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 22.2, Problem 8E
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If I pick a vertex on Luna's graph (pictured, Question 1), 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?
Suppose you have a graph with 100 nodes and 500 edges and you want to find the shortest path between two nodes using Dijkstra's algorithm. What is the time complexity of this operation?
you need to write the solutions in python and provide a brief explanation of your codes and the efficiency analysis with comments.
3. Consider a loop tree which is an undirected wighted graph formed by taking a binary tree and adding an edge from exactly one of the leaves to another node in the tree as follows: Letnbe the number of vertices in a loop tree. How long does it take Prim's or Kruskal's algorithms to find the minimum spanning tree in terms ofn? Devise a more efficient algorithm that takes an nxn adjacency weighted matrix as input, and finds the minimum spanning tree of a loop tree.
三 input1 - Not Defteri Dosya Düzenle 1078000000 700650060 800006400 060000000 050000021 006000000 004000000 060020000 000010000
output1 - Not Defteri Dosya Düzenle Görünü p 14873265
三 input2 - Not Defteri Dosya Düzenle Görünüm0210000200344010000000300000040005604005000000
Chapter 22 Solutions
Introduction to Algorithms
Ch. 22.1 - Prob. 1ECh. 22.1 - Prob. 2ECh. 22.1 - Prob. 3ECh. 22.1 - Prob. 4ECh. 22.1 - Prob. 5ECh. 22.1 - Prob. 6ECh. 22.1 - Prob. 7ECh. 22.1 - Prob. 8ECh. 22.2 - Prob. 1ECh. 22.2 - Prob. 2E
Ch. 22.2 - Prob. 3ECh. 22.2 - Prob. 4ECh. 22.2 - Prob. 5ECh. 22.2 - Prob. 6ECh. 22.2 - Prob. 7ECh. 22.2 - Prob. 8ECh. 22.2 - Prob. 9ECh. 22.3 - Prob. 1ECh. 22.3 - Prob. 2ECh. 22.3 - Prob. 3ECh. 22.3 - Prob. 4ECh. 22.3 - Prob. 5ECh. 22.3 - Prob. 6ECh. 22.3 - Prob. 7ECh. 22.3 - Prob. 8ECh. 22.3 - Prob. 9ECh. 22.3 - Prob. 10ECh. 22.3 - Prob. 11ECh. 22.3 - Prob. 12ECh. 22.3 - Prob. 13ECh. 22.4 - Prob. 1ECh. 22.4 - Prob. 2ECh. 22.4 - Prob. 3ECh. 22.4 - Prob. 4ECh. 22.4 - Prob. 5ECh. 22.5 - Prob. 1ECh. 22.5 - Prob. 2ECh. 22.5 - Prob. 3ECh. 22.5 - Prob. 4ECh. 22.5 - Prob. 5ECh. 22.5 - Prob. 6ECh. 22.5 - Prob. 7ECh. 22 - Prob. 1PCh. 22 - Prob. 2PCh. 22 - Prob. 3PCh. 22 - Prob. 4P
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- Design a nondeterministic polynomial-time algorithm for following problem: Given a graph G = (V, E), is there a spanning tree with exactly two leaves? Please give an analysis on correctness and running time of your algorithm.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_forwardSuppose you have a polynomial-time algorithm that, given a multigraph H, computes the number of spanning forests of H. Using this algorithm as a subroutine, design a polynomial-time algorithm that, given a weighted graph G, computes the number of minimum spanning trees of G.arrow_forward
- Given 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_forwardGiven an undirected weighted graph G with n nodes and m edges, and we have used Prim’s algorithm to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct. (Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)arrow_forwardLet G = (V, E) be an undirected and connected graph, where each edge (u, v) E E has a weight wt(u, v) > 0. Moreover, assume that all weights are equal. (a) G. The running time of your algorithm must be faster than the running time of Kruskal's algorithm. Design an algorithm to compute a minimum spanning tree (MST) of You must describe your algorithm in plain English (no pseudocode). You must write the running time of your algorithm and explain why you get this running time. (b) In at most 50 words, explain why your algorithm is correct.arrow_forward
- Use the high-level version of Kruskal's algorithm to find a minimum spanning tree for the following graph, showing the actions step-by-step.arrow_forwardApply Kruskal's algorithm to find a minimum spanning tree of the following graph. You need to do it step by step. Answer: here a 5 b () 3 1 d 4 C 2 6 earrow_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
- Let T be a binary search tree, and let z be a key. Give an efficient algorithm for finding the smallest key y in T such that y > r. Note that r may or may not be in T. Explain why your algorithm has the running time it does.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_forwardTrue or false: For graphs with negative weights, one workaround to be able to use Dijkstra’s algorithm (instead of Bellman-Ford) would be to simply make all edge weights positive; for example, if the most negative weight in a graph is -8, then we can simply add +8 to all weights, compute the shortest path, then decrease all weights by -8 to return to the original graph. Select one: True Falsearrow_forward
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