Pretty standard union-find setup once you see it.
Model the problem as dynamic connectivity: process friendship events in chronological order, maintaining connected components. Use Union-Find (Disjoint Set Union) with union by rank and path compression to efficiently merge components, and track the number of components to detect when it becomes 1.
Pro tip: Mention that if the events are not sorted by time, you must sort them first (O(m log m)), and clarify whether the input is guaranteed sorted. Also, discuss the trade-off between Union-Find and other approaches like BFS/DFS for each time step, emphasizing Union-Find's near-constant time per operation.
Ask if the events are sorted by timestamp, the range of n and number of events, and whether multiple events can have the same timestamp. This affects sorting and tie-breaking.
Select Union-Find (Disjoint Set Union) with path compression and union by rank/size for efficient merging and component counting.
Iterate through events sorted by time. For each event, union the two people if they are in different components, and decrement the component count accordingly.
After each union, if the number of components becomes 1, return the current timestamp. If all events are processed without reaching 1, return -1.
State time complexity: O(m α(n)) for processing after sorting, where m is number of events and α is the inverse Ackermann function. Space: O(n).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify that deletions make the problem dynamic and that a simple union-find is insufficient because it doesn't support edge removal. Then propose a robust approach like maintaining a dynamic connectivity structure (e.g., Euler Tour Trees or Link-Cut Trees) or using offline processing with divide-and-conquer over time and rollback union-find to find the earliest time when the graph becomes connected.
Pro tip: Mention that if the graph is guaranteed to become connected and stay connected, you can binary search on time using a dynamic connectivity check, but be prepared to discuss the trade-offs of different data structures.
Confirm that unfriend events remove edges, and we need the earliest time when the graph is connected (all nodes in one component). Ask if the graph is guaranteed to become connected eventually and if nodes can be isolated.
Explain that deletions break standard union-find, so we need a dynamic connectivity data structure or an offline algorithm that handles edge removals.
Describe an offline approach: process events in reverse time (treating deletions as additions) and use union-find to find the earliest time the graph becomes connected, or use a dynamic connectivity structure like Euler Tour Trees for online queries.
Compare approaches: offline reverse processing is simpler and O((V+E) α(V)) but requires all events upfront; online dynamic connectivity is more complex but handles streaming. Discuss memory and time complexity.
Consider cases where the graph never becomes connected, multiple components, or when the earliest time is before any deletions. Also discuss how to track the number of connected components.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.