I knew prefix sums going in but the 'maximize without exceeding' part tripped me up more than I expected.
First, clarify the problem constraints and edge cases, then explain the prefix sum approach with a balanced BST or sorted list to achieve O(n log n) time, and finally discuss the trade-offs and potential optimizations. Emphasize that while O(n) is not possible for arbitrary integers, the prefix sum method is optimal for this problem.
Pro tip: Mention that the problem is essentially finding the maximum prefix sum difference that does not exceed the target, and that using a balanced BST (like a TreeSet in Java) is a common interview solution. Also, note that if the array contains only non-negative numbers, a sliding window can achieve O(n), but for general integers, O(n log n) is the best possible.
Ask about constraints: Can the array contain negative numbers? What should be returned if no subarray sum is <= target? Is the subarray required to be contiguous? Confirm the expected time complexity.
Describe how to compute prefix sums and use a balanced BST to store previous prefix sums. For each prefix sum, find the smallest prefix sum >= current - target, and update the closest sum.
Choose a small array (e.g., [2, -1, 3, -2] with target 4) and demonstrate how the algorithm works step by step, showing the BST operations and the closest sum updates.
State that the time complexity is O(n log n) due to BST operations, and space is O(n). Discuss that O(n) is not achievable for arbitrary integers, but if all numbers are non-negative, a sliding window can achieve O(n).
Cover cases like empty array, target negative, all sums exceeding target, and mention that using a sorted list with binary search can also work but may have O(n) insertion in some languages.
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