Dijkstra’s Algorithm & Bellman-Ford Algorithm – Detailed overview

We then take a number of actions. First, we remove the node with the lowest cost from each step, update the distances between its neighbors, and, if necessary, move

Dijkstra’s Algorithm

One SSSP (Single Source Shortest Path) algorithm is Dijkstra’s. As a result, it determines the shortest route between a source node and every node in the graph. It is an important part of data structures and algorithms.

Although it is well known that Dijkstra’s algorithm functions with weighted graphs, it only does so when the edge weights are non-negative.

We’ll go into detail about this in a moment.

  • Theoretical Idea

In Dijkstra’s algorithm, we initially begin at a source node and set the distance to zero. The source node is then moved to a priority queue with zero cost.

We then take a number of actions. First, we remove the node with the lowest cost from each step, update the distances between its neighbors, and, if necessary, move them up the priority queue. Naturally, the cost of the extracted node plus the edge we just passed over is added to each of the nearby nodes as their corresponding new cost.

Until there are no more nodes to extract from the priority queue, we keep visiting every node. The calculated distances are then returned.

  • Proof of Concept

We always extract the node with the lowest cost in Dijkstra’s algorithm. Moreover, in the case of non-negative edges, we can demonstrate our strategy’s accuracy.

  • Limitations

We thereby demonstrated the Dijkstra algorithm’s superiority. For this, we assumed that all edges have weights that are not negative and, as a result, will always be non-negative.

The shortest paths are incorrectly determined using Dijkstra’s method when applied to graphs with negative weights. The rationale is that it can be detrimental, making it easy to go there for less money. Therefore, we are still determining the merits of picking the node with the lowest cost.

Advantages and Disadvantages

The main benefit of Dijkstra’s method is its nearly linearly low degree of complexity. However, Dijkstra’s algorithm cannot be applied when dealing with negative weights.

Additionally, employing Dijkstra’s technique is not a suitable choice when dealing with dense networks, where it is close, and we need to determine the shortest path between any two nodes.

Bellman-Ford Algorithm

The Bellman-Ford algorithm is one of the SSSP algorithms, just like Dijkstra’s algorithm. As a result, it determines the shortest route between a starting source node and every node in a weighted graph. The Bellman-Ford algorithm, however, is based on a different idea than Dijkstra’s.

  • Theoretical Idea

The source node’s distance is initialized to zero in the Bellman-Ford algorithm, but all other nodes’ distances are set to zero. We then carry out actions.

After several iterations, every node will have the correct distance, and the algorithm will be terminated.

  • Proof of Concept

The Bellman-Ford algorithm makes the assumption that all nodes will unquestionably have the right distances at the end of each stage. So let’s establish this premise.

Within the graph, any acyclic path may have at most nodes, indicating that it has edges. If a path has more edges than nodes, it has a cycle since there are more edges than nodes. As a result, it needs to make many trips to the same node. For more detail, refer to the data structure course and learn how to implement these algorithms in DSA problems

  • Limitations

So, we established that the Bellman-Ford algorithm provides the SSSP problem with the best solution. The first flaw in our argument is that cycling over a path might make it shorter!

When we have a cycle with a negative total sum of edges, that is the only situation in which this is true. However, it is rare to determine the shortest path in that instance because we can always find a shorter one by repeating the cycle. As a result, the meaning of the word “shortest path” is lost.

Advantages and Disadvantages

The Bellman-Ford algorithm’s key benefit is its ability to tolerate negative weights. The Bellman-Ford algorithm, however, is far more sophisticated than the Dijkstra algorithm. Moreover, since graphs with negative weights are typically considered exceptional instances,

Dijkstra’s technique has additional uses.

  1. Non-Negative Weights Example

Let’s examine how Dijkstra’s algorithm determines the shortest paths for a graph with non-negative weights.

Positive Weights

The priority queue is pushed to first, with its distance set to 0. Then, after extracting it, we visit its nearby neighbors and update their distances. Then, since it is closest to us, we extract it from the priority queue, update its neighbors, and then add them to the priority queue.

Since it has the shortest path, the following node is to be extracted. As previously, we update its nearby neighbors and, if necessary, move them up the list. The queue is then removed and has the proper shortest path.

  1. Negative Weights Example

Let’s look at an illustration of a graph with negative weights but no negative cycles.

Negative Weights

The sequence is shown by the red number close to each edge. We took three actions. We iterated over the edges in each step according to their sequence and updated the distances.

  1. Negative Cycles Example

Let’s now examine a case with negative cycles and discuss how the Bellman-Ford algorithm recognizes them.

Negative Cycles

We updated the distance from the first edge, the distance from the third edge, and the distance from the fifth edge in the first step. The distance from the second edge and the weight from the fifth edge were then revised. Finally, the weight of the fourth edge was updated third.

See if the Bellman-Ford algorithm can identify it after a few more cycles.

Negative Cycles 2

The first graph displays the distances that were determined after completing the steps. If we take one more step, we can see that the distance from the second edge and the distance from the fourth edge have been updated. As we continued iterations, we observed that nodes maintained shorter distances since they were inside the negative cycle.

Comparison

Compare and contrast the Bellman-Ford and Dijkstra algorithms:

We can observe that Dijkstra’s algorithm reduces the time complexity more effectively. However, we must use the Bellman-Ford algorithm when we have negative weights. The Bellman-Ford approach can also help us determine whether or not the graph has negative cycles.

Conclusion

We discussed an overview of Dijkstra’s and Bellman-Ford’s algorithms in this blog.Then, we enumerated each algorithm’s drawbacks, benefits, and limits. Finally, we contrasted their advantages and disadvantages. These algorithms are important to master if you want to solve technical problems in Data structures. You can learn more by joining the best course for data structures and algorithms, offered by Learnbay. Sign up and get started!

 

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