My first instinct was to panic internally because maximum independent set is NP-hard and I almost said that out loud.
Model the grid as a bipartite graph (checkerboard coloring) and recognize that the maximum independent set in a bipartite graph equals total vertices minus minimum vertex cover, which by König's theorem equals total vertices minus maximum matching. Since existing trees block adjacent cells, remove them and their neighbors, then compute maximum matching on the remaining grid to find the maximum number of additional trees.
Pro tip: Mention that for large grids, Hopcroft-Karp is preferred over simple DFS augmenting paths, and discuss the trade-off between exact matching and greedy heuristics for real-time systems.
Ask about grid size, number of existing trees, and whether diagonal adjacency matters. Confirm that only orthogonal adjacency is prohibited.
Color the grid like a chessboard. Each empty cell is a vertex; edges connect adjacent empty cells. Existing trees and their neighbors are removed.
The maximum number of non-adjacent cells (independent set) equals total empty cells minus minimum vertex cover, which equals total empty cells minus maximum matching.
Use Hopcroft-Karp for O(E√V) or simpler DFS-based augmenting paths for smaller grids. Return the size of the independent set.
Compare exact matching vs. greedy heuristics. Mention memory/time trade-offs and potential parallelization for NVIDIA GPUs.
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