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Data structures algorithms_tutorial

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This document is a tutorial on data structures and algorithms, aimed at computer science graduates and software professionals seeking to enhance their programming skills. It covers foundational concepts of data structures, algorithm analysis, and common programming practices using the C programming language. Additionally, it includes setup instructions for various operating systems and numerous practical examples to facilitate learning.

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Data structures algorithms_tutorial
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Data Structures & Algorithms AbouttheTutorial Data Structures are the programmatic way of storing data so that data can be used efficiently. Almost every enterprise application uses various types of data structures in one or the other way. This tutorial will give you a great understanding on Data Structures needed to understand the complexity of enterprise level applications and need of algorithms, and data structures. Audience This tutorial is designed for Computer Science graduates as well as Software Professionals who are willing to learn data structures and algorithm programming in simple and easy steps. After completing this tutorial you will be at intermediate level of expertise from where you can take yourself to higher level of expertise. Prerequisites Before proceeding with this tutorial, you should have a basic understanding of C programming language, text editor, and execution of programs, etc. CopyrightandDisclaimer © Copyright 2016 by Tutorials Point (I) Pvt. Ltd. All the content and graphics published in this e-book are the property of Tutorials Point (I) Pvt. Ltd. The user of this e-book is prohibited to reuse, retain, copy, distribute or republish any contents or a part of contents of this e-book in any manner without written consent of the publisher. We strive to update the contents of our website and tutorials as timely and as precisely as possible, however, the contents may contain inaccuracies or errors. Tutorials Point (I) Pvt. Ltd. provides no guarantee regarding the accuracy, timeliness or completeness of our website or its contents including this tutorial. If you discover any errors on our website or in this tutorial, please notify us at contact@tutorialspoint.com i
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Data Structures & Algorithms Compile&ExecuteOnline For most of the examples given in this tutorial you will find Try it option, so just make use of this option to execute your programs on the spot and enjoy your learning. Try the following example using the Try it option available at the top right corner of the following sample code box − #include <stdio.h> int main(){ /* My first program in C */ printf("Hello, World! n"); return 0; } ii
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Data Structures & Algorithms TableofContents About the Tutorial ............................................................................................................................................i Audience........................................................................................................................................................... i Prerequisites.....................................................................................................................................................i Copyright and Disclaimer .............................................................................................................................i Compile & Execute Online............................................................................................................................... ii Table of Contents ........................................................................................................................................... iii BASICS.........................................................................................................................................1 1. Overview .................................................................................................................................................2 Characteristics of a Data Structure..................................................................................................................2 Need for Data Structure ..................................................................................................................................2 Execution Time Cases ......................................................................................................................................3 Basic Terminology ...........................................................................................................................................3 2. Environment Setup ..................................................................................................................................4 Try it Option Online .........................................................................................................................................4 Local Environment Setup.................................................................................................................................4 Installation on UNIX/Linux...............................................................................................................................5 Installation on Mac OS.....................................................................................................................................5 Installation on Windows..................................................................................................................................6 ALGORITHM................................................................................................................................7 3. Algorithms ─ Basics...................................................................................................................................8 Characteristics of an Algorithm .......................................................................................................................8 How to Write an Algorithm? ...........................................................................................................................9 Algorithm Analysis.........................................................................................................................................10 Algorithm Complexity....................................................................................................................................11 Space Complexity ..........................................................................................................................................11 Time Complexity............................................................................................................................................11 4. Asymptotic Analysis...............................................................................................................................12 Asymptotic Notations....................................................................................................................................12 Common Asymptotic Notations ....................................................................................................................15 5. Greedy Algorithms.................................................................................................................................16 Counting Coins...............................................................................................................................................16 6. Divide & Conquer....................................................................................................................................18 Divide/Break..................................................................................................................................................18 Conquer/Solve...............................................................................................................................................18 Merge/Combine ............................................................................................................................................19 7. Dynamic Programming............................................................................................................................20 iii
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Data Structures & Algorithms DATA STRUCTURES ...................................................................................................................21 8. Basic Concepts .......................................................................................................................................22 Data Definition ..............................................................................................................................................22 Data Object....................................................................................................................................................22 Data Type.......................................................................................................................................................22 Basic Operations............................................................................................................................................23 9. Arrays ....................................................................................................................................................24 Array Representation ....................................................................................................................................24 Basic Operations............................................................................................................................................25 Insertion Operation .......................................................................................................................................25 Array Insertions .............................................................................................................................................27 Insertion at the Beginning of an Array ..........................................................................................................28 Insertion at the Given Index of an Array .......................................................................................................30 Insertion After the Given Index of an Array ..................................................................................................32 Insertion Before the Given Index of an Array................................................................................................34 Deletion Operation........................................................................................................................................36 Search Operation...........................................................................................................................................37 Update Operation..........................................................................................................................................39 LINKED LIST...............................................................................................................................41 10. Linked List ─ Basics..................................................................................................................................42 Linked List Representation ............................................................................................................................42 Types of Linked List .......................................................................................................................................42 Basic Operations............................................................................................................................................43 Insertion Operation .......................................................................................................................................43 Deletion Operation........................................................................................................................................44 Reverse Operation.........................................................................................................................................45 Linked List Program in C ................................................................................................................................46 11. Doubly Linked List...................................................................................................................................55 Doubly Linked List Representation................................................................................................................55 Basic Operations............................................................................................................................................55 Insertion Operation .......................................................................................................................................56 Deletion Operation........................................................................................................................................57 Insertion at the End of an Operation.............................................................................................................57 Doubly Linked List Program in C....................................................................................................................58 12. Circular Linked List..................................................................................................................................67 Singly Linked List as Circular ..........................................................................................................................67 Doubly Linked List as Circular ........................................................................................................................67 Basic Operations............................................................................................................................................67 Insertion Operation .......................................................................................................................................68 Deletion Operation........................................................................................................................................68 Display List Operation....................................................................................................................................69 Circular Linked List Program in C...................................................................................................................69 iv
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Data Structures & Algorithms STACK & QUEUE........................................................................................................................74 13. Stack......................................................................................................................................................75 Stack Representation.....................................................................................................................................75 Basic Operations............................................................................................................................................76 peek().............................................................................................................................................................76 isfull().............................................................................................................................................................77 isempty()........................................................................................................................................................77 Push Operation..............................................................................................................................................78 Pop Operation ...............................................................................................................................................79 Stack Program in C.........................................................................................................................................81 14. Expression Parsing ..................................................................................................................................84 Infix Notation.................................................................................................................................................84 Prefix Notation ..............................................................................................................................................84 Postfix Notation.............................................................................................................................................84 Parsing Expressions .......................................................................................................................................85 Postfix Evaluation Algorithm .........................................................................................................................86 Expression Parsing Using Stack......................................................................................................................86 15. Queue.....................................................................................................................................................92 Queue Representation ..................................................................................................................................92 Basic Operations............................................................................................................................................92 peek().............................................................................................................................................................93 isfull().............................................................................................................................................................93 isempty()........................................................................................................................................................94 Enqueue Operation .......................................................................................................................................95 Dequeue Operation.......................................................................................................................................96 Queue Program in C ......................................................................................................................................98 SEARCHING TECHNIQUES........................................................................................................102 16. Linear Search .......................................................................................................................................103 Linear Search Program in C .........................................................................................................................104 17. Binary Search........................................................................................................................................107 How Binary Search Works? .........................................................................................................................107 Binary Search Program in C .........................................................................................................................110 18. Interpolation Search .............................................................................................................................113 Positioning in Binary Search ........................................................................................................................113 Position Probing in Interpolation Search.....................................................................................................114 Interpolation Search Program in C ..............................................................................................................116 19. Hash Table ............................................................................................................................................118 Hashing........................................................................................................................................................118 Linear Probing..............................................................................................................................................119 Basic Operations..........................................................................................................................................120 Data Item.....................................................................................................................................................120 v
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Data Structures & Algorithms Hash Method...............................................................................................................................................120 Search Operation.........................................................................................................................................120 Insert Operation ..........................................................................................................................................121 Delete Operation.........................................................................................................................................122 Hash Table Program in C .............................................................................................................................123 SORTING TECHNIQUES............................................................................................................128 20. Sorting Algorithm..................................................................................................................................129 In-place Sorting and Not-in-place Sorting ...................................................................................................129 Stable and Not Stable Sorting......................................................................................................................129 Adaptive and Non-Adaptive Sorting Algorithm...........................................................................................130 Important Terms..........................................................................................................................................130 21. Bubble Sort Algorithm ..........................................................................................................................132 How Bubble Sort Works?.............................................................................................................................132 Bubble Sort Program in C ............................................................................................................................136 22. Insertion Sort........................................................................................................................................140 How Insertion Sort Works? .........................................................................................................................140 Insertion Sort Program in C .........................................................................................................................143 23. Selection Sort........................................................................................................................................147 How Selection Sort Works? .........................................................................................................................147 Selection Sort Program in C.........................................................................................................................150 24. Merge Sort Algorithm ...........................................................................................................................153 How Merge Sort Works? .............................................................................................................................153 Merge Sort Program in C .............................................................................................................................156 25. Shell Sort ..............................................................................................................................................158 How Shell Sort Works? ................................................................................................................................158 Shell Sort Program in C................................................................................................................................162 26. Quick Sort .............................................................................................................................................166 Partition in Quick Sort .................................................................................................................................166 Quick Sort Pivot Algorithm ..........................................................................................................................166 Quick Sort Pivot Pseudocode ......................................................................................................................167 Quick Sort Algorithm ...................................................................................................................................167 Quick Sort Pseudocode................................................................................................................................168 Quick Sort Program in C ..............................................................................................................................168 GRAPH DATA STRUCTURE.......................................................................................................172 27. Graphs ..................................................................................................................................................173 Graph Data Structure ..................................................................................................................................173 Basic Operations..........................................................................................................................................175 vi
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Data Structures & Algorithms 28. Depth First Traversal............................................................................................................................176 Depth First Traversal in C ............................................................................................................................179 29. Breadth First Traversal.........................................................................................................................184 Breadth First Traversal in C .........................................................................................................................186 TREE DATA STRUCTURE ..........................................................................................................192 30. Tree ......................................................................................................................................................193 Important Terms..........................................................................................................................................193 Binary Search Tree Representation.............................................................................................................194 Tree Node....................................................................................................................................................194 BST Basic Operations...................................................................................................................................195 Insert Operation ..........................................................................................................................................195 Search Operation.........................................................................................................................................197 Tree Traversal in C.......................................................................................................................................198 31. Tree Traversal .......................................................................................................................................204 In-order Traversal........................................................................................................................................204 Pre-order Traversal......................................................................................................................................205 Post-order Traversal....................................................................................................................................206 Tree Traversal in C.......................................................................................................................................207 32. Binary Search Tree................................................................................................................................213 Representation............................................................................................................................................213 Basic Operations..........................................................................................................................................214 Node............................................................................................................................................................214 Search Operation.........................................................................................................................................214 Insert Operation ..........................................................................................................................................215 33. AVL Trees.............................................................................................................................................217 AVL Rotations ..............................................................................................................................................218 34. Spanning Tree.......................................................................................................................................222 General Properties of Spanning Tree ..........................................................................................................222 Mathematical Properties of Spanning Tree.................................................................................................223 Application of Spanning Tree ......................................................................................................................223 Minimum Spanning Tree (MST)...................................................................................................................223 Minimum Spanning-Tree Algorithm............................................................................................................223 Kruskal's Spanning Tree Algorithm..............................................................................................................224 Prim's Spanning Tree Algorithm..................................................................................................................227 35. Heaps....................................................................................................................................................231 Max Heap Construction Algorithm..............................................................................................................232 Max Heap Deletion Algorithm.....................................................................................................................233 RECURSION.............................................................................................................................234 vii
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Data Structures & Algorithms 36. Recursion ─ Basics.................................................................................................................................235 Properties ....................................................................................................................................................235 Implementation...........................................................................................................................................236 Analysis of Recursion...................................................................................................................................236 Time Complexity..........................................................................................................................................236 Space Complexity ........................................................................................................................................237 37. Tower of Hanoi .....................................................................................................................................238 Rules ............................................................................................................................................................238 Algorithm.....................................................................................................................................................242 Tower of Hanoi in C .....................................................................................................................................245 38. Fibonacci Series ....................................................................................................................................249 Fibonacci Iterative Algorithm ......................................................................................................................250 Fibonacci Interactive Program in C..............................................................................................................250 Fibonacci Recursive Algorithm ....................................................................................................................252 Fibonacci Recursive Program in C................................................................................................................252 viii
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Data Structures & Algorithms Basics 1
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1. Overview Data Structure is a systematic way to organize data in order to use it efficiently. Following terms are the foundation terms of a data structure.  Interface − Each data structure has an interface. Interface represents the set of operations that a data structure supports. An interface only provides the list of supported operations, type of parameters they can accept and return type of these operations.  Implementation − Implementation provides the internal representation of a data structure. Implementation also provides the definition of the algorithms used in the operations of the data structure. CharacteristicsofaDataStructure  Correctness − Data structure implementation should implement its interface correctly.  Time Complexity − Running time or the execution time of operations of data structure must be as small as possible.  Space Complexity − Memory usage of a data structure operation should be as little as possible. NeedforDataStructure As applications are getting complex and data rich, there are three common problems that applications face now-a-days.  Data Search − Consider an inventory of 1 million(106) items of a store. If the application is to search an item, it has to search an item in 1 million(106) items every time slowing down the search. As data grows, search will become slower.  Processor Speed − Processor speed although being very high, falls limited if the data grows to billion records.  Multiple Requests − As thousands of users can search data simultaneously on a web server, even the fast server fails while searching the data. To solve the above-mentioned problems, data structures come to rescue. Data can be organized in a data structure in such a way that all items may not be required to be searched, and the required data can be searched almost instantly. 2
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Data Structures & Algorithms ExecutionTimeCases There are three cases which are usually used to compare various data structure's execution time in a relative manner.  Worst Case − This is the scenario where a particular data structure operation takes maximum time it can take. If an operation's worst case time is ƒ(n) then this operation will not take more than ƒ(n) time, where ƒ(n) represents function of n.  Average Case − This is the scenario depicting the average execution time of an operation of a data structure. If an operation takes ƒ(n) time in execution, then m operations will take mƒ(n) time.  Best Case − This is the scenario depicting the least possible execution time of an operation of a data structure. If an operation takes ƒ(n) time in execution, then the actual operation may take time as the random number which would be maximum as ƒ(n). BasicTerminology  Data − Data are values or set of values.  Data Item − Data item refers to single unit of values.  Group Items − Data items that are divided into sub items are called as Group Items.  Elementary Items − Data items that cannot be divided are called as Elementary Items.  Attribute and Entity − An entity is that which contains certain attributes or properties, which may be assigned values.  Entity Set − Entities of similar attributes form an entity set.  Field − Field is a single elementary unit of information representing an attribute of an entity.  Record − Record is a collection of field values of a given entity.  File − File is a collection of records of the entities in a given entity set. 3
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2. Environment Setup TryitOptionOnline You really do not need to set up your own environment to start learning C programming language. Reason is very simple, we already have set up C Programming environment online, so that you can compile and execute all the available examples online at the same time when you are doing your theory work. This gives you confidence in what you are reading and to check the result with different options. Feel free to modify any example and execute it online. Try the following example using the Try it option available at the top right corner of the sample code box − #include <stdio.h> int main(){ /* My first program in C */ printf("Hello, World! n"); return 0; } For most of the examples given in this tutorial, you will find Try it option, so just make use of it and enjoy your learning. LocalEnvironmentSetup If you are still willing to set up your environment for C programming language, you need the following two tools available on your computer, (a) Text Editor and (b) The C Compiler. Text Editor This will be used to type your program. Examples of few editors include Windows Notepad, OS Edit command, Brief, Epsilon, EMACS, and vim or vi. The name and the version of the text editor can vary on different operating systems. For example, Notepad will be used on Windows, and vim or vi can be used on Windows as well as Linux or UNIX. The files you create with your editor are called source files and contain program source code. The source files for C programs are typically named with the extension ".c". Before starting your programming, make sure you have one text editor in place and you have enough experience to write a computer program, save it in a file, compile it, and finally execute it. 4
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Data Structures & Algorithms The C Compiler The source code written in the source file is the human readable source for your program. It needs to be "compiled", to turn into machine language so that your CPU can actually execute the program as per the given instructions. This C programming language compiler will be used to compile your source code into a final executable program. We assume you have the basic knowledge about a programming language compiler. Most frequently used and free available compiler is GNU C/C++ compiler. Otherwise, you can have compilers either from HP or Solaris if you have respective Operating Systems (OS). The following section guides you on how to install GNU C/C++ compiler on various OS. We are mentioning C/C++ together because GNU GCC compiler works for both C and C++ programming languages. InstallationonUNIX/Linux If you are using Linux or UNIX, then check whether GCC is installed on your system by entering the following command from the command line − $ gcc -v If you have GNU compiler installed on your machine, then it should print a message such as the following − Using built-in specs. Target: i386-redhat-linux Configured with: ../configure --prefix=/usr ....... Thread model: posix gcc version 4.1.2 20080704 (Red Hat 4.1.2-46) If GCC is not installed, then you will have to install it yourself using the detailed instructions available at http://gcc.gnu.org/install/ This tutorial has been written based on Linux and all the given examples have been compiled on Cent OS flavor of Linux system. InstallationonMacOS If you use Mac OS X, the easiest way to obtain GCC is to download the Xcode development environment from Apple's website and follow the simple installation instructions. Once you have Xcode setup, you will be able to use GNU compiler for C/C++. Xcode is currently available at developer.apple.com/technologies/tools/ 5
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Data Structures & Algorithms InstallationonWindows To install GCC on Windows, you need to install MinGW. To install MinGW, go to the MinGW homepage, www.mingw.org, and follow the link to the MinGW download page. Download the latest version of the MinGW installation program, which should be named MinGW- <version>.exe. While installing MinWG, at a minimum, you must install gcc-core, gcc-g++, binutils, and the MinGW runtime, but you may wish to install more. Add the bin subdirectory of your MinGW installation to your PATH environment variable, so that you can specify these tools on the command line by their simple names. When the installation is complete, you will be able to run gcc, g++, ar, ranlib, dlltool, and several other GNU tools from the Windows command line. 6
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Data Structures & Algorithms Algorithm 7
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3. Algorithms ─ Basics Algorithm is a step-by-step procedure, which defines a set of instructions to be executed in a certain order to get the desired output. Algorithms are generally created independent of underlying languages, i.e. an algorithm can be implemented in more than one programming language. From the data structure point of view, following are some important categories of algorithms −  Search − Algorithm to search an item in a data structure.  Sort − Algorithm to sort items in a certain order.  Insert − Algorithm to insert item in a data structure.  Update − Algorithm to update an existing item in a data structure.  Delete − Algorithm to delete an existing item from a data structure. CharacteristicsofanAlgorithm Not all procedures can be called an algorithm. An algorithm should have the following characteristics −  Unambiguous − Algorithm should be clear and unambiguous. Each of its steps (or phases), and their inputs/outputs should be clear and must lead to only one meaning.  Input − An algorithm should have 0 or more well-defined inputs.  Output − An algorithm should have 1 or more well-defined outputs, and should match the desired output.  Finiteness − Algorithms must terminate after a finite number of steps.  Feasibility − Should be feasible with the available resources.  Independent − An algorithm should have step-by-step directions, which should be independent of any programming code. 8
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Data Structures & Algorithms How toWriteanAlgorithm? There are no well-defined standards for writing algorithms. Rather, it is problem and resource dependent. Algorithms are never written to support a particular programming code. As we know that all programming languages share basic code constructs like loops (do, for, while), flow-control (if-else), etc. These common constructs can be used to write an algorithm. We write algorithms in a step-by-step manner, but it is not always the case. Algorithm writing is a process and is executed after the problem domain is well-defined. That is, we should know the problem domain, for which we are designing a solution. 9
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Data structures algorithms_tutorial
Data Structures & Algorithms AbouttheTutorial Data Structures are the programmatic way of storing data so that data can be used efficiently. Almost every enterprise application uses various types of data structures in one or the other way. This tutorial will give you a great understanding on Data Structures needed to understand the complexity of enterprise level applications and need of algorithms, and data structures. Audience This tutorial is designed for Computer Science graduates as well as Software Professionals who are willing to learn data structures and algorithm programming in simple and easy steps. After completing this tutorial you will be at intermediate level of expertise from where you can take yourself to higher level of expertise. Prerequisites Before proceeding with this tutorial, you should have a basic understanding of C programming language, text editor, and execution of programs, etc. CopyrightandDisclaimer © Copyright 2016 by Tutorials Point (I) Pvt. Ltd. All the content and graphics published in this e-book are the property of Tutorials Point (I) Pvt. Ltd. The user of this e-book is prohibited to reuse, retain, copy, distribute or republish any contents or a part of contents of this e-book in any manner without written consent of the publisher. We strive to update the contents of our website and tutorials as timely and as precisely as possible, however, the contents may contain inaccuracies or errors. Tutorials Point (I) Pvt. Ltd. provides no guarantee regarding the accuracy, timeliness or completeness of our website or its contents including this tutorial. If you discover any errors on our website or in this tutorial, please notify us at contact@tutorialspoint.com i
Data Structures & Algorithms Compile&ExecuteOnline For most of the examples given in this tutorial you will find Try it option, so just make use of this option to execute your programs on the spot and enjoy your learning. Try the following example using the Try it option available at the top right corner of the following sample code box − #include <stdio.h> int main(){ /* My first program in C */ printf("Hello, World! n"); return 0; } ii
Data Structures & Algorithms TableofContents About the Tutorial ............................................................................................................................................i Audience........................................................................................................................................................... i Prerequisites.....................................................................................................................................................i Copyright and Disclaimer .............................................................................................................................i Compile & Execute Online............................................................................................................................... ii Table of Contents ........................................................................................................................................... iii BASICS.........................................................................................................................................1 1. Overview .................................................................................................................................................2 Characteristics of a Data Structure..................................................................................................................2 Need for Data Structure ..................................................................................................................................2 Execution Time Cases ......................................................................................................................................3 Basic Terminology ...........................................................................................................................................3 2. Environment Setup ..................................................................................................................................4 Try it Option Online .........................................................................................................................................4 Local Environment Setup.................................................................................................................................4 Installation on UNIX/Linux...............................................................................................................................5 Installation on Mac OS.....................................................................................................................................5 Installation on Windows..................................................................................................................................6 ALGORITHM................................................................................................................................7 3. Algorithms ─ Basics...................................................................................................................................8 Characteristics of an Algorithm .......................................................................................................................8 How to Write an Algorithm? ...........................................................................................................................9 Algorithm Analysis.........................................................................................................................................10 Algorithm Complexity....................................................................................................................................11 Space Complexity ..........................................................................................................................................11 Time Complexity............................................................................................................................................11 4. Asymptotic Analysis...............................................................................................................................12 Asymptotic Notations....................................................................................................................................12 Common Asymptotic Notations ....................................................................................................................15 5. Greedy Algorithms.................................................................................................................................16 Counting Coins...............................................................................................................................................16 6. Divide & Conquer....................................................................................................................................18 Divide/Break..................................................................................................................................................18 Conquer/Solve...............................................................................................................................................18 Merge/Combine ............................................................................................................................................19 7. Dynamic Programming............................................................................................................................20 iii
Data Structures & Algorithms DATA STRUCTURES ...................................................................................................................21 8. Basic Concepts .......................................................................................................................................22 Data Definition ..............................................................................................................................................22 Data Object....................................................................................................................................................22 Data Type.......................................................................................................................................................22 Basic Operations............................................................................................................................................23 9. Arrays ....................................................................................................................................................24 Array Representation ....................................................................................................................................24 Basic Operations............................................................................................................................................25 Insertion Operation .......................................................................................................................................25 Array Insertions .............................................................................................................................................27 Insertion at the Beginning of an Array ..........................................................................................................28 Insertion at the Given Index of an Array .......................................................................................................30 Insertion After the Given Index of an Array ..................................................................................................32 Insertion Before the Given Index of an Array................................................................................................34 Deletion Operation........................................................................................................................................36 Search Operation...........................................................................................................................................37 Update Operation..........................................................................................................................................39 LINKED LIST...............................................................................................................................41 10. Linked List ─ Basics..................................................................................................................................42 Linked List Representation ............................................................................................................................42 Types of Linked List .......................................................................................................................................42 Basic Operations............................................................................................................................................43 Insertion Operation .......................................................................................................................................43 Deletion Operation........................................................................................................................................44 Reverse Operation.........................................................................................................................................45 Linked List Program in C ................................................................................................................................46 11. Doubly Linked List...................................................................................................................................55 Doubly Linked List Representation................................................................................................................55 Basic Operations............................................................................................................................................55 Insertion Operation .......................................................................................................................................56 Deletion Operation........................................................................................................................................57 Insertion at the End of an Operation.............................................................................................................57 Doubly Linked List Program in C....................................................................................................................58 12. Circular Linked List..................................................................................................................................67 Singly Linked List as Circular ..........................................................................................................................67 Doubly Linked List as Circular ........................................................................................................................67 Basic Operations............................................................................................................................................67 Insertion Operation .......................................................................................................................................68 Deletion Operation........................................................................................................................................68 Display List Operation....................................................................................................................................69 Circular Linked List Program in C...................................................................................................................69 iv
Data Structures & Algorithms STACK & QUEUE........................................................................................................................74 13. Stack......................................................................................................................................................75 Stack Representation.....................................................................................................................................75 Basic Operations............................................................................................................................................76 peek().............................................................................................................................................................76 isfull().............................................................................................................................................................77 isempty()........................................................................................................................................................77 Push Operation..............................................................................................................................................78 Pop Operation ...............................................................................................................................................79 Stack Program in C.........................................................................................................................................81 14. Expression Parsing ..................................................................................................................................84 Infix Notation.................................................................................................................................................84 Prefix Notation ..............................................................................................................................................84 Postfix Notation.............................................................................................................................................84 Parsing Expressions .......................................................................................................................................85 Postfix Evaluation Algorithm .........................................................................................................................86 Expression Parsing Using Stack......................................................................................................................86 15. Queue.....................................................................................................................................................92 Queue Representation ..................................................................................................................................92 Basic Operations............................................................................................................................................92 peek().............................................................................................................................................................93 isfull().............................................................................................................................................................93 isempty()........................................................................................................................................................94 Enqueue Operation .......................................................................................................................................95 Dequeue Operation.......................................................................................................................................96 Queue Program in C ......................................................................................................................................98 SEARCHING TECHNIQUES........................................................................................................102 16. Linear Search .......................................................................................................................................103 Linear Search Program in C .........................................................................................................................104 17. Binary Search........................................................................................................................................107 How Binary Search Works? .........................................................................................................................107 Binary Search Program in C .........................................................................................................................110 18. Interpolation Search .............................................................................................................................113 Positioning in Binary Search ........................................................................................................................113 Position Probing in Interpolation Search.....................................................................................................114 Interpolation Search Program in C ..............................................................................................................116 19. Hash Table ............................................................................................................................................118 Hashing........................................................................................................................................................118 Linear Probing..............................................................................................................................................119 Basic Operations..........................................................................................................................................120 Data Item.....................................................................................................................................................120 v
Data Structures & Algorithms Hash Method...............................................................................................................................................120 Search Operation.........................................................................................................................................120 Insert Operation ..........................................................................................................................................121 Delete Operation.........................................................................................................................................122 Hash Table Program in C .............................................................................................................................123 SORTING TECHNIQUES............................................................................................................128 20. Sorting Algorithm..................................................................................................................................129 In-place Sorting and Not-in-place Sorting ...................................................................................................129 Stable and Not Stable Sorting......................................................................................................................129 Adaptive and Non-Adaptive Sorting Algorithm...........................................................................................130 Important Terms..........................................................................................................................................130 21. Bubble Sort Algorithm ..........................................................................................................................132 How Bubble Sort Works?.............................................................................................................................132 Bubble Sort Program in C ............................................................................................................................136 22. Insertion Sort........................................................................................................................................140 How Insertion Sort Works? .........................................................................................................................140 Insertion Sort Program in C .........................................................................................................................143 23. Selection Sort........................................................................................................................................147 How Selection Sort Works? .........................................................................................................................147 Selection Sort Program in C.........................................................................................................................150 24. Merge Sort Algorithm ...........................................................................................................................153 How Merge Sort Works? .............................................................................................................................153 Merge Sort Program in C .............................................................................................................................156 25. Shell Sort ..............................................................................................................................................158 How Shell Sort Works? ................................................................................................................................158 Shell Sort Program in C................................................................................................................................162 26. Quick Sort .............................................................................................................................................166 Partition in Quick Sort .................................................................................................................................166 Quick Sort Pivot Algorithm ..........................................................................................................................166 Quick Sort Pivot Pseudocode ......................................................................................................................167 Quick Sort Algorithm ...................................................................................................................................167 Quick Sort Pseudocode................................................................................................................................168 Quick Sort Program in C ..............................................................................................................................168 GRAPH DATA STRUCTURE.......................................................................................................172 27. Graphs ..................................................................................................................................................173 Graph Data Structure ..................................................................................................................................173 Basic Operations..........................................................................................................................................175 vi
Data Structures & Algorithms 28. Depth First Traversal............................................................................................................................176 Depth First Traversal in C ............................................................................................................................179 29. Breadth First Traversal.........................................................................................................................184 Breadth First Traversal in C .........................................................................................................................186 TREE DATA STRUCTURE ..........................................................................................................192 30. Tree ......................................................................................................................................................193 Important Terms..........................................................................................................................................193 Binary Search Tree Representation.............................................................................................................194 Tree Node....................................................................................................................................................194 BST Basic Operations...................................................................................................................................195 Insert Operation ..........................................................................................................................................195 Search Operation.........................................................................................................................................197 Tree Traversal in C.......................................................................................................................................198 31. Tree Traversal .......................................................................................................................................204 In-order Traversal........................................................................................................................................204 Pre-order Traversal......................................................................................................................................205 Post-order Traversal....................................................................................................................................206 Tree Traversal in C.......................................................................................................................................207 32. Binary Search Tree................................................................................................................................213 Representation............................................................................................................................................213 Basic Operations..........................................................................................................................................214 Node............................................................................................................................................................214 Search Operation.........................................................................................................................................214 Insert Operation ..........................................................................................................................................215 33. AVL Trees.............................................................................................................................................217 AVL Rotations ..............................................................................................................................................218 34. Spanning Tree.......................................................................................................................................222 General Properties of Spanning Tree ..........................................................................................................222 Mathematical Properties of Spanning Tree.................................................................................................223 Application of Spanning Tree ......................................................................................................................223 Minimum Spanning Tree (MST)...................................................................................................................223 Minimum Spanning-Tree Algorithm............................................................................................................223 Kruskal's Spanning Tree Algorithm..............................................................................................................224 Prim's Spanning Tree Algorithm..................................................................................................................227 35. Heaps....................................................................................................................................................231 Max Heap Construction Algorithm..............................................................................................................232 Max Heap Deletion Algorithm.....................................................................................................................233 RECURSION.............................................................................................................................234 vii
Data Structures & Algorithms 36. Recursion ─ Basics.................................................................................................................................235 Properties ....................................................................................................................................................235 Implementation...........................................................................................................................................236 Analysis of Recursion...................................................................................................................................236 Time Complexity..........................................................................................................................................236 Space Complexity ........................................................................................................................................237 37. Tower of Hanoi .....................................................................................................................................238 Rules ............................................................................................................................................................238 Algorithm.....................................................................................................................................................242 Tower of Hanoi in C .....................................................................................................................................245 38. Fibonacci Series ....................................................................................................................................249 Fibonacci Iterative Algorithm ......................................................................................................................250 Fibonacci Interactive Program in C..............................................................................................................250 Fibonacci Recursive Algorithm ....................................................................................................................252 Fibonacci Recursive Program in C................................................................................................................252 viii
Data Structures & Algorithms Basics 1
1. Overview Data Structure is a systematic way to organize data in order to use it efficiently. Following terms are the foundation terms of a data structure.  Interface − Each data structure has an interface. Interface represents the set of operations that a data structure supports. An interface only provides the list of supported operations, type of parameters they can accept and return type of these operations.  Implementation − Implementation provides the internal representation of a data structure. Implementation also provides the definition of the algorithms used in the operations of the data structure. CharacteristicsofaDataStructure  Correctness − Data structure implementation should implement its interface correctly.  Time Complexity − Running time or the execution time of operations of data structure must be as small as possible.  Space Complexity − Memory usage of a data structure operation should be as little as possible. NeedforDataStructure As applications are getting complex and data rich, there are three common problems that applications face now-a-days.  Data Search − Consider an inventory of 1 million(106) items of a store. If the application is to search an item, it has to search an item in 1 million(106) items every time slowing down the search. As data grows, search will become slower.  Processor Speed − Processor speed although being very high, falls limited if the data grows to billion records.  Multiple Requests − As thousands of users can search data simultaneously on a web server, even the fast server fails while searching the data. To solve the above-mentioned problems, data structures come to rescue. Data can be organized in a data structure in such a way that all items may not be required to be searched, and the required data can be searched almost instantly. 2
Data Structures & Algorithms ExecutionTimeCases There are three cases which are usually used to compare various data structure's execution time in a relative manner.  Worst Case − This is the scenario where a particular data structure operation takes maximum time it can take. If an operation's worst case time is ƒ(n) then this operation will not take more than ƒ(n) time, where ƒ(n) represents function of n.  Average Case − This is the scenario depicting the average execution time of an operation of a data structure. If an operation takes ƒ(n) time in execution, then m operations will take mƒ(n) time.  Best Case − This is the scenario depicting the least possible execution time of an operation of a data structure. If an operation takes ƒ(n) time in execution, then the actual operation may take time as the random number which would be maximum as ƒ(n). BasicTerminology  Data − Data are values or set of values.  Data Item − Data item refers to single unit of values.  Group Items − Data items that are divided into sub items are called as Group Items.  Elementary Items − Data items that cannot be divided are called as Elementary Items.  Attribute and Entity − An entity is that which contains certain attributes or properties, which may be assigned values.  Entity Set − Entities of similar attributes form an entity set.  Field − Field is a single elementary unit of information representing an attribute of an entity.  Record − Record is a collection of field values of a given entity.  File − File is a collection of records of the entities in a given entity set. 3
2. Environment Setup TryitOptionOnline You really do not need to set up your own environment to start learning C programming language. Reason is very simple, we already have set up C Programming environment online, so that you can compile and execute all the available examples online at the same time when you are doing your theory work. This gives you confidence in what you are reading and to check the result with different options. Feel free to modify any example and execute it online. Try the following example using the Try it option available at the top right corner of the sample code box − #include <stdio.h> int main(){ /* My first program in C */ printf("Hello, World! n"); return 0; } For most of the examples given in this tutorial, you will find Try it option, so just make use of it and enjoy your learning. LocalEnvironmentSetup If you are still willing to set up your environment for C programming language, you need the following two tools available on your computer, (a) Text Editor and (b) The C Compiler. Text Editor This will be used to type your program. Examples of few editors include Windows Notepad, OS Edit command, Brief, Epsilon, EMACS, and vim or vi. The name and the version of the text editor can vary on different operating systems. For example, Notepad will be used on Windows, and vim or vi can be used on Windows as well as Linux or UNIX. The files you create with your editor are called source files and contain program source code. The source files for C programs are typically named with the extension ".c". Before starting your programming, make sure you have one text editor in place and you have enough experience to write a computer program, save it in a file, compile it, and finally execute it. 4
Data Structures & Algorithms The C Compiler The source code written in the source file is the human readable source for your program. It needs to be "compiled", to turn into machine language so that your CPU can actually execute the program as per the given instructions. This C programming language compiler will be used to compile your source code into a final executable program. We assume you have the basic knowledge about a programming language compiler. Most frequently used and free available compiler is GNU C/C++ compiler. Otherwise, you can have compilers either from HP or Solaris if you have respective Operating Systems (OS). The following section guides you on how to install GNU C/C++ compiler on various OS. We are mentioning C/C++ together because GNU GCC compiler works for both C and C++ programming languages. InstallationonUNIX/Linux If you are using Linux or UNIX, then check whether GCC is installed on your system by entering the following command from the command line − $ gcc -v If you have GNU compiler installed on your machine, then it should print a message such as the following − Using built-in specs. Target: i386-redhat-linux Configured with: ../configure --prefix=/usr ....... Thread model: posix gcc version 4.1.2 20080704 (Red Hat 4.1.2-46) If GCC is not installed, then you will have to install it yourself using the detailed instructions available at http://gcc.gnu.org/install/ This tutorial has been written based on Linux and all the given examples have been compiled on Cent OS flavor of Linux system. InstallationonMacOS If you use Mac OS X, the easiest way to obtain GCC is to download the Xcode development environment from Apple's website and follow the simple installation instructions. Once you have Xcode setup, you will be able to use GNU compiler for C/C++. Xcode is currently available at developer.apple.com/technologies/tools/ 5
Data Structures & Algorithms InstallationonWindows To install GCC on Windows, you need to install MinGW. To install MinGW, go to the MinGW homepage, www.mingw.org, and follow the link to the MinGW download page. Download the latest version of the MinGW installation program, which should be named MinGW- <version>.exe. While installing MinWG, at a minimum, you must install gcc-core, gcc-g++, binutils, and the MinGW runtime, but you may wish to install more. Add the bin subdirectory of your MinGW installation to your PATH environment variable, so that you can specify these tools on the command line by their simple names. When the installation is complete, you will be able to run gcc, g++, ar, ranlib, dlltool, and several other GNU tools from the Windows command line. 6
Data Structures & Algorithms Algorithm 7
3. Algorithms ─ Basics Algorithm is a step-by-step procedure, which defines a set of instructions to be executed in a certain order to get the desired output. Algorithms are generally created independent of underlying languages, i.e. an algorithm can be implemented in more than one programming language. From the data structure point of view, following are some important categories of algorithms −  Search − Algorithm to search an item in a data structure.  Sort − Algorithm to sort items in a certain order.  Insert − Algorithm to insert item in a data structure.  Update − Algorithm to update an existing item in a data structure.  Delete − Algorithm to delete an existing item from a data structure. CharacteristicsofanAlgorithm Not all procedures can be called an algorithm. An algorithm should have the following characteristics −  Unambiguous − Algorithm should be clear and unambiguous. Each of its steps (or phases), and their inputs/outputs should be clear and must lead to only one meaning.  Input − An algorithm should have 0 or more well-defined inputs.  Output − An algorithm should have 1 or more well-defined outputs, and should match the desired output.  Finiteness − Algorithms must terminate after a finite number of steps.  Feasibility − Should be feasible with the available resources.  Independent − An algorithm should have step-by-step directions, which should be independent of any programming code. 8
Data Structures & Algorithms How toWriteanAlgorithm? There are no well-defined standards for writing algorithms. Rather, it is problem and resource dependent. Algorithms are never written to support a particular programming code. As we know that all programming languages share basic code constructs like loops (do, for, while), flow-control (if-else), etc. These common constructs can be used to write an algorithm. We write algorithms in a step-by-step manner, but it is not always the case. Algorithm writing is a process and is executed after the problem domain is well-defined. That is, we should know the problem domain, for which we are designing a solution. 9
Data Structures & Algorithms End of ebook preview If you liked what you saw… Buy it from our store @ https://store.tutorialspoint.com 10
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Data structures algorithms_tutorial

  • 2. Data Structures & Algorithms AbouttheTutorial Data Structures are the programmatic way of storing data so that data can be used efficiently. Almost every enterprise application uses various types of data structures in one or the other way. This tutorial will give you a great understanding on Data Structures needed to understand the complexity of enterprise level applications and need of algorithms, and data structures. Audience This tutorial is designed for Computer Science graduates as well as Software Professionals who are willing to learn data structures and algorithm programming in simple and easy steps. After completing this tutorial you will be at intermediate level of expertise from where you can take yourself to higher level of expertise. Prerequisites Before proceeding with this tutorial, you should have a basic understanding of C programming language, text editor, and execution of programs, etc. CopyrightandDisclaimer © Copyright 2016 by Tutorials Point (I) Pvt. Ltd. All the content and graphics published in this e-book are the property of Tutorials Point (I) Pvt. Ltd. The user of this e-book is prohibited to reuse, retain, copy, distribute or republish any contents or a part of contents of this e-book in any manner without written consent of the publisher. We strive to update the contents of our website and tutorials as timely and as precisely as possible, however, the contents may contain inaccuracies or errors. Tutorials Point (I) Pvt. Ltd. provides no guarantee regarding the accuracy, timeliness or completeness of our website or its contents including this tutorial. If you discover any errors on our website or in this tutorial, please notify us at [email protected] i
  • 3. Data Structures & Algorithms Compile&ExecuteOnline For most of the examples given in this tutorial you will find Try it option, so just make use of this option to execute your programs on the spot and enjoy your learning. Try the following example using the Try it option available at the top right corner of the following sample code box − #include <stdio.h> int main(){ /* My first program in C */ printf("Hello, World! n"); return 0; } ii
  • 4. Data Structures & Algorithms TableofContents About the Tutorial ............................................................................................................................................i Audience........................................................................................................................................................... i Prerequisites.....................................................................................................................................................i Copyright and Disclaimer .............................................................................................................................i Compile & Execute Online............................................................................................................................... ii Table of Contents ........................................................................................................................................... iii BASICS.........................................................................................................................................1 1. Overview .................................................................................................................................................2 Characteristics of a Data Structure..................................................................................................................2 Need for Data Structure ..................................................................................................................................2 Execution Time Cases ......................................................................................................................................3 Basic Terminology ...........................................................................................................................................3 2. Environment Setup ..................................................................................................................................4 Try it Option Online .........................................................................................................................................4 Local Environment Setup.................................................................................................................................4 Installation on UNIX/Linux...............................................................................................................................5 Installation on Mac OS.....................................................................................................................................5 Installation on Windows..................................................................................................................................6 ALGORITHM................................................................................................................................7 3. Algorithms ─ Basics...................................................................................................................................8 Characteristics of an Algorithm .......................................................................................................................8 How to Write an Algorithm? ...........................................................................................................................9 Algorithm Analysis.........................................................................................................................................10 Algorithm Complexity....................................................................................................................................11 Space Complexity ..........................................................................................................................................11 Time Complexity............................................................................................................................................11 4. Asymptotic Analysis...............................................................................................................................12 Asymptotic Notations....................................................................................................................................12 Common Asymptotic Notations ....................................................................................................................15 5. Greedy Algorithms.................................................................................................................................16 Counting Coins...............................................................................................................................................16 6. Divide & Conquer....................................................................................................................................18 Divide/Break..................................................................................................................................................18 Conquer/Solve...............................................................................................................................................18 Merge/Combine ............................................................................................................................................19 7. Dynamic Programming............................................................................................................................20 iii
  • 5. Data Structures & Algorithms DATA STRUCTURES ...................................................................................................................21 8. Basic Concepts .......................................................................................................................................22 Data Definition ..............................................................................................................................................22 Data Object....................................................................................................................................................22 Data Type.......................................................................................................................................................22 Basic Operations............................................................................................................................................23 9. Arrays ....................................................................................................................................................24 Array Representation ....................................................................................................................................24 Basic Operations............................................................................................................................................25 Insertion Operation .......................................................................................................................................25 Array Insertions .............................................................................................................................................27 Insertion at the Beginning of an Array ..........................................................................................................28 Insertion at the Given Index of an Array .......................................................................................................30 Insertion After the Given Index of an Array ..................................................................................................32 Insertion Before the Given Index of an Array................................................................................................34 Deletion Operation........................................................................................................................................36 Search Operation...........................................................................................................................................37 Update Operation..........................................................................................................................................39 LINKED LIST...............................................................................................................................41 10. Linked List ─ Basics..................................................................................................................................42 Linked List Representation ............................................................................................................................42 Types of Linked List .......................................................................................................................................42 Basic Operations............................................................................................................................................43 Insertion Operation .......................................................................................................................................43 Deletion Operation........................................................................................................................................44 Reverse Operation.........................................................................................................................................45 Linked List Program in C ................................................................................................................................46 11. Doubly Linked List...................................................................................................................................55 Doubly Linked List Representation................................................................................................................55 Basic Operations............................................................................................................................................55 Insertion Operation .......................................................................................................................................56 Deletion Operation........................................................................................................................................57 Insertion at the End of an Operation.............................................................................................................57 Doubly Linked List Program in C....................................................................................................................58 12. Circular Linked List..................................................................................................................................67 Singly Linked List as Circular ..........................................................................................................................67 Doubly Linked List as Circular ........................................................................................................................67 Basic Operations............................................................................................................................................67 Insertion Operation .......................................................................................................................................68 Deletion Operation........................................................................................................................................68 Display List Operation....................................................................................................................................69 Circular Linked List Program in C...................................................................................................................69 iv
  • 6. Data Structures & Algorithms STACK & QUEUE........................................................................................................................74 13. Stack......................................................................................................................................................75 Stack Representation.....................................................................................................................................75 Basic Operations............................................................................................................................................76 peek().............................................................................................................................................................76 isfull().............................................................................................................................................................77 isempty()........................................................................................................................................................77 Push Operation..............................................................................................................................................78 Pop Operation ...............................................................................................................................................79 Stack Program in C.........................................................................................................................................81 14. Expression Parsing ..................................................................................................................................84 Infix Notation.................................................................................................................................................84 Prefix Notation ..............................................................................................................................................84 Postfix Notation.............................................................................................................................................84 Parsing Expressions .......................................................................................................................................85 Postfix Evaluation Algorithm .........................................................................................................................86 Expression Parsing Using Stack......................................................................................................................86 15. Queue.....................................................................................................................................................92 Queue Representation ..................................................................................................................................92 Basic Operations............................................................................................................................................92 peek().............................................................................................................................................................93 isfull().............................................................................................................................................................93 isempty()........................................................................................................................................................94 Enqueue Operation .......................................................................................................................................95 Dequeue Operation.......................................................................................................................................96 Queue Program in C ......................................................................................................................................98 SEARCHING TECHNIQUES........................................................................................................102 16. Linear Search .......................................................................................................................................103 Linear Search Program in C .........................................................................................................................104 17. Binary Search........................................................................................................................................107 How Binary Search Works? .........................................................................................................................107 Binary Search Program in C .........................................................................................................................110 18. Interpolation Search .............................................................................................................................113 Positioning in Binary Search ........................................................................................................................113 Position Probing in Interpolation Search.....................................................................................................114 Interpolation Search Program in C ..............................................................................................................116 19. Hash Table ............................................................................................................................................118 Hashing........................................................................................................................................................118 Linear Probing..............................................................................................................................................119 Basic Operations..........................................................................................................................................120 Data Item.....................................................................................................................................................120 v
  • 7. Data Structures & Algorithms Hash Method...............................................................................................................................................120 Search Operation.........................................................................................................................................120 Insert Operation ..........................................................................................................................................121 Delete Operation.........................................................................................................................................122 Hash Table Program in C .............................................................................................................................123 SORTING TECHNIQUES............................................................................................................128 20. Sorting Algorithm..................................................................................................................................129 In-place Sorting and Not-in-place Sorting ...................................................................................................129 Stable and Not Stable Sorting......................................................................................................................129 Adaptive and Non-Adaptive Sorting Algorithm...........................................................................................130 Important Terms..........................................................................................................................................130 21. Bubble Sort Algorithm ..........................................................................................................................132 How Bubble Sort Works?.............................................................................................................................132 Bubble Sort Program in C ............................................................................................................................136 22. Insertion Sort........................................................................................................................................140 How Insertion Sort Works? .........................................................................................................................140 Insertion Sort Program in C .........................................................................................................................143 23. Selection Sort........................................................................................................................................147 How Selection Sort Works? .........................................................................................................................147 Selection Sort Program in C.........................................................................................................................150 24. Merge Sort Algorithm ...........................................................................................................................153 How Merge Sort Works? .............................................................................................................................153 Merge Sort Program in C .............................................................................................................................156 25. Shell Sort ..............................................................................................................................................158 How Shell Sort Works? ................................................................................................................................158 Shell Sort Program in C................................................................................................................................162 26. Quick Sort .............................................................................................................................................166 Partition in Quick Sort .................................................................................................................................166 Quick Sort Pivot Algorithm ..........................................................................................................................166 Quick Sort Pivot Pseudocode ......................................................................................................................167 Quick Sort Algorithm ...................................................................................................................................167 Quick Sort Pseudocode................................................................................................................................168 Quick Sort Program in C ..............................................................................................................................168 GRAPH DATA STRUCTURE.......................................................................................................172 27. Graphs ..................................................................................................................................................173 Graph Data Structure ..................................................................................................................................173 Basic Operations..........................................................................................................................................175 vi
  • 8. Data Structures & Algorithms 28. Depth First Traversal............................................................................................................................176 Depth First Traversal in C ............................................................................................................................179 29. Breadth First Traversal.........................................................................................................................184 Breadth First Traversal in C .........................................................................................................................186 TREE DATA STRUCTURE ..........................................................................................................192 30. Tree ......................................................................................................................................................193 Important Terms..........................................................................................................................................193 Binary Search Tree Representation.............................................................................................................194 Tree Node....................................................................................................................................................194 BST Basic Operations...................................................................................................................................195 Insert Operation ..........................................................................................................................................195 Search Operation.........................................................................................................................................197 Tree Traversal in C.......................................................................................................................................198 31. Tree Traversal .......................................................................................................................................204 In-order Traversal........................................................................................................................................204 Pre-order Traversal......................................................................................................................................205 Post-order Traversal....................................................................................................................................206 Tree Traversal in C.......................................................................................................................................207 32. Binary Search Tree................................................................................................................................213 Representation............................................................................................................................................213 Basic Operations..........................................................................................................................................214 Node............................................................................................................................................................214 Search Operation.........................................................................................................................................214 Insert Operation ..........................................................................................................................................215 33. AVL Trees.............................................................................................................................................217 AVL Rotations ..............................................................................................................................................218 34. Spanning Tree.......................................................................................................................................222 General Properties of Spanning Tree ..........................................................................................................222 Mathematical Properties of Spanning Tree.................................................................................................223 Application of Spanning Tree ......................................................................................................................223 Minimum Spanning Tree (MST)...................................................................................................................223 Minimum Spanning-Tree Algorithm............................................................................................................223 Kruskal's Spanning Tree Algorithm..............................................................................................................224 Prim's Spanning Tree Algorithm..................................................................................................................227 35. Heaps....................................................................................................................................................231 Max Heap Construction Algorithm..............................................................................................................232 Max Heap Deletion Algorithm.....................................................................................................................233 RECURSION.............................................................................................................................234 vii
  • 9. Data Structures & Algorithms 36. Recursion ─ Basics.................................................................................................................................235 Properties ....................................................................................................................................................235 Implementation...........................................................................................................................................236 Analysis of Recursion...................................................................................................................................236 Time Complexity..........................................................................................................................................236 Space Complexity ........................................................................................................................................237 37. Tower of Hanoi .....................................................................................................................................238 Rules ............................................................................................................................................................238 Algorithm.....................................................................................................................................................242 Tower of Hanoi in C .....................................................................................................................................245 38. Fibonacci Series ....................................................................................................................................249 Fibonacci Iterative Algorithm ......................................................................................................................250 Fibonacci Interactive Program in C..............................................................................................................250 Fibonacci Recursive Algorithm ....................................................................................................................252 Fibonacci Recursive Program in C................................................................................................................252 viii
  • 10. Data Structures & Algorithms Basics 1
  • 11. 1. Overview Data Structure is a systematic way to organize data in order to use it efficiently. Following terms are the foundation terms of a data structure.  Interface − Each data structure has an interface. Interface represents the set of operations that a data structure supports. An interface only provides the list of supported operations, type of parameters they can accept and return type of these operations.  Implementation − Implementation provides the internal representation of a data structure. Implementation also provides the definition of the algorithms used in the operations of the data structure. CharacteristicsofaDataStructure  Correctness − Data structure implementation should implement its interface correctly.  Time Complexity − Running time or the execution time of operations of data structure must be as small as possible.  Space Complexity − Memory usage of a data structure operation should be as little as possible. NeedforDataStructure As applications are getting complex and data rich, there are three common problems that applications face now-a-days.  Data Search − Consider an inventory of 1 million(106) items of a store. If the application is to search an item, it has to search an item in 1 million(106) items every time slowing down the search. As data grows, search will become slower.  Processor Speed − Processor speed although being very high, falls limited if the data grows to billion records.  Multiple Requests − As thousands of users can search data simultaneously on a web server, even the fast server fails while searching the data. To solve the above-mentioned problems, data structures come to rescue. Data can be organized in a data structure in such a way that all items may not be required to be searched, and the required data can be searched almost instantly. 2
  • 12. Data Structures & Algorithms ExecutionTimeCases There are three cases which are usually used to compare various data structure's execution time in a relative manner.  Worst Case − This is the scenario where a particular data structure operation takes maximum time it can take. If an operation's worst case time is ƒ(n) then this operation will not take more than ƒ(n) time, where ƒ(n) represents function of n.  Average Case − This is the scenario depicting the average execution time of an operation of a data structure. If an operation takes ƒ(n) time in execution, then m operations will take mƒ(n) time.  Best Case − This is the scenario depicting the least possible execution time of an operation of a data structure. If an operation takes ƒ(n) time in execution, then the actual operation may take time as the random number which would be maximum as ƒ(n). BasicTerminology  Data − Data are values or set of values.  Data Item − Data item refers to single unit of values.  Group Items − Data items that are divided into sub items are called as Group Items.  Elementary Items − Data items that cannot be divided are called as Elementary Items.  Attribute and Entity − An entity is that which contains certain attributes or properties, which may be assigned values.  Entity Set − Entities of similar attributes form an entity set.  Field − Field is a single elementary unit of information representing an attribute of an entity.  Record − Record is a collection of field values of a given entity.  File − File is a collection of records of the entities in a given entity set. 3
  • 13. 2. Environment Setup TryitOptionOnline You really do not need to set up your own environment to start learning C programming language. Reason is very simple, we already have set up C Programming environment online, so that you can compile and execute all the available examples online at the same time when you are doing your theory work. This gives you confidence in what you are reading and to check the result with different options. Feel free to modify any example and execute it online. Try the following example using the Try it option available at the top right corner of the sample code box − #include <stdio.h> int main(){ /* My first program in C */ printf("Hello, World! n"); return 0; } For most of the examples given in this tutorial, you will find Try it option, so just make use of it and enjoy your learning. LocalEnvironmentSetup If you are still willing to set up your environment for C programming language, you need the following two tools available on your computer, (a) Text Editor and (b) The C Compiler. Text Editor This will be used to type your program. Examples of few editors include Windows Notepad, OS Edit command, Brief, Epsilon, EMACS, and vim or vi. The name and the version of the text editor can vary on different operating systems. For example, Notepad will be used on Windows, and vim or vi can be used on Windows as well as Linux or UNIX. The files you create with your editor are called source files and contain program source code. The source files for C programs are typically named with the extension ".c". Before starting your programming, make sure you have one text editor in place and you have enough experience to write a computer program, save it in a file, compile it, and finally execute it. 4
  • 14. Data Structures & Algorithms The C Compiler The source code written in the source file is the human readable source for your program. It needs to be "compiled", to turn into machine language so that your CPU can actually execute the program as per the given instructions. This C programming language compiler will be used to compile your source code into a final executable program. We assume you have the basic knowledge about a programming language compiler. Most frequently used and free available compiler is GNU C/C++ compiler. Otherwise, you can have compilers either from HP or Solaris if you have respective Operating Systems (OS). The following section guides you on how to install GNU C/C++ compiler on various OS. We are mentioning C/C++ together because GNU GCC compiler works for both C and C++ programming languages. InstallationonUNIX/Linux If you are using Linux or UNIX, then check whether GCC is installed on your system by entering the following command from the command line − $ gcc -v If you have GNU compiler installed on your machine, then it should print a message such as the following − Using built-in specs. Target: i386-redhat-linux Configured with: ../configure --prefix=/usr ....... Thread model: posix gcc version 4.1.2 20080704 (Red Hat 4.1.2-46) If GCC is not installed, then you will have to install it yourself using the detailed instructions available at http://gcc.gnu.org/install/ This tutorial has been written based on Linux and all the given examples have been compiled on Cent OS flavor of Linux system. InstallationonMacOS If you use Mac OS X, the easiest way to obtain GCC is to download the Xcode development environment from Apple's website and follow the simple installation instructions. Once you have Xcode setup, you will be able to use GNU compiler for C/C++. Xcode is currently available at developer.apple.com/technologies/tools/ 5
  • 15. Data Structures & Algorithms InstallationonWindows To install GCC on Windows, you need to install MinGW. To install MinGW, go to the MinGW homepage, www.mingw.org, and follow the link to the MinGW download page. Download the latest version of the MinGW installation program, which should be named MinGW- <version>.exe. While installing MinWG, at a minimum, you must install gcc-core, gcc-g++, binutils, and the MinGW runtime, but you may wish to install more. Add the bin subdirectory of your MinGW installation to your PATH environment variable, so that you can specify these tools on the command line by their simple names. When the installation is complete, you will be able to run gcc, g++, ar, ranlib, dlltool, and several other GNU tools from the Windows command line. 6
  • 16. Data Structures & Algorithms Algorithm 7
  • 17. 3. Algorithms ─ Basics Algorithm is a step-by-step procedure, which defines a set of instructions to be executed in a certain order to get the desired output. Algorithms are generally created independent of underlying languages, i.e. an algorithm can be implemented in more than one programming language. From the data structure point of view, following are some important categories of algorithms −  Search − Algorithm to search an item in a data structure.  Sort − Algorithm to sort items in a certain order.  Insert − Algorithm to insert item in a data structure.  Update − Algorithm to update an existing item in a data structure.  Delete − Algorithm to delete an existing item from a data structure. CharacteristicsofanAlgorithm Not all procedures can be called an algorithm. An algorithm should have the following characteristics −  Unambiguous − Algorithm should be clear and unambiguous. Each of its steps (or phases), and their inputs/outputs should be clear and must lead to only one meaning.  Input − An algorithm should have 0 or more well-defined inputs.  Output − An algorithm should have 1 or more well-defined outputs, and should match the desired output.  Finiteness − Algorithms must terminate after a finite number of steps.  Feasibility − Should be feasible with the available resources.  Independent − An algorithm should have step-by-step directions, which should be independent of any programming code. 8
  • 18. Data Structures & Algorithms How toWriteanAlgorithm? There are no well-defined standards for writing algorithms. Rather, it is problem and resource dependent. Algorithms are never written to support a particular programming code. As we know that all programming languages share basic code constructs like loops (do, for, while), flow-control (if-else), etc. These common constructs can be used to write an algorithm. We write algorithms in a step-by-step manner, but it is not always the case. Algorithm writing is a process and is executed after the problem domain is well-defined. That is, we should know the problem domain, for which we are designing a solution. 9
  • 19. Data Structures & Algorithms End of ebook preview If you liked what you saw… Buy it from our store @ https://store.tutorialspoint.com 10