Asymptotic Analysis in JavaScript

Introduction to Asymptotic Analysis

Asymptotic analysis is the primary method we use to analyze algorithms. It focuses on:

• Measuring the order of growth of time taken by a program in terms of input size (n)
• Using notations like Θ (Theta), O (Big-O), and Ω (Omega)
• Being independent of programming language, machine, or test cases

Core Concepts

• Input Size (n): Number of elements/operations
• Growth Rate: How runtime/memory scales with n
• Machine Independence: Analysis valid across environments

Key Characteristics

Machine/Language Independent

• Doesn't depend on CPU speed or programming language
• Focuses purely on algorithmic efficiency

No Implementation Required

• Can analyze algorithms just from pseudocode
• Don't need to actually run the code

Worst-Case Focused

• Typically analyzes worst-case scenarios
• Provides guaranteed upper bounds

Time Complexity Analysis Examples

Constant Time Function (O(1))

function sumFormula(n) {
    return n * (n + 1) / 2;  // 1 multiplication, 1 addition, 1 division
}

Time Expression: C₁

• Always takes constant time regardless of input size
• Arithmetic operations (+, *, /) considered constant time

Linear Time Function (O(n))

function sumLoop(n) {
    let sum = 0;             // C₂
    for (let i = 1; i <= n; i++) {  // C₃ executed n times
        sum += i;             // C₄ executed n times
    }
    return sum;
}

Time Expression: C₂ + C₃n + C₄n = C₅n + C₆

• Grows linearly with input size
• Constants don't affect the growth rate

Quadratic Time Function (O(n²))

function sumNestedLoop(n) {
    let sum = 0;             // C₇
    for (let i = 1; i <= n; i++) {      // C₈ executed n times
        for (let j = 1; j <= i; j++) {  // C₉ executed n(n+1)/2 times
            sum += 1;        // C₁₀ executed n(n+1)/2 times
        }
    }
    return sum;
}

Time Expression: C₇ + C₈n + C₉(n²/2 + n/2) + C₁₀(n²/2 + n/2)

• Simplifies to C₁₁n² + C₁₂n + C₁₃
• Dominated by the n² term for large n

Memory Considerations

• Primitive types: Constant space
• Objects/Arrays: Linear space
• Recursion: Call stack space

Time Complexity Cheat Sheet 📋

Key Takeaways:

• Green Zone (Good) 🟢: O(1), O(log n), O(n)
• Yellow Zone (Caution) 🟡: O(n log n)
• Red Zone (Avoid) 🔴: O(n²), O(2ⁿ)

Analysis Process

1. Write time as algebraic expression
2. Find order of growth
3. Compare growth rates

Next Steps

1. Detailed explanation of asymptotic notations (Big-O, Θ, Ω)
2. How to analyze more complex algorithm complexities
3. Practical examples of algorithm analysis
4. Space complexity analysis