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Classic topological sort or DFS with a visited-state tracker.
Use DFS with a recursion stack to detect back edges, or Kahn's algorithm for topological sorting. Clearly state the time and space complexity, and discuss trade-offs between the two approaches.
Pro tip: Mention that Kahn's algorithm can also provide a topological order if no cycle exists, which is useful for dependency resolution. Also, clarify that the graph may not be connected, so you must check all nodes.
Confirm that the graph is directed, may have multiple components, and that we need to detect any cycle. Ask if self-loops or parallel edges are possible.
Select either DFS with recursion stack or Kahn's algorithm. Explain why you chose it, considering factors like ease of implementation, need for topological order, or memory constraints.
Describe the steps: For DFS, mark nodes as unvisited, visiting, or visited; if a visiting node is encountered, a cycle exists. For Kahn's, compute in-degrees, enqueue nodes with in-degree 0, and count processed nodes; if count < total nodes, a cycle exists.
State that both algorithms run in O(V+E) time and O(V) space. Mention that DFS uses recursion stack space, while Kahn's uses a queue.
Cover disconnected graphs, self-loops, and large graphs. Mention that if a cycle is found, you might want to output the cycle nodes, which DFS can do easily.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Clarify that any edge in a cycle can be removed to break it, but the choice depends on trade-offs like minimizing disruption or preserving important dependencies. Then propose a concrete algorithm: detect cycles using DFS or Kahn's algorithm, and during detection, identify a back edge in the DFS tree—removing that back edge guarantees acyclicity.
Pro tip: Mention that in real systems, you often want to remove the edge that is least critical or easiest to refactor, and that you can use topological sort to verify the result. This shows you think beyond the algorithm to practical impact.
Confirm whether the goal is to remove any edge that breaks all cycles, or to minimize disruption. This affects which edge you choose.
Use DFS with a recursion stack or Kahn's algorithm to detect cycles. During DFS, classify edges as tree, forward, back, or cross.
A back edge (an edge from a node to an ancestor in the DFS tree) always indicates a cycle. Removing any back edge breaks that cycle and makes the graph acyclic.
After removal, run a topological sort or cycle detection again to ensure no cycles remain.
Explain that if multiple cycles exist, removing one back edge may not break all cycles; you might need to remove multiple edges. Also consider edge importance.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.