Ee693 questionshomework

EE693
Data Structure & Algorithms
Home Work MCQ Questions
1) The number of distinct minimum spanning trees for the weighted graph below is ____
(A) 6
(B) 7
(C) 8
(D) 9
2) Consider the following rooted tree with the vertex P labeled as root
The order in which the nodes are visited during in-order traversal is
(A) SQPTRWUV
(B) SQPTURWV
(C) SQPTWUVR
(D) SQPTRUWV
3) The minimum number of arithmetic operations required to evaluate the polynomial P(X) = X5
+
4X3
+ 6X + 5 for a given value of X using only one temporary variable.
(A) 6
(B) 7
(C) 8
(D) 9
4) You have an array of n elements. Suppose you implement quicksort by always choosing the central
element of the array as the pivot. Then the tightest upper bound for the worst case performance is
(A) O(n2
)
(B) O(nLogn)

(C) Θ(nLogn)
(D) O(n3
)
5) Consider the pseudocode given below. The function DoSomething() takes as argument a pointer to
the root of an arbitrary tree represented by the leftMostChild-rightSibling representation. Each node
of the tree is of type treeNode.
typedef struct treeNode* treeptr;
struct treeNode {
treeptr leftMostChild, rightSibling;
};
int DoSomething (treeptr tree) {
int value = 0;
if (tree != NULL) {
if (tree->leftMostChild == NULL)
value = 1;
else
value = DoSomething(tree->leftMostChild);
value = value + DoSomething(tree->rightSibling);
}
return(value);
When the pointer to the root of a tree is passed as the argument to DoSomething, the value
returned by the function corresponds to the
(A) number of internal nodes in the tree.
(B) height of the tree.
(C) number of nodes without a right sibling in the tree.
(D) number of leaf nodes in the tree.
6) Suppose a polynomial time algorithm is discovered that correctly computes the largest clique in a
given graph. In this scenario, which one of the following represents the correct Venn diagram of the
complexity classes P, NP and NP Complete (NPC)?
(A) A
(B) B
(C) C
(D) D

7) Consider two strings A = “qpqrr” and B = “pqprqrp”. Let x be the length of the longest common
subsequence (not necessarily contiguous) between A and B and let y be the number of such longest
common subsequences between A and B. Then x + 10y = ___.
(A) 43
(B) 34
(C) 57
(D) 75
8) Suppose P, Q, R, S, T are sorted sequences having lengths 20, 24, 30, 35, 50 respectively. They are to
be merged into a single sequence by merging together two sequences at a time. The number of
comparisons that will be needed in the worst case by the optimal algorithm for doing this is ____.
(A) 231
(B) 358
(C) 573
(D) 475
9) Consider the tree arcs of a BFS traversal from a source node W in an unweighted, connected,
undirected graph. The tree T formed by the tree arcs is a data structure for computing.
(A) the shortest path between every pair of vertices.
(B) the shortest path from W to every vertex in the graph.
(C) the shortest paths from W to only those nodes that are leaves of T.
(D) the longest path in the graph
10) Consider the following pseudo code. What is the total number of multiplications to be performed?
D = 2
for i = 1 to n do
for j = i to n do
for k = j + 1 to n do
D = D * 3
(A) Half of the product of the 3 consecutive integers.
(B) One-third of the product of the 3 consecutive integers.
(C) One-sixth of the product of the 3 consecutive integers.
(D) None of the above.
11) Consider a hash table with 9 slots. The hash function is h(k) = k mod 9. The collisions are resolved
by chaining. The following 9 keys are inserted in the order: 5, 28, 19, 15, 20, 33, 12, 17, 10. The
maximum, minimum, and average chain lengths in the hash table, respectively, are
(A) 3, 0, and 1
(B) 3, 3, and 3
(C) 4, 0, and 1
(D) 3, 0, and 2
12) A priority queue is implemented as a Max-Heap. Initially, it has 5 elements. The level-order
traversal of the heap is: 10, 8, 5, 3, 2. Two new elements 1 and 7 are inserted into the heap in that
order. The level-order traversal of the heap after the insertion of the elements is:
(A) 10, 8, 7, 3, 2, 1, 5
(B) 10, 8, 7, 2, 3, 1, 5
(C) 10, 8, 7, 1, 2, 3, 5
(D) 10, 8, 7, 5, 3, 2, 1

13) Which one of the following correctly determines solution of the recurrence relation with T(1) = 1?
T(n) = 2T(n/2) + Logn
(A) Θ(n)
(B) Θ(nLogn)
(C) Θ(n*n)
(D) Θ(log n)
14) Suppose implementation supports an instruction REVERSE, which reverses the order of elements
on the stack, in addition to the PUSH and POP instructions. Which one of the following statements is
TRUE with respect to this modified stack?
(A) A queue cannot be implemented using this stack.
(B) A queue can be implemented where ENQUEUE takes a single instruction and DEQUEUE takes a
sequence of two instructions.
(C) A queue can be implemented where ENQUEUE takes a sequence of three instructions and
DEQUEUE takes a single instruction.
(D) A queue can be implemented where both ENQUEUE and DEQUEUE take a single instruction each.
15) Let G be a graph with n vertices and m edges. What is the tightest upper bound on the running
time on Depth First Search of G? Assume that the graph is represented using adjacency matrix.
(A) O(n)
(B) O(m+n)
(C) O(n2
)
(D) O(mn)
16) Consider the directed graph given below. Which one of the following is TRUE?
(A) The graph doesn’t have any topological ordering
(B) Both PQRS and SRPQ are topological ordering
(C) Both PSRQ and SPRQ are topological ordering
(D) PSRQ is the only topological ordering
17) Let P be a Quicksort Program to sort numbers in ascending order using the first element as pivot.
Let t1 and t2 be the number of comparisons made by P for the inputs {1, 2, 3, 4, 5} and {4, 1, 5, 3, 2}
respectively. Which one of the following holds?
(A) t1 = 5
(B) t1 < t2

(C) t1 > t2
(D) t1 = t2
18) Consider the following C function in which size is the number of elements in the array E:
The value returned by the function MyX is the
int MyX(int *E, unsigned int size)
{
int Y = 0;
int Z;
int i, j, k;
for (i = 0; i < size; i++)
Y = Y + E[i];
for (i = 0; i < size; i++)
for (j = i; j < size; j++)
{
Z = 0;
for (k = i; k <= j; k++)
Z = Z + E[k];
if (Z > Y)
Y = Z;
}
return Y;
}
(A) maximum possible sum of elements in any sub-array of array E.
(B) maximum element in any sub-array of array E.
(C) sum of the maximum elements in all possible sub-arrays of array E
(D) the sum of all the elements in the array E.
19) The preorder traversal sequence of a binary search tree is 30, 20, 10, 15, 25, 23, 39, 35, 42. Which
one of the following is the postorder traversal sequence of the same tree?
(A) 10, 20, 15, 23, 25, 35, 42, 39, 30
(B) 15, 10, 25, 23, 20, 42, 35, 39, 30
(C) 15, 20, 10, 23, 25, 42, 35, 39, 30
(D) 15, 10, 23, 25, 20, 35, 42, 39, 30
20) Consider the following function
int unknown(int n) {
int i, j, k = 0;
for (i = n/2; i <= n; i++)
for (j = 2; j <= n; j = j * 2)
k = k + n/2;
return k;
}
What is the returned value of the above function?
(A) Θ(n^2)
(B) Θ(n^2Logn)
(C) Θ(n^3)
(D) Θ(n^3Logn)

21) Which of the following statements is/are TRUE for an undirected graph?
P: Number of odd degree vertices is even
Q: Sum of degrees of all vertices is even
A) P Only
B) Q Only
C) Both P and Q
D) Neither P nor Q
22) Consider an undirected random graph of eight vertices. The probability that there is an edge
between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?
(A) 1/8
(B) 1
(C) 7
(D) 8
23) Which of the following statements are TRUE?
(1) The problem of determining whether there exists a cycle in an undirected graph is in P.
(2) The problem of determining whether there exists a cycle in an undirected graph is in NP.
(3) If a problem A is NP-Complete, there exists a non-deterministic polynomial time algorithm to solve
A.
(A) 1,2 and 3
(B) 1 and 2 only
(C) 2 and 3 only
(D) 1 and 3 only
24) Consider the directed graph shown in the figure below. There are multiple shortest paths between
vertices S and T. Which one will be reported by Dijstra?s shortest path algorithm? Assume that, in any
iteration, the shortest path to a vertex v is updated only when a strictly shorter path to v is
discovered.
(A) SDT
(B) SBDT
(C) SACDT
(D) SACET

25) Suppose a circular queue of capacity (n – 1) elements is implemented with an array of n elements.
Assume that the insertion and deletion operation are carried out using REAR and FRONT as array
index variables, respectively. Initially, REAR = FRONT = 0. The conditions to detect queue full and
queue empty are
(A) Full: (REAR+1) mod n == FRONT, empty: REAR == FRONT
(B) Full: (REAR+1) mod n == FRONT, empty: (FRONT+1) mod n == REAR
(C) Full: REAR == FRONT, empty: (REAR+1) mod n == FRONT
(D) Full: (FRONT+1) mod n == REAR, empty: REAR == FRONT
26) Assuming P != NP, which of the following is true ?
(A) NP-complete = NP
(B) NP-complete ∩ P = Φ
(C) NP-hard = NP
(D) P = NP-complete
27) An undirected graph G(V, E) contains n ( n > 2 ) nodes named v1 , v2 ,….vn. Two nodes vi , vj are
connected if and only if 0 < |i – j| <= 2. Each edge (vi, vj ) is assigned a weight i + j. A sample graph
with n = 4 is shown below.
What will be the cost of the minimum spanning tree (MST) of such a graph with n nodes?
(A) 1/12(11n^2 – 5n)
(B) n^2 – n + 1
(C) 6n – 11
(D) 2n + 1
28) Consider two binary operators ‘↑ ‘ and ‘↓’ with the precedence of operator ↓ being lower than
that of the ↑ operator. Operator ↑ is right associative while operator ↓ is left associative. Which one
of the following represents the parse tree for expression (7 ↓ 3 ↑ 4 ↑ 3 ↓ 2)?

29) A max-heap is a heap where the value of each parent is greater than or equal to the values of its
children. Which of the following is a max-heap?
30) Four matrices M1, M2, M3 and M4 of dimensions pxq, qxr, rxs and sxt respectively can be
multiplied is several ways with different number of total scalar multiplications. For example, when
multiplied as ((M1 X M2) X (M3 X M4)), the total number of multiplications is pqr + rst + prt. When
multiplied as (((M1 X M2) X M3) X M4), the total number of scalar multiplications is pqr + prs + pst.
If p = 10, q = 100, r = 20, s = 5 and t = 80, then the number of scalar multiplications needed is
A) 248000
B) 44000
C) 19000
D) 25000

31) Which of the given options provides the increasing order of asymptotic complexity of functions f1,
f2, f3 and f4?
f1(n) = 2^n
f2(n) = n^(3/2)
f3(n) = nLogn
f4(n) = n^(Logn)
A) f3, f2, f4, f1
B) f3, f2, f1, f4
C) f2, f3, f1, f4
D) f2, f3, f4, f1
32) We are given a set of n distinct elements and an unlabeled binary tree with n nodes. In how many
ways can we populate the tree with the given set so that it becomes a binary search tree?
A) 0
B) 1
C) n!
D) (1/(n+1)).2nCn
33) An algorithm to find the length of the longest monotonically increasing sequence of numbers in an
array A[0 :n-1] is given below.
Let Li denote the length of the longest monotonically increasing sequence starting at index i in the
array
which of the following statements is TRUE?
(A) The algorithm uses dynamic programming paradigm
(B) The algorithm has a linear complexity and uses branch and bound paradigm
(C) The algorithm has a non-linear polynomial complexity and uses branch and bound paradigm
(D) The algorithm uses divide and conquer paradigm.
34) consider a complete undirected graph with vertex set {0, 1, 2, 3, 4}. Entry Wij in the matrix W
below is the weight of the edge {i, j}.

What is the minimum possible weight of a spanning tree T in this graph such that vertex 0 is a leaf
node in the tree T?
(A) 7
(B) 8
(C) 9
(D) 10
35) In the graph given in question 1, what is the minimum possible weight of a path P from vertex 1 to
vertex 2 in this graph such that P contains at most 3 edges?
(A) 7
(B) 8
(C) 9
(D) 10
36) The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in
decreasing order. Which of the following sequences cannot be the degree sequence of any graph?
I. 7, 6, 5, 4, 4, 3, 2, 1
II. 6, 6, 6, 6, 3, 3, 2, 2
III. 7, 6, 6, 4, 4, 3, 2, 2
IV. 8, 7, 7, 6, 4, 2, 1, 1
(A) I and II
(B) III and IV
(C) IV only
(D) II and IV
38) A hash table of length 10 uses open addressing with hash function h(k)=k mod 10, and linear
probing. After inserting 6 values into an empty hash table, the table is as shown below.
Which one of the following choices gives a possible order in which the key values could have been
inserted in the table?

(A) 46, 42, 34, 52, 23, 33
(B) 34, 42, 23, 52, 33, 46
(C) 46, 34, 42, 23, 52, 33
(D) 42, 46, 33, 23, 34, 52
39) How many different insertion sequences of the key values using the same hash function and linear
probing will result in the hash table shown above?
(A) 10
(B) 20
(C) 30
(D) 40
40) Which one of the following is a key factor for preferring B-trees to binary search trees for indexing
database relations?
(a) Database relations have a large number of records
(b) Database relations are sorted on the primary key
(c) B-trees require less memory than binary search trees
(d) Data transfer form disks is in blocks.
41) How many distinct binary search trees can be created out of 4 distinct keys?
(a) 5
(b) 14
(c) 24
(d) 42
42) In a complete k-ary tree, every internal node has exactly k children. The number of leaves in such a
tree with n internal nodes is:
(a) nk
(b) (n – 1) k+ 1
(c) n( k – 1) + 1
(d) n(k – 1)
43) 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
44) 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

45) 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]
46) A set X can be represented by an array x[n] as follows:
Consider the following algorithm in which x,y and z are Boolean arrays of size n:
algorithm zzz(x[] , y[], z [])
{
int i;
for (i=O; i<n; ++i)
z[i] = (x[i] ^ ~y[i]) V (~x[i] ^ y[i])
}
The set Z computed by the algorithm is:
(A) (X Intersection Y)
(B) (X Union Y)
(C) (X-Y) Intersection (Y-X)
(D) (X-Y) Union (Y-X)
47) Let X be a problem that belongs to the class NP. Then which one of the following is TRUE?
(A) There is no polynomial time algorithm for X.
(B) If X can be solved deterministically in polynomial time, then P = NP.
(C) If X is NP-hard, then it is NP-complete.
(D) X may be undecidable.
48) Consider the polynomial p(x) = a0 + a1x + a2x^2 +a3x^3, where ai != 0, for all i. The minimum
number of multiplications needed to evaluate p on an input x is:
(A) 3
(B) 4
(C) 6
(D) 9
49) To implement Dijkstra’s shortest path algorithm on unweighted graphs so that it runs in linear
time, the data structure to be used is:
(A) Queue
(B) Stack
(C) Heap
(D) B-Tree

50) A 3-ary max heap is like a binary max heap, but instead of 2 children, nodes have 3 children. A 3-
ary heap can be represented by an array as follows: The root is stored in the first location, a[0], nodes
in the next level, from left to right, is stored from a[1] to a[3]. The nodes from the second level of the
tree from left to right are stored from a[4] location onward. An item x can be inserted into a 3-ary
heap containing n items by placing x in the location a[n] and pushing it up the tree to satisfy the heap
property.
Which one of the following is a valid sequence of elements in an array representing 3-ary max heap?
(A) 1, 3, 5, 6, 8, 9
(B) 9, 6, 3, 1, 8, 5
(C) 9, 3, 6, 8, 5, 1
(D) 9, 5, 6, 8, 3, 1
51) An implementation of a queue Q, using two stacks S1 and S2, is given below:
void insert(Q, x) {
push (S1, x);
}
void delete(Q){
if(stack-empty(S2)) then
if(stack-empty(S1)) then {
print(“Q is empty”);
return;
}
else while (!(stack-empty(S1))){
x=pop(S1);
push(S2,x);
}
x=pop(S2);
}
Let n insert and m (<=n) delete operations be performed in an arbitrary order on an empty queue Q.
Let x and y be the number of push and pop operations performed respectively in the process. Which
one of the following is true for all m and n?
(A) n+m <= x < 2n and 2m <= y <= n+m
(B) n+m <= x < 2n and 2m<= y <= 2n
(C) 2m <= x < 2n and 2m <= y <= n+m
(D) 2m <= x <2n and 2m <= y <= 2n

52) Consider the following graph:
Which one of the following cannot be the sequence of edges added, in that order, to a minimum
spanning tree using Kruskal’s algorithm?
(A) (a—b),(d—f),(b—f),(d—c),(d—e)
(B) (a—b),(d—f),(d—c),(b—f),(d—e)
(C) (d—f),(a—b),(d—c),(b—f),(d—e)
(D) (d—f),(a—b),(b—f),(d—e),(d—c)
53) The median of n elements can be found in O(n)time. Which one of the following is correct about
the complexity of quick sort, in which median is selected as pivot?
(A) Θ(n)
(B) Θ(nlogn)
(C) Θ(n^2)
(D) Θ(n^3)
54) Consider a binary max-heap implemented using an array. Which one of the following array
represents a binary max-heap?
(A) 25,12,16,13,10,8,14
(B) 25,14,13,16,10,8,12
(C) 25,14,16,13,10,8,12
(D) 25,14,12,13,10,8,16
55) What is the content of the array after two delete operations on the correct answer to the previous
question?
(A) 14,13,12,10,8
(B) 14,12,13,8,10
(C) 14,13,8,12,10
(D) 14,13,12,8,10
56) In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n)
time algorithm. What is the worst case time complexity of the quick sort?
(A) Θ(n)
(B) Θ(n Log n)
(C) Θ(n^2)
(D) Θ(n2
log n)

57) Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which
splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be
the number of comparisons required to sort n elements. Then
(A) T(n) <= 2T(n/5) + n
(B) T(n) <= T(n/5) + T(4n/5) + n
(C) T(n) <= 2T(4n/5) + n
(D) T(n) <= 2T(n/2) + n
58) Dijkstra’s single source shortest path algorithm when run from vertex a in the below graph,
computes the correct shortest path distance to
(A) only vertex a
(B) only vertices a, e, f, g, h
(C) only vertices a, b, c, d
(D) all the vertices
59) The following C function takes a single-linked list of integers as a parameter and rearranges the
elements of the list. The function is called with the list containing the integers 1, 2, 3, 4, 5, 6, 7 in the
given order. What will be the contents of the list after the function completes execution?
struct node
{
int value;
struct node *next;
};
void rearrange(struct node *list)
{
struct node *p, * q;
int temp;
if ((!list) || !list->next)
return;
p = list;
q = list->next;
while(q)
{
temp = p->value;
p->value = q->value;
q->value = temp;
p = q->next;
q = p?p->next:0;
}
}

(A) 1,2,3,4,5,6,7
(B) 2,1,4,3,6,5,7
(C) 1,3,2,5,4,7,6
(D) 2,3,4,5,6,7,1
60) The Breadth First Search algorithm has been implemented using the queue data structure. One
possible order of visiting the nodes of the following graph is
(A) MNOPQR
(B) NQMPOR
(C) QMNPRO
(D) QMNPOR
61) Consider the following functions:
f(n) = 2^n
g(n) = n!
h(n) = n^logn
Which of the following statements about the asymptotic behaviour of f(n), g(n), and h(n) is true?
(A) f(n) = O(g(n)); g(n) = O(h(n))
(B) f(n) = Ω(g(n)); g(n) = O(h(n))
(C) g(n) = O(f(n)); h(n) = O(f(n))
(D) h(n) = O(f(n)); g(n) = Ω(f(n))
62) The number of leaf nodes in a rooted tree of n nodes, with each node having 0 or 3 children is:
a) n/2
b) (n-1)/3
c) (n-1)/2
d) (2n+1)/3
63) A weight-balanced tree is a binary tree in which for each node. The number of nodes in the left
sub tree is at least half and at most twice the number of nodes in the right sub tree. The maximum
possible height (number of nodes on the path from the root to the farthest leaf) of such a tree on n
nodes is best described by which of the following?
a) log2 n
b) log4/3 n
c) log3 n
d) log3/2 n
64) Let w be the minimum weight among all edge weights in an undirected connected graph. Let e be
a specific edge of weight w . Which of the following is FALSE?
(A) There is a minimum spanning tree containing e.
(B) If e is not in a minimum spanning tree T, then in the cycle formed by adding e to T, all edges have
the same weight.

(C) Every minimum spanning tree has an edge of weight w .
(D) e is present in every minimum spanning tree.
65) An array of n numbers is given, where n is an even number. The maximum as well as the minimum
of these n numbers needs to be determined. Which of the following is TRUE about the number of
comparisons needed?
(A) At least 2n – c comparisons, for some constant c, are needed.
(B) At most 1.5n – 2 comparisons are needed.
(C) At least nLog2n comparisons are needed.
(D) None of the above.
66) In an unweighted, undirected connected graph, the shortest path from a node S to every other
node is computed most efficiently, in terms of time complexity by
(A) Dijkstra’s algorithm starting from S.
(B) Warshall’s algorithm
(C) Performing a DFS starting from S.
(D) Performing a BFS starting from S.
67) A complete n-ary tree is a tree in which each node has n children or no children. Let I be the
number of internal nodes and L be the number of leaves in a complete n-ary tree. If L = 41, and I = 10,
what is the value of n?
(A) 3
(B) 4
(C) 5
(D) 6
68) The height of a binary tree is the maximum number of edges in any root to leaf path. The
maximum number of nodes in a binary tree of height h is:
(A) 2^h -1
(B) 2^(h-1) – 1
(C) 2^(h+1) -1
(D) 2*(h+1)
69) The maximum number of binary trees that can be formed with three unlabeled nodes is:
(A) 1
(B) 5
(C) 4
(D) 3
70) Which of the following sorting algorithms has the lowest worst-case complexity?
(A) Merge sort
(B) Bubble sort
(C) Quick sort
(D) Selection sort
71) Suppose the numbers 7, 5, 1, 8, 3, 6, 0, 9, 4, 2 are inserted in that order into an initially empty
binary search tree. The binary search tree uses the usual ordering on natural numbers. What is the in-
order traversal sequence of the resultant tree?
(A) 7 5 1 0 3 2 4 6 8 9

(B) 0 2 4 3 1 6 5 9 8 7
(C) 0 1 2 3 4 5 6 7 8 9
(D) 9 8 6 4 2 3 0 1 5 7
72) Let S be a stack of size n >= 1. Starting with the empty stack, suppose we push the first n natural
numbers in sequence, and then perform n pop operations. Assume that Push and Pop operation take
X seconds each, and Y seconds elapse between the end of one such stack operation and the start of
the next operation. For m >= 1, define the stack-life of m as the time elapsed from the end of Push(m)
to the start of the pop operation that removes m from S. The average stack-life of an element of this
stack is
(A) n(X+ Y)
(B) 3Y + 2X
(C) n(X + Y)-X
(D) Y + 2X
73) Let G be an undirected connected graph with distinct edge weight. Let emax be the edge with
maximum weight and emin the edge with minimum weight. Which of the following statements is
false?
(a) Every minimum spanning tree of G must contain emin
(b) If emax is in a minimum spanning tree, then its removal must disconnect G
(c) No minimum spanning tree contains emax
(d) G has a unique minimum spanning tree
74) Let G be an undirected graph. Consider a depth-first traversal of G, and let T be the resulting
depth-first search tree. Let u be a vertex in G and let v be the first new (unvisited) vertex visited after
visiting u in the traversal. Which of the following statements is always true?
(a) {u,v} must be an edge in G, and u is a descendant of v in T
(b) {u,v} must be an edge in G, and v is a descendant of u in T
(c) If {u,v} is not an edge in G then u is a leaf in T
(d) If {u,v} is not an edge in G then u and v must have the same parent in T
75) Consider an undirected unweighted graph G. Let a breadth-first traversal of G be done starting
from a node r. Let d(r, u) and d(r, v) be the lengths of the shortest paths from r to u and v respectively,
in G. lf u is visited before v during the breadth-first traversal, which of the following statements is
correct?
a) d(r, u) < d (r, v)
b) d(r, u) > d(r, v)
c) d(r, u) <= d (r, v)
d) None of the above
76) How many undirected graphs (not necessarily connected) can be constructed out of a given set V=
{V 1, V 2,…V n} of n vertices ?
a) n(n-l)/2
b) 2^n
c) n!
d) 2^(n(n-1)/2)

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