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Get Full Access to Introduction To Algorithms - 3 Edition - Chapter 23 - Problem 23-1
Get Full Access to Introduction To Algorithms - 3 Edition - Chapter 23 - Problem 23-1

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# Second-best minimum spanning tree Let G D .V; E/ be an

ISBN: 9780262033848 130

## Solution for problem 23-1 Chapter 23

Introduction to Algorithms | 3rd Edition

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Problem 23-1

Second-best minimum spanning tree Let G D .V; E/ be an undirected, connected graph whose weight function is w W E ! R, and suppose that jEj jV j and all edge weights are distinct. We define a second-best minimum spanning tree as follows. Let T be the set of all spanning trees of G, and let T 0 be a minimum spanning tree of G. Then a second-best minimum spanning tree is a spanning tree T such that w.T / D minT 002T fT 0g fw.T 00/g. a. Show that the minimum spanning tree is unique, but that the second-best minimum spanning tree need not be unique. b. Let T be the minimum spanning tree of G. Prove that G contains edges .u; / 2 T and .x; y/ 62 T such that T f.u; /g [ f.x; y/g is a second-best minimum spanning tree of G. c. Let T be a spanning tree of G and, for any two vertices u; 2 V , let maxu; denote an edge of maximum weight on the unique simple path between u and in T . Describe an O.V 2/-time algorithm that, given T , computes maxu; for all u; 2 V . d. Give an efficient algorithm to compute the second-best minimum spanning tree of G.

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##### ISBN: 9780262033848

Introduction to Algorithms was written by and is associated to the ISBN: 9780262033848. This textbook survival guide was created for the textbook: Introduction to Algorithms, edition: 3. This full solution covers the following key subjects: spanning, tree, minimum, let, Best. This expansive textbook survival guide covers 35 chapters, and 151 solutions. The answer to “Second-best minimum spanning tree Let G D .V; E/ be an undirected, connected graph whose weight function is w W E ! R, and suppose that jEj jV j and all edge weights are distinct. We define a second-best minimum spanning tree as follows. Let T be the set of all spanning trees of G, and let T 0 be a minimum spanning tree of G. Then a second-best minimum spanning tree is a spanning tree T such that w.T / D minT 002T fT 0g fw.T 00/g. a. Show that the minimum spanning tree is unique, but that the second-best minimum spanning tree need not be unique. b. Let T be the minimum spanning tree of G. Prove that G contains edges .u; / 2 T and .x; y/ 62 T such that T f.u; /g [ f.x; y/g is a second-best minimum spanning tree of G. c. Let T be a spanning tree of G and, for any two vertices u; 2 V , let maxu; denote an edge of maximum weight on the unique simple path between u and in T . Describe an O.V 2/-time algorithm that, given T , computes maxu; for all u; 2 V . d. Give an efficient algorithm to compute the second-best minimum spanning tree of G.” is broken down into a number of easy to follow steps, and 216 words. The full step-by-step solution to problem: 23-1 from chapter: 23 was answered by , our top Engineering and Tech solution expert on 11/10/17, 05:55PM. Since the solution to 23-1 from 23 chapter was answered, more than 452 students have viewed the full step-by-step answer.

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