I knew the prefix/suffix product trick so the base case wasn't the problem.
Use a two-pass prefix and suffix product approach: first pass computes prefix products into the output array, second pass multiplies by suffix products while maintaining a running suffix product variable. This achieves O(n) time and O(1) extra space (output array not counted).
Pro tip: Clarify upfront that the output array is not considered extra space, and mention edge cases like zeros and negative numbers to show thoroughness. Also, discuss the trade-off between time and space, and why this approach is optimal for Amazon's scale.
Confirm that the output array doesn't count towards space complexity, and discuss handling of zeros, negative numbers, and empty input.
Describe how to compute prefix products in the first pass and then multiply by suffix products in the second pass using a running variable.
Use a small array (e.g., [1,2,3,4]) to illustrate how the output array is built step by step.
State that time complexity is O(n) and extra space is O(1), excluding the output array.
Mention that division is disallowed, and compare with other approaches like using logarithms or handling zeros separately.
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Caught me slightly off guard because I'd been thinking about the happy path.
First, clarify what 'many zeros' means in the context of your solution—whether it's a sparse array, zero-heavy input, or zeros affecting algorithm behavior. Then systematically walk through edge cases: all zeros, zeros at boundaries, zeros causing division by zero or infinite loops, and performance implications. Finally, explain how your solution handles or could be adapted to handle these cases robustly.
Pro tip: Demonstrate awareness of Amazon's leadership principles by proactively discussing how you would optimize for zero-heavy inputs (e.g., using sparse representations) and how you would test for such edge cases in production ML pipelines.
Ask or state assumptions about the array: size, data type, and what 'many zeros' implies (e.g., >50% zeros). Confirm whether zeros are valid data or missing values.
List ways zeros could break the solution: division by zero, incorrect averages, infinite loops in while conditions, or degraded performance due to unnecessary computations.
Enumerate cases: all zeros, zeros at start/end, alternating zeros, zeros in sorted arrays, and zeros affecting hash collisions or tree balancing. Explain expected behavior for each.
Describe modifications: early termination, skipping zeros, using sparse data structures, or adding guards. Discuss trade-offs in time/space complexity.
Conclude with how you would validate the solution with unit tests covering zero-heavy inputs and mention any monitoring for production ML systems.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Acknowledge the integer overflow risk and propose using logarithms to transform multiplicative operations into additive ones, then exponentiate back. Discuss the precision trade-offs: logarithms introduce floating-point errors, especially for large numbers, and exponentiation can amplify these errors, so consider using arbitrary-precision arithmetic or alternative methods when exact results are needed.
Pro tip: Mention that in practice, for very large numbers, you might use Python's built-in big integers or libraries like GMP, but understanding the logarithmic approach demonstrates deeper algorithmic knowledge. Also, highlight that in ML, such large numbers often appear in probabilities, so working in log-space is standard to avoid underflow/overflow.
Explain that directly computing large factorials, exponentials, or products can exceed integer limits, leading to overflow and incorrect results.
Describe how taking the logarithm converts multiplication into addition and exponentiation into multiplication, keeping numbers manageable.
Show how to compute the log of the desired quantity, sum or multiply as needed, and then exponentiate to get the final result.
Explain that logarithms and exponentiation introduce floating-point errors, which can be significant for large numbers, and that the result may be approximate.
Suggest using arbitrary-precision arithmetic (e.g., Python's int, Java's BigInteger) for exact results, or using higher precision floats, or staying in log-space if only comparisons are needed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.