I went straight to a sweep-line approach and started talking through a sorted event list, open events, active shard counts.
Clarify the problem by defining what 'overlapping assignments' means and the constraints, then propose a sweep-line algorithm with a priority queue to track active ranges and assign shards greedily, ensuring the limit is not exceeded. Discuss how to handle nested ranges and identical endpoints by sorting events carefully and using tie-breaking rules.
Pro tip: Mention that the greedy approach is optimal for minimizing the maximum overlap if we assign shards to the range with the earliest end time first, similar to interval scheduling. Also, highlight that using a balanced BST or segment tree can efficiently query and update overlaps.
Ask questions to understand the exact meaning of 'overlapping assignments', the limit, and whether shards can be reassigned. Confirm if ranges are inclusive/exclusive and if endpoints can be identical.
Propose a sweep-line algorithm: sort all start and end events, use a priority queue (min-heap) to track active ranges by end time, and a counter for current assignments. For each start event, assign a shard if under limit, else rebalance by moving a shard from the range with the latest end time.
Address fully nested ranges by ensuring the sweep processes starts before ends at the same point, and identical endpoints by using stable sorting or tie-breaking. Discuss how to handle ranges with zero length or when limit is zero.
State that sorting takes O(n log n) time, and each event is processed in O(log n) with heap operations, leading to O(n log n) overall time. Space complexity is O(n) for storing events and active ranges.
Compare with alternative approaches like segment trees or interval graphs, and discuss trade-offs between time and space, and whether the greedy assignment is optimal for minimizing maximum overlap.
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