Find a Lexicographically Optimal Robot Path with Charging Cells

Question

The question asks the candidate to develop an algorithm that finds a lexicographically optimal path for a robot in a grid, considering charging cells to minimize battery usage and movement steps.

Codemia · Accepted Answer

To solve this problem, we will use a modified breadth-first search (BFS) algorithm. The BFS will explore all possible paths from the start to the end point while keeping track of the number of charging cells used, the maximum battery required, and the total moves made. We will use a priority queue to ensure that we explore paths in lexicographical order based on the defined criteria.

1. Initialize a priority queue to store states as tuples of (charging_cells_used, max_battery_required, total_moves, current_position).  
2. Start from the initial position with zero charging cells used, zero battery required, and zero moves.  
3. While the queue is not empty, pop the state with the lowest priority.  
4. Check if the current position is the end position; if so, return the path.  
5. For each possible move (up, down, left, right), calculate the new state and push it to the queue if it has not been visited with a better score.

The time complexity of this algorithm is O(N log N), where N is the number of cells in the grid. This is due to the priority queue operations. The space complexity is O(N) for storing the visited states and the queue.

```python
from queue import PriorityQueue

def find_optimal_path(grid, start, end):
    rows, cols = len(grid), len(grid[0])
    directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
    pq = PriorityQueue()
    pq.put((0, 0, 0, start))  # (charging_cells, max_battery, moves, position)
    visited = set()

while not pq.empty():
        charging_cells, max_battery, moves, (r, c) = pq.get()
        if (r, c) == end:
            return moves
        for dr, dc in directions:
            nr, nc = r + dr, c + dc
            if 0 <= nr < rows and 0 <= nc < cols:
                new_charging_cells = charging_cells + (1 if grid[nr][nc] == 'C' else 0)
                new_max_battery = max(max_battery, new_charging_cells)
                pq.put((new_charging_cells, new_max_battery, moves + 1, (nr, nc)))
    return -1  # No path found
```

Consider scenarios such as:  
1. The grid is completely empty or filled with charging cells.  
2. The start and end positions are the same.  
3. There are no valid moves available due to obstacles.

Find a Lexicographically Optimal Robot Path with Charging Cells

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