Start by clarifying the problem constraints and edge cases, then propose a solution using prefix sums and binary search to achieve O(n) preprocessing and O(log n) query time. Explain the algorithm step-by-step, analyze its complexity, and discuss potential optimizations or alternative approaches like the alias method.
Pro tip: Mention that the prefix sum array should be built once and reused for multiple random picks, and highlight that using binary search on the prefix sums ensures efficient sampling even for large arrays. Also, briefly note that the alias method can achieve O(1) query time with O(n) preprocessing if queries are frequent.
Ask about input size, number of queries, whether weights can be zero or negative, and if the array can be modified. Confirm that the function should return an index with probability proportional to its weight.
Explain that you will compute a prefix sum array where prefix[i] = sum of weights from 0 to i. Then generate a random number between 0 and total sum, and use binary search to find the smallest index where prefix sum exceeds the random number.
Use a small example like weights = [1, 3, 2] to illustrate how the prefix sums are [1, 4, 6], and how a random number in [0,6) maps to indices with correct probabilities.
State that preprocessing takes O(n) time and O(n) space, and each query takes O(log n) time. Mention that if many queries are expected, the alias method can achieve O(1) query time with O(n) preprocessing.
Address cases like all weights zero (return any index or throw error), single element, and floating-point weights. Summarize the solution and its trade-offs.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Clarify the problem constraints (e.g., array size, element types, handling zeros) and then propose an O(n) time and O(n) space solution using prefix and suffix products. Explain that you will compute prefix products in one pass and suffix products in another, then multiply them to get the result without division.
Pro tip: Mention the trade-off between extra space and time: you can achieve O(1) extra space (excluding output) by using the output array to store prefix products first, then a running suffix product variable. This shows you optimize beyond the basic solution.
Ask about input size, possible zeros, negative numbers, and whether the output can be in-place. Confirm that division is not allowed and discuss how zeros affect the result.
Explain that for each index i, the result is the product of all elements before i (prefix) times the product of all elements after i (suffix). This avoids division and handles zeros naturally.
Describe two passes: first, compute prefix products and store in the output array; second, traverse from the right while maintaining a running suffix product, multiplying it into the output array.
State that time complexity is O(n) and space complexity is O(n) for the output array, but extra space can be O(1) if we reuse the output array for prefixes. Mention that this is optimal.
Walk through a small example (e.g., [1,2,3,4]) and an edge case with zeros (e.g., [0,1,2]) to verify correctness. Discuss how the algorithm handles multiple zeros.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.