Data Stuctures & Algorithms - Lecture Notes 1.pdf

UNIVERSIDAD DE DAGUPAN SCHOOL OF INFORMATION TECHNOLOGY EDUCATION ITC04 | DATA STRUCTURES & ALGORITHMS Lecture Notes 1 MODULE 1: Introduction to Data Structures and Algorithms in Java What i...

UNIVERSIDAD DE DAGUPAN SCHOOL OF INFORMATION TECHNOLOGY EDUCATION ITC04 | DATA STRUCTURES & ALGORITHMS Lecture Notes 1 MODULE 1: Introduction to Data Structures and Algorithms in Java What is Data Structures? A data structure is a specialized format for organizing, processing, retrieving, and storing data. More specifically, data structures are ways of storing and organizing data in a computer so that it can be used efficiently. They provide a means to manage large amounts of data for various applications, such as large databases and internet indexing services. Data structures can be classified into two broad categories: 1. Primitive Data Structures: o These are the basic data types that are available in most programming languages. Examples include: ▪ Integer ▪ Float ▪ Character ▪ Boolean 2. Non-Primitive Data Structures: o These are more complex data structures that are derived from primitive data types. They can be further classified into: ▪ Linear Data Structures: Elements are arranged in a sequential order. Examples include: ▪ Arrays ▪ Linked Lists ▪ Stacks ▪ Queues ▪ Non-Linear Data Structures: Elements are arranged in a hierarchical manner. Examples include: ▪ Trees (e.g., binary trees, AVL trees) ▪ Graphs ▪ Hash-based Data Structures: Use hash tables for efficient data retrieval. Examples include: ▪ Hash Tables ▪ Hash Maps Key Concepts Array: A collection of elements identified by index or key. Linked List: A linear collection of data elements, in which the linear order is not given by their physical placement in memory. Stack: A linear data structure which follows a particular order in which the operations are performed. The order may be LIFO (Last In First Out) or FILO (First In Last Out). Queue: A linear data structure that follows a particular order in which the operations are performed. The order is usually FIFO (First In First Out). Tree: A hierarchical structure that stores a collection of elements, with one element being the root and the rest being branches or leaves. Graph: A set of nodes connected by edges, used to represent various types of networks. Hash Table: A data structure that implements an associative array abstract data type, a structure that can map keys to values. Importance Efficiency: Proper use of data structures can significantly improve the efficiency of algorithms. Reusability: Standard data structures can be reused to solve different problems. Semester: 1st Semester | Academic Year: 2024-2025 1  Manageability: They help in managing and organizing data in a way that makes it easier to perform operations like insertion, deletion, and traversal. Characteristic of Data Structure The characteristics of data structures are crucial in understanding how they work and how to use them effectively. Here are some key characteristics: 1. Time Complexity Definition: The amount of time an algorithm takes to process an input and perform operations. Importance: Determines the efficiency of the data structure in terms of speed. Examples: Access time, insertion time, deletion time, search time. 2. Space Complexity Definition: The amount of memory a data structure uses. Importance: Determines the efficiency of the data structure in terms of memory usage. Examples: Memory needed for storage, memory overhead. 3. Dynamic vs. Static Static Data Structures: o Definition: Size and structure are fixed at compile-time. o Examples: Arrays. Dynamic Data Structures: o Definition: Size and structure can change during runtime. o Examples: Linked lists, dynamic arrays. 4. Linear vs. Non-Linear Linear Data Structures: o Definition: Elements are arranged sequentially. o Examples: Arrays, linked lists, stacks, queues. Non-Linear Data Structures: o Definition: Elements are arranged in a hierarchical manner. o Examples: Trees, graphs. 5. Homogeneous vs. Heterogeneous Homogeneous Data Structures: o Definition: All elements are of the same type. o Examples: Arrays (all elements are of the same data type). Heterogeneous Data Structures: o Definition: Elements can be of different types. o Examples: Structures in C, objects in Java. 6. Ordered vs. Unordered Ordered Data Structures: o Definition: Elements have a specific, predictable order. o Examples: Arrays, lists. Unordered Data Structures: o Definition: No specific order of elements. o Examples: Sets, hash tables. 7. Mutable vs. Immutable Mutable Data Structures: o Definition: Elements can be changed after the structure is created. o Examples: Lists in Python. Immutable Data Structures: o Definition: Elements cannot be changed once created. o Examples: Tuples in Python, strings in Java. 8. Direct Access vs. Sequential Access Direct Access: o Definition: Accessing elements directly via an index. o Examples: Arrays. Semester: 1st Semester | Academic Year: 2024-2025 2  Sequential Access: o Definition: Accessing elements in a sequential manner. o Examples: Linked lists. 9. Hierarchical Structure Definition: Organizes data in a tree-like structure, with one root and various levels of nodes. Importance: Efficient for operations like searching and sorting. Examples: Binary trees, AVL trees. 10. Hashing Definition: Converts data into a fixed-size value (hash) which acts as an index to an array. Importance: Provides fast access to data. Examples: Hash tables, hash maps. 11. Scalability Definition: Ability of a data structure to handle increasing amounts of data. Importance: Ensures performance remains efficient as data grows. Examples: Trees and graphs can be more scalable compared to arrays. Understanding these characteristics helps in choosing the appropriate data structure for a specific application, ensuring optimal performance in terms of both time and space complexity. Need for Data Structure Data structures are fundamental for efficient data management and algorithm implementation. Here are several key reasons why data structures are essential: 1. Efficient Data Management Organization: Properly organized data can be processed more efficiently, leading to faster operations. Examples: Arrays provide fast index-based access, while linked lists allow dynamic memory usage. 2. Optimized Performance Time Efficiency: Different data structures provide varying efficiencies for different operations such as searching, inserting, and deleting. Examples: Hash tables offer average O(1) time complexity for search, insertion, and deletion, while binary search trees provide O(log n) time complexity for these operations. 3. Memory Utilization Space Efficiency: Data structures help in efficiently managing memory, reducing waste. Examples: Linked lists use memory dynamically, allocating space as needed, unlike arrays which require predefined memory allocation. 4. Data Integrity and Manipulation Consistency: Ensuring data integrity through structured organization. Manipulation: Facilitating data manipulation and updating efficiently. Examples: Stacks ensure LIFO order, making them suitable for undo operations in software. 5. Complex Problem Solving Algorithm Implementation: Many algorithms rely on specific data structures for optimal performance. Examples: Dijkstra’s algorithm for shortest paths uses priority queues (a type of heap). 6. Reusability and Abstraction Modular Code: Data structures allow for reusable and modular code. Abstract Data Types (ADTs): Data structures provide the foundation for ADTs, which abstract away implementation details. Examples: Lists, queues, stacks as abstract data types. 7. Scalability Handling Large Data: Efficient data structures are crucial for handling large datasets and ensuring scalability. Examples: B-trees and databases use specific data structures to manage large volumes of data efficiently. 8. Improved Performance in Applications Real-Time Systems: Applications such as operating systems, database management systems, and network routing rely heavily on efficient data structures. Examples: Routing tables in networking, process scheduling in operating systems. Semester: 1st Semester | Academic Year: 2024-2025 3 9. Search and Retrieval Efficient Access: Data structures enable fast search and retrieval of data. Examples: Binary search trees allow efficient searching, insertion, and deletion of elements. 10. Data Relationships Hierarchy and Relationships: Representing hierarchical relationships and networks. Examples: Tree structures for representing hierarchical data, graph structures for networks. 11. Concurrency Control Multi-threading: Efficient data structures are essential in concurrent programming to ensure data integrity and consistency. Examples: Concurrent data structures like concurrent queues and stacks. 12. Data Analysis and Processing Big Data: Efficient data structures are crucial for processing and analyzing large datasets. Examples: Hadoop uses specific data structures for distributed storage and processing. In summary, data structures are vital for efficient data management, algorithm implementation, and overall performance optimization in software development. They enable the development of scalable, high-performance applications by providing efficient ways to store, organize, and manipulate data. What are Algorithms? Algorithms are step-by-step procedures or formulas for solving problems or performing tasks. They consist of a sequence of instructions that are followed to achieve a specific goal, such as performing a computation, processing data, or automating a process. Algorithms are fundamental to computer science and are used in various applications, from simple calculations to complex data processing and analysis. Key Characteristics of Algorithms 1. Definiteness o Definition: Each step of the algorithm is precisely defined. o Importance: Ensures that there is no ambiguity in the instructions. 2. Input o Definition: Algorithms have zero or more inputs, which are the data needed to perform the task. o Importance: Inputs provide the necessary data for the algorithm to process. 3. Output o Definition: Algorithms produce one or more outputs, which are the results of the processing. o Importance: Outputs are the end results that the algorithm is designed to achieve. 4. Finiteness o Definition: An algorithm must always terminate after a finite number of steps. o Importance: Ensures that the algorithm does not run indefinitely and eventually produces a result. 5. Effectiveness o Definition: Each step of the algorithm is simple enough to be carried out, in principle, by a person using a pencil and paper. o Importance: Ensures that the algorithm can be practically implemented and executed. Types of Algorithms 1. Simple Recursive Algorithms o Definition: Solve the base case directly and recursively solve the subproblem. o Examples: Factorial computation, Fibonacci sequence. 2. Divide and Conquer Algorithms o Definition: Divide the problem into smaller subproblems, solve each subproblem, and combine the solutions. o Examples: Merge sort, quicksort, binary search. 3. Dynamic Programming Algorithms o Definition: Solve complex problems by breaking them down into simpler subproblems and storing the results to avoid redundant computations. Semester: 1 Semester | Academic Year: 2024-2025 st 4  o Examples: Fibonacci sequence (using memoization), shortest path in graphs. 4. Greedy Algorithms o Definition: Make the locally optimal choice at each step with the hope of finding a global optimum. o Examples: Prim’s algorithm, Kruskal’s algorithm, Huffman coding. 5. Backtracking Algorithms o Definition: Build up a solution incrementally and abandon solutions that fail to satisfy the constraints. o Examples: N-Queens problem, solving mazes, generating permutations. 6. Branch and Bound Algorithms o Definition: Divide the problem into branches and use bounding functions to eliminate certain branches. o Examples: Travelling salesman problem, knapsack problem. 7. Heuristic Algorithms o Definition: Use practical methods or various tricks to produce solutions that are good enough for solving complex problems. o Examples: Genetic algorithms, simulated annealing, tabu search. Importance of Algorithms 1. Efficiency o Algorithms determine the efficiency of programs by optimizing the use of resources like time and space. o Examples: Sorting algorithms (quicksort, mergesort) provide efficient ways to sort data. 2. Problem Solving o Algorithms provide systematic approaches to solving problems. o Examples: Pathfinding algorithms (Dijkstra’s algorithm, A* algorithm) help in finding the shortest path in graphs. 3. Automation o Algorithms enable the automation of tasks, making processes faster and more reliable. o Examples: Search algorithms (binary search, depth-first search) automate data retrieval. 4. Scalability o Efficient algorithms are essential for handling large datasets and ensuring scalability of applications. o Examples: Database indexing algorithms (B-trees, hash indexing) ensure quick data retrieval in large databases. 5. Consistency and Accuracy o Algorithms provide consistent and accurate results by following a defined set of instructions. o Examples: Cryptographic algorithms (RSA, AES) ensure data security and integrity. 6. Optimization o Algorithms are used to find optimal solutions to problems, maximizing or minimizing certain criteria. o Examples: Linear programming algorithms optimize resource allocation and logistics. In summary, algorithms are the backbone of computer science and programming. They provide structured methods for solving problems, performing computations, and processing data efficiently and effectively. How to write an algorithm Writing an algorithm involves a systematic approach to solving a problem by breaking it down into a sequence of well- defined steps. Here’s a general process to write an algorithm: Steps to Write an Algorithm 1. Understand the Problem o Thoroughly read and comprehend the problem statement. o Identify the input, expected output, and constraints. 2. Define the Inputs and Outputs o Clearly specify the inputs that the algorithm will take. o Define what outputs the algorithm should produce. 3. Outline the Algorithm o Create a high-level overview or a rough draft of the steps needed to solve the problem. o Break down the problem into smaller, manageable sub-problems if necessary. Semester: 1st Semester | Academic Year: 2024-2025 5  4. Detailed Step-by-Step Instructions o Write each step in a clear and unambiguous manner. o Ensure each step is simple and can be easily understood. 5. Consider Edge Cases o Think about and include steps to handle any special or edge cases. o Ensure the algorithm covers all possible scenarios. 6. Optimize the Algorithm o Review the steps to find more efficient ways to perform them. o Optimize for time and space complexity where possible. 7. Test the Algorithm o Test the algorithm with various inputs, including typical, edge, and boundary cases. o Make sure it produces the correct output for all test cases. Example: Algorithm to Find the Largest Number in a List Problem Statement Given a list of numbers, find the largest number in the list. Inputs and Outputs Input: A list of numbers [a1, a2, a3,..., an] Output: The largest number in the list Algorithm Outline 1. Initialize a variable to hold the largest number. 2. Iterate through the list of numbers. 3. Compare each number with the current largest number. 4. Update the largest number if a larger number is found. 5. Return the largest number. Detailed Algorithm 1. Start 2. Input: A list of numbers L 3. Output: The largest number in the list highest_num 4. Solution o Step 1: Initialize highest_num to the first element of the list, highest_num = L o Step 2: For each number num in the list starting from the second element: 1. If num > highest_num, then update highest_num = num o Step 3: After the loop ends, max_num will hold the largest number in the list. 5. Return: highest_num 6. End Example in Pseudocode Algorithm FindLargestNumber Input: List of numbers L Output: Largest number max_num Step 1: Set highest_num to L Step 2: For each num in L starting from L If num > highest_num then Set highest_num to num Step 3: Return highest_num End Algorithm Semester: 1st Semester | Academic Year: 2024-2025 6 # Example usage: numbers = [3, 5, 7, 2, 8, 1] print(find_largest_number(numbers)) # Output: 8 Tips for Writing Good Algorithms Clarity: Ensure each step is clear and easy to understand. Modularity: Break down complex problems into simpler sub-problems or functions. Efficiency: Aim for optimal performance in terms of time and space complexity. Maintainability: Write the algorithm in a way that it can be easily modified or extended. Correctness: Ensure the algorithm handles all possible inputs correctly. By following these steps and tips, you can write effective and efficient algorithms to solve various computational problems. Algorithm complexity Algorithm complexity refers to the analysis of the performance of an algorithm in terms of time and space. It helps in understanding the efficiency of an algorithm and predicting its behavior as the input size grows. The two primary aspects of algorithm complexity are: 1. Time Complexity 2. Space Complexity Time Complexity Time complexity measures the amount of time an algorithm takes to complete as a function of the length of the input. It provides an upper bound on the running time of the algorithm for any input of size nnn. Types of Time Complexity 1. Best Case: The minimum time an algorithm takes to complete. It represents the most efficient scenario. 2. Average Case: The expected time an algorithm takes to complete, averaged over all possible inputs. 3. Worst Case: The maximum time an algorithm takes to complete. It represents the least efficient scenario and is often used for performance guarantees. Common Time Complexity Notations O(1): Constant time complexity. The algorithm takes the same amount of time regardless of the input size. O(log n): Logarithmic time complexity. The algorithm’s time grows logarithmically as the input size increases. O(n): Linear time complexity. The algorithm’s time grows linearly with the input size. O(n log n): Linearithmic time complexity. Common in efficient sorting algorithms like merge sort and heapsort. O(n^2): Quadratic time complexity. The algorithm’s time grows quadratically with the input size. Common in simple sorting algorithms like bubble sort, insertion sort, and selection sort. O(2^n): Exponential time complexity. The algorithm’s time doubles with each addition to the input size. O(n!): Factorial time complexity. The algorithm’s time grows factorially with the input size. Common in permutation-based algorithms. Space Complexity Space complexity measures the amount of memory an algorithm needs to run as a function of the length of the input. It includes the memory needed for the input, auxiliary variables, and any additional data structures used by the algorithm. Types of Space Complexity 1. Auxiliary Space: The extra space or temporary space used by the algorithm, excluding the space used by the input. 2. Total Space: The total space used by the algorithm, including the input space and auxiliary space. Semester: 1st Semester | Academic Year: 2024-2025 7 Common Space Complexity Notations O(1): Constant space complexity. The algorithm uses a fixed amount of space regardless of the input size. O(n): Linear space complexity. The algorithm’s space grows linearly with the input size. O(n^2): Quadratic space complexity. The algorithm’s space grows quadratically with the input size. Big O Notation Big O notation is used to describe the upper bound of an algorithm’s complexity, providing an asymptotic analysis of its performance. It helps in understanding the worst-case scenario of an algorithm’s running time and space requirements. Examples 1. Linear Search o Time Complexity: O(n) (worst-case) o Space Complexity: O(1) 2. Binary Search o Time Complexity: O(log n) (worst-case) o Space Complexity: O(1) for iterative version, O(log n) for recursive version due to call stack. 3. Merge Sort o Time Complexity: O(n log n) (worst-case) o Space Complexity: O(n) 4. Bubble Sort o Time Complexity: O(n^2) (worst-case) o Space Complexity: O(1) Analyzing Algorithm Complexity 1. Identify the Basic Operations: Determine the fundamental operations that significantly impact the algorithm’s running time and space usage. 2. Count the Operations: Count how many times the basic operations are executed as a function of the input size nnn. 3. Express the Complexity: Use Big O notation to express the time and space complexity in terms of nnn. Practical Considerations Scalability: Understanding algorithm complexity helps in choosing algorithms that scale well with increasing input sizes. Trade-offs: There is often a trade-off between time and space complexity. An algorithm that uses less time might use more space and vice versa. Real-world Performance: While theoretical analysis is crucial, actual performance can vary based on factors like hardware, compiler optimizations, and specific input characteristics. By analyzing the complexity of algorithms, developers can make informed decisions about which algorithms to use in different scenarios, ensuring efficient and effective solutions. Classification of Data Structures Data structures can be classified in various ways based on their characteristics and usage. Here are the primary classifications of data structures: 1. Primitive Data Structures Definition: Basic data types that are directly operated upon by machine instructions. Examples: Integers, Floats, Characters, Booleans. 2. Non-Primitive Data Structures Definition: More complex data structures derived from primitive data types. These can be further classified into: o Linear Data Structures o Non-Linear Data Structures Semester: 1st Semester | Academic Year: 2024-2025 8  ❖ Linear Data Structures o Definition: Elements are arranged in a sequential order, and each element is connected to its previous and next element. o Examples: ▪ Arrays: Fixed-size, homogeneous elements stored in contiguous memory locations. ▪ Linked Lists: Elements (nodes) where each node contains data and a reference to the next node. ▪ Stacks: Follows Last In First Out (LIFO) principle. ▪ Queues: Follows First In First Out (FIFO) principle. ❖ Non-Linear Data Structures o Definition: Elements are not arranged sequentially, and each element can be connected to multiple elements. o Examples: ▪ Trees: Hierarchical structure with a root node and child nodes (e.g., Binary Trees, AVL Trees, B- Trees). ▪ Graphs: Set of vertices (nodes) connected by edges (e.g., Directed Graphs, Undirected Graphs). 3. Dynamic vs. Static Data Structures Static Data Structures: Size is fixed at compile-time. o Examples: Arrays. Dynamic Data Structures: Size can change at runtime. o Examples: Linked Lists, Dynamic Arrays. 4. Homogeneous vs. Heterogeneous Data Structures Homogeneous Data Structures: All elements are of the same type. o Examples: Arrays. Heterogeneous Data Structures: Elements can be of different types. o Examples: Structures in C, objects in Java. 5. Hash-based Data Structures Definition: Uses hash functions to map keys to values for fast data retrieval. Examples: o Hash Tables: Stores key-value pairs and provides average O(1) time complexity for search, insertion, and deletion. o Hash Maps: Similar to hash tables but often provide more functionalities. 6. Indexed Data Structures Definition: Elements can be accessed directly via an index. Examples: o Arrays: Elements accessed by their index. o Strings: Arrays of characters with indexed access. 7. Composite or Compound Data Structures Definition: Combines multiple data types. Examples: o Structures (Structs): Group different data types together. o Classes: Define objects with attributes and methods. 8. File Structures Definition: Organizes data in a storage medium. Examples: o Sequential File Organization: Data is stored in a sequence. o Indexed File Organization: Uses an index to access data. o Direct or Hashed File Organization: Uses hash functions to determine the location of data. Semester: 1st Semester | Academic Year: 2024-2025 9 Examples and Use Cases 1. Arrays o Use Case: Accessing elements by index, implementing matrices. o Advantages: Simple and efficient for indexed access. o Disadvantages: Fixed size, inefficient for insertion/deletion. 2. Linked Lists o Use Case: Implementing queues, stacks, dynamic data structures. o Advantages: Dynamic size, efficient for insertion/deletion. o Disadvantages: Inefficient for indexed access, more memory usage due to pointers. 3. Stacks o Use Case: Implementing function calls, undo mechanisms. o Advantages: LIFO access, simple implementation. o Disadvantages: Limited access (only top element). 4. Queues o Use Case: Task scheduling, breadth-first search. o Advantages: FIFO access, simple implementation. o Disadvantages: Limited access (only front and rear elements). 5. Trees o Use Case: Hierarchical data representation, database indexing. o Advantages: Efficient searching, insertion, and deletion. o Disadvantages: Complex implementation and balancing. 6. Graphs o Use Case: Network representation, pathfinding algorithms. o Advantages: Flexible representation of connections. o Disadvantages: Complex implementation and traversal. 7. Hash Tables o Use Case: Fast data retrieval, implementing dictionaries. o Advantages: Average O(1) time complexity for basic operations. o Disadvantages: Possible collisions, requires good hash function. Choosing the Right Data Structure The choice of data structure depends on: Nature of the data: Whether it is linear or hierarchical. Operations needed: Access, insertion, deletion, searching. Efficiency requirements: Time and space complexity considerations. Memory usage: Available memory and overhead. Understanding the classification and characteristics of data structures helps in selecting the most appropriate one for a given problem, ensuring optimal performance and efficient resource utilization. Semester: 1st Semester | Academic Year: 2024-2025 10