Can you explain the Floyd-Warshall algorithm with an example?

Can you explain the Floyd-Warshall algorithm with an example?

Sure! The Floyd-Warshall algorithm is used to find the shortest paths in a weighted graph with positive or negative edge weights (but no negative cycles). For example, consider a graph with three vertices A, B, and C, with edges AB=1, AC=4, and BC=2. The algorithm will help us find the shortest paths between all pairs of vertices.

How does the algorithm work step by step?

The algorithm works in three main steps: 1) Initialize a distance matrix with direct distances between vertices. 2) Update the matrix by checking if a path through an intermediate vertex is shorter. 3) Repeat for all vertices as intermediates. This way, we ensure that all shortest paths are found.

Can you show me the distance matrix after each step?

Absolutely! Initially, the distance matrix is:

  A  B  C
A 0  1  4
B ∞  0  2
C ∞  ∞  0

After considering vertex B as an intermediate, it updates to:

  A  B  C
A 0  1  3
B ∞  0  2
C ∞  ∞  0

Finally, after considering vertex C, the matrix remains the same.

What are the time and space complexities of this algorithm?

The time complexity of the Floyd-Warshall algorithm is O(V^3), where V is the number of vertices. The space complexity is O(V^2) due to the distance matrix.