A Textbook of Data Structures and Algorithms, Volume 2: Mastering Nonlinear Data Structures

304 Pages·2022·27.716 MB·English

by  G.A. Vijayalakshmi Pai

A Textbook of Data Structures and Algorithms 2
One of the greatest lessons I have learnt in my life is
to pay as much attention to the means of work as to its end…
I have been always learning great lessons from that one principle,
and it appears to me that all the secret of success is there;
to pay as much attention to the means as to the end….
Let us perfect the means; the end will take care of itself.

– Swami Vivekananda
(Lecture Delivered at Los Angeles, California, January 4, 1900)
A Textbook of Data
Structures and Algorithms 2

Mastering Nonlinear Data Structures

G A Vijayalakshmi Pai
First published 2022 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Previous edition published in 2008 as “Data Structures and Algorithms: Concepts, Techniques and
Applications” by McGraw Hill Education (India) Pvt Ltd. © McGraw Hill Education (India) Pvt Ltd. 2008

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted
under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or
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ISTE Ltd John Wiley & Sons, Inc.
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© ISTE Ltd 2022
The rights of G A Vijayalakshmi Pai to be identified as the author of this work have been asserted by her
in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the

author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2022947241

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-891-7
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 8. Trees and Binary Trees . . . . . . . . . . . . . . . . . . . . . . . 1
8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
8.2. Trees: definition and basic terminologies . . . . . . . . . . . . . . . . . . 1
8.2.1. Definition of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
8.2.2. Basic terminologies of trees . . . . . . . . . . . . . . . . . . . . . . . . 2
8.3. Representation of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
8.4. Binary trees: basic terminologies and types . . . . . . . . . . . . . . . . . 6
8.4.1. Basic terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
8.4.2. Types of binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
8.5. Representation of binary trees . . . . . . . . . . . . . . . . . . . . . . . . . 8
8.5.1. Array representation of binary trees . . . . . . . . . . . . . . . . . . . 8
8.5.2. Linked representation of binary trees . . . . . . . . . . . . . . . . . . 10
8.6. Binary tree traversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
8.6.1. Inorder traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
8.6.2. Postorder traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.6.3. Preorder traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8.7. Threaded binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.7.1. Linked representation of a threaded binary tree . . . . . . . . . . . . 24
8.7.2. Growing threaded binary trees . . . . . . . . . . . . . . . . . . . . . . 24
vi A Textbook of Data Structures and Algorithms 2
8.8. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
8.8.1. Expression trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.8.2. Traversals of an expression tree . . . . . . . . . . . . . . . . . . . . . 27
8.8.3. Conversion of infix expression to postfix expression . . . . . . . . . 27
8.8.4. Segment trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.9. Illustrative problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Chapter 9. Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.2. Definitions and basic terminologies . . . . . . . . . . . . . . . . . . . . . 63
9.3. Representations of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.3.1. Sequential representation of graphs . . . . . . . . . . . . . . . . . . . 76
9.3.2. Linked representation of graphs . . . . . . . . . . . . . . . . . . . . . 80
9.4. Graph traversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.4.1. Breadth first traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.4.2. Depth first traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9.5.1. Single source shortest path problem . . . . . . . . . . . . . . . . . . . 87
9.5.2. Minimum cost spanning trees . . . . . . . . . . . . . . . . . . . . . . . 90
9.6. Illustrative problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Chapter 10. Binary Search Trees and AVL Trees . . . . . . . . . . . . . 115
10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10.2. Binary search trees: definition and operations . . . . . . . . . . . . . . . 115
10.2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10.2.2. Representation of a binary search tree . . . . . . . . . . . . . . . . . 116
10.2.3. Retrieval from a binary search tree . . . . . . . . . . . . . . . . . . . 117
10.2.4. Why are binary search tree retrievals more efficient
than sequential list retrievals? . . . . . . . . . . . . . . . . . . . . . . . . . . 118
10.2.5. Insertion into a binary search tree . . . . . . . . . . . . . . . . . . . 120
10.2.6. Deletion from a binary search tree . . . . . . . . . . . . . . . . . . . 122
10.2.7. Drawbacks of a binary search tree . . . . . . . . . . . . . . . . . . . 125
10.2.8. Counting binary search trees . . . . . . . . . . . . . . . . . . . . . . 128
10.3. AVL trees: definition and operations . . . . . . . . . . . . . . . . . . . . 130
10.3.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.3.2. Retrieval from an AVL search tree . . . . . . . . . . . . . . . . . . . 132
10.3.3. Insertion into an AVL search tree . . . . . . . . . . . . . . . . . . . 133
10.3.4. Deletion from an AVL search tree . . . . . . . . . . . . . . . . . . . 141
10.3.5. R category rotations associated with the delete operation . . . . . 146
10.3.6. L category rotations associated with the delete operation . . . . . 150
Contents vii
10.4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.4.1. Representation of symbol tables in compiler design . . . . . . . . . 151
10.5. Illustrative problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Chapter 11. B Trees and Tries . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.2. m-way search trees: definition and operations . . . . . . . . . . . . . . . 176
11.2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
11.2.2. Node structure and representation . . . . . . . . . . . . . . . . . . . 176
11.2.3. Searching an m-way search tree . . . . . . . . . . . . . . . . . . . . 178
11.2.4. Inserting into an m-way search tree . . . . . . . . . . . . . . . . . . 178
11.2.5. Deleting from an m-way search tree . . . . . . . . . . . . . . . . . . 179
11.2.6. Drawbacks of m-way search trees . . . . . . . . . . . . . . . . . . . 184
11.3. B trees: definition and operations . . . . . . . . . . . . . . . . . . . . . . 184
11.3.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
11.3.2. Searching a B tree of order m . . . . . . . . . . . . . . . . . . . . . . 186
11.3.3. Inserting into a B tree of order m . . . . . . . . . . . . . . . . . . . . 186
11.3.4. Deletion from a B tree of order m . . . . . . . . . . . . . . . . . . . 190
11.3.5. Height of a B tree of order m . . . . . . . . . . . . . . . . . . . . . . 194
11.4. Tries: definition and operations . . . . . . . . . . . . . . . . . . . . . . . 195
11.4.1. Definition and representation . . . . . . . . . . . . . . . . . . . . . . 195
11.4.2. Searching a trie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.4.3. Insertion into a trie . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.4.4. Deletion from a trie . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
11.4.5. Some remarks on tries . . . . . . . . . . . . . . . . . . . . . . . . . . 200
11.5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
11.5.1. File indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11.5.2. Spell checker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.6. Illustrative problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Chapter 12. Red-Black Trees and Splay Trees . . . . . . . . . . . . . . . 215
12.1. Red-black trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
12.1.1. Introduction to red-black trees . . . . . . . . . . . . . . . . . . . . . 215
12.1.2. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
12.1.3. Representation of a red-black tree . . . . . . . . . . . . . . . . . . . 219
12.1.4. Searching a red-black tree . . . . . . . . . . . . . . . . . . . . . . . . 220
12.1.5. Inserting into a red-black tree . . . . . . . . . . . . . . . . . . . . . . 220
12.1.6. Deleting from a red-black tree . . . . . . . . . . . . . . . . . . . . . 228
12.1.7. Time complexity of search, insert and delete operations
on a red-black tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
viii A Textbook of Data Structures and Algorithms 2
12.2. Splay trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
12.2.1. Introduction to splay trees . . . . . . . . . . . . . . . . . . . . . . . . 236
12.2.2. Splay rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
12.2.3. Some remarks on amortized analysis of splay trees . . . . . . . . . 242
12.3. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
12.4. Illustrative problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Summaries of other volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . 265