I knew the general idea (prefix sums plus binary search) but fumbled the implementation details under pressure.
Clarify the requirements: the function should return a city name with probability proportional to its population, and we can assume the dictionary is static. Use prefix sums to create cumulative intervals, then generate a random number in [0, total_population) and binary search to find the corresponding city. This gives O(n) preprocessing and O(log n) per query, which is efficient and scalable.
Pro tip: Mention that you can optimize space by storing cumulative sums in an array and using binary search, and discuss how to handle updates if the population data changes frequently (e.g., using a Fenwick tree for O(log n) updates).
Ask about the size of the dictionary, whether populations can change, and if the generator needs to be called multiple times. Confirm that probabilities are proportional to population.
Compute prefix sums of populations to create cumulative intervals. Store city names in an array parallel to the prefix sums for O(1) access after binary search.
Generate a random integer between 0 and total population - 1. Use binary search on the prefix sums to find the first index where prefix sum > random number, then return the corresponding city.
Discuss time complexity: O(n) preprocessing, O(log n) per query. Handle edge cases: empty dictionary, zero populations, and large populations causing overflow (use 64-bit integers).
Write unit tests to verify distribution by sampling many times and comparing frequencies to expected probabilities. Also test with edge cases like single city or all populations equal.
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