I jumped straight to thinking about dynamic programming and wasted a few minutes going down that road.
Clarify the problem constraints and edge cases, then propose an efficient algorithm that preprocesses the array to answer multiple queries quickly. Explain how sorting and prefix sums can be used to find the maximum number of elements that sum to at most each query value.
Pro tip: Mention that sorting the array is optimal because to maximize the count of elements under a sum constraint, you should always pick the smallest elements first. This greedy insight simplifies the problem and leads to an efficient solution.
Ask about constraints: array size, number of queries, value ranges, and whether elements can be negative. Confirm that subsequence means any subset of elements (order doesn't matter).
Explain that to maximize the number of elements under a sum limit, we should select the smallest elements first. Sorting the array enables this.
Sort the array and compute prefix sums. For each query, use binary search on the prefix sums to find the largest index where the sum is ≤ query value. The count is that index + 1.
Sorting takes O(n log n), prefix sums O(n), and each query O(log n) via binary search. Total O(n log n + q log n), which is efficient for large inputs.
Discuss cases where query is smaller than the smallest element (return 0), or larger than total sum (return n). Also consider negative numbers if allowed, which would require a different approach.
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