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, Problem 4P
Program Plan Intro
To show the time complexity big O ( V + E ) in the directed graph G ( V, E ) with minimum labels between the vertices or nodes.
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Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is
incident to at least one vertex in C. We say that a set of vertices I form an independent set if
no edge in G connects two vertices from I.
For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of
the 20 edges in the graph has at least one endpoint in C, and I = = [a, c, i, k] is an
independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik.
2a
In the example above, notice that each vertex belongs to the vertex cover C or the independent
set I. Do you think that this is a coincidence?
2b
In the above graph, clearly explain why the maximum size of an independent set is 5. In other
words, carefully explain why there does not exist an independent set with 6 or more vertices.
Be G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is
the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is
the greatest distance between two of its vertices.
a) Show that if A(G) 24 then A(G) <2.
b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no
neighbors.
The minimum vertex cover problem is stated as follows: Given an undirected graph
G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X
from V such that every edge (u, v) in E is incident on at least one vertex in X. In
other words you want to find a minimal subset of vertices that together touch all the
edges.
For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the
following graph:
a---b---c---g
d
e
Formulate the minimum vertex cover problem as a Genetic Algorithm or another
form of evolutionary optimization. You may use binary representation, OR any repre-
sentation that you think is more appropriate. you should specify:
• A fitness function. Give 3 examples of individuals and their fitness values if you
are solving the above example.
• A set of mutation and/or crossover and/or repair operators. Intelligent operators
that are suitable for this particular domain will earn more credit.
• A termination criterion for the…
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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