Optimizing Array Computation: Product of Array Except Self

Feb 132 min read

Introduction

In this blog, we will explore an efficient algorithm to compute the product of an array except for the current element, without using division. This problem is a classic example of how we can optimize calculations using prefix and postfix traversal.

Problem Statement

Given an array arr, we need to construct an output array where each element at index i is the product of all elements in arr except arr[i].

Approach

We solve this problem using the prefix and postfix product method, which allows us to achieve a time complexity of O(n) and a space complexity of O(1) (excluding the output array). The key idea is:

Compute the prefix product for each index.
Compute the postfix product and update the output array in a single pass.

Python Implementation

Below is the Python implementation of the algorithm:

def prod_except_self(arr: list) -> list:
    print("######### Product of Array Except Self ######")
    array_length = len(arr)
    output_array = [1] * array_length  

    # Compute prefix products
    for i in range(1, array_length):
        output_array[i] = output_array[i-1] * arr[i-1]
    print("After Prefix Computation:", output_array)

    # Compute postfix products and update the output array
    post_fix = 1
    for j in range(array_length - 1, -1, -1):
        output_array[j] *= post_fix
        post_fix *= arr[j]
    
    print("Final Output:", output_array)
    return output_array  

# Test Case
prod_except_self([1, 2, 3, 4, 10, 2, 5])

Explanation

Step 1: Prefix Computation

We iterate through the array and store the cumulative product of elements before the current index in the output_array.

Example: For [1, 2, 3, 4], after prefix computation:

[1, 1, 2, 6]

Step 2: Postfix Computation

We traverse the array in reverse and multiply each element in output_array with the cumulative product of elements after the current index.

Example:

[1, 1, 2, 6] -> [24, 12, 8, 6]

Complexity Analysis

Time Complexity: O(n) (One pass for prefix and one for postfix computation)
Space Complexity: O(1) (We modify the output array in place)

Conclusion

This optimized approach efficiently calculates the product of an array except for the current element without using division. It demonstrates the power of prefix-postfix computation in solving array problems efficiently.

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