My first instinct was DFS with memoization and that ended up being the right call.
Model the grid as a directed acyclic graph where edges go from higher to lower elevation, then use DFS with memoization to compute the longest path from each cell. Start from the given position and return the maximum length found.
Pro tip: Clarify whether the skier can revisit cells or if the path must be simple; the strictly lower elevation condition guarantees no cycles, so memoization is safe. Also, mention that you can optimize space by using a 2D array for memoization and early termination if the current path length plus remaining possible cells cannot exceed the best found.
Confirm that the skier can only move to strictly lower elevations, that diagonal moves are not allowed, and that the path length includes the starting cell. Ask if the grid can be modified or if extra space is allowed.
For a cell (i, j), the maximum path length is 1 + max of the lengths from all valid neighbors with lower elevation. If no such neighbor exists, the length is 1.
Use a memoization table (e.g., 2D array) to store computed lengths for each cell to avoid redundant work. Alternatively, sort cells by elevation and compute lengths in increasing order (dynamic programming).
Write a recursive function that checks the memo table first, then explores all four directions, updating the maximum length. Ensure to mark cells as visited or use the memo table to avoid recomputation.
Time complexity is O(m*n) because each cell is computed once. Space complexity is O(m*n) for the memo table. Discuss edge cases like a single cell, all equal elevations, or a strictly decreasing grid.
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Recognize this as a multi-source dynamic programming problem where each starting cell is a source. Precompute the score for all cells in a single pass using a multi-source BFS/DFS (or topological order if DAG), then answer each query in O(1) by lookup. Emphasize that this avoids redundant computation and scales well with many skiers.
Pro tip: Mention that you can further optimize by grouping skiers with the same starting cell or using memoization with a shared cache, and discuss trade-offs like memory vs. time. This shows you think about practical constraints and not just the algorithm.
Confirm the definition of 'score' (e.g., longest increasing path, max sum, etc.) and whether the grid is static. Ask about the number of skiers, grid size, and if multiple skiers can start at the same cell.
Recognize that computing scores from many starting points is equivalent to a multi-source traversal. If the score is based on longest path in a DAG (e.g., strictly decreasing heights), use topological sort or DFS with memoization to compute all scores in one pass.
Initialize a score array. For multi-source BFS/DFS, enqueue all starting cells with their initial scores, then propagate. For DAG, compute in topological order from sinks to sources. Ensure each cell's score is computed once.
After precomputation, each skier's score is simply the precomputed value at their starting cell. This gives O(1) per query after O(V+E) preprocessing.
Compare with recomputing per skier (O(K*(V+E)) vs O(V+E + K)). Mention memory usage, potential for parallelization, and handling dynamic updates if needed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.