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The base structure is Dijkstra with state (row, col, fuel), which I got to pretty quickly.
Model the problem as a shortest path on an expanded state space where each state is (row, col, fuel). Use Dijkstra's algorithm to find the minimum cost path, since edge weights (cell costs) are non-negative. Handle blocked cells, refuel cells, and fuel constraints during state transitions.
Pro tip: Mention that the state space size is O(m*n*K), so for large K you might need to optimize memory or consider alternative approaches like 0-1 BFS if costs are uniform. Also, clarify that refuel cells reset fuel to K upon entry, and blocked cells are impassable.
Each state is (r, c, f) where r and c are grid coordinates and f is remaining fuel (0 to K). The start state is (0,0,K) with initial cost equal to the cost of cell (0,0).
From (r,c,f), move to adjacent cells (up, down, left, right) if within bounds, not blocked, and f > 0. The new fuel is f-1, except if the destination is a refuel cell, then new fuel is K. The cost added is the cost of the destination cell.
Use Dijkstra's algorithm with a priority queue to explore states in increasing order of total cost. Maintain a distance array or map for each state to avoid revisiting with higher cost.
If the start or end cell is blocked, return -1. When the end cell is reached, return the accumulated cost. If the priority queue empties without reaching the end, return -1.
The number of states is O(m*n*K). Each state has up to 4 transitions, so time complexity is O(m*n*K log(m*n*K)) with a binary heap. Space complexity is O(m*n*K) for the distance array.
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This one I didn't have a clean answer for on the spot.
First, clarify the problem context and the role of K in the state space. Then, propose optimizations such as reducing the state space via mathematical insights, using advanced data structures, or applying algorithmic paradigms like divide-and-conquer or dynamic programming with state compression. Finally, discuss trade-offs and validate the approach with complexity analysis.
Pro tip: Demonstrate awareness of the practical constraints by mentioning that sometimes approximate or heuristic solutions are acceptable, and always relate optimizations back to the specific problem requirements.
Restate the problem to ensure understanding and identify why K causes the state space to explode. Ask clarifying questions if needed.
Consider if K can be reduced by mathematical properties (e.g., periodicity, monotonicity) or if the problem can be transformed to eliminate K.
Suggest techniques like state compression, sliding window, divide-and-conquer, or using advanced data structures (e.g., segment trees, Fenwick trees) to reduce time/space complexity.
Compare the optimized approach with the original, discussing time and space complexity improvements and any assumptions or limitations.
Walk through a small example or edge case to demonstrate the optimization works and handles large K efficiently.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.