How to Find the Product of an Array Except Self?

Finding the product of an array except for itself is a classic algorithm problem often encountered in technical interviews. The main challenge is to derive an output where each element at index i is the product of all numbers in the array except the one at index i. This problem tests not only your algorithm skills but also your ability to think in terms of time and space complexity.

Problem Statement

Given an integer array nums, return an array output where output[i] is equal to the product of all the elements of nums except nums[i]. You cannot use the division operation, and the solution must be efficient, ideally O(n) time complexity and O(1) space complexity if possible.

Example

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In this example:

• For index 0: The product is 2 * 3 * 4 = 24
• For index 1: The product is 1 * 3 * 4 = 12
• For index 2: The product is 1 * 2 * 4 = 8
• For index 3: The product is 1 * 2 * 3 = 6

Interview Questions

Question 1: Can you describe your approach to solving this problem?

A good way to solve this problem without using division is through two passes of the array.

1. Left Products: Create an output array that holds the product of all elements to the left of the current element.
2. Right Products: Traverse the array from the right to calculate the product of all elements to the right of the current element and multiply it with the value stored in the output array from the first step.

This method leverages two passes and avoids the necessity of additional space beyond the output array.

Example Answer

1. Create an output array initialized to 1.
2. Loop through the nums from left to right. For each index i, set output[i] to be the product of all previous elements.
3. Loop through the nums from right to left. For each index i, multiply output[i] by the product of all next elements.

Here’s a possible implementation in Python:

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Question 2: What is the time and space complexity of your solution?

The solution described has a time complexity of O(n), as it makes two passes through the input array, each taking O(n) time. The space complexity is O(1) if we consider the output array to be the only extra space used, as the output is included in the final answer.

Example Answer

• Time Complexity: O(n) because we traverse the list twice.
• Space Complexity: O(1) for the output array (not counting the input array); the algorithm only uses a few extra variables for multiplication.

Follow-up Question: What if division was allowed?

If division is allowed, the problem can be simplified to computing the total product of the array and then dividing by each element to get the result.

Example Answer

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In this case, the time complexity remains O(n), while the space complexity can often be reduced to O(1) if outputs do not count.

Showing a systematic approach to solving the product of array except self problem is often key in developer interviews. Presenting well-structured solutions while explaining your thought process can make a positive impression.