Spent the first few minutes trying to think of a greedy approach, like always scaling the cheapest bottleneck first.
Model the problem as maximizing the minimum throughput after allocating budget to scale services, where scaling multiplies throughput. Use binary search on the target throughput to check feasibility: for each service, compute the minimum scaling factor needed to reach the target, then sum the costs and compare with the budget. Optimize by considering discrete scaling options or continuous cost functions.
Pro tip: Mention that in practice, scaling often has diminishing returns and discrete steps, so a greedy approach may not be optimal; binary search on the answer is a robust technique for bottleneck optimization problems.
Clarify that total throughput is the minimum of all services' throughputs, and scaling a service multiplies its throughput by a factor. Define the budget and cost function for scaling each service.
For a given target throughput T, determine if it's possible to scale services so that each service's throughput ≥ T, and the total cost ≤ budget. This involves computing the required scaling factor for each service and summing costs.
Use binary search on T if the feasibility check is monotonic (higher T requires more budget). Alternatively, if scaling options are discrete, consider dynamic programming or greedy with priority queue, but binary search is often simpler.
For each service, given current throughput and cost function, compute the minimal cost to achieve at least T. Sum these costs and compare with budget. Optimize by precomputing or using mathematical formulas.
Discuss time complexity (e.g., O(N log(maxT)) for binary search) and handle cases where scaling is not possible (e.g., budget insufficient even for base throughput). Mention that if scaling factors are continuous, the problem may be solvable analytically.
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