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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 directed graph G(V,E)with (possibly negative) weighted edges, along with a specific node s鈭�Vand a tree T=(V,E'),E'鈯�E. Give an algorithm that checks whether T is a shortest-path tree for G with starting point s . Your algorithm should run in linear time.

Shortest path algorithms can be applied in currency trading. Let c1,c2,cn be various currencies; for instance, c1might be dollars, c2pounds, and c3 lire.

For any two currencies ci and cj , there is an exchange rate 蟿i,j; this means that you can purchase 蟿i,j units of currency cj in exchange for one unit of cj. These exchange rates satisfy the condition that rij.rji<1 so that if you start with a unit of currency cj, change it into currency and then convert back to currency localid="1658917254028" ci, you end up with less than one unit of currency ci (the difference is the cost of the transaction).

a. Give an efficient algorithm for the following problem: Given a set of exchange rates rij , and two currencies s and t , find the most advantageous sequence of currency exchanges for converting currency into currency . Toward this goal, you should represent the currencies and rates by a graph whose edge lengths are real numbers.

The exchange rates are updated frequently, rejecting the demand and supply of the various currencies. Occasionally the exchange rates satisfy the following property: there is a sequence of currencies ci1,ci2,.......ciksuch that ri1,ri2.i3,.........ri(k-1),ik,rik+1>1. This means that by starting with a unit of currency ci1and then successively converting it to currencies ci1,ci2.......cik, and finally back to ci1, you would end up with more than one unit of currency ci1 . Such anomalies Last only a fraction of a minute on the currency exchange, but they provide an opportunity for risk-free profits.

b. Give an efficientalgorithm for detecting the presence of such an anomaly. Use the graph representation you found above.

Generalized shortest-paths problem.In Internet routing, there are delays on lines but also, more significantly, delays at routers. This motivates a generalized shortest-paths problem.

Suppose that in addition to having edge lengths {Ie:e鈭�E} ,a graph also has vertex costs {cV:v鈭�V} . Now define the cost of a path to be the sum of its edge lengths, plusthe costs ofall vertices on the path (including the endpoints). Give an efficient algorithm for the followingproblem.

Input:A directed graph G={V,E} positive edge lengths Ie and positive vertex costs cv; a starting vertex s鈭�v.

Output:An array cost[.] such that for every vertex u,costu, is the least cost of any path from s to u (i.e., the cost of the cheapest path), under the defnition above.

Notice that cost[s]=c.

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 O|V3|.

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 .
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