A14993. PermutationForces

You have a permutation $p$ of integers from $1$ to $n$ .

You have a strength of $s$ and will perform the following operation some times:

- Choose an index $i$ such that $1 \leq i \leq |p|$ and $|i-p_i| \leq s$ .
- For all $j$ such that $1 \leq j \leq |p|$ and $p_i<p_j$ , update $p_j$ to $p_j-1$ .
- Delete the $i$ -th element from $p$ . Formally, update $p$ to $[p_1,\ldots,p_{i-1},p_{i+1},\ldots,p_n]$ .

It can be shown that no matter what $i$ you have chosen, $p$ will be a permutation of integers from $1$ to $|p|$ after all operations.

You want to be able to transform $p$ into the empty permutation. Find the minimum strength $s$ that will allow you to do so.

The first line of input contains a single integer $n$ ( $1 \leq n \leq 5 \cdot 10^5$ ) — the length of the permutation $p$ .

The second line of input conatains $n$ integers $p_1, p_2, \ldots, p_n$ ( $1 \leq p_i \leq n$ ) — the elements of the permutation $p$ .

It is guaranteed that all elements in $p$ are distinct.

Print the minimum strength $s$ required.

In the first test case, the minimum $s$ required is $1$ .

Here is how we can transform $p$ into the empty permutation with $s=1$ :

- In the first move, you can only choose $i=2$ as choosing any other value of $i$ will result in $|i-p_i| \leq s$ being false. With $i=2$ , $p$ will be changed to $[2,1]$ .
- In the second move, you choose $i=1$ , then $p$ will be changed to $[1]$ .
- In the third move, you choose $i=1$ , then $p$ will be changed to $[~]$ .

It can be shown that with $s=0$ , it is impossible to transform $p$ into the empty permutation.