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 27.2, Problem 1E
Program Plan Intro
To analyse the work, span, and parallelism of the computation dag computing P-SQAURE-MATRIX-MULTIPLY on 2X2 matrices.
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In Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…
Consider the graph in following. Suppose the nodes are stored in memory in a linear array DATA as follows: A, B, C, D, E, F, G, H, I, J, K, L, M
Find the path matrix P of graph using powers of the adjacency matrix A
Chapter 27 Solutions
Introduction to Algorithms
Ch. 27.1 - Prob. 1ECh. 27.1 - Prob. 2ECh. 27.1 - Prob. 3ECh. 27.1 - Prob. 4ECh. 27.1 - Prob. 5ECh. 27.1 - Prob. 6ECh. 27.1 - Prob. 7ECh. 27.1 - Prob. 8ECh. 27.1 - Prob. 9ECh. 27.2 - Prob. 1E
Ch. 27.2 - Prob. 2ECh. 27.2 - Prob. 3ECh. 27.2 - Prob. 4ECh. 27.2 - Prob. 5ECh. 27.2 - Prob. 6ECh. 27.3 - Prob. 1ECh. 27.3 - Prob. 2ECh. 27.3 - Prob. 3ECh. 27.3 - Prob. 4ECh. 27.3 - Prob. 5ECh. 27.3 - Prob. 6ECh. 27 - Prob. 1PCh. 27 - Prob. 2PCh. 27 - Prob. 3PCh. 27 - Prob. 4PCh. 27 - Prob. 5PCh. 27 - Prob. 6P
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- If I have a large number of edges that form and undirected graph and I want to store them in an adjacency matrix, what are some techniques I can use to efficiently build the matrix without using brute force techniques?arrow_forwardWrite pseudocode to find all pairs shortest paths using the technique used in Bellman-Ford's algorithm so that it will produce the same matrices like Floyd-Warshall algorithm produces. Also provide the algorithm to print the paths for a source vertex and a destination vertex. Describe the properties of the algorithm you provide and the run time for your algorithm in detail.arrow_forwardQ5. Solve the following classification problem with the perceptron rule. Apply each input vector in order, for as many repetitions as it takes to ensure that the problem is solved. Draw a graph of the problem only after you have found a solution. Use the initial weights and bias: [o o] W(0) = b(0) = 0.arrow_forward
- Specifications: You will create an implementation of this algorithm. Your driver program should provide a graph and a source vertex in the graph. Your implementation should use Dijkstra's Algorithm to determine the shortest path using adjacency matrix representation. Specifically, given a graph and a source vertex in the graph, find the shortest paths from source to all vertices in the given graph, using Dijkstra's Algorithm.arrow_forwardAssume for an undirected graph with 6 vertices and 6 edges that a vertex index requires 3 bytes, a pointer requires 5 bytes, and that edge weights require 3 bytes. Since the graph is undirected, each undirected edge is represented by two directed edges. Calculate the byte requirements for an adjacency matrix.arrow_forwardBellman-Ford algorithm Draw a graph G with weights of edges ranging from 3 to 9, is it possible to calculate the LONGEST PATH without altering the algorithm at all? Justify your answer by providing solid reasons.arrow_forward
- In the last homework, we gave an algorithm to check whether a graph has a triangle in anundirected graph in O(n2 +nm) time, which is Θ(n3) when the graph is dense. Show how given a graph G in adjacency matrix representation, it is possible to use Strassen's matrix multiplication algorithm to determine if the graph has a triangle in time o(n3). (You can use the algorithmwithout description or proof, but you must explain connection to the existence of a triangle in the graph).arrow_forwardDesign an adjacency Matrix of the alphabets of Uzair Bhatti . In accordance with the following conditions: If your name has repeated characters (e.g. character E, 2 times) then you will consider only 1 time. If your Name contains both G and S, then there will be an edge between them and one additional edge from W to each if W is also in your name. If N is the alphabet in your name, it will have an edge to A and S if it available in your name. If P is available then it will have an edge with L if it is available. If there is a blank space in full Name then it will be represented by “_”, and it must have an edge with all alphabets. (Note: Place on in 1 for Edge and 0 for No Edge) Sketch an undirected graph of the above designed adjacency matrixarrow_forwardP R O B L E M D E S C R I P T I O N :1. As described in the reading and lecture, an adjacency matrix for a graph with n verticesnumber 0, 1, …, n – 1 is an n by n array matrix such that matrix[i][j] is 1 (or true) ifthere is an edge from vertex i to vertex j, and 0 (or false) otherwise.2. In this assignment, you will implement the methods identified in the stub code below insupport of the Graph ADT that uses an adjacency matrix to represent an undirected,unweighted graph with no self-loops.import java.util.*; // for all needed JCF classespublic class Graph { private int[][] matrix;// the adjacency matrix of the graph. // Creates an n x n array with all values initialized to 0. public Graph(int n) {// your code here } // end constructor // This method returns the number of nodes in the graph. public int getNumVertices() {// your code here } // end getNumVertices // This method returns the number of edges in the graph. public int getNumEdges() {// your code here } // end getNumEdges //…arrow_forward
- Simplify the following k-map and compare it to boolean simplification. Answer both item.arrow_forwardConsider the following Digraph to answer the 1, 2 1. 2. C 1 3 4 S 3 5 G 3 Find all possible paths from the node S to T Draw the adjacency matrix of the graph, considering that the matrix will contain the weights of the edges?arrow_forwardFor the sample space S = {c,d,e,f,g,k,l}, identify the complement of A = {d,f,k,}.arrow_forward
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