Understanding O(n) vs O(1) Space Complexity in Programming

This content originally appeared on DEV Community and was authored by Grant Riordan

When developing, you may have heard terms like linear space O(n) and constant space O(1), or seen them completing coding challenges; but what do they actually mean?

These terms describe the space requirements of an algorithm and how those requirements grow as the input size increases. They’re part of what’s known as Big O notation, a mathematical way to describe algorithmic efficiency — both in terms of time and space. This post focuses on space complexity.

What Is Space Complexity?

Space complexity refers to the amount of memory an algorithm needs to run, including:

• Input space: Memory used to store the input itself.

• Auxiliary space: Extra memory used during processing (e.g., temporary variables, stacks, arrays, etc.).

In Big O notation, we usually refer to the auxiliary space, since input space is often considered a given.

O(n) Space – Linear Space

An algorithm with O(n) space complexity means the memory usage grows linearly with the input size. That is, if your input doubles, memory usage roughly doubles too.

Analogy

Imagine someone hands you a stack of n books (the input).
Now, you’re asked to copy each one onto a new stack.
For each book, you use a new page in your notebook.
The more books you get, the more pages you use.

Programming Example: Creating a List

def create_list(n):
    my_list = []
    for i in range(n):
        my_list.append(i)
    return my_list

Here, the list will contain n elements, so its space complexity is O(n).

Other Examples of O(n) Space

• Recursion (in some cases): Each recursive call consumes stack space. If a function calls itself n times before reaching a base case, the stack will have n frames — resulting in O(n) space.

def recursive_countdown(n):
    if n == 0:
        return
    recursive_countdown(n - 1)

Is O(n) Bad?

You may hear some developers say O(n) isn’t performant, especially when working with large datasets. That’s because growing memory usage can lead to:

• Memory overload
• Performance bottlenecks

When Is O(n) Acceptable?

Very often! If your problem requires storing all elements (like reading a file into memory or tracking all results), O(n) is completely reasonable.

O(1) Space – Constant Space

An algorithm that uses constant space (O(1)) requires a fixed amount of memory regardless of input size. Whether the input is 10, 1,000, or a million elements, memory usage doesn’t change significantly.

Analogy

Imagine you’re adding up numbers from a list. You don’t need to write all the numbers down. You just need one notepad to keep a running total.

Programming Example: Sum of Numbers

def sum_numbers(numbers):
    total = 0  # Only one variable is used
    for num in numbers:
        total += num
    return total

This function only uses a fixed number of variables, regardless of the size of the numbers list. So it has O(1) space complexity.

In-Place Algorithms

Algorithms that modify the input directly — without creating large additional data structures — often use constant auxiliary space.

Example: In-place sorting like selection sort or bubble sort (not merge sort).

Auxiliary Space

When talking about space complexity, you may hear the term auxiliary space — this refers to additional space an algorithm uses beyond the input itself.

Example Breakdown

• Input: an array of n elements → takes O(n) space (input space)
• Temporary variables: e.g., a counter, index → takes O(1) space (auxiliary)

If an algorithm creates a new array of size n, then that’s O(n) auxiliary space.

How to Think About Total Space

When evaluating an algorithm, consider both input and auxiliary space:

Total Space = O(Input Space) + O(Auxiliary Space)

But in Big O notation, we drop constants and lower-order terms, keeping only the dominant term.

Example 1: Input O(n), Auxiliary O(1)

def sum_array_elements(arr):
    # Input takes O(n) space
    total = 0  # Auxiliary O(1)
    for x in arr:
        total += x
    return total

my_data = [1, 2, 3, 4, 5] * 1000000  # n = 5,000,000
result = sum_array_elements(my_data)

• Input space: O(n)
• Auxiliary space: O(1)
• Total space: O(n) (since O(n) + O(1) → O(n))

Example 2: Input O(n), Auxiliary O(n)

def double_elements(arr):
    doubled = []  # New array, O(n) auxiliary space
    for x in arr:
        doubled.append(x * 2)
    return doubled

• Input space: O(n)
• Auxiliary space: O(n) (due to doubled)
• Total space: O(n) (because O(n) + O(n) = O(2n) → still O(n))

Analogy: Warehouse & Forklift

Imagine your program is a warehouse (memory).

• A truck delivers n boxes → that’s the input (O(n)).
• Inside, you use a forklift and a table (fixed-size tools) to process the boxes → that’s auxiliary space (O(1)).

Regardless of how many boxes arrive, the forklift and table stay the same size. The input dominates the memory needs, not the tools.

Final Thoughts

O(1) space means your algorithm is memory-efficient, often a key requirement in embedded systems, large-scale data processing, or performance-critical systems.

O(n) space is not bad — it’s often necessary and totally acceptable, especially when storing results or processing input data.

Understanding how memory usage scales helps you make better decisions when designing efficient, scalable code.

This content originally appeared on DEV Community and was authored by Grant Riordan