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 2E
Program Plan Intro
To give a implementation of Prim’s
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Give a linear time algorithm via pseudo code that takes as input a directed acyclic graph
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- Suppose are you given an undirected graph G = (V, E) along with three distinct designated vertices u, v, and w. Describe and analyze a polynomial time algorithm that determines whether or not there is a simple path from u to w that passes through v. [Hint: By definition, each vertex of G must appear in the path at most once.]arrow_forwardSay that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.arrow_forwardGiven a directed graph, write a function to determine if it contains a cycle. What is the time complexity of your solution?arrow_forward
- ........arrow_forwardDraw a simple, connected, weighted graph with 8 vertices and 16 edges, each with unique edge weights. Identify one vertex as a “start” vertex and illustrate a running of Dijkstra’s algorithm on this graph. Problem R-14.23 in the photoarrow_forwardA Hamiltonian path on a directed graph G = (V, E) is a path that visits each vertex in V exactly once. Consider the following variants on Hamiltonian path: (a) Give a polynomial-time algorithm to determine whether a directed graph G contains either a cycle or a Hamiltonian path (or both). Given a directed graph G, your algorithm should return true when a cycle or a Hamiltonian path or both and returns false otherwise. (b) Show that it is NP-hard to decide whether a directed graph G’ contains both a cycle and a Hamiltonian Path, by giving a reduction from the HAMILTONIAN PATH problem: given a graph G, decide whether it has a Hamiltonian path. (Recall that the HAMILTONIAN PATH problem is NP-complete.)arrow_forward
- My professor went over the proof on this slide and I don't really understand it. Can you explain it in detail step by step?arrow_forwardWe 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 ×n 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. We are given a weighted undirected graph G containing exactly 10 cycles. Write an algorithm to compute the MST of G. Your algorithm should have a runtime of O(V + E).arrow_forward
- Please help me with this practice problem in python : Implement two-level iterative method B = B_{TL} for graph Laplacian matrices. We want the symmetric B. Components: Given a graph, construct its graph Laplacian matrix. Then using Luby's algorithm, construct the P matrix that ensures a prescribed coarsening factor, e.g., 2, 4, or 8 times smaller number of coarse vertices. Since the graph Laplacian matrix is singular (it has the constants in its nullspace), to make it invertible, make its last row and columns zero, but keep the diagonal as it were (nonzero). The resulting modified graph Laplacian matrix A is invertible and s.p.d.. Form the coarse matrix A_c = P^TAP. To implement symmetric two-level cycle use one of the following M and M^T: (i) M is forward Gauss-Seidel, M^T - backward Gauss-Seidel (both corresponding to A) (ii) M = M^T - the ell_1 smoother. Compare the performance (convergence properties in terms of number of iterations) of B w.r.t. just using the smoother M…arrow_forwardWrite program for Kurskal’s algorithm to finding the MST of a weighted graph.arrow_forwardYou are given a weighted, undirected graph G = (V, E) which is guaranteed to be connected. Design an algorithm which runs in O(V E + V 2 log V ) time and determines which of the edges appear in all minimum spanning trees of G. Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain Englisharrow_forward
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