This is a Python program to implement the Bellman-Ford algorithm on a graph.

The problem is to find the shortest distance to all vertices from a source vertex in a weighted directed graph that can have negative edge weights. For the problem to be well-defined, there should be no cycles in the graph with a negative total weight.

1. Create classes for Graph and Vertex.

2. Create a function bellman-ford that takes a Graph object and a source vertex as arguments.

3. A dictionary distance is created with keys as the vertices in the graph and their value all set to infinity.

4. distance[source] is set to 0.

5. The algorithm proceeds by performing an update operation on each edge in the graph n – 1 times. Here n is the number of vertices in the graph.

6. The update operation on an edge from vertex i to vertex j is distance[j] = min(distance[j], distance[i] + weight(i, j)).

7. The dictionary distance is returned.

Here is the source code of a Python program to implement Bellman-Ford algorithm on a graph. The program output is shown below.

class Graph: def __init__(self): # dictionary containing keys that map to the corresponding vertex object self.vertices = {} def add_vertex(self, key): """Add a vertex with the given key to the graph.""" vertex = Vertex(key) self.vertices[key] = vertex def get_vertex(self, key): """Return vertex object with the corresponding key.""" return self.vertices[key] def __contains__(self, key): return key in self.vertices def add_edge(self, src_key, dest_key, weight=1): """Add edge from src_key to dest_key with given weight.""" self.vertices[src_key].add_neighbour(self.vertices[dest_key], weight) def does_edge_exist(self, src_key, dest_key): """Return True if there is an edge from src_key to 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 key corresponding to this vertex object.""" return self.key def add_neighbour(self, dest, weight): """Make this vertex point to dest with given edge weight.""" self.points_to[dest] = weight def get_neighbours(self): """Return all vertices pointed to by this vertex.""" return self.points_to.keys() def get_weight(self, dest): """Get weight of edge from this vertex to dest.""" return self.points_to[dest] def does_it_point_to(self, dest): """Return True if this vertex points to dest.""" return dest in self.points_to def bellman_ford(g, source): """Return distance where distance[v] is min distance from source to v. This will return a dictionary distance. g is a Graph object which can have negative edge weights. source is a Vertex object in g. """ 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(): distance[n] = min(distance[n], distance[v] + v.get_weight(n)) return distance g = Graph() print('Menu') print('add vertex <key>') print('add edge <src> <dest> <weight>') print('bellman-ford <source vertex key>') print('display') print('quit') while True: do = input('What would you like to do? ').split() 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('Vertex {} does not exist.'.format(src)) elif dest not in g: print('Vertex {} does not exist.'.format(dest)) else: if not g.does_edge_exist(src, dest): g.add_edge(src, dest, weight) else: print('Edge already exists.') elif operation == 'bellman-ford': key = int(do[1]) source = g.get_vertex(key) distance = bellman_ford(g, source) print('Distances from {}: '.format(key)) for v in distance: print('Distance to {}: {}'.format(v.get_key(), distance[v])) print() 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('(src={}, dest={}, weight={}) '.format(v.get_key(), dest.get_key(), w)) print() elif operation == 'quit': break

1. An instance of Graph is created.

2. A menu is presented to the user to perform various operations on the graph.

3. To find all shortest distances from a source vertex, bellman-ford is called on the graph and the source vertex.

Case 1: Menu add vertex <key> add edge <src> <dest> <weight> bellman-ford <source vertex key> display quit What would you like to do? add vertex 1 What would you like to do? add vertex 2 What would you like to do? add vertex 3 What would you like to do? add vertex 4 What would you like to do? add vertex 5 What would you like to do? add vertex 6 What would you like to do? add vertex 7 What would you like to do? add vertex 8 What would you like to do? add edge 1 2 10 What would you like to do? add edge 1 8 8 What would you like to do? add edge 2 6 2 What would you like to do? add edge 3 2 1 What would you like to do? add edge 3 4 1 What would you like to do? add edge 4 5 3 What would you like to do? add edge 5 6 -1 What would you like to do? add edge 6 3 -2 What would you like to do? add edge 7 2 -4 What would you like to do? add edge 7 6 -1 What would you like to do? add edge 8 7 1 What would you like to do? bellman-ford 1 Distances from 1: Distance to 5: 9 Distance to 6: 7 Distance to 7: 9 Distance to 2: 5 Distance to 1: 0 Distance to 8: 8 Distance to 3: 5 Distance to 4: 6 Case 2: Menu add vertex <key> add edge <src> <dest> <weight> bellman-ford <source vertex key> display quit What would you like to do? add vertex 1 What would you like to do? bellman-ford 1 Distances from 1: Distance to 1: 0 What would you like to do? add vertex 2 What would you like to do? bellman-ford 1 Distances from 1: Distance to 1: 0 Distance to 2: inf What would you like to do? add edge 1 2 2 What would you like to do? add vertex 3 What would you like to do? add edge 1 3 -1 What would you like to do? bellman-ford 1 Distances from 1: Distance to 1: 0 Distance to 3: -1 Distance to 2: 2 What would you like to do? add edge 3 2 2 What would you like to do? bellman-ford 1 Distances from 1: Distance to 1: 0 Distance to 3: -1 Distance to 2: 1 What would you like to do? quit

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