REDUCED ORDERED BINARY DECISION DIAGRAM

(ROBDD)

Devi Sivaraman12ECP026

M.Tech-VLSI

Binary Decision Diagrams

• Boolean function of m variables defines a Boolean space of 2m points

• The data structure to represent Boolean function should be compact and easy to manipulate

• A binary decision diagram (BDD) is a data structure that is used to represent a Boolean function, directly derivable from Shannon’s expansion

Shannon’s Expansion• Shannon's expansion is a method by which a Boolean

function can be represented by the sum of two sub-functions of the original

• For Example:

BINARY TREE REPRESENTATION

11

1

0

0

0

x

yy

( ) ( )

x x

xy x y xy x y

f xf x f

x yf y f x yf y f

xyfx yfxy

fx yf

ROBDD Flow

Example 1: XOR

Truth Table Binary Decision Tree

A B F

0 0 0

0 1 1

1 0 1

1 1 0

A

BB

0 1 1 0

0

0

1

1 0 1

Binary Decision Diagram

OBDDOrdered BDD (OBDD) : Input variables are ordered – each path

from root to sink visits nodes with labels (variables) in ascending order.

a

c c

b

0 1

orderedorder = a,c,b

a

b c

c

0 1

notordered

b

Ordered Binary Decision Tree (OBDT)

• This graph representation is called OBDT or OBDD (Ordered binary Decision Diagram) which has a directed tree structure

• Each vertex has two children. Two edges originated from a vertex are called high (positive co-factor) & low (negative co-factor)

OBDT to ROBDD ROBDD can be obtained from an OBDT (Ordered Binary

Decision Tree) by repeatedly applying following reduction rules (until none of the them can be applied anymore):

• Remove duplicate terminal (leaf) nodes• Remove duplicate non-terminal (internal) nodes• Remove nodes with redundant tests

Reduction rules of OBDD

Elimination rule

Reduction Rule

OBDT to ROBDD Rule 1: Collapse Leaf Nodes to remove redundant tests

It’s no longer a tree

OBDT to ROBDD Rule 2: Isomorphic sub-graphs

OBDT to ROBDD Final Representation

BDDs 15

ROBDDs• Directed acyclic graph (DAG)• One root node per function, two terminals 0, 1• Each node, two children, and a variable• Shannon co-factoring tree, except reduced and ordered

(ROBDD)Reduced:– any node with two identical children is removed– two nodes with isomorphic BDD’s are merged

Ordered: Co-factoring variables (splitting variables) always

follow the same order from a root to a terminalxi1

< xi2 < xi3

< … < xin

Example 2Binary Decision Tree

16

a c

b

f

a

b b

cccc

0 0 1 0 0 1 1 1

0 1

0 0 0

01

11 1

1

1

0

0Graph representation of a Boolean function.

Leaf nodes

Ordered Binary Decision Diagram (OBDD)

17

a c

b

f

a

b b

cc

0 1 0 0 1 1

0 1

0 0

01

1 1

10a

b b

cccc

0 0 1 0 0 1 1 1

0 1

0 0 0

01

11 1

1

1

0

0

TreeOBDD

OBDD With Different Input Ordering

18

a c

b

f

a

b b

cc

0 1 0 0 1 1

0 1

0 0

01

1 1

10

c

b b

a

0 1 0 1 1

0 1

0

0

1

1

0 a

0

1 0 1

Reduction: OBDD to ROBDD

19

a c

b

f

a

b b

cc

0 1 0 0 1 1

0 1

0 0

01

1 1

10

a

10

cc

b b

0

00 0

1

1

1 1

1

0

Properties of ROBDD• The 0- and 1-succesor of any node should not be identical• ROBDDs provide canonical representation of switching

functions• ROBDDs can be manipulated efficiently• ROBDDs representations are small for many important

switching functions• Isomorphic graph should not be there i.e. sub-graphs that

yields same information about function several times

Thank You