Asymptotic Notations Explained

1. Big O Notation – O (Upper Bound)

Big O Notation tells the Worst-Case performance of an algorithm. It is also known as the Upper Bound.

  • Meaning: It guarantees that the algorithm’s running time will never go above a certain limit. It shows how slow the algorithm can be in the worst case.
  • On Graph:
    • If a function f(n) has a time complexity O(g(n)), it means that after a point (n₀), f(n) will always remain below or on the line c · g(n).
    • Formula: f(n) ≤ c · g(n) for all n ≥ n₀
  • Example: If we say Quick Sort has a Worst-Case complexity of O(n²), it means its growth will not exceed n².
Big O Graph

2. Omega Notation – Ω (Lower Bound)

Omega Notation tells the Best-Case performance of an algorithm. It is also called the Lower Bound.

  • Meaning: It guarantees that the algorithm’s running time will never go below a certain limit. It represents how fast the algorithm can be.
  • On Graph:
    • If a function f(n) has a time complexity Ω(g(n)), it means that after a point (n₀), f(n) will always remain above or on the line c · g(n).
    • Formula: f(n) ≥ c · g(n) for all n ≥ n₀
  • Example: Quick Sort’s Best-Case complexity is Ω(n log n). This means it will not run faster than n log n.
Omega Graph

3. Theta Notation – Θ (Tight Bound)

Theta Notation represents the Average-Case or Exact Asymptotic Behavior of an algorithm. It is also known as the Tight Bound.

  • Meaning: It is the most precise notation. It guarantees that the algorithm’s running time will lie between two functions (c₁ · g(n) and c₂ · g(n)). This means the growth rate in both best and worst cases is almost the same.
  • On Graph:
    • If a function f(n) has a time complexity Θ(g(n)), then after a point (n₀), f(n) will always remain between c₁ · g(n) and c₂ · g(n>.
    • Formula: c₁ · g(n) ≤ f(n) ≤ c₂ · g(n) for all n ≥ n₀
  • Example: Merge Sort always has a complexity of Θ(n log n). This means its running time will always stay around n log n.
Theta Graph

Summary

Notation Big O (O) Omega (Ω) Theta (Θ)
Meaning Upper Bound Lower Bound Tight Bound
Case Worst-Case Best-Case Average-Case
Simple Meaning Will never exceed this Will never go below this Will stay around this

These notations help us understand how an algorithm grows or slows down when the input size increases.

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