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 2P
a.
Program Plan Intro
To prove the size of the maximum clique in given graph is equal to the size of the maximum clique in G.
b.
Program Plan Intro
To provide another example where bias due to under Coverage is likely to occur.
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True or false: let G be an arbitrary connected, undirected graph with a distinct cost c(e) on every edge e. suppose e* is the cheapest edge in G; that is, c(e*) <c(e) for every edge e is not equal to e*. Any minimum spanning tree T of G contains the edge e*
A path of length two is denoted by P2. If a graph G does not contain P2 as induced subgraph, then:
1- G must be a clique (i.e., a complete graph).
2- Every vertex of G must of degree one.
3- Every connected component of G must be a clique.
4- Every connected component of G must consist of at most two vertices.
Consider the sets of Aā, Aā, ..., Aā. The intersection graph of a collection of sets Aā, Aā, ..., Aā is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Construct the intersection graph for the sets given below.
Aā = {0, 2, 4, 6, 8), Aā = {0, 1, 2, 3, 4), Aā = {1,3,5,7,9), Aā = {5, 6, 7, 8, 9), Aā
= {0, 1,8,9}
Note: Aā {0, 2, 4, 6,8} represents vertex Aā (first vertex). This set defines the possible edge connections of the corresponding vertex.
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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- Let G be a connected graph that has exactly 4 vertices of odd degree: v1,v2,v3 and v4ā. Show that there are paths with no repeated edges from v1ā to v2ā, and from v3ā to v4ā, such that every edge in G is in exactly one of these paths.arrow_forward. Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.arrow_forwardLet G be a connected graph, and let T1, T2 be two spanning trees. Prove thatT1 can be transformed to T2 by a sequence of intermediate trees, each obtainedby deleting an edge from the previous tree and adding another.arrow_forward
- Consider a graph G that has k vertices and k ā2 connected components,for k ā„ 4. What is the maximum possible number of edges in G? Proveyour answer.arrow_forwardA clique in an undirected graph is a subgraph wherein every two nodes are connected by an edge. Consider the language: CLIQUE = {G, k : G = (V, E) is an undirected graph containing a clique of size k} Show that 3SAT ā¤p CLIQUEarrow_forwardLet G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ā V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-setarrow_forward
- Prove 1 For a graph G = (V, E), a forest F is any set of edges of G that doesnot contain any cycles. M = (E, F) where F = {F ā E : F is a forest of G} is amatroid.arrow_forwardLet 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.arrow_forwardFind the shortest path from S to other nodes, on the given directed acyclic graph.Graph: R ā A : 3 S ā A : 1 A ā C : 6 B ā D : 3 C ā E : 2R ā S : 2 S ā B : 2 B ā A : 4 C ā D : 1 D ā E : 1Ā Answer: Topological Ordering: __________________________ Ā Node Edge Relax? Update Ā Ā Ā Ā Ā Shortest Path from S: Length Path R Ā Ā S Ā Ā A Ā Ā B Ā Ā C Ā Ā D Ā Ā Earrow_forward
- Let G be a directed acyclic graph with exactly one source r such that for any other vertex v there exists a unique directed path from r to v. Let Gu be the undirected graph obtained by erasing the direction on each edge of G. Prove that (Gu,r) is a rooted tree.arrow_forwardWhat is the largest and what is the smallest possible cardinality of a matching in a bipartite graph G = <V, U, E> with n vertices in each vertex set V and U and at least n edges?arrow_forwardA 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).arrow_forward
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