# Solution Alignment The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests. ## Problem Restatement Part A: Implement a data structure for an ad-matching service that supports add(campaign), remove(campaign), and match(request), where a request has attributes (e.g., location, device, interests) and campaigns specify targeting predicates and budgets. Optimize for fast matching and frequent updates; discuss complexity and trade-offs. Part B: Given a weighted road graph with nonnegative edges, a depot, and N delivery stops, implement plan_route(graph, depot, stops) to produce a near-optimal route. Compare exact versus heuristic approaches (e.g., shortest paths plus TSP heuristics), analyze complexity, and handle practical constraints such as capacity or time windows. ## Recommended Approach Model the states explicitly and use BFS for unweighted shortest paths, Dijkstra for weighted non-negative paths, or topological DP for DAGs. Track visited states at the right granularity so cycles do not cause repeated work. ## Correctness The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call. ## Complexity BFS is O(V + E) time and O(V) space for a standard graph. Expanded-state problems multiply those bounds by the number of state dimensions. ## Edge Cases and Tests Disconnected graph, source equals target, cycles, duplicate edges, unreachable target, and whether the answer counts nodes, edges, moves, or transfers.