This was a lot to hold in your head at once.
Start by clarifying requirements and constraints, then propose a streaming algorithm like weighted reservoir sampling (A-ES) that handles extreme weights via log-space arithmetic. Explain the algorithm step-by-step, prove its unbiasedness, discuss numerical stability techniques, and outline a testing strategy including edge cases and statistical validation.
Pro tip: Emphasize that you would use log-space computations to avoid overflow/underflow with extreme weights, and mention that you'd validate the implementation using statistical tests like chi-squared or Kolmogorov-Smirnov on the sampling distribution.
Restate the problem: sample k items without replacement from a stream of up to 100M items with weights, O(k) memory, reproducibility, and handling of extreme weights and nonpositive weights. Ask clarifying questions about weight distribution, stream characteristics, and performance needs.
Propose a streaming algorithm such as weighted reservoir sampling (e.g., A-ES or Efraimidis-Spirakis) that maintains a reservoir of k items. Explain how it processes each item, updates the reservoir, and ensures proportional selection without replacement.
Describe how to handle extreme weights using log-space arithmetic (e.g., log of weights, log-sum-exp) to prevent overflow/underflow. Explain rejection of nonpositive weights and how to incorporate a seed for reproducibility.
Provide a proof sketch that each item is selected with probability proportional to its weight. Analyze time and space complexity: O(n) time, O(k) memory, and note that log-space operations add constant overhead.
Describe unit tests for edge cases (zero/negative weights, k > n, extreme weights), statistical tests (chi-squared, KS) to verify distribution, and reproducibility tests with seeds. Mention performance testing on large streams.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.