Dijkstras Search Algorithm in Python

In this tutorial, I will implement Dijkstras algorithm to find the shortest path in a grid and a graph. Dijkstras algorithm was created by Edsger W. Dijkstra, a programmer and computer scientist from the Netherlands. Dijkstras performs a uniform-cost search as it expands nodes in order of cost from the root node.

Dijkstras is an informed algorithm in searches as it uses an heuristic (cost so far), it starts at an initial start node and updates each neighbor node with the cost so far. The algorithm selects the neighbor with the lowest cost and continues to expand nodes until it reaches the goal node, this can be implemented by using a priority queue or by sorting the list of open nodes in ascending order. The algorithm favors nodes that are close to the starting point.

Dijkstras algorithm is complete and it will find the optimal solution, it may take a long time and consume a lot of memory in a large search space. The time complexity is O(n) in a grid and O(b^(c/m)) in a graph/tree with a branching factor (b), an optimal cost (c) and minimum cost (m). The branching factor is the average number of neighbor nodes that can be expanded from each node, the optimal cost is the cost of the optimal solution and the minimum cost is the lowest cost for a node.

Grid problem (maze)

I have created a simple maze (download it) with walls, a start point (@) and a goal point ($). Dijkstras algorithm is used to find the shortest path from the start node to a goal node by using the distance to the start node (g) as the heuristic.

# 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 

# Dijkstra search
def dijkstra_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:

        # Sort the open list to get the node with the lowest cost first
        open.sort()

        # Get the node with the lowest cost
        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

            # Generate heuristics (Manhattan distance)
            neighbor.g = abs(neighbor.position[0] - start_node.position[0]) + abs(neighbor.position[1] - start_node.position[1])
            neighbor.h = 0
            neighbor.f = neighbor.g

            # Check if neighbor is in open list and if it has a lower f value
            for node in open:
                if (neighbor == node and neighbor.f > node.f):
                    continue

             # 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():

    # 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 = dijkstra_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

The problem formulation is to find the shortest path from a departure city to a destination city, a map has been used to create connections between cities in the graph. Dijkstras algorithm uses a Graph class, a Node class and the distance to the departure city (start) as the heuristic to guide the search.

# 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))

# Dijkstra search
def dijkstra_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:

        # Sort the open list to get the node with the lowest cost first
        open.sort()

        # Get the node with the lowest cost
        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

            # Calculate cost so far
            neighbor.g = current_node.g + graph.get(current_node.name, neighbor.name)
            neighbor.h = 0
            neighbor.f = neighbor.g

            # Check if neighbor is in open list and if it has a lower f value
            for node in open:
                if (neighbor == node and neighbor.f > node.f):
                    continue

            # 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 = dijkstra_search(graph, 'Frankfurt', 'Ulm')
    print(path)
    print()

# Tell python to run main method
if __name__ == "__main__": main()
['Frankfurt: 0', 'Wurzburg: 111', 'Ulm: 294']

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