Almost Arithmetic Progression

Polycarp 喜欢等差数列，对于一个数列 $ [a_1,a_2,a_3,\dots,a_n] $，如果对于每个 $ i\,(1\le i < n) $，$ a_{i+1}-a_i $ 的值均相同，那么这个数列就是一个等差数列。例如，$ [42] $ ，$ [5,5,5] $ ，$ [2,11,20,29] $ 和 $ [3,2,1,0] $ 是等差数列，但是 $ [1,0,1] $，$ [1,3,9] $ 和 $ [2,3,1] $ 不是。 根据定义，任何长度为 $ 1 $ 或 $ 2 $ 的数列都是等差数列。 Polycarp 找到了一个只包含正整数的数列 $ [b_1,b_2,b_3,\dots,b_n] $ 。他只想最多将每个元素改变一次。换句话说，对于每个元素，有三种选项：自增 $ 1 $，自减 $ 1 $ 或者不变。 你需要找到对 $ b $ 中元素操作（每个只能操作一次）的最小次数，使 $ b $ 变成一个等差数列。 操作完成后的数列可以含有 $ 0 $ 。 **【输入格式】** 第 $ 1 $ 行有 $ 1 $ 个整数 $ n\,(1\le n\le 100\,000) $ ，表示 $ b $ 中元素的个数。 第 $ 2 $ 行包含一个数列 $ b_1,b_2,b_3,\dots,b_n \,(1\le b_i \le 10^9)$。 **【输出格式】** 如果无法使用上述的操作将数列变为等差数列，输出 $ -1 $ 。在另一种情况下，输出一个非负整数，表示要将 $ b $ 变为等差数列所需要改变的元素的个数。唯一允许的操作是元素的自增 $/$ 减（不能够对一个元素操作两次）。 **【说明/提示】** 在样例 $ 1 $ 中，Polycarp 应该将第 $ 1 $ 个元素加 $ 1 $ ，将第 $ 2 $ 个元素加 $ 1 $ ，将第 $ 3 $ 个元素加 $ 1 $ ，并且不改变第 $ 4 $ 个元素。所以，Polycarp 操作之后的数列为 $ [25,20,15,10] $ 。 在样例 $ 2 $ 中，Polycarp 不应进行操作，因为他的数列已经是等差数列了。 在样例 $ 3 $ 中，不可能通过上述的操作将数列变为等差数列。 在样例 $ 4 $ 中，Polycarp 应该只将第 $ 1 $ 个元素加 $ 1 $ 。操作之后的数列为 $ [0,3,6,9,12] $ 。

Polycarp likes arithmetic progressions. A sequence $ [a_1, a_2, \dots, a_n] $ is called an arithmetic progression if for each $ i $ ( $ 1 \le i < n $ ) the value $ a_{i+1} - a_i $ is the same. For example, the sequences $ [42] $ , $ [5, 5, 5] $ , $ [2, 11, 20, 29] $ and $ [3, 2, 1, 0] $ are arithmetic progressions, but $ [1, 0, 1] $ , $ [1, 3, 9] $ and $ [2, 3, 1] $ are not. It follows from the definition that any sequence of length one or two is an arithmetic progression. Polycarp found some sequence of positive integers $ [b_1, b_2, \dots, b_n] $ . He agrees to change each element by at most one. In the other words, for each element there are exactly three options: an element can be decreased by $ 1 $ , an element can be increased by $ 1 $ , an element can be left unchanged. Determine a minimum possible number of elements in $ b $ which can be changed (by exactly one), so that the sequence $ b $ becomes an arithmetic progression, or report that it is impossible. It is possible that the resulting sequence contains element equals $ 0 $ .

The first line contains a single integer $ n $ $ (1 \le n \le 100\,000) $ — the number of elements in $ b $ . The second line contains a sequence $ b_1, b_2, \dots, b_n $ $ (1 \le b_i \le 10^{9}) $ .

If it is impossible to make an arithmetic progression with described operations, print -1. In the other case, print non-negative integer — the minimum number of elements to change to make the given sequence becomes an arithmetic progression. The only allowed operation is to add/to subtract one from an element (can't use operation twice to the same position).

4
24 21 14 10

3

2
500 500

0

3
14 5 1

-1

5
1 3 6 9 12

1

In the first example Polycarp should increase the first number on $ 1 $ , decrease the second number on $ 1 $ , increase the third number on $ 1 $ , and the fourth number should left unchanged. So, after Polycarp changed three elements by one, his sequence became equals to $ [25, 20, 15, 10] $ , which is an arithmetic progression. In the second example Polycarp should not change anything, because his sequence is an arithmetic progression. In the third example it is impossible to make an arithmetic progression. In the fourth example Polycarp should change only the first element, he should decrease it on one. After that his sequence will looks like $ [0, 3, 6, 9, 12] $ , which is an arithmetic progression.