How to find the shortest cycle in a network?

To find the shortest cycle in a network, we can use the Floyd-Warshall algorithm.

The Floyd-Warshall algorithm is a dynamic programming algorithm that can be used to find the shortest path between all pairs of vertices in a weighted graph. It works by considering all possible intermediate vertices between any two vertices and updating the shortest path accordingly.

To adapt the Floyd-Warshall algorithm to find the shortest cycle in a network, we can modify the algorithm to consider cycles of length 1, 2, 3, ..., n-1, where n is the number of vertices in the network. We can then take the minimum of these cycle lengths to find the shortest cycle.

To implement this modification, we can add a third dimension to the distance matrix used in the Floyd-Warshall algorithm. The (i, j, k) entry of the distance matrix represents the shortest path from vertex i to vertex j using only vertices 1, 2, ..., k as intermediate vertices. We can then update the distance matrix using the following recurrence relation:

d(i, j, k) = min(d(i, j, k-1), d(i, k, k-1) + d(k, j, k-1))

where d(i, j, k-1) represents the shortest path from vertex i to vertex j using only vertices 1, 2, ..., k-1 as intermediate vertices, and d(i, k, k-1) + d(k, j, k-1) represents the path that goes through vertex k.

After running the modified Floyd-Warshall algorithm, we can iterate over all cycles of length 3 or greater and take the minimum length as the shortest cycle in the network. If there are no cycles of length 3 or greater, then the network is acyclic.