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 16, Problem 3P
(a)
Program Plan Intro
To prove for thegiven graph G = ( V, E ) which contains matrix M , the M is linearly independent if the set of edges are acyclic.
(b)
Program Plan Intro
To design an efficient
(c)
Program Plan Intro
To explain the condition that fails to hold the matriod condition for the graph G and associated system ( E, I ).
(d)
Program Plan Intro
To discuss that edges set without directed cycle contains linearly dependent column set of matrix M .
(e)
Program Plan Intro
To prove that satisfying the matriod condition for associated system of the graph G and linear independence of matrix M are not contradictory.
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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 graphĀ using powers of the adjacency matrix A
1. Consider the directed acyclic graph (DAG) D shown below.
(a) Write down the adjacency matrix A corresponding to the ordering of the vertices
given by alphabetical order: a, b, c, d, e, f, g.
(b) Find all permutations of the set of vertices {a, b, c, d, e, Ę, g} such that the asso-
ciated adjacency matrix is strictly lower triangular.
(c) Find the number of spanning trees rooted at vertex g.
Transitive ClosureThe transitive closure of a graph G = (V, E) is defined as the graph G = (V, E)with edge (u, v) ā E if there is a path between the vertices u and v in G. Thus, fora connected graph which has paths between every vertex pairs it has, its transitiveclosure is a complete graph. The connectivity matrix of a graph G is a matrix C withentries C[i, j] having a unity value if there exists a path between vertices i and jin the graph G. Finding the connectivity matrix of a graph G is basically findingthe adjacency matrix of its transitive closure. We will see other ways of finding theconnectivity matrices of directed and undirected graphs in Chap.8.Warshallās algorithm to find the transitive closure of a graph works similar to finding distances using Floyd-Warshall algorithm, however, logical and and logical oroperations are used instead of multiplication and addition performed during normalmatrix multiplication.Python ImplementationPython implementation of this algorithm
Chapter 16 Solutions
Introduction to Algorithms
Ch. 16.1 - Prob. 1ECh. 16.1 - Prob. 2ECh. 16.1 - Prob. 3ECh. 16.1 - Prob. 4ECh. 16.1 - Prob. 5ECh. 16.2 - Prob. 1ECh. 16.2 - Prob. 2ECh. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5E
Ch. 16.2 - Prob. 6ECh. 16.2 - Prob. 7ECh. 16.3 - Prob. 1ECh. 16.3 - Prob. 2ECh. 16.3 - Prob. 3ECh. 16.3 - Prob. 4ECh. 16.3 - Prob. 5ECh. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.4 - Prob. 1ECh. 16.4 - Prob. 2ECh. 16.4 - Prob. 3ECh. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.5 - Prob. 1ECh. 16.5 - Prob. 2ECh. 16 - Prob. 1PCh. 16 - Prob. 2PCh. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5P
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- We are given a graph G = (V, E); G could be a directed graph or undirected graph. Let M bethe adjacency matrix of G. Let n be the number of vertices so that the matrix M is n Ćn matrix. For anymatrix A, let us denote the element of i-th row and j-th column of the matrix A by A[i, j].1. Consider the square of the adjacency matrix M . For all i and j, show that M 2[i, j] is the number ofdifferent paths of length 2 from the i-th vertex to the j-th vertex. It should be explained or proved asclearly as possible.2. For any positive integer k, show that M k[i, j] is the number of different paths of length k from the i-th vertex to the j-th vertex. You may use induction on k to prove it.3. Assume that we are given a positive integer k. Design an algorithm to find the number of different paths of length k from the i-th vertex to j-th vertex for all pairs of (i, j). The time complexity of your algorithm should be O(n3 log k). You can get partial credits if you design an algorithm of O(n3k).arrow_forward3. 4. Given a directed acyclic graph G = (V, E) and two vertices s, te V, design an efficient algorithm that computes the number of different directed paths from s to t. Define the incidence matrix B of a directed graph with no self-loop to be an nxm matrix with rows indexed by vertices, column indexed by edges such that Bij = -1 1 0 if edge j leaves vertex i, if edge jenters vertex i, otherwise. Let BT be the transpose of matrix B. Find out what the entries of the n x n matrix BBT stand for.arrow_forwardGiven the following adjacency matrix representation of a graph to answer the followed questions. 0 1 1 0 1 100 10 A.. B. I C. D.ā. 1000 1 0 0 1 0 0 1 0 1 1 is it directed graph or not and why? ] How many vertices are there in the graph? How many edges are there in the graph? reRepresent the graph using an adjacency list.arrow_forward
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