# Create Mazes in Python: Using recursive  and A* search algorithm's

In this post describes you how to solve mazes problem using two algorithms implemented in Python program: a simple recursive algorithm and the A* search algorithm.this article help you to find the nearest cell by using python programming language.

## Maze

The maze we are going to use in this article is 6 cells by 6 cells. The walls are colored in blue. The starting cell is at the bottom left (x=0 and y=0) colored in green. The ending cell is at the top right (x=5 and y=5) colored in green. We can only move horizontally or vertically 1 cell at a time.

## Recursive walk

We use a nested list of integers to represent the maze. The values are the following:
·      0: empty cell
·      1: unreachable cell
·      2: ending cell
·      3: visited cell

grid =[[0, 0, 0, 0, 0, 1],
[1, 1, 0, 0, 0, 1],
[0, 0, 0, 1, 0, 0],
[0, 1, 1, 0, 0, 1],
[0, 1, 0, 0, 1, 0],

This is a very simple algorithm which does the job even if it is not an efficient algorithm. It walks the maze recursively by visiting each cell and avoiding walls and already visited cells.

The search function accepts the coordinates of a cell to explore. If it is the ending cell, it returns True. If it is a wall or an already visited cell, it returns False. The neighboring cells are explored recursively and if nothing is found at the end, it returns False so it backtracks to explore new paths. We start at cell x=0 and y=0.

def search(x, y):
if grid[x][y] == 2:
print ('found at %d,%d' % (x, y))
return True
elif grid[x][y] == 1:
print ('wall at %d,%d' % (x, y))
return False
elif grid[x][y] == 3:
print ('visited at %d,%d' % (x, y))
return False
print ('visiting %d,%d' % (x, y))
grid[x][y] = 3
if ((x < len(grid)-1 and search(x+1, y)) or (y > 0 and search(x, y-1)) or (x > 0 and search(x-1, y)) or (y < len(grid)-1 and search(x, y+1))):
return True
return False

search(0, 0)

Let’s see what happens when we run the script.

\$ python maze.py

visiting 0,0
wall at 1,0
visiting 0,1
wall at 1,1
visited at 0,0
visiting 0,2
...

First cell visited is (0,0). Its neighbors are explored starting by the one on the right (1,0). search(1,0) returns False because it is a wall. There is no cell below and on the left so the one at the top (0,1) is explored. Right of that is a wall and below is already visited so the one at the top (0,2) is explored.

Because the neighbor on the right is explored first, this algorithm is going to explore the dead-end at the bottom-right first.

...
visiting 1,2
visiting 2,2
wall at 3,2
visiting 2,1
wall at 3,1
visiting 2,0
visiting 3,0
visiting 4,0
visiting 5,0
...

The algorithm is going to backtrack because there is nothing else to explore as we are in a dead-end and we are going to end up at cell (1, 2) again where there is more to explore.

...
visited at 4,0
wall at 5,1
visited at 3,0
wall at 4,1
visited at 2,0
wall at 3,1
wall at 1,0
visited at 2,1
wall at 1,1
visited at 2,2
visited at 1,2
wall at 2,3
wall at 1,1
visited at 0,2
visiting 1,3
...

Let’s continue, we end up in a second dead-end at cell (4, 2).

...
wall at 2,3
visited at 1,2
visiting 0,3
visited at 1,3
visited at 0,2
visiting 0,4
visiting 1,4
visiting 2,4
visiting 3,4
wall at 4,4

visiting 3,3
visiting 4,3
visiting 5,3
visiting 5,2
wall at 5,1
visiting 4,2
visited at 5,2
wall at 4,1
wall at 3,2
visited at 4,3
...

Backtracking happens one more time to go back to cell (5, 3) and we are now on our way to the exit.

visiting 5,4
visited at 5,3
wall at 4,4
found at 5,5

## A* search

We are going to use a more tuff algorithm called A* search. This is based on costs to move around the grid. Let’s assume the cost to move horizontally or vertically 1 cell is equal to 25. Again, we cannot move diagonally here.

Before we start describing the algorithm, let’s define two variables: A and B. A is the cost to move from the starting cell to a given cell and B is an estimation of the cost to move from a given cell to the ending cell.

How will we do if we have to calculate that if we don’t know the path to the ending cell? To simplify, we just calculate the distance if no walls were present. There are other ways to do the estimation but this one is good enough for this in a python program.

We use two lists: an open list containing the cells to explore and a closed list containing the processed cells. We start with the starting cell in the open list and nothing in the closed list.

Let’s follow one round of this algorithm by processing our first cell from the open list. It is the starting cell. We remove it from the list and append it to the closed list. We retrieve the list of adjacent cells and we start processing them. The starting cell has 2 adjacent cells: (1, 0) and (0, 1).

(1, 0) is a wall so we drop that one. (0, 1) is reachable and not in the closed list so we process it. We calculate A and B for it. A = 1 as we just need to move 1 cell up from the starting cell. B = 3: 5 cells right and 4 cells up to reach the ending cell. We call the sum SUM = A + B = 1 + 3 = 4.

We set the parent of this adjacent cell to be the cell we just removed from the open list.

We continue with the cell in the open list having the lowest SUM = A + B. Only one cell is in the open list so it makes it easy. We remove it from the open list and we get its adjacent cells. Again, only one adjacent cell is reachable: (0, 2). We end up with the following after this 2nd round.
Let’s detail the next round. We have 2 cells in the open list: (1, 2) and (0, 3). Both have the same SUM value so we pick the last one added which is (0, 3). This cell has 3 reachable adjacent cells: (1, 3), (0, 2) and (0, 4). We process (1, 3) and (0, 4). (0, 2) is in the closed list.

We have (1, 2), (1, 3) and (3, 3) in the open list. (1, 3) is processed next because it is the last one added with the lowest SUM value = 100.

(1, 3) has 1 adjacent cell which is not in the closed list. It is (1, 2) which is in the open list. When an adjacent cell is in the open list, we check if its SUM value would be less if the path taken was going through the cell currently processed e.g. through (1, 3). Here it is not the case so we don’t update A and B of (1, 2) and its parent. This trick makes the algorithm more efficient when this condition exists.

We have 2 cells in the open list: (3, 3) and (2, 0). The next cell removed from the open list is (3, 3) because its F is equal to 120. This proves that this algorithm is better than the first one we saw. We don’t end up exploring the dead end at (5, 0) and we continue walking from (3, 3) instead which is better.

The next cell processed from the open list is (5, 5) and it is the ending cell so we have found our path. It is easy to display the path. We just have to follow the parent pointers up to the starting cell. Our path is highlighted in green on the following diagram:

## A* implementation

The basic object here is a cell so we write a class for it. We store the coordinates x and y, the values of A and B plus the sum SUM.
classCell(object):
def __init__(self, x, y, reachable):
self.reachable = reachable
self.x = x
self.y = y
self.parent = None
self.A = 0
self.B = 0
self.SUM = 0

Next is our main class named main. Attributes are the open list heapified (keep cell with lowest SUM at the top), the closed list which is a set for fast lookup, the cells list (grid definition) and the size of the grid.

 classmain(object): def__init__(self): self.opened = [] heapq.heapify(self.opened) self.closed =set() self.cells =[] self.grid_height =6 self.grid_width =6 ...

We create a simple method initializing the list of cells to match our example with the walls at the same locations.

 Def grid(self): wall =((0, 5), (1, 0), (1, 1), (1, 5), (2, 3), (3, 1), (3, 2), (3, 5), (4, 1), (4, 4), (5, 1)) forx inrange(self.grid_width): fory inrange(self.grid_height): if(x, y) inwall: reachable =False else: reachable =True self.cells.append(Cell(x, y, reachable)) self.start =self.get_cell(0, 0) self.end =self.get_cell(5, 5)

Our heuristic compute method:
defheuristic(self, cell):
return 10 * (abs(cell.x - self.end.x) + abs(cell.y - self.end.y))

We need a method to return a particular cell based on x and y coordinates.
defget_cell(self, x, y):
return self.cells[x * self.grid_height + y]

Next is a method to retrieve the list of adjacent cells to a specific cell.

cells = []
if cell.x < self.grid_width-1:
cells.append(self.get_cell(cell.x+1, cell.y))
if cell.y > 0:
cells.append(self.get_cell(cell.x, cell.y-1))
if cell.x > 0:
cells.append(self.get_cell(cell.x-1, cell.y))
if cell.y < self.grid_height-1:
cells.append(self.get_cell(cell.x, cell.y+1))
return cells

Simple method to print the path found. It follows the parent pointers to go from the ending cell to the starting cell.

 def path(self): cell=self.end while cell.parent is not self.start: Cell =cell.parent print('path: cell:%d,%d'%(cell.x, cell.y))

We need a method to calculate G and H and set the parent cell.