Understanding The Bellman-Ford Algorithm, With Examples

Key Points

• The Bellman-Ford Algorithm is ideal for finding the shortest path in graphs with negative values
• This algorithm is unique as it considers negative edge weights, making it suitable for time zones, financial transactions, and trade networks
• Although not the most efficient, its iterative nature allows it to navigate complex graphs

Do you need to find the shortest path between values in a graph? Having the right algorithm can make your life easier, and the Bellman-Ford Algorithm is one of the best. While not as efficient as others, it has unique qualities that allow it to navigate anything you throw at it.

So now you might wonder how to apply it to your graphs. In this article, we break down the algorithm and how it works. We talk about the aspects to which you might apply it and even provide the syntax to get you started. So let’s get into it and start finding the shortest path.

What Is the Bellman-Ford Algorithm?

Do you have a graph that includes negative values? To find the shortest path between each of these nodes, you might consider using the Bellman-Ford algorithm. This tool applies a distance array while repetitively exploring pathways to find the shortest distance. 

The algorithm takes into consideration negative edge weights, which makes it unique to other traversal algorithms. Here’s how it works:

1. Define the source vertex.
2. Go through all edges toward every existing vertex.
3. Record the shortest distance to each vertex from the source.
4. Repeat step 3 to ensure consideration of all possible paths.
5. Check for negative cycles.
6. Use the information to determine the shortest path.

How Is the Algorithm Used?

The Bellman-Ford algorithm is not the most efficient tool for traversing graphs due to its iterative nature. However, the function finds applications where negative edge weights exist. These aspects could include time zones, financial transactions, and trade networks.

When your weighted directed graph includes negative edge weights, it’s helpful to know how to apply the Bellman-Ford algorithm. We’ll help you get started with the program in Python as an example:

def bellman_ford(graph, source):
    # Step 1: Initialization
    distance = {}  # Distance dictionary to store shortest distances
    predecessor = {}  # Predecessor dictionary to store the preceding node in the shortest path

    # Set the distance of the source vertex to 0, and all others to infinity
    for vertex in graph:
        distance[vertex] = float('inf')
        predecessor[vertex] = None
    distance[source] = 0

    # Step 2: Relax edges repeatedly
    for _ in range(len(graph) - 1):
        for u, v, weight in graph.edges():
            if distance[u] + weight < distance[v]:
                distance[v] = distance[u] + weight
                predecessor[v] = u

    # Step 3: Check for negative cycles
    for u, v, weight in graph.edges():
        if distance[u] + weight < distance[v]:
            raise ValueError("Negative cycle detected. The graph contains a negative cycle.")

    # Step 4: Output the shortest distances and paths
    return distance, predecessor

Summary Table

The image featured at the top of this post is ©Vintage Tone/Shutterstock.com.