Arrays are fundamental in computer science, serving as the backbone for countless algorithms and data structures. Often, problems involving arrays require us to transform their elements based on specific rules. This article delves into an intriguing array transformation problem: repeatedly identifying adjacent numbers that share common factors (i.e., are non-coprime) and replacing them with their Least Common Multiple (LCM), continuing until no such adjacent pair remains. We’ll unpack the problem, clarify the underlying mathematical principles, detail an effective algorithmic approach, and discuss its efficiency.

The Core Problem Explained

You’re presented with an array of integers, nums. Your objective is to process this array by repeatedly performing a specific operation: locate any two adjacent numbers that are “non-coprime” (meaning their Greatest Common Divisor, or GCD, is greater than 1). Once such a pair is found, remove them and insert their Least Common Multiple (LCM) in their place. This iterative merging process continues until no adjacent non-coprime pairs can be found anywhere in the array. The final state of the array is your solution.

Decoding GCD and LCM

To tackle this challenge, a firm grasp of two elementary number theory concepts is crucial:

• Greatest Common Divisor (GCD): This is the largest positive integer that divides two or more integers without leaving a remainder. For example, the GCD of 12 and 18 is 6.
• Least Common Multiple (LCM): This is the smallest positive integer that is a multiple of two or more integers. For instance, the LCM of 12 and 18 is 36.
• The Non-Coprime Condition: Two numbers are considered non-coprime if their GCD is greater than 1. If their GCD is exactly 1, they are coprime. The relationship between GCD and LCM is often expressed as: LCM(a, b) = (a * b) / GCD(a, b).

Illustrative Example: A Step-by-Step Journey

Let’s trace the process with an example input: [6, 4, 3, 2, 7, 6, 2]

1. Initial Array: [6, 4, 3, 2, 7, 6, 2]
2. Examine (6, 4): Their GCD is 2 (since 2 > 1, they are non-coprime). Their LCM is (6 * 4) / 2 = 12.
   Array becomes: [12, 3, 2, 7, 6, 2]
3. Examine (12, 3): Their GCD is 3. Their LCM is (12 * 3) / 3 = 12.
   Array becomes: [12, 2, 7, 6, 2]
4. Examine (12, 2): Their GCD is 2. Their LCM is (12 * 2) / 2 = 12.
   Array becomes: [12, 7, 6, 2]
5. Examine (7, 6): Their GCD is 1. They are coprime, so no merge occurs.
6. Examine (6, 2): Their GCD is 2. Their LCM is (6 * 2) / 2 = 6.
   Array becomes: [12, 7, 6]
7. Final Check:
   • (12, 7): GCD is 1 (coprime).
   • (7, 6): GCD is 1 (coprime).
     No more adjacent non-coprime pairs exist.

Final Output: [12, 7, 6]

The Stack-Based Approach

The key to efficiently solving this problem lies in using a stack. A stack is a Last-In, First-Out (LIFO) data structure, perfect for maintaining the “current” sequence of numbers and checking newly formed adjacent pairs.

Here’s the algorithm:

1. Initialize an empty stack. This stack will store the numbers as they are processed and potentially merged.
2. Iterate through each number n in the input array nums:
   a. Push n onto the stack.
   b. Continuously check and merge: As long as the stack contains at least two elements, inspect the top two numbers.
   i. Calculate their GCD.
   ii. If their GCD is greater than 1 (meaning they are non-coprime):
   * Pop both numbers from the stack.
   * Calculate their LCM.
   * Push the calculated LCM back onto the stack.
   iii. If their GCD is 1 (coprime), then these two numbers cannot be merged, and the inner merging loop for the current element n terminates. Proceed to the next number in nums.
3. Return the stack: Once all numbers from the input array nums have been processed, the elements remaining in the stack represent the final reduced array.

This stack-based method ensures that whenever a new number is introduced or an LCM is formed, it’s immediately checked against its new adjacent neighbor (the element just below it in the stack), mimicking the continuous replacement process.

Performance Analysis

Understanding the efficiency of an algorithm is crucial.

• Time Complexity: O(N * log(Max_Value))
  • We iterate through the input array once, which is O(N) operations.
  • Within each iteration, we might perform multiple merges. However, the total number of merges across the entire process is bounded by O(N) because each merge reduces the number of elements in the conceptual array (or stack) by one.
  • The most computationally intensive part of each merge is calculating the GCD, which takes O(log(Max_Value)) time (using the Euclidean algorithm), where Max_Value is the largest possible number in the array (which can grow due to LCM calculations).
  • Combining these, the overall time complexity is O(N * log(Max_Value)).
• Space Complexity: O(N)
  • In the worst-case scenario, if no numbers are ever non-coprime (e.g., all prime numbers), the stack will hold all N elements from the input array.
  • Therefore, the space complexity is proportional to the number of elements in the input array.

Key Takeaways

• The problem elegantly combines fundamental array manipulation with number theory concepts like GCD and LCM.
• A stack proves to be an ideal data structure for maintaining the “live” sequence of elements and efficiently processing adjacent pairs, enabling a continuous merging mechanism.
• The final reduced array is unique, regardless of the order in which non-coprime pairs are merged, thanks to the properties of LCM and GCD.
• This problem serves as an excellent illustration of how carefully chosen data structures and mathematical principles can lead to an efficient solution for complex transformations.