Neko and Aki's Prank CodeForces - 1152D (括号序列,dp)

Posted 2021-11-05 uid001

大意: 将所有长度为2*n的合法括号序列建成一颗trie树, 求trie树上选出一个最大不相交的边集, 输出边集大小.

 

最大边集数一定不超过奇数层结点数. 这个上界可以通过从底层贪心达到, 所以就转化为求奇数层结点数.

然后就dp求出前$i$为‘(‘比‘)‘多j个的方案数, 奇数层且合法的时候统计一下贡献即可.

#include <iostream>
#include <iostream>
#include <algorithm>
#include <cstdio>
#include <math.h>
#include <set>
#include <map>
#include <queue>
#include <string>
#include <string.h>
#include <bitset>
#define REP(i,a,n) for(int i=a;i<=n;++i)
#define PER(i,a,n) for(int i=n;i>=a;--i)
#define hr putchar(10)
#define pb push_back
#define lc (o<<1)
#define rc (lc|1)
#define mid ((l+r)>>1)
#define ls lc,l,mid
#define rs rc,mid+1,r
#define x first
#define y second
#define io std::ios::sync_with_stdio(false)
#define endl ‘\n‘
#define DB(a) ({REP(__i,1,n) cout<<a[__i]<<‘ ‘;hr;})
using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
const int P = 1e9+7, INF = 0x3f3f3f3f;
ll gcd(ll a,ll b) {return b?gcd(b,a%b):a;}
ll qpow(ll a,ll n) {ll r=1%P;for (a%=P;n;a=a*a%P,n>>=1)if(n&1)r=r*a%P;return r;}
ll inv(ll x){return x<=1?1:inv(P%x)*(P-P/x)%P;}
inline int rd() {int x=0;char p=getchar();while(p<‘0‘||p>‘9‘)p=getchar();while(p>=‘0‘&&p<=‘9‘)x=x*10+p-‘0‘,p=getchar();return x;}
//head



const int N = 4e3+10;
int n, dp[N][N];

int main() {
	scanf("%d", &n);
	dp[0][0] = 1;
	int ans = 0;
	REP(i,0,2*n) {
		REP(j,0,i) {
			(dp[i+1][j+1] += dp[i][j]) %= P;
			if (j) (dp[i+1][j-1] += dp[i][j]) %= P;
			if (2*n-i>=j&&i%2==1) (ans += dp[i][j]) %= P;
		}
	}
	printf("%d\n", ans);
}