QUESTION 9 We are given a connected graph G with costs ci, ) on edges. Assume all costs are positive integors and that there are no ties (i e. no two edges have the sanvo cost costs are squared (meaning that edgo () now costs c'i, D (c(i, ))) the Minimum Spanning Tree (MST) remains the same tree. O True O False
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![QUESTION 9
We are given a connected graph G with costs ci, ) on edges. Assume all costs are positive integers and that there are no ties (ie. no two edges have the sanvo cost
costs are squarod (meaning that edge () now costs c'. ) = (c(, ))) the Minimum Spanning Tree (MST) remains the same tree.
O True
O False
QUESTION 10
e are given a connected graph G with costs c(i, ) on edges. Assume all costs are positive integers and that there are no ties (1.e. no two edges have the same ce
sts are squared (meaning that edge (3) now costs c'( 1) (ci, ) the Shortest Path Tree rooted at node A (SPT) remains the same tree.
True
False
QUESTION 11](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffdea7e45-267b-4145-bdd8-519bcfef8cef%2F88eed70f-b441-4788-a394-30efb420cc1f%2F1xzb9e_processed.jpeg&w=3840&q=75)
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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.Given an undirected graph G=(V,E) with positive edge weights, solve the following problem. Let SG(u,v) signify the length of the shortest path between u and v in the graph G for any two nodes u and v. Is the following statement correct or incorrect? SG(u,v)8G(u,w) denotes SG" (u,v)8G" (u,w), where G" denotes the graph produced by doubling each of G's edge weights.Consider a connected graph G with at least 4 edges that has all distinct edge weights. Which of the following properties must be true of a Minimum Spanning Tree (MST) of G? I. The MST must contain the shortest edge of G. II. The MST must contain the second-shortest edge of G. III. The MST can never contain the longest edge of G. O None OI Only OI and II Only I and III Only OI, II, and III
- Question 8. A connected graph G has 4 vertices and 4 edges of costs 1, 2, 3 and respectively 4. (a) Show that G is not a tree. (b) Show that G has a single minimum spanning tree. (Hint: For example, think how Prim's algorithm constructs the minimum spanning tree). (c) Draw a connected graph G with 4 nodes and 4 edges of costs 1, 2, 3, 4, respectively, so that the minimum spanning tree contains the edge of cost 4. The 4 nodes are below, you need to draw the 4 edges and indicate their cost. a O b d3. Given the following example of UAG graphs: 12 12 6. 10 21 5 3 5 3 5 \21 14 15 8 15 8. Graph A Graph B Graph C (degree(v) <= 1) a) Give implementation to find the shortest path b) Does your implementation take all considerations and could accept any kind of inputs? Explain c) Justify your choice by providing the approximating time/space complexity of each type graph. 3.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*
- We have undirected graph K, with two distinct vertices z, k. And let O be a minimum spanning tree of K. Prove that the path from z to k along O is a minmax path. Assume that K has distinct edge weights. (Assume for contradiction that the minmax path is not completely on the minimum spanning tree.) Note: A MinMax path in an undirected Graph K is a path between two vertices z, k that minimizes the maximum weight of the edges on the pat h.h.1. Prove the following known result on minimum spanning trees. Theorem. Let G = (V, E) be a graph and let w : E → R+ be a weight function. Let A be a subset of edges that is contained in some minimum spanning tree, and let (S, S) be a cut so that (S, S)N A = 0. Prove that if e is a light edge in (S, S), then e is safe for A. %3D 2. Let S = fi and a profit p; associated with it. Design a dynamic programing algorithm for computing a set of compatible activities of maximum profit. You should start with defining some terms and then write a recurrence relation for solving the problem. What is the time complexity of your algorithm? {1,2...n} be a set activities, where activity i E S has a start time si, a finish timeQ1a 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. 1b INDEPENDENT SET has the following Decision Problem: given a graph G and an integer k₁, does G have an independent set of size at least k₁? VERTEX COVER has the following Decision Problem: given a graph G and an integer k2, does G have a vertex cover of size at most k2? It is known that INDEPENDENT SET is NP-complete. Prove that INDEPENDENT SETWhich is true/false. Let G be a connected undirected weighted graph. Consider the following two statements. S₁: There exists a minimum weight edge in G which is present in every minimum spanning tree of G. S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree.3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)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').SEE MORE QUESTIONSRecommended textbooks for youDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill EducationDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education