Binary decision diagrams (BDD

1
Binary decision diagrams (BDDs)

• Compact representation of a logic function
• ROBDDs (reduced ordered BDDs) are a canonical
  representation equivalence of ROBDDs implies
  that the functions are identical
• Example f abc bd cd
• T then edge, E else edge
• Same variable ordering on
• each path a ? b ? c ? d
• (Ordered BDD)

Material taken mostly from G. Hachtel and F.
Somenzi, Logic Synthesis and Verification
Algorithms, Kluwer Academic Publishers,
Boston, MA, 1996.
2
Effect of variable ordering

• Size of diagram varies with variable ordering
• a ? d ? b ? c b ? c ? a ? d

b
a
E
T
E
T
d
d
c
E
T
E
T
T
b
b
E
a
E
E
E
T
T
d
T
c
c
T
E
E
E
T
T
1
0
1
0
3
Relation to the Shannon expansion

• Each node is basically a Shannon expansion
• f a fa a fa

f
fa
fa
4
Building a BDD from a Shannon expansion
f abc bd cd
b
T
E
fb ac cd
c
fbc d
T
fb d
E
fbc a
( and so on )
5
BDD as a compact truth table

• Truth table complete ordered binary tree
• Reduce this by combining isomorphic parts and
  removing redundant nodes (T,E point to same node)
  to get ROBDD

redundant
T
E
T
E
E
T
E
T
T
T
E
E
E
T
T
E
T
T
redundant
E
T
1
0
E
E
E
E
T
T
T
T
1
0
1
0
1
0
E
E
T
T
BDD shown earlier for the ordering b ? c ? a ? d
isomorphic
1
0
1
0
isomorphic
6
Multioutput BDDs

• (Notation solid line Then edge dashed line
  Else edge)
• Example F1 bc, F2 a bc

Combined Multioutput BDD
Separate BDDs
F2
F2
F1
F1
a
a
b
b
b
c
c
c
1
0
1
0
1
0
7
Compactness of BDDs

• BDDs are successful at compactly representing
  many common functions
• XOR is an example of a function with a large
  SOP/POS representation, but a very compact BDD
• Worst case O(2n) nodes
• Functions that require this many nodes do exist
• Can use multilevel techniques to represent BDDs
  more compactly (partitioned ROBDDs)

8
Operations on BDDs

• Given BDDs for functions f and g, can use
  Shannon expansion to see how operations are
  performed
• Assume variables v1, v2 vn
• f ltopgt g v1 (f ltopgt g)v1 v1 (f ltopgt g)v1
• v1 (fv1 ltopgt gv1) v1 (fv1 ltopgt gv1)
• Can now do this recursively
• Pictorially
• Identify identical subtrees as we come up the
  recursion tree using a hashing function

v1
E
T
(BDD for fv1 ltopgt gv1)
(BDD for fv1 ltopgt gv1)