Product of Array Except Self
Given an integer array `nums`, return an array `answer` such that `answer[i]` is equal to the product of all the elements of `nums` except `nums[i]`. Do not use division and solve in $O(n)$.
Why Interviewers Ask This
Amazon interviewers use this problem to evaluate a candidate's ability to optimize space complexity while adhering to strict constraints. They specifically test if you can recognize that division is forbidden, forcing you to think about prefix and suffix products. This assesses your proficiency in handling edge cases like zeros and your capacity to design an O(n) solution without auxiliary arrays for the final result.
How to Answer This Question
1. Clarify Constraints: Immediately confirm if the array contains zeros or negative numbers and reiterate the no-division rule to show attention to detail.
2. Define the Logic: Propose the two-pass approach where the left pass calculates cumulative products from the start, and the right pass multiplies by cumulative products from the end.
3. Optimize Space: Explicitly state your intention to reuse the output array as one of the product arrays to achieve O(1) extra space, a key expectation at Amazon.
4. Walk Through an Example: Trace the algorithm manually with a small array like [1, 2, 3, 4] to demonstrate how indices map to results before coding.
5. Implement and Verify: Write clean code, then immediately test boundary conditions like arrays with a single zero or all zeros to ensure robustness.
Key Points to Cover
- Explicitly confirming the 'no division' constraint early in the conversation
- Demonstrating the optimization of using the output array to save space
- Explaining the separation of left and right cumulative products clearly
- Tracing the algorithm step-by-step with a concrete numerical example
- Addressing edge cases involving zeros or negative numbers
Sample Answer
To solve the Product of Array Except Self efficiently without division, I will utilize a two-pass strategy that leverages prefix and suffix products. First, I initialize an answer array of the same length as the input. In the first pass, I iterate from left to right, storing the cumulative product of all elements preceding the current index. For example, if the input is [1, 2, 3, 4], the left pass yields [1, 1, 2, 6].
Next, I perform a second pass from right to left. I maintain a running variable for the suffix product. As I iterate backwards, I multiply the existing value in the answer array by this suffix variable and update the suffix variable by multiplying it with the current element. Continuing the example, the right pass updates the array to [24, 12, 8, 6].
This approach ensures we meet the O(n) time complexity requirement since we traverse the array exactly twice. Crucially, by reusing the output array to store the left products initially, we only need a constant amount of extra space for the suffix variable, satisfying the O(1) space constraint often emphasized in Amazon technical interviews. Finally, I would verify the solution handles edge cases, such as arrays containing zeros, where the logic naturally produces the correct result without special branching.
Common Mistakes to Avoid
- Attempting to use division to simplify the logic, which violates the core constraint
- Allocating two separate arrays for left and right products instead of optimizing space
- Failing to handle the edge case where the input array contains one or more zeros
- Starting the cumulative product calculation incorrectly, such as including the current element in its own product
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