Preparando MOJI
James Bond, Johnny's favorite secret agent, has a new mission. There are $$$n$$$ enemy bases, each of them is described by its coordinates so that we can think about them as points in the Cartesian plane.
The bases can communicate with each other, sending a signal, which is the ray directed from the chosen point to the origin or in the opposite direction. The exception is the central base, which lies at the origin and can send a signal in any direction.
When some two bases want to communicate, there are two possible scenarios. If they lie on the same line with the origin, one of them can send a signal directly to the other one. Otherwise, the signal is sent from the first base to the central, and then the central sends it to the second base. We denote the distance between two bases as the total Euclidean distance that a signal sent between them has to travel.
Bond can damage all but some $$$k$$$ bases, which he can choose arbitrarily. A damaged base can't send or receive the direct signal but still can pass it between two working bases. In particular, James can damage the central base, and the signal can still be sent between any two undamaged bases as before, so the distance between them remains the same. What is the maximal sum of the distances between all pairs of remaining bases that 007 can achieve by damaging exactly $$$n - k$$$ of them?
The first line contains two integers $$$n$$$ and $$$k$$$ $$$(2 \leq k \leq n \leq 5 \cdot 10^5)$$$ — the total number of bases and number of bases that have to remain, respectively.
Each of the next $$$n$$$ lines contains two integers $$$x$$$ and $$$y$$$ $$$(-10^9 \leq x, y \leq 10^9)$$$, $$$i$$$-th line contains coordinates of the $$$i$$$-th base. You can assume that no two points coincide and that one of them is $$$(0, 0)$$$.
You should output one number — the maximal possible sum of distances between all pairs of some $$$k$$$ from given bases. Your answer will be accepted if the absolute or relative error is less than $$$10^{-6}$$$.
6 2 0 0 1 1 2 2 3 3 0 1 0 2
6.24264069
6 5 0 0 1 1 2 2 3 3 0 1 0 2
32.62741700
13 10 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 0 -2 0 1 0 2
237.00000000
10 5 2 2 4 4 3 5 6 10 0 5 0 0 5 0 10 0 0 10 4 7
181.52406315
In the first example, in an optimal solution Bond doesn't destroy bases with indices $$$4$$$ and $$$6$$$ (marked in orange):
The following picture represents an optimal solution for the second example. These bases are are not destroyed: $$$2$$$, $$$3$$$, $$$4$$$, $$$5$$$, $$$6$$$ (marked in orange).
An optimal solution for the third test is visible in the picture. Only bases $$$3$$$, $$$4$$$, $$$5$$$ are destroyed. Again, the not destroyed bases are marked in orange.
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Provedor Codeforces
Código CF1361D
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Datas 09/05/2023 10:03:14
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