Problem:
Score: 500 points

Problem Statement

New AtCoder City has an infinite grid of streets, as follows:

At the center of the city stands a clock tower. Let (0, 0) be the coordinates of this point.

A straight street, which we will call East-West Main Street, runs east-west and passes the clock tower. It corresponds to the x-axis in the two-dimensional coordinate plane.

There are also other infinitely many streets parallel to East-West Main Street, with a distance of 1 between them. They correspond to the lines \ldots, y = -2, y = -1, y = 1, y = 2, \ldots in the two-dimensional coordinate plane.

A straight street, which we will call North-South Main Street, runs north-south and passes the clock tower. It corresponds to the y-axis in the two-dimensional coordinate plane.

There are also other infinitely many streets parallel to North-South Main Street, with a distance of 1 between them. They correspond to the lines \ldots, x = -2, x = -1, x = 1, x = 2, \ldots in the two-dimensional coordinate plane.

There are N residential areas in New AtCoder City. The i-th area is located at the intersection with the coordinates (X_i, Y_i) and has a population of P_i. Each citizen in the city lives in one of these areas.

The city currently has only two railroads, stretching infinitely, one along East-West Main Street and the other along North-South Main Street.

M-kun, the mayor, thinks that they are not enough for the commuters, so he decides to choose K streets and build a railroad stretching infinitely along each of those streets.

Let the walking distance of each citizen be the distance from his/her residential area to the nearest railroad.

M-kun wants to build railroads so that the sum of the walking distances of all citizens, S, is minimized.

For each K = 0, 1, 2, \dots, N, what is the minimum possible value of S after building railroads?

Constraints

1 \leq N \leq 15

-10 \ 000 \leq X_i \leq 10 \ 000

-10 \ 000 \leq Y_i \leq 10 \ 000

1 \leq P_i \leq 1 \ 000 \ 000

The locations of the N residential areas, (X_i, Y_i), are all distinct.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
X_1 Y_1 P_1
X_2 Y_2 P_2
: : :
X_N Y_N P_N

Output

Print the answer in N+1 lines.

The i-th line (i = 1, \ldots, N+1) should contain the minimum possible value of S after building railroads for the case K = i-1.

Sample Input 1

3
1 2 300
3 3 600
1 4 800

Sample Output 1

2900
900
0
0

When K = 0, the residents of Area 1, 2, and 3 have to walk the distances of 1, 3, and 1, respectively, to reach a railroad.

Thus, the sum of the walking distances of all citizens, S, is 1 \times 300 + 3 \times 600 + 1 \times 800 = 2900.

When K = 1, if we build a railroad along the street corresponding to the line y = 4 in the coordinate plane, the walking distances of the citizens of Area 1, 2, and 3 become 1, 1, and 0, respectively.

Then, S = 1 \times 300 + 1 \times 600 + 0 \times 800 = 900.

We have many other options for where we build the railroad, but none of them makes S less than 900.

When K = 2, if we build a railroad along the street corresponding to the lines x = 1 and x = 3 in the coordinate plane, all citizens can reach a railroad with the walking distance of 0, so S = 0. We can also have S = 0 when K = 3.

The figure below shows the optimal way to build railroads for the cases K = 0, 1, 2.

The street painted blue represents the roads along which we build railroads.

Sample Input 2

5
3 5 400
5 3 700
5 5 1000
5 7 700
7 5 400

Sample Output 2

13800
1600
0
0
0
0

The figure below shows the optimal way to build railroads for the cases K = 1, 2.

Sample Input 3

6
2 5 1000
5 2 1100
5 5 1700
-2 -5 900
-5 -2 600
-5 -5 2200

Sample Output 3

26700
13900
3200
1200
0
0
0

The figure below shows the optimal way to build railroads for the case K = 3.

Sample Input 4

8
2 2 286017
3 1 262355
2 -2 213815
1 -3 224435
-2 -2 136860
-3 -1 239338
-2 2 217647
-1 3 141903

Sample Output 4

2576709
1569381
868031
605676
366338
141903
0
0
0

The figure below shows the optimal way to build railroads for the case K = 4.

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include<bits/stdc++.h>
using namespace std;

typedef long long ll;

int n,x[15],y[15],p[15],D[15],d[15];
ll ans[16];

int main(){

    ios::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);

    cin>>n;
    for(int i=0;i<n;i++)cin>>x[i]>>y[i]>>p[i];
    memset(ans,0x3f,sizeof(ans));
    ans[0]=0;
    for(int i=0;i<n;i++){
        D[i]=min(abs(x[i]),abs(y[i]));
        ans[0]+=(ll)p[i]*D[i];
    }
    for(int s=0;s<1<<n;s++){
        for(int t=s;true;t=(t-1)&s){
            memcpy(d,D,sizeof(d));
            for(int i=0;i<n;i++){
                if(s>>i&1){
                    if(t>>i&1){
                        for(int j=0;j<n;j++){
                            d[j]=min(d[j],abs(x[i]-x[j]));
                        }
                    }else{
                        for(int j=0;j<n;j++){
                            d[j]=min(d[j],abs(y[i]-y[j]));
                        }
                    }
                }
            }
            ll cur=0;
            for(int i=0;i<n;i++){
                cur+=(ll)d[i]*p[i];
            }
            ans[__builtin_popcount(s)]=min(ans[__builtin_popcount(s)],cur);
            if(t==0)break;
        }
       /* {
            int t=0;
            memcpy(d,D,sizeof(d));
            for(int i=0;i<n;i++){
                if(s>>i&1){
                    // if(t>>i&1){
                    //     for(int j=0;j<n;j++){
                    //         d[j]=min(d[j],abs(x[i]-x[j]));
                    //     }
                    // }else{
                        for(int j=0;j<n;j++){
                            d[j]=min(d[j],abs(y[i]-y[j]));
                        }
                   // }
                }
            }
            ll cur=0;
            for(int i=0;i<n;i++){
                cur+=(ll)d[i]*p[i];
            }
            ans[__builtin_popcount(s)]=min(ans[__builtin_popcount(s)],cur);
        }*/
    }
    for(int i=0;i<=n;i++)cout<<ans[i]<<'\n';

    return 0;
}