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Adnan owns the biggest shoe store in Baku. A box containing $n$ pairs of shoes has just arrived at the store. Each pair consists of two shoes of the same size: a left and a right one. Adnan has put all of the $2n$ shoes in a row consisting of $2n$ positions numbered $0$ through $2n-1$ from left to right.
Adnan wants to rearrange the shoes into a valid arrangement. An arrangement is valid if and only if for every $i$ ($0 \leq i \leq n-1$), the following conditions hold:
For this purpose, Adnan can make a series of swaps.
In each swap, he selects two shoes that are adjacent at that moment and exchanges them (i.e., picks them up and puts each one on the former position of the other shoe).
Two shoes are adjacent if their positions differ by one.
Determine the minimum number of swaps that Adnan needs to perform in order to obtain a valid arrangement of the shoes.
You should implement the following procedure:
int64 count_swaps(int S)
Consider the following call:
count_swaps([2, 1, -1, -2])
Adnan can obtain a valid arrangement in $4$ swaps.
For instance, he can first swap shoes $1$ and $-1$, then $1$ and $-2$, then $-1$ and $-2$, and finally $2$ and $-2$. He would then obtain the following valid arrangement: $[-2, 2, -1, 1]$. It is not possible to obtain any valid arrangement with less than $4$ swaps. Therefore, the procedure should return $4$.
In the following example, all the shoes have the same size:
count_swaps([-2, 2, 2, -2, -2, 2])
Adnan can swap the shoes at positions $2$ and $3$ to obtain the valid arrangement $[-2, 2, -2, 2, -2, 2]$, so the procedure should return $1$.
$n = 1$
$n \leq 8$
All the shoes are of the same size.
All shoes at positions $0, \ldots, n-1$ are left shoes, and all shoes at positions $n, \ldots, 2n-1$ are right shoes. Also, for each $i$ ($0 \leq i \leq n-1$), the shoes at positions $i$ and $i+n$ are of the same size.
$n \leq 1000$
No additional constraints.
C++17, C++14, C++20, C++14 (Clang), C++17 (Clang), C++20 (Clang)