I knew this was a greedy problem pretty fast but fumbled explaining WHY the greedy choice is safe.
Clarify the problem as scheduling to maximize the number of on-time tasks, then propose a greedy algorithm using a max-heap to track selected tasks. Sort tasks by deadline, add each task's duration to the heap, and if the total time exceeds the current deadline, remove the longest task. This yields an O(n log n) solution; argue correctness via an exchange argument showing that at each step the greedy choice is safe.
Pro tip: Emphasize that the algorithm maintains a set of tasks that can be completed by the current deadline, and removing the longest task when over capacity is optimal because it frees the most time for future tasks. This demonstrates a deep understanding of greedy exchange arguments, which Amazon values.
Restate the problem: given n tasks with durations and deadlines, schedule them sequentially to maximize the number completed by their deadlines. Confirm that tasks are non-preemptive and can be done in any order.
Sort tasks by deadline. Iterate through tasks, adding each to a max-heap (by duration) and accumulating total time. If total time exceeds the current task's deadline, remove the task with the largest duration from the heap and subtract its duration from total time.
Sorting takes O(n log n). Each task is inserted and possibly removed from the heap once, each operation O(log n). Total time O(n log n), space O(n).
Use an exchange argument: at each step, the heap contains a maximum-size subset of tasks that can be completed by the current deadline. If adding a task causes infeasibility, removing the longest task yields a feasible set of the same size, and any optimal solution can be transformed to include this choice without reducing the count.
Consider tasks with equal deadlines, zero durations (if allowed), and the case where all tasks can be completed. Mention that the algorithm is optimal and efficient, but if tasks had weights, a different approach (e.g., dynamic programming) would be needed.
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