Solution Explanation

This problem involves finding the minimum cost 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. The solution uses Depth-First Search (DFS) to explore the grid and Dijkstra's algorithm (implemented using a min-heap priority queue) to find the shortest path.

Approach

1. Exploration with DFS: A DFS function is used to initially explore the reachable cells in the grid. It starts from an arbitrary point (assumed to be in the center of a large enough grid to encompass the actual grid). As it moves, it records the cost to each reachable cell. If the target cell is found during the DFS, its coordinates are stored. The canMove and move functions from the GridMaster API are used to determine if a move is possible and to retrieve the cost of that move, respectively.
2. Dijkstra's Algorithm: Dijkstra's algorithm is employed to find the minimum cost path from the starting cell (which is assumed to be the initial position of DFS) to the target cell found in the DFS step. A priority queue is used to efficiently select the cell with the minimum cost at each step. The cost to reach each cell is updated iteratively, ensuring that the shortest path is discovered.

Time and Space Complexity

• DFS: The DFS explores a portion of the grid that is potentially the entire grid, leading to a time complexity of O(MN), where M and N are the dimensions of the hidden grid. The space complexity of DFS is O(MN) in the worst case, due to the recursive calls. We use a relatively large grid (200x200) to ensure that it will encompass the actual hidden grid.

• Dijkstra's Algorithm: Dijkstra's algorithm using a min-heap has a time complexity of O(E log V), where E is the number of edges (at most 4 * M * N, representing the four possible moves from each cell) and V is the number of vertices (MN). Therefore, the time complexity is approximately O(MN log(MN)). The space complexity is O(MN) to store distances.

• Overall Complexity: Since both DFS and Dijkstra's Algorithm contribute O(MN log(MN)) time complexity , this is the overall time complexity. The overall space complexity is O(M*N) to store the grid, costs and distances. In our case the grid is 200x200, so the overall time and space complexity can be considered O(1) with respect to input size.

Code Explanation (Python)

The Python code implements the described approach:

# ... (GridMaster API interface, as provided in the problem statement) ...
 
class Solution(object):
    def findShortestPath(self, master: 'GridMaster') -> int:
        # ... (DFS and Dijkstra's implementation as described above) ...

The DFS function explores the grid, recording costs and finding the target. Dijkstra's algorithm then finds the shortest path using a priority queue (heapq in Python). The code uses a sufficiently large grid (200x200) to handle various grid sizes.

Code Explanation (Java)

The Java code follows a similar structure to the Python solution:

// ... (GridMaster API interface, as provided in the problem statement) ...
 
class Solution {
    // ... (DFS and Dijkstra's implementation as described above) ...
}

The Java implementation uses PriorityQueue for the min-heap, and the core logic is the same as the Python version. The constants (N, INF, dirs, etc.) are defined for clarity and efficiency.

This detailed explanation should help you understand the solution's approach, implementation, and complexity analysis. Remember that the interactive nature of the problem necessitates the use of the GridMaster API to interact with the hidden grid.