# Directed Acyclic Graphs | DAGs | Examples

Directed Acyclic Graph-

 Directed Acyclic Graph (DAG) is a special kind of Abstract Syntax Tree.

• Each node of it contains a unique value.
• It does not contain any cycles in it, hence called Acyclic.

## Optimization Of Basic Blocks-

 DAG is a very useful data structure for implementing transformations on Basic Blocks.

• A DAG is constructed for optimizing the basic block.
• A DAG is usually constructed using Three Address Code.
• Transformations such as dead code elimination and common sub expression elimination are then applied.

## Properties-

• Reachability relation forms a partial order in DAGs.
• Both transitive closure & transitive reduction are uniquely defined for DAGs.
• Topological Orderings are defined for DAGs.

## Applications-

DAGs are used for the following purposes-

• To determine the expressions which have been computed more than once (called common sub-expressions).
• To determine the names whose computation has been done outside the block but used inside the block.
• To determine the statements of the block whose computed value can be made available outside the block.
• To simplify the list of Quadruples by not executing the assignment instructions x:=y unless they are necessary and eliminating the common sub-expressions.

## Construction of DAGs-

Following rules are used for the construction of DAGs-

### Rule-01:

In a DAG,

• Interior nodes always represent the operators.
• Exterior nodes (leaf nodes) always represent the names, identifiers or constants.

### Rule-02:

While constructing a DAG,

• A check is made to find if there exists any node with the same value.
• A new node is created only when there does not exist any node with the same value.
• This action helps in detecting the common sub-expressions and avoiding the re-computation of the same.

### Rule-03:

The assignment instructions of the form x:=y are not performed unless they are necessary.

## Problem-01:

Consider the following expression and construct a DAG for it-

( a + b ) x ( a + b + c )

## Solution-

Three Address Code for the given expression is-

T1 = a + b

T2 = T1 + c

T3 = T1 x T2

Now, Directed Acyclic Graph is-

## NOTE

From the constructed DAG, we observe-

• The common sub-expression (a+b) has been expressed into a single node in the DAG.
• The computation is carried out only once and stored in the identifier T1 and reused later.

This illustrates how the construction scheme of a DAG identifies the common sub-expression and helps in eliminating its re-computation later.

## Problem-02:

Consider the following expression and construct a DAG for it-

( ( ( a + a ) + ( a + a ) ) + ( ( a + a ) + ( a + a ) ) )

## Solution-

Directed Acyclic Graph for the given expression is-

## Problem-03:

Consider the following block and construct a DAG for it-

(1) a = b x c

(2) d = b

(3) e = d x c

(4) b = e

(5) f = b + c

(6) g = f + d

## Solution-

Directed Acyclic Graph for the given block is-

## Problem-04:

Optimize the block in the Problem-03.

## Solution-

### Step-01:

Firstly, construct a DAG for the given block (already done above).

### Step-02:

Now, the optimized block can be generated by traversing the DAG.

• The common sub-expression e = d x c which is actually b x c (since d = b) is eliminated.
• The dead code b = e is eliminated.

The optimized block is-

(1) a = b x c

(2) d = b

(3) f = a + c

(4) g = f + d

## Problem-05:

Consider the following basic block-

B10:

S1 = 4 x I

S3 = S2[S1]

S4 = 4 x I

S6 = S5[S4]

S7 = S3 x S6

S8 = PROD + S7

PROD = S8

S9 = I + 1

I = S9

If I <= 20 goto L10

1. Draw a directed acyclic graph and identify local common sub-expressions.
2. After eliminating the common sub-expressions, re-write the basic block.

## Solution-

Directed Acyclic Graph for the given basic block is-

In this code fragment,

• 4 x I is a common sub-expression. Hence, we can eliminate because S1 = S4.
• We can optimize S8 = PROD + S7 and PROD = S8 as PROD = PROD + S7.
• We can optimize S9 = I + 1 and I = S9 as I = I + 1.

After eliminating S4, S8 and S9, we get the following basic block-

B10:

S1 = 4 x I

S3 = S2[S1]

S6 = S5[S1]

S7 = S3 x S6

PROD = PROD + S7

I = I + 1

If I <= 20 goto L10

To gain better understanding about Directed Acyclic Graphs,

Watch this Video Lecture

Next Article- Misc Problems On Directed Acyclic Graphs

Get more notes and other study material of Compiler Design.

Watch video lectures by visiting our YouTube channel LearnVidFun.

Summary
Article Name
Directed Acyclic Graphs | DAGs | Examples
Description
In Compiler design, Directed Acyclic Graph is a directed graph that does not contain any cycles in it. Directed Acyclic Graph Examples. Properties and Applications. Problems On Directed Acyclic Graphs.
Author
Publisher Name
Gate Vidyalay
Publisher Logo