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 23.2, Problem 4E
Program Plan Intro
To find the run time of Kruskal’s
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Suppose you have a graph with 100 nodes and 500 edges and you want to find the shortest path between two nodes using Dijkstra's algorithm. What is the time complexity of this operation?
Suppose 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?
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.
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- Consider the line from (5, 5) to (13, 9).Use the Bresenham’s line drawing algorithm to draw this line. You are expected to find out all the pixels of the line and draw it on a graph paperarrow_forwardOne can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…arrow_forwardGiven the following Graphs: Graph A: В 12c 1 4. E F G- H 4 J K 3 Graph B: Graph B is the undirected version of Graph A. 3.arrow_forward
- I have a directed graph with N nodes. How can I tell if the graph has a cycle? Your answer should be an algorithm in English. It is nice but not essential for you to give the name of the algorithm. You should describe how the algorithm works.arrow_forwardGiven a graph G, check whether can we color the vertices of a graph G using two colors such that no two adjacent vertices have the same color. Which algorithm will you apply for this problem and what would be the complexity?arrow_forwardBrad is provided with a graph containing X branches. And it's given that the xth branch has a weight of Px. And for the given condition y-I + Py-Px there exits an edge connecting the branches. You have to develop a Python code that finds the number of connected components in this graph. Example Input: 1 3 152 Result: 1arrow_forward
- Be a connected, undirected graph, G D.V; E/. To calculate a path across G that precisely travels each edge in E once in each direction, provide an O.V C E/-time method. Give you a lot of pennies; explain how you can navigate a maze using them.arrow_forwardDraw a 5 × 5 grid graph. How many edges does the n × n grid graphhave?arrow_forwardtreytarrow_forward
- ........arrow_forwardIn the figure below there is a weighted graph, dots represent vertices, links represent edges, and numbers represent edge weights. S 2 1 2 1 2 3 T 1 1 2 4 (a) Find the shortest path from vertex S to vertex T, i.e., the path of minimum weight between S and T. (b) Find the minimum subgraph (set of edges) that connects all vertices in the graph and has the smallest total weight (sum of edge weights). 2. 3.arrow_forwardRefer to the undirected graph provided below: B G What is the maximum length of a path in the graph? Give an example of a path of that length. What is the maximum length of a cycle in the graph? Give an examnple of a cycle of that length. Give an example of an open walk of length five in the graph that is a trail but not a path. Give an example of a closed walk of length four in the graph that is not a сircuit. Give an example of a circuit of length zero in the graph.arrow_forward
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