I'd seen topological sort problems before and thought I had it.
Start by clarifying the problem: it's a topological sort where each node has a DP value that can only be computed after all its prerequisites are processed. Use Kahn's algorithm with a queue, and when a node's in-degree becomes zero, compute its DP value based on its predecessors' values, then propagate to its successors. Emphasize that this is essentially dynamic programming on a DAG, and discuss handling cycles and multiple valid orders.
Pro tip: Mention that the DP constraint often requires aggregating values from all predecessors (e.g., max, sum), so you must store predecessor contributions or process edges carefully. Also, note that the order of processing nodes with zero in-degree doesn't affect the final DP values if the DP is well-defined, which is a key insight for correctness.
Ask questions to understand the exact DP constraint: what operation combines predecessor values? Are there multiple roots? Can there be cycles? This ensures you solve the right problem.
Represent the graph and define what DP value each node holds. Explain that the DP value for a node depends only on its predecessors' DP values, and it can be computed once all prerequisites are satisfied.
Select Kahn's algorithm (BFS-based) because it naturally tracks when a node's prerequisites are all processed (in-degree becomes zero). Alternatively, DFS-based topological sort can work but requires post-order processing.
When a node's in-degree reaches zero, compute its DP value by combining the DP values of its predecessors (e.g., take max, sum). Then, for each outgoing edge, update the successor's in-degree and possibly accumulate the predecessor's contribution.
Discuss cycles (detect if not all nodes processed), multiple valid topological orders (DP should be invariant), and time/space complexity (O(V+E) time, O(V) space).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.