pyyzDay5

大炮(DP)

T1 [COCI 2022/2023 #5] Slastičarnica

T2 AND Segments

T3 [XJTUPC 2024] 最后一块石头的重量

2个小结论

Dp+随机化

#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<cmath>
#include<algorithm>
#include<iomanip>
#include<bits/stdc++.h>
#define int long long
#define jiaa(a,b) {a+=b;if(a>=MOD) a-=MOD;}
#define jian(a,b) {a-=b;if(a<0) a+=MOD;}
using namespace std;
int ksm(int a,int b,int p){
	if(b==0) return 1;
	if(b==1) return a%p;
	int c=ksm(a,b/2,p);
	c=c*c%p;
	if(b%2==1) c=c*a%p;
	return c%p;	
}
inline int read()
{
	int x=0,f=1;char ch=getchar();
	while (ch<'0'||ch>'9'){if (ch=='-') f=-1;ch=getchar();}
	while (ch>='0'&&ch<='9'){x=x*10+ch-48;ch=getchar();}
	return x*f;
}
int a[10005];
bitset<1000005> dp[2];//注意数值有正有负，二倍数组
signed main()
{
	//freopen("filename.in", "r", stdin);
	//freopen("filename.out", "w", stdout);
	srand(time(0));
	int n=read();
	for(int i=1;i<=n;i++) a[i]=read();
	random_shuffle(a+1,a+n+1);
	dp[0][0]=1;
	for(int i=1;i<=n;i++){
		int f=i&1^1;
		int g=i&1;
		dp[g]=(dp[f]<<a[i])|(dp[f]>>a[i]);
		for(int j=a[i];j;j--){
			dp[g][a[i]-j]=dp[g][a[i]-j]|dp[f][j];
		}
	}
	for(int i=0;i<=1000000;i++){
		if(dp[n&1][i]){
			cout<<i<<'\n';
			return 0;
		}
	}
	return 0;
}

T4 The Maximum Prefix

计数Dp

反转进行转移

T5 [ARC170C] Prefix Mex Sequence

考虑有几种数

#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<cmath>
#include<algorithm>
#include<iomanip>
#include<bits/stdc++.h>
#define int long long
#define jiaa(a,b) {a+=b;if(a>=MOD) a-=MOD;}
#define jian(a,b) {a-=b;if(a<0) a+=MOD;}
using namespace std;
int ksm(int a,int b,int p){
	if(b==0) return 1;
	if(b==1) return a%p;
	int c=ksm(a,b/2,p);
	c=c*c%p;
	if(b%2==1) c=c*a%p;
	return c%p;	
}
inline int read()
{
	int x=0,f=1;char ch=getchar();
	while (ch<'0'||ch>'9'){if (ch=='-') f=-1;ch=getchar();}
	while (ch>='0'&&ch<='9'){x=x*10+ch-48;ch=getchar();}
	return x*f;
}
const int MOD=998244353;
int s[5005],ans;
int dp[5005][5005];
signed main()
{
	//freopen("filename.in", "r", stdin);
	//freopen("filename.out", "w", stdout);
	int n=read(),m=read();
	for(int i=1;i<=n;i++) s[i]=read();
	if(m>=n){
		ans=1;
		for(int i=1;i<=n;i++){
			if(!s[i]) ans=ans*m%MOD;
		}
		cout<<ans<<'\n';
		return 0;
	}
	dp[0][0]=1;
	for(int i=1;i<=n;i++){
		for(int j=1;j<=m+1;j++){
			if(s[i]) dp[i][j]=dp[i-1][j-1];
			else{
				dp[i][j]=dp[i-1][j]*j%MOD+dp[i-1][j-1]*(m-j+1)%MOD;
				dp[i][j]%=MOD;
			}
		}
	}
	for(int j=1;j<=m+1;j++){
		ans+=dp[n][j];
		ans%=MOD;
	}
	cout<<ans<<'\n';
	return 0;
}

T6 [COCI 2021/2022 #2] Magneti

#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<cmath>
#include<algorithm>
#include<iomanip>
#include<bits/stdc++.h>
#define int long long
#define jiaa(a,b) {a+=b;if(a>=MOD) a-=MOD;}
#define jian(a,b) {a-=b;if(a<0) a+=MOD;}
using namespace std;
const int MOD=1000000007;
int ksm(int a,int b,int p){
	if(b==0) return 1;
	if(b==1) return a%p;
	int c=ksm(a,b/2,p);
	c=c*c%p;
	if(b%2==1) c=c*a%p;
	return c%p;	
}
inline int read()
{
	int x=0,f=1;char ch=getchar();
	while (ch<'0'||ch>'9'){if (ch=='-') f=-1;ch=getchar();}
	while (ch>='0'&&ch<='9'){x=x*10+ch-48;ch=getchar();}
	return x*f;
}
int fac[5000006],infac[5000006],inv[5000006];
int C(int n,int m){
	return fac[n]*infac[m]%MOD*infac[n-m]%MOD;
}
int r[55],dp[55][55][10005];
signed main()
{
	//freopen("filename.in", "r", stdin);
	//freopen("filename.out", "w", stdout);
	int N=5000000;
	fac[0]=1;
	for(int i=1;i<=N;i++){
		fac[i]=fac[i-1]*i%MOD;
		fac[i]%=MOD;
	}
	infac[N]=ksm(fac[N],MOD-2,MOD);
	for(int i=N-1;i>=0;i--){
		infac[i]=infac[i+1]*(i+1)%MOD;
	}
	int n=read(),l=read();
	for(int i=1;i<=n;i++){
		r[i]=read();
	}
	sort(r+1,r+n+1);
	dp[0][0][0]=1;
	for(int i=1;i<=n;i++){
		for(int j=1;j<=i;j++){
			for(int k=1;k<=l;k++){
				dp[i][j][k]+=dp[i-1][j-1][k-1];
				dp[i][j][k]%=MOD;
				if(k>=r[i]) dp[i][j][k]+=2*j%MOD*dp[i-1][j][k-r[i]]%MOD;
				dp[i][j][k]%=MOD;
				if(k>2*r[i]) dp[i][j][k]+=j*(j+1)%MOD*dp[i-1][j+1][k-2*r[i]+1]%MOD;
				dp[i][j][k]%=MOD;
			}
		}
	}
	int ans=0;
	for(int i=1;i<=l;i++){
		ans+=dp[n][1][i]*C(l-i+n,n)%MOD;;
		ans%=MOD;
	}
	cout<<ans<<'\n';
	return 0;
}

T7 [ARC146E] Simple Speed

值域Dp

区间连续段DP

T8 [CSP-S 2021] 括号序列

开三个DP数组

T9 And Reachability

考虑拆位进行DP

#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<cmath>
#include<algorithm>
#include<iomanip>
#include<bits/stdc++.h>
#define int long long
#define jiaa(a,b) {a+=b;if(a>=MOD) a-=MOD;}
#define jian(a,b) {a-=b;if(a<0) a+=MOD;}
using namespace std;
int ksm(int a,int b,int p){
	if(b==0) return 1;
	if(b==1) return a%p;
	int c=ksm(a,b/2,p);
	c=c*c%p;
	if(b%2==1) c=c*a%p;
	return c%p;	
}
inline int read()
{
	int x=0,f=1;char ch=getchar();
	while (ch<'0'||ch>'9'){if (ch=='-') f=-1;ch=getchar();}
	while (ch>='0'&&ch<='9'){x=x*10+ch-48;ch=getchar();}
	return x*f;
}
int a[300005];
int pos[300005][25],dp[300005][25];
signed main()
{
	//freopen("filename.in", "r", stdin);
	//freopen("filename.out", "w", stdout);
	int n=read(),q=read();
	for(int i=1;i<=n;i++) a[i]=read();
	for(int i=1;i<=n;i++){
		for(int j=0;j<=20;j++){
			if(a[i-1]>>j&1){
				pos[i][j]=i-1;
			}
			else pos[i][j]=pos[i-1][j];
		}
	}
	for(int i=1;i<=n;i++){
		for(int j=0;j<=20;j++){
			for(int k=0;k<=20;k++){
				if(!(a[i]>>k&1)) continue;
				if(a[pos[i][k]]>>j&1) dp[i][j]=max(dp[i][j],pos[i][k]);
				dp[i][j]=max(dp[i][j],dp[pos[i][k]][j]);
			}
		}
	}
	while(q--){
		int x=read(),y=read();
		int ff=0;
		for(int j=0;j<=20;j++){
			if(dp[y][j]>=x&&a[x]>>j&1){
				cout<<"Shi"<<'\n';
				ff=1;
				break;
			}
		}
		if(!ff) cout<<"Fou"<<'\n';
	}
	return 0;
}

T10

预处理+倍增询问

T11 [NOI Online #3 提高组] 魔法值

转化成矩阵快速幂

二进制拆位

T12 「JZOI-1」旅行