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 + 10;
const int MOD = 1e9 + 7;
char s[N];
int dp[2][N];
int main() {
    int n;
    scanf("%d", &n); 
    scanf("%s", s+1);
    int now = 0;
    dp[now][0] = 1;
    for (int i = 1; i <= n; i++) {
        now ^= 1;
        if(s[i] == '(') {
            for(int i = 0; i < n; i++) {
                dp[now][i+1] = (dp[now^1][i+1] + dp[now^1][i]) % MOD;
            } 
            dp[now][0] = dp[now^1][0];
        }
        else {
            for (int i = 1; i <= n; i++) {
                dp[now][i-1] = (dp[now^1][i-1] + dp[now^1][i]) % MOD; 
            } 
            dp[now][n] = dp[now^1][n];
        }
    }
    printf("%d
", (dp[now][0] + MOD - 1) % MOD);
    return 0;
}