My first instinct was to just run Kahn's algorithm and call it done.
Clarify that enumerating all topological orderings is exponential in the worst case, then present a backtracking algorithm that repeatedly selects any node with in-degree zero, recurses, and backtracks. Discuss complexity and optimizations like using a priority queue for lexicographic order or pruning based on constraints.
Pro tip: Mention that the number of topological orderings can be exponential (e.g., an independent set of n nodes has n! orderings), so you should ask whether the interviewer wants all orderings or just a count, and if the graph is large, propose a randomized sampling approach.
Ask whether the graph is guaranteed acyclic, whether nodes are labeled, and if the output should be all orderings or just a count. Also discuss input size limits to choose between exact enumeration and approximation.
Describe maintaining in-degree counts and a set of available nodes (in-degree zero). At each step, pick an available node, add it to the current ordering, decrement in-degrees of its neighbors, and recurse; then backtrack.
State that the time complexity is O(V+E) per ordering and O(V! ) in the worst case. Mention optimizations: using a priority queue for lexicographic order, pruning if only a subset is needed, or using DP for counting.
For counting only, mention DP over subsets (O(2^V * V)) or inclusion-exclusion. For sampling, mention random topological orderings via random selection of available nodes.
Walk through a small DAG (e.g., 3 nodes) to demonstrate the algorithm. Discuss edge cases: empty graph, single node, disconnected components, and graphs with multiple valid orderings.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.