1. Use the graph below to answer the following questions. A 5 B H 10 G 10 25 20 50 5 5 30 15 5 E 15 C 20 F (a) Use Kruskal's Algorithm to produce a minimum spanning tree. Write down the edges that you choose in order of the algorithm. D

1. Use the graph below to answer the following questions. A 5 B H 10 G 10 25 20 50 5 5 30 15 5 E 15 C 20 F (a) Use Kruskal's Algorithm to produce a minimum spanning tree. Write down the edges that you choose in order of the algorithm. D

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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Concept explainers

Minimum Spanning Tree

A spanning tree is a sub-graph of a connected, undirected graph. It has all the vertices of a graph with the minimum possible number of edges. If a vertex is skipped, then it will not be a spanning tree. The edges of a spanning tree may or may not have weights.

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1) Find the minimum spanning tree of the following graph by applying Kruskal's Algorithm of the graph. (First create the graph and then draw the lines on the graph) 2) Find the shortest path from node/vertex 1 to other nodes by using Dijkstra Algorithm. Upload your answers as a Pictures by showing your steps for Kruskal and Dijkstra Algorithms. 0 1 2 3 4 5 6 7 8 0 0 4 0 0 7 0 0 0 0 1 4 0 5 0 8 0 0 0 0 2 0 5 0 11 6 0 0 0 0 Adjacency Matrix 3 0 0 11 0 9 10 0 0 0 4 7 8 6 9 0 0 0 0 0 5 0 0 0 10 0 0 7 0 13 6 0 0 0 0 0 7 0 5 0 7 0 0 0 0 0 0 5 0 4 8 0 0 0 0 0 13 0 4 0
4. a. Use Prim's algorithm starting at node A to compute the Minimum Spanning Tree (MST) of the following graph. In particular, write down the edges of the MST in the order in which Prim's algorithm adds them to the MST. Use the format (nodel; node2) to denote an edge. E 20 11 13 15 14 10 F 12 B b. For the same graph as above, write down the edges of the MST in the order in which Kruskal's algorithm adds them to the MST.
7. Consider the following graph: A 14 CO D 9 E C 12 10 17 b) What is the weight of the minimum spanning tree? B Apply Kruskal's algorithm to find the minimum spanning tree. Edges are sorted first by length, and in the event of a tie, by name, where the two letters are in alphabetical order. Use makeset (x), find (x), and union (x, y) to determine if there are cycles. a) Circle the edges that are part of the minimum spanning tree. AC, AD, AE, BC, BD, BE, DE c) Draw the tree that results from applying the union-find algorithm for cycle detection. When drawing the tree, put vertices with lower letters in the left subtrees so that all the vertices in a level are sorted alphabetically from left to right
8. For the weighted graph depicted below apply Prim's algorithm to find a minimal spanning tree. Show the steps of the algorithm, draw the resulting spanning tree and determine its total weight. 1 6 11 b d 10 8 7 3 13 a 20 24 11 f
Question 18 Consider the graph below. Use Prim's algorithm to find a minimal spanning tree of the graph rooted in vertex A. Note: enter your answer as a set of edges {E1, E2, ...} and write each edge as a pair of nodes between parentheses separate by a comma and one blank space e.g. (A, B) 6. D 3 4 1 F 4 C B
H F D 3 2 B G E Recall that Dijkstra's Algorithm asks you to label vertices as "current" as you visit them. List all 8 vertices in the order in which they become the "current" vertex. Write your answer as a sequence of vertices like ACDEFGHB. (Note that you can either start with A as the current vertex or start with B as the current vertex and execute the algorithm---this problem accepts either solution.) What is the shortest path from A to B? Write your answer as a sequence of vertices like ACDGEB.
Question 6. (a) Construct a minimum spanning tree of the graph in Figure 1 using Kruskal algorithm. Whenever you have a choice between several edges, use the alphabetical order. List the edges in the order in which they are introduced in the tree. (Note: I have written the first edge, you need to add the other ones.) (a, d), (b) Construct a minimum spanning tree of the graph in Figure 1 using Prim's algorithm starting at node a. Whenever you have a choice between several nodes, use the alphabetical order. List the edges in the order in which they are introduced in the tree. (Note: I have written the first edge, you need to add the other ones.) (a, d),
4. Please apply Kruskal's spanning tree algorithm in the graph below and find the minimum spanning tree (MST). Edge weights are in the adjacency matrix table. (you can list the edges in MST or draw the tree below) d b a e h f The adjacency matrix for the undirected weighted graph is mentioned below: a d e g h a 4 8 4 8 11 8. 7 4 2 d 7 14 e 9. 10 f 4 14 10 2 1 6. 8. 11 1 7 7
50. Use the Breadth-First Spanning Tree Search Algorithm to find a spanning tree of K,. Start from vertex h. Remember, we start with vertex h being labeled 0. b K6 a. Vertices that should be labeled as step 1: b. Vertices that should be labeled as step 2: c. Vertices that should be labeled as step 3: d. Edges that need to be removed to get a spanning tree:
Draw the minimum spanning tree for the following graph. Use induction to prove your result is a minimum spanning tree.
4. Consider the following graph. Apply Kruskal’s algorithm to find the minimum spanning tree and write the sequence of edges added to the tree. E B D A 15 5 I 2 4 3 6 10 5 H 12 TI F
13 D 20 Show the execution of Dijkstra's shortest path algorithm, starting from vertex A. Draw the graph. After each relaxation, you will cross the previous best length and assign a new best length. Which list below shows the list of "best lengths" of the node C? O a. 6,5 O b. 0,6,5 O c. 00,6 O d. 0,5