Lesson 6: Big O, Big Omega, and Big Theta Notation

Importance of Algorithm Efficiency

• Measures scalability of algorithms, especially for large datasets
• Focus on speed (growth in operations relative to input size)
• Memory requirements are also important but not the current focus

Input Size Representation

• Typically represented by N
• Discuss growth in terms of N

Types of Bounds

1. Big O Notation

   • Describes upper bound on the growth rate of a function
   • Used to express worst-case scenario of an algorithm's time complexity
   • Formal Definition: f(x) is in Big O of g(x) if there exist constants C and x₀ such that f(x) <= C * g(x) for all x > x₀

2. Big Omega Notation

   • Describes lower bound on the growth rate of a function
   • Used to express best-case scenario
   • Formal Definition: f(x) is in Big Omega of g(x) if there exist constants C and x₀ such that f(x) >= C * g(x) for all x > x₀

3. Big Theta Notation

   • Describes both upper and lower bounds
   • Represents average-case scenario where the growth rate is tightly bound
   • Formal Definition: f(x) is in Big Theta of g(x) if there exist constants C1, C2, and x₀ such that C1 * g(x) >= f(x) >= C2 * g(x) for all x > x₀

Usage in Algorithms

• Input size for algorithms is generally a positive integer (e.g., size of an array)
• Functions defined on integers, denoted by n

Integer Functions

• Big O for integer functions: f(n) is in Big O of g(n) if there exists C and n₀ such that f(n) <= C * g(n) for all n >= n₀
• Big Omega for integer functions: f(n) is in Big Omega of g(n) if there exists C and n₀ such that f(n) >= C * g(n) for all n >= n₀
• Big Theta for integer functions: f(n) is in Big Theta of g(n) if there exist C1, C2, and n₀ such that C1 * g(n) >= f(n) >= C2 * g(n) for all n >= n₀

Conclusion

• Visual impression and exact definitions provided
• Future lessons will apply these concepts to specific algorithms for clarity