Question

Give an algorithm that takes as input a directed graph with positive edge lengths, and returns the length of the shortest cycle in the graph (if the graph is acyclic, it should say so). Your algorithm should take time at most $\mathbf{O}\left|V^3\right|$.

Short answer

A directed graph with positive edge lengths, and returns the length of the shortest cycle in the graph take time at most$O\left|V^3\right|$ it proved by the Dijkstra’s algorithm, which is an application of a single source shortest path.

Step-by-step solution

Define the concept of the algorithm used for the directed graph.

Dijkstra algorithm is an application of a single source shortest path.

Dijkstra’s algorithm also known asthe SPF algorithm and is an algorithm for finding the shortest paths between thevertices in a graph. It returns a search tree for all the paths the given node can take. An acyclic graph is a directed graph that has no cycles. Its operation is performed in the minheap.

Time complexity of Dijkstra’s algorithm

Time complexity:

$$\begin{gathered} TC=V+VIogV+E+EIogV \\ TC=VIogV+E\log V \\ TC=O\left[\left(V+E\right)\log V\right] \end{gathered}$$

For finding the shortest path adjacent list and min heap may be used.

The time complexity is$TC=O\left[\left(V+E\right)\log V\right]$.

And, if the graph is acyclic then the complexity is:$O\left|V^3\right|$.

Design the Algorithm

Dijkstra algorithm applies on the graph for finding the single source shortest path.

A directed graph with positive edge lengths, and returns the length of the shortest cycle in the graph and the graph is acyclic, which takes time at most$O\left|V^3\right|$.

So, here the vertex A is the source vertex. now take a minheap as a data structure to evaluate a single source’s shortest path between the source and the destination.

From A the distance of A is zero and take the distance of vertex A from each and every vertex is infinity.

Now take A as the first vertex and evaluate the weight towards each vertex.

And choose the next vertex from the vertices which have minimum weight and select that node as the second vertex.

Then again evaluate the distance of it from every vertex and get the minimum weight of the node and consider it as the main node.

Through this, the series of the vertex arises.

here the vertex A is the source vertex. now take a minheap as a data structure for evaluatinga single source’s shortest path between the source and the destination.

From A the distance of A is zero and take the distance of vertex A from each and every vertex is infinity.

Select every vertex one by one and put it into the min heap is a data structure one by one as shown in the figure.

Hence, the shortest distance from the vertex A to vertex D is 13 .