Lexicographic Grid Travel.

To solve the Lexicographic Grid Travel problem, we need to find the optimal path considering both time and cost, prioritizing time first, then cost. The approach depends on whether the time and cost per step are uniform across all cells or vary based on specific cell values.

1. Uniform Time/Cost per Step: If the time and cost for each step are constant regardless of the cell, we can use Breadth-First Search (BFS). BFS naturally finds the shortest path in terms of the number of steps. The total time will be the number of steps multiplied by the uniform time per step, and the total cost will be the number of steps multiplied by the uniform cost per step. The tie-breaking condition is to choose the path with fewer steps if times are equal, which BFS handles directly. The objective is to minimize (total time, total cost) lexicographically.

2. Variable Time/Cost per Step: When time and cost vary per cell (provided as matrices), BFS is insufficient because the cost of traversing an edge depends on the destination node. In this scenario, Dijkstra's algorithm is the appropriate choice. For each of the four possible modes of transportation (bicycle, bike, car, bus), we run Dijkstra's algorithm independently. Dijkstra's algorithm will maintain the best-known lexicographical pair of (total time, total cost) to reach each cell. When exploring paths, we relax an edge if the new path to a cell offers a strictly better lexicographical pair (time, cost) than the current best for that cell. After running Dijkstra for each mode, we compare the final (total time, total cost) from the destination cell across all modes and select the overall best path.