DATA STRUCTURE
&
ALGORITHM
For
Computer Science
&
Information Technology
By
www.thegateacademy.com
Syllabus DSA
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Syllabus for Data Structures and Algorithms
Programming in C; Functions, Recursion, Parameter passing, Scope, Binding; Abstract data types,
Arrays, Stacks, Queues, Linked Lists, Trees, Binary search trees, Binary heaps.
Analysis, Asymptotic notation, Notions of space and time complexity, Worst and average case
analysis; Design: Greedy approach, Dynamic programming, Divide-and-conquer; Tree and graph
traversals, Connected components, Spanning trees, Shortest paths; Hashing, Sorting, Searching.
Analysis of GATE Papers
(Data Structures and Algorithms)
Year Percentage of marks Overall Percentage
2013 18.00
11.33%
2012 19.00
2011 13.0
2010 18.00
2009 4.67
2008 4.67
2007 4.67
2006 8.00
2005 7.33
2004 16.67
2003 10.67
Contents DSA
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CC OO NN TT EE NN TT SS
Chapters Page No.
#1. Data Structure and Algorithm Analysis 1 – 32 Assymptotic Notation 1 – 5
Algorithm Analysis 5 – 10
Notation of Abstract Data Types 10 – 14
Recurrence 14 – 17
Assignment 1 18 – 23
Assignment 2 24 – 27
Answer keys 28
Explanations 28 – 32
#2. Stacks and Queues 33 – 55 Stacks 33
Stack ADT Implementations 34 – 36
The Stack Purmutation 36 – 40
Running Time Analysis 40 – 41
Binary Expression Tree 41 – 45
Queue 45
Different Type of Queue Implementations 46 – 48
Assignment 1 49 – 51
Assignment 2 51 – 52
Answer keys 53
Explanations 53 – 55
#3. Trees 56 – 84 Extended Binary Tree 56
Binary Tree 56 – 58
Height Analysis 59 – 60
Binary Tree Construction Using Inorder 60 – 70
Assignment 1 71 – 75
Assignment 2 76 – 78
Answer keys 79
Explanations 79 - 84
#4. Height Balanced Trees (AVL Trees, B and B+) 85 – 113
AVL Trees 85 – 94
B – Tree 94 – 96
Maximizing B-Tree Degree 96 – 102
B+Tree 102 – 103
Contents DSA
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Maximizing B+ Tree Degree 103 – 104
Assignment 1 105 – 107
Assignment 2 107 – 108
Answer keys 109
Explanations 109 – 113
#5. Priority Queues (Heaps) 114 – 135 Introduction 114
Binary Heap 114 – 118
Array Representation of Binary Heap 118 – 119
MinHeap Vs MaxHeap 119
Basic Heap Operation 119 – 121
Building a Heap by Inserting Items One at the Time 121 – 125
Sum of the Height of All Nodes of a Perfect Binary Tree 125 – 126
Assignment 1 127 – 129
Assignment 2 130 – 131
Answer keys 132
Explanations 132 – 135
#6. Sorting Algorithms 136 – 149 Bubble Sort 136 – 137
Insertion Sort 137 – 139
Selection Sort 139 – 140
Merge Sort 140 – 141
Heap Sort 141
Quick Sort 141 – 142
Assignment 1 143 – 144
Assignment 2 145 – 146
Answer keys 147
Explanations 147 – 149
#7. Graph Algorithms 150 – 170 Important Definitions 150 – 151
Representation of Graphs 151
Single Source Shortest Path Algorithm 151 – 154
Minimum Spanning Tree 154 – 159
Assignment 1 160 – 163
Assignment 2 163 – 166
Answer keys 167
Explanations 167 – 170
#8. Dynamic Programming 171 – 194 Introduction 171
Idea of Dynamic Programming 171 – 172
Contents DSA
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Matrix Chain Multiplication Algorithm 172 – 175
Greedy Algorithm 175 – 183
NP-Completeness 183 – 186
Other NP- Complete Problems 186 – 189
Hashing 189 – 194
Module Test 195 – 209
Test Questions 195 – 205
Answer Keys 206
Explanations 206 – 209
Reference Books 210
Chapter-1 DSA
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CHAPTER 1
Data Structure and Algorithm Analysis
Once an algorithm is given for a problem and decided to be correct, then an important step is to determine how much in the way of resources, such as time or space, the algorithm will be required.
The analysis required to estimate use of these resources of an algorithm is generally a theoretical issue and therefore a formal framework is required. In this framework, we shall consider a normal computer as a model of computation that will have the standard repertoire of simple instructions like addition, multiplication, comparison and assignment, but unlike the case with real computer, it takes exactly one unit time unit to do anything (simple) and there are no fancy operations such as matrix inversion or sorting, that clearly cannot be done in one unit time. We also always assume infinite memory.
Asymptotic Notation
The asymptotic notations are used to represent the relative growth rate between functions.
Big–Oh
Represent upper bound on the running time and the memory being consumed by the algorithms. O(n) essentially conveys that the growth rate of running time/memory consumption rate will not be more than “n” for all inputs of size n for a given algorithm. However, it may be less than this.
More formally Big-Oh is defined as follows:
The function ( )f n O g n if and only if .f n c g n for all 0,n n n where 0,c n are
positive constants.
Thus, if ( )f n O g n statement is said to be true then the growth rate of function g(n) is
surely higher than/equal to f(n).
Example 1
3 2f n n
3 2 4n n for all 2n
3 2n O n Here 04, 2c n
Example 2
2 3 5f n n n
2 23 5 2n n n for 3n
Chapter-1 DSA
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2 23 5n n O n Here 02, 3c n
Example 3
23.4nf n n
23.4 5.4n nn
23.4 4n nn O for 1n
Example 4
23 2 4n n O n
Because here doesn’t exist any positive 0n and cso that Big-Oh equation gets satisfied.
Remarks:
For the function 4n+3,
4n+3 is O n
4n+3 is also 2O n and 3O n
Even though 4n+3 is 2O n and 3O n but the best answer for , 4n+3 is O n only, as
O n shows most tighter upper bound than the other in the question.
Big-Oh Properties
1. If f n is O g n then .a f n is also O g n
2. If f n is O g n and h n is O p n then max ,f n h n O g n p n
Example 5
2f n =n , h n =logn
2 2n logn O n
3. If f n is O g n and h n is O p n then .f n h n is .O g n p n
Chapter-1 DSA
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4. If f n is O g n and g n is O h n then f n is also O h n
5. log kn is logO n
6. If f n is any polynomial of degree m , 11 1 0. ....m m
m mf n a n a n a n a ,
then f n is mO n
In general one should remember order of the following functions which will help while solving the relative growth rate of more complicated functions.
2 3 k nO 1 ,O logn ,O n ,O nlogn ,O n ,O n ....O n ,O 2
All the functions are arranged in increasing order of growth rate.
If an algorithm has the time complexity 1O , then the time complexity is said to be constant,
that means running time is independent of input size.
Big Omega
Big Omega represents lower bound on the running time and the memory being consumed by the algorithms. Ω n essentially conveys that the growth rate of running time/memory consumption rate will not be less than “n” for all inputs of size n for a given algorithm. However, it may be greater than this.
More formally Big-Omega is defined as follows:
If f x and g x are any two functions and f x is ,g x
If .f x c g x for x k where c and k are any two positive constants.
Thus, if f x is g x statement is said to be true then the growth rate of function g(x) is
surely lower than/equal to f(x).
Example 6: f n n n n for n 2 2n y n for 1n
2 4n is Ω n here 2, 1c k
we can also say that 2 4n n for 1n then 1, 1c k
Chapter-1 DSA
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Remarks:
If f n is O g n , then g n is f n .
Example: 7
2n is 3O n
3n is 2n
Theta Notation ( )
Theta represents tightest bound on the running time and the memory being consumed by the algorithms. (n) essentially conveys that the growth rate of running time/memory consumption rate will be equal to “n” for all inputs of size n for a given algorithm. It actually conveys that both lower and upper bounds are equal.
More formally Theta is defined as follows:
If f x and g x are two functions, and
if .f x c g x for 0x x , then
f x g x here c and x0 are two positive constants.
Thus, if f x g x statement is said to be true then the growth rate of function g(x) is
surely equal to f(x) and not less or not more than f(x).
Example 8
f(n) = n2 + n + 1; g(n) = 5n2 + 1; h(n) = 2logn + n2
Then, f(n) = (g(n)) because both have same degree and hence will have same growth rate.
f(n) = (h(n)) statement is also true because both have same degree and hence will have same growth rates.
h(n) can be simplified as follows:
2logn is n only,
Let 2logn = n ---------> 1
By taking log on both sides in equation 1.
logn*loge2 = logen
Chapter-1 DSA
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Then, after simplifying the above equation
logn = logen/ loge2 = logn.
Remarks
If f x g x then g x is also f x
If f x g x we can say that f x is O g x and f x is
g x and also g x is O f x and g x is Ω f x
Algorithm Definition
An algorithm is a finite set of steps or instructions to accomplish a particular task represented in a step by step procedure. Algorithm possesses the following basic properties:
An algorithm may have some input.
An algorithm should produce at least one output.
Each statement should be clear without any ambiguity.
An algorithm in contrast to a program should terminate in a finite amount of time.
Algorithm Analysis
The following two components need to be analyzed for determining algorithm efficiency. If we have more than one algorithms for solving a problem then we really need to consider these two before utilizing one of them.
Running time complexity
The time required for running an algorithm.
Space complexity
The amount of space required at run-time by an algorithm for solving a given problem.
In general these measurements are expressed in terms of asymptotic notations, like Big-Oh, theta, Omega etc.
Therefore, h(n) = n + n2.