Product of Array Except Self

In many coding challenges, especially in array manipulation problems, one often encounters the need to compute the product of all elements in an array except the one at the current index. The productExceptSelf function is a classic example of such a problem, demonstrating an efficient approach without using division.

The Function

function productExceptSelf(nums: number[]): number[] {
    let res = new Array(nums.length).fill(nums[0]);

    let prefix = 1;
    for (let i = 0; i < nums.length; i++) {
        res[i] = prefix;
        prefix *= nums[i];
    }

    let postfix = 1;
    for (let i = nums.length - 1; i >= 0; i--) {
        res[i] *= postfix;
        postfix *= nums[i];
    }

    return res;
}

The "Product of Array Except Self" problem is a classic challenge often encountered in coding interviews. It tests a candidate's ability to manipulate arrays and optimize algorithms.

Problem Statement

Given an array nums of n integers where n > 1, the task is to return an array output such that output[i] is equal to the product of all the elements of nums except nums[i].

Constraints and Challenges

• No division operation is allowed in the solution.
• The goal is to achieve a linear runtime complexity, O(n).
• A straightforward nested loop approach leads to O(n²) time complexity, which is inefficient.

Techniques for Solution

1. Prefix and Postfix Products: This efficient approach involves two passes through the array:

   • Prefix Product: Traverse the array from left to right, calculating the cumulative product up to the current element. Store these products in a new array.
   • Postfix Product: Traverse the array from right to left, calculating the cumulative product from the end to the current element. Multiply these with the prefix products in the new array.

2. Left and Right Product Lists: Use two additional arrays to store products of all elements to the left and right of each element, and then multiply corresponding elements from these arrays for the final result. This method uses extra space.

3. Optimized Space Approach: Optimize the space complexity by using the output array for storing one of the lists (like left products) and a variable to track the right product during the second traversal.

Example

Consider nums = [1, 2, 3, 4]. The expected output is [24, 12, 8, 6].

• After the first pass (prefix): [1, 1, 2, 6].
• After the second pass (postfix): [24, 12, 8, 6].

This problem demonstrates the importance of array manipulation and optimizing solutions for better time and space efficiency.

References

https://www.enjoyalgorithms.com/blog/product-of-array-except-self