Knew it was a graph problem pretty fast, but I fumbled the cycle detection part.
Model the courses as a directed graph where edges represent prerequisites, then detect cycles using either Kahn's algorithm (BFS-based topological sort) or DFS with recursion stack. If a topological ordering exists, all courses can be completed; otherwise, a cycle indicates impossibility.
Pro tip: Clarify edge direction upfront (e.g., prerequisite → course) and mention that Kahn's algorithm naturally provides the order and detects cycles, which is often preferred in interviews for its iterative nature and easy complexity analysis.
Confirm input format (e.g., number of courses and list of prerequisite pairs) and define the graph: nodes are courses, directed edges represent prerequisites. Ask if there are any constraints like disconnected components or self-loops.
Select either Kahn's algorithm (BFS with in-degree tracking) or DFS with cycle detection. Explain why one is preferred (e.g., Kahn's avoids recursion depth issues and directly yields a topological order).
Describe the steps: for Kahn's, compute in-degrees, enqueue nodes with in-degree 0, process until queue empty, decrement in-degrees of neighbors. For DFS, mark nodes as unvisited, visiting, or visited, and detect back edges.
State time and space complexity: O(V+E) for both approaches. Discuss edge cases: empty input, no prerequisites, self-loop, disconnected graph, and large input sizes.
Conclude whether all courses can be completed based on cycle detection. Mention potential optimizations like early termination if a cycle is found or using iterative DFS to avoid stack overflow.
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