Question

Squares.Design and analyse an algorithm that takes as input an undirected graph G(V,E) and determines whether graph contains a simple cycle (that is, a cycle which doesn’t intersect itself) of length four. Its running time should be at most$\mathbf{O}\left(\left|V^3\right|\right)$ time.

You may assume that the input graph is represented either as an adjacency matrix or with adjacency lists, whichever makes your algorithm simpler.

Short answer

The algorithm covers a shortest distance from the vertex A to vertex$\mathrm{D}$ is$13$ and the graph will take$\mathrm{O}\left|{\mathrm{v}}^3\right|$ running time.

Step-by-step solution

Define the concept of the algorithm used for undirected graph

Dijkstra algorithm is an application of single source shortest path.

Dijkstra’s algorithm also known as SPF algorithm and is an algorithm for finding the shortest paths between the vertices 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 algorithm.

Time complexity:

$$\begin{gathered} \mathrm{TC}=\mathrm{V}+\mathrm{VlogV}+\mathrm{E}+\mathrm{ElogV} \\ \mathrm{TC}=\mathrm{VlogV}+\mathrm{ElogV} \\ \mathrm{TC}=\mathrm{O}\left[\left(\mathrm{V}+\mathrm{E}\right)\mathrm{logV}\right] \end{gathered}$$

When the given graph is complete graph then,

role="math" localid="1659328321979" $$\begin{gathered} E=V^2 \\ O(V^2+V^2.V) \\ O(V^2+V^3) \\ O\left(V^3\right) \\ TC=O\left|V^3\right| \end{gathered}$$

For finding the shortest path adjacent list and minheap both may use.

The time complexity is

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

Design the Algorithm.

Dijkstra algorithm apply 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 take 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 for evaluate single source 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.

Theinput graph is represented either as an adjacency matrix or with adjacency lists is used to stored the vertices of the graph.

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 as get the minimum weight of the node and consider it as the main node.

Through this the series of vertex are arises.

here the vertex A is the source vertex. now take a minheap as a data structure for evaluate single source 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 vertices is infinity.

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

Fig: Adjacency list of the given graph

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

And the graph will take $O\left|V^3\right|$time.