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 22.4, Problem 4E
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To show that if a graph holds a cycle then the topological sorting minimizes the no. of bad edges by creating the ordering of vertex.
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Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.
Part 2: Random GraphsA tournament T is a complete graph whose edges are all oriented. Given a completegraph on n vertices Kn, we can generate a random tournament by orienting each edgewith probability 12 in each direction.Recall that a Hamiltonian path is a path that visits every vertex exactly once. AHamiltonian path in a directed graph is a path that follows the orientations of thedirected edges (arcs) and visits every vertex exactly once. Some directed graphs havemany Hamiltonian paths.In this part, we give a probabilistic proof of the following theorem:Theorem 1. There is a tournament on n vertices with at least n!2n−1 Hamiltonian paths.For the set up, we will consider a complete graph Kn on n vertices and randomlyorient the edges as described above. A permutation i1i2 ...in of 1,2,...,n representsthe path i1 −i2 −···−in in Kn. We can make the path oriented by flipping a coin andorienting each edge left or right: i1 ←i2 →i3 ←···→in.(a) How many permutations of the vertices…
Consider an undirected
graph G with 100 nodes.
The maximum number of edges to be included
in G so that the graph is not connected is
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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- (a) Prove that for all n > 6, if a connected n-vertex graph has n+ 2 edges or fewer, then it is planar. (b) Prove that for all n> 6, there is a connected n-vertex graph with n +3 edges which is not planar.arrow_forwardBe 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.arrow_forwardDENSE-SUBGRAPH: Given a graph G and two integers m and n, does G have a set of m vertices with at least n edges between them? Prove that DENSE-SUBGRAPH is NP-complete.arrow_forward
- A Vertex Cover of an undirected graph G is a subset of the nodes of G,such that every edge of G touches one of the selected nodes.The VERTEX-COVER problem is to decide if a graph G has a vertex cover of size k.VERTEX-COVER = { <G,k> | G is an undirected graph with a k-node vertex cover }The VC3 problem is a special case of the VERTEX-COVER problem where the value of k is fixed at 3.VERTEX-COVER 3 = { <G> | G is an undirected graph with a 3-node vertex cover }Use parts a-b below to show that Vertex-Cover 3 is in the class P.a. Give a high-level description of a decider for VC3.A high-level description describes an algorithmwithout giving details about how the machine manages its tape or head.b. Show that the decider in part a runs in deterministic polynomial time.arrow_forwardRequired information NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the bipartite graph Km.n- Find the values of mand n if Km n has an Euler path. (Check all that apply.) Check All That Apply Km,n has an Euler path when both mand n are even. Km,n has an Euler path when both mand n are odd. Km, n has an Euler path if m=2 and n is odd. Km, n has an Euler path if n= 2 and m is odd. Km, n has an Euler path when m= n=1.arrow_forward4-Clique Problem The clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: https://en.wikipedia.org/wiki/Clique_problem You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a…arrow_forward
- Create a weighted connected graph with the following characteristics: The assigned number of vertices are alphabetically labeled starting with A. Vertex Ais not adjacent to vertex G. At leasttwovertices have a degree greater than 2. Allweights are greater than 0. • No weights are the same.arrow_forward8. When running the depth first search on a connected graph, which of the following are possible for the ending time of the starting vertex: a) It may have the smallest ending time. b) It will have the largest ending time. c) neither a or b d) both a and barrow_forwardIndependent set: Given a graph, find a largest subset of the nodesfor which there are no edges between any pair in the set. Give the bird-and-friendabstraction of a recursive backtracking algorithm for this problem. What do you askthe bird, and what do give your friend?arrow_forward
- Draw a connected graph with 7 nodes and 10 edges which has no cutnode nor bridge. Find by the use of reduction algorithm the number of spanning trees in this graph.arrow_forwardFind the minimal Spanning Tree for the following Graph:arrow_forwardGiven an undirected graph G = <V,E>, a vertex cover is a subset of vertices S V such that for each edge (u,v) belongs to E, either u S or v S or both. The Vertex Cover Problem is to find minimum size of the set S. Consider the following algorithm to Vertex Cover Problem: (1) Initialize the result as {} (2) Consider a set of all edges in given graph. Let the set be E’. (3) Do following while E’ is not empty ...a) Pick an arbitrary edge (u,v) from set E’ and add u and v to result ...b) Remove all edges from E which are either incident on u or v. (4) Return result. It claim that this algorithm is exact for undirected connected graphs. Is this claim True or False? Justify the answer.arrow_forward
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