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

The Dijkstra Algorithm and the Floyd-Warshall Algorithm are both popular algorithms used in graph theory and network analysis. While they serve similar purposes of finding the shortest path between nodes in a graph, there are several key differences between the two algorithms.

1. Purpose:
- Dijkstra Algorithm: The main objective of the Dijkstra Algorithm is to find the shortest path between a single source node and all other nodes in the graph. It is primarily used for solving the single-source shortest path problem.
- Floyd-Warshall Algorithm: The Floyd-Warshall Algorithm, on the other hand, aims to find the shortest path between all pairs of nodes in a graph. It is used to solve the all-pairs shortest path problem.

2. Approach:
- Dijkstra Algorithm: Dijkstra's algorithm uses a greedy approach, where it iteratively selects the node with the smallest distance from the source and updates the distances of its neighboring nodes. It maintains a priority queue or a min-heap to efficiently select the next node to visit.
- Floyd-Warshall Algorithm: The Floyd-Warshall Algorithm utilizes dynamic programming to find the shortest paths between all pairs of nodes. It builds a matrix of intermediate distances and updates it iteratively until it finds the shortest paths between all pairs of nodes.

3. Graph Type:
- Dijkstra Algorithm: The Dijkstra Algorithm works efficiently on graphs with non-negative edge weights. It may not produce correct results if the graph contains negative edge weights.
- Floyd-Warshall Algorithm: The Floyd-Warshall Algorithm can handle graphs with both positive and negative edge weights. However, it cannot handle graphs with negative cycles, as it may lead to an infinite loop.

4. Time Complexity:
- Dijkstra Algorithm: The time complexity of Dijkstra's algorithm is O((V + E) log V), where V is the number of vertices and E is the number of edges in the graph. It is more efficient for sparse graphs with fewer edges.
- Floyd-Warshall Algorithm: The time complexity of the Floyd-Warshall Algorithm is O(V^3), where V is the number of vertices in the graph. It is more efficient for dense graphs with many edges.

5. Memory Usage:
- Dijkstra Algorithm: The Dijkstra Algorithm requires additional data structures such as priority queues or min-heaps to store and update the distances of nodes. Therefore, it consumes more memory compared to the Floyd-Warshall Algorithm.
- Floyd-Warshall Algorithm: The Floyd-Warshall Algorithm only requires a 2D matrix to store the intermediate distances between all pairs of nodes. Hence, it consumes less memory compared to Dijkstra's algorithm.

In summary, the Dijkstra Algorithm is suitable for finding the shortest path from a single source to all other nodes, while the Floyd-Warshall Algorithm is used to find the shortest paths between all pairs of nodes. Dijkstra's algorithm is more efficient for sparse graphs with non-negative edge weights, while the Floyd-Warshall Algorithm can handle both positive and negative edge weights but not negative cycles. The time complexity and memory usage also differ between the two algorithms based on the characteristics of the graph.