Recursive functions in R Programming - Time & Space Complexity

Time Complexity: Recursive functions

O(n)

Understanding Time Complexity

When we use recursive functions, the program calls itself repeatedly. We want to know how the time it takes grows as the input gets bigger.

How many times does the function call itself and how does that affect the total work done?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

factorial <- function(n) {
  if (n <= 1) {
    return(1)
  } else {
    return(n * factorial(n - 1))
  }
}
    

This function calculates the factorial of a number by calling itself with smaller numbers until it reaches 1.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The function calls itself once per step, reducing n by 1 each time.
• How many times: It repeats this call n times until it reaches the base case.

How Execution Grows With Input

Each time we increase n by 1, the function does one more call. So the work grows steadily with n.

Input Size (n)	Approx. Operations
10	About 10 calls
100	About 100 calls
1000	About 1000 calls

Pattern observation: The number of calls grows directly with n, so doubling n doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time taken grows in a straight line with the size of the input number.

Common Mistake

[X] Wrong: "Recursive functions always take more time than loops because they call themselves multiple times."

[OK] Correct: Some recursive functions call themselves only once per step, just like a loop. So their time grows linearly, not more.

Interview Connect

Understanding how recursion affects time helps you explain your code clearly and choose the best approach. It shows you can think about how programs grow with input size.

Self-Check

"What if the recursive function called itself twice each time, like in a Fibonacci calculation? How would the time complexity change?"

Practice

1. What is the main purpose of a base case in a recursive function?

easy

A. To stop the recursion and prevent infinite calls

B. To call the function again with the same input

C. To increase the input size for the next call

D. To print the result of the recursion

Solution

1. Step 1: Understand recursion basics

   A recursive function calls itself repeatedly with smaller inputs.

2. Step 2: Identify the role of base case

   The base case stops the recursion to avoid infinite loops and eventual crash.

3. Final Answer:

   To stop the recursion and prevent infinite calls -> Option A

4. Quick Check:

   Base case = stop recursion [OK]

Hint: Base case always stops recursion to avoid infinite loops [OK]

Common Mistakes:

• Forgetting the base case causes infinite recursion
• Thinking base case calls function again
• Confusing base case with recursive step

2. Which of the following is the correct syntax to define a recursive function fact in R that calculates factorial?

easy

A. fact <- function(n) { repeat n * fact(n - 1) }

B. fact <- function(n) { while(n > 0) n * fact(n - 1) }

C. fact <- function(n) { if(n == 1) n + fact(n - 1) }

D. fact <- function(n) { if(n == 0) 1 else n * fact(n - 1) }

Solution

1. Step 1: Check base case and recursive call

   fact <- function(n) { if(n == 0) 1 else n * fact(n - 1) } correctly uses if condition for base case (n == 0) and recursive call with n-1.

2. Step 2: Identify syntax errors in other options

   Options B, C, and D misuse loops or have wrong base case or missing else.

3. Final Answer:

   fact <- function(n) { if(n == 0) 1 else n * fact(n - 1) } -> Option D

4. Quick Check:

   Correct base case and recursive call syntax [OK]

Hint: Base case uses if, recursive call uses else with smaller input [OK]

Common Mistakes:

• Using loops instead of recursion
• Missing else leads to errors
• Wrong base case condition

3. What is the output of this R code?

fib <- function(n) {
  if (n <= 1) return(n)
  else return(fib(n-1) + fib(n-2))
}
fib(4)

medium

A. 5

B. 4

C. 3

D. 7

Solution

1. Step 1: Trace fib(4) calls

   fib(4) = fib(3) + fib(2); fib(3) = fib(2) + fib(1); fib(2) = fib(1) + fib(0)

2. Step 2: Calculate values bottom-up

   fib(1)=1, fib(0)=0; fib(2)=1+0=1; fib(3)=1+1=2; fib(4)=2+1=3

3. Final Answer:

   3 -> Option C

4. Quick Check:

   Fibonacci(4) = 3 [OK]

Hint: Fibonacci(4) = 3 by adding previous two values [OK]

Common Mistakes:

• Confusing index starting at 0 or 1
• Adding wrong recursive calls
• Forgetting base cases

4. Identify the error in this recursive R function to sum numbers from 1 to n:

sum_to_n <- function(n) {
  if (n == 1) return(1)
  else return(n + sum_to_n(n))
}
sum_to_n(3)

medium

A. The recursive call uses the same n, causing infinite recursion

B. The function should use a loop instead

C. The base case is missing

D. The return statement is outside the function

Solution

1. Step 1: Check recursive call argument

   The function calls sum_to_n(n) instead of sum_to_n(n-1), so input never decreases.

2. Step 2: Understand effect of error

   This causes infinite recursion because n stays the same and base case is never reached.

3. Final Answer:

   The recursive call uses the same n, causing infinite recursion -> Option A

4. Quick Check:

   Recursive call must reduce input [OK]

Hint: Recursive call must use smaller input to reach base case [OK]

Common Mistakes:

• Calling function with same argument
• Missing base case
• Confusing recursion with loops

5. Write a recursive R function flatten_list that takes a nested list of numbers and returns a flat vector of all numbers.
What is the correct approach?

hard

A. Use recursion but forget to combine results from recursive calls

B. Check if input is a list; if yes, recursively flatten each element and combine; else return the element as vector

C. Return the input directly without any processing

D. Use a for loop to iterate and append elements without recursion

Solution

1. Step 1: Identify recursive structure

   Nested lists require checking each element; if it's a list, call flatten_list recursively.

2. Step 2: Combine results properly

   Combine all flattened elements into one vector to get a flat list of numbers.

3. Final Answer:

   Check if input is a list; if yes, recursively flatten each element and combine; else return the element as vector -> Option B

4. Quick Check:

   Recursion + combine results = flatten list [OK]

Hint: Recursively flatten each list element and combine results [OK]

Common Mistakes:

• Not combining recursive results
• Forgetting to check if element is a list
• Using loops only without recursion