IGNOU MMTE 2 SOLVED ASSIGNMENT

Ques 1.

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 ,1,5 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 works.

Ques 2.

Analyse the algorithm to find the upper bound for run time of the above algorith.

Ques 3.

With the help of an example, explain the following:
    i) Algorithm,
    ii) Input and output for an algorithm
    iii) Running time of an algorithm (5)
   b) Illustrate the operations of partition in quick sort on the array
    . 

Ques 4.

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. (5)
   b) Illustrate the operations of performing counter sort on the array . 

Ques 5.

 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. 

Ques 6.

Using bipartite checking algorithm find out whether the graph given below is bipartite or not. 

Ques 7.

Show step by step how the merge procedure of merge sort will merge the arrays ,1 ,3 ,4 ,7 11,9 13, 14, and ,2 ,5 ,6 10,8 12, . 

Ques 8.

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),(1,1),(−5,−2).

Ques 9.

  Find an optimal parenthesisation of a matrix chain product whose sequence of dimensions is (4,8,7,2,3).

Ques 10.

 Taking  v₁ as the root, find a spanning tree of the following graph using the breadth first search (BFS) algorithm.

Ques 11.

 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).

Ques 12.

Apply DFS algorithm to find a spanning tree of the following graph. Also find all the fundamental cycles of the spanning tree.

Ques 13.

 Use Kruskal's algorithm to find a minimal spanning tree in the graph given below

Ques 14.

 Use Dijkstra's algorithm to find the shortest paths in the graph given below with a as the source vertex.

Ques 15.

Construct a MAXHEAP tree on the following list of numbers

15, 22, 7, 10, 43, 52 13, 16,

Ques 16.

 Construct a binary search tree with the following keys, inserting them in the given order:
          110, 8, 19, 6, 9, 12, 32.

Ques 17.

 Construct the Huffman code tree for the set of frequencies in the table below:

Ques 18.

  Show the minimum heap after each of the following operations
          i)    Create with values 13, 46, 68, 15, 87, 4, 33, 21, 1, 79.
          ii)   Insertion of value 9.
          iii)  Update of value 1 by 71.

Ques 19.

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 ,1,5 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 works.

Ques 20.

Analyse the algorithm to find the upper bound for run time of the above algorith.

Ques 21.

With the help of an example, explain the following:
    i) Algorithm,
    ii) Input and output for an algorithm
    iii) Running time of an algorithm (5)
   b) Illustrate the operations of partition in quick sort on the array
    . 

Ques 22.

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. (5)
   b) Illustrate the operations of performing counter sort on the array . 

Ques 23.

 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. 

Ques 24.

Using bipartite checking algorithm find out whether the graph given below is bipartite or not. 

Ques 25.

Show step by step how the merge procedure of merge sort will merge the arrays ,1 ,3 ,4 ,7 11,9 13, 14, and ,2 ,5 ,6 10,8 12, . 

Ques 26.

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),(1,1),(−5,−2).

Ques 27.

  Find an optimal parenthesisation of a matrix chain product whose sequence of dimensions is (4,8,7,2,3).

Ques 28.

 Taking  v₁ as the root, find a spanning tree of the following graph using the breadth first search (BFS) algorithm.

Ques 29.

 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).

Ques 30.

Apply DFS algorithm to find a spanning tree of the following graph. Also find all the fundamental cycles of the spanning tree.

Ques 31.

 Use Kruskal's algorithm to find a minimal spanning tree in the graph given below

Ques 32.

 Use Dijkstra's algorithm to find the shortest paths in the graph given below with a as the source vertex.

Ques 33.

Construct a MAXHEAP tree on the following list of numbers

15, 22, 7, 10, 43, 52 13, 16,

Ques 34.

 Construct a binary search tree with the following keys, inserting them in the given order:
          110, 8, 19, 6, 9, 12, 32.

Ques 35.

 Construct the Huffman code tree for the set of frequencies in the table below:

Ques 36.

  Show the minimum heap after each of the following operations
          i)    Create with values 13, 46, 68, 15, 87, 4, 33, 21, 1, 79.
          ii)   Insertion of value 9.
          iii)  Update of value 1 by 71.

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