Went with union-find pretty quickly, which felt right.
Model the grid as a dynamic connectivity problem where each added land cell may merge with adjacent islands. Use Union-Find (Disjoint Set Union) to efficiently track and update the number of islands after each addition, or alternatively BFS/DFS with careful incremental updates. Start with an empty grid and maintain a running count of islands, incrementing when a new cell is added and decrementing for each successful union with an existing island.
Pro tip: Emphasize that Union-Find with path compression and union by rank gives near O(1) amortized time per operation, making it optimal for this incremental problem. Also mention that you can avoid revisiting the entire grid by only checking the four neighbors of the newly added cell.
Confirm the grid dimensions, whether land cells are added at arbitrary positions, and that islands are defined by 4-directional adjacency. Discuss edge cases like duplicate additions or out-of-bounds coordinates.
Select Union-Find (Disjoint Set Union) to efficiently manage connected components, or BFS/DFS if you prefer a simpler but potentially less efficient approach. Explain why Union-Find is ideal for incremental connectivity.
For each added land cell, increment the island count, then check its four neighbors. For each neighbor that is land, if it belongs to a different set, union the two sets and decrement the island count.
Code the Union-Find with path compression and union by rank. Ensure that each addition is processed in near-constant time by only examining the four adjacent cells.
State the time complexity: O(k * α(mn)) where k is the number of additions and α is the inverse Ackermann function. Walk through a small example to verify correctness.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the original solution's data structures and whether deletions are arbitrary or only of previously added cells. Then, extend the approach by adding a removal operation that updates the same state (e.g., union-find with size tracking, or a counter with boundary checks) and handles edge cases like isolated cells or merging/splitting components.
Pro tip: Mention that deletions are often harder than additions because they can split components; if the problem allows, process operations offline in reverse to turn deletions into additions, which reuses your original solution.
Restate the data structures and invariants used for adding land cells, such as union-find for connectivity or a counter for islands.
Explain why deletion is harder: removing a cell can split a connected component, which union-find cannot handle directly.
Describe how to adapt the solution: either use a dynamic connectivity structure, or if operations are known in advance, process them offline in reverse.
Discuss cases like deleting an isolated cell, deleting a cell that disconnects a component, and analyze time/space complexity of the modified approach.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.