Recursive function in assembly in ARM Architecture - Time & Space Complexity

Time Complexity: Recursive function in assembly

O(n)

Understanding Time Complexity

When we look at recursive functions in assembly, we want to understand how the number of steps grows as the input changes.

We ask: How many times does the function call itself and what does that mean for performance?

Scenario Under Consideration

Analyze the time complexity of the following ARM assembly recursive function.

    factorial:
      cmp r0, #1
      ble end_factorial
      push {lr}
      mov r1, r0
      sub r0, r0, #1
      bl factorial
      pop {lr}
      mul r0, r0, r1
    end_factorial:
      bx lr
    

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

Identify Repeating Operations

Look for repeated actions in the code.

• Primary operation: The function calls itself recursively.
• How many times: It calls itself once for each number down to 1.

How Execution Grows With Input

Each time the input number increases, the function calls itself more times.

Input Size (n)	Approx. Operations
5	5 calls
10	10 calls
20	20 calls

Pattern observation: The number of calls grows directly with the input number.

Final Time Complexity

Time Complexity: O(n)

This means the time to finish grows in a straight line as the input number gets bigger.

Common Mistake

[X] Wrong: "Recursive functions always take much longer than loops."

[OK] Correct: Some recursive functions, like this factorial, take time proportional to input size, just like loops do.

Interview Connect

Understanding how recursion affects time helps you explain your code clearly and shows you know how programs behave as inputs grow.

Self-Check

"What if the function called itself twice each time instead of once? How would the time complexity change?"

Practice

1. What is the main purpose of a recursive function in ARM assembly?

easy

A. To execute instructions only once

B. To call itself to solve smaller parts of a problem

C. To avoid using the stack

D. To run faster than iterative loops

Solution

1. Step 1: Understand recursion concept

   A recursive function calls itself to break down a problem into smaller parts.

2. Step 2: Apply to ARM assembly context

   In ARM assembly, recursion is used to solve problems by repeated self-calls with updated parameters.

3. Final Answer:

   To call itself to solve smaller parts of a problem -> Option B

4. Quick Check:

   Recursion means self-calling function [OK]

Hint: Recursion means function calls itself repeatedly [OK]

Common Mistakes:

• Thinking recursion avoids stack usage
• Confusing recursion with iteration
• Assuming recursion always runs faster

2. Which ARM assembly instruction is commonly used to return from a recursive function call?

easy

A. BL

B. MOV

C. BX LR

D. PUSH

Solution

1. Step 1: Identify return instruction

   In ARM assembly, BX LR is used to branch back to the address stored in the link register, effectively returning from a function.

2. Step 2: Confirm usage in recursion

   Recursive functions use BX LR to return after completing each call.

3. Final Answer:

   BX LR -> Option C

4. Quick Check:

   Return uses BX LR instruction [OK]

Hint: Return from function with BX LR instruction [OK]

Common Mistakes:

• Using BL to return instead of call
• Confusing PUSH with return
• Using MOV incorrectly for return

3. Consider this ARM assembly snippet for a recursive factorial function:

factorial:
    CMP R0, #1
    BLE end_factorial
    PUSH {LR}
    SUB R0, R0, #1
    BL factorial
    POP {LR}
    MUL R0, R0, R1
end_factorial:
    MOV R1, R0
    BX LR

What will be the value in R1 after calling factorial with R0 = 3?

medium

A. 0

B. 3

C. 1

D. 6

Solution

1. Step 1: Trace the recursive factorial

   The function saves LR, decrements R0, calls factorial recursively, restores LR, then multiplies R0 by R1.

2. Step 2: Calculate factorial for R0=3

   Call factorial(3): R0=3, not base case, PUSH LR, SUB R0=2, BL factorial(2). factorial(2): PUSH LR, SUB R0=1, BL factorial(1). factorial(1): base case, sets R1=1, returns. factorial(2): POP LR, MUL R0=1 * R1=1 =1, MOV R1=1, return. factorial(3): POP LR, MUL R0=2 * R1=1=2, MOV R1=2, return.

3. Step 3: Correction

   However, the MUL uses R0 and R1, but R0 was decremented before recursive call, so the multiplication is (n-1) * fact(n-1), which is incorrect for factorial.

4. Step 4: Final value

   Tracing carefully, the final R1 will be 2, which is incorrect factorial of 3 (should be 6). But the code as given will produce 2.

5. Final Answer:

   2 -> Option D

Hint: SUB decrements R0 before call; MUL uses decremented R0 causing incorrect factorial [Check carefully]

Common Mistakes:

• Assuming correct factorial computation (expecting 6)
• Confusing R0 and R1 values
• Missing base case return value

4. Identify the error in this recursive ARM assembly function snippet:

factorial:
    CMP R0, #1
    BLE end_factorial
    SUB R0, R0, #1
    BL factorial
    MUL R0, R0, R1
end_factorial:
    MOV R1, R0
    BX LR

What is the main problem?

medium

A. Missing PUSH and POP of LR causing return address loss

B. Incorrect base case comparison

C. Wrong multiplication order

D. Using MOV instead of MOVS

Solution

1. Step 1: Check stack usage for recursion

   Recursive calls must save LR on stack before calling to preserve return address.

2. Step 2: Identify missing PUSH/POP

   This code lacks PUSH {LR} before BL and POP {LR} after, risking corrupted return address.

3. Final Answer:

   Missing PUSH and POP of LR causing return address loss -> Option A

4. Quick Check:

   Save LR on stack in recursion [OK]

Hint: Always save LR on stack before recursive call [OK]

Common Mistakes:

• Ignoring stack save for LR
• Assuming base case is wrong
• Confusing MOV with MOVS

5. You want to write a recursive ARM assembly function to compute the nth Fibonacci number. Which of these is essential to correctly implement recursion here?

hard

A. Handle base cases to stop recursion and save LR on stack before recursive calls

B. Use only registers without stack to speed up recursion

C. Avoid base cases to allow infinite recursion for all n

D. Use unconditional jumps instead of BL for recursive calls

Solution

1. Step 1: Importance of base cases

   Base cases stop recursion to prevent infinite loops, essential for Fibonacci calculation.

2. Step 2: Save LR on stack before recursive calls

   Saving LR preserves return addresses during multiple recursive calls, avoiding crashes.

3. Final Answer:

   Handle base cases to stop recursion and save LR on stack before recursive calls -> Option A

4. Quick Check:

   Base cases + LR save = correct recursion [OK]

Hint: Always include base cases and save LR in recursion [OK]

Common Mistakes:

• Skipping base cases causing infinite recursion
• Not saving LR causing return errors
• Using jumps instead of BL for calls