I knew the answer involved a hash map from old nodes to new ones, but I fumbled the explanation of why you need it.
Use a depth-first search (DFS) or breadth-first search (BFS) to traverse the graph while maintaining a hash map from original nodes to their copies. For each visited node, create a copy if it doesn't exist, then recursively (or iteratively) copy its neighbors and add them to the copy's neighbor list. Return the copy of the starting node.
Pro tip: Clarify that the graph is connected and undirected, and mention that you'll handle cycles using the hash map to avoid infinite recursion. Also, discuss edge cases like a single node with no neighbors or an empty graph (though the problem states connected, so at least one node).
Confirm that the graph is connected and undirected, and that each node has a unique value (or not, but typically values are unique). Ask if the graph can have cycles or self-loops, and whether the input node is guaranteed to be non-null.
Decide between DFS (recursive or iterative) and BFS. Use a hash map (dictionary) to map original nodes to their copies, which also serves as the visited set to handle cycles.
Start from the given node. If the node is already in the hash map, return its copy. Otherwise, create a new node with the same value, add it to the map, then recursively copy each neighbor and append the copies to the new node's neighbors list.
Walk through a simple graph (e.g., two nodes connected) and a graph with a cycle. Verify that the deep copy is independent (modifying the copy does not affect the original). Also consider a single node with no neighbors.
State that the time complexity is O(N + E) where N is the number of nodes and E is the number of edges, since each node and edge is visited once. Space complexity is O(N) for the hash map and recursion stack (or queue for BFS).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.