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IGNOU MCS-211 (January 2025 – July 2025) Assignment Questions
Q1: a) Design and develop an efficient algorithm to find the list of prime numbers in the range 501 to 2000. What is the complexity of this algorithm?
b) Differentiate between Cubic-time and Factorial-time algorithms. Give example of one algorithm each for these two running times.
c) Write an algorithm to multiply two square matrices of order n*n. Also explain the time complexity of this algorithm.
d) What are asymptotic bounds for analysis of efficiency of algorithms? Why are asymptotic bounds used? What are their shortcomings? Explain the Big O and Big notation with the help of a diagram. Find the Big O-notation and Θnotation for the function:
𝑓(𝑛)= 100𝑛4 +1000𝑛3 +100000
e) Write and explain the Left to Right binary exponentiation algorithm. Demonstrate the use of this algorithm to compute the value of 329 (Show the steps of computation). Explain the worst-case complexity of this algorithm.
f) Write and explain the Bubble sort algorithm. Discuss its best and worst-case time complexity.
g) What are the uses of recurrence relations? Solve the following recurrence relations using the Master’s method
Q2: a) What is an Optimisation Problem? Explain with the help of an example. When would you use a Greedy Approach to solve optimisation problem? Formulate the Task Scheduling Problem as an optimisation problem and write a greedy algorithm to solve this problem. Also, solve the following fractional Knapsack problem using greedy approach. Show all the steps.
Suppose there is a knapsack of capacity 20 Kg and the following 6 items are to packed in it. The weight and profit of the items are as under:
(p1, p2,…, p6) = (30,16,18,20,10, 7)
(w1, w2,…, w6) = ( 5, 4, 6, 4, 5, 7)
Select a subset of the items that maximises the profit while keeping the total weight below or equal to the given capacity.
b) Assuming that data to be transmitted consists of only characters ‘a’ to ‘g’, design the Huffman code for the following frequencies of character data. Show all the steps of building a huffman tree. Also, show how a coded sequence using Huffman code can be decoded.
a:5, b:25, c:10, d:15, e:8, f:7, g:30
c) Explain the Merge procedure of the Merge Sort algorithm. Demonstrate the use of recursive Merge sort algorithm for sorting the following data of size 8: [19, 18, 16, 12, 11, 10, 9, 8]. Compute the complexity of Merge Sort algorithm.
d) Explain the divide and conquer approach of multiplying two large integers. Compute the time complexity of this approach. Also, explain the binary search algorithm and find its time complexity.
e) Explain the Topological sorting with the help of an example. Also, explain the algorithm of finding strongly connected components in a directed Graph.
Q3: Consider the following Graph:
a) Write the Prim’s algorithm to find the minimum cost spanning tree of a graph. Also, find the time complexity of Prim’s algorithm. Demonstrate the use of Kruskal’s algorithm and Prim’s algorithm to find the minimum cost spanning tree for the Graph given in Figure 1. Show all the steps.
b) Write the Dijkstra’s shortest path algorithm. Also, find the time complexity of this shortest path algorithm. Find the shortest paths from the vertex ‘A’ using Dijkstra’s shortest path algorithm for the graph given in Figure 1. Show all the steps of computation.
c) Explain the algorithm to find the optimal Binary Search Tree. Demonstrate this algorithm to find the Optimal Binary Search Tree for the following probability data (where pi represents the probability that the search will be for the key node ki, where as qi represents that the search is for dummy node di. Make suitable assumptions, if any)
d) Given the following sequence of chain multiplication of the matrices. Find the optimal way of multiplying these matrices:
Matrix Dimension
A1 10 × 15
A2 15 × 5
A3 5 × 20
A4 20 × 10
e) Explain the Rabin Karp algorithm for string matching with the help of an example. Find the time complexity of this algorithm.
Q4: a) Explain the term Decision problem with the help of an example. Define the following problems and identify if they are decision problem or optimisation problem? Give reasons in support of your answer.
(i) Travelling Salesman Problem
(ii) Graph Colouring Problem
(iii) 0-1 Knapsack Problem
b) What are P and NP class of Problems? Explain each class with the help of at least two examples.
c) Define the NP-Hard and NP-Complete problem. How are they different from each other. Explain the use of polynomial time reduction with the help of an example.
d) Define the following Problems:
(i) SAT Problem
(ii) Clique problem
(iii) Hamiltonian Cycle Problem
(iv) Subset Sum Problem
IGNOU MCS-211 (January 2024 – July 2024) Assignment Questions
Q1: a) Develop an efficient algorithm to find a list of prime numbers from 100 to 1000.
b) Differentiate between Polynomial-time and exponential-time algorithms. Give an example of one problem each for these two running times.
c) Using Horner’s rule, evaluate the polynomial p(x)= 2×5-5×4-3×2+15 at x=2. Analyse the computation time required for polynomial evaluation using Horner’s rule against the Brute force method.
d) State and explain the theorems for computing the bounds O, Ω and Θ. Apply these theorem to find the O-notation, Ω-notation and Θ-notation for the function: 𝑓(𝑛)= 10𝑛3 +18𝑛2 +1
e) Explain binary exponentiation for computing the value 519. Write the right-toleft binary exponentiation algorithm and show its working for the exponentiation 519. Also, find the worst-case complexity of this algorithm.
f) Write and explain the linear search algorithm and discuss its best and worstcase time complexity. Show the working of the linear search algorithm for the data: 12, 11, 3, 9, 15, 20, 18, 19, 13, 8, 2, 1, 16.
g) What is a recurrence relation? Solve the following recurrence relations using the Master’s method
Q2: a) What is a Greedy Approach to Problem-solving? Formulate the fractional Knapsack Problem as an optimisation problem and write a greedy algorithm to solve this problem. Solve the following fractional Knapsack problem using this algorithm. Show all the steps.
Suppose there is a knapsack of capacity 15 Kg and 6 items are to packed in it. The weight and profit of the items are as under:
(p1, p2,…, p6) = (3, 2, 4, 5, 1, 6)
(w1, w2,…, w6) = (2, 1, 2, 1, 5, 1)
b) What is the purpose of using Huffman Codes? Explain the steps of building a huffman tree. Design the Huffman codes for the following set of characters and their frequencies: a:15, e:19, s:5, d:6, f:4, g:7, h:8, t:10.
c) Expalin the Partition procedure of the Quick Sort algorithm. Use this procedure and quick sort algorithm to sort the following array of size 8: [12, 9, 17, 15, 23, 19, 16, 24]. Compute the worst case and best case complexity of Quick sort algorithm.
d) Explain the divide and conquer apprach of multiplying two matrices of large size. Also, explain the Strassen’s matric multiplication algorithm. Find the time complexity of both these approaches.
e) What is the use of Topological sorting? Write and explain the Topological sorting algorithm. Also, compute the time complexity for the topological sorting algorithm.
Q3: Consider the following Graph:
a) Write the Kruskal’s algorithm and Prim’s algorithm to find the minimum cost spanning tree of the graph given in Figure 1. Show all the steps of computation. Also, compute the time complexity of both the algorithms.
b) In the Figure 1, find the shortest path from the vertex ‘a’ using Dijkstra’s shortest path algorithm. Show all the steps of computation. Also, find the time complexity of the algorithm.
c) What is dynamic programming? What is the principle of Optimality? Use the dynamic programming approach to find the optimal sequence of chain multiplication of the following matrices:
d) Make all the possible Binary Search Trees for the key values 25, 50, 75.
e) Explain the Knuth Morris Pratt algorithm for string matching. Use this algorithm to find a pattern “algo” in the Text “From algae to algorithms”. Show all the steps. What is the time complexity of this algorithm.
Q4: a) What are decision problems and Optimisation problems? Differentiate the decision problems and Optimisation problems with the help of at least two problem statements of each.
b) Define P and NP class of Problems with the help of examples. How are P class of problem different from NP class of Problems.
c) What are NP-Hard and NP-Complete problem? What is the role of reduction? Explain with the help of an example.
d) Define the following Problems:
(i) 3-CNF SAT
(ii) Clique problem
(iii) Vertex cover problem
(iv) Graph Colouring Problem
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