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 35.2, Problem 4E
Program Plan Intro
To demonstrate that there exist a polynomial time estimation
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The low-degree spanning tree problem is as follows. Given a graph G and an integer k,
does G contain a spanning tree such that all vertices in the tree have degree at most k
(obviously, only tree edges count towards the degree)? For example, in the following graph,
there is no spanning tree such that all vertices have a degree at most three.
(a) Prove that the low-degree spanning tree problem is NP-hard with a reduction from
Hamiltonian path.
(b) Now consider the high-degree spanning tree problem, which is as follows. Given
a graph G and an integer k, does G contain a spanning tree whose highest degree
vertex is at least k? In the previous example, there exists a spanning tree with a
highest degree of 7. Give an efficient algorithm to solve the high-degree spanning
tree problem, and an analysis of its time complexity.
Computer Science
Consider the d-Independent Set problem:
Input: an undirected graph G = (V,E) such that every vertex has degree less or equal than d.
Output: The largest Independent Set.
Describe a polynomial time algorithm Athat approximates the optimal solution by a factor α(d). Your must
write the explicit value of α, which may depend on d. Describe your algorithm in words (no pseudocode) and
prove the approximation ratio α you are obtaining. Briefly explain why your algorithm runs in polytime.
• Problem 2
Given undirected graph G=(V,E), with |VM=n, |El=m, and three vertices a,b,c e V. Describe an efficient algorithm that determines whether there is a simple cycle in G that contains all of a,
b and c.
(No vertex appears more than once within a simple cycle.) Analyze the worst-case time complexity of your algorithm.
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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