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DEPARTMENT OF INFORMATION TECHNOLOGY
2020 – 2021 EVEN SEMESTER
GATE BOOKLET
DATA STRUCTURES
&
ALGORITHMS
(Questions From 2000 To 2020)
COMPILED BY
Dr. P. Subathra, Prof./IT
GATE SYLLABUS
CS Computer Science and Information Technology
Section1: Engineering Mathematics
Discrete Mathematics: Propositional and first order logic. Sets, relations, functions, partial orders and
lattices. Groups. Graphs: connectivity, matching, coloring. Combinatorics: counting, recurrence relations,
generating functions. Linear Algebra: Matrices, determinants, system of linear equations, eigenvalues and
eigenvectors, LU decomposition. Calculus: Limits, continuity and differentiability. Maxima and minima.
Mean value theorem. Integration. Probability: Random variables. Uniform, normal, exponential, poisson
and binomial distributions. Mean, median, mode and standard deviation. Conditional probability and
Bayes theorem.
Computer Science and Information Technology
Section 2: Digital Logic
Boolean algebra. Combinational and sequential circuits. Minimization. Number representations and
computer arithmetic (fixed and floating point).
Section 3: Computer Organization and Architecture
Machine instructions and addressing modes. ALU, data‐path and control unit. Instruction pipelining.
Memory hierarchy: cache, main memory and secondary storage; I/O interface (interrupt and DMA mode).
Section 4: Programming and Data Structures
Programming in C. Recursion. Arrays, Stacks, Queues, Linked Lists, Trees, Binary Search Trees,
Binary Heaps, Graphs.
Section 5: Algorithms
Searching, Sorting, Hashing. Asymptotic Worst Case Time and Space Complexity. Algorithm Design
Techniques: Greedy, Dynamic programming and Divide‐and‐Conquer. Graph Search, Minimum
Spanning Trees, Shortest Paths.
Section 6: Theory of Computation
Regular expressions and finite automata. Context-free grammars and push-down automata. Regular and
contex-free languages, pumping lemma. Turing machines and undecidability.
Section 7: Compiler Design
Lexical analysis, parsing, syntax-directed translation. Runtime environments. Intermediate code
generation.
Section 8: Operating System
Processes, threads, inter‐process communication, concurrency and synchronization. Deadlock. CPU
scheduling. Memory management and virtual memory. File systems.
CS8391 DATA STRUCTURES
OBJECTIVES:
To understand the concepts of ADTs
To learn linear data structures – lists, stacks, and queues
To understand sorting, searching and hashing algorithms
To apply Tree and Graph structures
UNIT I LINEAR DATA STRUCTURES – LIST
Abstract Data Types (ADTs) – List ADT – array-based implementation – linked list implementation ––
singly linked lists- circularly linked lists- doubly-linked lists – applications of lists –Polynomial
Manipulation – All operations (Insertion, Deletion, Merge, Traversal).
UNIT II LINEAR DATA STRUCTURES – STACKS, QUEUES
Stack ADT – Operations - Applications - Evaluating arithmetic expressions- Conversion of Infix to
postfix expression - Queue ADT – Operations - Circular Queue – Priority Queue - deQueue –
applications of queues.
UNIT III NON LINEAR DATA STRUCTURES – TREES
Tree ADT – tree traversals - Binary Tree ADT – expression trees – applications of trees – binary search
tree ADT –Threaded Binary Trees- AVL Trees – B-Tree - B+ Tree - Heap – Applications of heap.
UNIT IV NON LINEAR DATA STRUCTURES - GRAPHS Definition –
Representation of Graph – Types of graph - Breadth-first traversal - Depth-first traversal – Topological
Sort – Bi-connectivity – Cut vertex – Euler circuits – Applications of graphs.
UNIT V SEARCHING, SORTING AND HASHING TECHNIQUES
Searching- Linear Search - Binary Search. Sorting - Bubble sort - Selection sort - Insertion sort - Shell
sort – Radix sort. Hashing- Hash Functions – Separate Chaining – Open Addressing – Rehashing –
Extendible Hashing.
OUTCOMES:
At the end of the course, the student should be able to:
Implement abstract data types for linear data structures
Apply the different linear and non-linear data structures to problem solutions
Critically analyze the various sorting algorithms
CS8451 DESIGN AND ANALYSIS OF ALGORITHMS
OBJECTIVES:
To understand and apply the algorithm analysis techniques
To critically analyze the efficiency of alternative algorithmic solutions for the same problem
To understand different algorithm design techniques
To understand the limitations of Algorithmic power
UNIT I INTRODUCTION
Notion of an Algorithm – Fundamentals of Algorithmic Problem Solving – Important Problem Types –
Fundamentals of the Analysis of Algorithmic Efficiency –Asymptotic Notations and their properties.
Analysis Framework – Empirical analysis - Mathematical analysis for Recursive and Non-recursive
algorithms - Visualization
UNIT II BRUTE FORCE AND DIVIDE-AND-CONQUER
Brute Force – Computing an – String Matching - Closest-Pair and Convex-Hull Problems - Exhaustive
Search - Travelling Salesman Problem - Knapsack Problem - Assignment problem. Divide and Conquer
Methodology – Binary Search – Merge sort – Quick sort – Heap Sort - Multiplication of Large Integers –
Closest-Pair and Convex - Hull Problems.
UNIT III DYNAMIC PROGRAMMING AND GREEDY TECHNIQUE
Dynamic programming – Principle of optimality - Coin changing problem, Computing a Binomial
Coefficient – Floyd‘s algorithm – Multi stage graph - Optimal Binary Search Trees – Knapsack Problem
and Memory functions. Greedy Technique – Container loading problem - Prim‘s algorithm and Kruskal's
Algorithm – 0/1 Knapsack problem, Optimal Merge pattern - Huffman Trees.
UNIT IV ITERATIVE IMPROVEMENT The Simplex
Method - The Maximum-Flow Problem – Maximum Matching in Bipartite Graphs, Stable marriage
Problem.
UNIT V COPING WITH THE LIMITATIONS OF ALGORITHM POWER Lower - Bound
Arguments - P, NP NP- Complete and NP Hard Problems. Backtracking – n-Queen problem -
Hamiltonian Circuit Problem – Subset Sum Problem. Branch and Bound – LIFO Search and FIFO search
- Assignment problem – Knapsack Problem – Travelling Salesman Problem - Approximation Algorithms
for NP-Hard Problems – Travelling Salesman problem – Knapsack problem.
OUTCOMES:
At the end of the course, the students should be able to:
Design algorithms for various computing problems
Analyze the time and space complexity of algorithms
Critically analyze the different algorithm design techniques for a given problem
Modify existing algorithms to improve efficiency
GATE QUESTIONS - DATA STRUCTURES AND ALGORITHMS
INDEX
S. No. Name of the Topic Number of
Questions Page Number
1 Stacks 4 1
2 Queues 4 3
3 Linked Lists 5 5
4
Trees
4.1 Trees
4.2 Binary Trees
4.3 Binary Search Trees
4.4 n- ary Trees
4.5 B Trees
4.6 B+ Trees
4.7 Expression Trees
4.8 AVL Trees
28 7
5 Heaps
14 19
6
Graphs
6.1 Graph Traversal
6.2 Minimum Spanning Tree
6.3 Shortest Path
34 26
7 Searching 2 42
8 Sorting 12 43
9 Hashing 7 47
10 Asymptotic Worst Case Time
and Space Complexity 11 50
11
Algorithm Design Techniques:
11.1 Greedy Algorithms
11.2 Dynamic Programming
11.3 Divide‐and‐Conquer
Technique
8 54
TOTAL NUMBER OF QUESTIONS: 109
Faculty In - Charge GATE Coordinator HoD/IT
GATE – DATA STRUCTURES AND ALGORITHMS Page 1
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
STACK
1. The result evaluating the postfix expression 10 5 + 60 6 / * 8 – is GATE-CS-2015 (Set 3)
(A) 284
(B) 213
(C) 142
(D) 71
2. The following postfix expression with single digit operands is evaluated using a stack:
8 2 3 ^ / 2 3 * + 5 1 * -
Note that ^ is the exponentiation operator. The top two elements of the stack after the first * is
evaluated are: GATE CS 2007
(A) 6, 1
(B) 5, 7
(C) 3, 2
(D) 1, 5
GATE – DATA STRUCTURES AND ALGORITHMS Page 2
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
STACK
3. A single array A[1..MAXSIZE] is used to implement two stacks. The two stacks grow from
opposite ends of the array. Variables top1 and top2 (topl< top 2) point to the location of the
topmost element in each of the stacks. If the space is to be used efficiently, the condition for
“stack full” is (GATE CS 2004)
a) (top1 = MAXSIZE/2) and (top2 = MAXSIZE/2+1)
b) top1 + top2 = MAXSIZE
c) (top1= MAXSIZE/2) or (top2 = MAXSIZE)
d) top1= top2 -1
4. The best data structure to check whether an arithmetic expression has balanced
parentheses is a (GATE CS 2004)
a) queue
b) stack
c) tree
d) list
GATE – DATA STRUCTURES AND ALGORITHMS Page 3
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
QUEUE
5. 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.
6. A queue is implemented using a non-circular singly linked list. The queue has a head pointer
and a tail pointer, as shown in the figure. Let n denote the number of nodes in the queue. Let
‘enqueue’ be implemented by inserting a new node at the head, and ‘dequeue’ be
implemented by deletion of a node from the tail.
Which one of the following is the time complexity of the most time-efficient implementation
of ‘enqueue’ and ‘dequeue, respectively, for this data structure? GATE CS 2018
(A) Θ(1), Θ(1)
(B) Θ(1), Θ(n)
(C) Θ(n), Θ(1)
(D) Θ(n), Θ(n)
GATE – DATA STRUCTURES AND ALGORITHMS Page 4
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
CIRCULAR QUEUE
7. A Circular queue has been implemented using singly linked list where each node consists of a
value and a pointer to next node. We maintain exactly two
pointers FRONT and REAR pointing to the front node and rear node of queue. Which of the
following statements is/are correct for circular queue so that insertion and deletion operations
can be performed in O(1) i.e. constant time. GATE-CS-2017 (Set 2)
I. Next pointer of front node points to the rear node.
II. Next pointer of rear node points to the front node.
A. I only
B. II only
C. Both I and II
D. Neither I nor II
8. 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 (GATE CS 2012)
(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
GATE – DATA STRUCTURES AND ALGORITHMS Page 5
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
LINKED LIST
9. An unordered list contains n distinct elements. The number of comparisons to find an element
in this list that is neither maximum nor minimum is GATE-CS-2015 (Set 2)
(A) Θ(nlogn)
(B) Θ(n)
(C) Θ(logn)
(D) Θ(1)
10. In the worst case, the number of comparisons needed to search a singly linked list of
length n for a given element is (GATE CS 2002)
a) log 2 n
b) n/2
c) log 2 n – 1
d) n
11. Suppose each set is represented as a linked list with elements in arbitrary order. Which of the
operations among union, intersection, membership, cardinality will be the slowest? (GATE
CS 2004)
a) union only
b) intersection, membership
c) membership, cardinality
d) union, intersection
GATE – DATA STRUCTURES AND ALGORITHMS Page 6
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
LINKED LIST
12. The following C function takes a simply-linked list as input argument. It modifies the list by
moving the last element to the front of the list and returns the modified list. Some part of the
code is left blank.
typedef struct node
{
int value;
struct node *next;
}Node;
Node *move_to_front(Node *head)
{
Node *p, *q;
if ((head == NULL: || (head->next == NULL))
return head;
q = NULL; p = head;
while (p-> next !=NULL)
{
q = p;
p = p->next;
}
_______________________________
return head;
}
Choose the correct alternative to replace the blank line. GATE CS 2010
(A) q = NULL; p->next = head; head = p;
(B) q->next = NULL; head = p; p->next = head;
(C) head = p; p->next = q; q->next = NULL;
(D) q->next = NULL; p->next = head; head = p;
GATE – DATA STRUCTURES AND ALGORITHMS Page 7
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
TREES - STRUCTURE
13. Let T be a tree with 10 vertices. The sum of the degrees of all the vertices in T is _____.
GATE-CS-2017 (Set 1)
Note: This questions appeared as Numerical Answer Type.
(A) 18
(B) 19
(C) 20
(D) 21
14. The height of a tree is the length of the longest root-to-leaf path in it. The maximum and
minimum number of nodes in a binary tree of height 5 are GATE-CS-2015 (Set 1)
(A) 63 and 6, respectively
(B) 64 and 5, respectively
(C) 32 and 6, respectively
(D) 31 and 5, respectively
GATE – DATA STRUCTURES AND ALGORITHMS Page 8
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
BINARY TREES - STRUCTURE
15. A binary tree T has 20 leaves. The number of nodes in T having two children is GATE-CS-
2015 (Set 2)
(A) 18
(B) 19
(C) 17
(D) Any number between 10 and 20
16. Consider a binary tree T that has 200 leaf nodes. Then, the number of nodes in T that have
exactly two children are _________. GATE-CS-2015 (Set 3)
(A) 199
(B) 200
(C) Any number between 0 and 199
(D) Any number between 100 and 200
17. In a binary tree with n nodes, every node has an odd number of descendants. Every node is
considered to be its own descendant. What is the number of nodes in the tree that have exactly
one child? GATE CS 2010
(A) 0
(B) 1
(C) (n-1)/2
(D) n-1
GATE – DATA STRUCTURES AND ALGORITHMS Page 9
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
BINARY TREES - STRUCTURE
18. The maximum number of binary trees that can be formed with three unlabeled nodes is:
GATE CS 2007
(A) 1
(B) 5
(C) 4
(D) 3
19. 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: GATE CS 2007
(A) 2^h -1
(B) 2^(h-1) – 1
(C) 2^(h+1) -1
(D) 2*(h+1)
GATE – DATA STRUCTURES AND ALGORITHMS Page 10
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
BINARY TREES - TRAVERSAL
20. The inorder and preorder traversal of a binary tree are d b e a f c g and a b d e c f g,
respectively. The postorder traversal of the binary tree is: GATE CS 2007
(A) d e b f g c a
(B) e d b g f c a
(C) e d b f g c a
(D) d e f g b c a
21. Let LASPOST, LASTIN and LASTPRE denote the last vertex visited in a postorder, inorder
and preorder traversal. Respectively, of a complete binary tree. Which of the following is
always true? (GATE CS 2000)
(a) LASTIN = LASTPOST
(b) LASTIN = LASTPRE
(c) LASTPRE = LASTPOST
(d) None of the above
22. Consider the following nested representation of binary trees: (X Y Z) indicates Y and Z are
the left and right sub stress, respectively, of node X. Note that Y and Z may be NULL, or
further nested. Which of the following represents a valid binary tree? (GATE CS 2000)
(a) (1 2 (4 5 6 7))
(b) (1 (2 3 4) 5 6) 7)
(c) (1 (2 3 4)(5 6 7))
(d) (1 (2 3 NULL) (4 5))
GATE – DATA STRUCTURES AND ALGORITHMS Page 11
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
BINARY SEARCH TREES - CREATION
23. While inserting the elements 71, 65, 84, 69, 67, 83 in an empty binary search tree (BST) in
the sequence shown, the element in the lowest level is GATE-CS-2015 (Set 3)
A. 65
B. 67
C. 69
D. 83
GATE – DATA STRUCTURES AND ALGORITHMS Page 12
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
BINARY SEARCH TREES - STRUCTURE
24. Let T be a full binary tree with 8 leaves. (A full binary tree has every level full.) Suppose two
leaves aa and b of T are chosen uniformly and independently at random. The expected value
of the distance between aa and b in T (i.e., the number of edges in the unique path
between aa and b) is (rounded off to 2 decimal places) . (GATE 2019)
25. Let T be a binary search tree with 15 nodes. The minimum and maximum possible heights of
T are: GATE-CS-2017 (Set 1)
Note: The height of a tree with a single node is 0.
(A) 4 and 15 respectively
(B) 3 and 14 respectively
(C) 4 and 14 respectively
(D) 3 and 15 respectively
26. Breadth First Search (BFS) is started on a binary tree beginning from the root vertex. There is
a vertex t at a distance four from the root. If t is the n-th vertex in this BFS traversal, then the
maximum possible value of n is ________ GATE-CS-2016 (Set 2)
[This Question was originally a Fill-in-the-blanks Question]
(A) 15
(B) 16
(C) 31
(D) 32
GATE – DATA STRUCTURES AND ALGORITHMS Page 13
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
BINARY SEARCH TREES - TRAVERSAL
27. The preorder traversal of a binary search tree is 15, 10, 12, 11, 20, 18, 16, 19.Which one of
the following is the postorder traversal of the tree? (GATE 2019)
(A) 10, 11, 12, 15, 16, 18, 19, 20
(B) 11, 12, 10, 16, 19, 18, 20, 15
(C) 20, 19, 18, 16, 15, 12, 11, 10
(D) 19, 16, 18, 20, 11, 12, 10, 15
28. The preorder traversal of a binary search tree is 15, 10. 12, 11, 20, 18, 16, 19. Which one of
the following is the postorder traversal of the tree? (GATE 2020)
(a) 20, 19, 18, 16, 15, 12, 11, 10
(b) 10, 11, 12, 15, 16, 18, 19, 20
(c) 19, 16, 18, 20, 11, 12, 10, 15
(d) 11, 12, 10, 16, 19, 18, 20, 15
29. Postorder traversal of a given binary search tree, T produces the following sequence of keys
10, 9, 23, 22, 27, 25, 15, 50, 95, 60, 40, 29
Which one of the following sequences of keys can be the result of an in-order traversal of the
tree T? (GATE CS 2005)
a) 9, 10, 15, 22, 23, 25, 27, 29, 40, 50, 60, 95
b) 9, 10, 15, 22, 40, 50, 60, 95, 23, 25, 27, 29
c) 29, 15, 9, 10, 25, 22, 23, 27, 40, 60, 50, 95
d) 95, 50, 60, 40, 27, 23, 22, 25, 10, 9, 15, 29
30. 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? (GATE CS
2013)
(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
GATE – DATA STRUCTURES AND ALGORITHMS Page 14
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
N-ARY TREES - STRUCTURE
31. 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: GATE CS 2005
(a) nk
(b) (n – 1) k+ 1
(c) n( k – 1) + 1
(d) n(k – 1)
32. 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
GATE – DATA STRUCTURES AND ALGORITHMS Page 15
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Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
B TREES AND B+ TREES
33. Which one of the following is a key factor for preferring B-trees to binary search trees for
indexing database relations? GATE CS 2005
(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.
34. B+ Trees are considered BALANCED because GATE-CS-2016 (Set 2)
A. the lengths of the paths from the root to all leaf nodes are all equal.
B. the lengths of the paths from the root to all leaf nodes differ from each other by at
most 1.
C. the number of children of any two non-leaf sibling nodes differ by at most 1.
D. the number of records in any two leaf nodes differ by at most 1.
GATE – DATA STRUCTURES AND ALGORITHMS Page 16
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Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
B+ TREES
35. With reference to the B+ tree index of order 1 shown below, the minimum number of nodes
(including the root node) that must be fetched in order to satisfy the following query: “Get all
records with a search key greater than or equal to 7 and less than 15” is ________ GATE-CS-
2015 (Set 2)
(A) 4
(B) 5
(C) 6
(D) 7
36. Consider a B+-tree in which the maximum number of keys in a node is 5. What is the
minimum number of keys in any non-root node? (GATE CS 2010)
(A) 1
(B) 2
(C) 3
(D) 4
GATE – DATA STRUCTURES AND ALGORITHMS Page 17
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
EXPRESSION TREES & AVL TREES
37. Consider the expression tree shown. Each leaf represents a numerical value, which can either
be 0 or 1. Over all possible choices of the values at the leaves, the maximum possible value of
the expression represented by the tree is ___. GATE-CS-2014-(Set-2)
(A) 4
(B) 6
(C) 8
(D) 10
38. What is the worst case time complexity of inserting n2 elements into an AVL-tree
with n elements intially ?(GATE 2019 & GATE 2020)
GATE – DATA STRUCTURES AND ALGORITHMS Page 18
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
EXPRESSION TREES & AVL TREES
39. In a balanced binary search tree with n elements, what is the worst case time complexity of
reporting all elements in range [a, b]? Assume that the number of reported elements is k
(GATE 2020)
40. What is the maximum height of any AVL-tree with 7 nodes? Assume that the height of a tree
with a single node is 0. GATE-CS-2009
(A) 2
(B) 3
(C) 4
(D) 5
GATE – DATA STRUCTURES AND ALGORITHMS Page 19
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Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HEAP - TRAVERSAL
41. 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: GATE-
CS-2014-(Set-2)
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
42. 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: GATE CS 2005
(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
GATE – DATA STRUCTURES AND ALGORITHMS Page 20
Date : Hour : Mark : /10
Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HEAP - CREATION
43. Consider a max heap, represented by the array: 40, 30, 20, 10, 15, 16, 17, 8, 4. Now consider
that a value 35 is inserted into this heap. After insertion, the new heap is GATE-CS-2015
(Set 1)
(A) 40, 30, 20, 10, 15, 16, 17, 8, 4, 35
(B) 40, 35, 20, 10, 30, 16, 17, 8, 4, 15
(C) 40, 30, 20, 10, 35, 16, 17, 8, 4, 15
(D) 40, 35, 20, 10, 15, 16, 17, 8, 4, 30
GATE – DATA STRUCTURES AND ALGORITHMS Page 21
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Signature of the Student Mentor : Signature of Subject In – Charge:
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HEAP - IDENTIFICATION
44. The number of possible min-heaps containing each value from {1, 2, 3, 4, 5, 6, 7}exactly
once is _____. (GATE 2018)
45. Consider the following statements:
I. The smallest element in a max-heap is always at a leaf node
II. The second largest element in a max-heap is always a child of the root node
III. A max-heap can be constructed from a binary search tree in Θ(n)Θ(n) time
IV. A binary search tree can be constructed from a max-heap in Θ(n)Θ(n) time
Which of the above statements are TRUE? (GATE 2019)
46. Consider the array representation of a binary min-heap containing 1023 elements. The
minimum number of comparisons required to find the maximum in the heap ________.
(GATE 2020)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HEAP - IDENTIFICATION
47. Consider a binary max-heap implemented using an array. Which one of the following array
represents a binary max-heap? GATE-CS-2009
▪ 25,12,16,13,10,8,14
▪ 25,14,13,16,10,8,12
▪ 25,14,16,13,10,8,12
▪ 25,14,12,13,10,8,16
48. 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? GATE CS 2011
(A) A
(B) B
(C) C
(D) D
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HEAP – SEARCHING TIME COMPLEXITY
49. Consider the process of inserting an element into a Max Heap, where the Max Heap is
represented by an array. Suppose we perform a binary search on the path from the new leaf to
the root to find the position for the newly inserted element, the number of comparisons
performed is: (GATE CS 2007)
(A) Θ(logn)
(B) Θ(LogLogn )
(C) Θ(n)
(D) Θ(nLogn)
50. In a binary max heap containing n numbers, the smallest element can be found in time
(GATE CS 2006)
(A) O(n)
(B) O(logn)
(C) O(loglogn)
(D) O(1)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HEAP – MAXIMUM DEPTH
51. A complete binary min-heap is made by including each integer in [1, 1023] exactly once. The
depth of a node in the heap is the length of the path from the root of the heap to that node.
Thus, the root is at depth 0. The maximum depth at which integer 9 can appear is
_____________ GATE-CS-2016 (Set 2)
[This question was originally asked as Fill-in-the-Blanks question]
(A) 6
(B) 7
(C) 8
(D) 9
52. The number of possible min-heaps containing each value from {1, 2, 3, 4, 5, 6, 7} exactly
once is _______. GATE CS 2018
Note –This was Numerical Type question.
(A) 80
(B) 8
(C) 20
(D) 210
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HEAP – n ary HEAP
53. 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? GATE CS 2006
(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
54. Suppose the elements 7, 2, 10 and 4 are inserted, in that order, into the valid 3- ary max heap
found in the above question, Which one of the following is the sequence of items in the array
representing the resultant heap? GATE CS 2006
(A) 10, 7, 9, 8, 3, 1, 5, 2, 6, 4
(B) 10, 9, 8, 7, 6, 5, 4, 3, 2, 1
(C) 10, 9, 4, 5, 7, 6, 8, 2, 1, 3
(D) 10, 8, 6, 9, 7, 2, 3, 4, 1, 5
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS – NUMBER OF VERTICES/ EDGES/ GRAPHS
55. G is undirected graph with n vertices and 25 edges such that each vertex has degree at least 3.
Then the maximum possible value of n is ________ GATE-CS-2017 (Set 2)
(A) 4
(B) 8
(C) 16
(D) 24
56. Let G be a connected planar graph with 10 vertices. If the number of edges on each face is
three, then the number of edges in G is _______________. GATE-CS-2015 (Set 1)
(A) 24
(B) 20
(C) 32
(D) 64
57. 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 ? (GATE CS 2001)
a) n(n-l)/2
b) 2^n
c) n!
d) 2^(n(n-1)/2)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS –TRAVERSAL (BFS)
58. Breath First Search(BFS) has been implemented using queue data structure.
Which one of the following is a possible order of visiting the nodes in the graph above.
GATE-CS-2017 (Set 2)
(A) MNOPQR
(B) NQMPOR
(C) QMNROP
(D) POQNMR
59. 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 GATE CS 2008
(A) MNOPQR
(B) NQMPOR
(C) QMNPRO
(D) QMNPOR
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS –TRAVERSAL (BFS)
60. 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? (GATE CS 2001)
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
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS – TRAVERSAL (DFS)
61. Suppose depth first search is executed on the graph below starting at some unknown vertex.
Assume that a recursive call to visit a vertex is made only after first checking that the vertex
has not been visited earlier. Then the maximum possible recursion depth (including the initial
call) is _________. GATE-CS-2014-(Set-3)
(A) 17
(B) 18
(C) 19
(D) 20
62. Consider the following graph
Among the following sequences
I) a b e g h f
II) a b f e h g
III) a b f h g e
IV) a f g h b e
Which are depth first traversals of the above graph? (GATE CS 2003)
a) I, II and IV only
b) I and IV only
c) II, III and IV only
d) I, III and IV only
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS – GRAPHS – TRAVERSAL (DFS)
63. 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?
(GATE CS 2000)
(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
64. 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. CS-2014-(Set-1)
(A) O(n)
(B) O(m+n)
(C) O(n2)
(D) O(mn)
65. Let GG be an undirected complete graph on nn vertices, where nn > 2. Then, the number of
different Hamiltonian cycles in GG is equal to (GATE 2019)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS – TOPOLOGICAL SORT
66. Consider the following directed graph.
The number of different topological orderings of the vertices of the graph is GATE-CS-2016
(Set 1)
Note : This question was asked as Numerical Answer Type.
(A) 1
(B) 2
(C) 4
(D) 6
67. Consider the directed graph given below. Which one of the following is TRUE? CS-2014-
(Set-1)
(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
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS – MINIMUM SPANNING TREE- CONSTRUCTION
68. Consider the following graph:
Which one of the following is NOT the sequence of edges added to the minimum spanning tree
using Kruskal’s algorithm? GATE-CS-2009
(A) (b,e)(e,f)(a,c)(b,c)(f,g)(c,d)
(B) (b,e)(e,f)(a,c)(f,g)(b,c)(c,d)
(C) (b,e)(a,c)(e,f)(b,c)(f,g)(c,d)
(D) (b,e)(e,f)(b,c)(a,c)(f,g)(c,d)
69. Consider the following undirected graph G:
Choose a value for x that will maximize the number of minimum weight spanning trees
(MWSTs) of G. The number of MWSTs of G for this value of x is ______.(GATE 2018)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS – MINIMUM SPANNING TREE- CONSTRUCTION
70. Let G be any connected, weighted, undirected graph.
I. G has a unique minimum spanning tree, if no two edges of G have the same weight.
II. G has a unique minimum spanning tree, if, for every cut of G, there is a unique minimum-
weight edge crossing the cut. Which of the above two statements is/are TRUE?
71. Let G = (V, E ) be a weighted undirected graph and let T be a Minimum Spanning Tree
(MST) of G maintained using adjacency lists. Suppose a new weighted edge (u, v) ∈ V × V is
added to G. The worst case time complexity of determining if T is still an MST of the
resultant graph is (GATE 2020)
72. Consider a graph G = (V, E), where V = {v2, v2, ..., v100}, E = {(vi , vj )⏐1 ≤ i < j ≤ 100}
and weight of the edge (vi , vj ) is ⏐i – j⏐. The weight of minimum spanning tree of G is
________. (GATE 2020)
73. 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)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS –MINIMUM SPANNING TREE – NUMBER OF TREES
74. Let G be a complete undirected graph on 4 vertices, having 6 edges with weights being 1, 2,
3, 4, 5, and 6. The maximum possible weight that a minimum weight spanning tree of G can
have is. GATE-CS-2016 (Set 1)
[This Question was originally a Fill-in-the-Blanks question]
(A) 6
(B) 7
(C) 8
(D) 9
75. The number of distinct minimum spanning trees for the weighted graph below is ____
GATE-CS-2014-(Set-2)
(A) 4
(B) 5
(C) 6
(D) 7
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS –MINIMUM SPANNING TREE- EDGE WEIGHT
76. Consider the following undirected graph G:
Choose a value for x that will maximize the number of minimum weight spanning trees
(MWSTs) of G. The number of MWSTs of G for this value of x is _________ . GATE CS
2018
Note –This was Numerical Type question.
(A) 4
(B) 5
(C) 2
(D) 3
77. The graph shown below 8 edges with distinct integer edge weights. The minimum spanning
tree (MST) is of weight 36 and contains the edges: {(A, C), (B, C), (B, E), (E, F), (D, F)}.
The edge weights of only those edges which are in the MST are given in the figure shown
below. The minimum possible sum of weights of all 8 edges of this graph is
______________. GATE-CS-2015 (Set 1)
(A) 66
(B) 69
(C) 68
(D) 70
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS – MINIMUM SPANNING TREE – EDGE WEIGHT
78. Let G be a weighted connected undirected graph with distinct positive edge weights. If every
edge weight is increased by the same value, then which of the following statements is/are
TRUE? GATE-CS-2016 (Set 1)
P: Minimum spanning tree of G does not change
Q: Shortest path between any pair of vertices does not change
(A) P only
(B) Q only
(C) Neither P nor Q
(D) Both P and Q
79. Let G be connected undirected graph of 100 vertices and 300 edges. The weight of a
minimum spanning tree of G is 500. When the weight of each edge of G is increased by five,
the weight of a minimum spanning tree becomes ________. GATE-CS-2015 (Set 3)
(A) 1000
(B) 995
(C) 2000
(D) 1995
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS – COST OF MINIMUM SPANNING TREE
80. 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?
GATE CS 2011
(A) 1/12(11n^2 – 5n)
(B) n^2 – n + 1
(C) 6n – 11
(D) 2n + 1
81. Consider a weighted complete graph G on the vertex set {v1, v2, ..vn} such that the weight of
the edge (vi, vj) is 2|i-j|. The weight of a minimum spanning tree of G is: (GATE CS 2006)
(A) n — 1
(B) 2n — 2
(C) nC2
(D) 2
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS –MINIMUM SPANNING TREE
82. 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? GATE CS 2010
(A) 7
(B) 8
(C) 9
(D) 10
83. In the graph given in above question, 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? GATE CS 2010
(A) 7
(B) 8
(C) 9
(D) 10
84. G = (V, E) is an undirected simple graph in which each edge has a distinct weight, and e is a
particular edge of G. Which of the following statements about the minimum spanning trees
(MSTs) of G is/are TRUE GATE-CS-2016 (Set 1)
If e is the lightest edge of some cycle in G, then every MST of G includes e
If e is the heaviest edge of some cycle in G, then every MST of G excludes e
(A) I only
(B) II only
(C) both I and II
(D) neither I nor II
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GATE QUESTIONS DATA STRUCTURES AND ALGORITHMS
GRAPHS –SHORTEST PATH
85. Consider the weighted undirected graph with 4 vertices, where the weight of edge {i, j} g is
given by the entry Wij in the matrix W
The largest possible integer value of x, for which at least one shortest path between some pair
of vertices will contain the edge with weight x is ________ GATE-CS-2016 (Set 1)
Note : This question was asked as Numerical Answer Type.
(A) 8
(B) 12
(C) 10
(D) 11
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS –SHORTEST PATH- DIJKSTRA’S ALGORITHM
86. Suppose we run Dijkstra’s single source shortest-path algorithm on the following edge
weighted directed graph with vertex P as the source. In what order do the nodes get included
into the set of vertices for which the shortest path distances are finalized? (GATE CS 2004)
a) P, Q, R, S, T, U
b) P, Q, R, U, S, T
c) P, Q, R, U, T, S
d) P, Q, T, R, U, S
87. There are multiple routes to reach from node 1 to node 2, as shown in the network.
The cost of travel on an edge between two nodes is given in rupees. Nodes ‘a’ ‘b’, ‘c’, ‘d’, ‘e’
and ‘f’ are toll booths. The toll price at toll booths marked ‘a’ and ‘c’ is Rs. 200. and is Rs.
100 for the other toll booths. Which is the cheapest route from node 1 to node 2? (GATE
2020)
(a) 1-a-C-2
(b) 1-b-2
(c) 1-f-b-2
(d) 1-f-e-2
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
GRAPHS –SHORTEST PATH- DIJKSTRA’S ALGORITHM
88. 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
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
SEARCHING
89. The minimum number of comparisons required to find the minimum and the maximum of
100 numbers is ______________. CS-2014-(Set-1)
(A) 148
(B) 147
(C) 146
(D) 140
90. An element in an array X is called a leader if it is greater than all elements to the right of it in
X. The best algorithm to find all leaders in an array (GATE CS 2006)
(A) Solves it in linear time using a left to right pass of the array
(B) Solves it in linear time using a right to left pass of the array
(C) Solves it using divide and conquer in time 8(nlogn)
(D) Solves it in time 8(n2)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
SORTING- SELECTION SORT
91. Which one of the following is the tightest upper bound that represents the number of swaps
required to sort n numbers using selection sort? GATE CS 2013
(A) O(log n)
(B) O(n)
(C) O(nLogn)
(D) O(n^2)
92. What is the number of swaps required to sort n elements using selection sort, in the
worst case? GATE CS 2009
(A) Θ(n)
(B) Θ(n log n)
(C) Θ(n2 )
(D) Θ(nn2 log n)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
SORTING – MERGE SORT
93. 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 ____. GATE-CS-2014-(Set-2)
(A) 358
(B) 438
(C) 568
(D) 664
94. Assume that a mergesort algorithm in the worst case takes 30 seconds for an input of size 64.
Which of the following most closely approximates the maximum input size of a problem that
can be solved in 6 minutes? GATE-CS-2015 (Set 3)
(A) 256
(B) 512
(C) 1024
(D) 2048
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SORTING – QUICK SORT
95. An array of 25 distinct elements is to be sorted using quicksort. Assume that the pivot element
is chosen uniformly at random. The probability that the pivot element gets placed in the worst
possible location in the first round of partitioning (rounded off to 2 decimal places) is
________.(GATE 2019)
96. 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? CS-2014-(Set-1)
(A) t1 = 5
(B) t1 < t2
(C) t1 > t2
(D) t1 = t2
97. 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?
<pre> GATE-CS-2009
(A) (n)
(B) (nLogn)
(C) (n^2)
(D) (n^2 log n) </pre>
(A) A
(B) B
(C) C
(D) D
98. 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)
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
SORTING - COMPARISON
99. Assume that the algorithms considered here sort the input sequences in ascending order. If the
input is already in ascending order, which of the following are TRUE ? GATE-CS-2016 (Set
2)
I. Quicksort runs in Θ(n2) time
II. Bubblesort runs in Θ(n2) time
III. Mergesort runs in Θ(n) time
IV. Insertion sort runs in Θ(n) time
(A) I and II only
(B) I and III only
(C) II and IV only
(D) I and IV only
100. The worst case running times of Insertion sort, Merge sort and Quick sort,
respectively, are: GATE-CS-2016 (Set 1)
(A) Θ(n log n), Θ(n log n) and Θ(n2)
(B) Θ(n2), Θ(n2) and Θ(n Log n)
(C) Θ(n2), Θ(n log n) and Θ(n log n)
(D) Θ(n2), Θ(n log n) and Θ(n2)
101. Which one of the following in place sorting algorithms needs the minimum number
of swaps? (GATE CS 2006)
(A) Quick sort
(B) Insertion sort
(C) Selection sort
(D) Heap sort
102. Which of the following sorting algorithms has the lowest worst-case
complexity?GATE CS 2007
(A) Merge sort
(B) Bubble sort
(C) Quick sort
(D) Selection sort
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DATA STRUCTURES AND ALGORITHMS
HASHING
103. 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? GATE CS 2010
104. 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
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
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HASHING
105. Given the following input (4322, 1334, 1471, 9679, 1989, 6171, 6173, 4199) and the
hash function x mod 10, which of the following statements are true? (GATE CS 2004)
i. 9679, 1989, 4199 hash to the same value
ii. 1471, 6171 hash to the same value
iii. All elements hash to the same value
iv. Each element hashes to a different value
A. i only
B. ii only
C. i and ii only
D. iii or iv
106. The keys 12, 18, 13, 2, 3, 23, 5 and 15 are inserted into an initially empty hash table
of length 10 using open addressing with hash function h(k) = k mod 10 and linear probing.
What is the resultant hash table? GATE-CS-2009
(A) A
(B) B
(C) C
(D) D
107. Consider a hash table of size seven, with starting index zero, and a hash function (3x
+ 4)mod7. Assuming the hash table is initially empty, which of the following is the contents
of the table when the sequence 1, 3, 8, 10 is inserted into the table using closed hashing? Note
that ‘_’ denotes an empty location in the table. GATE CS 2007
(A) 8, _, _, _, _, _, 10
(B) 1, 8, 10, _, _, _, 3
(C) 1, _, _, _, _, _,3
(D) 1, 10, 8, _, _, _, 3
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
HASHING
108. Which one of the following hash functions on integers will distribute keys most
uniformly over 10 buckets numbered 0 to 9 for i ranging from 0 to 2020? GATE-CS-2015
(Set 2)
(A) h(i) =i2 mod 10
(B) h(i) =i3 mod 10
(C) h(i) = (11 ∗ i2) mod 10
(D) h(i) = (12 ∗ i) mod 10
109. Given a hash table T with 25 slots that stores 2000 elements, the load factor α for T is
__________ GATE-CS-2015 (Set 3)
(A) 80
(B) 0.0125
(C) 8000
(D) 1.25
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GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
ASYMPTOTIC WORST CASE TIME AND SPACE COMPLEXITY
110. What is the worst case time complexity of inserting n elements into an empty linked list, if
the linked list needs to be maintained in sorted order? (GATE 2020)
(a) Θ(n logn)
(b) Θ(n)
(c) Θ(1)
(d) Θ(n2)
111. Which one of the following correctly determines the solution of the recurrence relation with
T(1) = 1 where T(n) = 2T(n/2) + Log n ? GATE-CS-2014-(Set-2)
(A) Θ(n)
(B) Θ(nLogn)
(C) Θ(n*n)
(D) Θ(log n)
112. For parameters aa and bb, both of which are ω(1), T(n)=T(n1/a)+1,andT(b)=1.
ThenT(n)is (GATE 2019 & GATE 2020)
113. The recurrence relation capturing the optimal execution time of the Towers of Hanoi
problem with n discs is (GATE CS 2012)
(E) T(n) = 2T(n − 2) + 2
(F) T(n) = 2T(n − 1) + n
(G) T(n) = 2T(n/2) + 1
(H) T(n) = 2T(n − 1) + 1
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GATE QUESTIONS DATA STRUCTURES AND ALGORITHMS
ASYMPTOTIC WORST CASE TIME AND SPACE COMPLEXITY
114. The running time of an algorithm is represented by the following recurrence relation:
if n <= 3 then T(n) = n
else T(n) = T(n/3) + cn
Which one of the following represents the time complexity of the algorithm? <pre> GATE-
CS-2009
(A) (n)
(B) (n log n)
(C) (n^2)
(D) (n^2log n) </pre>
(A) A
(B) B
(C) C
(D) D
115. Consider the following C function.
int fun1 (int n)
{
int i, j, k, p, q = 0;
for (i = 1; i<n; ++i)
{
p = 0;
for (j=n; j>1; j=j/2)
++p;
for (k=1; k<p; k=k*2)
++q;
}
return q;
} Which one of the following most closely approximates the return value of the function fun1?
GATE-CS-2015 (Set 1)
(A) n3
(B) n (logn)2
(C) nlogn
(D) nlog(logn)
GATE QUESTIONS
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DATA STRUCTURES AND ALGORITHMS
ASYMPTOTIC WORST CASE TIME AND SPACE COMPLEXITY
116. Which of the given options provides the increasing order of asymptotic complexity of
functions f1, f2, f3 and f4? GATE CS 2011
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
117. Consider the following three claims
I (n + k)^m = Θ(n^m), where k and m are constants
II 2^(n + 1) = 0(2^n)
III 2^(2n + 1) = 0(2^n)
Which of these claims are correct? (GATE CS 2003)
(a) I and II
(b) I and III
(c) II and III
(d) I, II and III
GATE QUESTIONS
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DATA STRUCTURES AND ALGORITHMS
ASYMPTOTIC WORST CASE TIME AND SPACE COMPLEXITY (Number of Operations)
118. Consider the following pseudo code. What is the total number of multiplications to be
performed? CS-2014-(Set-1)
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.
119. There are n unsorted arrays: A1, A2, …, An. Assume that n is odd. Each of A1, A2,
…, An contains n distinct elements. There are no common elements between any two arrays.
The worst-case time complexity of computing the median of the medians of A1, A2, …, An is
(GATE 2019)
120. 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. GATE-
CS-2014-(Set-3)
(A) 6
(B) 7
(C) 8
(D) 9
GATE QUESTIONS
DATA STRUCTURES AND ALGORITHMS
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ALGORITHM DESIGN TECHNIQUES
121. Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the
edges in E are positive any distinct. Consider the following statements:
I. Minimum Spanning Tree of G is always unique.
II. Shortest path between any two vertices of G is always unique.
Which of the above statements is/are necessarily true? GATE-CS-2017 (Set 1)
(A) I only
(B) II only
(C) both I and II
(D) neither I and II
122. Let G be a simple undirected graph. Let TD be a depth first search tree of G.
Let TB be a breadth first search tree of G. Consider the following statements:
(I) No edge of G is a cross edge with respect to TD. (A cross edge in G is between two nodes
neither of which is an ancestor of the other in TD.)
(II) For every edge (u,v) of G, if u is at depth i and v is at depth j in TB, then |i-j| = 1.
Which of the statements above must necessarily be true? (GATE 2018)
123. Consider the weights and values of items listed below. Note that there is only one unit
of each item.
Item
number
Weight
(in Kgs)
Value
(in Rupees)
1 10 60
2 7 28
3 4 20
4 2 24
The task is to pick a subset of these items such that their total weight is no more than 11 Kg
sand their total value is maximized. Moreover, no item may be split. The total value of items
picked by an optimal algorithm is denoted by Vopt. A greedy algorithm sorts the items by
their value-to-weight ratios in descending order and packs them greedily, starting from the
first item in the ordered list. The total value of items picked by the greedy algorithm is
denoted by Vgreedy.
The value of Vopt − Vgreedy is ____________. (GATE 2018)
124. Consider a sequence of 14 elements: A = [−5, −10, 6, 3, −1, −2, 13, 4, −9, −1, 4, 12,
−3, 0]. The subsequence sum S(i,j)=∑jk=iA[k]S(i,j)=∑k=ijA[k] . Determine the maximum
GATE – DATA STRUCTURES AND ALGORITHMS Page 55
of S(i,j),S(i,j), where 0≤i≤j<14.0≤i≤j<14. (Divide and conquer approach may be used.)
(GATE 2019)
125. Consider the following table
Match the algorithm to design paradigms they are based on: GATE-CS-2017 (Set 1)
(A) P-(ii), Q-(iii), R-(i)
(B) P-(iii), Q-(i), R-(ii)
(C) P-(ii), Q-(i), R-(iii)
(D) P-(i), Q-(ii), R-(iii)
GATE QUESTIONS
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DATA STRUCTURES AND ALGORITHMS
ALGORITHM DESIGN TECHNIQUES
126. Match the following GATE-CS-2015 (Set 1)
List-I
A. Prim’s algorithm for minimum spanning tree
B. Floyd-Warshall algorithm for all pairs shortest paths
C. Mergesort
D. Hamiltonian circuit
List-II
1. Backtracking
2. Greed method
3. Dynamic programming
4. Divide and conquer
Codes: A B C D
(A) 3 2 4 1
(B) 1 2 4 3
(C) 2 3 4 1
(D) 2 1 3 4
127. Given below are some algorithms, and some algorithm design paradigms
List-I
A. Dijkstra’s Shortest Path
B. Floyd-Warshall algorithm to compute all pairs shortest path
C. Binary search on a sorted array
D. Backtracking search on a graph
List-II
1. Divide and Conquer
2. Dynamic Programming
3. Greedy design
4. Depth-first search
5. Breadth-first search
Match the above algorithms on the left to the corresponding design paradigm they follow
Codes: . GATE-CS-2015 (Set 2)
(A) 1 3 1 5
(B) 3 3 1 5
(C) 3 2 1 4
(D) 3 2 1 5
128. 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
GATE – DATA STRUCTURES AND ALGORITHMS Page 57
(A) Dijkstra’s algorithm starting from S. (GATE CS 2007)
(B) Warshall’s algorithm
(C) Performing a DFS starting from S.
(D) Performing a BFS starting from S.
GATE – DATA STRUCTURES AND ALGORITHMS Page 58
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