(to)
        return visited

import sys
input = sys.stdin.readline

N, M = map(int, input().split())
S = [input() for _ in range(N)]

mf = MaxFlow(N * M + 2)

dx = [1, 0, -1, 0]
dy = [0, 1, 0, -1]

res = [['.' for j in range(M)] for i in range(N)]

for i in range(N):
    for j in range(M):
        if S[i][j] == '#':
            res[i][j] = '#'
            continue
        if (i + j) % 2 == 0:
            mf.add_edge(N * M, i * M + j, 1)
            for k in range(4):
                ni = i + dx[k]
                nj = j + dy[k]
                if 0 <= ni < N and 0 <= nj < M and S[ni][nj] == '.':
                    mf.add_edge(i * M + j, ni * M + nj, 1)
        else:
            mf.add_edge(i * M + j, N * M + 1, 1)

print(mf.max_flow(M * N, M * N + 1))

for fr, to, cap, flow in mf.edges():
    if fr == N * M or to == N * M + 1 or flow == 0:
        continue
    si = fr // M
    sj = fr % M
    ti = to // M
    tj = to % M
    if si + 1 == ti:
        res[si][sj] = 'v'
        res[ti][tj] = '^'
    elif sj + 1 == tj:
        res[si][sj] = '>'
        res[ti][tj] = '<'
    elif si == ti + 1:
        res[si][sj] = '^'
        res[ti][tj] = 'v'
    else:
        res[si][sj] = '<'
        res[ti][tj] = '>'

for s in res:
    print(''.join(s))

Submission status: Accepted

Solution code:
from collections import deque

class MaxFlow():
    def __init__(self, n):
        self.n = n
        self.graph = [[] for _ in range(n)]
        self.pos = []

    def add_edge(self, fr, to, cap):
        assert 0 <= fr < self.n
        assert 0 <= to < self.n
        assert 0 <= cap
        m = len(self.pos)
        self.pos.append((fr, len(self.graph[fr])))
        self.graph[fr].append([to, len(self.graph[to]), cap])
        self.graph[to].append([fr, len(self.graph[fr]) - 1, 0])
        return m

    def get_edge(self, idx):
        m = len(self.pos)
        assert 0 <= idx < m
        to, rev, cap = self.graph[self.pos[idx][0]][self.pos[idx][1]]
        rev_to, rev_rev, rev_cap = self.graph[to][rev]
        return rev_to, to, cap + rev_cap, rev_cap

    def edges(self):
        m = len(self.pos)
        res = []
        for i in range(m):
            res.append(self.get_edge(i))
        return res

    def change_edge(self, idx, new_cap, new_flow):
        m = len(self.pos)
        assert 0 <= idx < m
        assert 0 <= new_flow <= new_cap
        to, rev, cap = self.graph[self.pos[idx][0]][self.pos[idx][1]]
        self.graph[self.pos[idx][0]][self.pos[idx][1]][2] = new_cap - new_flow
        self.graph[to][rev][2] = new_flow

    def dfs(self, s, v, up):
        if v == s: return up
        res = 0
        lv = self.level[v]
        for i in range(self.iter[v], len(self.graph[v])):
            to, rev, cap = self.graph[v][i]
            if lv <= self.level[to] or self.graph[to][rev][2] == 0: continue
            d = self.dfs(s, to, min(up - res, self.graph[to][rev][2]))
            if d <= 0: continue
            self.graph[v][i][2] += d
            self.graph[to][rev][2] -= d
            res += d
            if res == up: break
        return res

    def max_flow(self, s, t):
        return self.max_flow_with_limit(s, t, 2**63 - 1)

    def max_flow_with_limit(self, s, t, limit):
        assert 0 <= s < self.n
        assert 0 <= t < self.n
        flow = 0
        while flow < limit:
            self.level = [-1] * self.n
            self.level[s] = 0
            queue = deque()
            queue.append(s)
            while queue:
                v = queue.popleft()
                for to, rev, cap in self.graph[v]:
                    if cap == 0 or self.level[to] >= 0: continue
                    self.level[to] = self.level[v] + 1
                    if to == t: break
                    queue.append(to)
            if self.level[t] == -1: break
            self.iter = [0] * self.n
            while flow < limit:
                f = self.dfs(s, t, limit - flow)
                if not f: break
                flow += f
        return flow

    def min_cut(self, s):
        visited = [0] * self.n
        queue = deque()
        queue.append(s)
        while queue:
            p = queue.popleft()
            visited[p] = True
            for to, rev, cap in self.graph[p]:
                if cap and not visited[to]:
                    visited[to] = True
                    queue.append(to)
        return visited

import sys
input = sys.stdin.readline

N, M = map(int, input().split())
S = [input() for _ in range(N)]

mf = MaxFlow(N * M + 2)

dx = [1, 0, -1, 0]
dy = [0, 1, 0, -1]

res = [['.' for j in range(M)] for i in range(N)]

for i in range(N):
    for j in range(M):
        if S[i][j] == '#':
            res[i][j] = '#'
            continue
        if (i + j) % 2 == 0:
            mf.add_edge(N * M, i * M + j, 1)
            for k in range(4):
                ni = i + dx[k]
                nj = j + dy[k]
                if 0 <= ni < N and 0 <= nj < M and S[ni][nj] == '.':
                    mf.add_edge(i * M + j, ni * M + nj, 1)
        else:
            mf.add_edge(i * M + j, N * M + 1, 1)

print(mf.max_flow(M * N, M * N + 1))

for fr, to, cap, flow in mf.edges():
    if fr == N * M or to == N * M + 1 or flow == 0:
        continue
    si = fr // M
    sj = fr % M
    ti = to // M
    tj = to % M
    if si + 1 == ti:
        res[si][sj] = 'v'
        res[ti][tj] = '^'
    elif sj + 1 == tj:
        res[si][sj] = '>'
        res[ti][tj] = '<'
    elif si == ti + 1:
        res[si][sj] = '^'
        res[ti][tj] = 'v'
    else:
        res[si][sj] = '<'
        res[ti][tj] = '>'

for s in res:
    print(''.join(s))

Submission status: Accepted

Solution code:
class maxflow():
    def __init__(self, n):
        self.pos = []
        self.g = [[] for _ in range(n)]
        self.n = n
    
    def add_edge(self, v, w, cap):
        e1 = [w, 0, cap]
        e2 = [v, e1, 0]
        e1[1] = e2
        self.g[v].append(e1)
        self.g[w].append(e2)
    
    def flow(self, s, t, flow_limit = -1):
        if flow_limit < 0: flow_limit = 10 ** 100
        it = [0] * self.n
        def bfs():
            L = [-1] * self.n
            L[s] = 0
            Q = [s]
            while Q:
                v = Q.pop()
                for w, _, cap in self.g[v]:
                    if cap == 0 or L[w] >= 0: continue
                    L[w] = L[v] + 1
                    if w == t: return L
                    Q.append(w)
            return L
        
        def dfs(v, up):
            if v == s: return up
            res = 0
            lv = L[v]
            for i in range(it[v], len(self.g[v])):
                w, rev, cap = self.g[v][i]
                if lv <= L[w] or rev[2] == 0: continue
                
                d = dfs(w, min(up - res, rev[2]))
                if d <= 0: continue
                self.g[v][i][2] += d
                rev[2] -= d
                res += d
                if res == up: break
            return res
        
        flow = 0
        while flow < flow_limit:
            L = bfs()
            if L[t] == -1: break
            it = [0] * self.n
            while flow < flow_limit:
                f = dfs(t, flow_limit - flow)
                if not f: break
                flow += f
        return flow, self.g
    
    def min_cut(self, s):
        visited = [0] * self.n
        Q = [s]
        while Q:
            p = Q.pop()
            visited[p] = 1
            for e in self.g[p]:
                if e[2] and visited[e[0]] == 0:
                    visited[e[0]] = 1
                    Q.append(e[0])
        return visited

N, M = map(int, input().split())
mf = maxflow(N * M + 2)
A = [[a for a in input()] for _ in range(N)]
X = [[1 if a == "." else 0 for a in aa] for aa in A]
s = N * M
t = s + 1
for i in range(N):
    for j in range(M):
        if X[i][j]:
            if (i ^ j) & 1 == 0:
                mf.add_edge(s, i * M + j, 1)
            else:
                mf.add_edge(i * M + j, t, 1)

for i in range(N - 1):
    for j in range(M):
        if X[i][j] and X[i+1][j]:
            a = i * M + j
            b = a + M
            if (i ^ j) & 1 == 0:
                mf.add_edge(a, b, 1)
            else:
                mf.add_edge(b, a, 1)
for i in range(N):
    for j in range(M - 1):
        if X[i][j] and X[i][j+1]:
            a = i * M + j
            b = a + 1
            if (i ^ j) & 1 == 0:
                mf.add_edge(a, b, 1)
            else:
                mf.add_edge(b, a, 1)

fl, g = mf.flow(s, t)
# print("fl =", fl, g)
for v, gv in enumerate(g):
    if v >= N * M: continue
    for w, _, ca in gv:
        if w >= N * M: continue
        # if w > v: v, w = w, v
        vi, vj = v // M, v % M
        wi, wj = w // M, w % M
        if (vi ^ vj) % 2 == 0 and ca == 0:
            if v > w:
                vi, vj, wi, wj = wi, wj, vi, vj
            # print("v, w, ca =", v, w, ca)
            if vi == wi:
                A[vi][vj] = ">"
                A[wi][wj] = "<"
            else:
                A[vi][vj] = "v"
                A[wi][wj] = "^"

print(fl)
for x in A:
    print("".join(map(str, x)))