codeforces#P1842E. Tenzing and Triangle
Tenzing and Triangle
Description
There are $n$ pairwise-distinct points and a line $x+y=k$ on a two-dimensional plane. The $i$-th point is at $(x_i,y_i)$. All points have non-negative coordinates and are strictly below the line. Alternatively, $0 \leq x_i,y_i, x_i+y_i < k$.
Tenzing wants to erase all the points. He can perform the following two operations:
- Draw triangle: Tenzing will choose two non-negative integers $a$, $b$ that satisfy $a+b<k$, then all points inside the triangle formed by lines $x=a$, $y=b$ and $x+y=k$ will be erased. It can be shown that this triangle is an isosceles right triangle. Let the side lengths of the triangle be $l$, $l$ and $\sqrt 2 l$ respectively. Then, the cost of this operation is $l \cdot A$.
The blue area of the following picture describes the triangle with $a=1,b=1$ with cost $=1\cdot A$.
- Erase a specific point: Tenzing will choose an integer $i$ that satisfies $1 \leq i \leq n$ and erase the point $i$. The cost of this operation is $c_i$.
Help Tenzing find the minimum cost to erase all of the points.
The first line of the input contains three integers $n$, $k$ and $A$ ($1\leq n,k\leq 2\cdot 10^5$, $1\leq A\leq 10^4$) — the number of points, the coefficient describing the hypotenuse of the triangle and the coefficient describing the cost of drawing a triangle.
The following $n$ lines of the input the $i$-th line contains three integers $x_i,y_i,c_i$ ($0\leq x_i,y_i,x_i+y_i< k$, $1\leq c_i\leq 10^4$) — the coordinate of the $i$-th points and the cost of erasing it using the second operation. It is guaranteed that the coordinates are pairwise distinct.
Output a single integer —the minimum cost needed to erase all of the points.
Input
The first line of the input contains three integers $n$, $k$ and $A$ ($1\leq n,k\leq 2\cdot 10^5$, $1\leq A\leq 10^4$) — the number of points, the coefficient describing the hypotenuse of the triangle and the coefficient describing the cost of drawing a triangle.
The following $n$ lines of the input the $i$-th line contains three integers $x_i,y_i,c_i$ ($0\leq x_i,y_i,x_i+y_i< k$, $1\leq c_i\leq 10^4$) — the coordinate of the $i$-th points and the cost of erasing it using the second operation. It is guaranteed that the coordinates are pairwise distinct.
Output
Output a single integer —the minimum cost needed to erase all of the points.
4 6 1
1 2 1
2 1 1
1 1 1
3 2 6
6 7 1
4 2 1
3 3 1
5 1 4
3 2 5
4 1 1
0 6 4
10 4 100
0 0 1
0 1 1
0 2 50
0 3 200
1 0 1
1 1 1
1 2 1
2 0 200
2 1 200
3 0 200
4
4
355
Note
The picture of the first example:
Tenzing do the following operations:
- draw a triangle with $a=3,b=2$, the cost $=1\cdot A=1$.
- erase the first point, the cost $=1$.
- erase the second point, the cost $=1$.
- erase the third point, the cost $=1$.
The picture of the second example:
