The core idea isn't hard: build a prefix sum array over sorted cities, generate a random number in [0, total_population), binary search for it.
Clarify the problem requirements, then design a solution using prefix sums and binary search for efficient O(log n) sampling. Implement the class with a preprocessing step that builds the prefix sum array and a sampling method that generates a random number and finds the corresponding city. Discuss trade-offs and potential optimizations.
Pro tip: Mention that using binary search on prefix sums is optimal for repeated queries, and consider edge cases like zero population or empty map. Also, note that if the distribution is static, this approach is ideal; for dynamic updates, a Fenwick tree could be used.
Ask about input constraints, expected query frequency, and whether the population data can change. Confirm that probabilities should be proportional to population and that the method will be called many times.
Choose to store city names in an array and compute a prefix sum array of populations. This allows mapping a random number to a city via binary search.
In the constructor, iterate through the map, store city names and populations, and build the prefix sum array. Handle edge cases like empty map or zero total population.
Generate a random integer between 1 and total population (inclusive). Use binary search on the prefix sum array to find the index where the cumulative sum is >= the random number, then return the corresponding city.
Discuss time and space complexity: O(n) preprocessing, O(log n) per query, O(n) space. Mention potential optimizations like using a Fenwick tree for dynamic updates or alternative sampling methods.
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Two-pointer merge on sorted index lists for the base case, pretty standard.
Start by clarifying the sparse vector representation and constraints, then propose a two-pointer merge algorithm for the dot product. For the follow-up, discuss incremental computation using a hash map to track contributions and update only changed entries, while considering trade-offs like memory overhead and update frequency.
Pro tip: Emphasize that the optimal approach depends on the sparsity pattern and update frequency; showing awareness of these trade-offs demonstrates engineering maturity. Also, mention that in ML, sparse dot products are common in embedding lookups and feature interactions, so optimizing them can have real impact.
Confirm the input format: each vector is a list of (index, value) pairs, indices are sorted, and dimension is huge but sparse. Ask about value types (e.g., floats) and if indices are unique.
Use two pointers to traverse both lists simultaneously, advancing the pointer with the smaller index. When indices match, multiply values and add to the result. This runs in O(nnz1 + nnz2) time and O(1) extra space.
Discuss time and space complexity, and handle edge cases like empty vectors, no overlapping indices, and duplicate indices (if not guaranteed unique).
Propose maintaining a hash map from index to the product of values for overlapping indices, and a running sum. When an entry changes, update the map and adjust the sum by the difference. This gives O(1) update time per changed entry, assuming the map is kept in sync.
Compare the incremental approach with recomputation: incremental is faster for few changes but uses extra memory and requires tracking changes. Also mention potential optimizations like using a balanced BST if indices are dynamic, or batching updates.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.