Minimum number of operations to convert a given sequence into a Geometric Progression

Given a sequence of N elements, only three operations can be performed on any element at most one time. The operations are: 

1. Add one to the element.
2. Subtract one from the element.
3. Leave the element unchanged.

Perform any one of the operations on all elements in the array. The task is to find the minimum number of operations(addition and subtraction) that can be performed on the sequence, in order to convert it into a Geometric Progression. If it is not possible to generate a GP by performing the above operations, print -1.

Examples:  

Input: a[] = {1, 1, 4, 7, 15, 33} 
Output: The minimum number of operations are 4. 
Steps: 

1. Keep a₁ unchanged
2. Add one to a₂.
3. Keep a₃ unchanged
4. Subtract one from a₄.
5. Subtract one from a₅.
6. Add one to a₆.

The resultant sequence is {1, 2, 4, 8, 16, 32}

Input: a[] = {20, 15, 20, 15} 
Output: -1 
 

Approach The key observation to be made here is that any Geometric Progression is uniquely determined by only its first two elements (Since the ratio between each of the next pairs has to be the same as the ratio between this pair, consisting of the first two elements). Since only 3*3 permutations are possible. The possible combination of operations are (+1, +1), (+1, 0), (+1, -1), (-1, +1), (-1, 0), (-1, -1), (0, +1), (0, 0) and (0, -1). Using brute force all these 9 permutations and checking if they form a GP in linear time will give us the answer. The minimum of the operations which result in combinations which are in GP will be the answer. 

Below is the implementation of the above approach:  

Minimum Number of Operations are 2
The resultant sequence is:
8 20 50 125

Time Complexity : O(9*N)

Space Complexity: O(1)