What is the difference between the Dijkstra Algorithm and the Floyd-Warshall Algorithm?

The Dijkstra Algorithm and the Floyd-Warshall Algorithm are both used to solve the shortest path problem in graph theory, but they differ in their approach and the type of graphs they can handle efficiently.

1. Approach:
- Dijkstra Algorithm: It is a greedy algorithm that starts from a single source node and iteratively selects the node with the smallest distance from the source. It then updates the distances of its neighboring nodes and continues until all nodes have been visited or the destination node is reached.
- Floyd-Warshall Algorithm: It is a dynamic programming algorithm that considers all pairs of nodes in the graph. It iteratively updates the shortest path distances between any two nodes by considering intermediate nodes. It builds a matrix of shortest path distances for all pairs of nodes.

2. Graph Types:
- Dijkstra Algorithm: It is designed for solving the single-source shortest path problem, meaning it finds the shortest path from a single source node to all other nodes in a graph. It works efficiently for graphs with non-negative edge weights.
- Floyd-Warshall Algorithm: It is designed for solving the all-pairs shortest path problem, meaning it finds the shortest path between all pairs of nodes in a graph. It can handle graphs with both positive and negative edge weights, but it does not work efficiently for large graphs due to its time complexity.

3. Time Complexity:
- Dijkstra Algorithm: It has a time complexity of O((V + E) log V) when implemented using a priority queue, where V is the number of vertices and E is the number of edges in the graph.
- Floyd-Warshall Algorithm: It has a time complexity of O(V^3), where V is the number of vertices in the graph. This makes it less efficient for large graphs compared to Dijkstra's algorithm.

In summary, the main differences between the Dijkstra Algorithm and the Floyd-Warshall Algorithm lie in their approach, the type of shortest path problem they solve, the types of graphs they can handle efficiently, and their time complexity. Dijkstra's algorithm is suitable for finding the shortest path from a single source to all other nodes in a graph with non-negative edge weights, while Floyd-Warshall algorithm is used to find the shortest path between all pairs of nodes in a graph, including graphs with negative edge weights.