Question 1

a) Write an algorithm to find the first two largest numbers in an in array of integers. For example, given the input (2, 3, 7, −4, 5, 1), the algorithm should output 7, 5. State precisely a loop invariant for you algorithm. Prove that your loop invariant holds and hence conclude that your algorithm

Accepted Answer

The following analysis details an algorithm to efficiently identify the two largest numbers within an array of integers, along with a rigorous proof of its correctness using loop invariants and an examination of its runtime complexity. This approach is fundamental in algorithmic design, emphasizing both correctness and efficiency for typical data processing tasks.

Question 2

a) With the help of an example, explain the following: i) Algorithm, ii) Input and output for an algorithm iii) Running time of an algorithm b) Illustrate the operations of partition in quick sort on the array A = {4, 11, 9, 6, 13, 10, 7, 14, 5, 3, 8}.

Accepted Answer

Algorithms are fundamental to computer science, providing a systematic approach to problem-solving. This answer explores the core concepts of algorithms, their inputs, outputs, and running time, followed by a detailed illustration of the partition operation within the Quick Sort algorithm using a specific array example.

Question 3

a) Show step by step how you will construct a binary search tree by inserting keys in the order 12, 6, 4, 7, 8, 10, 7, 5, 13, 16, 11, starting from an empty binary tree. Explain step by step how you will delete the node with the key 13 from the binary tree. b) Illustrate the operations of performing

Accepted Answer

Binary Search Trees (BSTs) are fundamental data structures that organize data to facilitate efficient searching, insertion, and deletion operations, adhering to the property that all nodes in the left subtree are less than the root, and all nodes in the right subtree are greater than the root. Counter Sort, on the other hand, is a linear time sorting algorithm particularly efficient for sorting elements within a small integer range by counting the occurrences of each element.

**(a) Binary Searc

Question 4

a) Show the results of inserting the keys C, R, A, U, J, S, F, T, O, P, M, L, N, W, Q in order into an empty B-tree with minimum degree 2. Only draw the configurations of the tree just before some node must split, and also draw the final configuration. b) Using bipartite checking algorithm find out

Accepted Answer

This response addresses two fundamental algorithms in data structures and graph theory: B-tree insertion and bipartite graph checking. For B-trees, the insertion process involves maintaining balance and order, with specific rules for node splitting when capacity is exceeded, potentially leading to cascading splits. For bipartite graphs, a BFS-based 2-coloring algorithm effectively determines if a graph can be divided into two independent sets, indicating bipartiteness or the presence of odd-leng

Question 5

a) Show step by step how the merge procedure of merge sort will merge the arrays 1, 3, 4, 7, 9, 11, 13, 14 and 2, 5, 6, 8, 10, 12. b) For the following set of points, describe how the CLOSET-PAIR algorithm finds a closest pair of points: (1, 2), (2, −1), (3, 3), (−3, 2), (−3, 4), (−1, 2), (2, 3), (1

Accepted Answer

This response comprehensively addresses the given questions from MMTE-002 Design and Analysis of Algorithms, covering the merge procedure of merge sort, the CLOSET-PAIR algorithm, and optimal parenthesisation for matrix chain products. Each sub-question is answered step-by-step, adhering to specified word limits and formatting guidelines, demonstrating core algorithmic principles.

Question 6

a) Taking v₁ as the root, find a spanning tree of the following graph using the breadth first search (BFS) algorithm. b) Determine an LCS of (1, 1, 1, 1, 0, 1,1, 0, 1, 0) and (1, 1, 1, 0, 0, 1, 1, 0, 1, 0).

Accepted Answer

This response addresses two fundamental algorithm problems: constructing a Breadth-First Search (BFS) spanning tree and finding the Longest Common Subsequence (LCS) using dynamic programming. BFS systematically explores a graph layer by layer, ensuring the shortest path from the root in terms of edges, which is crucial for network routing and connectivity analysis.

Conversely, the LCS problem finds the longest sequence that can be obtained from two given sequences by deleting zero or more eleme

Question 7

a) Apply DFS algorithm to find a spanning tree of the following graph. Also find all the fundamental cycles of the spanning tree. b) Use Kruskal's algorithm to find a minimal spanning tree in the graph given below c) Use Dijkstra's algorithm to find the shortest paths in the graph given below with a

Accepted Answer

Graph algorithms are fundamental in computer science for solving various problems related to networks, paths, and connections. This response applies three essential graph algorithms – Depth First Search (DFS), Kruskal's algorithm, and Dijkstra's algorithm – to a given weighted, undirected graph. Each algorithm serves a distinct purpose: DFS for finding a spanning tree and fundamental cycles, Kruskal's for identifying a Minimum Spanning Tree, and Dijkstra's for determining the shortest paths from

Question 8

a) Construct a MAXHEAP tree on the following list of numbers 15, 22, 7, 10, 43, 52, 13, 16. b) Construct a binary search tree with the following keys, inserting them in the given order: 110, 8, 19, 6, 9, 12, 32.

Accepted Answer

This response comprehensively addresses the construction of both a MAXHEAP tree and a Binary Search Tree (BST) based on the provided lists of numbers. It meticulously details the step-by-step process for each construction, adhering to the fundamental principles and algorithms of these data structures, as typically covered in the MMTE-002 course. The explanations leverage the core properties of heaps and BSTs to illustrate how the final tree structures are achieved, ensuring clarity and accuracy

MMTE-002 Solved Assignment — January 2026 / July 2026

Design and Analysis of Algorithms | IGNOU Master of Science (Mathematics with Applications in Computer Science) (MSCMACS)

MMTE-002—January 2026 / July 2026

Design and Analysis of Algorithms

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MMTE-002 Assignment Questions — January 2026

Q2. a) With the help of an example, explain the following: i) Algorithm, ii) Input and output for an algorithm iii) Running time of an algorithm b) Illustrate the operations of partition in quick sort on the array A = {4, 11, 9, 6, 13, 10, 7, 14, 5, 3, 8}.

Q6. a) Taking v₁ as the root, find a spanning tree of the following graph using the breadth first search (BFS) algorithm. b) Determine an LCS of (1, 1, 1, 1, 0, 1,1, 0, 1, 0) and (1, 1, 1, 0, 0, 1, 1, 0, 1, 0).

Q8. a) Construct a MAXHEAP tree on the following list of numbers 15, 22, 7, 10, 43, 52, 13, 16. b) Construct a binary search tree with the following keys, inserting them in the given order: 110, 8, 19, 6, 9, 12, 32.