P3195 [HNOI2008]玩具装箱

[HNOI2008]玩具装箱

 

 f[i] =max( f[j]+  (s[i]-s[j] + i-j+1 )^2

 设 a[i] =fi + si ,b[i] =s[i]+i+1

 f[i] =max( f[j] + (a[i]-b[j])^2

 　　=max{ f[j]+a[j]^2+b[j]^2 -2*a[j]*b[j])

 

然后斜率优化： 考虑2个点j k, j<k ,  func(j)<func(k) ，得到斜率的柿子

 

 #include <iostream>
 #include <cmath>
 #include <cstdio>
 #include <algorithm>
 using namespace std;
  const int  N =5e4+2;
  #define int long long
  int f[N];
  int n,m, s[N];
  int q[N],hh,tt;
  
  int a(int i){
  	return s[i]+i;
  }
  int b(int i){
  	return a(i)+m+1;
  }
  int Y(int i){
  	return f[i]+b(i)*b(i);
  }
  long double slope(int i,int j){
  	return 1.0*(Y(i)-Y(j))/(b(i)-b(j));
  }
 signed main(){
 	cin>>n>>m;
 	int i;
 	for(i=1;i<=n;i++) cin>>s[i],s[i]+=s[i-1];
 	
 	hh=tt=1;
 	for(i=1;i<=n;++i){
 		while(hh<tt&&slope(q[hh],q[hh+1])<=2.0*a(i)) 
 		hh++;
 		f[i]=f[q[hh]]+(a(i)-b(q[hh]))*(a(i)-b(q[hh]));
 		while(hh<tt&&slope(q[tt-1],q[tt])>slope(q[tt],i))
 		tt--;
 		q[++tt]=i;
 	}
 	cout<<f[n]<<endl;
 }