Stack operations (PUSH, POP) in ARM Architecture - Time & Space Complexity

Time Complexity: Stack operations (PUSH, POP)

O(1)

Understanding Time Complexity

We want to understand how the time to add or remove items from a stack changes as the stack grows.

How does the time needed for PUSH and POP operations change with the number of items?

Scenario Under Consideration

Analyze the time complexity of the following ARM assembly code for PUSH and POP.

    PUSH {r0}       // Save register r0 on the stack
    POP {r0}        // Restore register r0 from the stack
    

This code saves a register value onto the stack and then restores it back.

Identify Repeating Operations

In this example, each PUSH or POP is a single instruction without loops or recursion.

• Primary operation: Single memory store (PUSH) or load (POP)
• How many times: Once per operation

How Execution Grows With Input

Each PUSH or POP takes the same amount of time no matter how many items are already on the stack.

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

Pattern observation: Time stays constant regardless of stack size.

Final Time Complexity

Time Complexity: O(1)

This means each PUSH or POP operation takes the same fixed time no matter how big the stack is.

Common Mistake

[X] Wrong: "PUSH and POP take longer as the stack grows because more items are involved."

[OK] Correct: Each operation only touches the top item, so the stack size does not affect the time.

Interview Connect

Knowing that stack operations run in constant time helps you explain efficient memory management and function calls in real programs.

Self-Check

"What if the stack was implemented using a linked list instead of memory addresses? How would the time complexity change?"

Practice

1. What does the PUSH instruction do in ARM architecture stack operations?

easy

A. It adds registers to the stack by decreasing the stack pointer.

B. It removes registers from the stack by increasing the stack pointer.

C. It clears the stack completely.

D. It duplicates the top value of the stack.

Solution

1. Step 1: Understand the role of PUSH

   PUSH adds data to the stack, which means it stores registers by moving the stack pointer downwards.

2. Step 2: Compare with POP

   POP removes data by moving the stack pointer upwards, opposite to PUSH.

3. Final Answer:

   It adds registers to the stack by decreasing the stack pointer. -> Option A

4. Quick Check:

   PUSH = add to stack [OK]

Hint: PUSH means add data, stack pointer moves down [OK]

Common Mistakes:

• Confusing PUSH with POP
• Thinking PUSH increases stack pointer
• Assuming PUSH clears the stack

2. Which of the following is the correct syntax to push registers R0 and R1 onto the stack in ARM assembly?

easy

A. PUSH [R0, R1]

B. PUSH (R0, R1)

C. PUSH {R0, R1}

D. PUSH R0 R1

Solution

1. Step 1: Recall ARM PUSH syntax

   ARM uses curly braces to list registers for PUSH, like PUSH {R0, R1}.

2. Step 2: Identify incorrect syntax

   Parentheses, spaces without braces, or square brackets are not valid for PUSH in ARM.

3. Final Answer:

   PUSH {R0, R1} -> Option C

4. Quick Check:

   Curly braces list registers in PUSH [OK]

Hint: Use curly braces {} to list registers in PUSH [OK]

Common Mistakes:

• Using parentheses instead of braces
• Listing registers without any brackets
• Using square brackets which are invalid

3. Given the following ARM instructions:
SUB SP, SP, #8
PUSH {R4, R5}
POP {R6, R7}
What happens to the stack pointer (SP) after these operations?

medium

A. SP decreases by 8, then decreases by 8, then increases by 8

B. SP decreases by 8, then increases by 8, then decreases by 8

C. SP decreases by 8, then decreases by 8, then increases by 16

D. SP decreases by 8, then decreases again by 8, then increases by 8

Solution

1. Step 1: Analyze SUB SP, SP, #8

   This subtracts 8 from SP, moving it down by 8 bytes.

2. Step 2: Understand PUSH {R4, R5}

   PUSH stores two registers, so SP decreases by another 8 bytes (4 bytes each register).

3. Step 3: Understand POP {R6, R7}

   POP removes two registers, so SP increases by 8 bytes.

4. Final Answer:

   SP decreases by 8, then decreases by 8, then increases by 8 -> Option A

5. Quick Check:

   SP moves down for SUB and PUSH, up for POP [OK]

Hint: PUSH decreases SP, POP increases SP by register size times count [OK]

Common Mistakes:

• Forgetting SUB changes SP before PUSH
• Assuming POP decreases SP
• Miscounting bytes per register

4. Identify the error in this ARM assembly snippet:
PUSH R0, R1
POP {R2, R3}

medium

A. PUSH cannot push more than one register at a time.

B. PUSH syntax is incorrect; registers must be in braces.

C. POP syntax is incorrect; registers must be in parentheses.

D. POP cannot pop into different registers.

Solution

1. Step 1: Check PUSH syntax

   PUSH requires registers to be enclosed in curly braces, so PUSH R0, R1 is invalid.

2. Step 2: Check POP syntax

   POP syntax with curly braces is correct, so POP {R2, R3} is valid.

3. Final Answer:

   PUSH syntax is incorrect; registers must be in braces. -> Option B

4. Quick Check:

   PUSH needs braces { } [OK]

Hint: Always use braces {} for multiple registers in PUSH/POP [OK]

Common Mistakes:

• Using commas without braces in PUSH
• Confusing parentheses with braces
• Thinking POP cannot pop multiple registers

5. You want to save registers R0, R1, and R2 on the stack, then restore them later. Which sequence correctly uses PUSH and POP to do this without corrupting the stack?

hard

A. PUSH {R2, R1, R0}...POP {R0, R1, R2}

B. PUSH {R0, R1, R2}...POP {R0, R1, R2}

C. PUSH {R0, R1, R2}...POP {R1, R2, R0}

D. PUSH {R0, R1, R2}...POP {R2, R1, R0}

Solution

1. Step 1: Understand stack order with PUSH

   PUSH stores registers in order: R0, then R1, then R2, with R2 at the lowest address.

2. Step 2: POP must restore in reverse order

   POP restores registers in reverse order of PUSH to correctly restore the original values.

3. Step 3: Check options

   PUSH {R0, R1, R2}
   ...
   POP {R2, R1, R0} pops in reverse order as pushed, preserving correct values.

4. Final Answer:

   PUSH {R0, R1, R2} ... POP {R2, R1, R0} -> Option D

5. Quick Check:

   POP order is reverse of PUSH order to restore correctly [OK]

Hint: POP registers in reverse order of PUSH to restore correctly [OK]

Common Mistakes:

• Reversing POP order incorrectly
• Changing register order between PUSH and POP
• Assuming POP order does not matter