I knew prefix sums were the move but fumbled explaining why you binary search instead of just scanning linearly.
Start by clarifying requirements: weights are positive, array is static, and we need O(1) query time after O(n) preprocessing. Then describe the prefix sum + binary search approach: compute cumulative weights, pick a random number in [0, totalWeight), and binary search for the first index where cumulative weight exceeds the random value. Discuss trade-offs like space vs. time and potential optimizations for large arrays.
Pro tip: Mention that you can use binary search on the prefix sums because they are sorted, and that this gives O(log n) per query. Also, note that if weights are integers, you could use a segment tree or Fenwick tree for dynamic updates, but for static weights, prefix sum is optimal.
Ask about weight types (positive, zero?), array size, frequency of queries, and whether weights can change. This shows you consider edge cases and scalability.
Propose storing prefix sums of weights. Explain that this allows mapping a uniform random number to an index via binary search.
Detail: compute total weight, generate random number r in [0, totalWeight), binary search prefix sums to find smallest index i such that prefix[i] > r. Return i.
State preprocessing O(n), query O(log n), space O(n). Compare with alternative approaches like linear scan O(n) per query or using a segment tree for dynamic updates.
Discuss zero weights, floating-point precision, and potential optimizations like using a Fenwick tree if updates are needed, or alias method for O(1) query with O(n) preprocessing.
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