Model the courses and prerequisites as a directed graph and use topological sorting to detect cycles and produce a valid order. Explain that if a cycle exists, no valid ordering is possible; otherwise, return the topological order. Discuss both Kahn's algorithm (BFS) and DFS-based approaches, and analyze time and space complexity.
Pro tip: Clarify edge direction upfront: an edge from prerequisite to course ensures correct ordering. Mention that Kahn's algorithm naturally detects cycles when the processed count is less than the total courses.
Represent each course as a node and each prerequisite as a directed edge from the prerequisite to the course. Build an adjacency list and compute in-degrees for all nodes.
Select either Kahn's algorithm (BFS with a queue) or DFS with cycle detection. Explain the trade-offs: Kahn's is iterative and easy to detect cycles; DFS uses recursion and can be simpler to code.
For Kahn's: repeatedly remove nodes with in-degree 0, add to order, and decrement in-degrees of neighbors. If the final order size is less than the number of courses, a cycle exists. For DFS: track visited and recursion stack to detect back edges.
If no cycle, return the topological order as the valid course sequence. If a cycle exists, return an empty list or indicate impossibility.
State that both approaches run in O(V + E) time and O(V + E) space, where V is the number of courses and E is the number of prerequisites.
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