Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 24.3, Problem 1E
Program Plan Intro
To implement the Dijkstra’s
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Draw an example of a graph which has 6 nodes and better to implement with a Adjacency Matrix.
The following table presents the implementation of Dijkstra's algorithm on the evaluated graph G with 8 vertices.
a) What do the marks (0, {a}) and (∞, {x}) in the 1st row of the table mean?
b) What do the marks marked in blue in the table mean?
c) Reconstruct all edges of the graph G resulting from the first 5 rows of the table of Dijkstra's algorithm.
d) How many different shortest paths exist in the graph G between the vertices a and g?
Programming with Python:
v Implement Depth First Search (Traversal)
v Implement Breadth First Traversal
For the implementation:
1. Create a directed graph with 10 Vertices
2. Add arbitrary edges among vertices to satisfy the directed graph
3. Run your program and show appropriate output (Traversal order)
4. Your implementation should be fully commented
Please, submit your source code and output (Screen Capture) too.
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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- Can you help me with this problem and can you do it step by step. question: Show that every graph with two or more nodes contains two nodes that have equal degrees.arrow_forwardConsider the following graph and Dijkstra's algorithm to find the shortest paths from the vertex A. (See image attached) The distances / weights in the incident edges at the vertex F are given in the table: edge: (B, F) (C, F) (E, F)distance: 8 11 4.5 At the end of the algorithm, what is the value of the distance for the vertex F?arrow_forwardlet a graph have vertices h,i,j,k,l,m,n,o and edge set {{h,i},{h,j},{h,k},{h,n},{h,o},{j,k},{j,l},{j,m},{k,l},{m,o},. on paper, draw the graph. then answer the questions below. a. what is the degree of vertex k? b. what is the degree of vertex h? c. how many connected components does the graph have?arrow_forward
- Give an example of an input graph that demonstrates that your solution above may find different cycles on the same graph, depending on the order of the edges in the implementation. Next, provide an algorithm that prints the length of the shortest cycle in the graph. Provide the pseudo-code and justify the runtime of O(V E + V 2 ).arrow_forwardImplement the dijkstra's algorithm on a directed graph from a given vertex. all edges have non-negative edge weights. Output the edge as they are added to the shortest path trees. Compute and print the weight of the shortest path to every reachable vertex. The source vertex is S. show the following: -program code -Screenshot of the output -Representation of the graph transversalarrow_forwardThe following table(picture) presents the implementation of the Dijkstra algorithm on the evaluated graph G with 8 vertices. What do the symbols (0, {a}) and (∞, {x}) in first row mean?arrow_forward
- (1)In a graph, the sum of the degrees of all vertices is equal to () times the number of edges of the graph. A．1/2 B．1 C．2 D．4 (2) In a directed graph, the sum of the in degrees of all vertices is equal to () times the sum of the out degrees of all vertices. A．1/2 B．1 C．2 D．4 (3) A directed graph with n vertices can have at most () edges. A．n B．n(n-1) C．n(n+1) D．n2 (4) When a connected graph with n vertices is represented by an adjacent distance matrix, the distance matrix has at least () nonzero elements. A．n B．2(n-1) C．n/2 D．n2 (5) G is a disconnected undirected graph with 28 edges, then the graph has at least () vertices. A．7 B．8 C．9 D．10arrow_forwardWrite a function in a directed graph represented by adjacency lists that returns true (1) if an edge exists between two provided vertices u and v and false (0) otherwise.arrow_forwardconstruct a bipartite graph with vertices L,M,N,O,P,Q that is a tree. what is the edge set? construct a connected bipartite graph with vertices L,M,N,O,P,Q that is not a tree. what is the edge set?arrow_forward
- True or false: For graphs with negative weights, one workaround to be able to use Dijkstra’s algorithm (instead of Bellman-Ford) would be to simply make all edge weights positive; for example, if the most negative weight in a graph is -8, then we can simply add +8 to all weights, compute the shortest path, then decrease all weights by -8 to return to the original graph. Select one: True Falsearrow_forwardSuppose a graph has 6 vertices of degree two, 12 vertices of degree three, and k vertices of degree 1. If the graph has 71 edges, then the value of k is _____.arrow_forwardSuppose you have a graph G with 6 vertices and 7 edges, and you are given the following information: The degree of vertex 1 is 3. The degree of vertex 2 is 4. The degree of vertex 3 is 2. The degree of vertex 4 is 3. The degree of vertex 5 is 2. The degree of vertex 6 is 2. What is the minimum possible number of cycles in the graph G?arrow_forward
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