Product of Array Except Self: Optimized Approach

The Product of Array Except Self is a popular coding problem that challenges your skills in array manipulation and efficient algorithm design. Instead of using division, you must find a smart way to compute the result. Handling prefix products, suffix products, zero handling, and ensuring constant space usage make this an exciting problem, especially in coding interviews.

In this lesson, we will explore the most efficient techniques, address important edge cases, and provide a step-by-step Python implementation.

Problem Statement

Given an array of integers, return a new array where each element at index i is the product of all numbers in the original array except the one at i, without using division.

Example:
Input: [1,2,3,4]
Output: [24,12,8,6]

Here, smart array manipulation and multiplication are necessary to achieve the desired output.

Brute Force Approach

The naive algorithm would involve nested loops:

• For each element, multiply all other elements except the current one.
• This results in a time complexity of O(n²), making it inefficient for large inputs.

Why it is inefficient:

• High time complexity
• Poor scalability
• Repeated multiplication operations

Thus, we must move toward better optimization.

Optimized Approach Using Prefix and Suffix Products

A powerful strategy involves precomputing prefix products and suffix products:

• Prefix products: Product of all elements before the current index.
• Suffix products: Product of all elements after the current index.

Algorithm Steps:

1. Create two auxiliary arrays: prefix and suffix.
2. Perform linear traversal to populate these arrays.
3. For each index, multiply the corresponding prefix and suffix values.

This technique reduces the time complexity to O(n), though initially, the space complexity is O(n) due to the extra arrays.

Further Optimization: Constant Space Approach

To achieve constant space (excluding the output array), we can avoid explicit prefix and suffix arrays:

• Initialize the output array with 1s.
• Perform a linear traversal from left to right to fill in the running prefix products.
• Perform another linear traversal from right to left, updating the output with the suffix products on the fly.

This dual-pass strategy fits well into divide and conquer thinking, breaking the array into parts and combining results efficiently.

Python Implementation

Here’s a clean Python implementation using constant extra space:

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def productExceptSelf(nums):
    n = len(nums)
    output = [1] * n
    
    # Prefix products
    prefix = 1
    for i in range(n):
        output[i] = prefix
        prefix *= nums[i]
        
    # Suffix products
    suffix = 1
    for i in range(n - 1, -1, -1):
        output[i] *= suffix
        suffix *= nums[i]
        
    return output

This code performs a linear traversal twice, achieving O(n) time complexity and O(1) space complexity aside from the output.

Edge Cases to Consider

When solving the Product of Array Except Self, proper zero handling is crucial:

• If there are multiple zeros, all products will be zero.
• If there is only one zero, only the index corresponding to the zero will have a non-zero product.

Other edge cases:

• Arrays with a single element
• Arrays containing very large numbers

Effective optimization must account for these situations.

Conclusion

The Product of Array Except Self problem is an excellent example of efficient array manipulation using prefix products and suffix products. By mastering divide and conquer thinking, linear traversal, and zero handling, you can significantly improve time complexity and space complexity while maintaining readable, efficient code.

Always focus on correct Python implementation, rigorous optimization, and careful consideration of edge cases to ace this problem in interviews.