When evaluating algorithms, we don’t just ask: “Does it work?” — we ask: “How well does it work, especially as inputs grow?” That’s where algorithm complexity and Big O notation come in. In this article, you’ll learn what algorithmic complexity means, how to interpret Big O notation, and how to analyze Python code with real-world examples.
“In my own learning journey, understanding Big O helped me debug slow code in a personal project. Once I realized my nested loops were causing O(n²) performance, replacing them saved hours of processing time.”
🔹 What Is Algorithm Complexity?
Algorithm complexity refers to how the time or space required by an algorithm changes as the input size increases. There are two main types:
Time Complexity: How long an algorithm takes to run.
Space Complexity: How much memory it needs.
🔹 What Is Big O Notation?
Big O notation describes the worst-case scenario performance of an algorithm using a function of input size n. It gives a high-level view of how the algorithm scales.
Big O
Name
Example
O(1)
Constant Time
Accessing arr[0]
O(log n)
Logarithmic
Binary Search
O(n)
Linear
Loop through list
O(n log n)
Linearithmic
Merge Sort
O(n²)
Quadratic
Nested loops
O(2ⁿ)
Exponential
Recursive Fibonacci
O(n!)
Factorial
Permutations
🔸 Why Big O Matters
Helps compare algorithms independently of hardware
Predicts performance at scale
Guides optimization and code efficiency
🔸 Python Example 1: Linear Time – O(n)
def linear_search(arr, target):
for i in arr:
if i == target:
return True
return False
# Complexity: O(n)
The function checks each element—time grows proportionally to input size.
No matter how big the list is, accessing an index takes the same time.
🔸 Python Example 3: Quadratic Time – O(n²)
def print_pairs(arr):
for i in arr:
for j in arr:
print(i, j)
# Complexity: O(n²)
Two nested loops → time grows with the square of input size.
🔸 How to Analyze Code for Big O
Identify loops: Each loop usually means linear growth.
Check nested loops: Often O(n²), O(n³), etc.
Consider recursion: Analyze the recursive tree.
Ignore constants: We focus on how it scales, not the exact time.
e.g., O(2n) → O(n)
🔹 Space Complexity Example
def make_list(n):
result = []
for i in range(n):
result.append(i)
return result
# Time: O(n), Space: O(n)
The list grows with input size, so both time and space are linear.
🧠 Tips to Improve Algorithm Complexity
Use efficient data structures (e.g., sets for O(1) lookups).
Replace recursion with iteration where possible.
Choose the right algorithm (e.g., binary search instead of linear search).
🔚 Conclusion
Big O notation is your lens into performance. Understanding it helps you choose better algorithms, write scalable code, and debug performance bottlenecks. With practice, recognizing O(n) from O(log n) will become second nature—and your code will thank you.
