Data Structures and Algorithms Guide

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
Cross, 10th
Main, Jayanagar 4th
Block, Bangalore-11
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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
 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
 Answer keys 53
#3. Trees 56 – 84
 Extended Binary Tree 56
 Binary Tree 56 – 58
 Height Analysis 59 – 60
 Binary Tree Construction Using Inorder 60 – 70
 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
 Answer keys 109
#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
 Answer keys 132
#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
 Answer keys 147
#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
 Answer keys 167
#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
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 2
3 5 2n n n   for 3n 

Chapter-1 DSA
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 2 2
3 5n n O n    Here 02, 3c n 
Example 3
  2
3.4n
f n n 
2
3.4 5.4n n
n 
 2
3.4 4n n
n O   for 1n 
Example 4
 2
3 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  2
O n and  3
O n
Even though 4n+3 is  2
O n and  3
O 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
   2
f n =n , h n =logn
 2 2
n 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 k
n is  logO n
6. If  f n is any polynomial of degree m,   1
1 1 0. ....m m
m mf n a n a n a n a
    ,
then f n is  m
O 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 n
O 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
2
n is  3
O n
3
n is  2
n
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

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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.

Data Structures and Algorithms Guide

Data Structures and Algorithms Guide

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