You have reached the end of the universe Graph and matrix of lengths Floyd-Warshall algorithm is a procedure, which is used to find the shorthest longest paths among all pairs of nodes in a graph, which does not contain any cycles of negative lenght. The main advantage of Floyd-Warshall algorithm is its simplicity. Description Floyd-Warshall algorithm uses a matrix of lengths as its input. If there is an edge between nodes and , than the matrix contains its length at the corresponding coordinates.

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A single execution of the algorithm will find the lengths summed weights of the shortest paths between all pair of vertices. With a little variation, it can print the shortest path and can detect negative cycles in a graph.

Floyd-Warshall is a Dynamic-Programming algorithm. These are adjacency matrices. The size of the matrices is going to be the total number of vertices.

The Distance Matrix is going to store the minimum distance found so far between two vertices. The Path Matrix is for regenerating minimum distance path between two vertices. This means the best way to come to vertex-v from vertex-u is to use the edge that connects v with u. And the distance from the vertex itself is 0. And the path[i][j] will be set to path[k][j], as it is better to go from i to k, and then k to j.

All the vertices will be selected as k. So it will remain unchanged. This is how we populate both matrices. The calculation for each step is shown here. For example, the shortest distance from 1 to 4 is 3 and the shortest distance between 4 to 3 is 2. This works because the path matrix stores the value of the vertex which shares the shortest path to v from any other node.

If any value on the diagonal is negative, that means there is a negative cycle in the graph.

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## Floydâ€“Warshall algorithm

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## Floyd-Warshall Algorithm

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