Clarify the problem constraints and define the cost function for a single shipment, then generalize to multiple shipments. For each shipment, compute the minimum tokens by considering the trade-off between adding capacity to the main warehouse versus the backup warehouses, ensuring the main capacity requirement is met first and excess does not count toward backup. Use a greedy or binary search approach to find the optimal allocation.
Pro tip: Demonstrate awareness of edge cases such as when the main warehouse already meets the requirement or when backup capacity is distributed across multiple warehouses. Also, mention that since costs reset, each shipment is independent, so you can solve each in isolation.
Restate the problem in your own words and ask clarifying questions to ensure you understand the inputs, outputs, and constraints. Confirm that each token adds 1 unit to any warehouse, costs reset per shipment, and excess main capacity cannot be used for backup.
For a single shipment, determine the minimum tokens needed. Let the main warehouse have current capacity C_main and required minimum R_main. If C_main < R_main, you must add at least R_main - C_main tokens to the main warehouse. Then, for backup, you need the sum of capacities of all other warehouses to be at least R_backup. If the current sum is S_backup, you need max(0, R_backup - S_backup) additional tokens distributed among backup warehouses. The total tokens is the sum of these two amounts, but note that adding tokens to the main warehouse beyond R_main does not help backup.
Check if there is a scenario where adding more than the minimum to the main warehouse could reduce total tokens? Since excess main capacity cannot spill over, adding extra to main never helps backup. However, if the main warehouse is also part of the backup set? Typically, the main warehouse is separate from backup warehouses. Clarify this. If the main warehouse is included in the backup sum, then adding to main could help both, but the problem states 'excess main warehouse capacity cannot spill over into the backup total', implying main is separate. So the minimal tokens is simply the sum of deficits.
Since costs reset between shipments, process each shipment independently. For each shipment, compute the tokens as described and sum them? Wait, the question asks for 'minimum tokens needed per shipment', so likely output an array of tokens per shipment. Clarify if total tokens across all shipments is needed or per shipment. The phrasing 'per shipment' suggests per shipment. So for each shipment, compute the minimum tokens and return a list.
Walk through a simple example to verify your logic. Consider edge cases: main capacity already sufficient, backup sum already sufficient, zero capacities, large numbers. Ensure your solution handles these efficiently.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.