In this article Asymptotic Notations in Algorithms Understand asymptotic notations used in algorithm analysis—Big-O for worst-case, Omega for best-case, and Theta for tight bound. Learn how these notations measure time and space complexity, compare algorithms, and analyze performance for large input sizes.
Asymptotic Notations in Algorithms
Asymptotic notations are used to measure and compare the efficiency of algorithms.
They help us understand how time complexity and space complexity grow as the input size increases (n → ∞).
The main objectives of using asymptotic notations are:
• To represent algorithm performance in a generalized way
• To compare different algorithms
• To analyze algorithms independent of programming language, hardware, or compiler
Asymptotic notations describe how an algorithm behaves as the input size becomes very large.
-
Big-O Notation (O) — Worst Case
Meaning:
Big-O represents the worst-case time complexity of an algorithm.
It shows the maximum time an algorithm may take as the input grows.
Examples:
- O(1) → Constant Time
• O(n) → Linear Time
• O(n²) → Quadratic Time
• O(log n) → Logarithmic Time
When is Big-O used?
- To represent the upper bound of an algorithm
• In worst-case performance
• To ensure the algorithm never exceeds this time limit
Examples:
Linear Search
Worst case: Element at the end or not present
→ Time = O(n)
Bubble Sort
Worst-case comparisons = n²
→ Time = O(n²)
-
Omega Notation (Ω) — Best Case
Meaning:
Omega (Ω) represents the best-case time complexity of an algorithm.
It shows minimum time an algorithm will take.
Examples:
- Ω(1) → Minimum constant time
• Ω(n) → Minimum linear time
• Ω(n²) → Minimum quadratic time
When is Omega used?
- To show the lower bound
• For best-case analysis
Examples:
Linear Search
If the element is at the first position
→ Time = Ω(1)
Bubble Sort
If the array is already sorted
→ Time = Ω(n)
-
Theta Notation (Θ) — Tight Bound
Meaning:
Theta (Θ) represents the exact or tight bound of an algorithm.
This means the complexity is the same in both upper (O) and lower (Ω) bounds.
It indicates the exact growth rate of the algorithm.
When is Theta used?
- When best-case and worst-case behavior are similar
• For precise average-case complexity
Examples:
- Θ(n) → Linear
• Θ(log n) → Logarithmic
• Θ(n²) → Quadratic
Binary Search:
• Best case = Ω(1)
• Worst case = O(log n)
• Tight bound / average case = Θ(log n)
Understanding O, Ω, and Θ Together
| Notation | Meaning | Represents |
| Big-O (O) | Upper bound | Worst case |
| Omega (Ω) | Lower bound | Best case |
| Theta (Θ) | Tight bound | Exact / average case |
Simple Example
for (i = 0; i < n; i++)
print(i);
- Worst case = O(n)
• Best case = Ω(n)
• Tight bound = Θ(n)
POP- Introduction to Programming Using ‘C’
OOP – Object Oriented Programming
DBMS – Database Management System
RDBMS – Relational Database Management System
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