Apply functions on matrices in R Programming - Time & Space Complexity

Time Complexity: Apply functions on matrices

O(n * m)

Understanding Time Complexity

When we apply functions on matrices in R, we want to know how the time to run the code changes as the matrix gets bigger.

We ask: How does the work grow when the matrix size grows?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

m <- matrix(1:10000, nrow=100, ncol=100)
result <- apply(m, 1, sum)  # sum each row

This code sums each row of a matrix with 100 rows and 100 columns.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Summing elements in each row of the matrix.
• How many times: The sum operation repeats once for each row (100 times here).

How Execution Grows With Input

As the matrix grows, the time to sum each row grows with the number of rows and columns.

Input Size (n x n)	Approx. Operations
10 x 10	100 sums (10 rows x 10 columns)
100 x 100	10,000 sums (100 rows x 100 columns)
1000 x 1000	1,000,000 sums (1000 rows x 1000 columns)

Pattern observation: The total work grows roughly by the square of the matrix size because each element is visited once.

Final Time Complexity

Time Complexity: O(n * m)

This means the time grows proportionally to the number of rows times the number of columns in the matrix.

Common Mistake

[X] Wrong: "Applying a function on rows only depends on the number of rows."

[OK] Correct: Each row operation processes all columns, so columns also affect the time.

Interview Connect

Understanding how function application scales on matrices helps you reason about data processing tasks and optimize code in real projects.

Self-Check

"What if we applied the function on columns instead of rows? How would the time complexity change?"

Practice

1. In R, what does the apply() function do when used on a matrix?

easy

A. It creates a new matrix with doubled values.

B. It converts the matrix into a vector.

C. It sorts the matrix elements in ascending order.

D. It applies a function to each row or column of the matrix.

Solution

1. Step 1: Understand the purpose of apply()

   The apply() function is designed to run a function on margins of an array or matrix.

2. Step 2: Identify what margins mean in a matrix

   Margin 1 means rows, margin 2 means columns. So apply() runs the function on each row or column.

3. Final Answer:

   It applies a function to each row or column of the matrix. -> Option D

4. Quick Check:

   apply() = function on rows/columns [OK]

Hint: Remember: MARGIN=1 rows, MARGIN=2 columns in apply() [OK]

Common Mistakes:

• Thinking apply() changes matrix shape automatically
• Confusing apply() with sorting or vector conversion
• Using apply() without specifying margin

2. Which of the following is the correct syntax to calculate the sum of each column in a matrix m using apply() in R?

easy

A. apply(m, 2, sum)

B. apply(m, sum, 2)

C. apply(m, 1, sum)

D. apply(sum, m, 2)

Solution

1. Step 1: Identify margin for columns

   In apply(), margin 2 means columns.

2. Step 2: Check function placement

   The function to apply (sum) is the third argument, so syntax is apply(m, 2, sum).

3. Final Answer:

   apply(m, 2, sum) -> Option A

4. Quick Check:

   sum columns = apply(m, 2, sum) [OK]

Hint: Use margin=2 for columns in apply() [OK]

Common Mistakes:

• Using margin=1 for columns instead of 2
• Swapping function and margin arguments
• Missing the function argument

3. What is the output of the following R code?

m <- matrix(1:6, nrow=2)
apply(m, 1, max)

medium

A. [1] 3 6

B. [1] 4 5

C. [1] 5 6

D. [1] 2 4

Solution

1. Step 1: Understand the matrix structure

   The matrix m is created with values 1 to 6, 2 rows: Row 1: 1, 3, 5 Row 2: 2, 4, 6

2. Step 2: Apply max function on rows (margin=1)

   Max of row 1 is 5, max of row 2 is 6.

3. Final Answer:

   [1] 5 6 -> Option C

4. Quick Check:

   max per row = 5, 6 [OK]

Hint: Check matrix layout before applying functions [OK]

Common Mistakes:

• Confusing rows and columns in margin
• Misreading matrix values
• Assuming apply returns matrix instead of vector

4. Identify the error in this R code snippet:

m <- matrix(1:9, nrow=3)
apply(m, 3, mean)

medium

A. apply() cannot be used with mean.

B. Margin 3 is invalid for a 2D matrix.

C. Matrix m is not created properly.

D. The function mean is not defined.

Solution

1. Step 1: Check matrix dimensions

   The matrix m has 2 dimensions (rows and columns).

2. Step 2: Validate margin argument

   Margin 3 refers to a third dimension, which does not exist in a 2D matrix.

3. Final Answer:

   Margin 3 is invalid for a 2D matrix. -> Option B

4. Quick Check:

   Margin must be 1 or 2 for matrices [OK]

Hint: Use only 1 or 2 as margin for 2D matrices [OK]

Common Mistakes:

• Using margin > 2 for 2D matrices
• Assuming apply works on non-existent dimensions
• Confusing apply() with array functions

5. You have a matrix m of numeric values. You want to create a new vector that contains the mean of each row but only for rows where the sum is greater than 10. Which R code correctly does this using apply()?

hard

A. apply(m, 1, function(x) if(sum(x) > 10) mean(x) else NA)[!is.na(.)]

B. apply(m, 1, function(x) if(sum(x) > 10) mean(x))

C. apply(m, 1, function(x) mean(x[sum(x) > 10]))

D. apply(m, 2, function(x) if(sum(x) > 10) mean(x))

Solution

1. Step 1: Understand the requirement

   We want mean of each row only if that row's sum is > 10, else exclude it.

2. Step 2: Analyze each option

   A returns NA for rows not >10 but does not remove them. C incorrectly filters inside mean. D uses margin=2 (columns) instead of rows.

3. Step 3: Correct approach

   apply(m, 1, function(x) if(sum(x) > 10) mean(x) else NA)[!is.na(.)] returns NA for rows with sum ≤ 10, then removes NAs with [!is.na(.)], giving only desired means.

4. Final Answer:

   apply(m, 1, function(x) if(sum(x) > 10) mean(x) else NA)[!is.na(.)] -> Option A

5. Quick Check:

   Filter rows by sum > 10, then mean [OK]

Hint: Use if-else inside apply and remove NA after [OK]

Common Mistakes:

• Using margin=2 instead of 1 for rows
• Not handling rows with sum ≤ 10 properly
• Trying to filter inside mean() incorrectly