Galaxy
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 262144/262144 K (Java/Others)Total Submission(s): 323 Accepted Submission(s): 71
Special Judge
Problem Description
Good news for us: to release the financial pressure, the government started selling galaxies and we can buy them from now on! The first one who bought a galaxy was Tianming Yun and he gave it to Xin Cheng as a present.
To be fashionable, DRD also bought himself a galaxy. He named it Rho Galaxy. There are n stars in Rho Galaxy, and they have the same weight, namely one unit weight, and a negligible volume. They initially lie in a line rotating around their center of mass.
Everything runs well except one thing. DRD thinks that the galaxy rotates too slow. As we know, to increase the angular speed with the same angular momentum, we have to decrease the moment of inertia.
The moment of inertia I of a set of n stars can be calculated with the formula
where wi is the weight of star i, di is the distance form star i to the mass of center.
As DRD’s friend, ATM, who bought M78 Galaxy, wants to help him. ATM creates some black holes and white holes so that he can transport stars in a negligible time. After transportation, the n stars will also rotate around their new center of mass. Due to financial pressure, ATM can only transport at most k stars. Since volumes of the stars are negligible, two or more stars can be transported to the same position.
Now, you are supposed to calculate the minimum moment of inertia after transportation.
To be fashionable, DRD also bought himself a galaxy. He named it Rho Galaxy. There are n stars in Rho Galaxy, and they have the same weight, namely one unit weight, and a negligible volume. They initially lie in a line rotating around their center of mass.
Everything runs well except one thing. DRD thinks that the galaxy rotates too slow. As we know, to increase the angular speed with the same angular momentum, we have to decrease the moment of inertia.
The moment of inertia I of a set of n stars can be calculated with the formula
where wi is the weight of star i, di is the distance form star i to the mass of center.
As DRD’s friend, ATM, who bought M78 Galaxy, wants to help him. ATM creates some black holes and white holes so that he can transport stars in a negligible time. After transportation, the n stars will also rotate around their new center of mass. Due to financial pressure, ATM can only transport at most k stars. Since volumes of the stars are negligible, two or more stars can be transported to the same position.
Now, you are supposed to calculate the minimum moment of inertia after transportation.
Input
The first line contains an integer T (T ≤ 10), denoting the number of the test cases.
For each test case, the first line contains two integers, n(1 ≤ n ≤ 50000) and k(0 ≤ k ≤ n), as mentioned above. The next line contains n integers representing the positions of the stars. The absolute values of positions will be no more than 50000.
For each test case, the first line contains two integers, n(1 ≤ n ≤ 50000) and k(0 ≤ k ≤ n), as mentioned above. The next line contains n integers representing the positions of the stars. The absolute values of positions will be no more than 50000.
Output
For each test case, output one real number in one line representing the minimum moment of inertia. Your answer will be considered correct if and only if its absolute or relative error is less than 1e-9.
Sample Input
2 3 2 -1 0 1 4 2 -2 -1 1 2
Sample Output
0 0.5
Source
题解及代码：
现场赛的时候，以为是一个贪心，策略搞了半天，也没能ac，绝望之时把o(nk)的暴力对拍程序交了一发，ac了.....无语，可能是时限开的比較大的缘故吧。
之后推了推公式，发现是一个很easy的数学题目，仅仅要o(n)的复杂度就能够了。
推导过程见：点击打开链接
#include <iostream>
#include <cstdio>
#include <algorithm>
using namespace std;
long long s[50050];
//对拍程序，请忽略
void solve(int n,int k)
{
double sum=0;
for(int i=1;i<=k;i++)
{
sum+=s[i];
}
int l=1,r=k;
double ans=-1;
for(int i=k;i<=n;i++)
{
double avg=sum/k,t=0;
for(int j=l;j<=r;j++)
{
t+=(s[j]-avg)*(s[j]-avg);
}
l++,r++;
sum=sum-s[l-1]+s[r];
if(i==k||ans>t) ans=t;
}
printf("%.12lf
",ans);
}
int main()
{
int cas,n,k;
scanf("%d",&cas);
while(cas--)
{
long long suml=0,sumr=0,ans=-1,t=0;
scanf("%d%d",&n,&k);
k=n-k;
for(int i=1;i<=n;i++)
{
scanf("%I64d",&s[i]);
}
if(k==0||k==1)
{
printf("0.000000000000
");
continue;
}
s[n+1]=0;
sort(s+1,s+n+1);
for(int i=1;i<=k;i++)
{
suml=suml+s[i],sumr=sumr+s[i]*s[i];
}
for(int i=k;i<=n;i++)
{
t=sumr*k-suml*suml;
if(i==k||ans>t) ans=t;
suml=suml-s[i-k+1]+s[i+1];
sumr=sumr-s[i-k+1]*s[i-k+1]+s[i+1]*s[i+1];
}
//solve(n,k);
printf("%.12lf
",(double)ans/(double)k);
}
return 0;
}