牛客网NOIP赛前集训营-普及组（第一场）C 括号

括号

思路：

dp

状态：dp[i][j]表示到i位置为止未匹配的 '(' 个数为j的方案数

状态转移:

如果s[i] == '('

dp[i][j] = dp[i-1][j] + dp[i-1][j-1]

如果s[i] == ')'

dp[i][j] = dp[i-1][j] + dp[i-1][j+1]

优化：用滚动数组优化掉i这一维度

代码：

#pragma GCC optimize(2)
#pragma GCC optimize(3)
#pragma GCC optimize(4)
#include<bits/stdc++.h>
using namespace std;
#define fi first
#define se second
#define pi acos(-1.0)
#define LL long long
#define mp make_pair
#define pb push_back
#define ls rt<<1, l, m
#define rs rt<<1|1, m+1, r
#define ULL unsigned LL
#define pll pair<LL, LL>
#define pii pair<int, int>
#define piii pair<pii, int>
#define mem(a, b) memset(a, b, sizeof(a))
#define fio ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define fopen freopen("in.txt", "r", stdin);freopen("out.txt", "w", stout);
//head
 
const int N = 1e4 + ;
const int MOD = 1e9 + ;
char s[N];
int dp[][N];
int main() {
    int n;
    scanf("%d", &n);
    scanf("%s", s+);
    int now = ;
    dp[now][] = ;
    for (int i = ; i <= n; i++) {
        now ^= ;
        if(s[i] == '(') {
            for(int i = ; i < n; i++) {
                dp[now][i+] = (dp[now^][i+] + dp[now^][i]) % MOD;
            }
            dp[now][] = dp[now^][];
        }
        else {
            for (int i = ; i <= n; i++) {
                dp[now][i-] = (dp[now^][i-] + dp[now^][i]) % MOD;
            }
            dp[now][n] = dp[now^][n];
        }
    }
    printf("%d\n", (dp[now][] + MOD - ) % MOD);
    return ;
}