Negative indexing in NumPy - Time & Space Complexity

Time Complexity: Negative indexing

O(1)

Understanding Time Complexity

We want to understand how fast operations run when using negative indexing in numpy arrays.

Specifically, how does the time to access elements change as the array grows?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
arr = np.arange(1000)
value = arr[-1]
value2 = arr[-10]
value3 = arr[-100]

This code accesses elements from the end of a numpy array using negative indexes.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Accessing an element by index in the array.
• How many times: Three times, each direct access to a single element.

How Execution Grows With Input

Accessing an element by negative index is just like positive index access; it directly finds the element.

Input Size (n)	Approx. Operations
10	3
100	3
1000	3

Pattern observation: The number of operations stays the same no matter how big the array is.

Final Time Complexity

Time Complexity: O(1)

This means accessing an element by negative index takes the same short time no matter the array size.

Common Mistake

[X] Wrong: "Negative indexing makes the program slower because it has to count from the end."

[OK] Correct: Numpy directly calculates the position internally, so it does not count through elements one by one.

Interview Connect

Knowing that negative indexing is fast helps you write clean and efficient code without worrying about speed.

Self-Check

"What if we accessed a slice using negative indexes instead of single elements? How would the time complexity change?"

Practice

1. What does negative indexing in a numpy array allow you to do?

easy

A. Sort the array in descending order

B. Access elements starting from the end of the array

C. Change the size of the array

D. Access elements only at the beginning of the array

Solution

1. Step 1: Understand negative indexing concept

   Negative indexing means counting elements from the end of the array, starting at -1 for the last element.

2. Step 2: Compare options with concept

   Only Access elements starting from the end of the array correctly describes accessing elements from the end using negative indexes.

3. Final Answer:

   Access elements starting from the end of the array -> Option B

4. Quick Check:

   Negative indexing = Access from end [OK]

Hint: Negative index -1 means last element [OK]

Common Mistakes:

• Thinking negative index changes array size
• Confusing negative indexing with sorting
• Assuming negative index accesses start only at beginning

2. Which of the following is the correct syntax to get the last element of a numpy array arr using negative indexing?

easy

A. arr[0]

B. arr[1]

C. arr[-0]

D. arr[-1]

Solution

1. Step 1: Recall negative index for last element

   In numpy, -1 is used to access the last element of an array.

2. Step 2: Check each option

   arr[-1] uses arr[-1] which correctly accesses the last element. Others do not.

3. Final Answer:

   arr[-1] -> Option D

4. Quick Check:

   Last element = arr[-1] [OK]

Hint: Use arr[-1] for last element [OK]

Common Mistakes:

• Using arr[-0] which is same as arr[0]
• Using positive index 1 which is second element
• Confusing zero index with negative zero

3. Given the numpy array arr = np.array([10, 20, 30, 40, 50]), what is the output of arr[-3]?

medium

A. 20

B. 40

C. 30

D. 50

Solution

1. Step 1: Understand negative index -3

   Negative index -3 means counting from the end: last is -1 (50), second last -2 (40), third last -3 (30).

2. Step 2: Identify element at index -3

   The element at arr[-3] is 30.

3. Final Answer:

   30 -> Option C

4. Quick Check:

   arr[-3] = 30 [OK]

Hint: Count backward: -1 last, -2 second last, -3 third last [OK]

Common Mistakes:

• Counting negative index from start instead of end
• Mixing up index -3 with positive index 3
• Off-by-one errors in counting

4. Consider the code snippet:

import numpy as np
arr = np.array([5, 10, 15, 20])
print(arr[-5])

What is the error in this code?

medium

A. IndexError: index -5 is out of bounds for axis 0 with size 4

B. SyntaxError due to negative index

C. Prints 5

D. Prints 20

Solution

1. Step 1: Check array size and index used

   Array has 4 elements, valid negative indexes are -1 to -4.

2. Step 2: Understand index -5 usage

   Index -5 is outside valid range, causing IndexError.

3. Final Answer:

   IndexError: index -5 is out of bounds for axis 0 with size 4 -> Option A

4. Quick Check:

   Negative index out of range = IndexError [OK]

Hint: Negative index must be >= -array length [OK]

Common Mistakes:

• Assuming negative index wraps around infinitely
• Thinking negative index causes syntax error
• Expecting last element instead of error

5. You have a 2D numpy array arr = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]). What does arr[-1, -2] return?

hard

A. 8

B. 7

C. 6

D. 9

Solution

1. Step 1: Understand negative indexing in 2D arrays

   For rows, -1 means last row: [7, 8, 9]. For columns, -2 means second last column.

2. Step 2: Identify element at arr[-1, -2]

   Last row is [7, 8, 9], second last column is element 8.

3. Final Answer:

   8 -> Option A

4. Quick Check:

   arr[-1, -2] = 8 [OK]

Hint: Negative indexes count from end in each dimension [OK]

Common Mistakes:

• Mixing row and column indexes
• Using positive indexes instead of negative
• Confusing which axis negative index applies to