Maximum Product Subarray
Given an integer array `nums`, find a contiguous subarray that has the largest product, and return the product.
Why Interviewers Ask This
Amazon interviewers ask this question to evaluate a candidate's ability to handle state management with negative numbers, which often breaks standard maximum subarray logic. It tests dynamic programming optimization skills and the capacity to maintain multiple states (max and min) simultaneously. The problem specifically probes whether you can identify edge cases like zeros and negative flips that cause simple greedy approaches to fail, ensuring you write robust code under pressure.
How to Answer This Question
1. Clarify requirements immediately by asking if the array is empty or contains only zeros, as Amazon values handling edge cases early. 2. Propose a brute-force O(n^2) solution first to establish a baseline, then explain why it is inefficient for large datasets. 3. Introduce the optimal O(n) approach using two variables: one tracking the maximum product ending at the current position and another tracking the minimum product, since a negative number can turn a small minimum into a large maximum. 4. Walk through the iteration logic step-by-step, explicitly showing how you swap max and min when encountering a negative number. 5. Conclude by discussing space complexity, emphasizing that you only need constant extra space by updating variables in place rather than using an array.
Key Points to Cover
- Explicitly handling the swap logic when a negative number is encountered
- Maintaining separate state for both maximum and minimum products
- Demonstrating understanding of why greedy approaches fail with negatives
- Achieving O(n) time complexity with O(1) space complexity
- Discussing edge cases like arrays containing zeros or single elements
Sample Answer
To solve the Maximum Product Subarray problem efficiently, I would first validate the input. If the array is null or empty, I'd return zero or handle it per specific constraints. A naive nested loop approach checking every subarray would take O(n^2) time, which is too slow for Amazon-scale data. Instead, I propose a single-pass Dynamic Programming solution with O(n) time and O(1) space. The core insight is that a negative number can flip a very small (negative) product into a very large positive one. Therefore, at each index, we must track both the local maximum and local minimum products ending there. We initialize these with the first element. As we iterate, if the current number is negative, we swap our max and min trackers because multiplying by a negative reverses their order. Then, for each element, the new max is the largest of the current element itself, the current element times the previous max, or the current element times the previous min. We update our global maximum result accordingly. For example, in [2, -3, 4], at index 1, the max becomes -3 and min becomes -6. At index 2, multiplying 4 by -6 gives -24, but 4 times -3 gives -12, so we reset max to 4. This ensures we capture the true peak even after negative flips.
Common Mistakes to Avoid
- Ignoring negative numbers and assuming the maximum product always comes from the largest positive sequence
- Using a brute-force O(n^2) approach without attempting to optimize for linear time
- Forgetting to swap the max and min trackers when the current number is negative
- Overlooking the case where the answer might be a single element or zero
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