The only difference between easy and hard versions is the length of the string. You can hack this problem if you solve it. But you can hack the previous problem only if you solve both problems.

Kirk has a binary string $s$ (a string which consists of zeroes and ones) of length $n$ and he is asking you to find a binary string $t$ of the same length which satisfies the following conditions: For any $l$ and $r$ ($1 \leq l \leq r \leq n$) the length of the longest non-decreasing subsequence of the substring $s_{l}s_{l+1} \ldots s_{r}$ is equal to the length of the longest non-decreasing subsequence of the substring $t_{l}t_{l+1} \ldots t_{r}$; The number of zeroes in $t$ is the maximum possible.

A non-decreasing subsequence of a string $p$ is a sequence of indices $i_1, i_2, \ldots, i_k$ such that $i_1 < i_2 < \ldots < i_k$ and $p_{i_1} \leq p_{i_2} \leq \ldots \leq p_{i_k}$. The length of the subsequence is $k$.

If there are multiple substrings which satisfy the conditions, output any.


-----Input-----

The first line contains a binary string of length not more than $10^5$.


-----Output-----

Output a binary string which satisfied the above conditions. If there are many such strings, output any of them.


-----Examples-----
Input
110

Output
010

Input
010

Output
010

Input
0001111

Output
0000000

Input
0111001100111011101000

Output
0011001100001011101000



-----Note-----

In the first example:  For the substrings of the length $1$ the length of the longest non-decreasing subsequnce is $1$;  For $l = 1, r = 2$ the longest non-decreasing subsequnce of the substring $s_{1}s_{2}$ is $11$ and the longest non-decreasing subsequnce of the substring $t_{1}t_{2}$ is $01$;  For $l = 1, r = 3$ the longest non-decreasing subsequnce of the substring $s_{1}s_{3}$ is $11$ and the longest non-decreasing subsequnce of the substring $t_{1}t_{3}$ is $00$;  For $l = 2, r = 3$ the longest non-decreasing subsequnce of the substring $s_{2}s_{3}$ is $1$ and the longest non-decreasing subsequnce of the substring $t_{2}t_{3}$ is $1$; 

The second example is similar to the first one.

Difficulty: HARD
Tags: String algorithms

Data structures

Mathematics

Greedy algorithms
Skill types: Data structures

Greedy algorithms

Starter code:

Solutions:
a = input() + '1'
n = len(a) - 1
s = [0] * (n + 1)
for q in range(n - 1, -1, -1):
	s[q] = s[q + 1]
	if a[q] == '1':
		s[q] += 1
d = [0] * (n + 1)
for q in range(n - 1, -1, -1):
	d[q] = max(s[q], d[q + 1])
	if a[q] == '0':
		d[q] = max(d[q], d[q + 1] + 1)
t = []
for q in range(n):
	if a[q] == '0':
		t.append('0')
	elif d[q] == d[q + 1] + 1:
		t.append('0')
	else:
		t.append('1')
print(''.join(t))


import sys
sys.setrecursionlimit(10 ** 5 + 1)
inf = int(10 ** 20)
max_val = inf
min_val = -inf
RW = lambda : sys.stdin.readline().strip()
RI = lambda : int(RW())
RMI = lambda : [int(x) for x in sys.stdin.readline().strip().split()]
RWI = lambda : [x for x in sys.stdin.readline().strip().split()]
given = input()[::-1]
zeros = 0
lens = len(given)
rets = ''
for i in range(lens):
	if given[i] == '0':
		zeros += 1
		rets += '0'
	elif zeros > 0:
		zeros -= 1
		rets += '1'
	else:
		rets += '0'
rets = rets[::-1]
print(rets)


pp = input()
if len(pp) == 1:
	print('0')
	return
z = 1 if pp[0] == '0' else 0
zc = [z]
l = 1
lndl = [l]
for p in pp[1:]:
	l = max(z + 1, l + (1 if p == '1' else 0))
	z += 1 if p == '0' else 0
	lndl.append(l)
	zc.append(z)
lnda = lndl[-1]
o = 1 if pp[-1] == '1' else 0
oc = [o]
l = 1
lndr = [l]
for p in reversed(pp[:-1]):
	l = max(o + 1, l + (1 if p == '0' else 0))
	o += 1 if p == '1' else 0
	lndr.append(l)
	oc.append(o)
oc.reverse()
lndr.reverse()
qq = []
ez = 0
if pp[0] == '1':
	if max(oc[1], lndr[1] + 1) != lnda:
		qq.append('1')
	else:
		qq.append('0')
		ez += 1
else:
	qq.append('0')
for (p, l, o, z, r) in zip(pp[1:-1], lndl, oc[2:], zc, lndr[2:]):
	if p == '1':
		if max(l + o, z + ez + 1 + r) != lnda:
			qq.append('1')
		else:
			qq.append('0')
			ez += 1
	else:
		qq.append('0')
qq.append('0')
print(''.join(qq))


s = input()
res = ['0'] * len(s)
min_dif = 0
length = len(s)
for i in range(length):
	if s[length - i - 1] == '0':
		min_dif = min([-1, min_dif - 1])
	else:
		if min_dif < 0:
			res[length - i - 1] = '1'
		min_dif = min([1, min_dif + 1])
print(''.join(res))


s = input()
ans = ['0' for i in range(len(s))]
st = []
for i in range(len(s)):
	if s[i] == '0':
		if len(st):
			ans[st.pop()] = '1'
	else:
		st.append(i)
print(''.join(ans))


s = input()
n = len(s)
tot = 0
S = list(s)
for i in range(n - 1, -1, -1):
	if s[i] == '0':
		tot += 1
	elif tot:
		tot -= 1
	else:
		S[i] = '0'
print(''.join(S))


import sys
input = sys.stdin.readline

def solve():
	i = 0
	s = list(input().strip())
	st = []
	for c in s:
		if c == '1':
			st.append(i)
		elif len(st) > 0:
			st.pop()
		i += 1
	for i in st:
		s[i] = '0'
	print(''.join(s))
solve()


s = input()
s = [*s]
c = m = 0
for i in range(len(s) - 1, -1, -1):
	c += -1 if s[i] == '0' else 1
	if c > m:
		s[i] = '0'
		m = c
print(''.join(s))


a = list(map(int, input()))
w = [0] * len(a)
q = []
for i in range(len(a)):
	if a[i] == 1:
		q.append(i)
	elif len(q) > 0:
		qw = q.pop()
		w[qw] = 1
print(*w, sep='')


S = input()
N = len(S)
p = q = 0
prv = S[0]
c = 0
C = []
if S[0] == '1':
	C.append(0)
for ch in S:
	if ch != prv:
		C.append(c)
		c = 1
		prv = ch
	else:
		c += 1
C.append(c)
ans = []
r = 0
L = len(C)
for i in range(L - 1, -1, -1):
	c = C[i]
	if i % 2:
		r += c
		if i != L - 1:
			m = max(min(c - 1, r), 0)
		else:
			m = max(min(c, r), 0)
		if m:
			r -= m
			ans.append('0' * m + '1' * (c - m))
		else:
			ans.append('1' * c)
	else:
		r -= c
		ans.append('0' * c)
ans.reverse()
print(*ans, sep='')


s = list(map(int, input()))
stack = []
for i in range(len(s)):
	if s[i] == 1:
		stack.append(i)
	elif len(stack) != 0:
		stack.pop()
for i in stack:
	s[i] = 0
s = list(map(str, s))
print(''.join(s))


s = input()
n = len(s)
b = [0] * n
l = []
m = []
j = 0
for i in s:
	if len(l) == 0:
		l.append(i)
		m.append(j)
	elif l[-1] == '1' and i == '0':
		l.pop()
		b[m.pop()] = 1
	else:
		l.append(i)
		m.append(j)
	j += 1
print(*b, sep='')


import os
import sys
from math import *
from collections import *
from bisect import *
from io import BytesIO, IOBase

def vsInput():
	sys.stdin = open('input.txt', 'r')
	sys.stdout = open('output.txt', 'w')
BUFSIZE = 8192

class FastIO(IOBase):
	newlines = 0

	def __init__(self, file):
		self._fd = file.fileno()
		self.buffer = BytesIO()
		self.writable = 'x' in file.mode or 'r' not in file.mode
		self.write = self.buffer.write if self.writable else None

	def read(self):
		while True:
			b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
			if not b:
				break
			ptr = self.buffer.tell()
			(self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr))
		self.newlines = 0
		return self.buffer.read()

	def readline(self):
		while self.newlines == 0:
			b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
			self.newlines = b.count(b'\n') + (not b)
			ptr = self.buffer.tell()
			(self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr))
		self.newlines -= 1
		return self.buffer.readline()

	def flush(self):
		if self.writable:
			os.write(self._fd, self.buffer.getvalue())
			(self.buffer.truncate(0), self.buffer.seek(0))

class IOWrapper(IOBase):

	def __init__(self, file):
		self.buffer = FastIO(file)
		self.flush = self.buffer.flush
		self.writable = self.buffer.writable
		self.write = lambda s: self.buffer.write(s.encode('ascii'))
		self.read = lambda : self.buffer.read().decode('ascii')
		self.readline = lambda : self.buffer.readline().decode('ascii')
(sys.stdin, sys.stdout) = (IOWrapper(sys.stdin), IOWrapper(sys.stdout))
input = lambda : sys.stdin.readline().rstrip('\r\n')
ALPHA = 'abcdefghijklmnopqrstuvwxyz'
M = 998244353
EPS = 1e-06

def Ceil(a, b):
	return a // b + int(a % b > 0)

def value():
	return tuple(map(int, input().split()))

def array():
	return [int(i) for i in input().split()]

def Int():
	return int(input())

def Str():
	return input()

def arrayS():
	return [i for i in input().split()]
s = [int(i) for i in input()]
n = len(s)
ans = [0] * n
have = 0
for i in range(n - 1, -1, -1):
	if not s[i]:
		have += 1
	elif have:
		have -= 1
		ans[i] = 1
print(*ans, sep='')


from sys import stdin
s = stdin.readline().strip()
dp = [0 for i in range(len(s) + 2)]
ons = [0 for i in range(len(s) + 2)]
zs = [0 for i in range(len(s) + 2)]
for i in range(len(s) - 1, -1, -1):
	if s[i] == '1':
		ons[i] += 1
	if i != len(s) - 1:
		ons[i] += ons[i + 1]
z = 0
for i in range(len(s) - 1, -1, -1):
	if s[i] == '1':
		dp[i] = max(1 + ons[i + 1], z)
	else:
		dp[i] = max(dp[i + 1] + 1, 1 + ons[i + 1])
		z = dp[i]
	zs[i] = z
ans = ''
for i in range(len(s)):
	if s[i] == '1':
		x = dp[i]
		y = 1 + dp[i + 1]
		if x == y:
			ans += '0'
		else:
			ans += '1'
	else:
		ans += '0'
print(ans)


s = input()[::-1]
ans = ''
c = 0
for i in s:
	if i == '0':
		ans += '0'
		c += 1
	elif c > 0:
		ans += '1'
		c -= 1
	else:
		ans += '0'
print(ans[::-1])


inn = lambda : int(input())
inl = lambda : list(map(int, input().split()))
inm = lambda : map(int, input().split())
DBG = True and False

def ddprint(x):
	if DBG:
		print(x)
s = input().strip()
n = len(s)
ones = []
stk = []
for i in range(n):
	if len(stk) > 0 and stk[-1][0] == '1' and (s[i] == '0'):
		ones.append(stk[-1][1])
		stk.pop()
	else:
		stk.append((s[i], i))
ones.sort()
ans = ''
cur = 0
for i in ones:
	for j in range(cur, i):
		ans += '0'
	ans += '1'
	cur = i + 1
for j in range(cur, n):
	ans += '0'
print(ans)


data = [int(i) for i in input()]
n = len(data)
minval = [0] * n
minval[-1] = 2 * data[-1] - 1
for i in range(n - 2, -1, -1):
	if data[i] == 1:
		minval[i] = min(1, minval[i + 1] + 1)
	else:
		minval[i] = min(-1, minval[i + 1] - 1)
ans = [i for i in data]
for i in range(n - 1, -1, -1):
	if minval[i] == 1:
		ans[i] = 0
ans2 = [str(i) for i in ans]
print(''.join(ans2))


s = input()
s = list(s)
l = len(s) - 1
z = 0
for i in range(l, -1, -1):
	if s[i] == '0':
		z += 1
	elif z:
		z -= 1
	else:
		s[i] = '0'
s = ''.join(s)
print(s)


import sys
input = sys.stdin.readline
s = list(input().rstrip())
n = len(s)
c = m = 0
for i in range(n)[::-1]:
	c += -1 if s[i] == '0' else 1
	if c > m:
		s[i] = '0'
		c = m
print(''.join(s))


s = input()
ans = []
for i in range(0, len(s)):
	ans.append(s[i])
c0 = 0
c1 = 0
for i in range(len(ans) - 1, -1, -1):
	if ans[i] == '0':
		c0 += 1
	elif ans[i] == '1' and c1 >= c0:
		ans[i] = '0'
	else:
		c1 += 1
ans = ''.join(ans)
print(ans)


s = input()
(a, b, c) = ([], [], [])
count = 0
for i in range(len(s) - 1, -1, -1):
	a.append(s[i])
for i in range(len(a)):
	if a[i] == '0':
		count += 1
		b.append('0')
	elif a[i] == '1' and count > 0:
		count -= 1
		b.append('1')
	elif a[i] == '1' and count == 0:
		b.append('0')
for i in range(len(b) - 1, -1, -1):
	c.append(b[i])
ans = ''.join(c)
print(ans)


s = str(input().strip())
t = list(s[::-1])
cnt = 0
for (i, v) in enumerate(t):
	if v == '0':
		cnt += 1
	elif cnt:
		cnt -= 1
	else:
		t[i] = '0'
print(''.join(t[::-1]))


line = input()
res = []
zer = 0
one = 0
for i in range(len(line)):
	ind = len(line) - 1 - i
	if line[ind] == '0':
		res.append('0')
		zer += 1
	else:
		if zer <= one:
			res.append('0')
			zer += 1
		else:
			res.append('1')
		one += 1
for i in range(len(res)):
	print(res[len(res) - 1 - i], end='')
print('\n', end='')


(v, t) = (0, '')
for c in input()[::-1]:
	if c < '1' or v:
		t += c
		v += 1 - (c < '1') * 2
	else:
		t += '0'
print(t[::-1])


st = input().rstrip()
a = []
for item in st:
	a.append(int(item))
a.reverse()
zero_count = 0
for (i, item) in enumerate(a):
	if item == 0:
		zero_count += 1
	elif zero_count > 0:
		zero_count -= 1
	else:
		a[i] = 0
a.reverse()
print(''.join([str(item) for item in a]))


s = list(map(int, input()))
ps = [0]
for i in s:
	if i == 0:
		ps.append(ps[-1] + 1)
	else:
		ps.append(ps[-1] - 1)
b = 0
maba = 0
sufmax = [-10 ** 9]
for i in range(len(ps) - 1, 0, -1):
	sufmax.append(max(sufmax[-1], ps[i]))
sufmax = sufmax[::-1]
ans = []
cnt = 0
if s[0] == 1:
	cnt += 1
else:
	ans.append('0')
for i in range(1, len(s)):
	if s[i] == 0 and s[i - 1] == 0:
		ans.append('0')
	elif s[i] == 1:
		cnt += 1
	else:
		maba = sufmax[i] - ps[i]
		maba = min(maba, cnt)
		for _ in range(cnt - maba):
			ans.append('0')
		for _ in range(maba):
			ans.append('1')
		cnt = 0
		ans.append('0')
for _ in range(len(s) - len(ans)):
	ans.append('0')
print(''.join(ans))


S = list(map(int, input().strip()))
N = len(S)
stack = []
for i in range(N):
	s = S[i]
	if s == 0 and stack and (stack[-1][0] == 1):
		stack.pop()
	else:
		stack.append((s, i))
T = S[:]
if stack:
	for i in tuple(map(list, zip(*stack)))[1]:
		T[i] = 0
print(''.join(map(str, T)))


s = str(input())
n = len(s)
ns = ['0' for i in range(n)]
st = []
for i in range(n):
	if s[i] == '0':
		if len(st):
			ns[st.pop()] = '1'
	else:
		st.append(i)
print(''.join(ns))


(v, ans) = (0, '')
for c in input()[::-1]:
	if c == '0' or v:
		ans += c
		v += 1 if c == '1' else -1
	else:
		ans += '0'
print(ans[::-1])


s = input()
t = list(s)
stack = []
for i in range(len(s)):
	if t[i] == '1':
		stack.append(i)
	elif len(stack):
		stack.pop()
for i in range(len(stack)):
	t[stack[i]] = '0'
print(''.join(t))


s = input()[::-1]
count = 0
n = len(s)
s1 = ''
for i in range(n):
	if s[i] == '0':
		count += 1
		s1 += '0'
	elif count > 0:
		count -= 1
		s1 += '1'
	else:
		s1 += '0'
s1 = s1[::-1]
print(s1)


s = [int(x) for x in list(input())]
n = len(s)
b = [0] * n
counter = 0
for i in range(n - 1, -1, -1):
	if s[i] == 0:
		counter += 1
	elif counter > 0:
		counter -= 1
	else:
		s[i] = 0
arr = ''
for item in s:
	arr += str(item)
print(arr)


def main():
	s = input()
	n = len(s)
	st = []
	flags = [False for i in range(n)]
	for i in range(n):
		if s[i] == '1':
			st.append(i)
		elif len(st) != 0:
			st.pop()
	for i in range(len(st)):
		flags[st[i]] = True
	for i in range(n):
		if flags[i] == True:
			print('0', end='')
		else:
			print(s[i], end='')
	print()
main()


l = list(input())
ct = 0
n = len(l)
for i in range(n - 1, -1, -1):
	if l[i] == '0':
		ct += 1
	elif ct != 0:
		ct -= 1
	else:
		l[i] = '0'
print(''.join(l))


INF = 10 ** 10

def main():
	print = out.append
	st = list(input())
	stk = []
	for (index, i) in enumerate(st):
		if i == '0' and len(stk) > 0 and (stk[-1][0] == '1'):
			stk.pop()
		else:
			stk.append([i, index])
	for li in stk:
		st[li[1]] = '0'
	print(''.join(st))
import sys
input = iter(sys.stdin.buffer.read().decode().splitlines()).__next__
out = []
get_int = lambda : int(input())
get_list = lambda : list(map(int, input().split()))
main()
print(*out, sep='\n')


s = input()
n = len(s)
if n == 1:
	print(0)
elif n == 2:
	if s == '10':
		print('10')
	else:
		print('00')
else:
	s_lis = 0
	s_max = 0
	s_zero = 0
	t_zero = 0
	stack_01 = []
	t = []
	for i in range(n):
		t.append('0')
		t_zero += 1
		if s[i] == '0':
			s_zero += 1
		if i > 0 and s[i - 1:i + 1] == '10':
			t[i - 1] = '1'
			t_zero -= 1
			stack_01.append(i - 2)
		if s_max <= int(s[i]):
			s_lis += 1
			s_max = max(s_max, int(s[i]))
		if s_lis <= s_zero:
			s_lis = s_zero
			s_max = 0
		if s_lis < t_zero:
			while t[stack_01[-1]] == '1' or s[stack_01[-1]] == '0' or t[stack_01[-1] - 1] == '1':
				stack_01.pop()
			t[stack_01[-1]] = '1'
			t_zero -= 1
			stack_01[-1] -= 1
	print(*t, sep='')


import sys
import math

def main():
	s = input()
	q = []
	for i in range(len(s)):
		if s[i] == '1':
			q.append(i)
		elif len(q) > 0:
			q.pop()
	result = []
	q = set(q)
	for i in range(len(s)):
		if i in q:
			result.append('0')
		else:
			result.append(s[i])
	print(''.join(result))

def __starting_point():
	main()
__starting_point()


s = input()
t = [0] * len(s)
stack = []
for i in range(len(s)):
	if s[i] == '1':
		stack.append(i)
	elif len(stack) > 0:
		pos = stack.pop()
		t[pos] = 1
print(''.join(map(str, t)))


s = input()
res = ['0'] * len(s)
stk = []
for i in range(len(s)):
	if s[i] == '0':
		if len(stk) > 0:
			res[stk[-1]] = '1'
			stk.pop()
	else:
		stk.append(i)
print(''.join(res))