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Model the courses and prerequisites as a directed graph, then perform a topological sort using either Kahn's algorithm (BFS with in-degree tracking) or DFS with cycle detection. If a cycle is detected, return an empty list; otherwise, return the topological ordering.
Pro tip: Mention that Kahn's algorithm naturally detects cycles when the number of processed nodes is less than n, and that it avoids recursion depth issues. Also, clarify that any valid topological order is acceptable, so you don't need to worry about a specific ordering.
Confirm that the input is a list of prerequisite pairs where [a, b] means b must be taken before a. Represent the courses as nodes and prerequisites as directed edges from b to a.
Decide between Kahn's algorithm (BFS with in-degree) or DFS with cycle detection. Both are O(V+E) time and space; choose the one you are most comfortable implementing.
For Kahn's: compute in-degrees, enqueue nodes with in-degree 0, then repeatedly dequeue and reduce in-degrees of neighbors. For DFS: perform DFS and add nodes to the result in post-order, while tracking visited and recursion stack to detect cycles.
If using Kahn's, check if the result size equals n; if not, a cycle exists, so return []. If using DFS, if a back edge is found, return []. Otherwise, return the topological order.
State that the time complexity is O(V+E) and space is O(V+E). Discuss edge cases: no prerequisites, disconnected graph, self-loop, and multiple valid orderings.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Break the problem into three distinct traversals: left boundary (excluding leaves), all leaves left-to-right, and right boundary (excluding leaves) collected bottom-up. Implement each traversal with careful condition checks to avoid duplicates and ensure correct ordering, then concatenate the results.
Pro tip: Clarify edge cases upfront, such as a single-node tree or skewed trees, and mention that the root is included only once even if it's also a leaf. This shows attention to detail and prevents off-by-one errors.
Confirm the exact definition of boundary nodes, especially edge cases like single-node trees and skewed trees. State that the root is always included, and leaves are only included in the leaves traversal.
Traverse from root's left child down the left side, adding nodes that are not leaves. Use a top-down approach, moving to left child if it exists, otherwise right child.
Perform a DFS (preorder) traversal, adding nodes that have no children. This ensures leaves are collected in left-to-right order.
Traverse from root's right child down the right side, adding non-leaf nodes, but store them in a stack or reverse the list to get bottom-up order.
Concatenate the three lists, ensuring no duplicates (e.g., root not added twice). Test with edge cases like empty tree, single node, and skewed trees.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.