The prefix sum plus binary search part came together fine, O(log n) per query, not too bad.
Start by explaining the standard approach: compute prefix sums of weights, then for each sample, generate a random number in [0, total weight) and binary search for the index. For millions of calls, precompute the prefix sums once and consider optimizations like the alias method for O(1) sampling, or parallelization if needed.
Pro tip: Mention that the alias method is often the best choice for static weights, but if weights change frequently, a segment tree or Fenwick tree can support updates and sampling in O(log n). Also, discuss the trade-off between preprocessing time and sampling speed.
Confirm whether weights are static or dynamic, and the expected number of samples. This determines the appropriate data structure and algorithm.
Explain the prefix sum + binary search method: precompute cumulative weights, then for each sample, pick a random target and binary search to find the index. Time complexity: O(n) preprocessing, O(log n) per sample.
Introduce the alias method for O(1) sampling after O(n) preprocessing, ideal for static weights. If weights change, consider a segment tree or Fenwick tree for O(log n) updates and sampling.
Compare preprocessing time, memory usage, and sampling speed. Mention numerical stability, handling zero weights, and potential parallelization or vectorization for large-scale sampling.
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