The PriorityQueue implementation is inefficient.

• add uses self.data.sort(key=self.f) which takes O(N log N) time for each addition, where N is the number of elements in the queue.
• pop uses self.data.pop(0) which takes O(N) time as it involves shifting all subsequent elements.

For search algorithms where the frontier can grow large, this quadratic complexity (O(N^2) or worse for add/pop operations over many iterations) will significantly degrade performance. Python's built-in heapq module provides an efficient min-heap implementation, allowing O(log N) for both add (push) and pop operations.

Consider replacing this custom implementation with heapq for better performance and scalability. For example:

import heapq
import itertools # To ensure stable sorting for equal f_values

class PriorityQueue:
    def __init__(self, f):
        self.data = []  # Stores (f_value, entry_id, node) tuples
        self.f = f
        self.counter = itertools.count() # Unique sequence numbers for tie-breaking

    def add(self, node):
        count = next(self.counter)
        heapq.heappush(self.data, (self.f(node), count, node))

    def pop(self):
        return heapq.heappop(self.data)[2] # Return the node

    def top(self):
        return self.data[0][2] # Return the node

    def is_empty(self):
        return len(self.data) == 0

The rows and cols variables are currently global. While this works for a single, fixed grid, it can lead to maintainability issues if you need to reuse the Problem class with different grid sizes or in a larger application where globals might conflict.

It's generally better practice to encapsulate grid dimensions within the Problem class itself. You can derive rows and cols from self.grid in the Problem's __init__ method and then use self.rows and self.cols throughout the class and in functions like BEST_FIRST_SEARCH.

Suggestion:

1. Update Problem.__init__:

   class Problem:
       def __init__(self, initial, goal, grid):
           self.initial = initial
           self.goal = goal
           self.grid = grid
           self.rows = len(grid)
           self.cols = len(grid[0]) if grid else 0 # Handle empty grid case

2. Update Problem.ACTIONS:

   # ...
           if 0 <= nr < self.rows and 0 <= nc < self.cols:
   # ...

3. Update BEST_FIRST_SEARCH:

   # ...
       reached = [[None for _ in range(problem.cols)] for _ in range(problem.rows)]
   # ...

This approach makes the Problem class more self-contained and robust.