Controversial Rounds (Codeforces 1398F)
Tags: Greedy, two pointers

Alice and Bob play a game. The game consists of several sets, and each set consists of several rounds. Each round is won either by Alice or by Bob, and the set ends when one of the players has won $$$x$$$ rounds in a row. For example, if Bob won five rounds in a row and $$$x = 2$$$, then two sets ends.You know that Alice and Bob have already played $$$n$$$ rounds, and you know the results of some rounds. For each $$$x$$$ from $$$1$$$ to $$$n$$$, calculate the maximum possible number of sets that could have already finished if each set lasts until one of the players wins $$$x$$$ rounds in a row. It is possible that the last set is still not finished — in that case, you should not count it in the answer.

Input: The first line contains one integer $$$n$$$ ($$$1 \le n \le 10^6$$$) — the number of rounds.The second line contains one string $$$s$$$ of length $$$n$$$ — the descriptions of rounds. If the $$$i$$$-th element of the string is 0, then Alice won the $$$i$$$-th round; if it is 1, then Bob won the $$$i$$$-th round, and if it is ?, then you don't know who won the $$$i$$$-th round.

Output: In the only line print $$$n$$$ integers. The $$$i$$$-th integer should be equal to the maximum possible number of sets that could have already finished if each set lasts until one of the players wins $$$i$$$ rounds in a row.

Note: Let's consider the first test case:  if $$$x = 1$$$ and $$$s = 110000$$$ or $$$s = 111000$$$ then there are six finished sets;  if $$$x = 2$$$ and $$$s = 110000$$$ then there are three finished sets;  if $$$x = 3$$$ and $$$s = 111000$$$ then there are two finished sets;  if $$$x = 4$$$ and $$$s = 110000$$$ then there is one finished set;  if $$$x = 5$$$ then there are no finished sets;  if $$$x = 6$$$ then there are no finished sets. 

Sample Inputs:
6
11?000

5
01?01

12
???1??????1?

Sample Outputs:
6 3 2 1 0 0

5 1 0 0 0

12 6 4 3 2 2 1 1 1 1 1 1

Solution (Python):
import sys
int1 = lambda x: int(x) - 1
p2D = lambda x: print(*x, sep='\n')

def II():
    return int(sys.stdin.readline())

def MI():
    return map(int, sys.stdin.readline().split())

def LI():
    return list(map(int, sys.stdin.readline().split()))

def LLI(rows_number):
    return [LI() for _ in range(rows_number)]

def SI():
    return sys.stdin.readline()[:-1]

def RLE(s):
    cc = []
    ww = []
    pc = s[0]
    w = 0
    for c in s:
        if c == pc:
            w += 1
        else:
            cc.append(pc)
            ww.append(w)
            w = 1
            pc = c
    cc.append(pc)
    ww.append(w)
    return (cc, ww)
n = II()
s = SI()
ans = []
(cc, ww) = RLE(s)
if len(cc) == 1 and cc[0] == '?':
    for x in range(1, n + 1):
        ans.append(n // x)
    print(*ans)
    exit()
if cc[0] == '?':
    ww[1] += ww[0]
    ww[0] = 0
if cc[-1] == '?':
    ww[-2] += ww[-1]
    ww[-1] = 0
for i in range(len(cc) - 2):
    if cc[i + 1] == '?' and cc[i] == cc[i + 2]:
        ww[i + 2] += ww[i + 1] + ww[i]
        ww[i + 1] = ww[i] = 0
aa = []
for (c, w) in zip(cc, ww):
    if w == 0:
        continue
    if len(aa) & 1 and c != '?':
        aa.append(0)
    aa.append(w)
aa.append(0)
ans.append(n)
for x in range(2, n + 1):
    bb = []
    r = 0
    pq = 0
    move = 0
    cur = 0
    for i in range(0, len(aa), 2):
        a = aa[i]
        q = aa[i + 1]
        cur += (r + a + q) // x
        r = min(q, (r + a + q) % x)
        if min(x, pq) + a + move + q <= x:
            if bb:
                b = bb.pop()
                bb[-1] += b
                bb.append(0)
            pq = 0
            move = q
        else:
            bb += [move + a, q]
            move = 0
            pq = q
    aa = bb
    ans.append(cur)
    if cur == 0:
        break
ans += [0] * (n - len(ans))
print(*ans)

Solution (Python):
import sys
sys.setrecursionlimit(10 ** 5)
int1 = lambda x: int(x) - 1
p2D = lambda x: print(*x, sep='\n')

def II():
    return int(sys.stdin.readline())

def MI():
    return map(int, sys.stdin.readline().split())

def LI():
    return list(map(int, sys.stdin.readline().split()))

def LLI(rows_number):
    return [LI() for _ in range(rows_number)]

def SI():
    return sys.stdin.readline()[:-1]

def RLE(s):
    cc = []
    ww = []
    pc = s[0]
    w = 0
    for c in s:
        if c == pc:
            w += 1
        else:
            cc.append(pc)
            ww.append(w)
            w = 1
            pc = c
    cc.append(pc)
    ww.append(w)
    return (cc, ww)
n = II()
s = SI()
ans = []
(cc, ww) = RLE(s)
if len(cc) == 1 and cc[0] == '?':
    for x in range(1, n + 1):
        ans.append(n // x)
    print(*ans)
    exit()
if cc[0] == '?':
    ww[1] += ww[0]
    ww[0] = 0
if cc[-1] == '?':
    ww[-2] += ww[-1]
    ww[-1] = 0
for i in range(len(cc) - 2):
    if cc[i + 1] == '?' and cc[i] == cc[i + 2]:
        ww[i + 2] += ww[i + 1] + ww[i]
        ww[i + 1] = ww[i] = 0
aa = []
for (c, w) in zip(cc, ww):
    if w == 0:
        continue
    if len(aa) & 1 and c != '?':
        aa.append(0)
    aa.append(w)
aa.append(0)
ans.append(n)
for x in range(2, n + 1):
    bb = []
    r = 0
    pq = 0
    move = 0
    cur = 0
    for i in range(0, len(aa), 2):
        a = aa[i]
        q = aa[i + 1]
        cur += (r + a + q) // x
        r = min(q, (r + a + q) % x)
        if min(x, pq) + a + move + q <= x:
            if bb:
                b = bb.pop()
                bb[-1] += b
                bb.append(0)
            pq = 0
            move = q
        else:
            bb += [move + a, q]
            move = 0
            pq = q
    aa = bb
    ans.append(cur)
    if cur == 0:
        break
ans += [0] * (n - len(ans))
print(*ans)

Solution (Python):
from bisect import bisect_left as bl
from bisect import bisect_right as br
from heapq import heappush, heappop
import math
from collections import *
from functools import reduce, cmp_to_key, lru_cache
import io, os
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
M = mod = 10 ** 9 + 7

def factors(n):
    return sorted(set(reduce(list.__add__, ([i, n // i] for i in range(1, int(n ** 0.5) + 1) if n % i == 0))))

def inv_mod(n):
    return pow(n, mod - 2, mod)

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

def st():
    return str(input().rstrip())[2:-1]

def val():
    return int(input().rstrip())

def li2():
    return [str(i)[2:-1] for i in input().rstrip().split()]

def li3():
    return [int(i) for i in st()]
n = val()
s = st()
pref0 = [0]
pref1 = [0]
for i in s:
    if i == '1':
        pref1.append(pref1[-1] + 1)
        pref0.append(pref0[-1])
    elif i == '0':
        pref0.append(pref0[-1] + 1)
        pref1.append(pref1[-1])
    else:
        pref1.append(pref1[-1] + 1)
        pref0.append(pref0[-1] + 1)
next0 = [-1]
next1 = [-1]
m1 = m2 = -1
for i in range(n):
    if s[i] == '1':
        m2 = i
    if s[i] == '0':
        m1 = i
    next0.append(m1)
    next1.append(m2)
flag = 0
last = float('inf')
for x in range(1, n + 1):
    if flag:
        ans = 0
    else:
        ans = start = 0
        while start + x <= n:
            if pref0[start + x] - pref0[start] == x:
                start += x
                ans += 1
            elif pref1[start + x] - pref1[start] == x:
                start += x
                ans += 1
            else:
                start = next0[start + x] if next1[start + x] > next0[start + x] else next1[start + x]
                start += 1
    if not ans:
        flag = 1
    last = ans
    print(ans, end=' ')