Sereja and Brackets
题目链接： CodeForces - 380C
Sereja has a bracket sequence s1, s2, ..., s**n, or, in other words, a string s of length n, consisting of characters "(" and ")".
Sereja needs to answer m queries, each of them is described by two integers l**i, r**i(1 ≤ l**i ≤ r**i ≤ n). The answer to the i-th query is the length of the maximum correct bracket subsequence of sequence sli, sli + 1, ..., sri. Help Sereja answer all queries.
You can find the definitions for a subsequence and a correct bracket sequence in the notes.
Input
The first line contains a sequence of characters s1, s2, ..., s**n (1 ≤ n ≤ 106) without any spaces. Each character is either a "(" or a ")". The second line contains integer m (1 ≤ m ≤ 105) — the number of queries. Each of the next m lines contains a pair of integers. The i-th line contains integers l**i, r**i (1 ≤ l**i ≤ r**i ≤ n) — the description of the i-th query.
Output
Print the answer to each question on a single line. Print the answers in the order they go in the input.
Examples
Input
())(())(())(71 12 31 21 128 125 112 10
Output
00210466
Note
A subsequence of length |x| of string s = s1s2... s|s| (where |s| is the length of string s) is string x = s**k1s**k2... s**k|x| (1 ≤ k1 < k2 < ... < k|x| ≤ |s|).
A correct bracket sequence is a bracket sequence that can be transformed into a correct aryphmetic expression by inserting characters "1" and "+" between the characters of the string. For example, bracket sequences "()()", "(())" are correct (the resulting expressions "(1)+(1)", "((1+1)+1)"), and ")(" and "(" are not.
For the third query required sequence will be «()».
For the fourth query required sequence will be «()(())(())».
题意：
给你一个只含有'(' 和')' 的字符串，
以及q个询问，每一个询问给你两个整数l和r，代表一个区间。对于每一个询问，让你输出区间中能选出最长的子序列是合法的括号序列的长度。
思路：
线段树+分治的思想来解决此问题。
我们线段树每一个区间维护以下信息：
1、区间中能选出最长的子序列是合法的括号序列的个数 num。
2、 区间中多余的'(' 字符的个数 a
3、区间中多余的')' 字符的个数 b
那么对于区间合并时，
num=左儿子的num+右儿子的num+min（左儿子的a，右儿子的b）
a=左儿子的a+右儿子的a - min（左儿子的a，右儿子的b）
b=左儿子的b+右儿子的b - min（左儿子的a，右儿子的b）
最后输出时，注意num个括号个数，*2才是长度。
细节见代码：
#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <queue>
#include <stack>
#include <map>
#include <set>
#include <vector>
#include <iomanip>
#define ALL(x) (x).begin(), (x).end()
#define sz(a) int(a.size())
#define all(a) a.begin(), a.end()
#define rep(i,x,n) for(int i=x;i<n;i++)
#define repd(i,x,n) for(int i=x;i<=n;i++)
#define pii pair<int,int>
#define pll pair<long long ,long long>
#define gbtb ios::sync_with_stdio(false),cin.tie(0),cout.tie(0)
#define MS0(X) memset((X), 0, sizeof((X)))
#define MSC0(X) memset((X), ' ', sizeof((X)))
#define pb push_back
#define mp make_pair
#define fi first
#define se second
#define eps 1e-6
#define gg(x) getInt(&x)
#define chu(x) cout<<"["<<#x<<" "<<(x)<<"]"<<endl
using namespace std;
typedef long long ll;
ll gcd(ll a, ll b) {return b ? gcd(b, a % b) : a;}
ll lcm(ll a, ll b) {return a / gcd(a, b) * b;}
ll powmod(ll a, ll b, ll MOD) {ll ans = 1; while (b) {if (b % 2) { ans = ans * a % MOD; } a = a * a % MOD; b /= 2;} return ans;}
inline void getInt(int *p);
const int maxn = 1000010;
const int inf = 0x3f3f3f3f;
/*** TEMPLATE CODE * * STARTS HERE ***/
struct node {
int l, r;
int num;
int a;// (
int b;// )
} segmeng_tree[maxn << 2];
char s[maxn];
int n;
int m;
void pushup(int rt)
{
int x = min(segmeng_tree[rt << 1].a, segmeng_tree[rt << 1 | 1].b);
segmeng_tree[rt].num = x + segmeng_tree[rt << 1].num + segmeng_tree[rt << 1 | 1].num;
segmeng_tree[rt].a = segmeng_tree[rt << 1].a + segmeng_tree[rt << 1 | 1].a - x;
segmeng_tree[rt].b = segmeng_tree[rt << 1].b + segmeng_tree[rt << 1 | 1].b - x;
}
void build(int rt, int l, int r)
{
segmeng_tree[rt].l = l;
segmeng_tree[rt].r = r;
if (l == r) {
segmeng_tree[rt].a = s[l] == '(';
segmeng_tree[rt].b = s[l] == ')';
segmeng_tree[rt].num = 0;
} else {
int mid = (l + r) >> 1;
build(rt << 1, l, mid);
build(rt << 1 | 1, mid + 1, r);
pushup(rt);
}
}
node ask(int rt, int l, int r)
{
if (segmeng_tree[rt].l >= l && segmeng_tree[rt].r <= r) {
return segmeng_tree[rt];
}
int mid = (segmeng_tree[rt].l + segmeng_tree[rt].r) >> 1;
if (r <= mid) {
return ask(rt << 1, l, r);
} else if (l > mid) {
return ask(rt << 1 | 1, l, r);
} else {
node res1 = ask(rt << 1, l, r);
node res2 = ask(rt << 1 | 1, l, r);
node res = res1;
int x = min(res1.a, res2.b);
res.num += x;
res.b += res2.b;
res.a += res2.a;
res.num += res2.num;
res.b -= x;
res.a -= x;
return res;
}
}
int main()
{
//freopen("D:\code\text\input.txt","r",stdin);
//freopen("D:\code\text\output.txt","w",stdout);
scanf("%s", s + 1);
n = strlen(s + 1);
build(1, 1, n);
scanf("%d", &m);
while (m--) {
int l, r;
scanf("%d %d", &l, &r);
printf("%d
", ask(1, l, r).num * 2);
}
return 0;
}
inline void getInt(int *p)
{
char ch;
do {
ch = getchar();
} while (ch == ' ' || ch == '
');
if (ch == '-') {
*p = -(getchar() - '0');
while ((ch = getchar()) >= '0' && ch <= '9') {
*p = *p * 10 - ch + '0';
}
} else {
*p = ch - '0';
while ((ch = getchar()) >= '0' && ch <= '9') {
*p = *p * 10 + ch - '0';
}
}
}