2D array indexing (row, col) in NumPy - Time & Space Complexity

Time Complexity: 2D array indexing (row, col)

O(1)

Understanding Time Complexity

We want to understand how long it takes to get a value from a 2D array using row and column indexes.

How does the time needed change when the array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np

arr = np.random.rand(1000, 1000)  # Create a 2D array with 1000 rows and 1000 columns
value = arr[500, 700]  # Access the element at row 500, column 700

This code creates a large 2D array and accesses one element by specifying its row and column.

Identify Repeating Operations

• Primary operation: Accessing a single element by its row and column index.
• How many times: Exactly once for this access.

How Execution Grows With Input

Getting one element from a 2D array takes the same time no matter how big the array is.

Input Size (n x n)	Approx. Operations
10 x 10	1
100 x 100	1
1000 x 1000	1

Pattern observation: The time to access one element stays the same even if the array grows larger.

Final Time Complexity

Time Complexity: O(1)

This means accessing an element by row and column is done in constant time, no matter the array size.

Common Mistake

[X] Wrong: "Accessing an element in a big 2D array takes longer because the array is large."

[OK] Correct: The computer calculates the element's position directly using the indexes, so it does not need to look through other elements.

Interview Connect

Knowing that 2D array indexing is fast helps you explain why some algorithms are efficient and others are slow. It shows you understand how data is stored and accessed.

Self-Check

"What if we tried to find a value by searching every element in the 2D array? How would the time complexity change?"

Practice

1. What does arr[2, 1] mean when arr is a 2D numpy array?

easy

A. Access the element in the 3rd row and 2nd column

B. Access the element in the 2nd row and 3rd column

C. Access the entire 2nd row

D. Access the entire 1st column

Solution

1. Step 1: Understand zero-based indexing

   In numpy, counting starts at 0, so index 2 means the 3rd row, and index 1 means the 2nd column.

2. Step 2: Interpret the indexing

   arr[2, 1] accesses the element at row 3, column 2.

3. Final Answer:

   Access the element in the 3rd row and 2nd column -> Option A

4. Quick Check:

   Row 2, Col 1 means 3rd row, 2nd column [OK]

Hint: Remember: rows and columns start counting at zero [OK]

Common Mistakes:

• Confusing row and column order
• Starting count from 1 instead of 0
• Thinking arr[2,1] returns a row or column

2. Which of the following is the correct syntax to access the element in the 1st row and 3rd column of a numpy 2D array arr?

easy

A. arr[1, 3]

B. arr[0, 2]

C. arr[3, 1]

D. arr[2, 0]

Solution

1. Step 1: Recall zero-based indexing

   The 1st row is index 0, and the 3rd column is index 2.

2. Step 2: Match indexes to syntax

   So, arr[0, 2] accesses the element in the 1st row and 3rd column.

3. Final Answer:

   arr[0, 2] -> Option B

4. Quick Check:

   Row 0, Col 2 = 1st row, 3rd column [OK]

Hint: Count rows and columns from zero for correct indexing [OK]

Common Mistakes:

• Using 1-based indexes like arr[1,3]
• Swapping row and column indexes
• Using parentheses instead of brackets

3. Given the numpy array:

import numpy as np
arr = np.array([[5, 10, 15], [20, 25, 30], [35, 40, 45]])
print(arr[1, 2])

What is the output?

medium

A. 30

B. 25

C. 20

D. 45

Solution

1. Step 1: Identify the element at row 1, column 2

   Row 1 is the second row: [20, 25, 30]. Column 2 is the third element in that row, which is 30.

2. Step 2: Confirm the printed value

   The code prints arr[1, 2], so the output is 30.

3. Final Answer:

   30 -> Option A

4. Quick Check:

   arr[1,2] = 30 [OK]

Hint: Remember row then column to find the element [OK]

Common Mistakes:

• Choosing element from wrong row or column
• Confusing zero-based indexing
• Reading the array incorrectly

4. What is wrong with this code snippet?

import numpy as np
arr = np.array([[1, 2], [3, 4]])
print(arr[2, 0])

medium

A. Prints 3 correctly

B. SyntaxError due to wrong brackets

C. IndexError because row 2 does not exist

D. Prints 1 correctly

Solution

1. Step 1: Check array shape and indexes

   The array has 2 rows (index 0 and 1). Index 2 for row is out of range.

2. Step 2: Understand error type

   Accessing arr[2, 0] causes an IndexError because row 2 does not exist.

3. Final Answer:

   IndexError because row 2 does not exist -> Option C

4. Quick Check:

   Row index 2 is invalid for 2-row array [OK]

Hint: Check array shape before indexing to avoid errors [OK]

Common Mistakes:

• Assuming indexing starts at 1
• Ignoring array shape
• Confusing error types

5. You have a 4x4 numpy array arr filled with numbers 1 to 16 in row-major order. Which code correctly changes the element in the last row and second column to 100?

hard

A. arr[3, -2] = 100

B. arr[4, 2] = 100

C. arr[-1, 2] = 100

D. arr[3, 1] = 100

Solution

1. Step 1: Identify last row and second column indexes

   For a 4x4 array, rows and columns are indexed 0 to 3. Last row is index 3, second column is index 1.

2. Step 2: Check each option

   arr[3, 1] = 100 uses arr[3, 1] which is correct. arr[4, 2] = 100 uses invalid index 4. arr[-1, 2] = 100 uses column index 2 (third column), not second. arr[3, -2] = 100 uses negative index -2 for column which is the third column, not second.

3. Final Answer:

   arr[3, 1] = 100 -> Option D

4. Quick Check:

   Last row = 3, second column = 1 [OK]

Hint: Use zero-based indexes: last row is size-1, second column is 1 [OK]

Common Mistakes:

• Using out-of-range indexes like 4
• Confusing negative indexes for columns
• Mixing row and column positions