np.cumsum() for cumulative sum in NumPy - Time & Space Complexity

Time Complexity: np.cumsum() for cumulative sum

O(n)

Understanding Time Complexity

We want to understand how the time taken by np.cumsum() changes as the input array grows.

Specifically, how does the work increase when the array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np

arr = np.array([1, 2, 3, 4, 5])
cum_sum = np.cumsum(arr)
print(cum_sum)

This code calculates the cumulative sum of the array elements, producing a new array where each element is the sum of all previous elements including itself.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Adding each element to a running total to build the cumulative sum.
• How many times: Once for each element in the array, from start to end.

How Execution Grows With Input

As the array size grows, the number of additions grows in a straight line with it.

Input Size (n)	Approx. Operations
10	10 additions
100	100 additions
1000	1000 additions

Pattern observation: The work grows evenly as the input size increases; doubling the input doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to compute the cumulative sum grows directly in proportion to the number of elements.

Common Mistake

[X] Wrong: "Since cumulative sum involves sums of sums, it must take much longer than just adding all elements once."

[OK] Correct: The function cleverly adds each element only once in order, so it does not repeat sums unnecessarily.

Interview Connect

Understanding how simple array operations scale helps you explain and optimize data processing tasks clearly and confidently.

Self-Check

"What if we computed cumulative sums on a 2D array along one axis? How would the time complexity change?"

Practice

1. What does the np.cumsum() function do when applied to a 1D numpy array?

easy

A. Calculates the cumulative sum of the elements step by step

B. Calculates the product of all elements

C. Sorts the array in ascending order

D. Finds the maximum value in the array

Solution

1. Step 1: Understand the function purpose

   np.cumsum() adds elements cumulatively, meaning each element is the sum of all previous elements plus itself.

2. Step 2: Compare with other options

   Options B, C, and D describe different operations: product, sorting, and max value, which are not what np.cumsum() does.

3. Final Answer:

   Calculates the cumulative sum of the elements step by step -> Option A

4. Quick Check:

   np.cumsum() = cumulative sum [OK]

Hint: Remember: cumsum means running total sum [OK]

Common Mistakes:

• Confusing cumulative sum with product
• Thinking it sorts the array
• Assuming it finds max or min values

2. Which of the following is the correct syntax to compute the cumulative sum of a numpy array arr?

easy

A. np.cumsum(arr)

B. All of the above

C. np.cumsum(arr, axis=0)

D. arr.cumsum()

Solution

1. Step 1: Check numpy function usage

   np.cumsum(arr) is the basic syntax to get cumulative sum of all elements in the array.

2. Step 2: Check array method and axis parameter

   arr.cumsum() is a valid method call on numpy arrays. Also, np.cumsum(arr, axis=0) computes cumulative sum along the first axis, valid for multi-dimensional arrays.

3. Final Answer:

   np.cumsum(arr) -> Option A

4. Quick Check:

   Basic syntax is np.cumsum(arr) [OK]

Hint: np and array method both work; axis optional [OK]

Common Mistakes:

• Thinking only np.cumsum(arr) works
• Ignoring axis parameter for multi-dimensional arrays
• Assuming cumsum is not an array method

3. What is the output of the following code?

import numpy as np
arr = np.array([2, 4, 6, 8])
cum_sum = np.cumsum(arr)
print(cum_sum)

medium

A. [2 8 14 22]

B. [2 4 6 8]

C. [2 6 12 20]

D. [20 18 14 8]

Solution

1. Step 1: Calculate cumulative sums stepwise

   Start with 2, then 2+4=6, then 6+6=12, then 12+8=20.

2. Step 2: Form the cumulative sum array

   The resulting array is [2, 6, 12, 20].

3. Final Answer:

   [2 6 12 20] -> Option C

4. Quick Check:

   Running total sums = [2, 6, 12, 20] [OK]

Hint: Add each element to sum of all before it [OK]

Common Mistakes:

• Confusing cumulative sum with original array
• Adding elements incorrectly
• Reversing the order of sums

4. The following code throws an error. What is the problem?

import numpy as np
arr = np.array([[1, 2], [3, 4]])
cum_sum = np.cumsum(arr, axis=2)
print(cum_sum)

medium

A. Syntax error in np.cumsum call

B. np.cumsum does not accept axis parameter

C. Array must be 1D for cumsum

D. Axis 2 does not exist for a 2D array

Solution

1. Step 1: Check array dimensions

   The array has shape (2, 2), so it has axes 0 and 1 only.

2. Step 2: Validate axis parameter

   Using axis=2 is invalid because axis 2 does not exist for this 2D array, causing an error.

3. Final Answer:

   Axis 2 does not exist for a 2D array -> Option D

4. Quick Check:

   Axis must be within array dimensions [OK]

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

Common Mistakes:

• Assuming axis parameter is optional but invalid
• Thinking cumsum only works on 1D arrays
• Confusing syntax error with axis error

5. Given a 2D numpy array representing daily sales for 3 products over 4 days:

sales = np.array([[5, 3, 2],
                  [4, 6, 1],
                  [7, 2, 3],
                  [3, 5, 4]])

Which code correctly computes the cumulative sales per product over the days?

hard

A. np.cumsum(sales)

B. np.cumsum(sales, axis=0)

C. np.cumsum(sales, axis=1)

D. np.cumsum(sales, axis=2)

Solution

1. Step 1: Understand the array shape and axis meaning

   The array shape is (4, 3): 4 days (rows), 3 products (columns). Axis 0 moves down rows (days), axis 1 moves across products.

2. Step 2: Choose axis for cumulative sum per product over days

   To get cumulative sales per product over days, sum down each column (axis=0).

3. Final Answer:

   np.cumsum(sales, axis=0) -> Option B

4. Quick Check:

   Cumulative sum down rows = axis 0 [OK]

Hint: Sum down rows (axis=0) for cumulative per product [OK]

Common Mistakes:

• Using axis=1 sums across products, not over days
• Ignoring axis parameter
• Using invalid axis=2 for 2D array