Question

Give an $\mathbf{O}{\left|V\right|}^{\mathbf{2}}$algorithm for the following task.

Input:An undirected graph $\mathbf{G}\mathbf{=}\left(V,E\right)$; edge lengths ${\mathbf{I}}_{\mathbf{e}}\mathbf{>}\mathbf{0}$;an edge $\mathbf{e}\mathbf{\in }\mathbf{E}$.

Output:The length of the shortest cycle containing edge e

Short answer

Algorithm:

Input: An undirected graph G=(V,E) ; edge lengths$1_e>0$ ;an edge$e\in E$ .

Output: The length of the shortest cycle containing edge

remove e to form $G\text{'}$

Compute shortest path between endpoints in $G\text{'}$

Add e to $G\text{'}$

Cycle completed

The runtime of the algorithm is $O{\left|V\right|}^2$

Step-by-step solution

Explain undirected graphs

Consider the graph with set od vertices and edges. In an undirected graph, edges are denoted by the straight line without arrows.

Step 2:Give an O|V|2 algorithm for the given task.

The algorithm is as follows:

Input: An undirected graph G=(V,E) ; edge lengths $1_e>0$;an edge $e\in E$.

Output: The length of the shortest cycle containing edge e

remove e to form $G\text{'}$

Compute shortest path between endpoints in G'

Add e to G'

Cycle completed

Remove e to form G' , then compute the shortest path between the endpoints of . Add to complete the cycle for the given input.

The runtime of the shortest path is $O\left(n^2\right)$. Any cycle including is a path between its two endpoints. Once e is removed, it suffices to minimize the length of the circle.

Thus, the runtime of the above algorithm is $O{\left|V\right|}^2$.