Design dynamic connectivity with alive nodes

Q: Design dynamic connectivity with alive nodes

This question evaluates understanding of dynamic graph data structures, connectivity algorithms, state management for node activation/deactivation, complexity analysis, and concurrent operation considerations.

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Question

Design a data structure to maintain an undirected graph of N nodes, where each node has a boolean 'alive' flag. Support the following operations efficiently for up to 100,000 operations: (

activate(i) / deactivate(i) to toggle a node's alive status; (
connect(u, v) that adds an edge only if both endpoints are currently alive; (
disconnect(u, v) to remove an existing edge; (
isConnected(u, v) that returns whether u and v are in the same connected component considering only alive nodes; and (
countComponents() over alive nodes. Describe data structures, time/space complexities, and how you would handle deletions and node deactivations (e.g., lazy deletion vs. fully dynamic connectivity). Discuss any concurrency considerations if these operations are called from multiple threads.

Design dynamic connectivity with alive nodes

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