Problem 5 R w. Let G be a connected weighted undirected graph with n nodes and m edges. The weight of edge e is represented as w. We define a minimum-maximum spanning tree to be a spanning tree T that minimizes the quantity Y = max w. A minimum-maximum spanning tree has the smallest maximum edge weight of all possible spanning tree. To show a statement is correct, you need to provide a proof. To show a statement is wrong, you only need to provide a counter-example. (a) maximum spanning tree for G. Prove that a minimum spanning tree of graph G is always a minimum-

Problem 5 R w. Let G be a connected weighted undirected graph with n nodes and m edges. The weight of edge e is represented as w. We define a minimum-maximum spanning tree to be a spanning tree T that minimizes the quantity Y = max w. A minimum-maximum spanning tree has the smallest maximum edge weight of all possible spanning tree. To show a statement is correct, you need to provide a proof. To show a statement is wrong, you only need to provide a counter-example. (a) maximum spanning tree for G. Prove that a minimum spanning tree of graph G is always a minimum-

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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b) Consider graph G=(V, E) with six nodes and eight edges. The edges have distinct integer weights. a 17 b 10 7 2 f 6 The minimum spanning tree of G is constructed by Kruskal's algorithm and is shown in tree MST below: a 17 101 10 7 C f d 6 The edge weights of only those edges that are in the MST are given in graph G. What is the minimum possible summation of edge weights for the edges (a, b), (e, d), (c, d)? (5 points) As you justify your answer for edge weights, please use graph G and apply Kruskal's algorithm to obtain the MST step by step (5 points). Show and explain all your work.
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 d
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. None I Only I and II Only I and III Only O1, II, and III
Given a directed graph w/ nonnegative edge lengths & two distinct nodes, a and z. Let X denote a shortest path from a to z. If we add 10 to the length of every edge in the graph, then: i) X definitely remains a shortest a-z path. ii) X definitely does not remain a shortest a-z path. iii) X might or might not remain a shortest a-z path (depending on the graph). iv) If X has only one edge, then X definitely remains a shortest a-z path. O i and iv only O iv only O None of the choices O i only O i and iv only O iii only O ii only
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3. 4. Given a directed acyclic graph G = (V, E) and two vertices s, te V, design an efficient algorithm that computes the number of different directed paths from s to t. Define the incidence matrix B of a directed graph with no self-loop to be an nxm matrix with rows indexed by vertices, column indexed by edges such that Bij = -1 1 0 if edge j leaves vertex i, if edge jenters vertex i, otherwise. Let BT be the transpose of matrix B. Find out what the entries of the n x n matrix BBT stand for.
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Let G be a graph, where each edge has a weight. A spanning tree is a set of edges that connects all the vertices together, so that there exists a path between any pair of vertices in the graph. A minimum-weight spanning tree is a spanning tree whose sum of edge weights is as small as possible. Last week we saw how Kruskal's Algorithm can be applied to any graph to generate a minimum-weight spanning tree. In this question, you will apply Prim's Algorithm on the graph below. You must start with vertex A. H 4 4 1 3 J 2 C 10 4 8 B 9 F 18 8 There are nine edges in the spanning tree produced by Prim's Algorithm, including AB, BC, and IJ. Determine the exact order in which these nine edges are added to form the minimum-weight spanning tree. 3.
Consider a directed graph G=(V,E) with n vertices, m edges, a starting vertex s∈V, real-valued edge lengths, and no negative cycles. Suppose you know that every shortest path in G from s to another vertex has at most k edges. How quickly can you solve the single-source shortest path problem? (Choose the strongest statement that is guaranteed to be true.) a) O(m+n) b) O(kn) c) O( km) d) O(mn)
Given the following graph, A C 3 1 E 4 В F 7 D 5 What is the next edge and vertex(s) to be included in the solution if a. we are using Prim's algorithm to find a minimum spanning tree and we have started with vertex E? Edge: Vertex(s): What is the next edge and vertex(s) to be included in the solution if b. we are using Kruskal's algorithm to find a minimum spanning tree Edge: Vertex(s): What is the next edge and vertex(s) to be included in the solution if c. we are using Dijkstra's shortest path algorithm to find shortest paths and we have started with vertex E, with C and F already included in the shortest path tree? Edge: Vertex(s):
Consider the problem of finding the length of a "longest" path in a weighted, not necessarily connected, dag. We assume that all weights are positive, and that a "longest" path is a path whose edge weights add up to the maximal possible value. For example, for the following graph, the longest path is of length 15: 9. 9. (h) 2 3 Use a dynamic programming approach to the problem of finding longest path in a weighted dag.
Let A, B, C, D be the vertices of a square with side length 100. If we want to create a minimum-weight spanning tree to connect these four vertices, clearly this spanning tree would have total weight 300 (e.g. we can connect AB, BC, and CD). But what if we are able to add extra vertices inside the square, and use these additional vertices in constructing our spanning tree? Would the minimum-weight spanning tree have total weight less than 300? And if so, where should these additional vertices be placed to minimize the total weight? Let G be a graph with the vertices A, B, C, D, and possibly one or more additional vertices that can be placed anywhere you want on the (two-dimensional) plane containing the four vertices of the square. Determine the smallest total weight for the minimum-weight spanning tree of G. Round your answer to the nearest integer. Attention: Please don't just copy these two following answers, which are not correct at all. Thank you.…