MCS-211 Solved Assignment July 2025 | Design and Analysis of Algorithms | IGNOU MCA_NEW

QQ1. This assignment has four questions (80 Marks). Answer all questions. The remaining 20 marks are for viva voce. You may use illustrations and diagrams to enhance the explanations. Please go through the guidelines regarding assignments given in the Programme guide for the presentation format.

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? (137 words)
b) Differentiate between Cubic-time and Factorial-time algorithms. Give example of one algorithm each for these two running times. (137 words)
c) Write an algorithm to multiply two square matrices of order n*n. Also explain the time complexity of this algorithm. (137 words)
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 -notation for the function: f(n)= 100n4 +1000n3 +100000 (137 words)
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. (137 words)
f) Write and explain the Bubble sort algorithm. Discuss its best and worst-case time complexity. (137 words)
g) What are the uses of recurrence relations? Solve the following recurrence relations using the Master's method (137 words)

Key Points:

Sieve of Eratosthenes finds primes up to N in O(N log log N) time.
Cubic-time (O(n³)) is feasible for moderate inputs; Factorial-time (O(n!)) is highly impractical.
Standard matrix multiplication for n x n matrices has O(n³) time complexity using three nested loops.
Asymptotic bounds (Big O, Big Omega) describe algorithm growth for large inputs, ignoring constants.

Answer: This assignment covers fundamental concepts and algorithms in Design and Analysis of Algorithms, focusing on efficiency, complexity analysis, and specific algorithmic techniques. The answers below provide detailed explanations for each sub-question, adhering to the specified word limits and formatting guidelines.

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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. (192 words)
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 (192 words)
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. (192 words)
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. (192 words)
e) Explain the Topological sorting with the help of an example. Also, explain the algorithm of finding strongly connected components in a directed Graph. (192 words)

Key Points:

Optimization Problems: Seek best solution (max/min objective) under given constraints.
Greedy Approach: Makes locally optimal choices, relies on optimal substructure for global optimum.
Fractional Knapsack: Solved greedily by prioritizing items with the highest profit-to-weight ratio.
Huffman Coding: Builds a frequency-based binary tree to create optimal variable-length prefix codes.

Answer: ## (a) Optimisation Problem and Greedy Approach An optimisation problem aims to find the best possible solution from a set of all feasible solutions, typically by maximizing or minimizing an objective function subject to certain constraints. For example, the Traveling Salesperson Problem seeks to find the shortest possible route that visits a set of cities exactly once and returns to the origin city, minimizing the total distance travelled. A greedy approach is used to solve optimisation proble...

QQ3. Consider the following Graph:

Key Points:

Graph representations (Adjacency Matrix, List) affect algorithm efficiency.
Dijkstra's finds single-source shortest paths in non-negative weighted graphs.
Dijkstra's uses a greedy approach with a priority queue for efficiency.
Algorithm initializes distances, iteratively relaxes edges, and updates paths.

Answer: Graphs are fundamental data structures within the realm of algorithm design and analysis, as extensively covered in MCS-211. They are powerful tools for representing relationships and connections between various entities, such as networks, transportation routes, or social structures. A comprehensive analysis of a graph involves understanding its representation, properties, and the application of appropriate algorithms to solve specific problems. Although the specific graph for analysis was not...

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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. (192 words)
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. (192 words)
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, whereas qi represents that the search is for dummy node di. Make suitable assumptions, if any) i | 0 | 1 | 2 | 3 | 4 -|---|---|---|---|--- Pi | | 0.10 | 0.15 | 0.20 | 0.10 qi | 0.05 | 0.10 | 0.10 | 0.10 | 0.10 (192 words)
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 (192 words)
e) Explain the Rabin Karp algorithm for string matching with the help of an example. Find the time complexity of this algorithm. (192 words)

Key Points:

Graph representations (Adjacency Matrix, List) affect algorithm efficiency.
Dijkstra's finds single-source shortest paths in non-negative weighted graphs.
Dijkstra's uses a greedy approach with a priority queue for efficiency.
Algorithm initializes distances, iteratively relaxes edges, and updates paths.

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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 (240 words)
b) What are P and NP class of Problems? Explain each class with the help of at least two examples. (240 words)
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. (240 words)
d) Define the following Problems: (i) SAT Problem (ii) Clique problem (iii) Hamiltonian Cycle Problem (iv) Subset Sum Problem (240 words)

Key Points:

Decision problems yield a 'Yes' or 'No' answer, while optimization problems seek the best solution (e.g., minimum or maximum value).
P class problems are solvable by a deterministic algorithm in polynomial time.
NP class problems have solutions that can be verified in polynomial time by a deterministic algorithm.
NP-Hard problems are at least as difficult as any NP problem; NP-Complete problems are in NP AND NP-Hard.

Answer: The field of computational complexity theory classifies problems based on the resources required to solve them, particularly time and space. This classification helps in understanding the inherent difficulty of problems and guides the development of efficient algorithms. Key concepts include decision problems, optimization problems, and complexity classes like P, NP, NP-Hard, and NP-Complete, along with techniques like polynomial time reduction. Decision problems are fundamental as many optimiz...

MCS-211 Solved Assignment — July 2025

Design and Analysis of Algorithms | IGNOU Master of Computer Applications (MCA_NEW) (MCA_NEW)

MCS-211—July 2025

Design and Analysis of Algorithms

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MCS-211 Assignment Questions — July 2025

QQ1. This assignment has four questions (80 Marks). Answer all questions. The remaining 20 marks are for viva voce. You may use illustrations and diagrams to enhance the explanations. Please go through the guidelines regarding assignments given in the Programme guide for the presentation format.

QQ2. N/A

QQ3. Consider the following Graph:

QQ3. N/A

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