Kahn's algorithm with in-degree tracking is probably the cleaner approach here since you get cycle detection for free when the queue empties before processing all nodes.
Model the courses as a directed graph where edges represent prerequisites, then use Kahn's algorithm (BFS-based topological sort) to detect cycles and produce an ordering. If the ordering includes all courses, return it; otherwise, report that no valid ordering exists due to a cycle.
Pro tip: Mention that Kahn's algorithm naturally detects cycles by comparing the output size to n, and that it's often preferred over DFS for its iterative nature and easier cycle detection. Also, discuss how to handle large inputs efficiently with O(V+E) time and space.
Confirm that courses are labeled 0 to n-1, prerequisites are given as pairs [a, b] meaning b must be taken before a, and that multiple valid orderings may exist. Ask about input size constraints and whether the graph is guaranteed to be connected.
Create an adjacency list for the directed graph and an array to track the in-degree (number of prerequisites) for each course. Iterate through the prerequisite pairs to populate both.
Add all courses with in-degree 0 to a queue (or list) as they have no prerequisites and can be taken first.
While the queue is not empty, dequeue a course, add it to the result order, and for each of its neighbors, decrement their in-degree. If any neighbor's in-degree becomes 0, enqueue it.
After processing, if the result order contains all n courses, return it as a valid ordering. Otherwise, a cycle exists, so return an empty array or indicate impossibility.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.