**Problem 8**An undirected weighted graph \( G \) is given below:### Graph Description:The graph consists of 6 vertices labeled a through f, and 9 edges. - **Vertices:** - 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 \( f \), but above vertex \( d \). - 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 \( c \).- **Edges and Weights:** - \( d \) and \( f \), weight 1 - \( d \) and \( e \), weight 4 - \( f \) and \( e \), weight 2 - \( e \) and \( a \), weight 2 - \( f \) and \( a \), weight 3 - \( e \) and \( c \), weight 5 - \( c \) and \( a \), weight 7 - \( c \) and \( b \), weight 5 - \( a \) and \( b \), weight 6### Task:(a) **Prim's Algorithm:** - Compute the minimum spanning tree for the graph using Prim’s algorithm starting at vertex \( a \). - Display the sequence of edge additions.(b) **Spanning Tree with Constraint:** - Determine the minimum weight spanning tree with the constraint that edge \(\{d, e\}\) is included.(c) **Generalization:** - Discuss how to adapt Prim’s algorithm to include a specific edge \(\{u, v\}\) in the spanning tree. This problem involves concepts of graph theory and provides practice with Prim's algorithm and constraint satisfaction in minimum spanning trees.

a. Given: Use the Prim's algorithm to compute the minimum spaning tree for the weighted graph. Start…

PROBLEM 8 An undirected weighted graph G is given below: 7 Figure 16: An undirecled weighted graph has 6 verlices, a lhrough f, and 9 edges. Verler d is on the left. Verler ſ is above and lo the right of verler d. Verlez e is below and lo the right of verler f, bul above verler d. Verler e is below and lo the right of verler e. Verler a is above verlez e and lo the right of verler c. Verlez b is below and to the right of verler a, bul above verler c. The edges belween the vertices and their weight are as follows: d and f, 1; d and e, 4; f and e, 2; e and a, 2; ƒ and a, 3; e and c, 5; c and a, 7; c and b, 5; and a and b, 6. (a) 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. (b) 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? (c) 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? 2.

PROBLEM 8 An undirected weighted graph G is given below: 7 Figure 16: An undirecled weighted graph has 6 verlices, a lhrough f, and 9 edges. Verler d is on the left. Verler ſ is above and lo the right of verler d. Verlez e is below and lo the right of verler f, bul above verler d. Verler e is below and lo the right of verler e. Verler a is above verlez e and lo the right of verler c. Verlez b is below and to the right of verler a, bul above verler c. The edges belween the vertices and their weight are as follows: d and f, 1; d and e, 4; f and e, 2; e and a, 2; ƒ and a, 3; e and c, 5; c and a, 7; c and b, 5; and a and b, 6. (a) 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. (b) 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? (c) 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? 2.

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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Question

**Problem 8**

An undirected weighted graph \( G \) is given below:

### Graph Description:
The graph consists of 6 vertices labeled a through f, and 9 edges.

- **Vertices:**
- 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 \( f \), but above vertex \( d \).
- 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 \( c \).

- **Edges and Weights:**
- \( d \) and \( f \), weight 1
- \( d \) and \( e \), weight 4
- \( f \) and \( e \), weight 2
- \( e \) and \( a \), weight 2
- \( f \) and \( a \), weight 3
- \( e \) and \( c \), weight 5
- \( c \) and \( a \), weight 7
- \( c \) and \( b \), weight 5
- \( a \) and \( b \), weight 6

### Task:
(a) **Prim's Algorithm:**
- Compute the minimum spanning tree for the graph using Prim’s algorithm starting at vertex \( a \).
- Display the sequence of edge additions.

(b) **Spanning Tree with Constraint:**
- Determine the minimum weight spanning tree with the constraint that edge \(\{d, e\}\) is included.

(c) **Generalization:**
- Discuss how to adapt Prim’s algorithm to include a specific edge \(\{u, v\}\) in the spanning tree.

This problem involves concepts of graph theory and provides practice with Prim's algorithm and constraint satisfaction in minimum spanning trees.

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