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 24.5, Problem 2E
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To explanation the BFS moreover proof of every shortest path and shortest path tress in Graph is unique.
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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…
Let G = (V, E) be an undirected graph with vertices V and edges E. Let w(e) denote the weight of e E E. Let T C E be a
spanning tree of G.
Select all of the following that imply that T is not a minimum spanning tree (MST) for G. Incorrect choices will be penalized.
There exists e'
(u, v) g T, u, v E V such that w(e') w(e').
O There exists e' g T such that w(e') w(e) for all e E E.
O There exists e'
(u, v) É T, u, v E V such that w(e') < w(e) for all e on the shortest path from u to v in T.
O There exists e E T, e' ¢ T with w(e) < w(e').
4. Let G (V, E) be a directed graph. Suppose we have performed a DFS traversal of
G, and for each vertex v, we know its pre and post numbers. Show the following:
(a) If for a pair of vertices u, v € V, pre(u) < pre(v) < post(v) < post(u), then there
is a directed path from u to v in G.
(b) If for a pair of vertices u, v € V, pre(u) < post(u) < pre(v) < post(v), then there
is no directed path from u to v in G.
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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- Consider a connected undirected graph G=(V,E) in which every edge e∈E has a distinct and nonnegative cost. Let T be an MST and P a shortest path from some vertex s to some other vertex t. Now suppose the cost of every edge e of G is increased by 1 and becomes ce+1. Call this new graph G′. Which of the following is true about G′ ? a) T must be an MST and P must be a shortest s - t path. b) T must be an MST but P may not be a shortest s - t path. c) T may not be an MST but P must be a shortest s - t path. d) T may not be an MST and P may not be a shortest s−t path. Pls use Kruskal's algorithm to reason about the MST.arrow_forwardWe are given a graph G = (V, E); G could be a directed graph or undirected graph. Let M bethe adjacency matrix of G. Let n be the number of vertices so that the matrix M is n ×n matrix. For anymatrix A, let us denote the element of i-th row and j-th column of the matrix A by A[i, j].1. Consider the square of the adjacency matrix M . For all i and j, show that M 2[i, j] is the number ofdifferent paths of length 2 from the i-th vertex to the j-th vertex. It should be explained or proved asclearly as possible.2. For any positive integer k, show that M k[i, j] is the number of different paths of length k from the i-th vertex to the j-th vertex. You may use induction on k to prove it.3. Assume that we are given a positive integer k. Design an algorithm to find the number of different paths of length k from the i-th vertex to j-th vertex for all pairs of (i, j). The time complexity of your algorithm should be O(n3 log k). You can get partial credits if you design an algorithm of O(n3k).arrow_forwardcomp sciencearrow_forward
- Let w be the minimum weight among all edge weights in an un-directed connected graph. Let e be a specific edge of weight w. Which of the following statements is/are TRUE/FALSE? Please discuss each of these statements on why that statement is True/False. a. Every minimum spanning tree has an edge of weight wb. If e is not in a minimum spanning tree T, then in the cycle formed by adding e to T, all edges have the same weightc. There is a minimum spanning tree containing e.d. e is present in every minimum spanning treearrow_forwardLet G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.arrow_forwardWe have a connected graph G=(V,E), and a specific vertex u∈V. Suppose we compute a depth-first search tree rooted at u, and obtain a tree T that includes all nodes of G. Suppose we then compute a breadth-first search tree rooted at u, and obtain the same tree T. Prove that G=T. (In other words, if T is both a depth-first search tree and a breadth-first search tree rooted at u, then G cannot contain any edges that do not belong to T.)arrow_forward
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