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LinkedIn·Data Scientist·Technical Phone Screen·Senior

Senior
May 2026

Summary

LinkedIn data scientist interview with a deep technical question on sampling algorithms. Single question but it had five sub-parts and felt more like a system design exercise than a coding screen. Came out of it feeling like I'd only really nailed two of the five parts.

Questions Asked (1)

Q1

Design an algorithm to sample from a categorical distribution over k outcomes using only a Uniform(0,1) random number generator, with O(k) preprocessing and O(1) sampling time. Walk through the build and sample logic, complexity, floating-point edge cases, support for incremental probability updates, and a statistical validation plan.

Algorithms & Data StructuresTechnical Trade-offsSystem Design
Author's notes

I knew the alias method from a stats course a few years back so the core pseudocode came out okay.

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AI HintsAI Generated

Suggested Approach

Start by explaining the inverse transform method: compute the cumulative distribution function (CDF) from the probabilities, then for each sample draw U ~ Uniform(0,1) and find the smallest index i such that CDF[i] >= U. Emphasize that preprocessing builds the CDF in O(k) time and sampling uses binary search for O(log k) time, but to achieve O(1) sampling, use the alias method or a precomputed lookup table if k is small. Then discuss floating-point precision, incremental updates, and validation.

Pro tip: Mention the alias method as the standard O(1) sampling technique, but also note that for small k, a simple linear scan or binary search on the CDF is often sufficient and easier to maintain; showing awareness of trade-offs between simplicity and optimal complexity impresses interviewers.

1. Clarify requirements and constraints

Confirm that k is fixed, probabilities sum to 1, and that O(k) preprocessing and O(1) sampling are required. Ask if probabilities can change dynamically or if k is small enough for simpler methods.

2. Describe the inverse transform method

Explain building the CDF array in O(k) time. For sampling, draw U ~ Uniform(0,1) and find the index via binary search (O(log k)) or linear scan (O(k)). Note that this does not meet O(1) sampling.

3. Introduce the alias method for O(1) sampling

Outline the alias method: preprocess probabilities into k buckets, each with a probability and an alias index. Sampling uses two uniform draws and O(1) lookup. Preprocessing is O(k).

4. Address floating-point edge cases and incremental updates

Discuss handling rounding errors (e.g., normalizing probabilities, using epsilon comparisons). For incremental updates, explain that the alias method requires O(k) rebuild, while a Fenwick tree can support O(log k) updates and O(log k) sampling.

5. Propose statistical validation

Suggest chi-square goodness-of-fit test, Kolmogorov-Smirnov test, or comparing empirical frequencies to expected probabilities with confidence intervals. Also mention using a large number of samples to detect bias.

Key Points to Mention

  • Inverse transform sampling and its O(log k) or O(k) sampling time
  • Alias method for O(1) sampling with O(k) preprocessing
  • Floating-point precision issues: normalization, epsilon comparisons, and avoiding cumulative error
  • Incremental probability updates: trade-offs between rebuild (O(k)) and dynamic data structures (e.g., Fenwick tree)
  • Statistical validation: chi-square test, KS test, and empirical frequency analysis
  • Space complexity: O(k) for CDF or alias tables

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.