1) A program P reads in 500 integers in the range [0..100] representing the scores of 500 students. It then prints the frequency of each score above 50. What would be the best way for P to store the frequencies?
(a) An array of 50 numbers
(b) An array of 100 numbers
(c) An array of 500 numbers
(d) A dynamically allocated array of 550 numbers
Answer (a)
An array of size 50 looks the best option to store number of students for each score. We need to store frequencies of scores above 50. We can ignore scores below 50 and to index the scores above 50, we can subtract 50 from the score value/
2) An undirected graph G has n nodes. Its adjacency matrix is given by an n × n square matrix whose (i) diagonal elements are 0‘s and (ii) non-diagonal elements are 1‘s. which one of the following is TRUE?
(a) Graph G has no minimum spanning tree (MST)
(b) Graph G has a unique MST of cost n-1
(c) Graph G has multiple distinct MSTs, each of cost n-1
(d) Graph G has multiple spanning trees of different costs
Answer (c)
If all non diagonal elements are 1, then every vertex is connected to every other vertex in the graph with an edge of weight 1. Such a graph has multiple distinct MSTs with cost n-1. See the below example.
The connected graph:
Below are three Minimum Spanning trees each of cost 2.0.
Minimum Spanning Tree 1
Minimum Spanning Tree 2
Minimum Spanning Tree 3
3) The time complexity of computing the transitive closure of a binary relation on a set of n elements is known to be:
a) O(n)
b) O(nLogn)
c) O(n^(3/2))
d) O(n^3)
Answer (d)
In mathematics, the transitive closure of a binary relation R on a set X is the smallest transitive relation on X that contains R. If the original relation is transitive, the transitive closure will be that same relation; otherwise, the transitive closure will be a different relation.
Warshall’s algorithm can be used to construct the Transitive closure of directed graphs (). In Warshall’s original formulation of the algorithm, the graph is unweighted and represented by a Boolean adjacency matrix. Then the addition operation is replaced by logical conjunction (AND) and the minimum operation by logical disjunction (OR).
References:
http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm
http://en.wikipedia.org/wiki/Transitive_closure
4. A Priority-Queue is implemented as a Max-Heap. Initially, it has 5 elements. The level-order traversal of the heap is given below:
10, 8, 5, 3, 2
Two new elements ”1‘ and ”7‘ are inserted in the heap in that order. The level-order traversal of the heap after the insertion of the elements is:
(a) 10, 8, 7, 5, 3, 2, 1
(b) 10, 8, 7, 2, 3, 1, 5
(c) 10, 8, 7, 1, 2, 3, 5
(d) 10, 8, 7, 3, 2, 1, 5
Answer (D)
Original Max-Heap is:
10
/ \
8 5
/ \
3 2
After Insertion of 1.
10
/ \
8 5
/ \ /
3 2 1
After Insertion of 7.
10
/ \
8 7
/ \ / \
3 2 1 5
5. The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,…,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?
(A) X[i, j] = X[i – 1, j] V X[i, j -ai]
(B) X[i, j] = X[i – 1, j] V X[i – 1, j – ai]
(C) X[i, j] = X[i – 1, j] V X[i, j – ai]
(D) X[i, j] = X[i – 1, j] V X[i -1, j – ai]
Answer (B)
6. X[I, j] (2 <= i <= n and ai <= j <= W), is true if any of the following is true 1) Sum of weights excluding ai is equal to j, i.e., if X[i-1, j] is true. 2) Sum of weights including ai is equal to j, i.e., if X[i-1, j-ai] is true so that we get (j – ai) + ai as j. 2. In question 1, which entry of the array X, if TRUE, implies that there is a subset whose elements sum to W?
(A) X[1, W]
(B) X[n ,0]
(C) X[n, W]
(D) X[n -1, n]
Answer (C)
If we get the entry X[n, W] as true then there is a subset of {a1, a2, .. an} that has sum as W.
Reference: http://en.wikipedia.org/wiki/Subset_sum_problem
8. Consider the following C program that attempts to locate an element x in an array Y[] using binary search. The program is erroneous.
1. f(int Y[10], int x) {2. int i, j, k;3. i = 0; j = 9;4. do {5. k = (i + j) /2;6. if( Y[k] < x) i = k; else j = k;7. } while(Y[k] != x && i < j);8. if(Y[k] == x) printf ("x is in the array ") ;9. else printf (" x is not in the array ") ;10. } |
On which of the following contents of Y and x does the program fail?
(A) Y is [1 2 3 4 5 6 7 8 9 10] and x < 10
(B) Y is [1 3 5 7 9 11 13 15 17 19] and x < 1
(C) Y is [2 2 2 2 2 2 2 2 2 2] and x > 2
(D) Y is [2 4 6 8 10 12 14 16 18 20] and 2 < x < 20 and x is even
Answer (C)
The above program doesn’t work for the cases where element to be searched is the last element of Y[] or greater than the last element (or maximum element) in Y[]. For such cases, program goes in an infinite loop because i is assigned value as k in all iterations, and i never becomes equal to or greater than j. So while condition never becomes false.