Algorithm Design and Analysis

Algorithm Design and Analysis is a fundamental discipline within computer science that focuses on the creation and evaluation of step-by-step procedures for solving computational problems. The "design" aspect involves employing various strategies and paradigms, such as divide and conquer, dynamic programming, and greedy approaches, to construct effective and correct algorithms. The "analysis" component provides a formal framework for measuring an algorithm's performance, primarily by quantifying its time complexity (how long it takes to run) and space complexity (how much memory it requires), often expressed using asymptotic notations like Big O to understand its efficiency as the input size grows.

1. Introduction to Algorithms
   1. Defining an Algorithm
      1. Formal Definition
      2. Properties of an Algorithm
         1. Finiteness
         2. Definiteness
         3. Input
         4. Output
         5. Effectiveness
      3. Algorithm Representation
         1. Pseudocode
         2. Flowcharts
         3. Natural Language Description
      4. Correctness and Efficiency
         1. Partial Correctness
         2. Total Correctness
         3. Termination
         4. Measuring Efficiency
   2. The Role of Algorithms in Computing
      1. Algorithms in Problem Solving
      2. Algorithms in Software Development
      3. Algorithms in Hardware Design
      4. Real-World Applications
   3. Models of Computation
      1. Random Access Machine (RAM) Model
         1. Basic Operations and Cost Model
         2. Word Size Assumptions
         3. Memory Hierarchy Considerations
      2. Turing Machine
         1. Components of a Turing Machine
         2. Computability and Decidability
         3. Church-Turing Thesis
      3. Other Models
         1. Pointer Machine
         2. Boolean Circuit Model
         3. Quantum Computing Model
   4. Review of Fundamental Data Structures
      1. Arrays and Linked Lists
         1. Static Arrays
         2. Dynamic Arrays
         3. Singly Linked Lists
         4. Doubly Linked Lists
         5. Circular Linked Lists
      2. Stacks and Queues
         1. Stack Operations
         2. Stack Applications
         3. Queue Operations
         4. Queue Applications
         5. Deques
         6. Priority Queues
      3. Trees and Heaps
         1. Binary Trees
         2. Binary Search Trees
         3. Balanced Trees
            1. AVL Trees
            2. Red-Black Trees
            3. B-Trees
         4. Heap Structure
         5. Heap Operations
         6. Heap Applications
      4. Hash Tables
         1. Hash Functions
         2. Collision Resolution Techniques
            1. Chaining
            2. Open Addressing
         3. Load Factor and Performance
         4. Applications of Hash Tables
      5. Graphs
         1. Directed Graphs
         2. Undirected Graphs
         3. Weighted Graphs
         4. Unweighted Graphs
         5. Graph Representations
            1. Edge List
            2. Adjacency Matrix
            3. Adjacency List