Spent the first ten minutes just re-reading the problem because the dual-counting on '11' models kept tripping me up.
First, categorize models into three groups: A-only, B-only, and both. Then, for each k, find the minimum cost to select at least k from A∪Both and at least k from B∪Both, ensuring the intersection (Both) is counted appropriately. Use sorting and prefix sums to efficiently compute the minimum cost for each k, or use a greedy approach with priority queues.
Pro tip: Clarify that models supporting both features are the most flexible and should be prioritized, but sometimes it's cheaper to use separate A-only and B-only models. Discuss trade-offs between time and space complexity, and mention that precomputing prefix sums can reduce per-k computation to O(1) after sorting.
Separate models into three lists: those supporting only A, only B, and both. Sort each list by cost ascending.
For each list, compute prefix sums of costs to quickly get the total cost of selecting the cheapest m models from that list.
For each possible number of both models selected (from 0 to total both), determine how many additional A-only and B-only models are needed to meet the requirement for a given k.
For each k, take the minimum over all valid choices of both models count, using prefix sums to get costs in O(1). If no valid combination exists, return -1.
The overall time complexity is O(n log n) due to sorting, and O(n^2) if iterating naively over both count and k, but can be optimized to O(n log n) or O(n) with two pointers or priority queues.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.