This is a connected components problem and I completely froze.
Model the servers as nodes and direct connections as edges in an undirected graph. The problem reduces to counting the number of connected components, which can be done using either BFS/DFS or Union-Find (Disjoint Set Union). Choose the method based on constraints and explain your reasoning.
Pro tip: At Amazon, interviewers value candidates who discuss trade-offs between approaches and consider edge cases like empty graphs or self-loops. Mention that Union-Find with path compression and union by rank is often more efficient for dynamic connectivity, but BFS/DFS is simpler if the graph is static.
Confirm that the network is undirected, connections are bidirectional, and isolated clusters are connected components. Ask about input format, constraints, and whether the graph can be modified.
Decide between BFS/DFS (traverse each component) and Union-Find (merge connected nodes). Consider time/space complexity and whether the graph is static or dynamic.
For BFS/DFS: iterate over all nodes, and for each unvisited node, perform traversal to mark its component and increment count. For Union-Find: initialize each node as its own parent, union connected nodes, then count unique roots.
State time and space complexity: BFS/DFS is O(V+E) time and O(V) space; Union-Find is O(E α(V)) time and O(V) space, where α is the inverse Ackermann function.
Walk through a small example, including edge cases like no edges (each node isolated) or all nodes connected (one cluster). Verify the count matches expectations.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.