Question 44

Given the basic block statements:

1. a = b + c

2. c = a + d

3. d = b + c

4. e = d - b

5. a = e + b

6. Trace Expression Values:

   • Initial values of variables: $b_0, c_0, d_0$.
   • From statement 1: $a_1 = b_0 + c_0$.
   • From statement 2: $c_1 = a_1 + d_0 = (b_0 + c_0) + d_0$.
   • From statement 3: $d_1 = b_0 + c_1 = b_0 + ((b_0 + c_0) + d_0)$.
   • From statement 4: $e_1 = d_1 - b_0 = (b_0 + c_1) - b_0 = c_1$.
     Thus $e_1$ evaluates identically to $c_1$ (pointing to the existing node for $c_1$).
   • From statement 5: $a_2 = e_1 + b_0 = c_1 + b_0 = b_0 + c_1 = d_1$.
     Thus $a_2$ evaluates identically to $d_1$ (pointing to the existing node for $d_1$).

7. DAG Structure:

   • Leaf nodes (3 nodes): $b_0, c_0, d_0$.
   • Interior operator nodes (3 nodes):
     • Node $n_1$ (+): left child $b_0$, right child $c_0$ (2 edges). Represents $a_1$.
     • Node $n_2$ (+): left child $n_1$, right child $d_0$ (2 edges). Represents $c_1$ and $e_1$.
     • Node $n_3$ (+): left child $b_0$, right child $n_2$ (2 edges). Represents $d_1$ and $a_2$.

8. Counts:

   • Total nodes = $3\text{ (leaves)} + 3\text{ (interior)} = 6\text{ nodes}$.
   • Total edges = $2 + 2 + 2 = 6\text{ edges}$.

This matches option A (6 and 6).