Data Structures and Algorithms made Easy (Cover Page)

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I would like to thank for encouraging me when I was at IIT Bombay. He is the first person who taught me the importance of and its . From that

day I keep on updating myself.

I would like to thank [ ] and [] for spending time in

reviewing this book and providing me the suggestions.

I would like to thank [! ""#, % ] for spending his valuable time in

reviewing the book and suggestions. Being a PhD holder in algorithms his review gave me

the confidence in the quality of the book.

I would like to thank and & [Founder’s of #'#('#(. ] for approaching

me for teaching of Algorithms and Data Structures at their GATE centers. They are the

primary reason for initiation of this book.

I would like to thank # [Founder of '. . ] for providing me the valuable suggestions in writing, reviewing and for providing coding solutions to some of the

problems.

Last but not least, I would like to thank Director’s of ,

[Director of '( ( '], #) * [Former Director of '!#] and + ) [) , (

, % ] for helping me and my family during our studies.

Also, my and colleagues have contributed greatly to the quality of this book. I thank

all of you for your help and suggestions.

Special thanks should go to my wife, , for her encouragement and help during writing

of this book.

-------- NNaarraassiimmhhaa KKaarruummaanncchhii





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Dear Reader,Dear Reader,Dear Reader,Dear Reader,

Please Hold onPlease Hold onPlease Hold onPlease Hold on! I know many people do not read preface. But I would like to strongly recommend reading preface of this book at least. This preface has

from regular prefaces.

As a , if you read the complete book with good understanding, I am sure you will

challenge the interviewer’s and that is the objective of this book.

If you read as an , you will give better lectures with easy go approach and as a

result your students will feel proud for selecting Computer Science / Information Technology

as their degree.

This book is very much useful for the of ( and during their

academic preparations. All the chapters of this book contain theory and their related

problems as many as possible. There a total of approximately 700 algorithmic puzzles and all

of them are with solutions.

Finally if you read as a preparing for competition exams [like

' # ( ('#(), /)/0 or any other exam for Computer

Science/Information Technology], then the content of this book covers the 1 topics in full details. While writing the book, an intense care has been taken to ensure that

the content should help students who are preparing for these kinds of exams.

In all the chapters you will see more importance given to problems and analyzing them

instead of concentrating more on theory. For each chapter, first you will see the basic

required theory and then followed by problems.

For many of the problems, solutions are provided with different complexities. First, we start with solution and slowly move towards the possible for

that problem. For each problem we will try to understand how much time the algorithm is

taking and how much memory the algorithm is taking.

It is that, at least one complete reading of this book is required to get full understanding of all the topics. In the subsequent readings, readers can directly go to any

chapter and refer.



Even though, enough readings were given for correcting the errors, due to human tendency,

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corrections, new problems and solutions. Also, please provide your valuable suggestions at:

"@ . .

Always remember the following quotes by 3 which will give you full energy

and confidence.

Wish you all the best. Have a nice reading.

--------NNaarraassiimmhhaa KKaarruummaanncchhii

All power is within you. You can do anything and everything. Believe in that.

-Swami Vivekananda

Stand up, be bold, be strong. Take the whole responsibility on your own shoulders, and

know that you are the creator of your own destiny.

-Swami Vivekananda



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Chapter 1 Introduction .................................................................................................. 33

Variables ................................................................................................................................... 33

Data types ................................................................................................................................. 33

System defined data types (Primitive data types) ............................................................... 34

User defined data types ........................................................................................................ 34

Data structure ........................................................................................................................... 35

Abstract Data Types (ADT’s) ................................................................................................... 35

Memory and variables .............................................................................................................. 36

Size of a variable ................................................................................................................... 37

Address of a variable ............................................................................................................ 37

Pointers ..................................................................................................................................... 38

Declaration of Pointers ........................................................................................................ 38

Pointers Usage ...................................................................................................................... 38

Pointer Manipulation ........................................................................................................... 39

Arrays and Pointers .............................................................................................................. 40

Dynamic Memory Allocation .............................................................................................. 41

Function Pointers ................................................................................................................. 42

Parameter Passing Techniques ................................................................................................ 42

Actual and Formal Parameters ............................................................................................ 42

Semantics of Parameter Passing ........................................................................................... 43

Language Support for Parameter Passing Techniques ........................................................ 43

Pass by Value ........................................................................................................................ 43

Pass by Result........................................................................................................................ 44

Pass by Value-Result ............................................................................................................ 46

Pass by reference (aliasing) .................................................................................................. 47

Pass by name ......................................................................................................................... 48

Binding ...................................................................................................................................... 48



Binding Times ....................................................................................................................... 49

Static binding (Early binding) .............................................................................................. 49

Dynamic binding (Late binding) ......................................................................................... 49

Scope ......................................................................................................................................... 50

Static scope ............................................................................................................................ 50

Dynamic scope ...................................................................................................................... 51

Storage classes ........................................................................................................................... 52

Auto storage class ................................................................................................................. 52

Extern storage class ............................................................................................................... 53

Register storage class ............................................................................................................ 59

Static storage class................................................................................................................. 59

Storage Organization ................................................................................................................ 60

Static Segment ....................................................................................................................... 61

Stack Segment ....................................................................................................................... 61

Heap Segment ....................................................................................................................... 63

Shallow copy versus Deep copy .............................................................................................. 65

Chapter 2 Analysis of Algorithms.................................................................................. 66

Introduction ............................................................................................................................. 66

What is an Algorithm? ............................................................................................................. 66

Why Analysis of Algorithms? ................................................................................................. 67

Goal of Analysis of Algorithms? .............................................................................................. 67

What is Running Time Analysis? ............................................................................................ 67

How to Compare Algorithms? ................................................................................................. 67

What is Rate of Growth? ......................................................................................................... 68

Commonly used Rate of Growths ........................................................................................... 68

Types of Analysis ...................................................................................................................... 69

Asymptotic notation? ............................................................................................................... 70

Big-O notation .......................................................................................................................... 70

Big-O Visualization .............................................................................................................. 71

Big-O examples ..................................................................................................................... 72

No uniqueness? ..................................................................................................................... 72



Omega-Ω notation.................................................................................................................... 73

Ω Examples............................................................................................................................ 73

Theta-θ notation ....................................................................................................................... 74

θ Examples ............................................................................................................................ 74

Important Notes .................................................................................................................... 75

Why is it called Asymptotic Analysis?.................................................................................... 75

Guidelines for asymptotic analysis? ........................................................................................ 76

Properties of notations ............................................................................................................. 78

Commonly used logarithms and summations ........................................................................ 79

Master Theorem for Divide and Conquer .............................................................................. 79

Problems on Master Theorem for Divide and Conquer......................................................... 80

Master Theorem for Subtract and Conquer Recurrences ...................................................... 82

Variant of subtraction and conquer master theorem ......................................................... 82

Problems on Algorithms Analysis ........................................................................................... 83

Chapter 3 Recursion and Backtracking ......................................................................... 99

Introduction ............................................................................................................................. 99

What is recursion? ................................................................................................................... 99

Why recursion? ........................................................................................................................ 99

Format of a recursive function ................................................................................................ 99

Recursion and Memory (Visualization) ................................................................................ 100

Recursion versus Iteration ..................................................................................................... 102

Recursion ............................................................................................................................ 102

Iteration ............................................................................................................................... 102

Notes On Recursion ............................................................................................................... 102

Example Algorithms of Recursion ........................................................................................ 102

Problems on recursion ........................................................................................................... 103

What is backtracking? ............................................................................................................ 104

Example Algorithms Of Backtracking .................................................................................. 104

Problems On Backtracking .................................................................................................... 105

Chapter 4 Linked Lists .................................................................................................. 107

What is a Linked List? ............................................................................................................ 107



Linked Lists ADT ................................................................................................................... 107

Main Linked Lists Operations ............................................................................................ 107

Auxiliary Linked Lists Operations ..................................................................................... 107

Why Linked Lists? .................................................................................................................. 107

Arrays Overview................................................................................................................. 108

Why constant time for accessing array elements?............................................................ 108

Advantages of arrays .......................................................................................................... 108

Disadvantages of arrays ...................................................................................................... 108

Dynamic Arrays .................................................................................................................. 109

Advantages of Linked Lists ................................................................................................ 109

Issues with Linked Lists (Disadvantages) .......................................................................... 109

Comparison of Linked Lists with Arrays and Dynamic Arrays ........................................... 110

Singly Linked Lists ................................................................................................................. 110

Basic Operations on a List .................................................................................................. 110

Traversing the Linked List ................................................................................................. 111

Singly Linked List Insertion .................................................................................................. 111

Inserting a node in Singly Linked List at the beginning .................................................. 112

Inserting a node in Singly Linked List at the ending ....................................................... 112

Inserting a node in Singly Linked List in the middle ....................................................... 113

Singly Linked List Deletion ................................................................................................... 115

Deleting the first node in Singly Linked List .................................................................... 115

Deleting the last node in Singly Linked List ..................................................................... 115

Deleting an intermediate node in Singly Linked List ....................................................... 116

Deleting Singly Linked List ................................................................................................... 118

Doubly Linked Lists ............................................................................................................... 118

Doubly Linked List Insertion ................................................................................................ 119

Inserting a node in Doubly Linked List at the beginning ................................................ 119

Inserting a node in Doubly Linked List at the ending ..................................................... 119

Inserting a node in Doubly Linked List in the middle ..................................................... 120

Doubly Linked List Deletion ................................................................................................. 122

Deleting the first node in Doubly Linked List .................................................................. 122



Deleting the last node in Doubly Linked List ................................................................... 123

Deleting an intermediate node in Doubly Linked List ..................................................... 123

Circular Linked Lists .............................................................................................................. 125

Counting nodes in a circular list ........................................................................................ 126

Printing the contents of a circular list ............................................................................... 127

Inserting a node at the end of a Circular Linked List ....................................................... 127

Inserting a node at front of a Circular Linked List ........................................................... 129

Deleting the last node in a circular list ............................................................................. 131

Deleting the first node in a circular list ............................................................................ 132

Applications of circular list ................................................................................................ 133

A Memory-Efficient Doubly Linked List .............................................................................. 133

Problems on Linked Lists ....................................................................................................... 135

Chapter 5 Stacks ........................................................................................................... 160

What is a Stack? ..................................................................................................................... 160

How are stacks used? .............................................................................................................. 160

Stack ADT ............................................................................................................................... 161

Main stack operations......................................................................................................... 161

Auxiliary stack operations .................................................................................................. 161

Exceptions ........................................................................................................................... 161

Applications ............................................................................................................................ 161

Implementation ...................................................................................................................... 162

Simple Array Implementation ........................................................................................... 162

Dynamic Array Implementation ....................................................................................... 164

Performance ........................................................................................................................ 167

Linked List Implementation .............................................................................................. 168

Performance ........................................................................................................................ 170

Comparison of Implementations ........................................................................................... 170

Comparing Incremental Strategy and Doubling Strategy ................................................ 170

Comparing Array Implementation and Linked List Implementation ............................. 170

Problems on Stacks ................................................................................................................ 171

Chapter 6 Queues ......................................................................................................... 197



What is a queue? .................................................................................................................... 197

How are queues used? ............................................................................................................ 197

Queue ADT ............................................................................................................................. 198

Main queue operations ....................................................................................................... 198

Auxiliary queue operations ................................................................................................ 198

Exceptions ............................................................................................................................... 198

Applications ............................................................................................................................ 198

Direct applications .............................................................................................................. 198

Indirect applications ........................................................................................................... 199

Implementation ...................................................................................................................... 199

Why Circular Arrays? ........................................................................................................ 199

Simple Circular Array Implementation ............................................................................ 200

Performance & Limitations ................................................................................................ 202

Dynamic Circular Array Implementation ......................................................................... 203

Performance ........................................................................................................................ 205

Linked List Implementation .............................................................................................. 205

Performance ........................................................................................................................ 207

Comparison of Implementations ....................................................................................... 208

Problems on Queues .............................................................................................................. 208

Chapter 7 Trees ............................................................................................................. 212

What is a tree? ........................................................................................................................ 212

Glossary ................................................................................................................................... 212

Binary Trees ............................................................................................................................ 213

Types of Binary Trees ......................................................................................................... 214

Properties of Binary Trees.................................................................................................. 215

Structure of Binary Trees ................................................................................................... 216

Operations on Binary Trees ............................................................................................... 217

Applications of Binary Trees .............................................................................................. 217

Binary Tree Traversals ........................................................................................................... 217

Traversal Possibilities ......................................................................................................... 218

Classifying the Traversals ................................................................................................... 218



Preorder Traversal .............................................................................................................. 219

Inorder Traversal ................................................................................................................ 221

Postorder Traversal ............................................................................................................. 222

Level order Traversal .......................................................................................................... 224

Problems on Binary Trees .................................................................................................. 225

Generic Trees (N-ary Trees) .................................................................................................. 252

Representation of generic trees ......................................................................................... 253

Problems on Generic Trees ................................................................................................ 254

Threaded Binary Tree Traversals [Stack or Queue less Traversals] .................................... 263

Issues with regular Binary Trees ........................................................................................ 263

Motivation for threaded binary trees ................................................................................ 263

Classifying threaded binary trees ...................................................................................... 264

Types of threaded binary trees .......................................................................................... 264

Threaded binary tree structure .......................................................................................... 264

Difference between Binary Tree and Threaded Binary Tree Structures ......................... 265

Finding Inorder Successor in Inorder Threaded Binary Tree .......................................... 266

Inorder Traversal in Inorder Threaded Binary Tree ........................................................ 267

Finding Preorder Successor in Inorder Threaded Binary Tree ........................................ 268

Preorder Traversal of Inorder Threaded Binary Tree ...................................................... 268

Insertion of Nodes in Inorder Threaded Binary Trees ..................................................... 269

Problems on Threaded binary Trees ................................................................................. 271

Expression Trees ..................................................................................................................... 273

Algorithm for Building Expression Tree from Postfix Expression .................................. 273

Example: .............................................................................................................................. 274

XOR Trees ............................................................................................................................... 276

Binary Search Trees (BSTs) .................................................................................................... 278

Why Binary Search Trees? ................................................................................................. 278

Binary Search Tree Property.............................................................................................. 278

Binary Search Tree Declaration ......................................................................................... 279

Operations on Binary Search Trees ................................................................................... 279

Important Notes on Binary Search Trees .......................................................................... 279



Finding an Element in Binary Search Trees ..................................................................... 280

Finding an Minimum Element in Binary Search Trees.................................................... 281

Finding an Maximum Element in Binary Search Trees ................................................... 282

Where is Inorder Predecessor and Successor? .................................................................. 283

Inserting an Element from Binary Search Tree ................................................................ 284

Deleting an Element from Binary Search Tree ................................................................. 285

Problems on Binary Search Trees ...................................................................................... 287

Balanced Binary Search Trees ............................................................................................... 299

Complete balanced binary search trees ............................................................................. 299

AVL (Adelson-Velskii and Landis) trees .............................................................................. 299

Properties of AVL Trees ..................................................................................................... 299

Minimum/Maximum Number of Nodes in AVL Tree...................................................... 300

AVL Tree declaration ......................................................................................................... 301

Finding height of an AVL tree ........................................................................................... 301

Rotations ............................................................................................................................. 302

Observation ......................................................................................................................... 302

Types of violations .............................................................................................................. 302

Single rotations ................................................................................................................... 303

Double rotations ................................................................................................................. 305

Insertion into an AVL tree ................................................................................................. 307

Problems on AVL Trees ..................................................................................................... 308

Other Variations in Trees ...................................................................................................... 314

Red-Black Trees .................................................................................................................. 314

Splay Trees .......................................................................................................................... 314

Augmented Trees ................................................................................................................ 315

Interval Trees ...................................................................................................................... 316

Chapter 8 Priority Queue and Heaps .......................................................................... 318

What is a Priority Queue? ..................................................................................................... 318

Priority Queue ADT............................................................................................................... 318

Main Priority Queues Operations ..................................................................................... 318

Auxiliary Priority Queues Operations .............................................................................. 319



Priority Queue Applications .................................................................................................. 319

Priority Queue Implementations .......................................................................................... 319

Unordered Array Implementation .................................................................................... 319

Unordered List Implementation ........................................................................................ 319

Ordered Array Implementation ........................................................................................ 320

Ordered List Implementation ............................................................................................ 320

Binary Search Trees Implementation ................................................................................ 320

Balanced Binary Search Trees Implementation ................................................................ 320

Binary Heap Implementation ............................................................................................ 320

Comparing Implementations ............................................................................................. 320

Heaps and Binary Heap .......................................................................................................... 321

What is a heap? ................................................................................................................... 321

Types of heaps? ................................................................................................................... 321

Binary Heaps .......................................................................................................................... 322

Representing Heaps ............................................................................................................ 322

Parent of a Node ................................................................................................................. 323

Children of a Node ............................................................................................................. 323

Getting the Maximum Element ......................................................................................... 323

Heapifying an Element ....................................................................................................... 323

Deleting an Element ........................................................................................................... 326

Inserting an Element .......................................................................................................... 326

Heapifying the Given Data (Array) ................................................................................... 328

Heapsort .............................................................................................................................. 329

Problems on Priority Queues [Heaps] .................................................................................. 330

Chapter 9 Disjoint Sets ADT ........................................................................................ 344

Introduction ........................................................................................................................... 344

Equivalence Relations and Equivalence Classes ................................................................... 344

Disjoint Sets ADT ................................................................................................................... 345

Applications ............................................................................................................................ 345

Tradeoffs in Implementing Disjoint Sets ADT operations ................................................... 346

Fast FIND Implementation (Quick FIND) ........................................................................ 346



Fast UNION Implementation (Quick UNION) ................................................................ 347

Fast UNION implementation (Slow FIND) .......................................................................... 347

Fast UNION implementations (Quick FIND) ....................................................................... 350

UNION by Size ................................................................................................................... 351

UNION by Height (UNION by Rank) ............................................................................... 352

Comparing UNION by Size and UNION by Height ......................................................... 353

Further Improving Fast UNION Implementations (Path Compression) ........................ 354

Summary ................................................................................................................................. 355

Problems on Disjoint Sets ...................................................................................................... 355

Chapter 10 Graph Algorithms ....................................................................................... 357

Introduction ........................................................................................................................... 357

Glossary ................................................................................................................................... 357

Applications of graphs ........................................................................................................... 361

Graph Representation ............................................................................................................ 361

Adjacency Matrix ............................................................................................................... 361

Adjacency List ..................................................................................................................... 363

Adjacency Set ...................................................................................................................... 365

Comparison of Graph Representations ............................................................................. 366

Graph Traversals .................................................................................................................... 366

Depth First Search [DFS] ................................................................................................... 366

Breadth First Search [BFS] ................................................................................................. 371

Comparing DFS and BFS .................................................................................................... 374

Topological Sort ...................................................................................................................... 375

Applications of Topological Sorting .................................................................................. 377

Shortest Path Algorithms....................................................................................................... 377

Shortest path in unweighted graph ................................................................................... 377

Shortest path in weighted graph [Dijkstra’s] .................................................................... 379

Bellman-Ford Algorithm .................................................................................................... 384

Overview of Shortest Path Algorithms ............................................................................. 385

Minimal Spanning Tree ......................................................................................................... 385

Prim's Algorithm ................................................................................................................ 386



Kruskal’s Algorithms .......................................................................................................... 387

Problems on Graph Algorithms ............................................................................................ 391

Chapter 11 Sorting .......................................................................................................... 427

What is sorting? ...................................................................................................................... 427

Why Sorting? .......................................................................................................................... 427

Classification ........................................................................................................................... 427

By number of comparisons ................................................................................................ 427

By number of swaps ........................................................................................................... 427

By Memory Usage............................................................................................................... 428

By Recursion ....................................................................................................................... 428

By Stability .......................................................................................................................... 428

By Adaptability ................................................................................................................... 428

Other classifications ............................................................................................................... 428

Internal Sort ........................................................................................................................ 428

External Sort ....................................................................................................................... 429

Bubble sort .............................................................................................................................. 429

Implementation .................................................................................................................. 429

Performance ........................................................................................................................ 430

Selection Sort .......................................................................................................................... 430

Algorithm ............................................................................................................................ 431

Implementation .................................................................................................................. 431

Performance ........................................................................................................................ 431

Insertion sort .......................................................................................................................... 431

Insertion advantages ........................................................................................................... 432

Algorithm ............................................................................................................................ 432

Implementation .................................................................................................................. 432

Example ............................................................................................................................... 433

Analysis ............................................................................................................................... 433

Performance ........................................................................................................................ 434

Comparisons to other sorting algorithms .......................................................................... 434

Shell sort ................................................................................................................................. 434



Implementation .................................................................................................................. 435

Analysis ............................................................................................................................... 435

Performance ........................................................................................................................ 436

Merge sort ............................................................................................................................... 436

Important Notes .................................................................................................................. 436

Implementation .................................................................................................................. 437

Analysis ............................................................................................................................... 438

Performance ........................................................................................................................ 438

Heapsort .................................................................................................................................. 438

Performance ........................................................................................................................ 439

Quicksort ................................................................................................................................ 439

Algorithm ............................................................................................................................ 439

Implementation .................................................................................................................. 439

Analysis ............................................................................................................................... 440

Performance ........................................................................................................................ 442

Randomized Quick sort ...................................................................................................... 442

Tree Sort ................................................................................................................................. 443

Performance ........................................................................................................................ 443

Comparison of sorting algorithms ......................................................................................... 443

Linear Sorting Algorithms ..................................................................................................... 444

Counting Sort ......................................................................................................................... 444

Bucket sort [or Bin Sort] ........................................................................................................ 446

Radix sort ................................................................................................................................ 446

Topological Sort ...................................................................................................................... 447

External Sorting ...................................................................................................................... 447

Problems on Sorting ............................................................................................................... 449

Chapter 12 Searching ..................................................................................................... 467

What is searching? ................................................................................................................. 467

Why Searching? ..................................................................................................................... 467

Types of Searching ................................................................................................................. 467

Unordered Linear Search ....................................................................................................... 467



Sorted/Ordered Linear Search ............................................................................................... 468

Binary Search.......................................................................................................................... 469

Comparing Basic Searching Algorithms ............................................................................... 470

Symbol Tables and Hashing ................................................................................................... 470

String Searching Algorithms ................................................................................................. 470

Problems on Searching .......................................................................................................... 471

Chapter 13 Selection Algorithms [Medians] ................................................................. 507

What are Selection Algorithms? ........................................................................................... 507

Linear minimum/maximum algorithms ................................................................................ 507

Selection by sorting ................................................................................................................ 507

Partition-based general selection algorithm [Quick Select] ................................................ 508

Linear general selection algorithm - Median of Medians algorithm .................................. 508

Finding the k smallest elements in sorted order .................................................................. 508

Problems on Selection Algorithms ........................................................................................ 508

Chapter 14 Symbol Tables .............................................................................................. 522

Introduction ........................................................................................................................... 522

What are Symbol Tables? ...................................................................................................... 522

Symbol Table implementations ............................................................................................. 523

Unordered Aarray Implementation ................................................................................... 523

Ordered [Sorted] Array Implementation .......................................................................... 523

Unordered Linked List Implementation ........................................................................... 523

Ordered Linked List Implementation ............................................................................... 523

Binary Search Trees Implementation ................................................................................ 524

Balanced Binary Search Trees Implementation ................................................................ 524

Ternary Search Implementation ........................................................................................ 524

Hashing Implementation ................................................................................................... 524

Comparison Symbol Ttable Implementations ...................................................................... 524

Chapter 15 Hashing ........................................................................................................ 525

What is Hashing? ................................................................................................................... 525

Why Hashing? ........................................................................................................................ 525

HashTable ADT ...................................................................................................................... 525



Understanding Hashing ......................................................................................................... 525

If Arrays Are There Why Hashing? .................................................................................. 526

Components in Hashing ........................................................................................................ 527

Hash Table .............................................................................................................................. 528

Hash Function ........................................................................................................................ 528

How to Choose Hash Function? ............................................................................................ 528

Characteristics of Good Hash Functions ........................................................................... 528

Load Factor ............................................................................................................................. 529

Collisions ................................................................................................................................ 529

Collision Resolution Techniques ........................................................................................... 529

Separate Chaining ............................................................................................................... 529

Open Addressing ................................................................................................................ 530

Comparing Collision Resolution Techniques ....................................................................... 530

Comparisons: Linear Probing 45. Double Hashing .......................................................... 530

Comparisons: Open Addressing 45. Separate Chaining ................................................... 530

Comparisons: Open Addressing methods ........................................................................ 531

How Hashing Gets O(1) Complexity? ................................................................................... 531

Hashing Techniques ............................................................................................................... 532

Static Hashing ..................................................................................................................... 532

Dynamic Hashing ............................................................................................................... 532

Problems for which Hash Tables are not Suitable ............................................................... 532

Problems on Hashing ............................................................................................................. 532

Chapter 16 String Algorithms ........................................................................................ 548

Introduction ........................................................................................................................... 548

String Matching Algorithms .................................................................................................. 548

Brute Force Method ............................................................................................................... 549

Robin-Karp String Matching Algorithm .............................................................................. 549

Selecting Hash Function .................................................................................................... 550

Step by Step explanation .................................................................................................... 551

String Matching with Finite Automata ................................................................................ 551

Finite Automata .................................................................................................................. 551



How Finite Automata Works? ........................................................................................... 552

Important Notes for Constructing the Finite Automata .................................................. 552

Matching Algorithm ........................................................................................................... 552

KMP Algorithm ...................................................................................................................... 553

Filling Prefix Table ............................................................................................................. 553

Matching Algorithm ........................................................................................................... 555

Boyce-Moore Algorithm ........................................................................................................ 558

Data structures for Storing Strings ........................................................................................ 559

Hash Tables for Strings .......................................................................................................... 559

Binary Search Trees for Strings ............................................................................................. 559

Issues with Binary Search Tree Representation ............................................................... 560

Tries ........................................................................................................................................ 560

What is a Trie? .................................................................................................................... 560

Why Tries? .......................................................................................................................... 561

Inserting a String in Trie .................................................................................................... 561

Searching a String in Trie ................................................................................................... 562

Issues with Tries Representation ....................................................................................... 562

Ternary Search Trees ............................................................................................................. 563

Ternary Search Trees Declaration ..................................................................................... 563

Inserting strings in TST ...................................................................................................... 564

Searching in Ternary Search Trees .................................................................................... 566

Displaying All Words of Ternary Search Tree ................................................................. 567

Finding Maximum Length of Words in TST .................................................................... 568

Comparing Binary Search Trees, Tries and Ternary Search Trees ...................................... 568

Suffix Trees ............................................................................................................................. 569

Prefix and Suffix ................................................................................................................. 569

Observation ......................................................................................................................... 569

What is a Suffix Tree? ........................................................................................................ 569

The Construction of Suffix Trees ....................................................................................... 570

Applications of Suffix Trees ............................................................................................... 573

Problems on Strings ............................................................................................................... 573



Chapter 17 Algorithms Design Techniques .................................................................. 581

Introduction ........................................................................................................................... 581

Classification ........................................................................................................................... 581

Classification by Implementation Method ........................................................................... 581

Recursion or Iteration ........................................................................................................ 581

Procedural or Declarative (Non-Procedural) .................................................................... 582

Serial or Parallel or Distributed ......................................................................................... 582

Deterministic or Non-Deterministic ................................................................................. 582

Exact or Approximate ......................................................................................................... 582

Classification by Design Method ........................................................................................... 582

Greedy Method ................................................................................................................... 582

Divide and Conquer ........................................................................................................... 583

Dynamic Programming ...................................................................................................... 583

Linear Programming .......................................................................................................... 583

Reduction [Transform and Conquer] ................................................................................ 583

Other Classifications .............................................................................................................. 584

Classification by Research Area ......................................................................................... 584

Classification by Complexity ............................................................................................. 584

Randomized Algorithms .................................................................................................... 584

Branch and Bound Enumeration and Backtracking ......................................................... 584

Chapter 18 Greedy Algorithms ...................................................................................... 585

Introduction ........................................................................................................................... 585

Greedy strategy ...................................................................................................................... 585

Elements of Greedy Algorithms ............................................................................................ 585

Greedy choice property ...................................................................................................... 585

Optimal substructure .......................................................................................................... 585

Does Greedy Works Always? ................................................................................................ 586

Advantages and Disadvantages of Greedy Method .............................................................. 586

Greedy Applications ............................................................................................................... 586

Understanding Greedy Technique ........................................................................................ 586

Huffman coding algorithm ................................................................................................ 587



Problems on greedy algorithms ............................................................................................. 591

Chapter 19 Divide and Conquer Algorithms ................................................................ 603

Introduction ........................................................................................................................... 603

What is Divide and Conquer Strategy? ................................................................................. 603

Does Divide and Conquer Work Always? ............................................................................ 603

Divide and Conquer Visualization ........................................................................................ 603

Understanding Divide and Conquer ..................................................................................... 604

Advantages of Divide and Conquer ...................................................................................... 605

Disadvantages of Divide and Conquer .................................................................................. 606

Master Theorem ..................................................................................................................... 606

Divide and Conquer Applications ......................................................................................... 606

Problems on Divide and Conquer ......................................................................................... 607

Chapter 20 Dynamic Programming ............................................................................... 625

Introduction ........................................................................................................................... 625

What is Dynamic Programming Strategy? ........................................................................... 625

Can Dynamic Programming Solve Any Problem? ............................................................... 625

Dynamic Programming Approaches ..................................................................................... 626

Bottom-up dynamic programming .................................................................................... 626

Top-down dynamic programming..................................................................................... 626

Bottom-up versus Top-down programming ..................................................................... 626

Examples of Dynamic Programming Algorithms ................................................................. 626

Understanding Dynamic Programming ................................................................................ 627

Fibonacci Series .................................................................................................................. 627

Observations ....................................................................................................................... 630

Factorial of a number ......................................................................................................... 630

Problems on Dynamic Programming .................................................................................... 632

Chapter 21 Complexity Classes ...................................................................................... 684

Introduction ........................................................................................................................... 684

Polynomial/exponential time ................................................................................................ 684

What is Decision Problem? ................................................................................................... 685

Decision Procedure ................................................................................................................ 685



What is a Complexity Class? .................................................................................................. 685

Types of Complexity Classes .................................................................................................. 686

P Class ................................................................................................................................. 686

NP Class ............................................................................................................................... 686

Co-NP Class ........................................................................................................................ 686

Relationship between P, NP and Co-NP ........................................................................... 686

NP-hard Class ..................................................................................................................... 687

NP-complete Class .............................................................................................................. 687

Relationship between P, NP Co-NP, NP-Hard and NP-Complete.................................. 688

Does P==NP? ....................................................................................................................... 688

Reductions .............................................................................................................................. 689

Important NP-Complete Problems (Reductions) ............................................................. 690

Problems on Complexity Classes ........................................................................................... 692

Chapter 22 Miscellaneous Concepts .............................................................................. 696

Introduction ........................................................................................................................... 696

Hacks on Bitwise Programming ............................................................................................ 696

Bitwise AND ....................................................................................................................... 696

Bitwise OR .......................................................................................................................... 696

Bitwise Exclusive-OR ......................................................................................................... 697

Bitwise Left Shift ................................................................................................................ 697

Bitwise Right Shift .............................................................................................................. 697

Bitwise Complement .......................................................................................................... 698

Checking whether K-th bit is set or not ........................................................................... 698

Setting K-th bit ................................................................................................................... 698

Clearing K-th bit ................................................................................................................. 698

Toggling K-th bit ................................................................................................................ 699

Toggling Rightmost One bit .............................................................................................. 699

Isolating Rightmost One bit ............................................................................................... 699

Isolating Rightmost Zero bit .............................................................................................. 700

Checking Whether Number is Power of 2 not ................................................................. 700

Multiplying Number by Power of 2 .................................................................................. 701



Dividing Number by Power of 2 ....................................................................................... 701

Finding Modulo of a Given Number ................................................................................. 701

Reversing the Binary Number ........................................................................................... 701

Counting Number of One’s in number ............................................................................. 702

Creating for Mask for Trailing Zero’s ................................................................................ 704





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