You are given two independent coding problems. ### Problem 1: Maximize Pipeline Throughput A production pipeline has `n` services connected in series. Service `i` initially has one instance. One instance of service `i` can process `throughput[i]` requests per second. You may buy additional identical instances for any service; each extra instance of service `i` costs `cost[i]`. You have a total budget `B`. The capacity of service `i` after buying `k` extra instances is `(k + 1) * throughput[i]`. Because the services are connected in series, the overall pipeline throughput is the minimum capacity among all services. Return the maximum integer overall throughput that can be achieved without exceeding the budget. Clarifications: - You may buy any nonnegative integer number of extra instances for each service. - The total cost of all purchased instances must be at most `B`. - `n`, `throughput`, `cost`, and `B` may be large enough that trying every allocation is not feasible. ### Problem 2: Minimum Triggers to Absorb All Balls You are given `n` balls on a 2D integer grid. Ball `i` is located at `(x[i], y[i])`. You are also given an integer distance threshold `d`. If you manually trigger one ball, it absorbs itself and may start a chain reaction: - Two balls can directly absorb each other if they are in the same row and their horizontal distance is at most `d`, or if they are in the same column and their vertical distance is at most `d`. - Absorption is transitive: if ball `A` can absorb ball `B`, and ball `B` can absorb ball `C`, then triggering `A` eventually absorbs `C` as well. Return the minimum number of manual triggers required to absorb all balls. Clarifications: - A trigger can be applied only to a ball that has not already been absorbed. - The answer is the number of connected components under the row/column distance relationship. - The input size may be large, so comparing every pair of balls directly may be too slow.