Ancient Books

문제

The city of Tehran is home to the National Library of Iran. The main treasure of this library is located in a long hall with a row of $nn$ tables, labeled $00$ through $n-1n-1$ from left to right. On each table there is one ancient handwritten book being displayed. These books are ordered based on their ages, which makes it hard for the visitors to search for books by title. So, the library manager has decided to sort the books in alphabetical order of their titles.

Aryan, a librarian, is going to do the job. He has created a list $pp$ of length $nn$, containing different integers from $00$ to $n-1n-1$. This list describes the changes needed to rearrange the books into alphabetical order: for all , the book that is currently on table $ii$ should be moved to table $p[i]p[i]$.

Aryan starts sorting the books at table $ss$. He wants to return to the same table after finishing the job. Since the books are very valuable, he cannot carry more than one book at any time. While sorting the books Aryan will perform a sequence of actions. Each of those actions has to be one of the following:

• If he is not carrying a book and there is a book on the table he is at, he can pick up this book.
• If he is carrying a book and there is another book on the table he is at, he can switch the book he is carrying with the one on the table.
• If he is carrying a book and he is at an empty table, he can put the book he is carrying on the table.
• He can walk to any table. He may carry a single book while he does so.

For all $0\le i,j\le n-10\; \backslash leq\; i,\; j\; \backslash leq\; n\; -\; 1$, the distance between tables $ii$ and $jj$ is precisely $\mathrm{\mid }j-i\mathrm{\mid }|j-i|$ meters. Your task is to compute the minimum total distance Aryan needs to walk in order to sort all the books.