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: Time Limit Exceeded)

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

#define F first
#define S second
#define pb push_back
#define sz(c) (int)c.size()
#define vint vector<int>
#define vll vector<ll>
#define fr(it, a, b) \
    for (int it = (a); it < (b); it++)
typedef pair<int, int> pi;
typedef long long ll;
typedef long double lf;
const ll mod = 1e9 + 7;
ll cost(ll n, vector<pi> &p, vll &c, vll &x, vll &y)
{
    ll ans = 0;
    for (int i = 0; i < n; i++)
    {
        pi u = p[i];
        ll mn = 1e18;
        for (auto xi : x)
        {
            mn = min(mn, llabs(xi - u.F));
        }
        for (auto yi : y)
        {
            mn = min(mn, llabs(yi - u.S));
        }
        ans += mn * c[i];
    }
    return ans;
}
int32_t main()
{
    // freopen("input.txt", "r", stdin);
    // freopen("output.txt", "w", stdout);
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    ll n;
    cin >> n;
    vector<pi> p(n);
    vector<ll> c(n), val(n + 1, 1e18);
    fr(i, 0, n)
    {
        cin >> p[i].F >> p[i].S >> c[i];
    }
    bitset<20> a, b;
    ll l = 1 << n;
    for (int m = 0; m < l; ++m)
    {
        for (int s = m; s >= 0; s = (s - 1) & m)
        {
            // ... s and m...
            vector<ll> x(1, 0), y(1, 0);
            a = m, b = s;
            for (int i = 0; i < n; i++)
            {
                if (a[i])
                {
                    if (b[i])
                    {
                        x.pb(p[i].F);
                    }
                    else
                    {
                        y.pb(p[i].S);
                    }
                }
            }
            val[a.count()] = min(val[a.count()], cost(n, p, c, x, y));
            if (s == 0)
            {
                break;
            }
        }
    }
    fr(i, 0, n + 1)
    {
        cout << val[i] << "\n";
    }
    return 0;
}