Took me a second to realize this was just cycle detection in a directed graph with extra packaging.
Model the packages and dependencies as a directed graph, then detect cycles using either Kahn's algorithm (topological sort) or DFS with recursion stack. Clearly explain the graph representation, the cycle detection method, and analyze time and space complexity.
Pro tip: Mention that Kahn's algorithm is often preferred in production systems because it naturally handles large graphs and can detect cycles while producing an installation order. Also note that real package managers must handle version constraints and conflicts, so this is a simplified model.
Confirm that dependencies are directed edges (A -> B means A depends on B), and that we need to check if a valid installation order exists. Ask about input format (adjacency list, edge list) and constraints.
Represent packages as nodes and dependencies as directed edges. Use an adjacency list for efficiency, and optionally compute in-degrees for Kahn's algorithm.
Explain either Kahn's algorithm (repeatedly remove nodes with in-degree 0) or DFS with a recursion stack (detect back edges). Describe how each detects cycles.
Trace the algorithm on a small graph, e.g., A->B, B->C, C->A (cycle) and A->B, B->C (no cycle), to demonstrate correctness.
State time and space complexity (O(V+E) for both algorithms). Discuss edge cases: empty graph, self-loop, disconnected components, and large graphs.
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