Problem:
Score : 800 points

Problem Statement

Given is a string S, where each character is 0, 1, or ?.

Consider making a string S' by replacing each occurrence of ? with 0 or 1 (we can choose the character for each ? independently).
Let us define the unbalancedness of S' as follows:

(The unbalancedness of S') = \max \{ The absolute difference between the number of occurrences of 0 and 1 between the l-th and r-th character of S (inclusive) :\ 1 \leq l \leq r \leq |S|\}

Find the minimum possible unbalancedness of S'.

Constraints

1 \leq |S| \leq 10^6

Each character of S is 0, 1, or ?.

Input

Input is given from Standard Input in the following format:

S

Output

Print the minimum possible unbalancedness of S'.

Sample Input 1

0??

Sample Output 1

1

We can make S'= 010 and achieve the minimum possible unbalancedness of 1.

Sample Input 2

0??0

Sample Output 2

2

Sample Input 3

??00????0??0????0?0??00??1???11?1?1???1?11?111???1

Sample Output 3

4

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

//by yjz
#include<bits/stdc++.h>
using namespace std;
#define FF first
#define SS second
#define PB push_back
#define MP make_pair
#ifndef LOCAL
#define cerr if(0)cout
#endif
typedef long long ll;
//My i/o stream
struct fastio
{
	char s[100000];
	int it, len;
	fastio() {it=len=0;}
	inline char get()
	{
		if (it<len) return s[it++]; it=0;
		len=fread(s, 1, 100000, stdin);
		if (len==0) return EOF; else return s[it++];
	}
	bool notend()
	{
		char c=get();
		while (c==' '||c=='\n') c=get();
		if (it>0) it--;
		return c!=EOF;
	}
}_buff;
#define geti(x) x=getnum()
#define getii(x,y) geti(x),geti(y)
#define getiii(x,y,z) getii(x,y),geti(z)
#define puti(x) putnum(x),putchar(' ')
#define putii(x,y) puti(x),puti(y)
#define putiii(x,y,z) putii(x,y),puti(z)
#define putsi(x) putnum(x),putchar('\n')
#define putsii(x,y) puti(x),putsi(y)
#define putsiii(x,y,z) putii(x,y),putsi(z)
inline ll getnum()
{
	ll r=0; bool ng=0; char c; c=_buff.get();
	while (c!='-'&&(c<'0'||c>'9')) c=_buff.get();
	if (c=='-') ng=1, c=_buff.get();
	while (c>='0'&&c<='9') r=r*10+c-'0', c=_buff.get();
	return ng?-r:r;
}
template <class T> inline void putnum(T x)
{
	if (x<0) putchar('-'), x=-x;
	register short a[20]={}, sz=0;
	while (x) a[sz++]=x%10, x/=10;
	if(sz==0) putchar('0');
	for (int i=sz-1; i>=0; i--) putchar('0'+a[i]);
}
inline char getreal() {char c=_buff.get(); while (c<=32) c=_buff.get(); return c;}

const int mod = 1e9+7;
ll qpow(ll x, ll k) {return k==0? 1: 1ll*qpow(1ll*x*x%mod,k>>1)*(k&1?x:1)%mod;}

const int maxn = 4000111;
struct node
{
	ll mn, mx, lz;
}a[maxn];
#define ls p<<1
#define rs p<<1|1
void update(int p)
{
	a[p].mx = max(a[ls].mx, a[rs].mx);
	a[p].mn = min(a[ls].mn, a[rs].mn);
}
void pushdown(int p)
{
	if (a[p].lz)
	{
		a[ls].mx += a[p].lz;
		a[ls].mn += a[p].lz;
		a[ls].lz += a[p].lz;
		a[rs].mx += a[p].lz;
		a[rs].mn += a[p].lz;
		a[rs].lz += a[p].lz;
		a[p].lz = 0;
	}
}
void add(int x, int y, ll v, int l, int r, int p)
{
	if (x<=l&&r<=y)
	{
		a[p].mx += v;
		a[p].mn += v;
		a[p].lz += v;
		return;
	}
	pushdown(p);
	int m = l+r>>1;
	if (x<=m) add(x, y, v, l, m, ls);
	if (m<y) add(x, y, v, m+1, r, rs);
	update(p);
}
int query(int x, int y, int l, int r, int p)
{
	if (x<=l&&r<=y) return a[p].mx;
	pushdown(p);
	int m = l+r>>1;
	if (x<=m&&m<y) return max(query(x, y, l, m, ls), query(x, y, m+1, r, rs));
	else if (x<=m) return query(x, y, l, m, ls);
	else return query(x, y, m+1, r, rs);
}
char s[1000111];
int n;
set<int> st, st2;
const ll B = 1e9;
int ans = 1e9;
void upd()
{
	ans = min(ans, (int)(a[1].mx/B-a[1].mn/B));
//	printf("%d %s\n", (int)(a[1].mx/B-a[1].mn/B), s+1);
}
int get()
{
	return min(ans, (int)(a[1].mx/B-a[1].mn/B));
}
int main()
{
	scanf("%s", s+1);
	n = strlen(s+1);
	int c = n;
	add(0, 0, n*B, 0, n, 1);
	for (int i=1; i<=n; i++)
	{
		if (s[i]=='1') c++; else c--;
		if (s[i]=='?') st.insert(i);
		add(i, i, c*B+i, 0, n, 1);
	}
	upd();
	int MX = a[1].mx/B;
	vector<int> v;
	for (int i=1; i<=n; i++)
	{
		if (s[i]=='?')
		{
			add(i, n, 2*B, 0, n, 1);
			if (a[1].mx/B>MX)
			{
				add(i, n, -2*B, 0, n, 1);
			}
			else v.PB(i);
		}
	}
	upd();
	for (auto x : v) add(x, n, -2*B, 0, n, 1);
	for (int i=1; i<=n; i++)
	{
		if (s[i]=='?')
		{
			add(i, n, 2*B, 0, n, 1);
			if (a[1].mx/B>MX+1)
			{
				add(i, n, -2*B, 0, n, 1);
			}
		}
	}
	upd();
	cout<<ans<<endl;
	return 0;
}