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Data Structure and Algorithms Using C++
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Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Publishers at Scrivener Martin Scrivener (martin@scrivenerpublishing.com) Phillip Carmical (pcarmical@scrivenerpublishing.com)
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Data Structure and Algorithms Using C++ A Practical Implementation Edited by Sachi Nandan Mohanty ICFAI Foundation For Higher Education, Hyderabad, India and Pabitra Kumar Tripathy Kalam Institute of Technology, Berhampur, India
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This edition first published 2021 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2021 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or other- wise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley prod- ucts visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no rep- resentations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchant- ability or fitness for a particular purpose. No warranty may be created or extended by sales representa- tives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further informa- tion does not mean that the publisher and authors endorse the information or services the organiza- tion, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-75054-3 Cover image: Pixabay.Com Cover design by Russell Richardson Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines Printed in the USA 10 9 8 7 6 5 4 3 2 1
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v Contents Preface xi 1 Introduction to Data Structure 1 1.1 Definition and Use of Data Structure 1 1.2 Types of Data Structure 2 1.3 Algorithm 3 1.4 Complexity of an Algorithm 6 1.5 Efficiency of an Algorithm 7 1.6 Asymptotic Notations 8 1.7 How to Determine Complexities 9 1.8 Questions 13 2 Review of Concepts of ‘C++’ 15 2.1 Array 15 2.1.1 One-Dimensional Array 16 2.1.2 Multi-Dimensional Array 17 2.1.3 String Handling 20 2.2 Function 26 2.2.1 User Defined Functions 26 2.2.2 Construction of a Function 27 2.2.3 Actual Argument and Formal Argument 31 2.2.4 Call by Value and Call by Reference 32 2.2.5 Default Values for Parameters 34 2.2.6 Storage Class Specifiers 35 2.3 Pointer 37 2.3.1 Declaration of a Pointer 37 2.3.2 Initialization of a Pointer 37 2.3.3 Arithmetic With Pointer 38 2.3.4 Passing of a Pointer to Function 39 2.3.5 Returning of a Pointer by Function 40 2.3.6 C++ Null Pointer 41
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vi Contents 2.4 Structure 42 2.4.1 The typedef Keyword 46 2.5 Questions 47 3 Sparse Matrix 49 3.1 What is Sparse Matrix 49 3.2 Sparse Matrix Representations 49 3.3 Algorithm to Represent the Sparse Matrix 51 3.4 Programs Related to Sparse Matrix 52 3.5 Why to Use Sparse Matrix Instead of Simple Matrix? 56 3.6 Drawbacks of Sparse Matrix 57 3.7 Sparse Matrix and Machine Learning 57 3.8 Questions 58 4 Concepts of Class 59 4.1 Introduction to CLASS 59 4.2 Access Specifiers in C++ 60 4.3 Declaration of Class 60 4.4 Some Manipulator Used In C++ 62 4.5 Defining the Member Functions Outside of the Class 64 4.6 Array of Objects 64 4.7 Pointer to Object 66 4.8 Inline Member Function 67 4.9 Friend Function 69 4.9.1 Simple Friend Function 69 4.9.2 Friend With Inline Substitution 70 4.9.3 Granting Friendship to Another Class (Friend Class) 71 4.9.4 More Than One Class Having the Same Friend Function 73 4.10 Static Data Member and Member Functions 75 4.11 Constructor and Destructor 78 4.11.1 Constructor 78 4.11.1.1 Empty Constructor 79 4.11.1.2 Default Constructor 79 4.11.1.3 Parameterized Constructors 80 4.11.1.4 Copy Constructor 81 4.11.2 Destructor 83 4.12 Dynamic Memory Allocation 84 4.13 This Pointer 86 4.14 Class Within Class 87 4.15 Questions 89
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Contents vii 5 Stack 91 5.1 STACK 91 5.2 Operations Performed With STACK 91 5.3 ALGORITHMS 93 5.4 Applications of STACK 96 5.5 Programming Implementations of STACK 106 5.6 Questions 126 6 Queue 129 6.1 Queue 129 6.2 Types of Queue 129 6.3 Linear Queue 129 6.4 Circular Queue 134 6.5 Double Ended Queue 138 6.6 Priority Queue 139 6.7 Programs 142 6.8 Questions 165 7 Linked List 167 7.1 Why Use Linked List? 167 7.2 Types of Link List 167 7.3 Single Link List 168 7.4 Programs Related to Single Linked List 177 7.4.1 /* Creation of a Linked List */ 177 7.4.2 /* Insert a Node Into a Simple Linked List at the Beginning */ 178 7.4.3 /* Insert a Node Into a Simple Linked List at the End of the List */ 180 7.4.4 /* Insert a Node Into a Simple Linked List When the Node Is Known */ 182 7.4.5 /* Insert a Node Into a Simple Linked List Information Is Known and Put After Some Specified Node */ 184 7.4.6 /* Deleting the First Node From a Simple Linked List */ 187 7.4.7 /* Deleting the Last Node From a Simple Linked List */ 189 7.4.8 /* Deleting a Node From a Simple Linked List When Node Number Is Known */ 191 7.4.9 Deleting a Node From a Simple Linked List When Information of a Node Is Given 193
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viii Contents 7.4.10 /* SEARCH A NODE INTO A SIMPLE LINKED LIST WITH INFORMATION IS KNOWN*/ 197 7.4.11 /* Sorting a Linked List in Ascending Order */ 199 7.4.12 /* Reversing a Linked List */ 202 7.4.13 Program for Student Data Using Linked List 203 7.5 Double Link List 210 7.6 Programs on Double Linked List 216 7.6.1 /* Creation of Double Linked List */ 216 7.6.2 /* Inserting First Node in the Doubly Linked List */ 218 7.6.3 /*Inserting a Node in the Doubly Linked List When Node Number Is Known*/ 220 7.6.4 /*Inserting a Node in the Doubly Linked List When Information Is Known*/ 223 7.6.5 /* Delete First Node From a Double Linked List */ 226 7.6.6 /*Delete the Last Node From the Double Linked List*/ 229 7.7 Header Linked List 231 7.7.1 /* Inserting a Node Into a Header Linked List */ 233 7.8 Circular Linked List 235 7.9 Application of Linked List 239 7.9.1 Addition of Two Polynomial 239 7.9.2 /* Polynomial With Help of Linked List */ 240 7.9.3 Program for Linked Queue 241 7.9.4 Program for Linked Stack 243 7.10 Garbage Collection and Compaction 245 7.11 Questions 247 8 TREE 249 8.1 Tree Terminologies 249 8.2 Binary Tree 251 8.3 Representation of Binary Tree 253 8.3.1 Array Representation of a Tree 253 8.3.2 Linked List Representation of a Tree 254 8.4 Operations Performed With the Binary Tree 254 8.4.1 /*Creation of a Tree*/ 255 8.5 Traversing With Tree 256 8.5.1 /* Binary Tree Traversal */ 259 8.6 Conversion of a Tree From Inorder and Preorder 262
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Contents ix 8.7 Types of Binary Tree 265 8.8 Expression Tree 265 8.9 Binary Search Tree 268 8.10 Height Balanced Tree (AVL Tree) 272 8.11 Threaded Binary Tree 277 8.12 Heap Tree 279 8.13 Huffman Tree 282 8.14 Decision Tree 286 8.15 B-Tree 287 8.16 B + Tree 292 8.17 General Tree 293 8.18 Red–Black Tree 293 8.19 Questions 294 9 Graph 295 9.1 Graph Terminologies 295 9.2 Representation of Graph 301 9.3 Traversal of Graph 305 9.3.1 Breadth First Search (BFS) 305 9.3.2 Depth First Search 311 9.4 Spanning Tree 315 9.4.1 Kruskal Algorithm 315 9.4.2 Prim’s Algorithm 318 9.5 Single Source Shortest Path 322 9.5.1 Bellman–Ford Algorithm 323 9.5.2 Dijkstra’s Algorithm 327 9.6 All Pair Shortest Path 335 9.7 Topological Sorting 345 9.8 Questions 347 10 Searching and Sorting 349 10.1 Linear Search 349 10.2 Binary Search 351 10.3 Bubble Sort 355 10.4 Selection Sort 359 10.5 Insertion Sort 361 10.6 Merge Sort 363 10.7 Quick Sort 366 10.8 Radix Sort 369 10.9 Heap Sort 372 10.10 Questions 389
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x Contents 11 Hashing 391 11.1 Hash Functions 391 11.2 Collisions 393 11.3 Collision Resolution Methods 393 11.4 Clustering 394 11.5 Questions 395 Index 397
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xi Preface This book was designed to serve as a textbook for undergraduate engi- neering students across all disciplines and postgraduate level courses in computer applications. Young researchers working on efficient data storage and related applications will also find it to be a helpful reference source to guide them in the newly established techniques of this rapidly growing research field. Dr. Sachi Nandan Mohanty and Prof. Pabitra Kumar Tripathy December 2020 Welcome to the first edition of Data Structures and Algorithms Using C++. A data structure is the logical or mathematical arrangement of data in memory. To be effective, data has to be organized in a manner that adds to the efficiency of an algorithm and also describe the relationships between these data items and the operations that can be performed on these items. The choice of appropriate data structures and algorithms forms the funda- mental step in the design of an efficient program. Thus, a deep understand- ing of data structure concepts is essential for students who wish to work on the design and implementation of system software written in C++, an object-oriented programming language that has gained popularity in both academia and industry. Therefore, this book was developed to provide comprehensive and logical coverage of data structures like stacks, queues, linked lists, trees and graphs, which makes it an excellent choice for learn- ing data structures. The objective of the book is to introduce the concepts of data structures and apply these concepts in real-life problem solving. Most of the examples presented resulted from student interaction in the classroom. This book utilizes a systematic approach wherein the design of each of the data structures is followed by algorithms of different operations that can be performed on them and the analysis of these algorithms in terms of their running times.
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1 1 Introduction to Data Structure 1.1 Definition and Use of Data Structure Data structure is the representation of the logical relationship existing between individual elements of data. In other words the data structure is a way of organizing all data items that considers not only the elements stored but also their relationship to each other. Data structure specifies • Organization of data • Accessing methods • Degree of associativity • Processing alternatives for information The data structures are the building blocks of a program and hence the selection of a particular data structure stresses on • The data structures must be rich enough in structure to reflect the relationship existing between the data, and • The structure should be simple so that we can process data effectively whenever required. In mathematically Algorithm + Data Structure = Program Finally we can also define the data structure as the “Logical and mathe- matical model of a particular organization of data” Sachi Nandan Mohanty and Pabitra Kumar Tripathy. Data Structure and Algorithms Using C++: A Practical Implementation, (1–14) © 2021 Scrivener Publishing LLC
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2 1.2 Types of Data Structure Data structure can be broadly classified into two categories as Linear and Non-Linear DATA STRUCTURE LINEAR NON LINEAR ARRAY STACKQUEUE TREE GRAPH TABLES SETS Linear Data Structures In linear data structures, values are arranged in linear fashion. Arrays, linked lists, stacks, and queues are the examples of linear data structures in which values are stored in a sequence. Non-Linear Data Structure This type is opposite to linear. The data values in this structure are not arranged in order. Tree, graph, table, and sets are the examples of non- linear data structure. Operations Performed in Data Structure In data structure we can perform the operations like • Traversing • Insertion • Deletion • Merging • Sorting • Searching Data Structure and Algorithms Using C++
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Introduction to Data Structure 3 1.3 Algorithm The step by step procedure to solve a problem is known as the ALGORITHM. An algorithm is a well-organized, pre-arranged, and defined computational module that receives some values or set of values as input and provides a single or set of values as out put. These well-defined computational steps are arranged in sequence, which processes the given input into output. An algorithm is said to be accurate and truthful only when it provides the exact wanted output. The efficiency of an algorithm depends on the time and space complex- ities. The complexity of an algorithm is the function which gives the run- ning time and/or space in terms of the input size. Steps Required to Develop an Algorithm • Finding a method for solving a problem. Every step of an algorithm should be defined in a precise and in a clear man- ner. Pseudo code is also used to describe an algorithm. • The next step is to validate the algorithm. This step includes all the steps in our algorithm and should be done manually by giving the required input, perform the required steps including in our algorithm and should get the required amount of output in a finite amount of time. • Finally implement the algorithm in terms of programming language. Mathematical Notations and Functions Floor and Ceiling Functions Floor function returns the greatest integer that does not exceed the number. Ceiling function returns the least integer that is not less than the number. no denotes the floor function no denotes the ceil function Ex : 5.23 = 5 5.23 = 6
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4 Remainder Function To find the remainder “mod” function is being used as A mod B To find the Integer and Absolute value of a number INT(5.34) = 5 This statement returns the integer part of the number INT(- 6.45) = 6 This statement returns the absolute as well as the integer portion of the number Summation Symbol To add a series of number as a1+ a2 + a3 +…………+ an the symbol Σ is used n Σ ai i=1 Factorial of a Number The product of the positive integers from 1 to n is known as the factorial of n and it is denoted as n!. 0! = 1 Algorithemic Notations While writing the algorithm the comments are provided with in [ ]. The assignment should use the symbol “: =” instead of “=” For Input use Read : variable name For output use write : message/variable name The control structures can also be allowed to use inside an algorithm but their way of approaching will be some what different as Simple If If condition, then: Statements [end of if structure] Data Structure and Algorithms Using C++
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Introduction to Data Structure 5 If…else If condition, then: Statements Else : Statements [end of if structure] If…else ladder If condition1, then: Statements Else If condition2, then: Statements Else If condition3, then: Statements Else If conditionN, then: Statements Else: Statements [end of if structure] LOOPING CONSTRUCT Repeat for var = start_value to end_value by step_value Statements [end of loop] Repeat while condition: Statements [end of loop] Ex : repeat for I = 1 to 10 by 2 Write: i [end of loop] OUTPUT 1 3 5 7 9
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6 1.4 Complexity of an Algorithm The complexity of programs can be judged by criteria such as whether it satisfies the original specification task, whether the code is readable. These factors affect the computing time and storage requirement of the program. Space Complexity The space complexity of a program is the amount of memory it needs to run to completion. The space needed by a program is the sum of the fol- lowing components: • A fixed part that includes space for the code, space for sim- ple variables and fixed size component variables, space for constants, etc. • A variable part that consists of the space needed by com- ponent variables whose size is dependent on the particular problem instance being solved, and the stack space used by recursive procedures. Time Complexity The time complexity of a program is the amount of computer time it needs to run to completion. The time complexity is of two types such as • Compilation time • Runtime The amount of time taken by the compiler to compile an algorithm is known as compilation time. During compilation time it does not calculate for the executable statements, it calculates only the declaration statements and checks for any syntax and semantic errors. The run time depends on the size of an algorithm. If the number of instructions in an algorithm is large, then the run time is also large, and if the number of instructions in an algorithm is small, then the time for exe- cuting the program is also small. The runtime is calculated for executable statements and not for declaration statements. Data Structure and Algorithms Using C++
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Introduction to Data Structure 7 Suppose space is fixed for one algorithm then only run time will be con- sidered for obtaining the complexity of algorithm, these are • Best case • Worst case • Average case Best Case Generally, most of the algorithms behave sometimes in best case. In this case, algorithm searches the element for the first time by itself. For example: In linear search, if it finds the element for the first time by itself, then it behaves as the best case. Best case takes shortest time to exe- cute, as it causes the algorithms to do the least amount of work. Worst Case In worst case, we find the element at the end or when searching of elements fails. This could involve comparing the key to each list value for a total of N comparisons. For example in linear search suppose the element for which algorithm is searching is the last element of array or it is not available in array then algorithm behaves as worst case. Average Case Analyzing the average case behavior algorithm is a little bit complex than the best case and worst case. Here, we take the probability with a list of data. Average case of algorithm should be the average number of steps but since data can be at any place, so finding exact behavior of algorithm is difficult. As the volume of data increases, the average case of algorithm behaves like the worst case of algorithm. 1.5 Efficiency of an Algorithm Efficiency of an algorithm can be determined by measuring the time, space, and amount of resources it uses for executing the program. The amount of time taken by an algorithm can be calculated by finding the number of steps the algorithm executes, while the space refers to the number of units it requires for memory storage.
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8 1.6 Asymptotic Notations The asymptotic notations are the symbols which are used to solve the dif- ferent algorithms and the notations are • Big Oh Notation (O) • Little Oh Notation (o) • Omega Notation (W) • Theta Notation (q) Big Oh (O) Notation This Notation gives the upper bound for a function to within a constant factor. We write f(n) = O(g(n)) if there are +ve constants n0 and C such that to the right of n0, the value of f(n) always lies on or below Cg(n) Omega Notation (W) This notation gives a lower bound for a function to with in a constant fac- tor. We write f(n) = Ωg(n) if there are positive constants n0 and C such that to the right of n0 the value of f(n) always lies on or above Cg(n) Theta Notation (q) This notation bounds the function to within constant factors. We say f(n) = θg(n) if there exists +ve constants n0, C1 and C2 such that to the right of n0 the value of f(n) always lies between c1g(n) and c2(g(n)) inclusive. Little Oh Notation (o) F(n) = o(g(n)) iff f(n) = O(g(n)) and f(n) != Ωg(n). Introduction An important question is: How efficient is an algorithm or piece of code? Efficiency covers lots of resources, including: CPU (time) usage Memory usage Disk usage Network usage Data Structure and Algorithms Using C++