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Model the courses and prerequisites as a directed graph where edges point from prerequisite to dependent course. Then, determine if the graph is a DAG by attempting a topological sort using either Kahn's algorithm (BFS) or DFS cycle detection. If a topological ordering exists, all courses can be completed; otherwise, a cycle prevents completion.
Pro tip: Clarify edge direction upfront (b → a for [a, b]) to avoid off-by-one errors, and mention that Kahn's algorithm naturally detects cycles by counting processed nodes. Also, discuss how to handle disconnected graphs and large inputs efficiently.
Confirm that prerequisite pairs [a, b] mean b must be taken before a, and ask about constraints like n up to 10^5, possible duplicate edges, or self-loops. Discuss edge cases: no prerequisites, empty course list, or a single course.
Represent courses as nodes 0 to n-1 and each prerequisite pair as a directed edge from b to a. Use an adjacency list for efficient traversal and an in-degree array to track prerequisites count for each course.
Select either Kahn's algorithm (BFS-based topological sort) or DFS with recursion stack. Explain the trade-offs: Kahn's is iterative and avoids recursion depth issues; DFS can be simpler but may need iterative implementation for large graphs.
For Kahn's: initialize a queue with courses having in-degree 0, process them while decrementing in-degrees of neighbors, and count processed courses. If count equals n, no cycle; else, cycle exists. For DFS: track visited and recursion stack states to detect back edges.
State that both approaches run in O(n + m) time and O(n + m) space, where m is the number of prerequisite pairs. Conclude whether all courses can be completed based on cycle detection result.
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