Reduced ordered binary decision diagram

1. REDUCED ORDERED BINARY
DECISION DIAGRAM
(ROBDD)
Devi Sivaraman
12ECP026
M.Tech-VLSI

2. 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

3. 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:

4. 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 y
fxy
fx y
f

5. ROBDD Flow

6. 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

7. Binary Decision Diagram

8. OBDD
Ordered 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
ordered
order = a,c,b
a
b c
c
0 1
not
ordered
b

9. 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)

10. 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

11. Reduction rules of OBDD
Elimination rule
Reduction Rule

12. OBDT to ROBDD
Rule 1: Collapse Leaf Nodes to remove redundant tests
It’s no longer a tree

13. OBDT to ROBDD
Rule 2: Isomorphic sub-graphs

14. OBDT to ROBDD
Final Representation

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 terminal
xi1
< xi2
< xi3
< … < xin
BDDs 15

16. Example 2
Binary 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

17. 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
1
0
a
b b
cccc
0 0 1 0 0 1 1 1
0 1
0 0 0
01
11 1
1
1
0
0
Tree
OBDD

18. 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
1
0
c
b b
a
0 1 0 1 1
0 1
0
0
1
1
0 a
0
1 0 1

19. 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
1
0
a
10
cc
b b
0
0
0 0
1
1
1 1
1
0

20. 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

21. Thank You