What is the complexity of Floyd warshall algorithm?

The Floyd-Warshall algorithm is a classic example of dynamic programming used to find the shortest paths between all pairs of vertices in a weighted graph. It's a powerful algorithm that works for both directed and undirected graphs, and handles negative weights as well. The algorithm operates in a systematic manner, progressively building up the solution by considering intermediate vertices between each pair of vertices, and determining if a shorter path can be found by going through that intermediate vertex.

The core of the Floyd-Warshall algorithm involves three nested loops. The outer loop iterates through each vertex in the graph, treating it as an intermediate vertex. The two inner loops iterate through all pairs of vertices, checking and updating the shortest path between them if a shorter path is found through the intermediate vertex. Due to this triple nested loop structure, the time complexity of the Floyd-Warshall algorithm is often expressed as

O(n3)

where

n is the number of vertices in the graph.

While the time complexity might seem high, the Floyd-Warshall algorithm's ability to solve the all-pairs shortest path problem in a straightforward and understandable manner makes it a valuable tool in the realm of graph theory and network analysis. The space complexity of the algorithm is O(n2)

as it requires a two-dimensional matrix to store the shortest path distances between all pairs of vertices.

The matrix used by the Floyd-Warshall algorithm is initialized with the direct distances between vertices, and is progressively updated through the algorithm's iterations. Each cell in the matrix ultimately contains the shortest distance between the corresponding pair of vertices.

In practical scenarios, the Floyd-Warshall algorithm can be used in various domains including routing protocols in networking, travel itinerary planning, and in many applications where optimizing routes through networks is crucial.

Despite its cubic time complexity, the Floyd-Warshall algorithm's ability to handle negative weights and its straightforward implementation makes it a popular choice for the all-pairs shortest path problem, especially when the graph has a relatively small number of vertices, or when a precise and comprehensive solution is required over performance.

In conclusion, the Floyd-Warshall algorithm is a compelling, albeit computationally intensive, method to solve the all-pairs shortest path problem. Its cubic time complexity might be a deterrent for extremely large graphs, yet its robustness and simplicity keep it relevant in many practical situations where understanding and optimizing network pathways are essential.