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 26.3, Problem 4E
Program Plan Intro

To explain that there exists a perfect matching in graph G if and only if |A||N(A)| for every subset AL .

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Consider an undirected graph G = (V;E). An independent set is a subset I   V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let  (G) denote the size of the largest maximal independent set in G.   1) What is  (G) if G is a complete graph on n vertices? What if G is a cycle on n vertices?
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).
Consider an undirected graph G = (V;E). An independent set is a subset I   V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let  (G) denote the size of the largest maximal independent set in G.   One way of trying to avoid this dependence on ordering is the use of randomized algorithms. Essentially, by processing the vertices in a random order, you can potentially avoid (with high probability) any particularly bad orderings. So consider the following randomized algorithm for constructing independent sets:   @ First, starting with an empty set I, add each vertex of G to I independently with probability p.   @ Next, for any edges with both vertices in I, delete one of the two vertices from I (at random).  @  Note - in this second step,…
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