Minimize steps to make all Array elements equal by cyclically shifting a segment or replacing prefix by maximum

Given an array arr[] consisting of N positive integers, the task is to print the minimum number of steps required to make the array such that all elements are equal by performing the following operations on the array any number of times (possibly 0).

• Operation-1: Select any prefix arr[1…k] such that max (arr[1], arr[2], …, arr[k]) = arr[k] and set arr[i] = arr[k] for all i in range [1,k−1].
• Operation-2: Select any segment [L, R] and shift the segment cyclically to the left keeping other elements unchanged.

Examples:

Input: arr[] = {1, 2, 12, 20, 18, 19}
Output: 2
Explanation: It can be solved in two steps:
In the first step, apply operation 2 by choosing L = 4 and R = 6 yields the sequence {1, 2, 12, 18, 19, 20}. 
In the second step, apply operation 1 on the prefix arr{1, 2, 12, 18, 19, 20} which converts arr into {20, 20, 20, 20, 20} where all elements are equal.

Input: arr[] = {2, 2, 2, 2}
Output: 0
Explanation: the elements in the array are all same.

Approach: The given problem can be solved by using the following observation:

• If all the elements are the same no operation needs to be done.
• If the maximum of the array is at last then only performing operation 1 for the whole array makes all the elements equal.
• If the maximum is somewhere before the last position, say j. Then performing operation 2 in segment [j, N] and after that operation 1 for the prefix [1, N] makes all the elements equal.

So based on the above observation, the answer can be 0, 1, or 2. Follow the steps mentioned below.

• Iterate the array and find the maximum from the array.
• If all the elements of the array are the same then return 0.
• If the maximum is present at the end of the array then the answer is 1 as shown in the observation.
• If these two are not the case then the answer is 2.

Below is the implementation of the above approach:

2

Time Complexity: O(N)
Auxiliary Space: O(1)