Arithmetic Operations
Description
You are given an array of integers $a_1, a_2, \ldots, a_n$.
You can do the following operation any number of times (possibly zero):
- Choose any index $i$ and set $a_i$ to any integer (positive, negative or $0$).
What is the minimum number of operations needed to turn $a$ into an arithmetic progression? The array $a$ is an arithmetic progression if $a_{i+1}-a_i=a_i-a_{i-1}$ for any $2 \leq i \leq n-1$.
The first line contains a single integer $n$ ($1 \le n \le 10^5$).
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^5$).
Print a single integer: the minimum number of operations needed to turn $a$ into an arithmetic progression.
Input
The first line contains a single integer $n$ ($1 \le n \le 10^5$).
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^5$).
Output
Print a single integer: the minimum number of operations needed to turn $a$ into an arithmetic progression.
Samples
9
3 2 7 8 6 9 5 4 1
6
14
19 2 15 8 9 14 17 13 4 14 4 11 15 7
10
10
100000 1 60000 2 20000 4 8 16 32 64
7
4
10000 20000 10000 1
2
Note
In the first test, you can get the array $a = [11, 10, 9, 8, 7, 6, 5, 4, 3]$ by performing $6$ operations:
- Set $a_3$ to $9$: the array becomes $[3, 2, 9, 8, 6, 9, 5, 4, 1]$;
- Set $a_2$ to $10$: the array becomes $[3, 10, 9, 8, 6, 9, 5, 4, 1]$;
- Set $a_6$ to $6$: the array becomes $[3, 10, 9, 8, 6, 6, 5, 4, 1]$;
- Set $a_9$ to $3$: the array becomes $[3, 10, 9, 8, 6, 6, 5, 4, 3]$;
- Set $a_5$ to $7$: the array becomes $[3, 10, 9, 8, 7, 6, 5, 4, 3]$;
- Set $a_1$ to $11$: the array becomes $[11, 10, 9, 8, 7, 6, 5, 4, 3]$.
$a$ is an arithmetic progression: in fact, $a_{i+1}-a_i=a_i-a_{i-1}=-1$ for any $2 \leq i \leq n-1$.
There is no sequence of less than $6$ operations that makes $a$ an arithmetic progression.
In the second test, you can get the array $a = [-1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38]$ by performing $10$ operations.
In the third test, you can get the array $a = [100000, 80000, 60000, 40000, 20000, 0, -20000, -40000, -60000, -80000]$ by performing $7$ operations.