Solution Explanation for Shortest Path in a Hidden Grid

This problem involves finding the shortest path in a hidden grid using a GridMaster API. We don't know the grid's dimensions or the locations of the start and target cells. We only have access to the GridMaster's methods: canMove, move, and isTarget.

The solution uses a two-stage approach:

Stage 1: DFS (Depth-First Search) for Exploration

1. Initialization: We assume the robot starts at (0,0). We use a set called vis to track visited cells and a variable target to store the coordinates of the target cell once found.

2. Recursive DFS: The dfs function recursively explores the grid. For each direction ('U', 'D', 'L', 'R'), it checks if a move is possible using canMove. If possible and the cell hasn't been visited, it marks the cell as visited (vis.add), makes the move (master.move), and recursively calls dfs from the new position. If isTarget() returns true, the target is found, its coordinates are stored in target, and the recursion unwinds.

3. Backtracking: After exploring a direction, the robot moves back to its previous position using the opposite direction (master.move(opposite_direction)). This ensures that all reachable cells are explored.

Stage 2: BFS (Breadth-First Search) for Shortest Path

1. BFS Initialization: Once the target is found (if target is not None), we perform a BFS starting from (0,0) to find the shortest path. vis is cleared (excluding the starting point) to be reused for BFS.

2. BFS Traversal: The BFS uses a queue. In each iteration, we process all nodes at the current distance from the starting point. If the target is found, the current distance (ans) is the shortest path length.

3. Shortest Path: The BFS guarantees that the first time we reach the target, we have found the shortest path.

Time and Space Complexity Analysis

• Time Complexity: The DFS explores all reachable cells in the grid. In the worst case, this could involve visiting every cell, resulting in O(mn) time complexity, where 'm' and 'n' are the grid's dimensions. The BFS also has a time complexity of O(mn) in the worst case, as it might need to visit all cells. Therefore, the overall time complexity is O(m*n).

• Space Complexity: The vis set stores the visited cells, which in the worst case can contain all the cells in the grid, leading to O(mn) space complexity. The queue used in BFS can also store up to O(mn) elements in the worst case. The overall space complexity is O(m*n).

Code Examples (Python, Java, C++)

The code examples are provided in the original response. They implement the described two-stage approach using DFS for exploration and BFS for finding the shortest path. The code handles edge cases such as the target being unreachable. Note that the specific implementation details (like the use of deque for the queue in Python or ArrayDeque in Java) can slightly affect the constants but not the overall time and space complexity.