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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ou are given a directed graph G = (V, E) and two vertices s and t. Moreover, each edge of this graph is colored either blue or red. Your goal is to find whether there is at least one path from s to t such that all red edges in this path appear after all blue edges (the path may not contain any blue edges or any red edges, but if it has both types of edges, all red edges should appear after all blue edges). Design and analyze an algorithm for solving this problem in O(n + m) time.
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Problem 3: Complete Graph [a] Prove by mathematical induction that the complete graph Kn has n(n-1)/2 edges. [b] Is K4 planar? Explain why or why not. [c] Is K5 bi-partite? Explain why or why not. [d] Is K6 Eulerian? Explain why or why not.
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