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Understanding The Bellman-Ford Algorithm, With Examples

# 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

What is the Bellman-Ford algorithm?

The Bellman-Ford algorithm is a graph algorithm that finds the shortest path from a source vertex to all other vertices in a graph, even when there are negative edge weights.

How does the Bellman-Ford algorithm work?

The Bellman-Ford algorithm works by iteratively relaxing the edges of a graph, repeatedly improving the estimates of the shortest path distances from a source vertex to all other vertices. It achieves this by comparing the current distance of each vertex to the source with the distance through neighboring vertices, updating the distances if a shorter path is found.

What is a weighted directed graph?

A weighted directed graph is a graph where each edge has a numerical value assigned to it, indicating a weight or cost associated with traversing that edge in a specific direction.

What are negative edge weights?

Negative edge weights are numerical values assigned to edges in a graph that represent costs or distances that are less than zero, indicating obstacles, penalties, or unfavorable conditions associated with traversing those edges.

#### Drew Baker, Author for History-Computer

Drew lives off-grid using a self-built solar array. He supports a nomadic lifestyle writing about solar energy, spaceflight, and anything travel-related. When he's not at the library, you'll typically find him chasing waterfalls despite all the warnings against it. You can connect with Drew on his Instagram.