Valera has 2·n cubes, each cube contains an integer from 10 to 99. He arbitrarily chooses n cubes and puts them in the first heap. The remaining cubes form the second heap. 

Valera decided to play with cubes. During the game he takes a cube from the first heap and writes down the number it has. Then he takes a cube from the second heap and write out its two digits near two digits he had written (to the right of them). In the end he obtained a single fourdigit integer — the first two digits of it is written on the cube from the first heap, and the second two digits of it is written on the second cube from the second heap.

Valera knows arithmetic very well. So, he can easily count the number of distinct fourdigit numbers he can get in the game. The other question is: how to split cubes into two heaps so that this number (the number of distinct fourdigit integers Valera can get) will be as large as possible?


-----Input-----

The first line contains integer n (1 ≤ n ≤ 100). The second line contains 2·n space-separated integers a_{i} (10 ≤ a_{i} ≤ 99), denoting the numbers on the cubes.


-----Output-----

In the first line print a single number — the maximum possible number of distinct four-digit numbers Valera can obtain. In the second line print 2·n numbers b_{i} (1 ≤ b_{i} ≤ 2). The numbers mean: the i-th cube belongs to the b_{i}-th heap in your division.

If there are multiple optimal ways to split the cubes into the heaps, print any of them.


-----Examples-----
Input
1
10 99

Output
1
2 1 

Input
2
13 24 13 45

Output
4
1 2 2 1 



-----Note-----

In the first test case Valera can put the first cube in the first heap, and second cube — in second heap. In this case he obtain number 1099. If he put the second cube in the first heap, and the first cube in the second heap, then he can obtain number 9910. In both cases the maximum number of distinct integers is equal to one.

In the second test case Valera can obtain numbers 1313, 1345, 2413, 2445. Note, that if he put the first and the third cubes in the first heap, he can obtain only two numbers 1324 and 1345.

Difficulty: HARD
Tags: Constructive algorithms

Sorting

Combinatorics

Implementation

Greedy algorithms

Mathematics
Skill types: Sorting

Greedy algorithms

Starter code:

Solutions:
def s():
	input()
	l = [[] for _ in range(100)]
	a = list(map(int, input().split()))
	for (i, v) in enumerate(a):
		l[v].append(i)
	c = 0
	cc = 0
	fs = [0, 0]
	for i in l:
		if len(i) == 0:
			continue
		if len(i) == 1:
			a[i[0]] = c + 1
			fs[c] += 1
			c = 1 - c
			continue
		fs[c] += 1
		fs[c - 1] += 1
		for e in i[:len(i) // 2]:
			a[e] = 1 + cc
		for e in i[len(i) // 2:]:
			a[e] = 2 - cc
		if len(i) % 2 == 1:
			cc = 1 - cc
	print(fs[0] * fs[1])
	print(*a)
s()


n = int(input())
a = list(map(int, input().split()))
proc = sorted([(a.count(i), i) for i in range(10, 100) if a.count(i) > 0])
(left, right) = ([], [])
for (cnt, val) in proc:
	half = cnt // 2
	left.extend([val] * half)
	right.extend([val] * half)
	if cnt % 2 == 1:
		if len(left) < len(right):
			left.append(val)
		else:
			right.append(val)
print(len(set(left)) * len(set(right)))
cnt = [0 for i in range(100)]
for i in range(2 * n):
	if left.count(a[i]) == cnt[a[i]]:
		print('2', end=' ')
	else:
		print('1', end=' ')
		cnt[a[i]] += 1


n = 2 * int(input())
t = list(map(int, input().split()))
(q, p, c) = ([0] * n, [0] * 100, [0, 0])
for i in t:
	p[i] += 1
for i in p:
	if i:
		c[i == 1] += 1
k = c[1] // 2
print((c[0] + k) * (c[0] + c[1] - k))
r = sorted([(p[j], j, i) for (i, j) in enumerate(t)])
for i in r[0:n:2]:
	q[i[2]] = '1'
for i in r[1:n:2]:
	q[i[2]] = '2'
print(' '.join(q))


from collections import defaultdict as defdict
n = int(input())
d = defdict(int)
a = list(map(int, input().split()))
for i in a:
	d[i] += 1
b = set((i for i in a if d[i] > 1))
c = [i for i in a if d[i] == 1]
mx = len(b) + len(c) // 2
d = defdict(int)
answ = [None] * (n * 2)
cnt_diff_1 = 0
cnt_diff_2 = 0
for (i, x) in enumerate(a):
	if cnt_diff_1 < mx:
		if x in b:
			if d[x] == 0:
				answ[i] = '1'
				d[x] += 1
				cnt_diff_1 += 1
for (i, x) in enumerate(a):
	if cnt_diff_1 < mx:
		if x not in b:
			answ[i] = '1'
			cnt_diff_1 += 1
cnt1 = cnt_diff_1
cnt2 = 0
for (i, x) in enumerate(a):
	if answ[i] == '1':
		continue
	if x in b:
		if d[x] == 0:
			answ[i] = '2'
			d[x] = 2
			cnt_diff_2 += 1
			cnt2 += 1
		elif d[x] == 1:
			answ[i] = '2'
			d[x] += 1
			cnt_diff_2 += 1
			cnt2 += 1
		elif cnt1 < n:
			answ[i] = '1'
			cnt1 += 1
		else:
			answ[i] = '2'
			cnt2 += 1
	elif cnt2 < n:
		answ[i] = '2'
		cnt2 += 1
		cnt_diff_2 += 1
	else:
		answ[i] = '1'
		cnt1 += 1
		cnt_diff_1 += 1
print(cnt_diff_1 * cnt_diff_2)
print(' '.join(answ))


N = int(input())
Nums = list(map(int, input().split()))
Good = [1] * (N * 2)
Amounts = [0] * 100
(Mono, Duo) = (0, 0)
for Num in Nums:
	if Amounts[Num] == 0:
		Mono += 1
	elif Amounts[Num] == 1:
		Duo += 1
		Mono -= 1
	Amounts[Num] += 1
Flag = Mono % 2
Duo_Flag = False
Counts = [0] * 100
for i in range(2 * N):
	Num = Nums[i]
	if Amounts[Num] == 1:
		if Flag:
			Good[i] = 1
		else:
			Good[i] = 2
		Flag = not Flag
	else:
		if Counts[Num] == 0:
			Good[i] = 1
		elif Counts[Num] == 1:
			Good[i] = 2
		else:
			if Duo_Flag:
				Good[i] = 1
			else:
				Good[i] = 2
			Duo_Flag = not Duo_Flag
		Counts[Num] += 1
print((Duo + Mono // 2) * (Duo + (Mono + 1) // 2))
print(*Good)


n = int(input())
a = list(map(int, input().split()))
R = range(100)
c = [[] for _ in [0] * 100]
for i in range(n * 2):
	c[a[i]].append(i)
d = [0] * 200
heap = 1
z = [0, 0, 0]
for i in R:
	if len(c[i]) == 1:
		z[heap] += 1
		d[c[i][0]] = heap
		heap = 3 - heap
for i in R:
	if len(c[i]) > 1:
		z[1] += 1
		z[2] += 1
		while len(c[i]):
			d[c[i].pop()] = heap
			heap = 3 - heap
print(z[1] * z[2])
print(' '.join(map(str, d[:n * 2])))


n = int(input())
a = list(map(int, input().split()))
g = [[] for i in range(100)]
for i in range(2 * n):
	g[a[i]].append(i)
x = [0, 0]
cur = 1
mus = []
ans = [0] * 2 * n
for i in range(10, 100):
	if len(g[i]) == 1:
		ans[g[i][0]] = cur
		x[cur - 1] += 1
		cur = 3 - cur
	if len(g[i]) >= 2:
		ans[g[i][0]] = 1
		ans[g[i][1]] = 2
		x[0] += 1
		x[1] += 1
		for j in range(2, len(g[i])):
			mus.append(g[i][j])
for i in range(len(mus)):
	ans[mus[i]] = 2 - i % 2
print(x[0] * x[1])
print(*ans)


n = int(input())
a = list(map(int, input().split()))
c = [0] * 100
for i in a:
	c[i] += 1
x = [0] * 100
y = [0] * 100
j = 0
for i in range(100):
	if c[i] == 1:
		[x, y][j][i] += 1
		j = 1 - j
for i in range(100):
	if c[i] != 1:
		x[i] += c[i] // 2
		y[i] += c[i] // 2
		if c[i] % 2 == 1:
			[x, y][j][i] += 1
			j = 1 - j
xk = len(list(filter(lambda it: it, x)))
yk = len(list(filter(lambda it: it, y)))
print(xk * yk)
zs = [None] * (2 * n)
for i in range(2 * n):
	if x[a[i]] > 0:
		x[a[i]] -= 1
		zs[i] = 1
	else:
		assert y[a[i]] > 0
		y[a[i]] -= 1
		zs[i] = 2
print(*zs)


from collections import *
n = int(input())
a = list(map(int, input().split()))
c = Counter(a)
for i in range(2 * n):
	a[i] = [a[i], i]
a.sort()
ans = [0 for i in range(2 * n)]
(da, db) = ({}, {})
f = 0
for i in range(2 * n):
	if c[a[i][0]] == 1:
		continue
	ans[a[i][1]] = f + 1
	if f & 1:
		da[a[i][0]] = 1
	else:
		db[a[i][0]] = 1
	f ^= 1
for i in range(2 * n):
	if ans[a[i][1]] != 0:
		continue
	ans[a[i][1]] = f + 1
	if f & 1:
		da[a[i][0]] = 1
	else:
		db[a[i][0]] = 1
	f ^= 1
print(len(da) * len(db))
print(*ans)


from math import *
from random import *
n = int(input()) * 2
A = list(map(int, input().split()))
amount = [0] * 101
B = []
for i in range(n):
	if amount[A[i]] < 2:
		amount[A[i]] += 1
		B += [(A[i], i)]
B.sort()
(x, y) = ([], [])
for i in range(len(B)):
	if i % 2 == 0:
		x.append(B[i][1])
	else:
		y.append(B[i][1])
lolka = 0
aaa = 0
print(len(x) * len(y))
for i in range(n):
	if i in x:
		lolka += 1
		aaa += 1
		print(1, end=' ')
	elif i in y:
		print(2, end=' ')
	elif len(x) - lolka + aaa < n // 2:
		print(1, end=' ')
		aaa += 1
	else:
		print(2, end=' ')
print()


(n, t) = (2 * int(input()), map(int, input().split()))
d = [[] for i in range(100)]
for (i, j) in enumerate(t):
	d[j].append(i)
(y, x) = ([], False)
(p, q) = ([], ['1'] * n)
for i in d[10:100]:
	if i:
		if len(i) == 1:
			if x:
				y.append(i[0])
			x = not x
		else:
			y.append(i[0])
			p += i[2:]
(k, l) = (len(p), len(y))
print(l * (n - k - l))
for i in y:
	q[i] = '2'
for i in p[k // 2:] if x else p[:k // 2]:
	q[i] = '2'
print(' '.join(q))


n = int(input())
arr = list(map(int, input().split()))
adj = [[] for i in range(100)]
for i in range(2 * n):
	adj[arr[i]].append(i)
res = [0] * (2 * n)
(mul, curr) = ([], 1)
x = [0, 0]
for i in range(10, 100):
	if len(adj[i]) == 1:
		res[adj[i][0]] = curr
		x[curr - 1] += 1
		curr = 3 - curr
	if len(adj[i]) > 1:
		res[adj[i][0]] = 1
		res[adj[i][1]] = 2
		x[0] += 1
		x[1] += 1
		for j in range(2, len(adj[i])):
			mul.append(adj[i][j])
for i in range(len(mul)):
	res[mul[i]] = curr
	curr = 3 - curr
print(x[0] * x[1])
print(*res)


from sys import stdin
n = int(stdin.readline())
a = [int(x) for x in stdin.readline().split()]
a = sorted([(a[x], x) for x in range(n * 2)])
group = {}
for (x, ind) in a:
	if x in group:
		group[x].append(ind)
	else:
		group[x] = [ind]
g2 = []
for x in group:
	g2.append([len(group[x]), group[x]])
g2.sort()
left = 0
right = 0
board = [0 for x in range(n * 2)]
ind = 0
for (x, l) in g2:
	if x == 1:
		ind += 1
		if left <= right:
			left += 1
			board[l[0]] = '1'
		else:
			right += 1
			board[l[0]] = '2'
	else:
		break
if right > left:
	turn = True
else:
	turn = False
for (x, l) in g2[ind:]:
	left += 1
	right += 1
	if x % 2 == 1:
		last = l.pop()
		if turn:
			board[last] = '1'
		else:
			board[last] = '2'
		turn = not turn
	for n in l[::2]:
		board[n] = '1'
	for n in l[1::2]:
		board[n] = '2'
print(left * right)
print(' '.join(board))


def readln():
	return tuple(map(int, input().split()))
(n,) = readln()
b = readln()
cnt = [0] * 101
for v in b:
	cnt[v] += 1
f = []
s = []
x = []
y = []
for i in range(10, 101):
	if cnt[i] > 1:
		f.extend([i] * (cnt[i] // 2))
		s.extend([i] * (cnt[i] // 2))
		if cnt[i] % 2:
			y.append(i)
		cnt[i] = 0
	elif cnt[i] == 1:
		x.append(i)
f.extend(x[:len(x) // 2])
f.extend(y[:n - len(f)])
s.extend(x[len(x) // 2:])
if len(s) < n:
	s.extend(y[-n + len(s):])
print(len(set(f)) * len(set(s)))
ans = [1] * 2 * n
for v in s:
	for i in range(2 * n):
		if b[i] == v and ans[i] == 1:
			ans[i] = 2
			break
print(*ans)


n = int(input())
arr = list(map(int, input().split()))
g = [[] for i in range(100)]
for i in range(2 * n):
	g[arr[i]].append(i)
x = 0
y = 0
curr = 1
flip = 2
for i in range(10, 100):
	if len(g[i]) == 1:
		arr[g[i][0]] = curr
		if curr == 1:
			x += 1
		else:
			y += 1
		curr = 3 - curr
	if len(g[i]) > 1:
		arr[g[i][0]] = 1
		arr[g[i][1]] = 2
		x += 1
		y += 1
		for j in range(2, len(g[i])):
			arr[g[i][j]] = flip
			flip = 3 - flip
print(x * y)
print(*arr)