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5. (10 pts) Give a concrete example of a directed and weighted graph G and two...
Consider the following weighted, directed graph G. There are 7 vertices and 10 edges. The edge list E is as follows:
Consider the following weighted, directed graph G. There are 7 vertices and 10 edges. The edge list E is as follows:The Bellman-Ford algorithm makes |V|-1 = 7-1 = 6 passes through the edge list E. Each pass relaxes the edges in the order they appear in the edge list. As with Dijkstra's algorithm, we record the current best known cost D[V] to reach each vertex V from the start vertex S. Initially D[A]=0 and D[V]=+oo for all the other vertices...
a. (15 marks) i (7 marks) Consider the weighted directed graph below. Carry out the steps...
a. (15 marks) i (7 marks) Consider the weighted directed graph below. Carry out the steps of Dijkstra's shortest path algorithm as covered in lectures, starting at vertex S. Consequently give the shortest path from S to vertex T and its length 6 A 2 3 4 S T F ii (2 marks) For a graph G = (V, E), what is the worst-case time complexity of the version of Dijkstra's shortest path algorithm examined in lectures? (Your answer should...
6. Dijkstra's Algorithm assumes that all edge weights in a given weighted directed graph G =...
6. Dijkstra's Algorithm assumes that all edge weights in a given weighted directed graph G = (VAE) are nonnegative. However, if we apply Dijkstra's Algorithm to the graph G where the edge weights may be negative, Dijkstra's Algorithm may produce incorrect answers. Show such an example where Dijkstra's Algorithm may produce incorrect answers. Then, explain why such incorrect answers happen. (15 pts]
Consider the problem of finding the shortest paths in a weighted directed graph using Dijkstra's algorithm.
Consider the problem of finding the shortest paths in a weighted directed graph using Dijkstra's algorithm. Denote the set of vertices as V, the number of vertices as |V|, the set of edges as E, and the number of edges as |E|. Answer the following questions.Below is a pseudo-code of the algorithm that computes the length c[v] of the shortest path from the start node s to each node v. Answer code to fill in the blank _______ .
Which of the following graph algorithms is designed specifically to accept negative edge weights? Check all...
Which of the following graph algorithms is designed specifically to accept negative edge weights? Check all that apply. a. topological sort b. Dijkstra's algorithm c. Bellman-Ford algorithm d. unweighted shortest path algorithm
Give an efficient algorithm that takes a directed graph G = (V, E) and two vertices...
Give an efficient algorithm that takes a directed graph G = (V, E) and two vertices u, v E V, and determines if there are at least two edge-disjoint paths in G from u to v. i.e., your algorithm should determine whether there are at least two paths from u to v in G that have no edges in common. Argue your algorithm's correctness and analyze its time complexity.
1. Consider a directed graph with distinct and non-negative edge lengths and a source vertex s....
1. Consider a directed graph with distinct and non-negative edge lengths and a source vertex s. Fix a destination vertex t, and assume that the graph contains at least one s-t path. Which of the following statements are true? [Check all that apply.] ( )The shortest (i.e., minimum-length) s-t path might have as many as n−1 edges, where n is the number of vertices. ( )There is a shortest s-t path with no repeated vertices (i.e., a "simple" or "loopless"...
5. Here are the vertices and edges of directed graph G: V= {2.6.c.de.f} E= {ab, ac,...
5. Here are the vertices and edges of directed graph G: V= {2.6.c.de.f} E= {ab, ac, af ca. bc. be.bf. cd, ce, de, df). Weights: w(ab) = 2 w(ac) = 5, w(af) = 10, w(ca) = 2. w(be) = 2. w(be) = 10, w(bf) = 11. w(cd)= 9. w(ce) = 7. w(de) = 2. w(df) = 2. a. Draw the Graph. This is a directed, weighted graph so you need to include arrows and weights. You can insert a pic...
Give an example of a graph G with at least 10 vertices such that the greedy...
Give an example of a graph G with at least 10
vertices such that the greedy 2-approximation algorithm for
Vertex-Cover given below is guaranteed to produce a suboptimal
vertex cover.
Algorithm Vertex CoverApprox(G): Input: A graph G Output: A small vertex cover C for G while G still has edges do select an edge e (v, w) of G add vertices v and w to C for each edge f incident to v or w do remove f from G...
Question 6 Let G be the weighted graph (a) Use Dijkstra's algorithm to find the shortest path from A to F. You can...
Question 6 Let G be the weighted graph (a) Use Dijkstra's algorithm to find the shortest path from A to F. You can do all the work on a single diagram, but, to show that you have used the algorithm correctly, if an annotation needs updating do not erase itjust put a line through it and write the new annotation above that b) In what order are the vertices added to the tree? (c) Notice that the algorithm does not,...
a. (15 marks) i (7 marks) Consider the weighted directed graph below. Carry out the steps of Dijkstra's shortest path algorithm as covered in lectures, starting at vertex S. Consequently give the shortest path from S to vertex T and its length 6 A 2 3 4 S T F ii (2 marks) For a graph G = (V, E), what is the worst-case time complexity of the version of Dijkstra's shortest path algorithm examined in lectures? (Your answer should...
6. Dijkstra's Algorithm assumes that all edge weights in a given weighted directed graph G = (VAE) are nonnegative. However, if we apply Dijkstra's Algorithm to the graph G where the edge weights may be negative, Dijkstra's Algorithm may produce incorrect answers. Show such an example where Dijkstra's Algorithm may produce incorrect answers. Then, explain why such incorrect answers happen. (15 pts]
Give an efficient algorithm that takes a directed graph G = (V, E) and two vertices u, v E V, and determines if there are at least two edge-disjoint paths in G from u to v. i.e., your algorithm should determine whether there are at least two paths from u to v in G that have no edges in common. Argue your algorithm's correctness and analyze its time complexity.
5. Here are the vertices and edges of directed graph G: V= {2.6.c.de.f} E= {ab, ac, af ca. bc. be.bf. cd, ce, de, df). Weights: w(ab) = 2 w(ac) = 5, w(af) = 10, w(ca) = 2. w(be) = 2. w(be) = 10, w(bf) = 11. w(cd)= 9. w(ce) = 7. w(de) = 2. w(df) = 2. a. Draw the Graph. This is a directed, weighted graph so you need to include arrows and weights. You can insert a pic...
Give an example of a graph G with at least 10
vertices such that the greedy 2-approximation algorithm for
Vertex-Cover given below is guaranteed to produce a suboptimal
vertex cover.
Algorithm Vertex CoverApprox(G): Input: A graph G Output: A small vertex cover C for G while G still has edges do select an edge e (v, w) of G add vertices v and w to C for each edge f incident to v or w do remove f from G...
Question 6 Let G be the weighted graph (a) Use Dijkstra's algorithm to find the shortest path from A to F. You can do all the work on a single diagram, but, to show that you have used the algorithm correctly, if an annotation needs updating do not erase itjust put a line through it and write the new annotation above that b) In what order are the vertices added to the tree? (c) Notice that the algorithm does not,...
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