Arithmetic Operations

5000ms

1048576K

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 $$$2 \leq i \leq n-1$$$.

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.

Sample Input:

9
3 2 7 8 6 9 5 4 1

Sample Output:

Sample Input:

14
19 2 15 8 9 14 17 13 4 14 4 11 15 7

Sample Output:

Sample Input:

10
100000 1 60000 2 20000 4 8 16 32 64

Sample Output:

Sample Input:

4
10000 20000 10000 1

Sample Output:

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.

Informação

Provedor Codeforces

Código CF1654E