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 29.2, Problem 3E
Program Plan Intro
To write a linear program for which the solution has the property that
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Model the following problem as a network flow problem and write down the corresponding
linear program: Find the maximum number of disjoint paths from the vertex A to the verlex D.'
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A
Write a 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. For the pseudocode consider the following definition of the graph - Given a weighted directed graph, G = (V, E) with a weight function wthat maps edges to real-valued weights. w(u, v) denotes the weight of an edge (u, v). Assume vertices are labeled using numbers from1 to n if there are n vertices.
Given a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/vertices
Chapter 29 Solutions
Introduction to Algorithms
Ch. 29.1 - Prob. 1ECh. 29.1 - Prob. 2ECh. 29.1 - Prob. 3ECh. 29.1 - Prob. 4ECh. 29.1 - Prob. 5ECh. 29.1 - Prob. 6ECh. 29.1 - Prob. 7ECh. 29.1 - Prob. 8ECh. 29.1 - Prob. 9ECh. 29.2 - Prob. 1E
Ch. 29.2 - Prob. 2ECh. 29.2 - Prob. 3ECh. 29.2 - Prob. 4ECh. 29.2 - Prob. 5ECh. 29.2 - Prob. 6ECh. 29.2 - Prob. 7ECh. 29.3 - Prob. 1ECh. 29.3 - Prob. 2ECh. 29.3 - Prob. 3ECh. 29.3 - Prob. 4ECh. 29.3 - Prob. 5ECh. 29.3 - Prob. 6ECh. 29.3 - Prob. 7ECh. 29.3 - Prob. 8ECh. 29.4 - Prob. 1ECh. 29.4 - Prob. 2ECh. 29.4 - Prob. 3ECh. 29.4 - Prob. 4ECh. 29.4 - Prob. 5ECh. 29.4 - Prob. 6ECh. 29.5 - Prob. 1ECh. 29.5 - Prob. 2ECh. 29.5 - Prob. 3ECh. 29.5 - Prob. 4ECh. 29.5 - Prob. 5ECh. 29.5 - Prob. 6ECh. 29.5 - Prob. 7ECh. 29.5 - Prob. 8ECh. 29.5 - Prob. 9ECh. 29 - Prob. 1PCh. 29 - Prob. 2PCh. 29 - Prob. 3PCh. 29 - Prob. 4PCh. 29 - Prob. 5P
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- 3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)arrow_forward3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1arrow_forwardProblem 3: Consider the following weighted graph: 1. Find the length of the shortest path between a and z. 2. Can you think of an algorithm to find the length of a shortest path between all pairs of vertices in a weighted connected simple graph? Show your work.arrow_forwardFloyd warshall algorithm java program. Find the shortest paths between all vertices in a graph using dynamic programming. The matrix and number of vertices as the input(using the scanner), and the shortest path matrix as the output.arrow_forwardConsider an undirected graph with n nodes and m edges. The goal is to find a path between two specified nodes u and v that maximizes the minimum weight of any edge along the path. Assume that all edge weights are positive and distinct. Design an algorithm to solve this problem with a time complexity of O(m log n).arrow_forwardUsing C++ programming language solve the following problem: You are given NN points in DD-dimensional space. The ih point has D coordinates - x, 1,x,2..,Xı,D. Consider a weighted undirected complete graph with these NN points as its vertices. Weight of the edge between points ii and j is |xi, 1–xj, 1|+|xi,2–xj,2|+...+|xi,D-xj,D||xi, 1–xj,1|+|xi,2–xj,2|+ ...+|xi,D-xj,D]. Find the weight of the maximum spanning tree of this graph. Input: 22 11 22 Output: 2arrow_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_forwardLet G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.arrow_forwardGiven an undirected graph G = <V,E>, a vertex cover is a subset of vertices S V such that for each edge (u,v) belongs to E, either u S or v S or both. The Vertex Cover Problem is to find minimum size of the set S. Consider the following algorithm to Vertex Cover Problem: (1) Initialize the result as {} (2) Consider a set of all edges in given graph. Let the set be E’. (3) Do following while E’ is not empty ...a) Pick an arbitrary edge (u,v) from set E’ and add u and v to result ...b) Remove all edges from E which are either incident on u or v. (4) Return result. It claim that this algorithm is exact for undirected connected graphs. Is this claim True or False? Justify the answer.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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