Introduction to Algorithms
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, Problem 4P

a.

Program Plan Intro

To show that a maximal matching need not be a maximum matching by drawing an undirected graph G.

b.

Program Plan Intro

To show that the linear-time greedy algorithm for maximal matching is a 2-approximation algorithm for maximum matching.

c.

Program Plan Intro

To show that the size of a maximum matching in G is a lower bound on the size of any vertex cover for G.

d.

Program Plan Intro

To prove that the number of bins used by the first-fit heuristic is never more than 2S .

e.

Program Plan Intro

To prove an approximation ratio of 2 for the first-fit heuristic.

f.

Program Plan Intro

To give an efficient implementation of the first-fit heuristic, and analyze its running time.

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A directed graph G= (V,E) consists of a set of vertices V, and a set of edges E such that each element e in E is an ordered pair (u,v), denoting an edge directed from u to v. In a directed graph, a directed cycle of length three is a triple of vertices (x,y,z) such that each of (x,y) (y,z) and (z,x) is an edge in E. Write a Mapreduce algorithm whose input is a directed graph presented as a list of edges (on a file in HDFS), and whose output is the list of all directed cycles of length three in G. Write the pseudocode for the mappers/reducers methods. Also, assuming that there are M mappers, R reducers, m edges and n vertices -- analyze the (upper-bound of the) communication cost(s).
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The clique problem is finding cliques in a diagram. A clique is a set of vertices that are adjacent to each other. The 4-clique is a set of four knots all connected. So, in this example of the 4-clique problem, we have a graph with 7 vertices. A brute force algorithm searched all possible combinations of four vertices and found a set that formed a clique. If you want to understand more about it, the problem (and if possible read on). Note that the clique problem is NP-complete, so deterministic search is not practical for large graph sizes. This makes it an ideal candidate for evolutionary exploration. In this problem, we have to assume that we are given the problem of implementing the 4-clique problem as an evolutionary algorithm for an arbitrary graph with an arbitrary number of vertices (an n-vertex graph). If 4 cliques are found, the algorithm succeeds. 1. Provide an algebraic expression, in terms of n, for the size of the phenotypic search space (the number of possible…
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