Recursion on Arrays and Strings in DSA C++ - Time & Space Complexity

Time Complexity: Recursion on Arrays and Strings

O(n)

Understanding Time Complexity

When we use recursion on arrays or strings, we call the same function repeatedly on smaller parts.

We want to know how the total work grows as the input size grows.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int sumArray(int arr[], int n) {
  if (n == 0) return 0;
  return arr[n - 1] + sumArray(arr, n - 1);
}
    

This function calculates the sum of all elements in the array using recursion.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive call reducing array size by 1 each time.
• How many times: Exactly n times, once for each element.

How Execution Grows With Input

Each recursive call does a small fixed amount of work plus calls itself on a smaller array.

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

Pattern observation: The work grows directly with input size; doubling input doubles work.

Final Time Complexity

Time Complexity: O(n)

This means the time to complete grows in a straight line with the size of the array.

Common Mistake

[X] Wrong: "Recursion here is faster because it skips some elements."

[OK] Correct: Each element is still processed once, so recursion does not skip work; it just organizes it differently.

Interview Connect

Understanding recursion time helps you explain your code clearly and choose the best approach in interviews.

Self-Check

"What if the recursion called itself twice each time, like in a binary tree traversal? How would the time complexity change?"

Practice

1. What does a recursive function on an array typically do to process the array elements?

easy

A. It processes one element and calls itself on the rest of the array

B. It processes all elements at once without calling itself

C. It uses loops instead of calling itself

D. It only processes the last element of the array

Solution

1. Step 1: Understand recursion on arrays

   A recursive function on an array usually handles one element and then calls itself on the remaining elements.

2. Step 2: Compare options

   It processes one element and calls itself on the rest of the array matches this behavior. Options A, C, and D describe non-recursive or incorrect methods.

3. Final Answer:

   It processes one element and calls itself on the rest of the array -> Option A

4. Quick Check:

   Recursion on arrays = process one element + recurse [OK]

Hint: Remember recursion breaks problem into smaller parts [OK]

Common Mistakes:

• Thinking recursion processes all elements at once
• Confusing recursion with loops
• Assuming recursion only processes last element

2. Which of the following is the correct base case for a recursive function that sums elements of an integer array arr of size n?

easy

A. if (n == 0) return 0;

B. if (n == 1) return arr[1];

C. if (arr == NULL) return 0;

D. if (n < 0) return 0;

Solution

1. Step 1: Identify base case for recursion on array size

   The base case should stop recursion when no elements are left, which is when n == 0.

2. Step 2: Check each option

   if (n == 0) return 0; returns 0 when n == 0, which is correct. if (n == 1) return arr[1]; incorrectly accesses arr[1] instead of arr[0]. if (arr == NULL) return 0; checks pointer but not size. if (n < 0) return 0; uses invalid negative size.

3. Final Answer:

   if (n == 0) return 0; -> Option A

4. Quick Check:

   Base case for empty array = return 0 [OK]

Hint: Base case usually when size is zero [OK]

Common Mistakes:

• Using wrong index in base case
• Checking pointer instead of size
• Allowing negative sizes

3. What is the output of the following C++ code?

int sumArray(int arr[], int n) {
  if (n == 0) return 0;
  return arr[n-1] + sumArray(arr, n-1);
}

int main() {
  int arr[] = {1, 2, 3, 4};
  int result = sumArray(arr, 4);
  std::cout << result;
  return 0;
}

medium

A. Compilation error

B. 24

C. 0

D. 10

Solution

1. Step 1: Trace recursive calls

   sumArray(arr,4) = arr[3] + sumArray(arr,3) = 4 + sumArray(arr,3) sumArray(arr,3) = arr[2] + sumArray(arr,2) = 3 + sumArray(arr,2) sumArray(arr,2) = arr[1] + sumArray(arr,1) = 2 + sumArray(arr,1) sumArray(arr,1) = arr[0] + sumArray(arr,0) = 1 + 0 = 1

2. Step 2: Calculate total sum

   Total = 4 + 3 + 2 + 1 = 10

3. Final Answer:

   10 -> Option D

4. Quick Check:

   Sum of {1,2,3,4} = 10 [OK]

Hint: Sum recursion adds last element + sum of rest [OK]

Common Mistakes:

• Confusing index n-1 with n
• Forgetting base case returns 0
• Assuming sum is product

4. Find the error in this recursive function that counts occurrences of character c in string s:

int countChar(string s, char c, int index) {
  if (index == s.length()) return 0;
  if (s[index] == c) return 1 + countChar(s, c, index++);
  else return countChar(s, c, index++);
}

medium

A. Missing check for empty string

B. Base case condition is wrong

C. Using post-increment (index++) causes infinite recursion

D. Function should return bool, not int

Solution

1. Step 1: Analyze recursive calls with index++

   Using index++ passes the current index value before increment, so recursive call uses same index repeatedly.

2. Step 2: Identify infinite recursion cause

   Because index does not increase in recursive calls, base case is never reached, causing infinite recursion.

3. Final Answer:

   Using post-increment (index++) causes infinite recursion -> Option C

4. Quick Check:

   Post-increment in recursion argument = infinite loop [OK]

Hint: Use pre-increment (++index) or index+1 in recursion call [OK]

Common Mistakes:

• Using post-increment in recursive call argument
• Ignoring base case correctness
• Confusing return types

5. Given a string s, write a recursive function to check if it is a palindrome (reads same forward and backward). Which approach below correctly implements this?

hard

A. Reverse string using loop and compare with original

B. Check if first and last chars match, then recurse on substring excluding them

C. Count matching characters from start only

D. Use recursion to count characters and compare counts

Solution

1. Step 1: Understand palindrome check recursively

   To check palindrome recursively, compare first and last characters; if they match, recurse on substring excluding them.

2. Step 2: Evaluate options

   Check if first and last chars match, then recurse on substring excluding them matches this logic. Reverse string using loop and compare with original uses loops, not recursion. Options C and D do not correctly check palindrome property.

3. Final Answer:

   Check if first and last chars match, then recurse on substring excluding them -> Option B

4. Quick Check:

   Palindrome recursion = compare ends + recurse inside [OK]

Hint: Compare ends, recurse inside substring [OK]

Common Mistakes:

• Using loops instead of recursion
• Only checking one side of string
• Counting chars instead of comparing