AtCoder Grand Contest 037 - Sorting a Grid

We have a grid with N rows and M columns of squares. Each integer from 1 to NM is written in this grid once. The number written in the square at the i-th row from the top and the j-th column from the left is A_{ij}.

You need to rearrange these numbers as follows:

1. First, for each of the N rows, rearrange the numbers written in it as you like.
2. Second, for each of the M columns, rearrange the numbers written in it as you like.
3. Finally, for each of the N rows, rearrange the numbers written in it as you like.



After rearranging the numbers, you want the number written in the square at the i-th row from the top and the j-th column from the left to be M\times (i-1)+j. Construct one such way to rearrange the numbers. The constraints guarantee that it is always possible to achieve the objective.

Constraints

* 1 \leq N,M \leq 100
* 1 \leq A_{ij} \leq NM
* A_{ij} are distinct.

Input

Input is given from Standard Input in the following format:


N M
A_{11} A_{12} ... A_{1M}
:
A_{N1} A_{N2} ... A_{NM}


Output

Print one way to rearrange the numbers in the following format:


B_{11} B_{12} ... B_{1M}
:
B_{N1} B_{N2} ... B_{NM}
C_{11} C_{12} ... C_{1M}
:
C_{N1} C_{N2} ... C_{NM}


Here B_{ij} is the number written in the square at the i-th row from the top and the j-th column from the left after Step 1, and C_{ij} is the number written in that square after Step 2.

Examples

Input

3 2
2 6
4 3
1 5


Output

2 6
4 3
5 1
2 1
4 3
5 6


Input

3 4
1 4 7 10
2 5 8 11
3 6 9 12


Output

1 4 7 10
5 8 11 2
9 12 3 6
1 4 3 2
5 8 7 6
9 12 11 10

Source: atcoder | Difficulty: UNKNOWN_DIFFICULTY


Solutions:
from collections import defaultdict, deque
(N, M) = map(int, input().split())
A = [[0] * M for _ in range(N)]
B = [[0] * M for _ in range(N)]
C = [[0] * M for _ in range(N)]
G = defaultdict(lambda : defaultdict(int))
F = defaultdict(lambda : defaultdict(int))
rows = [0] * (N * M + 1)
for i in range(N):
	A_list = list(map(int, input().split()))
	A[i] = A_list
	for a in A_list:
		G[i + 1][N + (a + M - 1) // M] += 1
		G[N + (a + M - 1) // M][i + 1] = 0
for t1 in range(M):
	for i in range(1, N + 1):
		G[0][i] = 1
		F[0][i] = 0
		G[N + i][2 * N + 1] = 1
		F[N + i][2 * N + 1] = 0
	for t2 in range(N):
		queue = deque([0])
		searched = [0] * (2 * N + 2)
		searched[0] = 1
		parent = [0] * (2 * N + 2)
		while len(queue) > 0:
			p = queue.pop()
			for q in G[p].keys():
				if (G[p][q] > 0 or F[q][p] > 0) and searched[q] == 0:
					parent[q] = p
					searched[q] = 1
					queue.append(q)
			if searched[2 * N + 1] == 1:
				break
		path = [2 * N + 1]
		while True:
			path.append(parent[path[-1]])
			if path[-1] == 0:
				break
		for i in range(len(path) - 1, 0, -1):
			(p, q) = (path[i], path[i - 1])
			if G[p][q] > 0:
				G[p][q] -= 1
				F[p][q] += 1
			elif F[q][p] > 0:
				F[q][p] -= 1
				G[q][p] += 1
	for i in range(1, N + 1):
		ji = 0
		for j in range(N + 1, 2 * N + 1):
			if F[i][j] == 1:
				ji = j
				break
		F[i][ji] -= 1
		for a in A[i - 1]:
			if N + (a + M - 1) // M == ji:
				A[i - 1].remove(a)
				B[i - 1][t1] = a
				break
for j in range(M):
	c_j = list(sorted([B[i][j] for i in range(N)]))
	for i in range(N):
		C[i][j] = c_j[i]
for i in range(N):
	print(*B[i])
for i in range(N):
	print(*C[i])


import sys
readline = sys.stdin.readline
import collections

class Dinic:

	def __init__(self, vnum):
		self.edge = [[] for i in range(vnum)]
		self.n = vnum
		self.inf = float('inf')

	def addedge(self, st, en, c):
		self.edge[st].append([en, c, len(self.edge[en])])
		self.edge[en].append([st, 0, len(self.edge[st]) - 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 getmf(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
(N, M) = map(int, readline().split())
A = [list(map(int, readline().split())) for _ in range(N)]
B = [[None] * M for _ in range(N)]
st = 2 * N
en = 2 * N + 1
used = set()
for num in range(M):
	T = Dinic(2 * N + 2)
	for i in range(N):
		T.addedge(st, i, 1)
		T.addedge(N + i, en, 1)
		for j in range(M):
			aij = A[i][j]
			if aij not in used:
				T.addedge(i, N + (aij - 1) // M, 1)
	T.getmf(st, en)
	for i in range(N):
		candi = [e for (e, cost, _) in T.edge[i + N] if cost == 1][0]
		for j in range(M):
			if A[candi][j] not in used and (A[candi][j] - 1) // M == i:
				used.add(A[candi][j])
				B[candi][num] = A[candi][j]
				break
C = list(map(list, zip(*B)))
C = [sorted(c) for c in C]
C = list(map(list, zip(*C)))
for b in B:
	print(*b)
for c in C:
	print(*c)


from collections import defaultdict
from collections import deque
route = []

def Ford_Fulkerson_Func(s, g, lines, cost):
	global route
	N = len(cost)
	ans = 0
	queue = deque([[s, float('inf')]])
	ed = [True] * N
	ed[s] = False
	route = [0] * N
	route[s] = -1
	while queue:
		(now, flow) = queue.pop()
		for nex in lines[now]:
			if ed[nex]:
				flow = min(cost[now][nex], flow)
				route[nex] = now
				queue.append([nex, flow])
				ed[nex] = False
				if nex == g:
					ans += flow
					break
		else:
			continue
		break
	else:
		return (False, ans)
	t = g
	s = route[t]
	while s != -1:
		cost[s][t] -= flow
		if cost[s][t] == 0:
			lines[s].remove(t)
		if cost[t][s] == 0:
			lines[t].add(s)
		cost[t][s] += flow
		t = s
		s = route[t]
	return (True, ans)

def Ford_Fulkerson(s, g, lines, cost):
	ans = 0
	while True:
		(fl, nans) = Ford_Fulkerson_Func(s, g, lines, cost)
		if fl:
			ans += nans
			continue
		else:
			break
	return ans
from sys import stdin
import sys
sys.setrecursionlimit(10000)
(N, M) = map(int, stdin.readline().split())
B = [[None] * M for i in range(N)]
Bs = [[True] * M for i in range(N)]
A = []
for loop in range(N):
	a = list(map(int, stdin.readline().split()))
	A.append(a)
for loop in range(M):
	lines = defaultdict(set)
	cost = [[0] * (2 * N + 2) for i in range(2 * N + 2)]
	for i in range(N):
		lines[2 * N].add(i)
		cost[2 * N][i] = 1
	for i in range(N, 2 * N):
		lines[i].add(2 * N + 1)
		cost[i][2 * N + 1] = 1
	for i in range(N):
		for j in range(len(A[i])):
			lines[i].add((A[i][j] - 1) // M + N)
			cost[i][(A[i][j] - 1) // M + N] = 1
	Ford_Fulkerson(2 * N, 2 * N + 1, lines, cost)
	ans = []
	for i in range(N):
		for j in range(len(A[i])):
			if cost[i][(A[i][j] - 1) // M + N] == 0:
				ans.append(A[i][j])
				del A[i][j]
				break
	for i in range(N):
		B[i][loop] = ans[i]
for i in B:
	print(*i)
C = [[None] * M for i in range(N)]
for j in range(M):
	tmp = []
	for i in range(N):
		tmp.append(B[i][j])
	tmp.sort()
	for i in range(N):
		C[i][j] = tmp[i]
for i in C:
	print(*i)