← Hudson River Trading Interview Insights
Classic problem but I fumbled the delete for a second.
Combine a dynamic array (list) with a hash map that stores each element's index in the array. Insert appends to the array and records the index; delete swaps the target element with the last element, updates the moved element's index, then removes the last element; getRandom picks a random index from the array. This achieves O(1) average time for all operations.
Pro tip: Mention that this design assumes unique elements; if duplicates are allowed, you can store a set of indices per value or use a different approach. Also, highlight that the swap-with-last trick is key to O(1) deletion and that you must handle edge cases like deleting the last element or the only element.
Ask whether elements are unique, whether duplicates are allowed, and if the data structure needs to support other operations. Confirm that O(1) average time is required for all three operations.
Explain that you will use a dynamic array to store elements for O(1) random access and a hash map to map each element to its index in the array for O(1) lookup.
Describe insert: append to array, add to map. Describe delete: swap target with last element, update map for swapped element, remove last element from array and map. Describe getRandom: generate random index and return array element.
State that each operation runs in O(1) average time due to hash map operations and array indexing. Space complexity is O(n) for storing n elements.
Mention handling of deleting the last element, deleting the only element, and how to adapt if duplicates are allowed (e.g., using a set of indices per value).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying the requirements: weights are positive, updates may occur, and getRandom should be O(1) or O(log n). Then present the prefix sum + binary search approach as the standard solution, and discuss trade-offs with alternatives like the alias method or segment trees for dynamic updates.
Pro tip: Mention that the alias method gives O(1) getRandom but requires O(n) preprocessing and is harder to update; for dynamic weights, a Fenwick tree with binary search is a good compromise. This shows you understand practical trade-offs beyond the textbook solution.
Ask about weight properties (positive, zero, negative), frequency of updates, and expected time complexity for getRandom and update operations.
Explain that you can precompute prefix sums of weights, generate a random number in [0, totalWeight), and binary search to find the corresponding value. This gives O(n) preprocessing, O(log n) getRandom, and O(n) update.
If updates are frequent, replace the prefix sum array with a Fenwick tree (binary indexed tree) to support O(log n) updates and O(log n) getRandom via binary search on the tree.
For static weights, the alias method achieves O(1) getRandom with O(n) preprocessing, but updates are expensive. Compare trade-offs.
Summarize time/space complexity for each approach and discuss handling zero weights, floating-point precision, and large datasets.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Didn't think through this carefully enough.
Clarify the data structure and context (e.g., hash table, balanced BST, or database index) before proposing a solution. Then discuss how to modify insertion logic to handle duplicates, such as chaining in hash tables or allowing equal keys in BSTs, and analyze the impact on search, deletion, and performance.
Pro tip: Mention that the choice depends on whether duplicates should be allowed, counted, or rejected, and that in trading systems, duplicate handling must be deterministic and low-latency. Also, consider concurrency and memory overhead.
Ask whether duplicates should be stored, counted, or rejected, and identify the data structure and operations (insert, search, delete) involved.
Select an approach: e.g., chaining with a list per bucket in a hash table, allowing equal keys in a BST with a count, or using a multiset.
Adjust the insertion algorithm to handle duplicates, such as appending to a chain, incrementing a count, or inserting to the right subtree for equal keys.
Discuss effects on time/space complexity, search and deletion behavior, and any necessary changes to other operations.
Address scenarios like many duplicates, concurrency, and performance in high-frequency trading contexts.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Short answer: update the prefix sum structure, which is O(log n) with a Fenwick tree.
Clarify the data structure and the meaning of 'weight' (e.g., in a graph, weighted union-find, or priority queue), then discuss the trade-offs of updating the weight after insertion. Propose an efficient solution that balances time complexity and implementation simplicity, such as lazy updates or maintaining auxiliary structures.
Pro tip: Mention that in many cases, updating a weight after insertion can be handled lazily—deferring the actual update until the weight is needed—which often simplifies the code and improves performance. This shows you think about real-world trade-offs beyond textbook solutions.
Ask clarifying questions to understand the data structure (e.g., graph, heap, union-find) and what 'weight' represents. Confirm whether updates are frequent and if queries are interleaved.
Determine the required time complexity for updates and queries, and whether the structure must remain balanced or ordered. Consider if the weight can be updated in place or if it affects ordering.
Outline an approach: for example, in a priority queue, use a decrease-key operation or lazy deletion; in a graph, update the edge weight and possibly recompute shortest paths if needed. Explain the steps clearly.
Compare the proposed solution with alternatives (e.g., eager vs. lazy updates, rebuilding vs. incremental updates) in terms of time, space, and code complexity.
Summarize why your chosen approach is optimal for the given scenario, and mention any edge cases or potential pitfalls.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
I said run a bunch of samples and compare to expected frequencies, maybe use a chi-squared test.
Start by clarifying the expected distribution and the implementation details, then outline a statistical testing strategy that includes both theoretical analysis and empirical validation. Use a combination of unit tests for edge cases, chi-squared goodness-of-fit tests for distribution matching, and large-scale simulations to verify probabilities converge to expected values.
Pro tip: Emphasize the importance of setting a significance level and understanding Type I/II errors; also mention that for trading firms, demonstrating awareness of performance constraints and the need for deterministic tests in CI is crucial.
Confirm the expected probability distribution (e.g., uniform, weighted) and review the getRandom implementation to understand its algorithm and potential biases.
Choose appropriate statistical tests such as chi-squared goodness-of-fit or Kolmogorov-Smirnov, and define null and alternative hypotheses with a significance level (e.g., α = 0.05).
Generate a large number of samples (e.g., 1e6) and compute the empirical distribution; compare it to the expected distribution using the chosen test.
Verify behavior with edge inputs (e.g., empty range, single element) and ensure reproducibility by seeding the random number generator for deterministic tests.
Interpret p-values and effect sizes; if the test fails, investigate potential bugs or biases and refine the implementation or test parameters.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.