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The task is to find the maximum product difference between two pairs in an integer array. The product difference between two pairs (a, b) and (c, d) is defined as (a * b) - (c * d).

Given an integer array nums, choose four distinct indices i, j, k, l such that the product difference between pairs (nums[i], nums[j]) and (nums[k], nums[l]) is maximized.

Return the maximum such product difference.

Clarifying Questions

1. Can the array nums contain negative numbers?
2. What is the minimum length of the array nums?

For the problem, let’s assume:

1. The array can contain negative numbers.
2. The array will have at least 4 elements, as having fewer elements would make it impossible to form two pairs.

Strategy

To find the maximum product difference:

1. Sort the array nums.
2. The two largest numbers in the sorted array (i.e., the last two elements of the sorted array) will form the pair for the maximum product.
3. The two smallest numbers (i.e., the first two elements of the sorted array) will form the pair for the minimum product.
4. Compute the product difference using these pairs.

Code

def maxProductDifference(nums):
    # Step 1: Sort the array
    nums.sort()
    
    # Step 2: Compute the product of the two largest elements
    max_product = nums[-1] * nums[-2]
    
    # Step 3: Compute the product of the two smallest elements
    min_product = nums[0] * nums[1]
    
    # Step 4: Calculate the product difference
    product_difference = max_product - min_product
    
    return product_difference

# Example usage:
nums = [5, 6, 2, 7, 4]
print(maxProductDifference(nums))  # Output: 34

Time Complexity

• Sorting the array will take (O(n \log n)), where (n) is the number of elements in the array.
• Accessing the four specific elements and computing the products each take (O(1)).

Thus, the overall time complexity is (O(n \log n)).

Space Complexity

• The space complexity is (O(1)) if we ignore the space used by the input and output, as we only use a constant amount of extra space for variable storage. The sorting operation may take additional space depending on the implementation, but this is typically (O(n)) for the timsort algorithm used in Python.

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