# Python Program to Implement Bellman Ford Algorithm

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

Problem Description

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.

Problem Solution

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.

Program/Source Code

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 = {}

"""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."""

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('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
if operation == 'add':
suboperation = do
if suboperation == 'vertex':
key = int(do)
if key not in g:
else:
elif suboperation == 'edge':
src = int(do)
dest = int(do)
weight = int(do)
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):
else:

elif operation == 'bellman-ford':
key = int(do)
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```
Program Explanation

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.

Runtime Test Cases
```Case 1:
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:
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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