CS302 Final Exam - Algorithms Design

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Questions and Answers

What is the primary purpose of LU decomposition in matrix algebra?

To solve linear equations efficiently (correct)

To find eigenvalues of a matrix

To transform a matrix into its diagonal form

To compute the determinant of a matrix

Which property must a matrix possess to have an LU decomposition?

It must be symmetric

It must have a determinant of 1

It must be diagonal

It must be a square matrix (correct)

How are two strings SI and SJ considered to be rotations of each other?

They must have the same length (correct)

They must contain entirely different characters

They must both start with the same character

One string must be a substring of the other

What is the time complexity requirement for the algorithm that checks if one string is a rotation of another?

O(n) Signup and view all the answers

What initial condition is set for the weights of the edges in a graph when using the Floyd-Warshall algorithm for transitive closure?

Assign all edges a weight of 1 Signup and view all the answers

What condition must be satisfied to confirm the presence of transitive closure using the Floyd-Warshall algorithm?

All values in matrix D must be less than n Signup and view all the answers

In the context of algorithms, what does the term 'transitive closure' refer to?

The ability to reach one vertex from another through intermediate vertices Signup and view all the answers

Which of the following is a characteristic of a lower triangular matrix?

All elements above the main diagonal are zero Signup and view all the answers

Which algorithm uses a greedy approach to find the Minimum Spanning Tree (MST)?

Kruskal's algorithm Signup and view all the answers

What type of algorithm is the 0/1 Knapsack when solved using Brute Force?

Brute Force Signup and view all the answers

Which statement accurately describes P, NP, and NP-Complete problems?

P problems are solvable in polynomial time, NP problems are verifiable in polynomial time, and NP-Complete problems are the hardest in NP. Signup and view all the answers

Which algorithm provides a guaranteed optimal solution for the shortest path in a weighted graph?

Dijkstra's algorithm Signup and view all the answers

In the context of set cover, what is the primary strategy used by the GREEDY-SET-COVER algorithm?

Selects the set that covers the most elements. Signup and view all the answers

Which of the following algorithms is categorized under Dynamic Programming?

Matrix Chain Multiplication using Dynamic Programming Signup and view all the answers

Which of these statements about greedy algorithms is correct?

Greedy algorithms make decisions based on immediate benefits. Signup and view all the answers

Which algorithm can potentially produce different results based on the order of elements in the input?

Kruskal's algorithm Signup and view all the answers

Study Notes

CS302 Final Exam - 18th December 2017

Course: Design and Analysis of Algorithms
Instructor: Dr. Muhammad Atif Tahir, Subhash Sagar, and Zeshan Khan
Exam Duration: 180 minutes
Total Marks: 50

Question 1: LU Decomposition

Part a: Given a matrix equation AX = C, showing LU Decomposition, implies separate lower (L) and upper (U) triangular matrix equations lead to a solution.
Part b: Determine if a given matrix allows LU Decomposition.

Question 2: String Rotation

Problem: Determine if one string is a rotation of another.
Analysis: Design an algorithm for this.
Time Complexity: Should be O(n) or less (where n is the string length).

Question 3: Transitive Closure of a Graph

Method: Use Floyd-Warshall algorithm to find transitive closure.
Steps/Procedure:
- Assign a weight of 1 to all edges.
- Run the Floyd-Warshall algorithm.
- Check if all values in the resulting matrix are less than the number of nodes (vertices) in the graph.
Decision: If the conditions met, the graph has a transitive closure.

Question 4: Algorithm Analysis and Classification

Task: Complete a table classifying algorithms as Dynamic Programming, Greedy, or neither.
Algorithms Included: 0/1 Knapsack (DP and Brute Force), Breadth-First Search, Depth-First Search, Kruskal's/Prim's algorithms, Dijkstra's algorithm, Bellman-Ford algorithm, Matrix Chain Multiplication (DP and Brute Force).

Question 5: Greedy vs Dynamic Programming

Part a: Explain the essential difference between greedy algorithms and dynamic programming.
Part b: Define P, NP, NP-Complete problems, and "P = NP" concept.

Question 6: Set Cover Algorithm

Problem: Apply the Greedy Set Cover algorithm to a specific set of words.
Input: A set of words.
Output: The set cover produced by the greedy algorithm.

Question 7: Function Analysis for Algorithm Efficiency

Problem: To analyze the correctness of statements about functions based on Big-O notations.
Content: T or F questions, to determine the validity of function statements related to analyzing function efficiency and classifications based on functions like O(f(n)), Ω(f(n)), Θ(f(n)).

Question 8: Minimum Spanning Tree (MST) Weight

Problem: Discuss the weight of a Minimum Spanning Tree (MST).
Conditions: In case of a graph where one edge has a negative weight.

Question 9: Matrix Chain Multiplication

Problem: Find the optimal parenthesization of a set of matrices using Matrix Chain Multiplication algorithm.
Input: Dimensions of matrices A1, A2, A3, A4, A5.
Output: The optimal order of multiplication, likely presented in a matrix format, and explained using the Matrix Chain Multiplication algorithm.

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Description

Test your understanding of algorithms in this CS302 final exam. Questions cover LU Decomposition, string rotation, and transitive closure using the Floyd-Warshall algorithm. Each question challenges your ability to apply key concepts and analyze algorithm efficiency.