How to Implement Johnson’s Algorithm in Python

Submitted by razormist on Monday, March 30, 2026 - 23:09.

In this tutorial, we will learn "How to implement Johnson’s Algorithm in Python". The main objective is to understand and implement the algorithm effectively. This guide will walk you step by step through the process, making it easier to follow and apply. By the end of this tutorial, you will have a solid understanding of how Johnson’s Algorithm works in Python, helping you strengthen your problem-solving abilities and improve your overall coding skills in data structure implementation.

This topic is straightforward and easy to understand. By following the instructions provided, you will be able to complete it with ease. The program will guide you step by step through the process of implementing Johnson’s Algorithm. So, let’s dive into the coding process and start implementing the solution to gain a deeper understanding of graph algorithms in Python.

Getting Started:

First you will have to download & install the Python IDLE's, here's the link for the Integrated Development And Learning Environment for Python https://www.python.org/downloads/.

Creating Main Function

This is the main function of the application. The following code will display a simple GUI in terminal console that will display program. To do this, simply copy and paste these blocks of code into the IDLE text editor.

class Graph:
    def __init__(self):
        self.vertices = {}
 
    def add_vertex(self, key):
        self.vertices[key] = Vertex(key)
 
    def get_vertex(self, key):
        return self.vertices[key]
 
    def __contains__(self, key):
        return key in self.vertices
 
    def add_edge(self, src_key, dest_key, weight=1):
        self.vertices[src_key].add_neighbour(self.vertices[dest_key], weight)
 
    def does_edge_exist(self, src_key, dest_key):
        return self.vertices[src_key].does_it_point_to(self.vertices[dest_key])
 
    def __len__(self):
        return len(self.vertices)
 
    def __iter__(self):
        return iter(self.vertices.values())
 
 
class Vertex:
    def __init__(self, key):
        self.key = key
        self.points_to = {}
 
    def get_key(self):
        return self.key
 
    def add_neighbour(self, dest, weight):
        self.points_to[dest] = weight
 
    def get_neighbours(self):
        return self.points_to.keys()
 
    def get_weight(self, dest):
        return self.points_to[dest]
 
    def set_weight(self, dest, weight):
        self.points_to[dest] = weight
 
    def does_it_point_to(self, dest):
        return dest in self.points_to
 
 
def bellman_ford(g, source):
    distance = dict.fromkeys(g, float('inf'))
    distance[source] = 0
 
    for _ in range(len(g) - 1):
        for v in g:
            for n in v.get_neighbours():
                if distance[v] + v.get_weight(n) < distance[n]:
                    distance[n] = distance[v] + v.get_weight(n)
 
    # NEGATIVE CYCLE CHECK
    for v in g:
        for n in v.get_neighbours():
            if distance[v] + v.get_weight(n) < distance[n]:
                print("Graph contains a negative weight cycle!")
                return None
 
    return distance
 
 
def dijkstra(g, source):
    unvisited = set(g)
    distance = dict.fromkeys(g, float('inf'))
    distance[source] = 0
 
    while unvisited:
        closest = min(unvisited, key=lambda v: distance[v])
        unvisited.remove(closest)
 
        for neighbour in closest.get_neighbours():
            if neighbour in unvisited:
                new_distance = distance[closest] + closest.get_weight(neighbour)
                if new_distance < distance[neighbour]:
                    distance[neighbour] = new_distance
 
    return distance
 
 
def johnson(g):
    # Step 1: Add temporary vertex q
    g.add_vertex('q')
    q = g.get_vertex('q')
 
    for v in list(g):  # avoid modifying during iteration
        if v.get_key() != 'q':
            g.add_edge('q', v.get_key(), 0)
 
    # Step 2: Bellman-Ford
    bell_dist = bellman_ford(g, q)
    if bell_dist is None:
        return None
 
    # Step 3: Reweight edges
    for v in g:
        for n in v.get_neighbours():
            w = v.get_weight(n)
            v.set_weight(n, w + bell_dist[v] - bell_dist[n])
 
    # Step 4: Remove q safely
    del g.vertices['q']
 
    # Step 5: Run Dijkstra for each vertex
    distance = {}
    for v in g:
        distance[v] = dijkstra(g, v)
 
    # Step 6: Restore original weights
    for v in g:
        for n in v.get_neighbours():
            w = v.get_weight(n)
            v.set_weight(n, w + bell_dist[n] - bell_dist[v])
 
    return distance
 
 
# MAIN PROGRAM
g = Graph()
 
while True:
    print("\n========== Johnson’s Algorithm ==========")
    print("Menu")
    print("add vertex <key>")
    print("add edge <src> <dest> <weight>")
    print("johnson")
    print("display")
    print("quit")
 
    do = input('What would you like to do? ').split()
    if not do:
        continue
 
    operation = do[0]
 
    if operation == 'add':
        suboperation = do[1]
 
        if suboperation == 'vertex':
            key = int(do[2])
            if key not in g:
                g.add_vertex(key)
            else:
                print('Vertex already exists.')
 
        elif suboperation == 'edge':
            src = int(do[2])
            dest = int(do[3])
            weight = int(do[4])
 
            if src not in g:
                print(f'Vertex {src} does not exist.')
            elif dest not in g:
                print(f'Vertex {dest} does not exist.')
            else:
                if not g.does_edge_exist(src, dest):
                    g.add_edge(src, dest, weight)
                else:
                    print('Edge already exists.')
 
    elif operation == 'johnson':
        distance = johnson(g)
        if distance:
            print('Shortest distances:')
            for start in g:
                for end in g:
                    print(f'{start.get_key()} -> {end.get_key()} = {distance[start][end]}')
 
    elif operation == 'display':
        print('Vertices:', end=' ')
        for v in g:
            print(v.get_key(), end=' ')
        print()
 
        print('Edges:')
        for v in g:
            for dest in v.get_neighbours():
                w = v.get_weight(dest)
                print(f'(src={v.get_key()}, dest={dest.get_key()}, weight={w})')
 
    elif operation == 'quit':
        print("Exiting program...")
        break
 
    else:
        print("Invalid command.")

This Python program implements Johnson’s algorithm for finding the shortest paths between all pairs of vertices in a weighted graph, including graphs with negative edge weights but no negative cycles. It defines `Graph` and `Vertex` classes to represent the graph structure, storing vertices and weighted edges. The program combines Bellman-Ford (to detect negative cycles and compute vertex potentials) and Dijkstra’s algorithm (to efficiently compute shortest paths after reweighting edges). Users interact with the program via a command-line menu to add vertices and edges, display the graph, run Johnson’s algorithm to compute all-pairs shortest paths, and exit. The algorithm ensures correct shortest distances are reported even in the presence of negative edges, while preventing negative weight cycles from producing invalid results.

Output:

There you have it we successfully created How to Implement Johnson’s Algorithm in Python. I hope that this simple tutorial help you to what you are looking for. For more updates and tutorials just kindly visit this site. Enjoy Coding!

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