Python – Determine the Minimum Operations Required to Make an Array a Palindrome

This tutorial explains how to determine the minimum number of operations required to transform an array into a palindrome. In this problem, an operation is defined as merging two adjacent elements by summing them up. The goal is to make the array read the same forwards and backwards with the fewest possible operations.

Problem Statement

Given an array of positive integers, you are allowed to perform the following operation: merge two adjacent elements (i.e., replace them with their sum). Determine the minimum number of merge operations required so that the array becomes a palindrome.

Sample Input and Output

Example 1:

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# Sample Input
arr = [3, 2, 3]

# Sample Output
# The array is already a palindrome, so no operations are needed.
Output: 0

Example 2:

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# Sample Input
arr = [1, 4, 5, 1]

# Sample Output
# Merge 4 and 5 to get [1, 9, 1], which is a palindrome.
Output: 1

Solution Approach

The problem can be solved using the two-pointer technique with the following steps:

  1. Initialize two pointers: one at the start (left) and one at the end (right) of the array.
  2. Compare the elements:
    • If arr[left] is equal to arr[right], move both pointers inward (left++ and right--).
    • If arr[left] is less than arr[right], merge arr[left] with the adjacent element (arr[left+1]). Increase the merge operation count and move the left pointer to the right.
    • If arr[left] is greater than arr[right], merge arr[right] with the adjacent element (arr[right-1]). Increase the merge operation count and move the right pointer to the left.
  3. Repeat: Continue the process until the left pointer is no longer less than the right pointer.
  4. Result: The total count of merge operations is the answer.

Python Program

The following Python program implements the above approach using a two-pointer technique:

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def min_operations_to_palindrome(arr):
    """
    Determine the minimum merge operations required to make the array a palindrome.
    
    Parameters:
        arr (list): A list of positive integers.
        
    Returns:
        int: The minimum number of operations required.
    """
    operations = 0
    left, right = 0, len(arr) - 1

    # Process the array until the two pointers meet
    while left < right:
        # If the current elements are equal, move both pointers inward
        if arr[left] == arr[right]:
            left += 1
            right -= 1
        # If the left element is smaller, merge it with its adjacent element
        elif arr[left] < arr[right]:
            arr[left + 1] += arr[left]
            left += 1
            operations += 1
        # If the right element is smaller, merge it with its adjacent element
        else:
            arr[right - 1] += arr[right]
            right -= 1
            operations += 1

    return operations

# Testing the function with sample inputs

# Example 1
arr1 = [3, 2, 3]
print("Input:", [3, 2, 3])
print("Minimum operations required:", min_operations_to_palindrome(arr1))
# Expected Output: 0

# Example 2
arr2 = [1, 4, 5, 1]
print("\nInput:", [1, 4, 5, 1])
print("Minimum operations required:", min_operations_to_palindrome(arr2))
# Expected Output: 1

Conclusion

This tutorial demonstrated how to determine the minimum number of merge operations required to make an array a palindrome. We used a two-pointer approach, which is efficient with a time complexity of O(n) and works in-place. This problem is a common interview question in data structures and algorithms, and understanding it will help you build a strong foundation for solving similar problems.