Start by clarifying the requirements and constraints, then explain the prefix sum and binary search approach to achieve O(log k) per mint. Walk through the algorithm step-by-step, including how to handle random number generation and edge cases, and discuss trade-offs like initialization cost and precision.
Pro tip: Mention that you can precompute the cumulative weights in the constructor to make mint() efficient, and discuss how to handle zero weights or floating-point precision issues. Also, consider using a binary search that returns the first index where the cumulative sum exceeds the random target.
Ask about input size, weight types (integers/floats), whether weights can be zero, and if the distribution needs to be exact. Confirm that O(log k) per mint is required and that the constructor can take O(k) time.
Store the cumulative sum of weights in an array (prefix sums) during construction. This allows O(1) access to the total weight and enables binary search.
Generate a random number between 0 and total weight, then binary search the prefix sums to find the smallest index where the cumulative sum exceeds the random number. Return that index.
Explain that construction is O(k) and mint is O(log k). Discuss trade-offs: memory O(k), potential precision issues with floating-point weights, and how to handle zero weights.
Mention testing with edge cases: all weights zero, single element, large k, and verifying distribution via simulation. Also consider randomness source and security if needed.
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I went with a Fenwick tree and explained point updates and prefix queries both in O(log k).
Start by clarifying the problem: we need a data structure that supports weighted random selection (mint) in O(log k) and point updates (update(i, delta)) in O(log k). Propose a Fenwick tree (Binary Indexed Tree) over the weights, which supports prefix sum queries and point updates in O(log k), and use binary search on prefix sums to select an index proportional to weight. Justify why this meets the complexity and is simpler than a segment tree.
Pro tip: Mention that a Fenwick tree uses less memory and has smaller constants than a segment tree, but if the problem requires range updates or more complex queries, a segment tree might be more flexible. Also, note that floating-point precision can be an issue; consider using integers or a balanced BST if weights are dynamic and large.
Confirm that mint() selects an index with probability proportional to its weight, and update(i, delta) changes the weight of index i by delta. Ensure both operations must be O(log k) and discuss any constraints on k, weight types, and update frequency.
Use a Fenwick tree to store weights, supporting point updates and prefix sum queries in O(log k). For mint(), generate a random number between 0 and total weight, then binary search on the Fenwick tree to find the smallest index where prefix sum >= random value.
Describe how to perform the binary search on the Fenwick tree: start from the highest power of two, accumulate sums, and narrow down to the target index. This takes O(log k) time.
Update the Fenwick tree at index i by adding delta, propagating the change to all relevant nodes. This also takes O(log k) time.
Compare with a segment tree: both give O(log k) for both operations, but Fenwick is simpler and more memory-efficient. Mention that if weights can be negative or if we need range updates, a segment tree might be better. Also note that if k is small, a simple array with linear scan might suffice.
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This one tripped me up more than I expected.
Clarify the data structures for weights and NFT IDs, then design an algorithm that efficiently samples distinct NFTs while updating weights after each draw. Discuss trade-offs between different approaches (e.g., cumulative sum + binary search vs. Fenwick tree) and handle edge cases like t exceeding available NFTs.
Pro tip: Mention that updating weights after each draw is crucial for fairness and can be optimized with a Fenwick tree to achieve O(log n) per draw, but also consider the simpler O(n) approach if t is small. Also, discuss how to handle precision issues with floating-point weights.
Ask about the data types (e.g., integer vs. float weights), the expected size of t and the NFT pool, and whether weights can become zero or negative. Confirm that after each draw, the selected NFT is removed and weights are updated (e.g., set to zero or re-normalized).
Decide between a simple array with cumulative sums and binary search (O(n) update, O(log n) draw) or a Fenwick tree (O(log n) update and draw). Discuss the trade-offs based on expected t and n.
For each draw, generate a random number in [0, totalWeight), find the corresponding NFT, record it, then update the weight (e.g., set to 0) and adjust the total weight. Ensure the algorithm correctly handles the removal of the selected NFT from future draws.
Consider cases where t > number of NFTs, all weights are zero, or weights are very small. Discuss how to handle floating-point precision (e.g., using integers by scaling) and whether to throw an error or return fewer NFTs.
State the time and space complexity of your solution. Walk through a small example to verify correctness, and mention potential optimizations or alternative approaches.
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Start by explaining how to make randomness reproducible using a seeded PRNG, then outline a property-based testing strategy that validates distributional properties and edge cases. Emphasize the importance of deterministic seeding for reproducibility and the use of statistical tests to verify correctness.
Pro tip: Mention that seeding should be explicit and isolated per test to avoid flakiness, and that property-based tests should be complemented with unit tests for edge cases to ensure comprehensive coverage.
Describe how to inject a seed into the random number generator, ensuring that the same seed produces the same sequence of random choices. Highlight the use of a dedicated PRNG instance per experiment or test.
List key properties to test: distribution uniformity (e.g., chi-squared test), sum of probabilities equals 1, and that selection respects weights. Use a property-based testing library like Hypothesis or QuickCheck.
Detail how to handle zero weights (should never be selected), extremely large weights (should not cause overflow or bias), all-equal weights (should yield uniform distribution), and k=1 (should return a single element).
Explain how to run multiple trials with different seeds and use statistical tests (e.g., chi-squared, Kolmogorov-Smirnov) to confirm the distribution matches expectations within a confidence interval.
Mention trade-offs between test runtime and statistical confidence, and how to balance thoroughness with CI/CD constraints. Suggest using fixed seeds for reproducibility in CI.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.