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Model the grid as a weighted graph where each cell is a node and edges connect adjacent cells. For each transportation mode, run Dijkstra's algorithm to find the shortest path from start to destination, using time as the primary weight and cost as the tiebreaker. Compare the results across modes and return the path with the minimum time, breaking ties by minimum cost.
Pro tip: Clarify whether the path must be strictly single-mode or if switching modes is allowed; if single-mode, precompute the cost and time for each mode separately. Also, consider using a priority queue that orders by (time, cost) to handle ties efficiently.
Confirm that the path must use a single transportation mode throughout, and that movement is only horizontal/vertical. Ask about tie-breaking rules and whether the start/destination cells have mode labels.
Represent each cell as a node. For each mode, define edge weights as (time, cost) for moving between adjacent cells, using the mode's cost and time arrays. Obstacles are impassable.
For each transportation mode, run Dijkstra's algorithm from start to destination, using time as the primary key and cost as the secondary key. Keep track of the best path for each mode.
Among the modes that reach the destination, select the one with the smallest total time. If there's a tie in time, choose the one with the smallest total cost.
Discuss time complexity: O(M * N log(MN)) per mode, where MxN is grid size. Mention potential optimizations like early termination or bidirectional search.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Model the problem as a shortest path on an expanded state graph where each node is (location, mode, switches_used). Apply Dijkstra's algorithm with edge weights that include the switch penalty when the mode changes, and prune states exceeding the switch limit. Then discuss how to optimize for large graphs and the trade-offs between state space size and solution accuracy.
Pro tip: Explicitly state that the state space grows as O(V * M * K) where V is locations, M is modes, and K is max switches, and mention that you can use a priority queue with lazy deletion to handle the increased state count efficiently. This shows you understand the scalability implications before being asked.
Represent each state as (node, current_mode, switches_used). Explain that this captures all necessary information to make optimal decisions without violating the switch limit.
For each edge, add transitions that keep the same mode (cost = edge weight) and transitions that change mode (cost = edge weight + penalty), but only if switches_used < K.
Run Dijkstra from the start state (source, initial_mode, 0) to any state at the destination. Use a priority queue and track the minimum cost to each state.
Prune any state where switches_used would exceed K. Alternatively, if K is small, you can run a layered Dijkstra where each layer corresponds to the number of switches used.
Discuss time and space complexity: O((V*M*K) log(V*M*K)) time and O(V*M*K) space. Mention that if K is large, the state space explodes, and consider alternative approaches like A* or bidirectional search.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.