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 Rust (1.42.0) (judge verdict: Time Limit Exceeded)

#![allow(non_snake_case,unused_imports,dead_code)]
use std::str::FromStr;
use std::collections::{HashMap,HashSet,VecDeque,BinaryHeap};
use std::cmp::{max,min,Reverse,Ordering};
use std::time::{Duration, Instant};
use rand::Rng;

use proconio::input;
use proconio::marker::{Chars,Usize1,Isize1};

fn main() {

    input!(
        N:usize,
        mut XYP:[(isize,isize,isize);N],
    );
    XYP.sort_by_key(|x| -x.2);

    /*
    let N = 15usize;
    let XYP = (0..N).map(|n| ((n+1) as isize,(n+1) as isize,1000000isize)).collect::<Vec<_>>();
    */

    let INF = 1e15 as isize;
    let mut ans = vec![INF;N+1];

    let mut V = vec![0;N];
    let mut X = vec![0;N+1];
    let mut Y = vec![0;N+1];

    for i in 0..(3usize.pow(N as u32)){
        for i in 0..N{
            V[i] = 0;
            X[i] = 0;
            Y[i] = 0;
        }
        X[N] = 0;
        Y[N] = 0;
        let mut iX = 1;
        let mut iY = 1;
        let mut n = 0;
        let mut k = 1;
        for j in 0..N{
            let v = i/k%3;
            V[j] = v;
            k *= 3;
            match v {
                1 => {
                    X[iX] = XYP[j].0;
                    iX += 1;
                    n += 1;
                }
                2 => {
                    Y[iY] = XYP[j].1;
                    iY += 1;
                    n += 1;
                }
                _ => {}
            }
        }
        let mut curr = 0;
        for j in 0..N{
            if V[j]!=0{continue;}
            let (a,b,p) = XYP[j];
            let mut tmp = 10000;
            
            for i in 0..iX{
                let x = X[i];
                tmp = min(tmp,(a-x).abs());
            }
            for i in 0..iY{
                let y = Y[i];
                tmp = min(tmp,(b-y).abs());
            }

            curr += tmp * p;
            if curr >= ans[n] {break;}
        }
        ans[n] = min(ans[n],curr);
    }

    for out in ans{
        println!("{}",out);
    }

}