np.sum() and axis parameter in NumPy - Time & Space Complexity

Time Complexity: np.sum() and axis parameter

O(n * m)

Understanding Time Complexity

We want to understand how the time it takes to add numbers using np.sum() changes as the size of the data grows.

Specifically, how does choosing different axes affect the work done?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np

arr = np.random.rand(1000, 500)
sum_all = np.sum(arr)
sum_axis0 = np.sum(arr, axis=0)
sum_axis1 = np.sum(arr, axis=1)

This code creates a 2D array and sums all elements, or sums along rows or columns.

Identify Repeating Operations

Look at what repeats when summing.

• Primary operation: Adding each number in the array.
• How many times: Once for each element in the array, no matter which axis is chosen.

How Execution Grows With Input

As the array gets bigger, the number of additions grows in direct proportion.

Input Size (n x m)	Approx. Operations
10 x 10	100 additions
100 x 100	10,000 additions
1000 x 1000	1,000,000 additions

Pattern observation: Doubling the size in each dimension roughly quadruples the work because total elements increase.

Final Time Complexity

Time Complexity: O(n * m)

This means the time to sum grows directly with the total number of elements in the array.

Common Mistake

[X] Wrong: "Summing along an axis is faster because it looks like fewer sums happen."

[OK] Correct: Even when summing along one axis, every element is still added once, so the total work stays the same.

Interview Connect

Knowing how array operations scale helps you write efficient code and explain your choices clearly in real projects or interviews.

Self-Check

What if we used np.sum() on a 3D array with the axis parameter? How would the time complexity change?