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Chapter 4: Q12E (page 133)

Give an O|V|2algorithm for the following task.

Input:An undirected graph G=(V,E); edge lengths Ie>0;an edge e鈭�E.

Output:The length of the shortest cycle containing edge e

Short Answer

Expert verified

Algorithm:

Input: An undirected graph G=(V,E) ; edge lengths1e>0 ;an edgee鈭�E .

Output: The length of the shortest cycle containing edge

remove e to form G'

Compute shortest path between endpoints in G'

Add e to G'

Cycle completed

The runtime of the algorithm is OV2

Step by step solution

01

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.

02

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 1e>0;an edge e鈭�E.

Output: The length of the shortest cycle containing edge e

remove e to form G'

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 On2. 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 OV2.

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Most popular questions from this chapter

You are given a strongly connected directed graph G=(V,E) with positive edge weights along with a particularv0鈭�V . Give an efficient algorithm for finding shortest paths between all pairs of nodes, with the one restriction that these paths must all pass throughv0 .

Shortest paths are not always unique: sometimes there are two or more different paths with the minimum possible length. Show how to solve the following problem in O((|V|+|E|)log|V|)time.

Input:An undirected graph G=(V,E);edge lengths le>0; starting vertex s鈭�V.

Output:A Boolean array for each node u , the entry usp[u]should be true if and only if there is a unique shortest path s to u (Note:usp[s]=true)

In cases where there are several different shortest paths between two nodes (and edges have varying length),the most convenient of these paths is often the one with fewest edges. Forinstance, if nodes represent cities and edge lengths represent costs of flying between cities, theremight be many ways to get from cityto city t which all have the same cost. The mostconvenientof these alternatives is the one which involves the fewest stopovers. Accordingly, for a specific starting node S , define

bestu=minimum number of edges in a shortest path from S to u .

In the example below, thebestvalues for nodes S,A,B,C,D,E,Fare 0,1,1,1,2,2,3, respectively.

Give an efficient algorithm for the following problem.

Input:Graph G=V,E; positive edge lengths le; starting node s鈭�V.

Output: The values of bestu should be set for all nodesu鈭�V

Question: Often there are multiple shortest paths between two nodes of a graph. Give a linear-time algorithm for the following task.

Input: Undirected graph G = (V , E )with unit edge lengths; nodesu,v鈭�V

Output: The number of distinct shortest paths from utov.

Suppose we want to run Dijkstra鈥檚 algorithm on a graph whose edge weights are integers in the range 0,1,........,W, where Wis a relatively small number.
(a) Show how Dijkstra鈥檚 algorithm can be made to run in time

O(W|V|+|E|)

(b) Show an alternative implementation that takes time just .

O((|V|+|E|)logW)
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