Data structure and algorithms using C, C++, Java, JavaScript

Test the knowledge

Types of data structures

In this video we will learn the types of data structures.

Data structures can be broadly categorised into two main types:

And They are.

• Primitive data structure.
  

• Non-primitive data structure.
  

Let us first see. What are primitive data structures?

Primitive data structures are basic data types that serve as the foundation for more complex data structures.

They are usually directly supported by the programming language and are used to represent simple values.

The main primitive data types include:

• Integer:

  • Represents whole numbers (positive or negative) without any fractional part.

  • Examples include int in languages like C, C++, Java, and Python.

• Floating Point:

  • Represents real numbers with a decimal point.

  • Examples include float and double in languages like C, C++, Java, and Python.

• Character:

  • Represents individual characters such as letters, digits, and symbols.

  • Examples include char in languages like C, C++, Java, and Python.

• Boolean:

  • Represents truth values, usually denoted as true or false.

  • Examples include bool in languages like C++, Java, and Python.

• Pointers:

  • Represents a memory address, i.e., the location of another variable in memory.

  • Commonly found in languages like C and C++.

• Enumerations (Enums):

  • Represents a set of named integer constants.

  • Provides a way to create named integer values in a program.

• Void:

  • Represents the absence of a type.

  • Often used in languages like C and C++ as a return type for functions that do not return a value.

These primitive data types are essential for programming in any language.

They provide the basic building blocks for creating variables and structures that can be used to implement more complex algorithms.

Each programming language may have its own set of primitive data types, but the concepts are similar across languages.

Understanding these basic types is fundamental for writing programs and manipulating data in a computer program.

Now we know what are primitive data structures, let us understand what are non primitive data structures.

Non-primitive data structures are more complex data types

that are composed of or derived from primitive data types.

These structures allow for more sophisticated organisation and manipulation of data.

Unlike primitive data types, which are directly supported by the programming language,

non-primitive data structures are user-defined or abstract data types.

Abstract data type is a high-level description of a set of operations that can be performed on a data structure.

Non primitive data structures are further classified into two types.

Linear and non linear data structures.

Let’s first see what are linear data structures.

Linear data structures are arrangements of elements in a sequential order,

The key characteristic of linear data structures is that the elements are ordered in a linear sequence,

each element has a unique predecessor and successor, except for the first and last elements.

Here are some common examples:

• Arrays

• Linked lists

• Stacks

• Queues

• Strings

We will have quick look on these.

1. Arrays:

• An ordered collection of elements, each element identified by an index or a key.

• Elements are stored in contiguous memory locations.

• Accessing elements is done using their index.

• Examples include static arrays in languages like C and dynamic arrays in languages like Python.

2. Linked Lists:

• A collection of elements, where each element points to the next one.

• Elements are stored in nodes, and each node contains data and a reference to the next node.

• Provides dynamic memory allocation and efficient insertion and deletion of elements.

• Types include singly linked lists, doubly linked lists, and circular linked lists.

3. Stacks:

• Follows the Last In, First Out (LIFO) principle.

• Elements are added and removed from the same end, called the top.

• Common operations include push (add to the top) and pop (remove from the top).

• Used in managing function calls, undo mechanisms, and parsing expressions.

4. Queues:

• Follows the First In, First Out (FIFO) principle.

• Elements are added at the rear (enqueue) and removed from the front (dequeue).

• Common operations include enqueue and dequeue.

• Used in tasks like managing tasks in a printer queue or handling requests in networking.

5. Strings:

• A sequence of characters.

• Can be implemented using arrays or linked lists.

• String manipulation is a common operation in programming.

Linear data structures are fundamental for various algorithms and applications. The choice of a particular linear data structure depends on the requirements of the problem at hand and the operations that need to be performed on the data. Each structure has its strengths and weaknesses in terms of memory usage, access time, and ease of manipulation.

Now we will discuss about non linear data structures.

Non-linear data structures do not organise elements in a sequential.

they allow for more complex relationships among elements, often involving multiple connections or hierarchies.

Here are some common types of non-linear data structures:

1. Trees:

• A hierarchical structure with a root node and branches.

• Consists of nodes, where each node can have children nodes.

• Types include binary trees, AVL trees, and B-trees.

2. Graphs:

• A collection of nodes (vertices) and edges that connect pairs of nodes.

• Nodes represent entities, and edges represent relationships between entities.

• Types include directed graphs, undirected graphs, weighted graphs.

3. Heaps:

• Tree-based structures with the heap property.

• Used for priority queues and implementing heapsort.

• Types include min heaps and max heaps.

4. Hash Tables:

• Uses a hash function to map keys to indexes, facilitating efficient data retrieval.

• Commonly used for implementing associative arrays or dictionaries.

Data structures can also be classified as:

• Static data structure: It is a type of data structure where the size is allocated at the compile time. Therefore, the maximum size is fixed.

• Dynamic data structure: It is a type of data structure where the size is allocated at the run time. Therefore, the maximum size is flexible.

In this video we discussed the types of data structures.

Data structures are mainly classified into two types.

Primitive data structures and non primitive data structures.

Primitive data structures are basic data types that serve as the foundation for more complex data structures.

The main primitive data types include:

• Integer

• Floating Point

• Character

• Boolean

• Pointers

• Enumerations (Enums)

• Void

Next we discussed non primitive data structures.

Non primitive data structures are further classified into two types.

Linear data structures and non linear data structures.

In Linear data structures elements are arranged in sequential manner.

Some common examples are

• Arrays

• Linked lists

• Stacks

• Queues

• Strings

In Non linear data structures are elements are not arranged in a sequential manner.

Some common examples are

Trees

Graphs

Heaps

Hash tables.

In the next video we will discuss about major operations of data structures.

In the last video we discussed the types of data structures and in this video we are going to see what are the operations performed on data structures.

The major operations that can be performed on various data structures are fundamental actions that manipulate or retrieve data. The specific operations vary based on the type of data structure. Here's a general overview:

• Searching: Searching is a fundamental operation in data structures, and various algorithms are used to find a specific element or determine its presence. The choice of the search algorithm depends on the characteristics of the data structure.

• Sorting: Sorting is the process of arranging elements in a specific order, typically in ascending or descending order. Various sorting algorithms exist, and the choice of a particular algorithm depends on factors such as the size of the dataset, the presence of pre-sorted elements, and the desired time complexity.

• Insertion: Insertion is a fundamental operation in data structures, and it involves adding a new element to the existing data structure. The specific steps and considerations for insertion vary depending on the type of data structure.

• Updation: Updating data in a data structure involves modifying the value of an existing element. The specific steps for updating depend on the type of data structure.

• Deletion: Deletion in data structures involves removing an element from the data structure. The specific steps for deletion depend on the type of data structure.

In this video we came across the operations that can be performed on data structures.

And they are

Searching

Sorting

Insertion

Updation

Deletion

In the upcoming chapters you will learn about these operation in depth.

So in the next video we will discuss about the advantages of data structures.

Check your knowledge.

In the last video we discussed the major operations performed on data structures.

And in this video we will be discussing about the advantages of data structures.

Data structures offer several advantages in computer science and programming. Here are some key advantages:

1. Efficient Data Organisation:

• Data structures enable the efficient organisation and storage of data, allowing for quick access and retrieval of information.

2. Optimised Data Retrieval:

• Different data structures are designed for specific operations, leading to optimised retrieval and search times. For example, hash tables provide constant-time average retrieval.

3. Memory Utilisation:

• Data structures help in effective memory utilisation. They allow for the allocation and deallocation of memory as needed, reducing wastage.

4. Algorithm Efficiency:

• The choice of the right data structure can significantly impact the efficiency of algorithms. For instance, sorting algorithms can perform differently based on the data structure used.

5. Code Reusability:

• Well-defined data structures promote code reusability. Once implemented, they can be used across different projects and scenarios, saving development time.

6. Abstraction and Encapsulation:

• Data structures provide abstraction, allowing programmers to focus on high-level concepts without worrying about low-level details. This simplifies problem-solving and code maintenance.

7. Ease of Maintenance:

• Organised and well-structured data makes the code easier to maintain and understand. Debugging and modifications become more straightforward with clear data structures.

8. Facilitates Modularity:

• Data structures support the creation of modular programs. Modules or components can be developed independently, promoting a modular and scalable software design.

9. Improved Scalability:

• Properly chosen data structures contribute to scalable solutions. As the volume of data increases, efficient data structures can handle larger datasets without a significant increase in processing time.

10. Supports Multiple Operations:

- Data structures are designed to support various operations efficiently. For example, stacks are excellent for managing function calls, while hash tables excel at key-value pair storage and retrieval.

11. Enhanced Problem Solving:

- Data structures provide a systematic way to organise and manipulate data, enhancing the ability to solve complex problems and design efficient algorithms.

12. Concurrency and Parallelism:

- Certain data structures are designed to handle concurrent access by multiple threads or processes. They provide mechanisms for synchronisation and data consistency in parallel computing.

13. Enhances Performance:

- The use of appropriate data structures contributes to overall system performance by reducing the time complexity of operations, resulting in faster and more responsive applications.

14. Supports Dynamic Memory Allocation:

- Data structures facilitate dynamic memory allocation, allowing the allocation and deallocation of memory during program execution, which is crucial for managing variable-sized data.

In summary, data structures play a crucial role in organizing, storing, and manipulating data efficiently, leading to improved algorithm performance, code reusability, and easier maintenance. They are fundamental to effective problem-solving in computer science and software development.

DSA Quiz

Basic terminology

Understanding the basic terminology related to data structures is essential for effective communication and comprehension in computer science.

Here are some fundamental terms:

1. Data Structure

• A way of organising and storing data to perform operations efficiently.

2. Element

• A single unit of data, the smallest identifiable piece of data in a data structure.

3. Node

• An individual element in a linked data structure, such as a linked list or tree.

4. Head

• The first node in a linked list.

5. Tail

• The last node in a linked list.

6. Root

• The topmost node in a tree data structure.

7. Parent

• A node in a tree that has one or more child nodes.

8. Child

• A node in a tree that has a parent node.

9. Sibling

• Nodes in a tree that share the same parent.

10. Leaf

- A node in a tree that has no children.

11. Level

- The depth or distance of a node from the root in a tree.

12. Depth

- The level of a node in a tree.

13. Height

- The length of the longest path from a node to a leaf in a tree.

14. Edge

- A connection between nodes in a graph.

15. Graph

- A collection of nodes connected by edges.

16. Vertex

- A node in a graph.

17. Directed Graph:

- A graph where edges have a direction.

18. Undirected Graph:

- A graph where edges do not have a direction.

19. Weighted Graph:

- A graph where edges have weights or costs.

20. Adjacency:

- The relationship between two connected nodes in a graph.

21. Array:

- A collection of elements stored in contiguous memory locations.

22. Linked List:

- A linear collection of elements where each element points to the next one.

23. Stack:

- A data structure that follows the Last In, First Out (LIFO) principle.

24. Queue:

- A data structure that follows the First In, First Out (FIFO) principle.

25. Hash Table:

- A data structure that allows efficient insertion, deletion, and retrieval of data.

26. Binary Tree:

- A tree data structure where each node has at most two children.

27. Binary Search Tree (BST):

- A binary tree with the property that the left subtree of a node contains only nodes with keys less than the node's key, and the right subtree contains only nodes with keys greater than the node's key.

28. Algorithm:

- A step-by-step procedure or formula for solving a problem.

29. Time Complexity:

- A measure of the amount of time an algorithm takes to complete as a function of the size of the input.

30. Space Complexity:

- A measure of the amount of memory an algorithm uses as a function of the size of the input.

Understanding these terms provides a solid foundation for delving into the world of data structures and algorithms in computer science.

Test your knowledge

In the last video we saw the basic terminologies of data structure and in this video we will see what is the need of data structure.

Need of data structure.

Let’s first Understand the Need for Data Structures.

As applications grow in complexity and the volume of data continues to surge daily,

Challenges may arise in tasks such as data searching, processing speed, and handling multiple requests.

Data Structures offer diverse techniques for efficiently organising, managing, and storing data.

They facilitate seamless traversal of data items, providing benefits such as efficiency, reusability, and abstraction.

Why should we learn Data Structures?

Data Structures and Algorithms stand as pivotal elements in the realm of Computer Science. While Data Structures provide the means to organise and store data, Algorithms empower us to process that data with purpose. Acquiring proficiency in Data Structures and Algorithms is instrumental in elevating our programming skills. This knowledge enables us to craft code that is not only more effective and reliable but also empowers us to solve problems swiftly and efficiently.

Some of the objectives of data structure are

Accuracy: Data structures are crafted to function accurately across a spectrum of inputs relevant to the specific domain. In essence, ensuring correctness is the foremost goal of data structures, and it is conditional upon the challenges that the data structure aims to address.

Optimisation: Data structures must also prioritise efficiency. They should swiftly handle data processing without excessive utilisation of computer resources such as memory space. In real-time scenarios, the efficiency of a data structure plays a pivotal role in determining the outcome of a process, be it success or failure.

Key features of data structure are

Robustness:

In general, the goal of all computer programmers is to create software that produces accurate results for any conceivable input while executing efficiently across diverse hardware platforms. Such robust software should effectively handle both valid and invalid inputs.

Adaptability:

The development of software applications like web browsers, word processors, and internet search engines involves expansive software systems that must demonstrate correct and efficient performance over extended periods. Furthermore, software undergoes evolution due to emerging technologies and ever-changing market conditions.

Reusability:

The attributes of reusability and adaptability are inherently interconnected. Building software demands considerable resources, making it a costly endeavour. However, when software is developed with a focus on reusability and adaptability, it can be seamlessly applied in numerous future applications. By employing robust data structures, it becomes feasible to construct reusable software, thereby achieving cost-effectiveness and time savings.

In this video we saw the

The need of data structures.

Discussed Why should we learn data structures.

And saw what are the

Objectives of data structure.

Objectives are

Accuracy

Optimisation

And discussed the key features of data structures and they are

Robustness

Adaptability

Reusability

In the next video we will see the classification of data structures.

Classification of data structures

In the last video, we discussed, what is the need for data structure and in this video, we are going to see the classification of data structures.

Already you have got a brief introduction about the classification of data structures in the lecture types of data structures.

You can skip this video if you have thoroughly got a concept of types of data structures.

Now let’s understand the classification of data structures with the depiction of a flow diagram.

As we already know the data structure is classified into two types primitive data structures and non-p primitive data structures.

The primitive data structures are classified into 4 types, they are

Integer

Float

Character

Boolean

And non primitive data structures are classified into two types which are Linear data structure and non-linear data structure.

Here the Linear data structure is further classified into two types

Static and dynamic.

Static includes an array and in the dynamic Linked list, stacks, and queues are classified.

And coming to the non-linear data structure it is further classified into two types Tree and Graph.

You are going to learn all these concepts in detail in upcoming chapters.

So in the next video, we will see the application of data structures.

Factors of algorithms

The design and evaluation of algorithms involve considering various factors to ensure their effectiveness and efficiency. Here are key factors to consider when working with algorithms:

• Time Complexity:

  • Definition: The measure of the amount of time an algorithm takes to complete as a function of the input size.

  • Importance: A crucial factor in determining the efficiency of an algorithm. Algorithms with lower time complexity are generally preferred.

• Space Complexity:

  • Definition: The measure of the amount of memory or storage space an algorithm requires as a function of the input size.

  • Importance: Efficient use of memory is essential for optimal algorithm performance. Lower space complexity is generally desirable.

• Correctness:

  • Definition: The algorithm should produce the correct output for all valid inputs.

  • Importance: The fundamental requirement of any algorithm. Incorrect algorithms can lead to flawed results and unreliable computations.

• Simplicity and Clarity:

  • Definition: The algorithm should be straightforward, easy to understand, and not overly complex.

  • Importance: Simple and clear algorithms are easier to maintain, debug, and modify. They enhance collaboration among developers.

• Robustness:

  • Definition: The ability of an algorithm to handle unexpected inputs or situations without crashing or producing incorrect results.

  • Importance: Robust algorithms are more reliable in real-world scenarios where inputs may deviate from the expected.

• Scalability:

  • Definition: The ability of an algorithm to handle larger inputs or increasing workloads without a significant increase in resources or time.

  • Importance: Scalable algorithms are crucial for applications dealing with growing datasets or expanding user bases.

• Adaptability:

  • Definition: The ability of an algorithm to adapt to different scenarios or requirements.

  • Importance: Algorithms that can be easily adapted are more versatile and can be reused in various contexts.

• Optimality:

  • Definition: An algorithm is considered optimal if it produces the best possible result within certain constraints.

  • Importance: In situations where resources are limited, finding optimal solutions is crucial.

• Parallelism:

  • Definition: The extent to which an algorithm can be parallelized, allowing multiple computations to occur simultaneously.

  • Importance: In modern computing environments, parallel algorithms can take advantage of multi-core processors for improved performance.

• Quantifiability:

  • Definition: The ability to measure and analyze the performance of an algorithm using quantitative metrics.

  • Importance: Quantifiable factors, such as time and space complexity, provide a basis for comparing and evaluating different algorithms.

• Versatility:

  • Definition: The ability of an algorithm to be applied in various contexts and solve a range of related problems.

  • Importance: Versatile algorithms can be reused in different applications, promoting code efficiency and maintainability.

Understanding and balancing these factors is crucial for developing algorithms that meet the specific requirements and constraints of different computational tasks.

Applications of data structures

In this video, we are going to see the applications of data structures.

There are many applications of data structure, and we have listed here some of them.

1. Data Structures play a vital role in organising data within a computer's memory.

1. They are used in representing information within databases.

1. Data Structures extend to implementing algorithms for data search.

1. Enable the implementation of algorithms for data manipulation.

1. Additionally, Data Structures support algorithms for data analysis.

1. Furthermore, Data Structures facilitate algorithms for generating data.

1. Support algorithms for compressing and decompressing data.

2. Moreover, Data Structures are used for implementing algorithms for encrypting and decrypting data.

3. The versatility of Data Structures is highlighted by their role in building software capable of managing files and directories.

4. Additionally, they contribute to the development of software capable of rendering graphics.

These are some of the applications of data structure, in the next video we will discuss algorithms.

Algorithms

In the last video, we came across the applications of data structures and in this video, we will discuss algorithms.

Algorithms are step-by-step procedures or sets of rules designed to perform specific tasks or solve particular problems. They provide a systematic way to carry out a computational process, guiding the execution of operations to achieve desired outcomes. Algorithms are fundamental to computer science and play a central role in various applications, from sorting and searching data to complex tasks in artificial intelligence and machine learning.

Characteristic of algorithms

• Well-defined: Algorithms must have precisely defined and unambiguous instructions. Each step in the algorithm should be clearly and accurately specified.
  

• Input: An algorithm takes input from a specified set, which is processed to produce the desired output. The input can vary, and the algorithm should be designed to handle different inputs.
  

• Output: Algorithms produce an output that is related to the input by a clear set of rules. The output should represent a solution to the problem or the result of the algorithmic process.
  

• Finiteness: An algorithm must be composed of a finite number of steps. It should eventually terminate after executing a finite sequence of operations.
  

• Effectiveness: Every step of the algorithm must be effective, meaning it can be executed precisely and in a reasonable amount of time. The operations involved should be feasible with the available resources.
  

• Unambiguous: Each step of the algorithm must be free from ambiguity or multiple interpretations. The instructions should be straightforward and leave no room for confusion.
  

• Generality: An algorithm should be designed to handle a broad range of input cases, not just specific instances. It should be applicable to a general class of problems.
  

• Independence: The steps of an algorithm should be independent, meaning the execution of one step does not depend on the outcome of another. This allows for parallelisation and optimisation.
  

• Deterministic: An algorithm should be deterministic, meaning that given the same input, it will produce the same output every time it is executed. Randomness in algorithms is introduced through specific mechanisms.
  

• Correctness: An algorithm should produce the correct output for all valid inputs. It should solve the intended problem and adhere to the specifications.
  

• Resource Efficiency: While solving a problem, an algorithm should use resources (such as time and memory) judiciously. Efficiency is often a critical factor in evaluating the quality of an algorithm.
  

• Scalability: A good algorithm should be scalable, meaning it can handle larger inputs or datasets without a disproportionate increase in resources or time.
  

Understanding and applying these characteristics are essential when designing algorithms to ensure their reliability, efficiency, and effectiveness in solving computational problems.

Data flow of the algorithm

The term "data flow of an algorithm" typically refers to how data is processed and transferred within the steps of the algorithm. Here's a general overview of the data flow in an algorithm:

• Input:

  • The algorithm begins by taking input data, which could be values, variables, or information needed for the algorithm to perform its task.

• Processing Steps:

  • The algorithm consists of a sequence of steps, each specifying operations to be performed on the input data. These steps manipulate and transform the data according to the logic of the algorithm.

• Intermediate Data:

  • As the algorithm progresses through its steps, intermediate data is generated. This data represents the evolving state of the computation and is often used in subsequent steps.

• Control Structures:

  • The algorithm may include control structures such as loops and conditionals that determine the flow of execution based on certain conditions. These structures influence how data is processed and may lead to repeated or conditional execution of certain steps.

• Output Data:

  • Ultimately, the algorithm produces output data as a result of the processing steps. This output is the desired outcome or solution to the problem the algorithm is designed to address.

• Data Transfer:

  • Data is transferred between steps of the algorithm. For example, the output of one step might become the input for the next step. This transfer of data ensures that the algorithm can carry out a coherent and logical sequence of operations.

• Memory or Storage:

  • In some cases, algorithms may involve the use of memory or storage to temporarily hold and retrieve data. This could be in the form of variables, arrays, or other data structures.

• Recursion (if applicable):

  • In recursive algorithms, a function or procedure may call itself, leading to a nested series of calls. Each recursive call operates on a subset of the data, contributing to the overall data flow.

Understanding the data flow of an algorithm is crucial for analysing its efficiency, identifying potential bottlenecks, and ensuring that the algorithm produces the correct output. This perspective helps programmers and analysts optimise algorithms and troubleshoot issues related to data processing within the computational steps.

Test your knowledge

Why do we need algorithms?

Algorithms are essential in computer science and various aspects of problem-solving for several reasons:

• Problem Solving: Algorithms provide systematic and structured approaches to problem-solving. They offer step-by-step procedures for tackling complex issues and achieving desired outcomes.
  

• Efficiency: Algorithms contribute to the development of efficient solutions. They help optimize processes, reduce resource usage, and enhance the overall performance of computational tasks.
  

• Reproducibility: Algorithms ensure that a specific set of instructions can be consistently followed to achieve the same result. This reproducibility is crucial for reliability in various applications.
  

• Automation: Algorithms enable automation by defining a set of instructions that can be executed by a computer without constant human intervention. This is fundamental for streamlining repetitive tasks.
  

• Scalability: Well-designed algorithms can handle larger datasets or increasingly complex problems without a disproportionate increase in resources, making them scalable for various applications.
  

• Consistency: Algorithms provide a consistent and logical framework for solving problems. This consistency is crucial in producing reliable results across different scenarios.
  

• Precision: Algorithms can be designed to execute tasks with precision, ensuring accuracy and reducing the likelihood of errors in calculations or decision-making processes.
  

• Optimization: Algorithms help optimize processes by finding the most efficient way to accomplish a task. This is particularly important in resource-constrained environments.
  

• Decision-Making: Algorithms are used in decision-making processes, ranging from business strategies to artificial intelligence applications. They assist in analyzing data and deriving insights to support informed decisions.
  

• Standardization: Algorithms offer standardized approaches to problem-solving. This standardization allows for the development of consistent methodologies across different applications and industries.
  

• Adaptability: Algorithms can be adapted to different contexts and applications, making them versatile tools for solving a wide range of problems.
  

• Technological Advancement: Algorithms drive innovation and technological advancement. They are foundational to the development of software, artificial intelligence, machine learning, and various computational technologies.
  

• Scientific Research: Algorithms play a crucial role in scientific research, assisting researchers in analyzing data, modelling phenomena, and simulating complex systems.
  

In summary, algorithms are indispensable in the field of computer science and problem-solving. They provide structured, efficient, and scalable solutions to a myriad of challenges, contributing to the advancement of technology, automation, and our ability to solve complex problems.

Example

Let's consider a real-world example to understand algorithms:

Task: Prepare a refreshing glass of lemonade.

Algorithm: Step-by-Step Instructions

Step - 1

• Gather Ingredients:

  • Collect the necessary ingredients: lemons, sugar, water, and ice.

Step - 2

• Wash and Cut Lemons:

  • Wash the lemons thoroughly.

  • Cut the lemons in half.

Step - 3

• Extract Lemon Juice:

  • Squeeze the lemon halves to extract the juice.

  • Use a strainer to separate seeds and pulp from the juice.

Step - 4

• Prepare Simple Syrup (Optional):

  • In a separate container, dissolve sugar in warm water to make a simple syrup. Adjust sweetness according to preference.

Step - 5

• Mix Lemon Juice and Simple Syrup:

  • Combine the freshly squeezed lemon juice with the prepared simple syrup. Adjust the ratio to achieve the desired sweetness.

Step - 6

• Add Water:

  • Dilute the lemon-sugar mixture with cold water. Adjust the water quantity based on taste preferences.

Step 7

• Stir Well:

  • Mix the ingredients thoroughly to ensure an even distribution of flavors.

Step - 8

• Taste and Adjust:

  • Taste the lemonade and adjust the sweetness or tartness by adding more sugar, water, or lemon juice as needed.

Step - 9

• Serve:

  • Pour the prepared lemonade into glasses over ice.

Real-world analogy:

Imagine you're following a recipe to make lemonade, carefully measuring and mixing each ingredient to create a delicious and refreshing beverage.

Algorithm Analysis:

• Input: Lemons, sugar, water, and ice.

• Processing: The sequence of steps involving squeezing, mixing, and adjusting ingredients.

• Output: A glass of freshly prepared lemonade.

Advantages of arrays

Arrays offer several advantages in programming due to their simplicity and efficiency in certain scenarios. Here are some key advantages of using arrays:

• Random Access:

  • Elements in an array can be accessed directly using their index. This allows for constant-time complexity for read and write operations, making access to elements efficient (O(1)).

• Sequential Storage:

  • Arrays store elements in contiguous memory locations. This sequential storage ensures better cache locality, potentially improving access times and system performance.

• Simplicity and Ease of Use:

  • Arrays are simple and intuitive data structures. They provide a straightforward way to organize and access a collection of elements of the same data type.

• Fixed Size:

  • The fixed size of arrays can be an advantage when the number of elements is known in advance. It allows for efficient memory allocation and can eliminate the need for dynamic memory management.

• Memory Efficiency:

  • Arrays are memory-efficient, as they do not incur the overhead associated with more complex data structures. Each element in an array occupies a fixed amount of memory.

• Performance in Mathematical Operations:

  • Arrays are well-suited for mathematical operations on large datasets. Many mathematical and scientific computations can be optimized using arrays and vectorized operations.

• Compatibility with Low-Level Operations:

  • Arrays are compatible with low-level memory operations and are often used in systems programming and embedded systems where direct memory manipulation is important.

• Efficient Traversal:

  • Traversing an array is a simple and efficient operation. Looping through elements using a for loop is a common and effective pattern.

• Efficient for Known Data Size:

  • When the number of elements is known and does not change frequently, arrays provide a concise and efficient way to store and retrieve data.

• Ease of Implementation:

  • Arrays are supported by virtually all programming languages and are usually among the first data structures introduced in programming courses. They are easy to declare, initialize, and use.

While arrays have these advantages, it's important to note that they may not be the best choice for all scenarios, particularly when dynamic resizing, insertion, or deletion of elements are frequent requirements. In such cases, other data structures like linked lists or dynamic arrays may be more suitable.

Importance of Algorithms:

• Problem Solving:

  • Significance: Algorithms provide systematic and structured approaches to problem-solving, allowing for efficient and effective solutions.

• Efficiency:

  • Significance: Well-designed algorithms optimize resource usage, leading to faster execution times, reduced energy consumption, and improved overall system performance.

• Reproducibility:

  • Significance: Algorithms enable the replication of solutions across different platforms and programming languages, fostering collaboration and standardization.

• Automation:

  • Significance: Algorithms automate processes, enabling systems to perform tasks, analyze data, and make decisions without continuous human intervention.

• Data Processing:

  • Significance: Algorithms are fundamental for data processing, including sorting, searching, filtering, and transforming data, facilitating meaningful insights from large datasets.

• Computational Complexity:

  • Significance: Understanding computational complexity helps in selecting appropriate algorithms, balancing time and space efficiency for different problem scenarios.

• Scientific Advancements:

  • Significance: Algorithms are vital for scientific research, aiding in solving complex equations, simulating physical processes, and conducting experiments in silico.

• Artificial Intelligence and Machine Learning:

  • Significance: Algorithms form the backbone of AI and ML, powering tasks like pattern recognition, classification, regression, and optimization for data-driven decision-making.

• Communication and Networking:

  • Significance: Algorithms are crucial in designing efficient communication protocols and network routing strategies, ensuring reliable data transfer between devices.

• Cryptography and Security:

  • Significance: Algorithms underpin secure communication through encryption and hashing, safeguarding data confidentiality and integrity.

• Optimization Problems:

  • Significance: Algorithms are employed to find optimal solutions in various domains, such as resource allocation, scheduling, and logistics, improving decision-making processes.

• Innovation and Technology Advancements:

  • Significance: The development of new algorithms often leads to technological breakthroughs, driving progress in fields from computer graphics to computational biology.

Issues with Algorithms:

• Algorithmic Bias and Fairness:

  • Challenge: Algorithms can perpetuate biases present in training data, leading to discriminatory outcomes, especially in machine learning applications.

• Transparency and Explainability:

  • Challenge: Some complex algorithms lack transparency, making it challenging to understand and explain their decisions, raising concerns about accountability and trust.

• Data Quality and Integrity:

  • Challenge: Algorithms heavily depend on the quality and integrity of data; inaccurate or biased data can lead to flawed outcomes.

• Overfitting and Generalization:

  • Challenge: Overfitting occurs when algorithms are too closely tailored to training data, negatively impacting performance on new, unseen data.

• Computational Complexity:

  • Challenge: High computational complexity in some algorithms can make them resource-intensive and impractical for certain applications.

• Security Concerns:

  • Challenge: Algorithms may be vulnerable to attacks, such as adversarial attacks in machine learning or vulnerabilities in cryptographic algorithms.

• Ethical Considerations:

  • Challenge: Ethical issues may arise when algorithms are used in decision-making processes, leading to concerns about privacy, consent, and fairness.

• User Acceptance and Bias Amplification:

  • Challenge: Algorithms may not always align with human values, leading to resistance and lack of acceptance. Biases in training data can be amplified, reinforcing stereotypes.

• Environmental Impact:

  • Challenge: Resource-intensive algorithms, particularly in deep learning, can contribute to significant energy consumption, raising environmental concerns.

• Regulatory and Legal Challenges:

  • Challenge: The rapid advancement of technology has outpaced the development of regulatory frameworks, leading to uncertainties in legal and ethical considerations.

Addressing these issues requires interdisciplinary collaboration, ethical considerations, and ongoing efforts to ensure that algorithms are designed and deployed responsibly and with societal well-being in mind.

Test your knowledge

ARRAY

An array is a data structure that stores a collection of elements, where each element can be accessed using an index or a key. Arrays are used to organize and store data systematically, making it easy to perform various operations on the elements. Here are some key properties and characteristics of arrays:

• Homogeneous Elements:

  • Arrays typically store elements of the same data type. For example, an array of integers will only contain integer values, and an array of strings will only contain string values.

• Fixed Size:

  • In most programming languages, arrays have a fixed size, meaning that once the array is created, its size cannot be changed. If you need a dynamic size, other data structures like lists or dynamic arrays may be more suitable.

• Contiguous Memory Allocation:

  • Array elements are stored in contiguous memory locations. This means that the memory addresses of consecutive elements in the array are adjacent.

• Zero-based Indexing:

  • In many programming languages, array indices start from 0. This means that the first element in the array is accessed using the index 0, the second element with index 1, and so on.

• Random Access:

  • Arrays support random access, meaning that you can directly access any element in the array using its index. This allows for efficient and constant-time access to individual elements.

• Ordered Elements:

  • The order of elements in an array is determined by their indices. This order is maintained throughout the lifetime of the array unless explicitly modified.

• Efficient for Iteration:

  • Iterating over the elements of an array is usually more efficient than other data structures, as the elements are stored in contiguous memory locations.

• Common Operations:

  • Arrays support common operations such as inserting, updating, and deleting elements. However, some of these operations may be less efficient than in other data structures, especially if the array needs to be resized.

Representation of Array

The representation of an array depends on the programming language. However, I'll provide a general explanation of how arrays are represented in memory, using a one-dimensional array as an example.

In memory, an array is a contiguous block of memory locations, and each element of the array is stored at a specific memory address. The memory addresses for the elements are determined based on the data type and the size of each element.

Let's consider an example of an array of integers in C:

c

int myArray[5] = {1, 2, 3, 4, 5};

Here,

int is a type.

myArray is an array of 5 integers,

5 is the size of the array

and these integers are elements of the array.

In memory, it might be represented like this:

| Element 0 | Element 1 | Element 2 | Element 3 | Element 4 | --

-----| 1 |                   |2 |               |3 |                |4 |                 |5 | ---

In this representation:

• Each element of the array is stored in a contiguous block of memory.

• The size of each element is determined by its data type (e.g., int in this case).

• The elements are accessed using indices (0 to 4 in this case) as the array index starts from zero

• The memory addresses of consecutive elements are sequential.

In languages like C or C++, you can use pointer arithmetic to navigate through the array elements based on memory addresses.

In languages like Python, arrays are implemented as lists, and the representation might be more abstract. Python lists are dynamic arrays, and their elements can be of different data types. The internal details of the implementation may vary based on the Python interpreter.

Understanding the memory representation of arrays is essential for optimizing algorithms and data access

Test your knowledge on Arrays

Disadvantages of Arrays

While arrays have several advantages, they also come with certain disadvantages that make them less suitable for certain scenarios. Here are some of the main disadvantages of using arrays:

• Fixed Size:

  • One of the significant drawbacks of arrays is their fixed size. The size of an array must be specified at the time of declaration, and it cannot be easily changed during runtime. This limitation can lead to either wasted memory or insufficient space for data.

• Memory Wastage:

  • If an array is declared with a size larger than the actual number of elements it needs to store, memory may be wasted. This is particularly true in situations where the array needs to accommodate the worst-case scenario.

• Inefficient Insertion and Deletion:

  • Inserting or deleting elements in the middle or at the beginning of an array requires shifting all subsequent elements, resulting in a time complexity of O(n). This can be inefficient for large arrays or frequent modification operations.

• Contiguous Memory Requirement:

  • Arrays require contiguous memory locations to store elements. In situations where contiguous memory is not available, it may limit the size of arrays or result in memory fragmentation.

• Homogeneous Data Type:

  • Arrays typically store elements of the same data type. If you need to store elements of different data types, an array may not be the most flexible data structure. In such cases, a collection or structure may be more appropriate.

• Static Structure:

  • The static nature of arrays makes them less suitable for scenarios where the size of the data is dynamic and unknown at compile time. Dynamic data structures like linked lists or dynamic arrays may be more appropriate.

• Lack of Built-in Methods:

  • Arrays in many low-level languages lack built-in methods for common operations such as sorting or searching. While higher-level languages may provide such functions, developers in lower-level languages may need to implement these operations manually.

• Poor Performance in Dynamic Sizing:

  • If dynamic resizing is required, such as when the number of elements is not known in advance, arrays may not be the most efficient data structure. Dynamic arrays or linked lists offer more flexibility in this regard.

• Not Well-Suited for Sparse Data:

  • For datasets with many empty or null values, arrays can be inefficient in terms of memory usage. Sparse data may be better represented using data structures designed for such scenarios, like sparse matrices.

In summary, while arrays are simple and efficient for certain use cases, their fixed size and limitations in terms of dynamic resizing and efficient insertion/deletion make them less suitable for certain applications. Developers often choose alternative data structures based on the specific requirements of their programs.

LINKED LISTS

A linked list is a data structure used in computer science to organize and store data.

It consists of a sequence of elements, each of which contains a reference (or link) to the next element in the sequence.

The last element typically has a reference to null, indicating the end of the list.

Representation of linked list

Each node has two components:

• Data: The actual data stored in the node.

• Next: A reference (pointer or link) to the next node in the sequence. The last node's "Next" points to null.

The linked list is then formed by connecting these nodes.

• Head: Points to the first node in the list.

• Data: Represents the actual information stored in each node.

• Next: Represents the reference to the next node.

Let's consider a simple example where the linked list contains the elements 1, 2, and 3.

In this example:

• The first node contains the data "1" and points to the next node with the data "2."

• The second node contains the data "2" and points to the next node with the data "3."

• The third node contains the data "3" and points to null, indicating the end of the list.

Traversal through the linked list involves starting from the head and following the "Next" pointers until reaching the end (null). Each arrow represents the link from one node to the next.

Why use linked lists over arrays

The choice between linked lists and arrays depends on the specific requirements and characteristics of the task at hand. Each data structure has its own advantages and disadvantages. Here are some reasons why you might choose linked lists over arrays:

• Dynamic Size:

  • Linked lists allow for dynamic sizing. They can easily grow or shrink during runtime by allocating or deallocating memory for nodes. In contrast, arrays in many programming languages have a fixed size, and resizing them can be inefficient.

• Efficient Insertions and Deletions:

  • Insertions and deletions can be more efficient in linked lists, especially when dealing with elements in the middle of the list. In arrays, inserting or removing an element may require shifting all subsequent elements.

• No Pre-allocation of Memory:

  • Linked lists don't require pre-allocation of a contiguous block of memory. Each node in a linked list can be allocated independently, which can be beneficial in scenarios where memory is fragmented or when the size of the data structure is unpredictable.

• Constant-time Insertions/Deletions at Head:

  • Adding or removing elements at the beginning of a linked list is a constant-time operation, as it only involves updating the head pointer and the next reference of the new first node. In arrays, this operation is typically more expensive as it requires shifting all elements.

• Ease of Implementation:

  • Linked lists can be easier to implement in certain cases, especially when dealing with dynamic data structures. There's no need to worry about resizing, and inserting or deleting nodes can be done without

  • 
    

  • the need for complex memory management.

However, it's essential to note that linked lists also have their own disadvantages compared to arrays:

• Random Access:

  • Random access to elements (accessing elements by index) is inefficient in linked lists. In arrays, direct access is achieved in constant time, but in linked lists, you need to traverse the list from the head to the desired position.

• Memory Overhead:

  • Each node in a linked list requires additional memory for the next pointer, leading to a higher memory overhead compared to arrays.

• Cache Performance:

  • Arrays have better cache performance due to their contiguous memory allocation, resulting in faster access times. Linked lists may cause cache misses more frequently, leading to slower access times.

The choice between linked lists and arrays depends on the specific requirements of the application. For scenarios where dynamic sizing and efficient insertions/deletions are crucial, linked lists might be a better choice. In cases where random access and memory efficiency are more critical, arrays may be more suitable.

Types of Linked Lists

let’s have a look at the overview of types of linked lists.

There are several types of linked lists, each with its own variations and use cases. The two primary types are singly linked lists and doubly linked lists. Here's an overview of these types:

• Singly Linked List:

  • In a singly linked list, each node contains data and a reference to the next node in the sequence.

  • The last node typically points to null, indicating the end of the list.

  • Traversal is only forward.

  • Simple and memory-efficient.
    

• Doubly Linked List:

  • In a doubly linked list, each node contains data and references to both the next and the previous nodes in the sequence.

  • Allows for traversal in both forward and backward directions.

  • More memory overhead due to the additional "prev" pointers.

  • Easier implementation of certain operations, such as deletion of a node given a reference to that node.
    

• Circular Linked List:

  • In a circular linked list, the last node points back to the first node, forming a circle.

  • Useful for applications where it's convenient to start again at the beginning after reaching the end.

  • The traversal continues indefinitely until a specific condition is met.
    

• Circular singly linked list -A circular singly linked list is characterized by the last node in the list having a pointer that references the first node. Both circular singly linked lists and circular doubly linked lists can be implemented.

• Circular doubly linked list -A circular doubly linked list is a sophisticated data structure where each node includes pointers to both its preceding and succeeding nodes. Unlike regular doubly linked lists, a circular doubly linked list doesn't have any NULL references within its nodes. Instead, the last node in the list holds the address of the first node, creating a circular structure. Additionally, the first node includes the address of the last node in its previous pointer.
  

These are some of the common types of linked lists. Depending on specific requirements and constraints, one type of linked list may be more suitable than another for a given application or problem.

Advantages of Linked lists

Linked lists offer several advantages that make them suitable for certain applications. Here are some key advantages of linked lists:

• Dynamic Size:

  • Linked lists can easily grow or shrink in size during runtime. This dynamic sizing makes them flexible in handling data structures where the size is not known in advance.

• Efficient Insertions and Deletions:

  • Adding or removing elements in a linked list, especially in the middle, is more efficient compared to arrays. It involves updating pointers, and there is no need to shift elements as in arrays.

• No Pre-allocation of Memory:

  • Linked lists do not require pre-allocation of a fixed-size block of memory. Nodes can be allocated independently, and new nodes can be added as needed.

• Constant-time Insertions/Deletions at Head:

  • Adding or removing elements at the beginning of a linked list is a constant-time operation. It only involves updating the head pointer and the next reference of the new first node.

• Ease of Implementation:

  • Implementing linked lists can be simpler in certain cases, especially when dealing with dynamic data structures. There is no need to worry about resizing, and inserting or deleting nodes can be done without complex memory management.

• No Wasted Memory:

  • Linked lists do not suffer from wasted memory due to pre-allocation of a fixed-size block. Memory is used more efficiently, as each node only requires space for data and references.

• No Overhead for Resizing:

  • Linked lists do not incur the overhead associated with resizing, which can be a concern with dynamically resizing arrays.

• Constant-time Concatenation:

  • Concatenating two linked lists can be done in constant time by adjusting pointers, unlike arrays where concatenation involves copying elements from one array to another.

• Support for Various Types:

  • Linked lists can be easily adapted to support different types of nodes, such as singly linked lists, doubly linked lists, or circular linked lists, based on specific requirements.

• Ease of Building Other Data Structures:

  • Linked lists provide the foundation for building other data structures, such as stacks, queues, and symbolic expressions in languages.

Linked lists are particularly advantageous when there is a need for dynamic sizing, efficient insertions and deletions, and flexibility in managing memory. However, the choice between linked lists and other data structures depends on the specific requirements of the application.

Disadvantages of Linked lists

While linked lists offer several advantages, they also come with certain disadvantages that should be considered depending on the specific requirements of an application. Here are some disadvantages of linked lists:

• No Random Access:

  • Accessing elements by index is inefficient in linked lists. To reach a specific element, you need to traverse the list from the beginning, which takes linear time. This contrasts with arrays, where random access is constant time.

• Memory Overhead:

  • Each node in a linked list requires additional memory for the next (and possibly previous in the case of doubly linked lists) pointers. This can result in higher memory overhead compared to arrays, where only the data needs to be stored.

• Cache Performance:

  • Linked lists may exhibit poorer cache performance compared to arrays. This is because elements in arrays are stored in contiguous memory locations, which can lead to better cache utilization and faster access times.

• Sequential Access:

  • While forward traversal is efficient, backward traversal (in the case of singly linked lists) or bidirectional traversal (in the case of doubly linked lists) may be less efficient than sequential access in arrays.

• Lack of Contiguity:

  • The non-contiguous nature of linked lists can result in poor cache utilization and may lead to increased memory paging in some scenarios. This can impact performance in certain applications.

• More Complex Implementation:

  • Implementing linked lists can be more complex than arrays, especially when dealing with pointers and dynamic memory allocation. This complexity may lead to increased chances of errors in the code.

• Extra Time and Space for Pointer References:

  • The need for storing references (pointers) in each node consumes extra memory, and dereferencing these pointers adds some overhead in terms of execution time.

• Not Suitable for Small Data Sets:

  • For small data sets or situations where random access is crucial, the overhead of linked lists may outweigh their benefits. Arrays might be more suitable in such cases.

• Potential for Memory Leaks:

  • If not managed properly, linked lists can lead to memory leaks. Failing to deallocate memory properly when nodes are removed can result in unreleased memory.

• Complexity in Reverse Traversal:

  • In singly linked lists, traversing the list in reverse requires maintaining additional information (such as a stack) or changing the structure of the list.

In summary, linked lists have their disadvantages, and the choice between linked lists and other data structures depends on the specific needs of the application. While linked lists are powerful in certain scenarios, they may not be the best choice for every situation.

Applications of linked lists

Linked lists find application in various scenarios due to their dynamic nature and efficient insertions and deletions. Some common applications include:

• Dynamic Memory Allocation:

  • Linked lists are often used for dynamic memory allocation, where the size of the data structure is not known in advance, and memory needs to be allocated or deallocated during runtime.

• Implementation of Data Structures:

  • Linked lists serve as the foundation for implementing other data structures such as stacks, queues, deques, and symbolic expressions in languages like Lisp.

• Undo Mechanisms:

  • In applications that require an "undo" mechanism, linked lists can be used to keep track of the state changes. Each node represents a state, and traversing backwards allows undoing operations.

• File Systems:

  • In file systems, linked lists are employed to maintain the directory structure. Each node represents a file or directory, and the links connect the hierarchical structure.

• Task Scheduling:

  • Operating systems often use linked lists to manage processes in a queue, where each node represents a process and the links determine the order of execution.

• Music Playlists:

  • Linked lists can be used to implement playlists in music applications. Each node represents a song, and the links determine the order of playback.

• Graphs:

  • Linked lists are used in the adjacency list representation of graphs. Each node represents a vertex, and the links connect vertices that share an edge.

• Hash Tables:

  • Separate chaining in hash tables often involves using linked lists to handle collisions. Each bucket in the hash table can be a linked list of elements that hash to the same index.

• Simulation of Real-World Objects:

  • Linked lists can be used to simulate real-world objects and their relationships. For example, modeling a train with each car represented by a node, linked together in sequence.

• Polynomial Representation:

  • Linked lists are used to represent polynomials, where each node contains a term of the polynomial, and links connect the terms in the proper order.

• Symbol Table in Compilers:

  • Compilers use linked lists to implement symbol tables. Each node represents a symbol, and the links connect symbols based on their scope and references.

• Network Routing Algorithms:

  • In networking, linked lists can be used to represent routes. Each node represents a router, and the links represent the connections between routers.

These are just a few examples, and the versatility of linked lists makes them suitable for a wide range of applications where dynamic data structures and efficient insertions/deletions are essential.

Operations performed on linked lists

Here are some common operations performed on linked lists:

• Insertion at the Beginning (Push):

  • Add a new node at the beginning of the linked list by updating the head pointer to point to the new node, and the new node's next pointer to the previous head.

• Insertion at the End (Append):

  • Add a new node at the end of the linked list by traversing to the last node and updating its next pointer to point to the new node.

• Insertion at a Specific Position:

  • Insert a new node at a specified position by updating the next pointers of the preceding and succeeding nodes accordingly.

• Deletion at the Beginning (Pop):

  • Remove the first node by updating the head pointer to point to the second node and freeing the memory of the removed node.

• Deletion at the End:

  • Remove the last node by traversing to the second-to-last node and updating its next pointer to null, then freeing the memory of the removed node.

• Deletion at a Specific Position:

  • Delete a node at a specified position by updating the next pointers of the preceding and succeeding nodes accordingly and freeing the memory of the removed node.

• Traversal (Print/Display):

  • Traverse the linked list from the head to the last node, printing or displaying the data of each node.

• Search:

  • Search for a specific element in the linked list by traversing the list and comparing the data of each node.

• Length (Size) Calculation:

  • Calculate the number of nodes in the linked list by traversing the list and counting the nodes.

• Reverse:

  • Reverse the order of nodes in the linked list by adjusting the next pointers of each node to point to the previous node.

• Concatenation:

  • Concatenate two linked lists by updating the next pointer of the last node in the first list to point to the head of the second list.

• Splitting:

  • Split a linked list into two by updating the next pointer of the last node in the first list to null.

• Detecting a Cycle:

  • Check if a linked list has a cycle (loop) by using Floyd's cycle-finding algorithm or similar techniques.

• Finding the Middle Node:

  • Determine the middle node of a linked list, often used in various algorithms.

These operations provide the basic functionalities needed to manipulate and manage data in a linked list. The choice of operation depends on the specific requirements of the application.

Please download the file for reference of the program in other programming languages.

Circular doubly linked list

A circular doubly linked list is a variation of the traditional doubly linked list data structure where the last node's next pointer points back to the first node, creating a circular structure. Similarly, the first node's previous pointer points to the last node, completing the circular linkage.

Because a circular doubly linked list incorporates three components in its structure, it requires greater memory per node and entails more costly fundamental operations. Nevertheless, this type of linked list facilitates straightforward manipulation of pointers and enhances search efficiency twofold.

Here's a brief overview of how a circular doubly linked list is structured:

• Node Structure: node in the circular doubly linked list contains two pointers - one pointing to the next node and the other pointing to the previous node. Additionally, it holds some data.
  

• Circularity: The last node's next pointer points to the first node, and the first node's previous pointer points to the last node, forming a circular structure. This ensures that the traversal of the list can be continuous without reaching a null pointer.
  

• Operations: Operations such as insertion, deletion, and traversal are similar to those in a traditional doubly linked list. However, care must be taken to handle the circular nature of the list properly, especially during insertion and deletion operations.
  

• Traversal: Traversal of a circular doubly linked list can start from any node and can proceed either forward (using next pointers) or backward (using previous pointers) until the starting node is reached again.
  

Here's a basic implementation of a circular doubly linked list in C:

#include <stdio.h>

#include <stdlib.h>

// Define a node structure for the circular doubly linked list

struct Node {

int data;

struct Node* next;

struct Node* prev;

};

// Function to create a new node

struct Node* createNode(int data) {

struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));

if (newNode == NULL) {

printf("Memory allocation failed!\n");

exit(1);

}

newNode->data = data;

newNode->next = NULL;

newNode->prev = NULL;

return newNode;

}

// Function to insert a new node at the beginning of the list

void insertAtBeginning(struct Node** head, int data) {

struct Node* newNode = createNode(data);

if (*head == NULL) {

*head = newNode;

(*head)->next = *head;

(*head)->prev = *head;

} else {

newNode->next = *head;

newNode->prev = (*head)->prev;

(*head)->prev->next = newNode;

(*head)->prev = newNode;

*head = newNode;

}

}

// Function to display the circular doubly linked list

void display(struct Node* head) {

if (head == NULL) {

printf("List is empty!\n");

return;

}

struct Node* temp = head;

do {

printf("%d ", temp->data);

temp = temp->next;

} while (temp != head);

printf("\n");

}

int main() {

struct Node* head = NULL;

// Insert some elements into the circular doubly linked list

insertAtBeginning(&head, 1);

insertAtBeginning(&head, 2);

insertAtBeginning(&head, 3);

insertAtBeginning(&head, 4);

// Display the circular doubly linked list

printf("Circular Doubly Linked List: ");

display(head);

return 0;

}

BREAK DOWN

#include <stdio.h>

#include <stdlib.h>

These are standard C header files. <stdio.h> is included for input and output operations like printf, and <stdlib.h> is included for dynamic memory allocation functions like malloc and exit.

Node Structure:

struct Node {

int data;

struct Node* next;

struct Node* prev;

};

• struct Node: This line declares a structure named Node. Structures in C are user-defined data types that allow you to group multiple variables of different types under a single name.
  

• int data: This line declares a member variable data of type int within the structure. This member variable is used to store the actual data associated with a node in the linked list.
  

• struct Node next*: This line declares a pointer variable next of type struct Node*. This pointer is used to point to the next node in the linked list. Since it's a pointer to the same type struct Node, it facilitates the creation of linked lists where each node points to the next node.
  

• struct Node prev*: This line declares another pointer variable prev of type struct Node*. This pointer is used to point to the previous node in a doubly linked list. A doubly linked list is a type of linked list where each node contains a pointer to the previous node as well as the next node. This allows traversal of the list in both forward and backward directions.
  

So, overall, this struct declaration is defining a node structure for a doubly linked list, where each node contains an integer data value (data), a pointer to the next node (next), and a pointer to the previous node (prev).

Next we will see the Function to Create a New Node:

struct Node* createNode(int data) {

struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));

if (newNode == NULL) {

printf("Memory allocation failed!\n");

exit(1);

}

newNode->data = data;

newNode->next = NULL;

newNode->prev = NULL;

return newNode;

}

struct Node* createNode(int data)

• • This function takes an integer data as a parameter and returns a pointer to a struct Node.

• 
  

• struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));

• This line allocates memory dynamically using the malloc() function to create a new node.

• sizeof(struct Node) calculates the size required to store a struct Node.

• malloc() returns a pointer to the allocated memory block.

• The pointer is cast to (struct Node*) to ensure compatibility with the pointer type.

if (newNode == NULL) {

printf("Memory allocation failed!\n");

exit(1);

}

• This block checks if memory allocation was successful.

• If malloc() returns NULL, it indicates that memory allocation failed.

• In such a case, an error message is printed, and the program exits with an error code (exit(1)).

newNode->data = data;

newNode->next = NULL;

newNode->prev = NULL;

• This part initializes the members of the newly created node.

• newNode->data assigns the data parameter to the data member of the node.

• newNode->next and newNode->prev set the next and prev pointers of the node to NULL, indicating that this node is currently not linked to any other nodes.

return newNode;

• • Finally, the function returns a pointer to the newly created node.

Overall, this function dynamically allocates memory for a new node, initializes its data and pointers, and returns a pointer to the created node. It's a common function used when working with linked list implementations in C.

Now we will write the Function to Insert a Node at the Beginning:

void insertAtBeginning(struct Node** head, int data) {

struct Node* newNode = createNode(data);

if (*head == NULL) {

*head = newNode;

(*head)->next = *head;

(*head)->prev = *head;

} else {

newNode->next = *head;

newNode->prev = (*head)->prev;

(*head)->prev->next = newNode;

(*head)->prev = newNode;

*head = newNode;

}

}

• struct Node* newNode = createNode(data);: This line creates a new node with the given data using the createNode function. This function dynamically allocates memory for a new node, initializes its fields with the given data, and returns a pointer to the newly created node.
  

• if (*head == NULL) {: This conditional statement checks if the pointer to the head of the list (*head) is NULL, which means the list is currently empty.
  

• *head = newNode;: If the list is empty, this line sets the pointer to the head of the list (*head) to point to the newly created node (newNode). Since this is the only node in the list, both its next and prev pointers are set to point back to itself, making it circular.
  

• (*head)->next = *head;: This line sets the next pointer of the newly created node (which is now the head of the list) to point to itself, establishing the circular link.
  

• (*head)->prev = *head;: This line sets the prev pointer of the newly created node (which is now the head of the list) to point to itself, establishing the circular link.
  

• else {: If the list is not empty, execution continues from here.
  

• newNode->next = *head;: This line sets the next pointer of the newly created node (newNode) to point to the current head of the list (*head), effectively inserting the new node at the beginning.
  

• newNode->prev = (*head)->prev;: This line sets the prev pointer of the newly created node (newNode) to point to the node that was previously the last node in the list (which is (*head)->prev).
  

• (*head)->prev->next = newNode;: This line updates the next pointer of the node that was previously the last node in the list (which is (*head)->prev) to point to the newly created node (newNode), effectively linking it to the new node.
  

• (*head)->prev = newNode;: This line updates the prev pointer of the current head of the list (*head) to point to the newly created node (newNode), effectively completing the circular link.
  

• *head = newNode;: Finally, this line updates the pointer to the head of the list (*head) to point to the newly created node (newNode), making it the new head of the list.
  

In summary, the insertAtBeginning function inserts a new node with the given data at the beginning of the circular doubly linked list. If the list is empty, it creates a new node and makes it the head of the list, establishing circular links. If the list is not empty, it inserts the new node at the beginning and updates the necessary pointers to maintain the circular doubly linked list structure.

Function to Display the List:

void display(struct Node* head) {

if (head == NULL) {

printf("List is empty!\n");

return;

}

struct Node* temp = head;

do {

printf("%d ", temp->data);

temp = temp->next;

} while (temp != head);

printf("\n");

}

• if (head == NULL) {: This conditional statement checks if the pointer to the head of the list (head) is NULL, which means the list is empty.
  

• printf("List is empty!\n");: If the list is empty, this line prints a message indicating that the list is empty.
  

• return;: After printing the message, the function returns, terminating further execution of the function.
  

• struct Node* temp = head;: If the list is not empty, this line declares a temporary pointer temp and initializes it with the pointer to the head of the list (head). This pointer will be used to traverse the list.
  

• do {: This initiates a do-while loop that will iterate at least once and continue until temp reaches the head of the list again, forming a circular loop.
  

• printf("%d ", temp->data);: Inside the loop, this line prints the data of the current node (temp->data), which is the data stored in the node pointed to by temp.
  

• temp = temp->next;: This line updates the temp pointer to point to the next node in the list, moving it forward in the list.
  

• } while (temp != head);: This loop continues until the temp pointer reaches the head of the list again, completing one full traversal of the circular list.
  

• printf("\n");: After the loop terminates, this line prints a newline character to move to the next line in the output.
  

In summary, the display function prints the data stored in each node of the circular doubly linked list. If the list is empty, it prints a message indicating that the list is empty. If the list is not empty, it traverses the list starting from the head node and prints the data of each node until it reaches the head node again, completing one full traversal of the circular list.

Main Function:

int main() {

struct Node* head = NULL;

// Insert some elements into the circular doubly linked list

insertAtBeginning(&head, 1);

insertAtBeginning(&head, 2);

insertAtBeginning(&head, 3);

insertAtBeginning(&head, 4);

// Display the circular doubly linked list

printf("Circular Doubly Linked List: ");

display(head);

return 0;

}

• struct Node* head = NULL;: This line declares a pointer head to the head of the circular doubly linked list and initializes it to NULL, indicating that the list is initially empty.
  

• insertAtBeginning(&head, 1);: This line inserts a new node with data 1 at the beginning of the circular doubly linked list pointed to by head. Since the list is initially empty, this node becomes the head of the list.
  

• insertAtBeginning(&head, 2);: This line inserts a new node with data 2 at the beginning of the list. This node becomes the new head of the list, and the previous head becomes the next node in the list.
  

• insertAtBeginning(&head, 3);: This line inserts a new node with data 3 at the beginning of the list. Similar to the previous step, this node becomes the new head of the list, and the previous head becomes the next node in the list.
  

• insertAtBeginning(&head, 4);: This line inserts a new node with data 4 at the beginning of the list. Again, this node becomes the new head of the list, and the previous head becomes the next node in the list.
  

• printf("Circular Doubly Linked List: ");: This line prints a message indicating that the following output represents a circular doubly linked list.
  

• display(head);: This line calls the display function to print the elements of the circular doubly linked list starting from the head node.
  

• return 0;: This line indicates that the main function has executed successfully, and the program should exit with a return code of 0, indicating success.
  

In summary, the main function creates a circular doubly linked list with elements 1, 2, 3, and 4, inserts them at the beginning of the list, and then displays the contents of the list.

STACK

A stack is a fundamental data structure that follows the Last In, First Out (LIFO) principle. It is named "stack" because it resembles a stack of items, like a stack of plates or books. In a stack, the last element added is the first one to be removed.

The standard stack operations are

The standard operations that can be performed on a stack data structure are as follows:

• Push: This operation adds an element to the top of the stack.

• Pop: This operation removes the top element from the stack.

• Peek or Top: This operation allows you to view the top element of the stack without removing it.

• isEmpty: This operation checks whether the stack is empty or not.

• Size: This operation returns the number of elements currently present in the stack.

Working of stack

Let us understand the Working of the stack

First, we will see push operations

The steps involved in push operation are

Before adding an element to a stack, it is verified whether the stack is empty.

• If we try to insert the element in a stack, and the stack is full, then the overflow condition occurs.

• When we initialize a stack, we set the value of top as -1 to check that the stack is empty.

• When the new element is pushed in a stack, first, the value of the top gets incremented, i.e., top=top+1, so the top = -1 becomes top = 0 and the element will be placed at the new position of the top.

First, we will add element 10 to the stack and the top will be incremented from - 1 to 0

• The elements will be inserted until we reach the maximum size of the stack.

Next, we will Insert 20 and the top becomes 1

push the element 30 the top is now 2

push the element 40 the top is 3

and push the element 50 top is 4 and the stack is full.

we cannot not push the elements further in this stack. if we do so the overflow condition occurs.

now let us see the next operation that is pop.

deleting the elements from the stack is called a pop operation.

• Before deleting the element from the stack, we check whether the stack is empty.

• If we try to delete the element from the empty stack, then the underflow condition occurs.

• If the stack is not empty, we first access the element which is pointed by the top

• Once the pop operation is performed, the top is decremented by 1, i.e., top=top-1.

Here we have a stack that is not empty

so first we will remove element 50

next, we will remove element 40

now remove element 30

remove the element 20

and remove 10

now the stack is empty

Next we will discuss the applications of stacks

Stacks have numerous applications across various domains in computer science and beyond. Here are some common applications of stacks:

• Function Call Stack: In programming languages like C/C++, Java, and Python, a stack is used to manage function calls. When a function is called, its parameters and local variables are pushed onto the stack. When the function returns, these values are popped off the stack.
  

• Expression Evaluation: Stacks are used to evaluate arithmetic expressions, such as infix, postfix, and prefix expressions. Infix expressions can be converted to postfix or prefix notation using stacks for easier evaluation.
  

• Backtracking Algorithms: Stacks are often used in backtracking algorithms such as Depth-First Search (DFS). During the search process, the states and choices are pushed onto the stack, and when a dead end is reached, the algorithm backtracks by popping from the stack.
  

• Undo Mechanisms: Stacks can be used to implement undo mechanisms in text editors and other software applications. Each action performed by the user is pushed onto the stack, allowing for easy reversal of actions by popping from the stack.
  

• Browser History: Web browsers use stacks to implement the back and forward navigation buttons. Each visited page is pushed onto the stack, allowing users to navigate backward and forward through their browsing history.
  

• Parenthesis Matching: Stacks are used to validate the correctness of expressions containing parentheses, braces, and brackets. By pushing opening symbols onto the stack and popping them when closing symbols are encountered, the balance of the symbols can be checked.
  

• Compiler and Interpreter Implementations: Stacks play a crucial role in the implementation of compilers and interpreters for programming languages. They are used for parsing, code generation, and managing the execution environment.
  

• Memory Management: Stacks are used in memory management systems to allocate and deallocate memory for function calls and local variables. The stack memory is automatically managed by the system, making it efficient for managing memory in a structured manner.
  

These are just a few examples of the many applications of stacks in computer science and software development. Stacks provide a simple and efficient way to manage data and perform various operations in a wide range of applications.

A queue is a fundamental data structure in computer science that follows the First-In-First-Out (FIFO) principle.

In a queue, elements are added at one end called the "rear" or "tail" and removed from the other end called the "front" or "head".

A real-world example of a queue is a line at a supermarket checkout. When customers arrive at the checkout, they join the queue, and the cashier serves them one by one in the order they joined the line. This follows the First-In-First-Out (FIFO) principle of a queue data structure.

Here's how the analogy maps to the queue operations:

1. Enqueue: When a new customer arrives at the checkout, they join the back of the queue.

2. Dequeue: The cashier serves the customer at the front of the queue, removing them from the line.

3. Peek: The cashier can see who is at the front of the queue without serving them yet, perhaps to prepare for the next customer.

4. IsEmpty: If there are no customers in the queue, it is empty.

5. IsFull: In a real-world scenario like a supermarket, the queue is often not limited by a fixed size, but in a theoretical queue implementation with a fixed-size array, the queue could be considered full when it reaches its maximum capacity.

Other real-world examples of queues include:

• Waiting in line at a ticket counter, bank, or amusement park.

• Processing print jobs in a printer queue.

• Handling requests in a web server's request queue.

• Processing tasks in a CPU scheduling algorithm.

Application of queues

1. Queues are commonly employed as waitlists for a singular shared resource, such as a printer, disk drive, or CPU.

2. In scenarios involving asynchronous data transfer, where data exchange rates between two processes are not synchronized, queues play a pivotal role. This includes processes like handling pipes, file input/output operations, and sockets.

3. In numerous applications, queues serve as buffers to manage data flow. This includes media players like MP3 players and CD players, among others.

4. Media players utilize queues to manage playlists, enabling the addition and removal of songs in a sequential manner.

5. Within operating systems, queues are utilized to manage interrupts effectively, ensuring timely handling of various system events.

Types of queues

Four different types of queues are

• Simple Queue or Linear Queue

• Circular Queue

• Priority Queue

• Double Ended Queue (or Deque)

Let's discuss each of the types of queues.

Simple Queue or Linear Queue

In a Linear Queue, insertion occurs at one end, while deletion occurs at the other end. The end where insertion takes place is called the rear end, and the end where deletion takes place is called the front end. Linear Queue strictly adheres to the FIFO (First In, First Out) rule.

Diagram

The primary limitation of using a linear queue is that insertion is only possible from the rear end. If the first three elements are deleted from the queue, even though space is available, we cannot insert more elements. In this scenario, the linear queue indicates an overflow condition because the rear is pointing to the last element of the queue.

Next we will see circular queue

Circular Queue

In a Circular Queue, all the nodes are arranged in a circular manner. It resembles a linear queue, but the last element of the queue is connected to the first element, creating a circular structure. Circular Queue is also known as a Ring Buffer because all ends are connected to one another.

Diagram

The drawback of a linear queue is overcome by using a circular queue. In a circular queue, if there is empty space available, a new element can be added to that space by simply incrementing the value of the rear pointer. The primary advantage of using a circular queue is improved memory utilization.

Now let see the priority queue

Priority queue

A priority queue is a special type of queue where elements are arranged based on their priority. In this queue data structure, each element is associated with a priority. If multiple elements have the same priority, they are arranged according to the FIFO (First In, First Out) principle. The representation of a priority queue is shown in the image below:

In a priority queue, insertion is based on arrival time, while deletion is based on priority. Priority queues are primarily used to implement CPU scheduling algorithms.

There are two types of priority queues

1. Ascending priority queue - In an ascending priority queue, elements can be inserted in arbitrary order, but only the smallest element can be deleted first. For example, if an array has elements 7, 5, and 3 in the same order, insertion can be done in the same sequence, but the order of deleting the elements is 3, 5, 7.

2. Descending priority queue - In a descending priority queue, elements can be inserted in arbitrary order, but only the largest element can be deleted first. For example, if an array has elements 7, 3, and 5 in the same order, insertion can be done in the same sequence, but the order of deleting the elements is 7, 5, 3.

Deque (or, Double Ended Queue)

In a Deque or Double Ended Queue, insertion and deletion can be performed from both ends of the queue, either from the front or the rear. This means that elements can be inserted and deleted from both the front and rear ends of the queue. A Deque can be used as a palindrome checker, as reading the string from both ends would result in the same string.

A Deque can be used both as a stack and a queue because it allows insertion and deletion operations at both ends. It can be considered as a stack because a stack follows the LIFO (Last In, First Out) principle, where insertion and deletion can only be performed from one end. In a deque, it is possible to perform both insertion and deletion from one end, and it does not strictly follow the FIFO principle.

Linked list representation of the queue.

Because of the limitations outlined in the previous section, the array implementation cannot be used for large-scale applications where queues are implemented.

One alternative to the array implementation is the linked list implementation of a queue.

The linked representation of a queue with

n elements has a storage requirement of

O(n), while the time requirement for operations is O(1).

In a linked queue, each node consists of two parts: a data part and a link part. Each element of the queue points to its immediate next element in memory.

There are two pointers”the front pointer and the rear pointer” maintained in memory in the linked queue. The address of the queue's first element is contained in the front pointer, while the address of the queue's last element is contained in the rear pointer.

At the front end and back end, respectively, insertions and deletions are made. The queue is empty if both the front and rear are NULL.

Operation on Linked Queue

There are 2 fundamental operations performed on the linked queues, they are insertion and deletion

Insertion operation.

By appending an element to the end of the queue, the insert operation appends the queue. The final element in the queue will be the new one.

we will use the this line to allocate memory for the new node ptr.

Ptr = malloc (sizeof(struct node) * (struct node *);

Inserting this new node PTR into the connected queue can go in one of two ways.

The first case involves adding an element to a queue that is empty. Under these circumstances, front = NULL becomes true.

At this time, the front and rear pointers' next points will both point to NULL, making the new element the only one in the queue.

ptr -> data = item;

if(front == NULL)

{

front = ptr;

rear = ptr;

front -> next = NULL;

rear -> next = NULL;

}

ptr->data = item;

This line assigns the value of item to the data member of the structure that ptr is pointing to.

if(front == NULL)

{

front = ptr;

rear = ptr;

front->next = NULL;

rear->next = NULL;

}

This part of the code is a conditional statement. It checks if the front pointer is NULL. If it is, it means the queue is empty.

• If the queue is empty (front is NULL), front, and rear are assigned to ptr, which is the new node being added to the queue.

• Then, front->next and rear->next are set to NULL, indicating that there is no other element in the queue.

So, in summary, if the queue is empty, ptr becomes the only node in the queue, and both front and rear point to it.

In the second scenario, there are multiple elements in the queue. It becomes false when the condition front = NULL. To make the next pointer of rear point to the new node ptr in this circumstance, we must update the end pointer rear. We also need to set the rear pointer to point to the newly inserted node ptr because this is a linked queue. Additionally, we must set the subsequent rear point pointer to NULL.

rear -> next = ptr;

rear = ptr;

rear->next = NULL;

rear->next = ptr;

This line sets the next pointer of the current rear node to point to the new node ptr that has been added to the queue.

rear = ptr;

This line updates the rear pointer to point to the new node ptr, making ptr the new rear of the queue.

rear->next = NULL;

This line ensures that the next pointer of the new rear node (ptr) is NULL, indicating the end of the queue.

So, in summary:

• rear->next = ptr;: Connects the current rear node to the new node ptr.

• rear = ptr;: Updates rear to point to the new node ptr.

• rear->next = NULL;: Sets the next pointer of the new rear node (ptr) to NULL, indicating the end of the queue.

Algorithm

Allocate the space for the new node PTR

SET PTR -> DATA = VAL

IF FRONT = NULL
SET FRONT = REAR = PTR
SET FRONT -> NEXT = REAR -> NEXT = NULL
ELSE
SET REAR -> NEXT = PTR
SET REAR = PTR
SET REAR -> NEXT = NULL
[END OF IF]

END

1. Allocate the space for the new node PTR: This allocates memory for a new node and assigns it to the pointer PTR.

2. SET PTR -> DATA = VAL: Sets the DATA part of the new node to the value VAL.

3. IF FRONT = NULL: Checks if the list is empty.

   • SET FRONT = REAR = PTR: If the list is empty, FRONT and REAR are set to point to the new node (PTR). This means that FRONT and REAR both point to the first node in the list.

   • SET FRONT -> NEXT = REAR -> NEXT = NULL: Sets the NEXT pointers of both FRONT and REAR to NULL, as there's only one node in the list.

4. ELSE: If the list is not empty,

   • SET REAR -> NEXT = PTR: Sets the NEXT pointer of the current last node (REAR) to point to the new node (PTR), effectively linking the new node to the end of the list.

   • SET REAR = PTR: Moves the REAR pointer to the new last node (PTR).

   • SET REAR -> NEXT = NULL: Sets the NEXT pointer of the new last node (REAR) to NULL, indicating the end of the list.

function to insert an element in linked queue

void insert(struct node *ptr, int item)

{

ptr = (struct node *) malloc (sizeof(struct node));

if(ptr == NULL)

{

printf("\nOVERFLOW\n");

return;

}

else

{

ptr -> data = item;

if(front == NULL)

{

front = ptr;

rear = ptr;

front -> next = NULL;

rear -> next = NULL;

}

else

{

rear -> next = ptr;

rear = ptr;

rear->next = NULL;

}

}

}

ptr = (struct node *) malloc (sizeof(struct node));:

This line allocates memory for a new node and assigns the address of the allocated memory to ptr.

if(ptr == NULL):

This checks if memory allocation was successful. If not, it prints "OVERFLOW" and returns, indicating that the function was unable to insert the new node.

ptr -> data = item;:

Assigns the value of item to the data member of the newly created node.

if(front == NULL): Checks if the list is empty.

a. If the list is empty, front and rear are set to the newly created node ptr. Also, front -> next and rear -> next are set to NULL.

b. If the list is not empty, the new node is added after rear. rear -> next is updated to point to the new node ptr, and rear is updated to point to the new node.

Deletion Operation

The element that is initially placed out of all the queue elements is removed during a deletion operation. First, we must determine whether or not the list is empty. If the list is empty, the condition front == NULL becomes true; in this scenario, we just print underflow to the console and quit.

Else, it will delete the element that is pointed by the pointer front.

For this reason, copy the node pointed by the front pointer into the pointer ptr.

Now, move the front pointer, point to its next node and free the node pointed by the node ptr. This is done by using the following statements.

These statements are utilised to do this.

ptr = front;

front = front -> next;

free(ptr);

ptr = front;

Here, ptr is a pointer that will be used to temporarily store the address of the node that is being removed.

front = front -> next;

This line moves the front pointer to the next node, effectively removing the first node from the linked list.

free(ptr);

Finally, this line deallocates the memory occupied by the node that was removed.

In summary, this code segment removes the first node from the linked list and deallocates the memory occupied by that node.

Algorithm DeleteFromQueue()

if front is NULL

print "Queue is empty"

return

end if

temp = front

front = front -> next

free(temp)

if front is NULL

rear = NULL

end if

End Algorithm

1. Check if the queue is empty:
   We check if the front pointer is NULL. If it is NULL, it means the queue is empty. In this case, we print an error message and exit the function.

2. Create a temporary pointer and point it to the front of the queue:
   We create a temporary pointer temp and set it to point to the same node as front.

3. Move the front pointer to the next node:
   We move the front pointer to the next node in the queue, effectively removing the first node from the queue.

4. Free the memory occupied by the node pointed to by temp:
   We use the free() function to deallocate the memory occupied by the node pointed to by temp, which is the node we are removing from the queue.

5. Check if the queue is now empty:
   After deleting the node, we check if the queue is now empty. If the front pointer is NULL, it means the queue is empty. In this case, we set both the front and rear pointers to NULL.

Algorithm DeleteFromQueue()

if front is NULL

print "Queue is empty"

return

end if

// Create a temporary pointer and point it to the front of the queue

temp = front

// Move the front pointer to the next node

front = front -> next

// Free the memory occupied by the node pointed to by temp

free(temp)

// If the queue is now empty, set both front and rear to NULL

if front is NULL

rear = NULL

end if

End Algorithm

C function to delete an element from linked queue

void delete (struct node *ptr)

{

if(front == NULL)

{

printf("\nUNDERFLOW\n");

return;

}

else

{

ptr = front;

front = front -> next;

free(ptr);

}

}

1. Check if the queue is empty:

   • The function first checks if the queue is empty. This is done by checking if the front pointer is NULL. If it is NULL, it means the queue is empty, and the function prints "UNDERFLOW" and returns.

2. Delete the first node:

   • If the queue is not empty, it proceeds to delete the first node.

   • It assigns the front pointer to the ptr pointer. This step is not necessary because ptr is a local variable and changing it won't affect the caller's front pointer.

3. Move the front pointer to the next node:

   • It moves the front pointer to the next node in the queue, effectively removing the first node from the queue.

4. Free the memory occupied by the node:

   • It deallocates the memory occupied by the node pointed to by ptr using the free() function.

Program for implementation of a circular queue using an array in C:

#include <stdio.h>

#include <stdlib.h>

#define MAX_SIZE 5

// Structure to represent a circular queue

struct CircularQueue {

int items[MAX_SIZE];

int front, rear;

};

// Function to create a new circular queue

struct CircularQueue* createQueue() {

struct CircularQueue* queue = (struct CircularQueue*)malloc(sizeof(struct CircularQueue));

queue->front = -1;

queue->rear = -1;

return queue;

}

// Function to check if the queue is full

int isFull(struct CircularQueue* queue) {

if ((queue->front == 0 && queue->rear == MAX_SIZE - 1) || (queue->rear == (queue->front - 1) % (MAX_SIZE - 1))) {

return 1;

}

return 0;

}

// Function to check if the queue is empty

int isEmpty(struct CircularQueue* queue) {

return queue->front == -1;

}

// Function to add an element to the queue

void enqueue(struct CircularQueue* queue, int value) {

if (isFull(queue)) {

printf("Queue is full\n");

return;

}

if (isEmpty(queue)) {

queue->front = queue->rear = 0;

} else {

queue->rear = (queue->rear + 1) % MAX_SIZE;

}

queue->items[queue->rear] = value;

printf("%d enqueued to queue\n", value);

}

// Function to remove an element from the queue

int dequeue(struct CircularQueue* queue) {

int item;

if (isEmpty(queue)) {

printf("Queue is empty\n");

return -1;

}

if (queue->front == queue->rear) {

item = queue->items[queue->front];

queue->front = queue->rear = -1;

} else {

item = queue->items[queue->front];

queue->front = (queue->front + 1) % MAX_SIZE;

}

return item;

}

// Function to display the elements of the queue

void display(struct CircularQueue* queue) {

if (isEmpty(queue)) {

printf("Queue is empty\n");

return;

}

int i;

printf("Queue elements are: ");

if (queue->rear >= queue->front) {

for (i = queue->front; i <= queue->rear; i++)

printf("%d ", queue->items[i]);

} else {

for (i = queue->front; i < MAX_SIZE; i++)

printf("%d ", queue->items[i]);

for (i = 0; i <= queue->rear; i++)

printf("%d ", queue->items[i]);

}

printf("\n");

}

int main() {

struct CircularQueue* queue = createQueue();

enqueue(queue, 1);

enqueue(queue, 2);

enqueue(queue, 3);

enqueue(queue, 4);

enqueue(queue, 5);

enqueue(queue, 6);

display(queue);

printf("Dequeued element: %d\n", dequeue(queue));

printf("Dequeued element: %d\n", dequeue(queue));

display(queue);

enqueue(queue, 7);

enqueue(queue, 8);

display(queue);

return 0;

}

Double ended queue

A double-ended queue, often abbreviated as deque (pronounced "deck"), is a data structure that allows insertion and deletion of elements from both the front and the back. It is similar to a queue, but it allows insertion and deletion at both ends, making it more flexible.

Operations supported by deque are

1. Insertion:

   • Insertion at the front (push_front)

   • Insertion at the back (push_back)

2. Deletion:

   • Deletion from the front (pop_front)

   • Deletion from the back (pop_back)

3. Access:

   • Accessing the front element (front)

   • Accessing the back element (back)

Deques can be implemented using various data structures such as arrays, linked lists, or dynamic arrays.

Types of deque

There are two types of deque

• Input restricted queue

• Output restricted queue

Input restricted Queue

In input restricted queue, insertion operation can be performed at only one end, while deletion can be performed from both ends.

Output restricted Queue

In output restricted queue, deletion operation can be performed at only one end, while insertion can be performed from both ends.

Applications of deque

Deques are very versatile data structures and have numerous applications in computer science and real-world scenarios. Some of the common applications of deques include:

Implementing a Queue or a Stack:

Deques can be used to implement both queues and stacks efficiently. Using a deque, you can implement a queue that supports insertion and deletion from both ends, making it more flexible than a simple queue.

Sliding Window Problems:

Deques are often used to solve sliding window problems efficiently. Sliding window problems involve finding the maximum (or minimum) value of all contiguous subarrays of size k in an array of size n.

Undo Operations in Text Editors:

Deques can be used to implement the undo functionality in text editors efficiently. You can maintain a deque of the previous states of the text document and easily undo/redo operations.

Breadth-First Search (BFS):

Deques can be used to implement the queue data structure required for BFS traversal of graphs. BFS involves visiting all the nodes of a graph level by level.

Task Scheduling:

Deques can be used in task scheduling algorithms where tasks need to be scheduled based on their priority or other criteria.

Memory Management:

Deques are used in memory management algorithms, such as the buddy memory allocation technique, which divides memory into partitions to satisfy allocation requests.

Implementation of deque in C

#include <stdio.h>

#include <stdlib.h>

#define MAX_SIZE 100

typedef struct {

int data[MAX_SIZE];

int front, rear;

} Deque;

// Function to initialize deque

void initDeque(Deque *dq) {

dq->front = -1;

dq->rear = -1;

}

// Function to check if deque is empty

int isEmpty(Deque *dq) {

return (dq->front == -1 && dq->rear == -1);

}

// Function to check if deque is full

int isFull(Deque *dq) {

return ((dq->rear + 1) % MAX_SIZE == dq->front);

}

// Function to add element at front end of deque

void addFront(Deque *dq, int value) {

if (isFull(dq)) {

printf("Deque is full. Cannot add element.\n");

return;

}

if (isEmpty(dq)) {

dq->front = dq->rear = 0;

} else {

dq->front = (dq->front - 1 + MAX_SIZE) % MAX_SIZE;

}

dq->data[dq->front] = value;

}

// Function to add element at rear end of deque

void addRear(Deque *dq, int value) {

if (isFull(dq)) {

printf("Deque is full. Cannot add element.\n");

return;

}

if (isEmpty(dq)) {

dq->front = dq->rear = 0;

} else {

dq->rear = (dq->rear + 1) % MAX_SIZE;

}

dq->data[dq->rear] = value;

}

// Function to delete element from front end of deque

void deleteFront(Deque *dq) {

if (isEmpty(dq)) {

printf("Deque is empty. Cannot delete element.\n");

return;

}

if (dq->front == dq->rear) {

dq->front = dq->rear = -1;

} else {

dq->front = (dq->front + 1) % MAX_SIZE;

}

}

// Function to delete element from rear end of deque

void deleteRear(Deque *dq) {

if (isEmpty(dq)) {

printf("Deque is empty. Cannot delete element.\n");

return;

}

if (dq->front == dq->rear) {

dq->front = dq->rear = -1;

} else {

dq->rear = (dq->rear - 1 + MAX_SIZE) % MAX_SIZE;

}

}

// Function to get the front element of deque

int getFront(Deque *dq) {

if (isEmpty(dq)) {

printf("Deque is empty.\n");

return -1;

}

return dq->data[dq->front];

}

// Function to get the rear element of deque

int getRear(Deque *dq) {

if (isEmpty(dq)) {

printf("Deque is empty.\n");

return -1;

}

return dq->data[dq->rear];

}

// Function to display elements of deque

void display(Deque *dq) {

if (isEmpty(dq)) {

printf("Deque is empty.\n");

return;

}

printf("Deque elements: ");

int i = dq->front;

while (i != dq->rear) {

printf("%d ", dq->data[i]);

i = (i + 1) % MAX_SIZE;

}

printf("%d\n", dq->data[i]);

}

int main() {

Deque dq;

initDeque(&dq);

addFront(&dq, 10);

addRear(&dq, 20);

addFront(&dq, 30);

addRear(&dq, 40);

display(&dq); // Output: Deque elements: 30 10 20 40

deleteFront(&dq);

deleteRear(&dq);

display(&dq); // Output: Deque elements: 10 20

printf("Front element: %d\n", getFront(&dq)); // Output: Front element: 10

printf("Rear element: %d\n", getRear(&dq)); // Output: Rear element: 20

return 0;

}

PRIORITY QUEUE

A priority queue is a data structure that stores elements with associated priorities and allows for efficient retrieval of the element with the highest (or lowest) priority. It's similar to a regular queue, but each element has a priority value, and the element with the highest priority is always at the front of the queue.

Only comparable elements are supported by the priority queue, therefore the elements are either sorted in ascending or descending order.

Assume, for instance, that we have values 1, 3, 4, 8, 14, and 22 added to a priority queue and that the values are arranged from least to greatest. As a result, number 1 would have the highest priority while number 22 would have the lowest.

characteristics of a priority queue:

1. Ordered Elements: Elements in a priority queue are ordered based on their priority. Higher priority elements are dequeued before lower priority elements.
   

2. Support for Arbitrary Types: Priority queues can store elements of any data type, as long as they can be compared.
   

3. Priority-based Retrieval: The element with the highest priority is dequeued first. The priority can be defined based on a predefined order or using a custom comparator function.
   

4. No Order Guarantees: Unlike regular queues, there is no guarantee about the order in which elements are dequeued if they have the same priority.
   

5. Efficient Insertion and Removal: Priority queues are designed to efficiently insert and remove elements while maintaining the order based on priority.
   

Types of Priority Queue

There are two types of priority queue:

Ascending Order Priority Queue: In an ascending order priority queue, elements with lower priority numbers are considered higher priority. For instance, if we have numbers from 1 to 5 arranged in ascending order (1, 2, 3, 4, 5), the smallest number (1) is given the highest priority in the queue.

Descending order priority queue:

In a descending order priority queue, higher priority is assigned to numbers with higher values. For instance, if we consider the numbers 1 through 5 arranged in descending order (5, 4, 3, 2, 1), the highest priority is given to the largest number, which in this case is 5.

Program to Implementation of priority queue

The priority queue can be implemented in four different ways: using arrays, linked lists, the heap data structure, or binary search trees.

Here's a simple implementation of a priority queue in C using an array. This implementation assumes that lower numerical priority values indicate higher priority.

c

#include <stdio.h>

#include <stdlib.h>

#define MAX 100

typedef struct {

int data;

int priority;

} Element;

typedef struct {

Element elements[MAX];

int size;

} PriorityQueue;

void initPriorityQueue(PriorityQueue *pq) {

pq->size = 0;

}

void swap(Element *a, Element *b) {

Element temp = *a;

*a = *b;

*b = temp;

}

void heapifyUp(PriorityQueue *pq, int index) {

if (index && pq->elements[(index - 1) / 2].priority < pq->elements[index].priority) {

swap(&pq->elements[(index - 1) / 2], &pq->elements[index]);

heapifyUp(pq, (index - 1) / 2);

}

}

void heapifyDown(PriorityQueue *pq, int index) {

int left = 2 * index + 1;

int right = 2 * index + 2;

int largest = index;

if (left < pq->size && pq->elements[left].priority > pq->elements[largest].priority) {

largest = left;

}

if (right < pq->size && pq->elements[right].priority > pq->elements[largest].priority) {

largest = right;

}

if (largest != index) {

swap(&pq->elements[index], &pq->elements[largest]);

heapifyDown(pq, largest);

}

}

void enqueue(PriorityQueue *pq, int data, int priority) {

if (pq->size == MAX) {

printf("Priority Queue is full\n");

return;

}

pq->elements[pq->size].data = data;

pq->elements[pq->size].priority = priority;

pq->size++;

heapifyUp(pq, pq->size - 1);

}

int dequeue(PriorityQueue *pq) {

if (pq->size == 0) {

printf("Priority Queue is empty\n");

return -1;

}

int data = pq->elements[0].data;

pq->elements[0] = pq->elements[--pq->size];

heapifyDown(pq, 0);

return data;

}

int peek(PriorityQueue *pq) {

if (pq->size == 0) {

printf("Priority Queue is empty\n");

return -1;

}

return pq->elements[0].data;

}

int isEmpty(PriorityQueue *pq) {

return pq->size == 0;

}

int main() {

PriorityQueue pq;

initPriorityQueue(&pq);

enqueue(&pq, 10, 2);

enqueue(&pq, 30, 3);

enqueue(&pq, 20, 1);

enqueue(&pq, 40, 4);

while (!isEmpty(&pq)) {

printf("Dequeued element: %d\n", dequeue(&pq));

}

return 0;

}

The tree is a data structure representing hierarchical data with connected nodes.

If we want to show an organisation’s employees concerning their designation in a hierarchical form then it can be shown like this.

This is an organisation hierarchy in which the topmost position is of CEO Mr Karan who has 2 direct reports Ankita and Vikas.

Ankita has 3 direct reports Ishita, Rakshita, and Reem on the other side Vikas has 2 reporters Rahul and Sudesh.

Sudesh also has 2 reporters bharti and Krishna

at last, bharti has one reporter that is harsh.

Tree is an efficient way of storing the data in a hierarchical way

A tree data structure can be defined as a collection of entities called nodes, which are interconnected to represent or simulate a hierarchical structure.

A tree data structure is non-linear because its elements are not stored sequentially. Instead, it organizes elements in a hierarchical structure, arranging them across multiple levels.

In a tree data structure, the topmost node is called the root node. Each node holds some data, which can be of any type. In the example above, the nodes contain employee names, so the data type would be a string.

Each node contains data and a link or reference to other nodes, which are called children.

now let us see some of the basic terms used in the tree structure.

As you can see each node in a tree here is labeled with some number. Each arrow depicted in the figure above represents a connection between two nodes.

Root: The root node is the uppermost node in the tree hierarchy, meaning it is the node without a parent. In the structure shown here, the node numbered 1 is the root node of the tree. When a node is directly connected to another node, this forms a parent-child relationship.

Child node: If a node is a descendant of another node, it is referred to as a child node.

Parent: If a node contains any sub-nodes, it is referred to as the parent of those sub-nodes.

Sibling: Nodes that share the same parent are known as siblings.

Leaf Node: A node in a tree that does not have any child nodes is called a leaf node. It is the bottom-most node of the tree. A general tree can have any number of leaf nodes, which are also known as external nodes.

Internal Nodes: A node that has at least one child node is known as an internal node.

Ancestor Node: An ancestor of a node is any predecessor on the path from the root to that node. The root node has no ancestors. In the tree depicted in the image above, nodes 1, 2, and 5 are the ancestors of node 10.

Descendant: The immediate successor of a given node is known as its descendant. In the figure above, node 10 is a descendant of node 5.

Properties of Tree data structure

**Recursive Data Structure**: A tree is often referred to as a recursive data structure because it can be defined recursively. In a tree, the distinguished node is the root node, which contains links to the roots of its subtrees. For example, in the figure below, the left subtree is shown in yellow, and the right subtree is shown in red. The left subtree can be further divided into smaller subtrees, each shown in different colours. Recursion involves breaking down a problem into smaller, self-similar problems, and this recursive property of the tree data structure is utilized in various applications.

**Number of Edges**: In a tree with \( n \) nodes, there are \( n-1 \) edges. Each arrow in the structure represents a link or path. Every node, except the root node, has at least one incoming link called an edge. Each parent-child relationship is represented by one link.

**Depth of Node \( x \)**: The depth of node \( x \) is defined as the length of the path from the root to node \( x \). Each edge in the path contributes one unit to the length. Therefore, the depth of node \( x \) can also be described as the number of edges between the root node and node \( x \). The root node has a depth of 0.

**Height of Node \( x \)**: The height of node \( x \) is defined as the length of the longest path from node \( x \) to a leaf node.

Applications of Tree Data Structure

Hierarchical Data Organization

• File Systems: Trees are used to represent directories and files in a hierarchical structure. Each directory is a node with subdirectories or files as its children.

• XML/HTML Data Parsing: Documents are parsed into trees, where each tag represents a node in the tree, making it easy to traverse and manipulate structured data.

Search Operations

• Binary Search Trees (BST): Used for efficient searching, insertion, and deletion. They keep data sorted, enabling quick access (O(log n) average time complexity).

• Tries: A special tree structure used for fast prefix-based search, commonly seen in autocomplete systems and spell-checkers.

Databases and Indexing

• B-trees/B+ Trees: Commonly used in databases to store and retrieve large datasets efficiently, particularly for indexing operations.

• Database Indexes: Trees help in creating indexes that improve the speed of data retrieval operations.

Compilers and Expression Evaluation

• Abstract Syntax Trees (AST): In compilers, code is converted into an abstract syntax tree for parsing and optimization before generating machine code.

• Expression Trees: Represent mathematical expressions where leaves represent operands and internal nodes represent operators, used for evaluating arithmetic expressions.

Artificial Intelligence and Machine Learning

• Decision Trees: A widely used algorithm in machine learning for classification and regression tasks. They model decisions and their possible consequences.

• Game Trees: Used in AI to simulate potential moves in games like chess or checkers, analyzing different strategies based on possible outcomes.

Network Algorithms

• Spanning Trees: Used in network routing to ensure data is transmitted efficiently without redundancy, forming the basis for protocols like Ethernet and wireless communication.

• Routing Protocols: Many routing algorithms use trees to find the shortest path between nodes in a network.

Compression Algorithms

• Huffman Coding: A tree-based algorithm used in file compression (e.g., ZIP, GZIP) where characters are encoded using variable-length codes based on their frequency.

• File Compression: Trees enable efficient representation of data that helps reduce file sizes without losing information.

Graphics and User Interface Design

• Scene Graphs: In computer graphics, trees represent hierarchical relationships between objects in a scene, helping manage complex rendering tasks.

• DOM Trees: In web development, the Document Object Model (DOM) represents the structure of an HTML or XML document, allowing dynamic interaction with the content.

Peer-to-Peer Networks

• Distributed Hash Tables (DHT): Trees help organize data in decentralized networks, ensuring efficient data distribution and lookup in systems like blockchain and file-sharing platforms.

Biological and Phylogenetic Trees

• Phylogenetic Trees: Used in bioinformatics to model the evolutionary relationships among species, representing how they diverged from common ancestors.

• Genealogical Trees: Trees represent family histories, tracking relationships over generations.

Internet and Web

• Search Engine Algorithms: Many search engines rely on tree-like structures (e.g., tries or suffix trees) to index and quickly retrieve web pages based on search queries.

These examples highlight the broad use of trees across different domains, making them one of the most versatile data structures in computing.

Types of Tree data structures

There are several types of tree data structures, each designed for specific use cases.

we will just list out the types of trees here, and will learn one by one in detail, in upcoming videos.

1. **Binary Tree**

- A tree where each node has at most two children, referred to as the left and right child.

2. **Binary Search Tree (BST)**

- A special type of binary tree where for each node:

- The left child contains values less than the node.

- The right child contains values greater than the node.

3. **AVL Tree**

- A self-balancing binary search tree where the height difference between the left and right subtrees of any node is at most 1.

- It ensures O(log n) time complexity for insertion, deletion, and searching.

4. **Red-Black Tree**

- A balanced binary search tree where each node has an additional color attribute (either red or black) to ensure the tree remains balanced after insertions and deletions.

- The tree ensures O(log n) operations for searching, inserting, and deleting.

5. **B-Tree**

- A self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time.

- Commonly used in databases and file systems.

6. **Heap**

- A complete binary tree that satisfies the heap property:

- **Max-Heap**: Parent nodes are always greater than or equal to their children.

- **Min-Heap**: Parent nodes are always less than or equal to their children.

- Typically used to implement priority queues.

7. **Trie (Prefix Tree)**

- A tree used to store strings where each node represents a character of the string.

- It is widely used for efficient retrieval of keys in dictionaries, such as autocomplete systems.

8. **Segment Tree**

- A tree used for storing intervals or segments.

- It allows answering range queries efficiently (like sum, minimum, maximum) and can also be updated dynamically.

9. **Fenwick Tree (Binary Indexed Tree)**

- A data structure that provides efficient methods for calculating cumulative frequencies or sums and supports updates in logarithmic time.

- Often used in scenarios like frequency analysis and range queries.

10. **Suffix Tree**

- A compressed trie that represents all suffixes of a given string.

- It is mainly used in string processing algorithms for pattern matching and finding the longest common substring.

11. **k-ary Tree**

- A tree where each node can have up to **k** children.

- For example, a **3-ary tree** allows each node to have up to three children.

12. **Spanning Tree**

- A subgraph of a connected graph that includes all vertices with the minimum possible number of edges, forming a tree.

- Commonly used in network routing algorithms.

13. **Binary Heap**

- A binary tree-based data structure where every parent node is either greater than (max-heap) or less than (min-heap) its children.

14. **Ternary Tree**

- A tree data structure where each node has at most three children.

TREE Traversal

The process of visiting the nodes is known as tree traversal. There are three types traversals used to visit a node:

• Inorder traversal

• Preorder traversal

• Postorder traversal

### 1. In-Order Traversal

**Definition:** In in-order traversal, nodes are recursively visited in the following order:

1. Left subtree

2. Root node

3. Right subtree

**Purpose:** For binary search trees (BSTs), in-order traversal retrieves the nodes in ascending order. This is useful for obtaining a sorted list of values.

**Implementation:**

```c

void inorderTraversal(Node* root) {

if (root != NULL) {

inorderTraversal(root->left); // Visit left subtree

printf("%d ", root->value); // Visit root node

inorderTraversal(root->right); // Visit right subtree

}

}

```

**Example Tree:**

```

4

/ \

2 6

/ \ / \

1 3 5 7

```

In the in order traversal first it visits the left subtree so it will first visit 1 and then root node of this tree that 2 and then right one that is 3,

so the next it will visit node 4, 5,6,and 7

so the.

**In-Order Traversal Output:** `1 2 3 4 5 6 7`

### 2. Pre-Order Traversal

**Definition:** In pre-order traversal, nodes are recursively visited in the following order:

1. Root node

2. Left subtree

3. Right subtree

**Purpose:** Pre-order traversal is useful for creating a copy of the tree or for prefix notation in expression trees.

**Implementation:**

void preorderTraversal(Node* root) {

if (root != NULL) {

printf("%d ", root->value); // Visit root node

preorderTraversal(root->left); // Visit left subtree

preorderTraversal(root->right); // Visit right subtree

}

}

```

**Example Tree:**

4

/ \

2 6

/ \ / \

1 3 5 7

```

**Pre-Order Traversal Output:** `4 2 1 3 6 5 7`

### 3. Post-Order Traversal

**Definition:** In post-order traversal, nodes are recursively visited in the following order:

1. Left subtree

2. Right subtree

3. Root node

**Purpose:** Post-order traversal is useful for deleting or freeing nodes in the tree, as it processes all children before their parent. It is also used for postfix notation in expression trees.

**Implementation:**

```c

void postorderTraversal(Node* root) {

if (root != NULL) {

postorderTraversal(root->left); // Visit left subtree

postorderTraversal(root->right); // Visit right subtree

printf("%d ", root->value); // Visit root node

}

}

```

**Example Tree:**

```

4

/ \

2 6

/ \ / \

1 3 5 7

```

**Post-Order Traversal Output:** `1 3 2 5 7 6 4`

### Summary

- **In-Order Traversal**: Visits nodes in left subtree → root → right subtree. Useful for BSTs to get values in sorted order.

- **Pre-Order Traversal**: Visits nodes in root → left subtree → right subtree. Useful for creating tree copies and prefix notation.

- **Post-Order Traversal**: Visits nodes in left subtree → right subtree → root. Useful for tree deletions and postfix notation.

Each traversal method has its own use cases and is chosen based on the specific needs of the algorithm or operation being performed on the tree.

BINARY SEARCH TREE

A binary search tree organizes elements by following a specific order., the left child's value must be less than the parent node, while the right child's value must be greater than the parent node. This rule is applied recursively to both the left and right subtrees of the root.

Let's understand the concept of Binary search tree with an example.

In the figure, we can see that the root node is 40. All nodes in the left subtree are smaller than the root, while all nodes in the right subtree are larger than the root.

Similarly, the left child of the root is greater than its own left child and smaller than its right child, adhering to the binary search tree property. Therefore, the tree in the image is a valid binary search tree.

Now, suppose we change the value of node 35 to 55. In that case, check whether the tree will still satisfy the binary search tree property or not.

In this tree, the root node has a value of 40, which is greater than its left child (30) but smaller than the right child of 30 (55). This violates the binary search tree property. Hence, the tree is not a valid binary search tree.

Advantages of binary search tree (BST)

1. **Efficient Searching**:

- BST allows for efficient search operations with a time complexity of O(log n) in a balanced tree. It eliminates half of the search space at each step.

2. **Efficient Insertion and Deletion**:

- In a balanced BST, inserting or deleting a node can also be done in O(log n) time, which is much faster than operations in unsorted lists.

3. **Dynamic Structure**:

- Unlike arrays, a BST doesn't require resizing as it can dynamically grow or shrink with insertions and deletions.

4. **Sorted Order Traversal**:

- An in-order traversal of a BST produces elements in sorted order, making it useful for applications that require data to be accessed in a sorted sequence.

5. **Space Efficiency**:

- A BST requires minimal space beyond the storage of its nodes, with no need for extra storage like in hash tables (for hash functions) or linked lists.

6. **Easy Data Retrieval**:

- BSTs are useful when you need quick access to both minimum and maximum values, as these can be found at the leftmost and rightmost nodes, respectively.

7. **Supports Range Queries**:

- BSTs allow for efficient querying of a range of values between a given lower and upper bound, which is not feasible in unsorted data structures.

8. **Flexible Structure**:

- Variants of BSTs (like AVL trees, Red-Black trees) maintain balance, ensuring efficient performance even when data is added in a way that could otherwise make the tree unbalanced.

BSTs combine the benefits of both dynamic and ordered data structures, making them ideal for many applications involving search, insertion, and sorted data retrieval.

Example of creating a binary search tree

Here's an example of how to create a Binary Search Tree (BST) with the following sequence of values: `50, 30, 70, 20, 40, 60, 80`.

### Steps to Create a Binary Search Tree:

1. **Insert 50**: This will be the root node.

2. **Insert 30**: Since 30 is less than 50, it becomes the left child of 50.

3. **Insert 70**: Since 70 is greater than 50, it becomes the right child of 50.

4. **Insert 20**: Since 20 is less than 50 and also less than 30, it becomes the left child of 30.

5. **Insert 40**: Since 40 is less than 50 but greater than 30, it becomes the right child of 30.

6. **Insert 60**: Since 60 is greater than 50 but less than 70, it becomes the left child of 70.

7. **Insert 80**: Since 80 is greater than 50 and also greater than 70, it becomes the right child of 70.

### Final Binary Search Tree:

```

50

/ \

30 70

/ \ / \

20 40 60 80

```

This tree satisfies the Binary Search Tree property where:

- The left child is smaller than the parent.

- The right child is greater than the parent.

The AVL Tree, developed by G.M. Adelson-Velsky and E.M. Landis in 1962, is named in honor of its inventors.

An AVL Tree is a height-balanced binary search tree, where each node has a balance factor determined by subtracting the height of its right subtree from that of its left subtree.

A tree is considered balanced if every node's balance factor falls between -1 and 1. If any node’s balance factor goes beyond this range, the tree becomes unbalanced and requires adjustments to restore balance.

The balance factor of a node k is calculated as:

Balance Factor (k) = height of left subtree of k − height of right subtree of k

If a node’s balance factor is 1, it indicates that the left subtree is one level taller than the right subtree.

Operations on an AVL Tree

Since an AVL tree is a type of binary search tree, all standard BST operations—such as searching and traversing—are carried out the same way and do not disrupt the AVL tree's balance properties. However, insertion and deletion operations can disrupt this balance, so they must be carefully managed to restore the AVL property when necessary.

Insertion

Insertion in AVL tree is performed in the same way as it is performed in a binary search tree. However, it may lead to violation in the AVL tree property and therefore the tree may need balancing. The tree can be balanced by applying rotations.

Deletion

Deletion can also be performed in the same way as it is performed in a binary search tree. Deletion may also disturb the balance of the tree therefore, various types of rotations are used to rebalance the tree.

Why Use an AVL Tree?

An AVL tree maintains control over the height of a binary search tree, preventing it from becoming skewed.

In a binary search tree with height h, the time complexity for operations is O(h).

However, if the tree becomes skewed (in the worst case), this can extend to O(n). By keeping the height around log n, an AVL tree ensures that each operation has an upper bound of O(log n), where n is the number of nodes.

AVL Rotations

We perform rotation in AVL tree only in case if Balance Factor is other than -1, 0, and 1. There are basically four types of rotations which are as follows:

1. L L rotation: Inserted node is in the left subtree of left subtree of A

2. R R rotation : Inserted node is in the right subtree of right subtree of A

3. L R rotation : Inserted node is in the right subtree of left subtree of A

4. R L rotation : Inserted node is in the left subtree of right subtree of A

Consider node A as the node where the balance factor deviates from -1, 0, or 1.

The first two rotations, LL and RR, are single rotations, while the other two, LR and RL, are double rotations. For a tree to become unbalanced, its minimum height must be at least 2. Let’s explore each type of rotation.

1. RR Rotation

When a binary search tree becomes unbalanced due to a node being inserted into the right subtree of the right subtree of node A, an RR rotation is performed. This is a counterclockwise rotation applied to the edge below a node with a balance factor of -2.

In the example , node A has a balance factor of -2 because node C was inserted into the right subtree of A's right subtree. To restore balance, we perform an RR rotation on the edge below A.

1. LL Rotation

When a binary search tree becomes unbalanced due to a node being inserted into the left subtree of the left subtree of node C, an LL rotation is performed. This clockwise rotation is applied to the edge below a node with a balance factor of 2.

In the example, node C has a balance factor of 2 because node A was inserted into the left subtree of C's left subtree. To restore balance, we perform an LL rotation on the edge below C.

3. LR Rotation

Double rotations are more complex than single rotations, as previously explained. An LR rotation involves performing an RR rotation followed by an LL rotation. This means we first apply an RR rotation to the subtree and then perform an LL rotation on the entire tree. Here, the "entire tree" refers to the first node along the path of the inserted node that has a balance factor other than -1, 0, or 1.

Let us understand each and every step very clearly:

A node B has been inserted into the right subtree of A the left subtree of C, because of which C has become an unbalanced node having balance factor 2. This case is L R rotation where: Inserted node is in the right subtree of left subtree of C

As LR rotation = RR + LL rotation, hence RR (anticlockwise) on subtree rooted at A is performed first. By doing RR rotation, node A, has become the left subtree of B.

After performing RR rotation, node C is still unbalanced, i.e., having balance factor 2, as inserted node A is in the left of left of C

Now we perform LL clockwise rotation on full tree, i.e. on node C. node C has now become the right subtree of node B, A is left subtree of B

Balance factor of each node is now either -1, 0, or 1, i.e. BST is balanced now.

1. RL Rotation

As previously mentioned, double rotations are a bit more complex than single rotations explained earlier. An RL rotation involves an LL rotation followed by an RR rotation. This means we first perform an LL rotation on the subtree, then an RR rotation on the entire tree. Here, "entire tree" refers to the first node along the path of the inserted node that has a balance factor other than -1, 0, or 1

A node B has been inserted into the left subtree of C the right subtree of A, because of which A has become an unbalanced node having balance factor - 2. This case is RL rotation where: Inserted node is in the left subtree of right subtree of A

As RL rotation = LL rotation + RR rotation, hence, LL (clockwise) on subtree rooted at C is performed first. By doing RR rotation, node C has become the right subtree of B.

After performing LL rotation, node A is still unbalanced, i.e. having balance factor -2, which is because of the right-subtree of the right-subtree node A.

Now we perform RR rotation (anticlockwise rotation) on full tree, i.e. on node A. node C has now become the right subtree of node B, and node A has become the left subtree of B.

Balance factor of each node is now either -1, 0, or 1, i.e., BST is balanced now.

A B-Tree is a specialized m-way tree commonly used for efficient disk access. In a B-Tree of order m, each node can contain a maximum of m-1 keys and have up to m children. Its primary advantage lies in its ability to store a large number of keys within a single node, including large key values, while maintaining a relatively small tree height.

A B tree of order m contains all the properties of an M way tree. In addition, it contains the following properties.

1. Every node in a B-Tree contains at most m children.

2. Every node in a B-Tree except the root node and the leaf node contain at least m/2 children.

3. The root nodes must have at least 2 nodes.

4. All leaf nodes must be at the same level.

It is not necessary that, all the nodes contain the same number of children but, each node must have m/2 number of nodes.

While performing some operations on B Tree, any property of B Tree may violate such as number of minimum children a node can have. To maintain the properties of B Tree, the tree may split or join.

Operations

Searching :

Searching in B Trees is similar to that in Binary search tree. For example, if we search for an item 49 in the following B Tree. The process will something like following :

1. Compare item 49 with root node 78. since 49 < 78 hence, move to its left sub-tree.

2. Since, 40<49<56, traverse right sub-tree of 40.

3. 49>45, move to right. Compare 49.

4. match found, return.

Searching in a B tree depends upon the height of the tree. The search algorithm takes O(log n) time to search any element in a B tree.

Inserting

Insertions are done at the leaf node level. The following algorithm needs to be followed in order to insert an item into B Tree.

1. Traverse the B Tree in order to find the appropriate leaf node at which the node can be inserted.

2. If the leaf node contain less than m-1 keys then insert the element in the increasing order.

3. Else, if the leaf node contains m-1 keys, then follow the following steps.

   • Insert the new element in the increasing order of elements.

   • Split the node into the two nodes at the median.

   • Push the median element upto its parent node.

   • If the parent node also contain m-1 number of keys, then split it too by following the same steps.

Insert the node 8 into the B Tree of order 5 shown in the following image.

8 will be inserted to the right of 5, therefore insert 8.

The node, now contain 5 keys which is greater than (5 -1 = 4 ) keys. Therefore split the node from the median i.e. 8 and push it up to its parent node shown as follows.

Deletion

Deletion is also performed at the leaf nodes. The node which is to be deleted can either be a leaf node or an internal node. Following algorithm needs to be followed in order to delete a node from a B tree.

1. Locate the leaf node.

2. If there are more than m/2 keys in the leaf node then delete the desired key from the node.

3. If the leaf node doesn't contain m/2 keys then complete the keys by taking the element from eight or left sibling.

   • If the left sibling contains more than m/2 elements then push its largest element up to its parent and move the intervening element down to the node where the key is deleted.

   • If the right sibling contains more than m/2 elements then push its smallest element up to the parent and move intervening element down to the node where the key is deleted.

4. If neither of the sibling contain more than m/2 elements then create a new leaf node by joining two leaf nodes and the intervening element of the parent node.

5. If parent is left with less than m/2 nodes then, apply the above process on the parent too.

If the the node which is to be deleted is an internal node, then replace the node with its in-order successor or predecessor. Since, successor or predecessor will always be on the leaf node hence, the process will be similar as the node is being deleted from the leaf node.

Example 1

Delete the node 53 from the B Tree of order 5 shown in the following figure.

53 is present in the right child of element 49. Delete it.

Now, 57 is the only element which is left in the node, the minimum number of elements that must be present in a B tree of order 5, is 2. it is less than that, the elements in its left and right sub-tree are also not sufficient therefore, merge it with the left sibling and intervening element of parent i.e. 49.

The final B tree is shown as follows.

Application of B tree

B tree is used to index the data and provides fast access to the actual data stored on the disks since, the access to value stored in a large database that is stored on a disk is a very time consuming process.

Searching an un-indexed and unsorted database containing n key values needs O(n) running time in worst case. However, if we use B Tree to index this database, it will be searched in O(log n) time in worst case.

B+ Tree is an extension of B Tree which allows efficient insertion, deletion and search operations.

In B Tree, Keys and records both can be stored in the internal as well as leaf nodes. Whereas, in B+ tree, records (data) can only be stored on the leaf nodes while internal nodes can only store the key values.

The leaf nodes of a B+ tree are linked together in the form of a singly linked lists to make the search queries more efficient.

B+ Tree are used to store the large amount of data which can not be stored in the main memory. Due to the fact that, size of main memory is always limited, the internal nodes (keys to access records) of the B+ tree are stored in the main memory whereas, leaf nodes are stored in the secondary memory.

Advantages of B+ Tree

1. Records can be fetched in equal number of disk accesses.

2. Height of the tree remains balanced and less as compare to B tree.

3. We can access the data stored in a B+ tree sequentially as well as directly.

4. Keys are used for indexing.

5. Faster search queries as the data is stored only on the leaf nodes.