PermutationForces
Description
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.
Input
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.
Output
Print the minimum strength $s$ required.
Samples
3
3 2 1
1
1
1
0
10
1 8 4 3 7 10 6 5 9 2
1
Note
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.