I am going to implement breadth-first search (BFS) for a grid and a graph in this tutorial. Breadth-first search is an algorithm used in the field of AI to find a path from one point to another.

Breadth-first search is an uninformed algorithm, it blindly searches toward a goal on the breadth. BFS starts from an initial node (start) and expands neighbor nodes on the breadth, this is implemented by using a FIFO-queue (First In First Out). BFS is complete as it not will get stuck in an infinite loop if there is a goal node in the search space.

Breadth-first search is guaranteed to find the optimal solution, but it may take time and consume a lot of memory. The time complexity is **O(n)** in a grid and **O(b^d)** in a graph/tree with a branching factor (b) and a depth (d). The branching factor is the average number of neighbor nodes that can be expanded from each node and the depth is the average number of levels in a graph/tree. BFS is not as memory efficient as depth-first search in trees. BFS can be used if the search space not is to large and when it is important to find an optimal solution.

## Grid problem (maze)

I have created a simple maze (download it) with walls, a start and a goal. BFS is used to find the shortest path to the goal, the algorithm uses a Node class. The BFS algorithm has an array for open nodes and an array for closed nodes, it will loop until it finds a goal node or until the list of open nodes is empty. The code and the output is shown below.

```
# This class represents a node
class Node:
# Initialize the class
def __init__(self, position:(), parent:()):
self.position = position
self.parent = parent
self.g = 0 # Distance to start node
self.h = 0 # Distance to goal node
self.f = 0 # Total cost
# Compare nodes
def __eq__(self, other):
return self.position == other.position
# Sort nodes
def __lt__(self, other):
return self.f < other.f
# Print node
def __repr__(self):
return ('({0},{1})'.format(self.position, self.f))
# Draw a grid
def draw_grid(map, width, height, spacing=2, **kwargs):
for y in range(height):
for x in range(width):
print('%%-%ds' % spacing % draw_tile(map, (x, y), kwargs), end='')
print()
# Draw a tile
def draw_tile(map, position, kwargs):
# Get the map value
value = map.get(position)
# Check if we should print the path
if 'path' in kwargs and position in kwargs['path']: value = '+'
# Check if we should print start point
if 'start' in kwargs and position == kwargs['start']: value = '@'
# Check if we should print the goal point
if 'goal' in kwargs and position == kwargs['goal']: value = '$'
# Return a tile value
return value
# Breadth-first search (BFS)
def breadth_first_search(map, start, end):
# Create lists for open nodes and closed nodes
open = []
closed = []
# Create a start node and an goal node
start_node = Node(start, None)
goal_node = Node(end, None)
# Add the start node
open.append(start_node)
# Loop until the open list is empty
while len(open) > 0:
# Get the first node (FIFO)
current_node = open.pop(0)
# Add the current node to the closed list
closed.append(current_node)
# Check if we have reached the goal, return the path
if current_node == goal_node:
path = []
while current_node != start_node:
path.append(current_node.position)
current_node = current_node.parent
#path.append(start)
# Return reversed path
return path[::-1]
# Unzip the current node position
(x, y) = current_node.position
# Get neighbors
neighbors = [(x-1, y), (x+1, y), (x, y-1), (x, y+1)]
# Loop neighbors
for next in neighbors:
# Get value from map
map_value = map.get(next)
# Check if the node is a wall
if(map_value == '#'):
continue
# Create a neighbor node
neighbor = Node(next, current_node)
# Check if the neighbor is in the closed list
if(neighbor in closed):
continue
# Everything is green, add the node if it not is in open
if (neighbor not in open):
open.append(neighbor)
# Return None, no path is found
return None
# The main entry point for this module
def main():
# Get a map (grid)
map = {}
chars = ['c']
start = None
end = None
width = 0
height = 0
# Open a file
fp = open('annytab\\ai_search_algorithms\\data\\maze.in', 'r')
# Loop until there is no more lines
while len(chars) > 0:
# Get chars in a line
chars = [str(i) for i in fp.readline().strip()]
# Calculate the width
width = len(chars) if width == 0 else width
# Add chars to map
for x in range(len(chars)):
map[(x, height)] = chars[x]
if(chars[x] == '@'):
start = (x, height)
elif(chars[x] == '$'):
end = (x, height)
# Increase the height of the map
if(len(chars) > 0):
height += 1
# Close the file pointer
fp.close()
# Find the closest path from start(@) to end($)
path = breadth_first_search(map, start, end)
print()
print(path)
print()
draw_grid(map, width, height, spacing=1, path=path, start=start, goal=end)
print()
print('Steps to goal: {0}'.format(len(path)))
print()
# Tell python to run main method
if __name__ == "__main__": main()
```

```
#################################################################################
#.#...#....$....#...................#...#.........#.......#.............#.......#
#.#.#.#.###+###.#########.#########.#.#####.#####.#####.#.#.#######.###.#.#####.#
#...#.....#+++#.#.........#.#.....#.#...#...#...#.......#.#.#.......#.#.#.#...#.#
#############+#.#.#########.#.###.#.###.#.###.#.#.#######.###.#######.#.#.#.#.#.#
#+++++++++++#+#...#.#.....#...#...#...#.#.#.#.#...#...#.......#.......#.#.#.#.#.#
#+#########+#+#####.#.#.#.#.###.#####.#.#.#.#.#####.#.#########.###.###.###.#.#.#
#+#........+#+++#...#.#.#.#...#.....#.#.#.#...#.#...#.......#.....#.#...#...#...#
#+#########+#.#+###.#.#.#####.###.#.#.#.#.#.###.#.#########.#####.#.#.###.#####.#
#+#+++++++#+#.#+++#...#.#.....#.#.#.#...#.#.....#.#.....#.#...#...#.......#...#.#
#+#+#####+#+#.###+#####.#.#####.#.#.###.#.#######.###.#.#.###.#.###########.#.#.#
#+++#+++#+#+#...#+++++#.#.......#.#.#...#.....#...#...#.....#.#.#...#...#...#...#
#####+#+#+#+#########+#.#######.#.###.#######.#.###.#########.###.#.#.#.#.#######
#+++++#+++#+#+++++++++#.......#.#...#.#.#.....#.#.....#.......#...#.#.#.#.#.....#
#+#########+#+#########.###.###.###.#.#.#.###.#.#.###.#.#######.###.#.###.#.###.#
#+++#.#+++++#+++#.....#.#.#...#.#.#.....#...#.#.#...#.#...#...#...#.#.#...#...#.#
###+#.#+#####.#+#.#.###.#.###.#.#.#####.###.###.#####.###.#.#.#.###.#.#.#####.#.#
#+++#+++#.....#+#.#.#...#...#.....#...#.#...#...........#.#.#...#...#.......#.#.#
#+###+#########+#.#.#.###.#.#####.#.#.###.###.###########.#.#####.#########.###.#
#+#..+++++++++++#.#.......#.#...#.#.#...#.#...#.#.......#.......#.#...#.....#...#
#+#.#############.#########.#.#.###.###.#.#.###.#.#####.#.#######.#.#.#.#####.#.#
#+#.#+++++++++++#.#.#.#.....#.#.....#...#.#.....#...#.#.#.#.#...#.#.#.#.#.....#.#
#+###+#########+#.#.#.#######.#######.###.#####.###.#.#.#.#.###.#.#.#.#.#####.#.#
#+++++#+++#+++++#...#.........#.....#...#.....#...#...#.#.....#.#...#.#.#.....#.#
#.#####+#+#+#######.###########.#######.#.#######.###.#.###.###.#####.#.#.#####.#
#.....#+#+#+++#...#.#+++++++#.........#.#...#.......#.#.#...#...#.....#.#.#...#.#
#######+#+###+#.###.#+#####+#.#####.###.#.#.#.#######.#.#####.###.#####.#.###.#.#
#+++++++#+#+++#.....#+#...#+#...#.#.....#.#.#.#.#.....#...#...#...#.....#...#.#.#
#+#######+#+#.#####.#+###.#+###.#.#######.#.#.#.#.#######.#.###.#.###.#####.#.#.#
#+#.#+++++#+#.#+++#.#+++#.#+++#...#.#...#.#...#.#.....#.#...#...#...#.......#...#
#+#.#+#####+#.#+#+#####+#.###+###.#.#.#.#.#####.#####.#.#####.#####.#########.###
#+#..+#..+++#.#+#+#+++#+++#.#+#...#...#.#.#...#.....#...#.#...#...#.....#...#.#.#
#+###+###+#.###+#+#+#+###+#.#+#.#######.#.#.#.#####.###.#.#.###.#.#####.###.#.#.#
#+++#+++#+#.#+++#+#+#+++#+#.#+#.#.......#...#.........#.#...#...#.#...#...#.#...#
#.#+###+#+#.#+###+#+###+#+#.#+#.###.###.###########.###.#.###.###.###.###.#.###.#
#.#+++#+#+#.#+++#+++#+++#+#.#+#.....#...#...#.....#.#...#.....#.....#.#...#...#.#
#.###+#+#+#####+#####+#.#+#.#+#######.###.#.#####.#.#.#############.#.#.###.#.#.#
#...#+#+++#+++#+++++#+#.#+#.#+#+++#...#.#.#.......#.#.#...#...#...#...#.#.#.#...#
###.#+#####+#+#####+#+###+#.#+#+#+#.###.#.#########.#.#.#.#.#.#.#.#####.#.#.#####
#...#+++++++#+++++++#+++++..#+++#+++++++@...........#...#...#...#.......#.......#
#################################################################################
Steps to goal: 339
```

## Graph problem

This graph problem is about finding the shortest path from one city to another city, a map has been used to create connections between cities. The BFS algorithm uses a Graph class and a Node class, it has a list of open nodes and a list of closed nodes. The BFS algorithm will find the shortest path to the goal. The code and the output is shown below.

```
# This class represent a graph
class Graph:
# Initialize the class
def __init__(self, graph_dict=None, directed=True):
self.graph_dict = graph_dict or {}
self.directed = directed
if not directed:
self.make_undirected()
# Create an undirected graph by adding symmetric edges
def make_undirected(self):
for a in list(self.graph_dict.keys()):
for (b, dist) in self.graph_dict[a].items():
self.graph_dict.setdefault(b, {})[a] = dist
# Add a link from A and B of given distance, and also add the inverse link if the graph is undirected
def connect(self, A, B, distance=1):
self.graph_dict.setdefault(A, {})[B] = distance
if not self.directed:
self.graph_dict.setdefault(B, {})[A] = distance
# Get neighbors or a neighbor
def get(self, a, b=None):
links = self.graph_dict.setdefault(a, {})
if b is None:
return links
else:
return links.get(b)
# Return a list of nodes in the graph
def nodes(self):
s1 = set([k for k in self.graph_dict.keys()])
s2 = set([k2 for v in self.graph_dict.values() for k2, v2 in v.items()])
nodes = s1.union(s2)
return list(nodes)
# This class represent a node
class Node:
# Initialize the class
def __init__(self, name:str, parent:str):
self.name = name
self.parent = parent
self.g = 0 # Distance to start node
self.h = 0 # Distance to goal node
self.f = 0 # Total cost
# Compare nodes
def __eq__(self, other):
return self.name == other.name
# Sort nodes
def __lt__(self, other):
return self.f < other.f
# Print node
def __repr__(self):
return ('({0},{1})'.format(self.position, self.f))
# Breadth-first search (BFS)
def breadth_first_search(graph, start, end):
# Create lists for open nodes and closed nodes
open = []
closed = []
# Create a start node and an goal node
start_node = Node(start, None)
goal_node = Node(end, None)
# Add the start node
open.append(start_node)
# Loop until the open list is empty
while len(open) > 0:
# Get the first node (FIFO)
current_node = open.pop(0)
# Add the current node to the closed list
closed.append(current_node)
# Check if we have reached the goal, return the path
if current_node == goal_node:
path = []
while current_node != start_node:
path.append(current_node.name + ': ' + str(current_node.g))
current_node = current_node.parent
path.append(start_node.name + ': ' + str(start_node.g))
# Return reversed path
return path[::-1]
# Get neighbours
neighbors = graph.get(current_node.name)
# Loop neighbors
for key, value in neighbors.items():
# Create a neighbor node
neighbor = Node(key, current_node)
# Check if the neighbor is in the closed list
if(neighbor in closed):
continue
# Check if neighbor is in open list and if it has a lower f value
if(neighbor in open):
continue
# Calculate cost so far
neighbor.g = current_node.g + graph.get(current_node.name, neighbor.name)
# Everything is green, add neighbor to open list
open.append(neighbor)
# Return None, no path is found
return None
# The main entry point for this module
def main():
# Create a graph
graph = Graph()
# Create graph connections (Actual distance)
graph.connect('Frankfurt', 'Wurzburg', 111)
graph.connect('Frankfurt', 'Mannheim', 85)
graph.connect('Wurzburg', 'Nurnberg', 104)
graph.connect('Wurzburg', 'Stuttgart', 140)
graph.connect('Wurzburg', 'Ulm', 183)
graph.connect('Mannheim', 'Nurnberg', 230)
graph.connect('Mannheim', 'Karlsruhe', 67)
graph.connect('Karlsruhe', 'Basel', 191)
graph.connect('Karlsruhe', 'Stuttgart', 64)
graph.connect('Nurnberg', 'Ulm', 171)
graph.connect('Nurnberg', 'Munchen', 170)
graph.connect('Nurnberg', 'Passau', 220)
graph.connect('Stuttgart', 'Ulm', 107)
graph.connect('Basel', 'Bern', 91)
graph.connect('Basel', 'Zurich', 85)
graph.connect('Bern', 'Zurich', 120)
graph.connect('Zurich', 'Memmingen', 184)
graph.connect('Memmingen', 'Ulm', 55)
graph.connect('Memmingen', 'Munchen', 115)
graph.connect('Munchen', 'Ulm', 123)
graph.connect('Munchen', 'Passau', 189)
graph.connect('Munchen', 'Rosenheim', 59)
graph.connect('Rosenheim', 'Salzburg', 81)
graph.connect('Passau', 'Linz', 102)
graph.connect('Salzburg', 'Linz', 126)
# Make graph undirected, create symmetric connections
graph.make_undirected()
# Run search algorithm
path = breadth_first_search(graph, 'Frankfurt', 'Ulm')
print(path)
print()
# Tell python to run main method
if __name__ == "__main__": main()
```

`['Frankfurt: 0', 'Wurzburg: 111', 'Ulm: 294']`