This is a Python program to implement Johnson’s algorithm on a directed graph to find the shortest distance between all pairs of vertices.

The problem is to find the shortest distance between all pairs of vertices 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 johnson that takes a Graph object g as argument.

3. It returns a dictionary distance where distance[u][v] is the minimum distance from vertex u to v.

4. The algorithm works by first adding a new vertex q to the graph g.

5. This vertex q is made to point to all other vertices wit zero-weight edges.

6. The Bellman-Ford algorithm is run on the graph with source vertex q to find the shortest distance from q to all other vertices. This is stored in bell_dist where bell_dist[v] is the shortest distance from q to v.

7. Modify the graph’s weight function and set it to w(u, v) = w(u, v) + bell_dist(u) – bell_dist(v).

8. Remove the vertex q from the graph.

9. Run Dijkstra’s algorithm on each source vertex in the graph to find the shortest distance from each source vertex to all other vertices in this modified graph.

10. These shortest distances are stored in distance where distance[u][v] is the shortest distance from u to v.

11. Add (bell_dist[v] – bell_dist[u]) to distance[u][v] for each pair of vertices u, v to get the shortest distances for the original graph.

12. Correct the weights in the graph by adding (bell_dist[v] – bell_dist[u]) to weight(u, v) for each edge (u, v) in the graph so that the graph is no longer modified.

Here is the source code of a Python program to implement Johnson’s algorithm on a directed 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 set_weight(self, dest, weight): """Set weight of edge from this vertex to dest.""" self.points_to[dest] = weight def does_it_point_to(self, dest): """Return True if this vertex points to dest.""" return dest in self.points_to def johnson(g): """Return distance where distance[u][v] is the min distance from u to v. distance[u][v] is the shortest distance from vertex u to v. g is a Graph object which can have negative edge weights. """ # add new vertex q g.add_vertex('q') # let q point to all other vertices in g with zero-weight edges for v in g: g.add_edge('q', v.get_key(), 0) # compute shortest distance from vertex q to all other vertices bell_dist = bellman_ford(g, g.get_vertex('q')) # set weight(u, v) = weight(u, v) + bell_dist(u) - bell_dist(v) for each # edge (u, v) 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]) # remove vertex q # This implementation of the graph stores edge (u, v) in Vertex object u # Since no other vertex points back to q, we do not need to worry about # removing edges pointing to q from other vertices. del g.vertices['q'] # distance[u][v] will hold smallest distance from vertex u to v distance = {} # run dijkstra's algorithm on each source vertex for v in g: distance[v] = dijkstra(g, v) # correct distances for v in g: for w in g: distance[v][w] += bell_dist[w] - bell_dist[v] # correct weights in original graph 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 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 def dijkstra(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. source is a Vertex object in g. """ unvisited = set(g) distance = dict.fromkeys(g, float('inf')) distance[source] = 0 while unvisited != set(): # find vertex with minimum distance closest = min(unvisited, key=lambda v: distance[v]) # mark as visited unvisited.remove(closest) # update distances for neighbour in closest.get_neighbours(): if neighbour in unvisited: new_distance = distance[closest] + closest.get_weight(neighbour) if distance[neighbour] > new_distance: distance[neighbour] = new_distance return distance g = Graph() print('Menu') print('add vertex <key>') print('add edge <src> <dest> <weight>') print('johnson') 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 == 'johnson': distance = johnson(g) print('Shortest distances:') for start in g: for end in g: print('{} to {}'.format(start.get_key(), end.get_key()), end=' ') print('distance {}'.format(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('(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 shortest distances between all pairs, johnson is called to get the dictionary distance.

4. The distances between each pair of vertices are then displayed.

Case 1: Menu add vertex <key> add edge <src> <dest> <weight> johnson 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 edge 1 2 3 What would you like to do? add edge 1 3 8 What would you like to do? add edge 1 5 -4 What would you like to do? add edge 2 5 7 What would you like to do? add edge 2 4 1 What would you like to do? add edge 3 2 4 What would you like to do? add edge 4 3 -5 What would you like to do? add edge 4 1 2 What would you like to do? add edge 5 4 6 What would you like to do? johnson Shortest distances: 1 to 1 distance 0 1 to 2 distance 1 1 to 3 distance -3 1 to 4 distance 2 1 to 5 distance -4 2 to 1 distance 3 2 to 2 distance 0 2 to 3 distance -4 2 to 4 distance 1 2 to 5 distance -1 3 to 1 distance 7 3 to 2 distance 4 3 to 3 distance 0 3 to 4 distance 5 3 to 5 distance 3 4 to 1 distance 2 4 to 2 distance -1 4 to 3 distance -5 4 to 4 distance 0 4 to 5 distance -2 5 to 1 distance 8 5 to 2 distance 5 5 to 3 distance 1 5 to 4 distance 6 5 to 5 distance 0 What would you like to do? quit Case 2: python 226__graph_johnson.py Menu add vertex <key> add edge <src> <dest> <weight> johnson 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? johnson Shortest distances: 1 to 1 distance 0 1 to 2 distance inf 2 to 1 distance inf 2 to 2 distance 0 What would you like to do? add edge 1 2 100 What would you like to do? add vertex 3 What would you like to do? add edge 2 3 -50 What would you like to do? add edge 1 3 60 What would you like to do? johnson Shortest distances: 1 to 1 distance 0 1 to 2 distance 100 1 to 3 distance 50 2 to 1 distance inf 2 to 2 distance 0 2 to 3 distance -50 3 to 1 distance inf 3 to 2 distance inf 3 to 3 distance 0 What would you like to do? quit

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