Snowflake SnowPro Core

Explanation

This question is about register allocation in compiler optimization, specifically determining the minimum number of registers needed for the given code sequence. Let's analyze the code:

t1 := a - mem[...]
t2 := d / mem[...]
t3 := t2 + mem[...]
t4 := t1 + t3

t1 := a - mem[...]
t2 := d / mem[...]
t3 := t2 + mem[...]
t4 := t1 + t3

We need to track the live ranges of variables:

1. Line 1: t1 is defined using a and mem[...]. After this line, t1 is live.
2. Line 2: t2 is defined using d and mem[...]. After this line, both t1 and t2 are live.
3. Line 3: t3 is defined using t2 and mem[...]. After this line, t1 and t3 are live (since t2 is no longer needed after being used to compute t3).
4. Line 4: t4 is defined using t1 and t3. After this line, only t4 is live.

Register usage analysis:

• At line 2, we need registers for: t1, a, d, and two memory locations (but memory locations don't need registers if accessed directly)
• At line 3, we need registers for: t1, t2, and a memory location
• At line 4, we need registers for: t1, t3

The key insight is that we can reuse registers when variables are no longer live:

1. Register 1: holds t1 (from line 1 to line 4)
2. Register 2: holds t2 (from line 2 to line 3)
3. Register 3: holds t3 (from line 3 to line 4)
4. Register 4: used for temporary values (memory operands, intermediate results)

However, we can optimize further:

• After line 3, t2 is no longer needed, so its register can be reused
• We need to hold t1 and t3 simultaneously for line 4
• We also need registers for the source operands in each operation

Minimum registers calculation: Using graph coloring/register allocation:

1. Build interference graph
2. Variables that are live simultaneously cannot share the same register
3. Maximum number of simultaneously live variables determines minimum registers needed

Live variable analysis:

• After line 1: {t1}
• After line 2: {t1, t2}
• After line 3: {t1, t3}
• After line 4: {t4}

Maximum simultaneously live variables = 2 (either {t1, t2} or {t1, t3})

However, we also need to consider that operations require registers for their operands. Looking at the operations:

1. t1 := a - mem[...] - needs registers for a and memory operand
2. t2 := d / mem[...] - needs registers for d and memory operand
3. t3 := t2 + mem[...] - needs registers for t2 and memory operand
4. t4 := t1 + t3 - needs registers for t1 and t3

Since memory operands can be accessed directly without needing registers (in many architectures), and a and d are source values that might be in registers or memory, the critical constraint is that we need to hold t1 and t3 simultaneously for the final addition.

Therefore, the minimum number of registers needed is 4 (Option C).

This allows us to:

• Have one register for t1
• One register for t2 (which becomes t3)
• One register for temporary values
• One register for the result t4

Why not fewer?

• 2 registers (Option E): Not enough because we need to hold t1 and t3 simultaneously
• 3 registers (Option D): Might work with careful scheduling, but typically insufficient for all operations
• 4 registers (Option C): Minimum practical number
• 5 or 6 registers (Options B and A): More than the minimum required