Algorithms

An algorithm is a finite, well-defined, step-by-step set of instructions or rules designed to solve a specific problem or perform a computation. As a cornerstone of computer science, algorithms provide the logical foundation for how software works, dictating the precise sequence of actions a computer must take to process data and achieve a desired outcome, from sorting a list of names to finding the shortest route on a map. The efficiency and design of an algorithm are critically dependent on the data structures it manipulates, making the study of algorithms and data structures a fundamental and interconnected field.

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. Examples of Algorithms
      4. Non-Examples of Algorithms
      5. Algorithm vs. Program
      6. Algorithm vs. Heuristic
   2. The Role of Algorithms in Computing
      1. Algorithms in Problem Solving
      2. Algorithms in Software Development
      3. Algorithms in Everyday Life
      4. Historical Perspective
      5. Modern Applications
   3. Algorithm Specification
      1. Pseudocode Conventions
         1. Syntax and Structure
         2. Variable Declaration
         3. Assignment Statements
         4. Control Structures
            1. Sequential Execution
            2. Conditional Statements
            3. Loop Constructs
         5. Function Definitions
         6. Comments and Documentation
         7. Readability and Clarity
      2. Flowcharts
         1. Flowchart Symbols
            1. Terminal Symbols
            2. Process Symbols
            3. Decision Symbols
            4. Connector Symbols
            5. Input/Output Symbols
         2. Flowchart Construction
         3. Interpreting Flowcharts
         4. Advantages and Limitations
      3. Stepwise Refinement
         1. Top-Down Design
         2. Modular Decomposition
         3. Refinement Process
   4. Mathematical Foundations for Analysis
      1. Asymptotic Notations
         1. Big-O Notation
            1. Formal Definition
            2. Mathematical Properties
            3. Common Big-O Classes
               1. Constant Time O(1)
               2. Logarithmic Time O(log n)
               3. Linear Time O(n)
               4. Linearithmic Time O(n log n)
               5. Quadratic Time O(n²)
               6. Cubic Time O(n³)
               7. Exponential Time O(2ⁿ)
               8. Factorial Time O(n!)
            4. Interpreting Big-O
            5. Rules for Big-O Calculations
         2. Omega Notation
            1. Formal Definition
            2. Use Cases
            3. Relationship to Big-O
         3. Theta Notation
            1. Formal Definition
            2. Tight Bounds
            3. Use Cases
         4. Little-o and Little-omega Notations
            1. Definitions
            2. Strict Bounds
            3. Differences from Big-O and Omega
      2. Analysis of Algorithms
         1. Time Complexity
            1. Measuring Input Size
            2. Counting Primitive Operations
            3. Basic Operations
            4. Worst-Case Analysis
            5. Average-Case Analysis
            6. Best-Case Analysis
            7. Amortized Analysis
         2. Space Complexity
            1. Auxiliary Space
            2. Input Space
            3. Types of Memory Usage
            4. Trade-offs with Time Complexity
         3. Empirical Analysis
            1. Benchmarking
            2. Profiling
            3. Statistical Analysis
      3. Recurrence Relations
         1. Formulating Recurrences
            1. Identifying Recursive Structure
            2. Base Cases
            3. Recursive Cases
         2. Substitution Method
            1. Guessing the Solution
            2. Proving by Induction
            3. Strong Induction
         3. Recursion-Tree Method
            1. Drawing Recursion Trees
            2. Calculating Level Costs
            3. Summing Costs
            4. Geometric Series
         4. Master Theorem
            1. Statement of the Theorem
            2. Three Cases
            3. Application to Common Recurrences
            4. Limitations of the Master Theorem
            5. Extended Master Theorem
      4. Mathematical Tools
         1. Summations
            1. Arithmetic Series
            2. Geometric Series
            3. Harmonic Series
         2. Logarithms
            1. Change of Base
            2. Logarithmic Identities
         3. Probability Theory Basics
            1. Random Variables
            2. Expected Value
            3. Probability Distributions