# 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 Given a scene or dependency graph, implement topological sort, BFS/DFS, and shortest path (Dijkstra). Discuss time/space complexity, memory layouts (CSR vs adjacency lists), and how you would parallelize traversals on CPU vs GPU while avoiding contention and ensuring determinism in a test framework. ## Recommended Approach Model each reachable configuration as a graph state and choose the traversal by edge cost: BFS for unweighted shortest paths, Dijkstra for non-negative weighted paths, or topological DP for DAGs. Track visited states at the correct 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.