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An undirected weighted graph G is given below: 12 2 6 Figure 16: An undirected weighted graph has 6 vertices, a through , and 9 edges. Verter d is on the left. Verter f is above and to the right of verter d. Verter e is below and to the right of verter f, but above vertez d. Verter e is belou and to the right of vertez e. Vertez a is above verter e and to the right of verter c. Verter b is below and to the right of vertez a, but above vertez e. The edges between the vertices and their weight are as follows: d and f. 1; d and e, 4: f and e, 2; e and a, 2: f and a, 3; e and c, 5; e and a, 7; e and b, 5; and a and b, 6. Use Prim's algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex a. Show the order in which the edges are added to the tree. What is the minimum weight spanning tree for the weighted graph in the previous question subject to the condition that edge {d, e} is in the span- ning tree? How would you generalize this idea? Suppose you are given a graph G and a particular edge {u, v} in the graph. How would you alter Prim's algorithm to find the minimum spanning tree subject to the condition that {u, v} is in the tree?

An undirected weighted graph G is given below: 12 2 6 Figure 16: An undirected weighted graph has 6 vertices, a through , and 9 edges. Verter d is on the left. Verter f is above and to the right of verter d. Verter e is below and to the right of verter f, but above vertez d. Verter e is belou and to the right of vertez e. Vertez a is above verter e and to the right of verter c. Verter b is below and to the right of vertez a, but above vertez e. The edges between the vertices and their weight are as follows: d and f. 1; d and e, 4: f and e, 2; e and a, 2: f and a, 3; e and c, 5; e and a, 7; e and b, 5; and a and b, 6. Use Prim's algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex a. Show the order in which the edges are added to the tree. What is the minimum weight spanning tree for the weighted graph in the previous question subject to the condition that edge {d, e} is in the span- ning tree? How would you generalize this idea? Suppose you are given a graph G and a particular edge {u, v} in the graph. How would you alter Prim's algorithm to find the minimum spanning tree subject to the condition that {u, v} is in the tree?

Database System Concepts
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
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### Figure 16 Description: An undirected weighted graph \( G \) is provided with 6 vertices labeled \( a \) through \( f \), and 9 edges. Detailed coordinates are as follows: - Vertex \( d \) is on the left. - Vertex \( f \) is above and to the right of vertex \( d \). - Vertex \( e \) is below and to the right of vertex \( f \), above vertex \( c \). - Vertex \( c \) is below and to the right of vertex \( e \). - Vertex \( a \) is above vertex \( e \) and to the right of vertex \( c \). - Vertex \( b \) is below and to the right of vertex \( a \), but above vertex \( c \). The edges and weights are: - \( a, b: 1 \) - \( f, d: 1 \) - \( d, e: 2 \) - \( e, a: 2 \) - \( f, a: 3 \) - \( e, c: 5 \) - \( c, f: 5 \) - \( b, c: 5 \) - \( d, b: 6 \) ### Instructions: 1. **Use Prim’s Algorithm:** - Compute the minimum spanning tree for the weighted graph starting with vertex \( a \). - Show the sequence in which edges are added to the tree. 2. **Minimum Weight Spanning Tree:** - Determine the minimum weight spanning tree given that edge \(\{d, e\}\) is included in the tree. 3. **Generalization:** - Consider a graph \( G \) with an edge \(\{u, v\}\). - Outline how to modify Prim's algorithm to find a minimum spanning tree including the edge \(\{u, v\}\). --- This content is designed for educational purposes to help understand graph algorithms, specifically focusing on calculating minimum spanning trees using Prim's algorithm in various scenarios.
Transcribed Image Text:### Figure 16 Description: An undirected weighted graph \( G \) is provided with 6 vertices labeled \( a \) through \( f \), and 9 edges. Detailed coordinates are as follows: - Vertex \( d \) is on the left. - Vertex \( f \) is above and to the right of vertex \( d \). - Vertex \( e \) is below and to the right of vertex \( f \), above vertex \( c \). - Vertex \( c \) is below and to the right of vertex \( e \). - Vertex \( a \) is above vertex \( e \) and to the right of vertex \( c \). - Vertex \( b \) is below and to the right of vertex \( a \), but above vertex \( c \). The edges and weights are: - \( a, b: 1 \) - \( f, d: 1 \) - \( d, e: 2 \) - \( e, a: 2 \) - \( f, a: 3 \) - \( e, c: 5 \) - \( c, f: 5 \) - \( b, c: 5 \) - \( d, b: 6 \) ### Instructions: 1. **Use Prim’s Algorithm:** - Compute the minimum spanning tree for the weighted graph starting with vertex \( a \). - Show the sequence in which edges are added to the tree. 2. **Minimum Weight Spanning Tree:** - Determine the minimum weight spanning tree given that edge \(\{d, e\}\) is included in the tree. 3. **Generalization:** - Consider a graph \( G \) with an edge \(\{u, v\}\). - Outline how to modify Prim's algorithm to find a minimum spanning tree including the edge \(\{u, v\}\). --- This content is designed for educational purposes to help understand graph algorithms, specifically focusing on calculating minimum spanning trees using Prim's algorithm in various scenarios.
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