Prefix sum plus binary search, pretty standard.
Start by clarifying the problem constraints (e.g., array size, dynamic updates, memory limits) and then present the prefix sum + binary search solution as the standard approach. Discuss time/space complexity and possible optimizations or alternatives like the alias method if appropriate.
Pro tip: Mention that for very large arrays or frequent sampling, the alias method can achieve O(1) sampling time after O(n) preprocessing, but it's more complex to implement. Also, highlight the importance of handling edge cases like zero weights or empty arrays.
Ask about array size, whether weights can be zero or negative, if the distribution changes over time, and any memory or time constraints.
Propose the prefix sum + binary search method: compute cumulative sums, generate a random number between 0 and total sum, then binary search for the index.
State that preprocessing takes O(n) time and O(n) space, each sample takes O(log n) time. Mention alternatives like the alias method for O(1) sampling with O(n) preprocessing.
Discuss handling zero weights (skip or ensure they are never selected), empty array, and floating-point precision issues.
Write clean code, possibly using binary search (e.g., bisect in Python), and test with simple cases to verify probabilities.
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First approach was just normalize upfront, divide each weight by the total.
Start by clarifying that unnormalized weights are common in practice (e.g., due to floating-point errors, missing data, or raw scores) and that the core issue is ensuring they represent valid probabilities. Then present at least two distinct approaches: normalization (dividing by the sum) and using a weighted sampling method that inherently handles unnormalized weights (e.g., the alias method or rejection sampling). Discuss trade-offs such as computational cost, numerical stability, and whether the weights are used for sampling or for computing expectations.
Pro tip: Mention that in production systems like LinkedIn's feed ranking, weights often come from model scores that are not normalized, and normalizing them can be expensive if done naively; instead, techniques like the Gumbel-max trick or the alias method can sample efficiently without explicit normalization. This shows you understand both theory and scalable engineering.
Ask whether the weights are used for sampling, averaging, or as probabilities in a model. Confirm if negative weights or zeros are possible, as this affects the choice of method.
Compute the sum of all weights and divide each weight by the sum. This yields a valid probability distribution but can be numerically unstable if the sum is very large or small, and requires a full pass over the data.
Use algorithms like the alias method (O(1) sampling after O(n) preprocessing) or rejection sampling (e.g., Gumbel-max trick) that work directly with unnormalized weights. These avoid explicit normalization and can be more efficient for large-scale systems.
Compare methods on time/space complexity, numerical stability, and suitability for streaming data. Mention handling of zero weights (e.g., they get zero probability) and negative weights (which require shifting or different techniques).
Summarize which approach you would choose based on the scenario, e.g., normalization for small batches, alias method for repeated sampling from a fixed distribution, or Gumbel-max for differentiable sampling.
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First, clarify the rejection sampling setup: you have unnormalized weights w_i and you sample uniformly from indices, then accept with probability w_i / max_j w_j. The expected number of trials is the reciprocal of the acceptance probability, which is (sum w_i) / (n * max w_i). Then discuss how this scales with the total sum and the maximum weight.
Pro tip: Mention that if the weights are highly skewed, the expected number of trials can be very large, and suggest alternative methods like the alias method or Walker's algorithm for efficient sampling.
State the exact algorithm: sample an index uniformly from 1 to n, then accept it with probability w_i / max_j w_j. This ensures the accepted index is proportional to w_i.
The overall acceptance probability is the average of w_i / max w_j over all i, which equals (sum w_i) / (n * max w_j).
Since each trial is independent with success probability p, the expected number of trials is 1/p = (n * max w_j) / (sum w_i).
If the total sum S = sum w_i, then the expected number of trials is n * max w_j / S. This shows it is inversely proportional to S and directly proportional to n and the maximum weight.
Note that if max w_j is close to the average weight, the expected trials is near 1; if skewed, it can be large. Mention that for large n or skewed weights, more efficient methods like the alias method are preferred.
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This one opened up into a mini system design conversation.
Start by clarifying the problem context: what operations are needed (sampling, updates, queries) and what constraints exist (memory, latency). Then propose a hybrid data structure that exploits sparsity, such as a compressed sparse representation or a hash map for non-zero entries, and discuss trade-offs in time, memory, preprocessing, and update cost.
Pro tip: Emphasize that the optimal choice depends on the read/write ratio and whether the distribution is static or dynamic; mentioning real-world ML scenarios like embedding tables or recommendation systems shows practical insight.
Ask about the operations (sampling, updates, queries), frequency, latency requirements, and memory limits to tailor the solution.
Recognize that probability mass is concentrated on few indices and suggest sparse representations like hash maps, sorted arrays of (index, weight), or compressed sparse row (CSR) format.
Compare time complexity for sampling and updates, memory overhead, preprocessing cost (e.g., building alias tables), and update cost (e.g., rebalancing).
Propose a solution that combines dense and sparse structures, or uses caching for frequent indices, and discuss when to switch strategies based on workload.
Conclude with the best approach for the given scenario, highlighting how it balances the trade-offs and meets the requirements.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.