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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consider the following undirected graph.
Draw out the Minimum Spanning Tree of the above graph.
Draw 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 photo
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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- Be G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.arrow_forwardGiven 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/verticesarrow_forwardWe are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…arrow_forward
- Design 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_forwardRequired information NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the bipartite graph Km.n- Find the values of mand n if Km n has an Euler path. (Check all that apply.) Check All That Apply Km,n has an Euler path when both mand n are even. Km,n has an Euler path when both mand n are odd. Km, n has an Euler path if m=2 and n is odd. Km, n has an Euler path if n= 2 and m is odd. Km, n has an Euler path when m= n=1.arrow_forwardConsider the following graph G. B E H. A G Find 6(G)= (G)= K(G)= number of edge-disjoint AD-paths= and number of vertex-disjoint AD-paths=arrow_forward
- How many spanning trees does the following graph have?arrow_forwardDraw a connected graph with 7 nodes and 10 edges which has no cutnode nor bridge. Find by the use of reduction algorithm the number of spanning trees in this graph.arrow_forwardGiven a graph that is a tree (connected and acyclic). (1) 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 true a. p is the longest path in the graph b. p is the shortest path in the graph c. p can be calculated in time linear in the number of edges/vertices a,c a,b a,b,c b.carrow_forward
- Given the following example of UAG graphs: 1)- In Java, give implementation to find the shortest path for graph 2 21 5 4. 3.arrow_forwardHow many edges does a graph have if its degree sequence is 2, 4, 4, 5, 3?A. Draw a graph with the above listed sequence.B. Is it possible to draw an Euler Circuit with such a sequence of vertex degrees?Is it possible to draw an Euler Path? If yes, to either of these questions, draw the a graph that supports your answer.arrow_forwardIf I pick a vertex on the graph, at random, and run Prim's Algorithm, let X be the number of edges in the minimum spanning tree that results. What is E[X] + 14?arrow_forward
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