Bellman Ford’s Algorithm

Bellman Ford algorithm helps us find the shortest path from a vertex to all other vertices of a weighted graph.

It is similar to Dijstra’s Algorithm but it can work with graphs in which edges can have negative weights.

Why would one ever have edges with negative weights in real life?

Negative weight edges might seem useless at first but they can explain a lot of phenomena like cashflow, the heat released/absorbed in a chemical reaction, etc.

For instance, if there are different ways to reach from one chemical A to another chemical B, each method will have sub-reactions involving both heat dissipation and absorption.

If we want to find the set of reactions where minimum energy is required, then we will need to be able to factor in the heat absorption as negative weights and heat dissipation as positive weights.

Why do we need to be careful with negative weights?

Negative weight edges can create negative weight cycles i.e. a cycle that will reduce the total path distance by coming back to the same point.

Negative weight cycles can give an incorrect result when trying to find out the shortest path

Shortest path algorithms like Dijkstra’s Algorithm that aren’t able to detect such a cycle can give an incorrect result because they can go through a negative weight cycle and reduce the path length.

How Bellman Ford’s algorithm works

Bellman Ford algorithm works by overestimating the length of the path from the starting vertex to all other vertices. Then it iteratively relaxes those estimates by finding new paths that are shorter than the previously overestimated paths.

By doing this repeatedly for all vertices, we can guarantee that the result is optimized.

Step-1 for Bellman Ford’s algorithmStep-2 for Bellman Ford’s algorithmStep-3 for Bellman Ford’s algorithmStep-4 for Bellman Ford’s algorithmStep-5 for Bellman Ford’s algorithmStep-6 for Bellman Ford’s algorithm

Bellman Ford Pseudocode

We need to maintain the path distance of every vertex. We can store that in an array of size v, where v is the number of vertices.

We also want to be able to get the shortest path, not only know the length of the shortest path. For this, we map each vertex to the vertex that last updated its path length.

Once the algorithm is over, we can backtrack from the destination vertex to the source vertex to find the path.

function bellmanFord(G, S)

  for each vertex V in G

    distance[V] <- infinite

      previous[V] <- NULL

  distance[S] <- 0

  for each vertex V in G

    for each edge (U,V) in G

      tempDistance <- distance[U] + edge_weight(U, V)

      if tempDistance < distance[V]

        distance[V] <- tempDistance

        previous[V] <- U

  for each edge (U,V) in G

    If distance[U] + edge_weight(U, V) < distance[V}

      Error: Negative Cycle Exists

  return distance[], previous[]

Bellman Ford vs Dijkstra

Bellman Ford’s algorithm and Dijkstra’s algorithm are very similar in structure. While Dijkstra looks only to the immediate neighbors of a vertex, Bellman goes through each edge in every iteration.

Python Examples

# Bellman Ford Algorithm in Python

class Graph:

    def __init__(self, vertices):

        self.V = vertices   # Total number of vertices in the graph

        self.graph = []     # Array of edges

    # Add edges

    def add_edge(self, s, d, w):

        self.graph.append([s, d, w])

    # Print the solution

    def print_solution(self, dist):

        print(“Vertex Distance from Source”)

        for i in range(self.V):

            print(“{0}\t\t{1}”.format(i, dist[i]))

    def bellman_ford(self, src):

        # Step 1: fill the distance array and predecessor array

        dist = [float(“Inf”)] * self.V

        # Mark the source vertex

        dist[src] = 0

        # Step 2: relax edges |V| – 1 times

        for _ in range(self.V – 1):

            for s, d, w in self.graph:

                if dist[s] != float(“Inf”) and dist[s] + w < dist[d]:

                    dist[d] = dist[s] + w

        # Step 3: detect negative cycle

        # if value changes then we have a negative cycle in the graph

        # and we cannot find the shortest distances

        for s, d, w in self.graph:

            if dist[s] != float(“Inf”) and dist[s] + w < dist[d]:

                print(“Graph contains negative weight cycle”)

                return

        # No negative weight cycle found!

        # Print the distance and predecessor array

        self.print_solution(dist)

g = Graph(5)

g.add_edge(0, 1, 5)

g.add_edge(0, 2, 4)

g.add_edge(1, 3, 3)

g.add_edge(2, 1, 6)

g.add_edge(3, 2, 2)

g.bellman_ford(0)

Bellman Ford’s Complexity

Time Complexity

Space Complexity

And, the space complexity is O(V).

Bellman Ford’s Algorithm Applications

1. For calculating shortest paths in routing algorithms
2. For finding the shortest path