Table Of ContentData Structures
And
Algorithmic
Thinking With Python
-To All My Readers And Parents
By
Narasimha Karumanchi
Concepts Problems Interview Questions
Copyright© 2020 by (cid:1829)(cid:1853)(cid:1870)(cid:1857)(cid:1857)(cid:1870)(cid:1839)(cid:1867)(cid:1866)(cid:1863).(cid:1855)(cid:1867)(cid:1865)
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ISBN978-8192107592
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Acknowledgements
(cid:1839)(cid:1867)(cid:1872)ℎ(cid:1857)(cid:1870) and (cid:1832)(cid:1853)(cid:1872)ℎ(cid:1857)(cid:1870), it is impossible to thank you adequately for everything you have done, from loving me unconditionally to raising me in
a stable household, where your persistent efforts and traditional values taught your children to celebrate and embrace life. I could not have
asked for better parents or role-models. You showed me that anything is possible with faith, hard work and determination.
This book would not have been possible without the help of many people. I would like to express my gratitude to all of the people who
provided support, talked things over, read, wrote, offered comments, allowed me to quote their remarks and assisted in the editing,
proofreading and design. In particular, I would like to thank the following individuals:
(cid:1839)(cid:1867)ℎ(cid:1853)(cid:1866) (cid:1839)(cid:1873)(cid:1864)(cid:1864)(cid:1853)(cid:1868)(cid:1873)(cid:1856)(cid:1861), IIT Bombay, Architect, dataRPM Pvt. Ltd.
(cid:1840)(cid:1853)(cid:1874)(cid:1861)(cid:1866) (cid:1837)(cid:1873)(cid:1865)(cid:1853)(cid:1870) (cid:1836)(cid:1853)(cid:1861)(cid:1871)(cid:1875)(cid:1853)(cid:1864), Senior Consultant, Juniper Networks Inc.
(cid:1846)(cid:1870)(cid:1861)(cid:1866)(cid:1853)(cid:1872)ℎ (cid:1837)(cid:1867)(cid:1877)(cid:1853), Software Developer, TinMen
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M-Tech, (cid:1835)(cid:1835)(cid:1846) (cid:1828)(cid:1867)(cid:1865)(cid:1854)(cid:1853)(cid:1877)
Founder, (cid:1829)(cid:1853)(cid:1870)(cid:1857)(cid:1857)(cid:1870)(cid:1839)(cid:1867)(cid:1866)(cid:1863).(cid:1855)(cid:1867)(cid:1865)
Preface
Dear Reader,
Please hold on! I know many people typically do not read the Preface of a book. But I strongly recommend that you read this particular
Preface.
The study of algorithms and data structures is central to understanding what computer science is all about. Learning computer science is not
unlike learning any other type of difficult subject matter. The only way to be successful is through deliberate and incremental exposure to the
fundamental ideas. A beginning computer scientist needs practice so that there is a thorough understanding before continuing on to the more
complex parts of the curriculum. In addition, a beginner needs to be given the opportunity to be successful and gain confidence. This textbook
is designed to serve as a text for a first course on data structures and algorithms. In this book, we cover abstract data types and data structures,
writing algorithms, and solving problems. We look at a number of data structures and solve classic problems that arise. The tools and
techniques that you learn here will be applied over and over as you continue your study of computer science.
It is not the main objective of this book to present you with the theorems and proofs on (cid:1856)(cid:1853)(cid:1872)(cid:1853) (cid:1871)(cid:1872)(cid:1870)(cid:1873)(cid:1855)(cid:1872)(cid:1873)(cid:1870)(cid:1857)(cid:1871) and (cid:1853)(cid:1864)(cid:1859)(cid:1867)(cid:1870)(cid:1861)(cid:1872)ℎ(cid:1865)(cid:1871). I have followed
a pattern of improving the problem solutions with different complexities (for each problem, you will find multiple solutions with different,
and reduced, complexities). Basically, it’s an enumeration of possible solutions. With this approach, even if you get a new question, it will
show you a way to think about the possible solutions. You will find this book useful for interview preparation, competitive exams preparation,
and campus interview preparations.
As a (cid:1862)(cid:1867)(cid:1854) (cid:1871)(cid:1857)(cid:1857)(cid:1863)(cid:1857)(cid:1870), if you read the complete book, I am sure you will be able to challenge the interviewers. If you read it as an (cid:1861)(cid:1866)(cid:1871)(cid:1872)(cid:1870)(cid:1873)(cid:1855)(cid:1872)(cid:1867)(cid:1870), it
will help you to deliver lectures with an approach that is easy to follow, and as a result your students will appreciate the fact that they have
opted for Computer Science / Information Technology as their degree.
This book is also useful for (cid:1831)(cid:1866)(cid:1859)(cid:1861)(cid:1866)(cid:1857)(cid:1857)(cid:1870)(cid:1861)(cid:1866)(cid:1859) (cid:1856)(cid:1857)(cid:1859)(cid:1870)(cid:1857)(cid:1857) (cid:1871)(cid:1872)(cid:1873)(cid:1856)(cid:1857)(cid:1866)(cid:1872)(cid:1871) and (cid:1839)(cid:1853)(cid:1871)(cid:1872)(cid:1857)(cid:1870)(cid:1871) (cid:1856)(cid:1857)(cid:1859)(cid:1870)(cid:1857)(cid:1857) (cid:1871)(cid:1872)(cid:1873)(cid:1856)(cid:1857)(cid:1866)(cid:1872)(cid:1871) during their academic preparations. In all the
chapters you will see that there is more emphasis on problems and their analysis rather than on theory. In each chapter, you will first read
about the basic required theory, which is then followed by a section on problem sets. In total, there are approximately 700 algorithmic
problems, all with solutions.
If you read the book as a (cid:1871)(cid:1872)(cid:1873)(cid:1856)(cid:1857)(cid:1866)(cid:1872) preparing for competitive exams for Computer Science / Information Technology, the content covers (cid:1853)(cid:1864)(cid:1864)
the (cid:1870)(cid:1857)(cid:1869)(cid:1873)(cid:1861)(cid:1870)(cid:1857)(cid:1856) topics in full detail. While writing this book, my main focus was to help students who are preparing for these exams.
In all the chapters you will see more emphasis on problems and analysis rather than on theory. In each chapter, you will first see the basic
required theory followed by various problems.
For many problems, (cid:1865)(cid:1873)(cid:1864)(cid:1872)(cid:1861)(cid:1868)(cid:1864)(cid:1857) solutions are provided with different levels of complexity. We start with the (cid:1854)(cid:1870)(cid:1873)(cid:1872)(cid:1857) (cid:1858)(cid:1867)(cid:1870)(cid:1855)(cid:1857) solution and slowly
move toward the (cid:1854)(cid:1857)(cid:1871)(cid:1872) (cid:1871)(cid:1867)(cid:1864)(cid:1873)(cid:1872)(cid:1861)(cid:1867)(cid:1866) possible for that problem. For each problem, we endeavor to understand how much time the algorithm takes
and how much memory the algorithm uses.
It is recommended that the reader does at least one complete reading of this book to gain a full understanding of all the topics that are
covered. Then, in subsequent readings you can skip directly to any chapter to refer to a specific topic. Even though many readings have been
done for the purpose of correcting errors, there could still be some minor typos in the book. If any are found, they will be updated
at (cid:1875)(cid:1875)(cid:1875).(cid:1829)(cid:1853)(cid:1870)(cid:1857)(cid:1857)(cid:1870)(cid:1839)(cid:1867)(cid:1866)(cid:1863).(cid:1855)(cid:1867)(cid:1865). You can monitor this site for any corrections and also for new problems and solutions. Also, please provide
your valuable suggestions at: (cid:1835)(cid:1866)(cid:1858)(cid:1867)@(cid:1829)(cid:1853)(cid:1870)(cid:1857)(cid:1857)(cid:1870)(cid:1839)(cid:1867)(cid:1866)(cid:1863).(cid:1855)(cid:1867)(cid:1865).
I wish you all the best and I am confident that you will find this book useful.
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M-Tech, (cid:1835)(cid:1835)(cid:1846) (cid:1828)(cid:1867)(cid:1865)(cid:1854)(cid:1853)(cid:1877)
Founder, (cid:1829)(cid:1853)(cid:1870)(cid:1857)(cid:1857)(cid:1870)(cid:1839)(cid:1867)(cid:1866)(cid:1863).(cid:1855)(cid:1867)(cid:1865)
Data Structure Operations Cheat Sheet
Space
Data Average Case Time Complexity Worst Case Time Complexity
Complexity
Structure
Accessing Accessing
Name Search Insertion Deletion Search Insertion Deletion Worst Case
(cid:1866)(cid:3047)(cid:3035)element (cid:1866)(cid:3047)(cid:3035)element
Arrays O(1) O((cid:1866)) O((cid:1866)) O((cid:1866)) O(1) O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866))
Stacks O((cid:1866)) O((cid:1866)) O(1) O(1) O((cid:1866)) O((cid:1866)) O(1) O(1) O((cid:1866))
Queues O((cid:1866)) O((cid:1866)) O(1) O(1) O((cid:1866)) O((cid:1866)) O(1) O(1) O((cid:1866))
Binary
O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866))
Trees
Binary
Search O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866))
Trees
Balanced
Binary
O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866))
Search
Trees
Hash
N/A O(1) O(1) O(1) N/A O((cid:1866)) O((cid:1866)) O((cid:1866)) O((cid:1866))
Tables
Note: For best case operations, the time complexities are O(1).
Sorting Algorithms Cheat Sheet
Space
Sorting Time Complexity
Complexity Is Sorting Class
Algorithm Remarks
Best Average Worst Stable? Type
Name Worst Case
Case Case Case
Bubble Sort O((cid:1866)) O((cid:1866)(cid:2870)) O((cid:1866)(cid:2870)) O(1) Yes Comparison Not a preferred sorting algorithm.
In the best case (already sorted),
Insertion Sort O((cid:1866)) O((cid:1866)(cid:2870)) O((cid:1866)(cid:2870)) O(1) Yes Comparison
every insert requires constant time
Even a perfectly sorted array
Selection Sort O((cid:1866)(cid:2870)) O((cid:1866)(cid:2870)) O((cid:1866)(cid:2870)) O(1) Yes Comparison
requires scanning the entire array
On arrays, it requires O((cid:1866)) space;
Merge Sort O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)) Yes Comparison and on linked lists, it requires
constant space
By using input array as storage for
Heap Sort O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O(1) No Comparison the heap, it is possible to achieve
constant space
Randomly picking a pivot value
can help avoid worst case
Quick Sort O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)(cid:2870)) O((cid:1864)(cid:1867)(cid:1859)(cid:1866)) No Comparison
scenarios such as a perfectly
sorted array.
Performing inorder traversal on
Tree Sort O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)(cid:1864)(cid:1867)(cid:1859)(cid:1866)) O((cid:1866)(cid:2870)) O((cid:1866)) Yes Comparison
the balanced binary search tree.
(cid:1863) is the range of the non-negative
Counting Sort O((cid:1866)+(cid:1863)) O((cid:1866)+(cid:1863)) O((cid:1866)+(cid:1863)) O((cid:1863)) Yes Linear
key values.
Bucket sort is stable, if the
Bucket Sort O((cid:1866)+(cid:1863)) O((cid:1866)+(cid:1863)) O((cid:1866)(cid:2870)) O((cid:1866)) Yes Linear underlying sorting algorithm is
stable.
Radix sort is stable, if the
Radix Sort O((cid:1856)(cid:1866)) O((cid:1856)(cid:1866)) O((cid:1856)(cid:1866)) O((cid:1856)+(cid:1866)) Yes Linear underlying sorting algorithm is
stable.
Table of Contents
0. Organization of Chapters ------------------------------------------------------------------------------------------------------------------------- 15
0.1 What Is This Book About? --------------------------------------------------------------------------------------------------------------------------------- 15
0.2 Should I Buy This Book? ----------------------------------------------------------------------------------------------------------------------------------- 15
0.3 Organization of Chapters------------------------------------------------------------------------------------------------------------------------------------- 15
0.4 Some Prerequisites --------------------------------------------------------------------------------------------------------------------------------------------- 18
1. Introduction -------------------------------------------------------------------------------------------------------------------------------------------- 19
1.1 Variables ------------------------------------------------------------------------------------------------------------------------------------------------------------ 19
1.2 Data Types --------------------------------------------------------------------------------------------------------------------------------------------------------- 19
1.3 Data Structures --------------------------------------------------------------------------------------------------------------------------------------------------- 20
1.4 Abstract Data Types (ADTs) ------------------------------------------------------------------------------------------------------------------------------- 20
1.5 What is an Algorithm? ---------------------------------------------------------------------------------------------------------------------------------------- 20
1.6 Why the Analysis of Algorithms? ------------------------------------------------------------------------------------------------------------------------- 20
1.7 Goal of the Analysis of Algorithms ----------------------------------------------------------------------------------------------------------------------- 20
1.8 What is Running Time Analysis? ------------------------------------------------------------------------------------------------------------------------- 20
1.9 How to Compare Algorithms------------------------------------------------------------------------------------------------------------------------------- 21
1.10 What is Rate of Growth? ----------------------------------------------------------------------------------------------------------------------------------- 21
1.11 Commonly Used Rates of Growth --------------------------------------------------------------------------------------------------------------------- 21
1.12 Types of Analysis ---------------------------------------------------------------------------------------------------------------------------------------------- 22
1.13 Asymptotic Notation ----------------------------------------------------------------------------------------------------------------------------------------- 22
1.14 Big-O Notation ------------------------------------------------------------------------------------------------------------------------------------------------- 22
1.15 Omega-Ω Notation -------------------------------------------------------------------------------------------------------------------------------------------- 23
1.16 Theta- Notation---------------------------------------------------------------------------------------------------------------------------------------------- 24
1.17 Why is it called Asymptotic Analysis?----------------------------------------------------------------------------------------------------------------- 25
1.18 Guidelines for Asymptotic Analysis -------------------------------------------------------------------------------------------------------------------- 25
1.19 Simplifying properties of asymptotic notations ---------------------------------------------------------------------------------------------------- 26
1.21 Commonly used Logarithms and Summations ---------------------------------------------------------------------------------------------------- 26
1.21 Master Theorem for Divide and Conquer Recurrences --------------------------------------------------------------------------------------- 27
1.22 Divide and Conquer Master Theorem: Problems & Solutions ----------------------------------------------------------------------------- 27
1.23 Master Theorem for Subtract and Conquer Recurrences------------------------------------------------------------------------------------- 28
1.24 Variant of Subtraction and Conquer Master Theorem----------------------------------------------------------------------------------------- 28
1.25 Method of Guessing and Confirming ----------------------------------------------------------------------------------------------------------------- 28
1.26 Amortized Analysis ------------------------------------------------------------------------------------------------------------------------------------------- 29
1.27 Algorithms Analysis: Problems & Solutions -------------------------------------------------------------------------------------------------------- 30
2. Recursion and Backtracking--------------------------------------------------------------------------------------------------------------------- 41
2.1 Introduction ------------------------------------------------------------------------------------------------------------------------------------------------------- 41
2.2 What is Recursion? --------------------------------------------------------------------------------------------------------------------------------------------- 41
2.3 Why Recursion? ------------------------------------------------------------------------------------------------------------------------------------------------- 41
2.4 Format of a Recursive Function --------------------------------------------------------------------------------------------------------------------------- 41
2.5 Recursion and Memory (Visualization) ----------------------------------------------------------------------------------------------------------------- 42
2.6 Recursion versus Iteration ----------------------------------------------------------------------------------------------------------------------------------- 42
2.7 Notes on Recursion -------------------------------------------------------------------------------------------------------------------------------------------- 42
2.8 Example Algorithms of Recursion------------------------------------------------------------------------------------------------------------------------ 43
2.9 Recursion: Problems & Solutions ------------------------------------------------------------------------------------------------------------------------- 43
2.10 What is Backtracking?--------------------------------------------------------------------------------------------------------------------------------------- 43
2.11 Example Algorithms of Backtracking ----------------------------------------------------------------------------------------------------------------- 44
2.12 Backtracking: Problems & Solutions------------------------------------------------------------------------------------------------------------------- 44
3. Linked Lists -------------------------------------------------------------------------------------------------------------------------------------------- 47
3.1 What is a Linked List? ---------------------------------------------------------------------------------------------------------------------------------------- 47
3.2 Linked Lists ADT----------------------------------------------------------------------------------------------------------------------------------------------- 47
3.3 Why Linked Lists? --------------------------------------------------------------------------------------------------------------------------------------------- 47
3.4 Arrays Overview ------------------------------------------------------------------------------------------------------------------------------------------------- 47
3.5 Comparison of Linked Lists with Arrays and Dynamic Arrays ------------------------------------------------------------------------------- 48
3.6 Singly Linked Lists ---------------------------------------------------------------------------------------------------------------------------------------------- 49
3.7 Doubly Linked Lists ------------------------------------------------------------------------------------------------------------------------------------------- 54
3.8 Circular Linked Lists ------------------------------------------------------------------------------------------------------------------------------------------- 58
3.9 A Memory-efficient Doubly Linked List --------------------------------------------------------------------------------------------------------------- 62
3.11 Skip Lists ---------------------------------------------------------------------------------------------------------------------------------------------------------- 67
3.12 Linked Lists: Problems & Solutions ------------------------------------------------------------------------------------------------------------------- 70
4. Stacks ----------------------------------------------------------------------------------------------------------------------------------------------------- 90
4.1 What is a Stack? ------------------------------------------------------------------------------------------------------------------------------------------------- 90
4.2 How Stacks are Used ------------------------------------------------------------------------------------------------------------------------------------------ 90
4.3 Stack ADT --------------------------------------------------------------------------------------------------------------------------------------------------------- 90
4.4 Applications ------------------------------------------------------------------------------------------------------------------------------------------------------- 91
4.5 Implementation -------------------------------------------------------------------------------------------------------------------------------------------------- 91
4.6 Comparison of Implementations ------------------------------------------------------------------------------------------------------------------------- 95
4.7 Stacks: Problems & Solutions ------------------------------------------------------------------------------------------------------------------------------ 95
5. Queues ------------------------------------------------------------------------------------------------------------------------------------------------- 111
5.1 What is a Queue? --------------------------------------------------------------------------------------------------------------------------------------------- 111
5.2 How are Queues Used -------------------------------------------------------------------------------------------------------------------------------------- 111
5.3 Queue ADT ----------------------------------------------------------------------------------------------------------------------------------------------------- 111
5.4 Exceptions -------------------------------------------------------------------------------------------------------------------------------------------------------- 111
5.5 Applications ----------------------------------------------------------------------------------------------------------------------------------------------------- 111
5.6 Implementation ------------------------------------------------------------------------------------------------------------------------------------------------ 112
5.7 Queues: Problems & Solutions -------------------------------------------------------------------------------------------------------------------------- 116
6. Trees ---------------------------------------------------------------------------------------------------------------------------------------------------- 125
6.1 What is a Tree? ------------------------------------------------------------------------------------------------------------------------------------------------ 125
6.2 Glossary ----------------------------------------------------------------------------------------------------------------------------------------------------------- 125
6.3 Binary Trees ----------------------------------------------------------------------------------------------------------------------------------------------------- 126
6.4 Types of Binary Trees --------------------------------------------------------------------------------------------------------------------------------------- 126
6.5 Properties of Binary Trees--------------------------------------------------------------------------------------------------------------------------------- 126
6.6 Binary Tree Traversals -------------------------------------------------------------------------------------------------------------------------------------- 128
6.7 Generic Trees (N-ary Trees) ------------------------------------------------------------------------------------------------------------------------------ 146
