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CS 325 - Homework Assignment 5

1. (3 points) Demonstrate Prim’s algorithm on the graph below by showing the steps in subsequent

graphs as shown in Figures 23.5 on page 635 of the text. What is the weight of the minimum spanning

tree? Start at vertex a.

2. (6 points) Consider an undirected graph G=(V,E) with nonnegative edge weights w(u,v)?0. Suppose

that you have computed a minimum spanning tree G, and that you have also computed shortest paths

to all vertices from vertex s?V. Now suppose each edge weight is increased by 1: the new weights

w’(u,v) = w(u,v) + 1.

(a) Does the minimum spanning tree change? Give an example it changes or prove it cannot change.

(b) Do the shortest paths change? Give an example where they change or prove they cannot change.

3. (4 points)In the bottleneck-path problem, you are given a graph G with edge weights, two vertices s

and t and a particular weight W; your goal is to find a path from s to t in which every edge has at least

weight W.

(a) Describe an efficient algorithm to solve this problem.

(b) What is the running time of youor algorithm.

CS 325 - Homework Assignment 5

4. (5 points) Below is a list of courses and prerequisites for a factious CS degree.

Course Prerequisite

CS 150 None

CS 151 CS 150

CS 221 CS 151

CS 222 CS 221

CS 325 CS 221

CS 351 CS 151

CS 370 CS 151

CS 375 CS 151, CS 222

CS 401 CS 375, CS 351, CS 325, CS 222

CS 425 CS 325, CS 222

MATH 200 None

MATH 201 MATH 200

(a) Draw a directed acyclic graph (DAG) that represents the precedence among the courses.

(b) Give a topological sort of the graph.

(c) If you are allowed to take multiple courses at one time as long as there is no prerequisite conflict,

find an order in which all the classes can be taken in the fewest number of terms.

(d) Determine the length of the longest path in the DAG. How did you find it? What does this represent?

CS 325 - Homework Assignment 5

5. (12 points) Suppose there are two types of professional wrestlers: “Babyfaces” (“good guys”) and

“Heels” (“bad guys”). Between any pair of professional wrestlers, there may or may not be a rivalry.

Suppose we have n wrestlers and we have a list of r pairs of rivalries.

(a) Give pseudocode for an efficient algorithm that determines whether it is possible to designate some

of the wrestlers as Babyfaces and the remainder as Heels such that each rivalry is between a Babyface

and a Heel. If it is possible to perform such a designation, your algorithm should produce it.

(b) What is the running time of your algorithm?

(c) Implement: Babyfaces vs Heels.

Input: Input is read in from a file specified in the command line at run time. The file contains the

number of wrestlers, n, followed by their names, the number of rivalries r and rivalries listed in pairs.

Note: The file only contains one list of rivalries

Output: Results are outputted to the terminal.

? Yes, if possible followed by a list of the Babyface wrestlers and a list of the Heels .

? No, if impossible.

Sample Input file:

Heels: Biggs Duke

Submit a copy of your files including a README file that explains how to compile and run your code in a

ZIP file to TEACH.