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?

Practice

1. What does the axis parameter in np.sum() control?

easy

A. The data type of the output array.

B. The direction along which the sum is calculated (rows or columns).

C. The size of the array to sum.

D. The number of times the sum is repeated.

Solution

1. Step 1: Understand the role of axis in np.sum()

   The axis parameter tells numpy whether to sum across rows (axis=1) or columns (axis=0).

2. Step 2: Clarify what axis does not affect

   It does not change data type, size, or repetition of sums.

3. Final Answer:

   The direction along which the sum is calculated (rows or columns). -> Option B

4. Quick Check:

   axis controls sum direction [OK]

Hint: Axis decides row or column sum direction [OK]

Common Mistakes:

• Thinking axis changes data type
• Confusing axis with array size
• Assuming axis repeats sums

2. Which of the following is the correct syntax to sum all elements of a 2D numpy array arr?

easy

A. np.sum(arr)

B. np.sum(arr, axis=1, 0)

C. np.sum(arr, axis=2)

D. np.sum(arr, axis='all')

Solution

1. Step 1: Recall default behavior of np.sum()

   Without axis, np.sum(arr) sums all elements in the array.

2. Step 2: Check invalid axis values

   axis=2 is invalid for 2D arrays, axis=1,0 is incorrect syntax, axis='all' is invalid.

3. Final Answer:

   np.sum(arr) -> Option A

4. Quick Check:

   Default np.sum sums all elements [OK]

Hint: No axis sums all elements [OK]

Common Mistakes:

• Using invalid axis values
• Passing multiple axis values incorrectly
• Using string instead of integer for axis

3. Given the array arr = np.array([[1, 2, 3], [4, 5, 6]]), what is the output of np.sum(arr, axis=0)?

medium

A. [5 7 9]

B. [3 7]

C. [6 15]

D. [1 2 3 4 5 6]

Solution

1. Step 1: Understand axis=0 sums columns

   Summing along axis=0 adds elements vertically: (1+4), (2+5), (3+6).

2. Step 2: Calculate sums for each column

   1+4=5, 2+5=7, 3+6=9, so result is [5 7 9].

3. Final Answer:

   [5 7 9] -> Option A

4. Quick Check:

   Sum columns = [5 7 9] [OK]

Hint: axis=0 sums columns vertically [OK]

Common Mistakes:

• Confusing axis=0 with axis=1
• Adding rows instead of columns
• Expecting a scalar output

4. What is wrong with this code snippet?

import numpy as np
arr = np.array([[1, 2], [3, 4]])
result = np.sum(arr, axis=2)

medium

A. The array is not defined correctly.

B. np.sum() cannot be used on 2D arrays.

C. The axis value 2 is invalid for a 2D array.

D. The import statement is missing.

Solution

1. Step 1: Check array dimensions

   The array has shape (2, 2), so valid axis values are 0 or 1.

2. Step 2: Validate axis parameter

   axis=2 is out of bounds for this 2D array, causing an error.

3. Final Answer:

   The axis value 2 is invalid for a 2D array. -> Option C

4. Quick Check:

   Axis must be less than array dimensions [OK]

Hint: Axis must be less than array dimensions [OK]

Common Mistakes:

• Using axis larger than array dimensions
• Thinking np.sum can't handle 2D arrays
• Ignoring error messages

5. You have a 3D numpy array arr = np.array([[[1,2],[3,4]], [[5,6],[7,8]]]). What is the shape of the result when you run np.sum(arr, axis=1)?

hard

A. (1, 2)

B. (3, 2)

C. (2, 4)

D. (2, 2)

Solution

1. Step 1: Understand the shape of arr

   arr has shape (2, 2, 2): 2 blocks, each with 2 rows and 2 columns.

2. Step 2: Summing along axis=1

   axis=1 sums over the second dimension (rows), reducing it from 2 to 1, so shape changes from (2, 2, 2) to (2, 2).

3. Final Answer:

   (2, 2) -> Option D

4. Quick Check:

   Sum axis=1 reduces dimension 2 to 1 [OK]

Hint: Sum along axis removes that dimension [OK]

Common Mistakes:

• Confusing which axis is summed
• Expecting shape to stay same
• Mixing up dimensions in 3D arrays