1.4 Time & Space Complexity – Big O, Big Θ, Big Ω

When learning Data Structures and Algorithms (DSA), one of the most important concepts to understand is time complexity and space complexity. These terms help developers evaluate how efficient an algorithm is, especially when dealing with large inputs.

In this article, we’ll break down time & space complexity, explain Big O, Big Θ, and Big Ω notations, and see why they matter in programming.

What is Time Complexity?

Time Complexity measures how the execution time of an algorithm grows with the size of the input.

👉 Example:

• Searching for a name in an unsorted list of 1,000 entries may take 1,000 steps.

• If the list grows to 1,000,000 entries, it may take 1,000,000 steps.

Instead of measuring in seconds (which depends on the computer’s speed), we use asymptotic notation like O(n), O(log n), O(n²) to express how time increases as input size grows.

What is Space Complexity?

Space Complexity measures how much memory (RAM) an algorithm requires during execution.

This includes:

• Memory for input data

• Auxiliary variables

• Function call stack

• Temporary storage

👉 Example:

• A simple loop through an array uses O(1) extra space.

• A recursive algorithm like quicksort may use O(log n) space for the call stack.

Why Do We Use Asymptotic Notation?

Asymptotic notations provide a way to compare algorithms independent of machine speed or programming language.

There are three main notations used to analyze algorithms:

1. Big O Notation (O) – Worst Case

Big O notation represents the upper bound of an algorithm. It shows the worst-case performance (maximum time or space needed).

👉 Examples:

• Linear Search → O(n)

• Binary Search → O(log n)

• Bubble Sort → O(n²)

If an algorithm is O(n²), it means in the worst case, execution time grows proportionally to the square of the input size.

2. Big Θ Notation (Θ) – Average Case

Big Theta (Θ) notation represents the tight bound of an algorithm. It describes the average case performance.

👉 Example:

• For Linear Search, Θ(n) means that on average, we might find the element after checking half of the list.

Big Θ is useful when we want to understand expected efficiency rather than the worst case.

3. Big Ω Notation (Ω) – Best Case

Big Omega (Ω) notation represents the lower bound of an algorithm. It describes the best-case scenario (minimum time or space needed).

👉 Example:

• For Linear Search, Ω(1) means if the element is the very first item, the algorithm finishes in constant time.

Common Time Complexities

Here are some typical complexities in algorithms:

• O(1) → Constant time (e.g., accessing an array element)

• O(log n) → Logarithmic time (e.g., Binary Search)

• O(n) → Linear time (e.g., Linear Search)

• O(n log n) → Efficient sorting algorithms (Merge Sort, Quick Sort)

• O(n²) → Nested loops (Bubble Sort, Insertion Sort)

• O(2ⁿ) → Exponential time (recursive algorithms like Fibonacci without DP)

👉 Rule of Thumb: Lower complexity = faster algorithm for larger inputs.

Example Problem:

Find the maximum element in an array of size n

• Best Case (Ω): Ω(1) – If the maximum is the first element.

• Average Case (Θ): Θ(n/2) ≈ Θ(n) – On average, we check half the elements.

• Worst Case (O): O(n) – We may need to check all elements.

Conclusion

Understanding time complexity and space complexity is essential for writing efficient programs.

• Big O → Worst case

• Big Θ → Average case

• Big Ω → Best case

By analyzing algorithms with these notations, developers can make informed decisions about which approach will be most efficient for their applications.

👉 Remember: Writing correct code is important, but writing efficient code is what makes you a great developer.