What is the difference between the Dijkstra Algorithm and the Bellman-Ford Algorithm?

The Dijkstra Algorithm and the Bellman-Ford Algorithm are both popular algorithms used to find the shortest path in a graph. However, there are some key differences between the two:

1. Approach:
- Dijkstra Algorithm: It is a greedy algorithm that starts from a source node and iteratively selects the node with the smallest distance, updating the distances of its neighboring nodes. It uses a priority queue to efficiently select the next node.
- Bellman-Ford Algorithm: It is a dynamic programming algorithm that iterates over all edges multiple times, relaxing them to find the shortest path. It does not require a priority queue and can handle negative edge weights.

2. Negative Edge Weights:
- Dijkstra Algorithm: It does not work correctly with negative edge weights. If there are negative weights, it may produce incorrect results or go into an infinite loop.
- Bellman-Ford Algorithm: It can handle negative edge weights and can detect negative cycles in the graph. However, it has a higher time complexity compared to Dijkstra's algorithm.

3. Time Complexity:
- Dijkstra Algorithm: It has a time complexity of O((V + E) log V), where V is the number of vertices and E is the number of edges.
- Bellman-Ford Algorithm: It has a time complexity of O(V * E), where V is the number of vertices and E is the number of edges.

4. Space Complexity:
- Dijkstra Algorithm: It has a space complexity of O(V), where V is the number of vertices, as it requires a priority queue and a distance array.
- Bellman-Ford Algorithm: It has a space complexity of O(V), where V is the number of vertices, as it requires a distance array.

In summary, the Dijkstra Algorithm is more efficient for graphs with non-negative edge weights and does not handle negative edge weights well. On the other hand, the Bellman-Ford Algorithm can handle negative edge weights and detect negative cycles, but it has a higher time complexity. The choice between the two algorithms depends on the specific requirements and characteristics of the graph being analyzed.