First, clarify the problem constraints and define the objective function. Then, model the problem as a dynamic program where the state captures the last T-1 selections to enforce the sliding window constraint. For the reusable case, recognize it as a variant of the unbounded knapsack with window constraints and discuss potential optimizations like greedy with priority queues or linear programming relaxations.
Pro tip: Emphasize the trade-offs between exact DP solutions and scalable heuristics, especially for large-scale ad serving systems where real-time constraints matter. Mention that in practice, you might use a sliding window with a heap to maintain top candidates, but be aware of the optimality gap.
Restate the problem: given a sequence of ads with revenues, select a subset such that in any window of T consecutive time units, the sum of revenues of selected ads is < M. For the reusable case, ads can be selected multiple times. Define variables and constraints clearly.
For the at-most-once case, define DP state as the last T-1 decisions (selected or not) to enforce the window constraint. Transition by considering selecting the current ad or not, updating the window sum. For the reusable case, the state remains similar but transitions allow multiple selections of the same ad, possibly leading to an unbounded knapsack-like DP.
Discuss the time and space complexity: O(n * 2^(T-1)) for the at-most-once case, which is exponential in T. For large T, consider alternative approaches like greedy with a max-heap or linear programming. For the reusable case, note that the problem may be NP-hard in general, but for small T or special cases, DP works.
In a real-world ad serving system, exact solutions may be infeasible due to scale. Discuss heuristics: e.g., greedy selection based on revenue density, or using a sliding window to maintain a set of candidates and selecting the highest revenue ad that doesn't violate the constraint. Mention that for the reusable case, one might use a priority queue to repeatedly select the best ad until the constraint is tight.
Compare the DP solution with heuristics in terms of optimality and runtime. Discuss potential extensions: non-uniform window sizes, multiple constraints, or online arrival of ads. Conclude with a recommendation based on the problem scale and requirements.
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