Problem Description: 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
.