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 24.5, Problem 1E
Program Plan Intro
To find various shortest path treesin a Directed Graph.
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You are given a simple connected undirected graph with N nodes(or vertices) for computing a Minimum Spanning Tree (MST) of the graph. Write down the upper and lower bound of the possible number of edges of that MST.
Now consider the same for a “simple disconnected undirected graph”. What is the lower and upper bound now?
Let G be a directed acyclic graph with exactly one source r such that for any other vertex v there exists a unique directed path from r to v. Let Gu be the undirected graph obtained by erasing the direction on each edge of G. Prove that (Gu,r) is a rooted tree.
G = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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- Let the graph G be a cycle of n nodes in which x edges have the weight 100 and y edges have weight 200. How many minimum spanning trees does G have?arrow_forwardLet G be a connected graph, and let T1, T2 be two spanning trees. Prove thatT1 can be transformed to T2 by a sequence of intermediate trees, each obtainedby deleting an edge from the previous tree and adding another.arrow_forwardDesign an algorithm for finding a maximum spanning tree—a spanning tree with the largest possible edge weight—of a weighted connected graph. (Look at Kruskal’s Algorithm for supplemental information)arrow_forward
- Let G be a connected graph that has exactly 4 vertices of odd degree: v1,v2,v3 and v4. Show that there are paths with no repeated edges from v1 to v2, and from v3 to v4, such that every edge in G is in exactly one of these paths.arrow_forwardConsider a graph G that is comprised only of non-negative weight edges such that (u, v) € E, w(u, w) > 0. Is it possible for Bellman-Ford and Dijkstra's algorithm to produce different shortest path trees despite always producing the same shortest-path weights? Justify your answer.arrow_forwardGiven the graph below, what should be the souce node such that in finding the shortest path tree, the result would be the same whe the minimum spanning tree is searched?arrow_forward
- let us take any standard graph G=(v,e) and let us pretend each edge is the same exact weight. let us think about a minimum spanning tree of the graph G, called T = (V, E' ). under each part a and b illustrate then show that a) s a unique path between u and v in T for all u, v ∈ V . b) tree T is not unique. provide proofarrow_forwardWhich is true/false. Let G be a connected undirected weighted graph. Consider the following two statements. S₁: There exists a minimum weight edge in G which is present in every minimum spanning tree of G. S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree.arrow_forwardFind the shortest path from S to other nodes, on the given directed acyclic graph.Graph: R → A : 3 S → A : 1 A → C : 6 B → D : 3 C → E : 2R → S : 2 S → B : 2 B → A : 4 C → D : 1 D → E : 1 Answer: Topological Ordering: __________________________ Node Edge Relax? Update Shortest Path from S: Length Path R S A B C D Earrow_forward
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